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1507.07370,"{'1507.07370-1-0-0': 'In this paper we study the relation between two notions of largeness that apply to a set of positive integers, namely d and k, as introduced by Host and Kra [CITATION].', '1507.07370-1-0-1': 'We prove that any d set is necessarily k where [MATH] is effectively bounded in terms of [MATH].', '1507.07370-1-0-2': 'This partially resolves a conjecture of Host and Kra.', '1507.07370-1-1-0': 'Combinatorial properties of Nil-Bohr sets', '1507.07370-1-2-0': '# Introduction', '1507.07370-1-3-0': 'Among the basic problems in additive combinatorics is the study of various notions of largeness which may apply to a set of integers.', '1507.07370-1-3-1': 'In this paper we are specifically interested in one such notion, namely that of being a d set, or a set of recurrence times for a [MATH]-step nilrotation (see [REF]).', '1507.07370-1-4-0': 'The study of these sets was pioneered by Host and Kra [CITATION], with later developments due to Huang, Shao and Ye [CITATION].', '1507.07370-1-4-1': 'In [CITATION], it was realised that d sets bear a striking relation to a purely combinatorial class of k sets (see [REF]).', '1507.07370-1-4-2': 'Namely, it was shown that a d set is piecewise- d. Here, we prove the reverse implication, although in a weaker form.', '1507.07370-1-5-0': 'Even though a proper motivation for our results requires more context, we are able to express some of them in relatively basic terms.', '1507.07370-1-5-1': 'Our first result is the following.', '1507.07370-1-6-0': 'of [MATH] are the integers [MATH].', '1507.07370-1-6-1': 'It is customary to denote elements of [MATH] by lowercase Greek letters [MATH].', '1507.07370-1-6-2': 'Bootstrapping (a slight modification of) the above result, we obtain a recurrence statement for nilrotations.', '1507.07370-1-7-0': 'Let [MATH] be a [MATH]-step nilpotent Lie group and let [MATH] be a cocompact, discrete subgroup.', '1507.07370-1-7-1': 'Fix [MATH], an open neighbourhood [MATH], as well as a sequence [MATH] of positive integers.', '1507.07370-1-8-0': 'Then, there exists [MATH] with gaps bounded by [MATH], such that [MATH].', '1507.07370-1-9-0': '## Bohr sets', '1507.07370-1-10-0': 'The notion of a [MATH] (or [MATH]) set is classical and well-studied.', '1507.07370-1-10-1': 'A set [MATH] is said to be a [MATH] set if it contains the preimage of an open, non-empty set [MATH] through the natural embedding of [MATH] in the Bohr compactification of [MATH], usually denoted [MATH].', '1507.07370-1-10-2': 'Accordingly, [MATH] is a [MATH] set if additionally [MATH].', '1507.07370-1-11-0': 'While very satisfying from the categorical point of view, the above definition gives limited idea of what a [MATH] set looks like.', '1507.07370-1-11-1': 'A more concrete description is possible.', '1507.07370-1-11-2': 'Namely, a set is [MATH] if it contains a non-empty set of the form [MATH] where [MATH] and [MATH] is open; [MATH] is [MATH] if additionally [MATH].', '1507.07370-1-11-3': 'Hence, [MATH] sets can be viewed dynamically as a return-times sets for the point [MATH], where the dynamics are given by [MATH].', '1507.07370-1-12-0': '(Note that we construe being a [MATH] set as a notion of largeness, hence above we only insist on containment, rather than equality.', '1507.07370-1-12-1': 'In context when precise structure is important, different definitions are used, see e.g. [CITATION].)', '1507.07370-1-13-0': '## Nil-Bohr sets', '1507.07370-1-14-0': 'With the advent of higher-order Fourier analysis, a natural analogue of the class of [MATH] sets has come into view.', '1507.07370-1-14-1': 'The role of the circle rotations in classical Fourier analysis is now played by nilrotations, which we presently define.', '1507.07370-1-15-0': 'Suppose that [MATH] is a [MATH]-step nilpotent Lie group, and let [MATH] be a cocompact and discrete subgroup.', '1507.07370-1-15-1': 'Here, by cocompact we simply mean that the quotient space [MATH] should be compact.', '1507.07370-1-15-2': 'The space [MATH] is a [MATH]-step nilmanifold and carries a natural action of [MATH], given by [MATH].', '1507.07370-1-15-3': 'There exists a unique Haar measure [MATH] on the Borel [MATH]-algebra [MATH], which is preserved by all translations [MATH].', '1507.07370-1-15-4': 'Hence, for any [MATH], translation by [MATH] is a measure-preserving transformation of [MATH].', '1507.07370-1-15-5': 'We call any such system [MATH] a [MATH]-step nilrotation.', '1507.07370-1-16-0': 'We now define a set [MATH] to be a d set, in analogy to the abelian case, if it contains a non-empty set of the form [EQUATION] where [MATH] is a [MATH]-step nilmanifold, and [MATH] is open.', '1507.07370-1-16-1': 'If additionally [MATH] then [MATH] is a d set.', '1507.07370-1-17-0': 'A useful example to keep in mind are sets of the form: [EQUATION] where [MATH] is a polynomial with at least one irrational non-constant coefficient.', '1507.07370-1-17-1': 'In general, such sets are d.', '1507.07370-1-17-2': 'If additionally [MATH] then they are d. (This can be seen by a classical construction, which is discussed for instance in [CITATION].)', '1507.07370-1-18-0': '## IP sets', '1507.07370-1-19-0': 'Another classical notion of largeness which is relevant to us is [MATH].', '1507.07370-1-19-1': 'For a sequence [MATH], we define the set of finite sums of [MATH], [EQUATION]', '1507.07370-1-19-2': 'For brevity of notation, it is convenient to define in this context [MATH] for [MATH]; this is consistent with the natural inclusion [MATH].', '1507.07370-1-20-0': 'A set [MATH] is said to be an [MATH] set if there exists a sequence [MATH] such that [MATH].', '1507.07370-1-20-1': 'Once again, we remark that since we view [MATH] as a notion of largeness, we only require [MATH] to contain a set of finite sums (as opposed to being equal to such a set).', '1507.07370-1-20-2': 'This is consistent with usage e.g. in [CITATION], but different from the original definition in [CITATION].', '1507.07370-1-21-0': '## SG sets', '1507.07370-1-22-0': 'In analogy to the [MATH] sets [MATH], we define for [MATH] the sets [MATH], where the index sets are additionally required to have bounded gaps.', '1507.07370-1-22-1': 'Let [MATH] denote the set of finite sets of integers [MATH], whose gaps are bounded by [MATH]; we might call such sets [MATH]-syndetic.', '1507.07370-1-22-2': 'In other words, we require that for any [MATH], either [MATH], or there exists [MATH] with [MATH].', '1507.07370-1-22-3': 'We allow the degenerate case [MATH], where [MATH] (up to identification [MATH]).', '1507.07370-1-22-4': 'For a sequence of integers [MATH] we put [EQUATION]', '1507.07370-1-22-5': 'In analogy with [MATH] sets, we define a set [MATH] to be a k set if it contains a set of the form [MATH] for some sequence [MATH].', '1507.07370-1-22-6': 'To the best of our knowledge, this definition first appears in [CITATION].', '1507.07370-1-22-7': 'We have an obvious chain of inclusions [MATH], whence [MATH].', '1507.07370-1-23-0': 'The simplest non-degenerate example, though possibly a misleading one, is when [MATH].', '1507.07370-1-23-1': 'Then [MATH] consists precisely of the consecutive sums [MATH], and it has been noted that [MATH] where [MATH].', '1507.07370-1-23-2': 'It is not difficult to see that conversely, any set of the form [MATH] as above can be expressed as [MATH] for some sequence [MATH].', '1507.07370-1-23-3': 'Thus, 1 sets coincide with the well-studied class of [MATH] sets (see also [CITATION]).', '1507.07370-1-24-0': '## Dual classes', '1507.07370-1-25-0': 'For a class [MATH] of subsets of [MATH], we define the dual class [MATH] by declaring that [MATH] if and only if for any [MATH] the sets [MATH] and [MATH] intersect non-trivially: [MATH] (see e.g. [CITATION]).', '1507.07370-1-26-0': 'Specifically, we define the class k, consisting of the sets [MATH] such that for any choice of integers [MATH], there exists some [MATH] with [MATH].', '1507.07370-1-26-1': 'We note the reversed chain of inclusions: [MATH].', '1507.07370-1-27-0': 'It is clear by definition that for any class [MATH], the dual class [MATH] is closed under taking supersets.', '1507.07370-1-27-1': 'The operation of taking the dual reverses the inclusion: if [MATH] then [MATH].', '1507.07370-1-27-2': 'If [MATH] is partition regular, then [MATH] is easily seen to be closed under finite intersections, but [MATH] will not generally be partition regular [CITATION].', '1507.07370-1-27-3': 'If additionally [MATH] then [MATH] [CITATION].', '1507.07370-1-27-4': 'We cite the latter two facts merely to provide context, they are not used at any point.', '1507.07370-1-28-0': '## Nil-Bohr sets vs. SG sets', '1507.07370-1-29-0': 'As noted earlier, there is a somewhat unexpected connection between the a priori unrelated notions of [MATH] sets and [MATH] sets.', '1507.07370-1-30-0': 'Following the usual convention, we say that a set [MATH] is piecewise d if there exists a thick set [MATH] (i.e. such that [MATH] contains arbitrarily long intervals) and a d set [MATH] such that [MATH].', '1507.07370-1-30-1': 'It was proved in [CITATION] that any [MATH] set is piecewise d. (In fact, the result proved there is slightly stronger, with a more rigid notion of ""strongly piecewise"".)', '1507.07370-1-30-2': 'The following question arises naturally:', '1507.07370-1-31-0': 'Is any d set a d set?', '1507.07370-1-32-0': 'The main purpose of this paper is to answer a weaker variant of this question.', '1507.07370-1-32-1': 'The reader will have no problem checking that the following is merely a succinct restatement of Theorem [REF].', '1507.07370-1-33-0': 'Any d set is k, provided that [MATH].', '1507.07370-1-34-0': 'We note that a weaker variant of theorem, with k replaced with [MATH], is true for much simpler (or at least better studied) reasons.', '1507.07370-1-34-1': 'It can be checked that any nilrotation is distal [CITATION].', '1507.07370-1-34-2': 'More explicitly, for any nilrotation [MATH] with metric [MATH], for any [MATH] there exists [MATH] such that if [MATH] are two points with [MATH] then for all [MATH] also [MATH].', '1507.07370-1-34-3': 'On the other hand, for any topological dynamical system [MATH] distality is equivalent to the condition that for any [MATH] and any open neighbourhood [MATH], the set [MATH] is [MATH] [CITATION].', '1507.07370-1-34-4': 'Since any d is (a superset of) a set of precisely this form, the claim follows.', '1507.07370-1-34-5': '(A similar argument can be found in [CITATION].)', '1507.07370-1-35-0': '## Notation', '1507.07370-1-36-0': 'By [MATH] we denote the set [MATH]; in particular [MATH].', '1507.07370-1-36-1': 'We put [MATH].', '1507.07370-1-37-0': 'By [MATH] we denote the partial semigroup of the finite, non-empty subsets of integers, where the operation is the disjoint union.', '1507.07370-1-37-1': 'We also put [MATH].', '1507.07370-1-37-2': 'Hence, whenever the symbol [MATH] is used for [MATH] it is implicitly assumed that [MATH] and [MATH] need to be disjoint.', '1507.07370-1-38-0': 'If [MATH] is a group equipped with a metric, then [MATH] denotes the distance from [MATH] to [MATH].', '1507.07370-1-38-1': 'In particular, for [MATH], [MATH] denotes the distance of [MATH] from the closest integer.', '1507.07370-1-39-0': 'Standard asymptotic notation, such as [MATH] and [MATH], is occasionally used.'}","{'1507.07370-2-0-0': 'In this paper we study the relation between two notions of largeness that apply to a set of positive integers, namely d and k, as introduced by Host and Kra [CITATION].', '1507.07370-2-0-1': 'We prove that any d set is necessarily k where [MATH] is effectively bounded in terms of [MATH].', '1507.07370-2-0-2': 'This partially resolves a conjecture of Host and Kra.', '1507.07370-2-1-0': '[Nil-Bohr sets and combinatorics]Combinatorial properties of Nil-Bohr sets', '1507.07370-2-2-0': '# Introduction', '1507.07370-2-3-0': 'Among the basic problems in additive combinatorics is the study of various notions of largeness which may apply to a set of integers.', '1507.07370-2-3-1': 'In this paper we are specifically interested in one such notion, namely that of being a d set, or a set of recurrence times for a [MATH]-step nilrotation (see Section [REF]).', '1507.07370-2-4-0': 'The study of these sets was pioneered by Host and Kra [CITATION], with later developments due to Huang, Shao and Ye [CITATION], Tu [CITATION], and Bergelson and Leibman [CITATION].', '1507.07370-2-4-1': 'In [CITATION], it was realised that d sets bear a striking relation to a purely combinatorial class of k sets (see Section [REF]).', '1507.07370-2-4-2': 'Namely, it was shown that a d set is (strongly) piecewise- d. Here, we prove the reverse implication, although in a weaker form.', '1507.07370-2-5-0': 'Even though a proper motivation for our results requires more context, we are able to express some of them in relatively basic terms.', '1507.07370-2-5-1': 'Our first result is the following.', '1507.07370-2-6-0': 'of [MATH] are the integers [MATH].', '1507.07370-2-7-0': 'Let [MATH] be a [MATH]-step nilpotent Lie group and let [MATH] be a cocompact, discrete subgroup.', '1507.07370-2-7-1': 'Fix [MATH], an open neighbourhood [MATH], as well as a sequence [MATH] of positive integers.', '1507.07370-2-8-0': 'Then, there exists [MATH] with gaps bounded by [MATH], such that [MATH].', '1507.07370-2-9-0': 'The bound [MATH] results from an inductive argument.', '1507.07370-2-9-1': 'By a marginally more careful computation, this could be improved to [MATH], but we sacrifice this inconsequential improvement for the sake of readability.', '1507.07370-2-9-2': 'The optimal value is believed to be [MATH], but our argument does not yield this conclusion.', '1507.07370-2-10-0': 'Finally, we point out that the recent paper of Bergelson and Leibman [CITATION] proves highly relevant results.', '1507.07370-2-10-1': 'It follows as a special case of Theorem 0.3 in [CITATION] that, in the situation of Theorem [REF], there exists a set [MATH] such that [MATH], for some [MATH].', '1507.07370-2-10-2': 'Hence, trivially, we may ensure that [MATH] has gaps bounded in terms of [MATH] and [MATH]; however, our result gives a good bound on the gaps, which also is uniform in [MATH].', '1507.07370-2-10-3': 'Similarly, a version of Theorem [REF] can be read off Theorem 0.5 in [CITATION].', '1507.07370-2-11-0': '## Bohr sets', '1507.07370-2-12-0': 'The notion of a [MATH] (or [MATH]) set is classical and well-studied.', '1507.07370-2-12-1': 'A set [MATH] is said to be a [MATH] set if it contains the preimage of an open, non-empty set [MATH] through the natural embedding of [MATH] in the Bohr compactification of [MATH], usually denoted [MATH].', '1507.07370-2-12-2': 'Accordingly, [MATH] is a [MATH] set if additionally [MATH].', '1507.07370-2-13-0': 'While very satisfying from the categorical point of view, the above definition gives limited idea of what a [MATH] set looks like.', '1507.07370-2-13-1': 'A more concrete description is possible.', '1507.07370-2-13-2': 'Namely, a set is [MATH] if it contains a non-empty set of the form [MATH] where [MATH] and [MATH] is open; [MATH] is [MATH] if additionally [MATH].', '1507.07370-2-13-3': 'Hence, [MATH] sets can be viewed dynamically as a return-times sets for the point [MATH], where the dynamics are given by [MATH].', '1507.07370-2-14-0': '(Note that we construe being a [MATH] set as a notion of largeness, hence above we only insist on containment, rather than equality.', '1507.07370-2-14-1': 'In context when precise structure is important, different definitions are used, see e.g. [CITATION].)', '1507.07370-2-15-0': '## Nil-Bohr sets', '1507.07370-2-16-0': 'With the advent of higher-order Fourier analysis, a natural analogue of the class of [MATH] sets has come into view.', '1507.07370-2-16-1': 'The role of the circle rotations in classical Fourier analysis is now played by nilrotations, which we presently define.', '1507.07370-2-17-0': 'Suppose that [MATH] is a [MATH]-step nilpotent Lie group, and let [MATH] be a cocompact and discrete subgroup.', '1507.07370-2-17-1': 'Here, by cocompact we simply mean that the quotient space [MATH] should be compact.', '1507.07370-2-17-2': 'The space [MATH] is a [MATH]-step nilmanifold and carries a natural action of [MATH], given by [MATH].', '1507.07370-2-17-3': 'There exists a unique Haar measure [MATH] on the Borel [MATH]-algebra [MATH], which is preserved by all translations [MATH].', '1507.07370-2-17-4': 'Hence, for any [MATH], translation by [MATH] is a measure-preserving transformation of [MATH].', '1507.07370-2-17-5': 'We call any such system [MATH] a [MATH]-step nilrotation.', '1507.07370-2-18-0': 'We now define a set [MATH] to be a d set, in analogy to the abelian case, if it contains a non-empty set of the form [EQUATION] where [MATH] is a [MATH]-step nilmanifold, and [MATH] is open.', '1507.07370-2-18-1': 'If additionally [MATH] then [MATH] is a d set.', '1507.07370-2-19-0': 'A useful example to keep in mind are sets of the form [EQUATION] where [MATH] is a polynomial with at least one irrational non-constant coefficient.', '1507.07370-2-19-1': 'In general, such sets are d.', '1507.07370-2-19-2': 'If additionally [MATH] then they are d. (This can be seen by a classical construction, which is discussed for instance in [CITATION].)', '1507.07370-2-20-0': '## IP sets', '1507.07370-2-21-0': 'Another classical notion of largeness which is relevant to us is [MATH].', '1507.07370-2-21-1': 'For a sequence [MATH], we define the set of finite sums of [MATH], [EQUATION]', '1507.07370-2-21-2': 'For brevity of notation, it is convenient to define in this context [MATH] for [MATH]; this is consistent with the natural inclusion [MATH].', '1507.07370-2-22-0': 'A set [MATH] is said to be an [MATH] set if there exists a sequence [MATH] such that [MATH].', '1507.07370-2-22-1': 'Once again, we remark that since we view [MATH] as a notion of largeness, we only require [MATH] to contain a set of finite sums (as opposed to being equal to such a set).', '1507.07370-2-22-2': 'This is consistent with usage e.g. in [CITATION], but different from the original definition in [CITATION].', '1507.07370-2-23-0': '## SG sets', '1507.07370-2-24-0': 'In analogy to the [MATH] sets [MATH], we define for [MATH] the sets [MATH], where the index sets are additionally required to have bounded gaps.', '1507.07370-2-24-1': 'Let [MATH] denote the set of finite sets of integers [MATH] whose gaps are bounded by [MATH]; we might call such sets [MATH]-syndetic.', '1507.07370-2-24-2': 'In other words, we require that for any [MATH], either [MATH], or there exists [MATH] with [MATH].', '1507.07370-2-24-3': 'We allow the degenerate case [MATH], where [MATH] (up to identification [MATH]).', '1507.07370-2-24-4': 'For a sequence of integers [MATH] we put [EQUATION]', '1507.07370-2-24-5': 'In analogy with [MATH] sets, we define a set [MATH] to be a k set if it contains a set of the form [MATH] for some sequence [MATH].', '1507.07370-2-24-6': 'To the best of our knowledge, this definition first appears in [CITATION].', '1507.07370-2-24-7': 'We have an obvious chain of inclusions [MATH], whence [MATH].', '1507.07370-2-25-0': 'The simplest non-degenerate example, though possibly a misleading one, is when [MATH].', '1507.07370-2-25-1': 'Then [MATH] consists precisely of the consecutive sums [MATH], and it has been noted that [MATH] where [MATH].', '1507.07370-2-25-2': 'It is not difficult to see that conversely, any set of the form [MATH] as above can be expressed as [MATH] for some sequence [MATH].', '1507.07370-2-25-3': 'Thus, 1 sets coincide with the well-studied class of [MATH] sets (see also [CITATION]).', '1507.07370-2-26-0': '## Dual classes', '1507.07370-2-27-0': 'For a class [MATH] of subsets of [MATH], we define the dual class [MATH] by declaring that [MATH] if and only if for any [MATH] the sets [MATH] and [MATH] intersect non-trivially: [MATH] (see e.g. [CITATION]).', '1507.07370-2-28-0': 'Specifically, we define the class k, consisting of the sets [MATH] such that for any choice of integers [MATH], there exists some [MATH] with [MATH].', '1507.07370-2-28-1': 'We note the reversed chain of inclusions: [MATH].', '1507.07370-2-29-0': 'It is clear by definition that for any class [MATH], the dual class [MATH] is closed under taking supersets.', '1507.07370-2-29-1': 'The operation of taking the dual reverses the inclusion: if [MATH] then [MATH].', '1507.07370-2-29-2': 'If [MATH] is partition regular, then [MATH] is easily seen to be closed under finite intersections, but [MATH] will not generally be partition regular [CITATION].', '1507.07370-2-29-3': 'If additionally [MATH] then [MATH] [CITATION].', '1507.07370-2-29-4': 'We cite the latter two facts merely to provide context, they are not used at any point.', '1507.07370-2-30-0': '## Nil-Bohr sets vs. SG sets', '1507.07370-2-31-0': 'As noted earlier, there is a somewhat unexpected connection between the a priori unrelated notions of [MATH] sets and [MATH] sets.', '1507.07370-2-32-0': 'Following the usual convention, we say that a set [MATH] is piecewise d if there exists a thick set [MATH] (i.e. such that [MATH] contains arbitrarily long intervals) and a d set [MATH] such that [MATH].', '1507.07370-2-32-1': 'It was proved in [CITATION] that any [MATH] set is piecewise d. (In fact, the result proved there is stronger, with a more rigid notion of ""strongly piecewise"".)', '1507.07370-2-32-2': 'The following question arises naturally:', '1507.07370-2-33-0': 'Is any d set a d set?', '1507.07370-2-34-0': 'The main purpose of this paper is to answer a weaker variant of this question.', '1507.07370-2-34-1': 'The reader will have no problem checking that the following is merely a succinct restatement of Theorem [REF].', '1507.07370-2-35-0': 'Any d set is k, provided that [MATH].', '1507.07370-2-36-0': 'We note that a weaker variant of theorem, with k replaced with [MATH], is true for much simpler (or at least better studied) reasons.', '1507.07370-2-36-1': 'It can be checked that any nilrotation is distal ([CITATION]; see also [CITATION],[CITATION]).', '1507.07370-2-36-2': 'More explicitly, for any nilrotation [MATH] with metric [MATH], for any [MATH] there exists [MATH] such that if [MATH] are two points with [MATH] then for all [MATH] also [MATH].', '1507.07370-2-36-3': 'On the other hand, for any topological dynamical system [MATH] distality is equivalent to the condition that for any [MATH] and any open neighbourhood [MATH], the set [MATH] is [MATH] [CITATION].', '1507.07370-2-36-4': 'Since any d is (a superset of) a set of precisely this form, the claim follows.', '1507.07370-2-36-5': '(Essentially the same argument can be found in [CITATION], and in [CITATION].)', '1507.07370-2-37-0': '## Notation', '1507.07370-2-38-0': 'By [MATH] we denote the set [MATH]; in particular [MATH].', '1507.07370-2-38-1': 'We put [MATH].', '1507.07370-2-39-0': 'By [MATH] we denote the partial semigroup of the finite, non-empty subsets of integers, where the operation is the disjoint union.', '1507.07370-2-39-1': 'We also put [MATH].', '1507.07370-2-39-2': 'Hence, whenever the symbol [MATH] is used for [MATH] it is implicitly assumed that [MATH] and [MATH] need to be disjoint.', '1507.07370-2-40-0': 'If [MATH] is a group equipped with a metric, then [MATH] denotes the distance from [MATH] to [MATH].', '1507.07370-2-40-1': 'In particular, for [MATH], [MATH] denotes the distance of [MATH] from the closest integer.', '1507.07370-2-41-0': 'Standard asymptotic notation, such as [MATH] and [MATH], is occasionally used: [MATH] if [MATH] for an absolute constant [MATH], and [MATH] if [MATH].'}","[['1507.07370-1-15-0', '1507.07370-2-17-0'], ['1507.07370-1-15-1', '1507.07370-2-17-1'], ['1507.07370-1-15-2', '1507.07370-2-17-2'], ['1507.07370-1-15-3', '1507.07370-2-17-3'], ['1507.07370-1-15-4', '1507.07370-2-17-4'], ['1507.07370-1-15-5', '1507.07370-2-17-5'], ['1507.07370-1-25-0', '1507.07370-2-27-0'], ['1507.07370-1-5-0', '1507.07370-2-5-0'], ['1507.07370-1-5-1', '1507.07370-2-5-1'], ['1507.07370-1-32-0', '1507.07370-2-34-0'], ['1507.07370-1-32-1', '1507.07370-2-34-1'], ['1507.07370-1-22-0', '1507.07370-2-24-0'], ['1507.07370-1-22-2', '1507.07370-2-24-2'], ['1507.07370-1-22-3', '1507.07370-2-24-3'], ['1507.07370-1-22-4', 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'1507.07370-2-7-0'], ['1507.07370-1-7-1', '1507.07370-2-7-1'], ['1507.07370-1-10-0', '1507.07370-2-12-0'], ['1507.07370-1-10-1', '1507.07370-2-12-1'], ['1507.07370-1-10-2', '1507.07370-2-12-2'], ['1507.07370-1-34-0', '1507.07370-2-36-0'], ['1507.07370-1-34-2', '1507.07370-2-36-2'], ['1507.07370-1-34-3', '1507.07370-2-36-3'], ['1507.07370-1-34-4', '1507.07370-2-36-4'], ['1507.07370-1-22-1', '1507.07370-2-24-1'], ['1507.07370-1-3-1', '1507.07370-2-3-1'], ['1507.07370-1-4-0', '1507.07370-2-4-0'], ['1507.07370-1-4-1', '1507.07370-2-4-1'], ['1507.07370-1-4-2', '1507.07370-2-4-2'], ['1507.07370-1-30-1', '1507.07370-2-32-1'], ['1507.07370-1-17-0', '1507.07370-2-19-0'], ['1507.07370-1-34-1', '1507.07370-2-36-1'], ['1507.07370-1-34-5', '1507.07370-2-36-5']]","[['1507.07370-1-15-0', '1507.07370-2-17-0'], ['1507.07370-1-15-1', '1507.07370-2-17-1'], ['1507.07370-1-15-2', '1507.07370-2-17-2'], ['1507.07370-1-15-3', '1507.07370-2-17-3'], ['1507.07370-1-15-4', '1507.07370-2-17-4'], ['1507.07370-1-15-5', '1507.07370-2-17-5'], ['1507.07370-1-25-0', '1507.07370-2-27-0'], ['1507.07370-1-5-0', '1507.07370-2-5-0'], ['1507.07370-1-5-1', '1507.07370-2-5-1'], ['1507.07370-1-32-0', '1507.07370-2-34-0'], ['1507.07370-1-32-1', '1507.07370-2-34-1'], ['1507.07370-1-22-0', '1507.07370-2-24-0'], ['1507.07370-1-22-2', '1507.07370-2-24-2'], ['1507.07370-1-22-3', '1507.07370-2-24-3'], ['1507.07370-1-22-4', '1507.07370-2-24-4'], ['1507.07370-1-22-5', '1507.07370-2-24-5'], ['1507.07370-1-22-6', '1507.07370-2-24-6'], ['1507.07370-1-22-7', '1507.07370-2-24-7'], ['1507.07370-1-38-0', '1507.07370-2-40-0'], ['1507.07370-1-38-1', '1507.07370-2-40-1'], ['1507.07370-1-3-0', '1507.07370-2-3-0'], ['1507.07370-1-16-0', '1507.07370-2-18-0'], ['1507.07370-1-16-1', '1507.07370-2-18-1'], ['1507.07370-1-26-0', '1507.07370-2-28-0'], ['1507.07370-1-26-1', '1507.07370-2-28-1'], ['1507.07370-1-27-0', '1507.07370-2-29-0'], ['1507.07370-1-27-1', '1507.07370-2-29-1'], ['1507.07370-1-27-2', '1507.07370-2-29-2'], ['1507.07370-1-27-3', '1507.07370-2-29-3'], ['1507.07370-1-27-4', '1507.07370-2-29-4'], ['1507.07370-1-12-0', '1507.07370-2-14-0'], ['1507.07370-1-12-1', '1507.07370-2-14-1'], ['1507.07370-1-11-0', '1507.07370-2-13-0'], ['1507.07370-1-11-1', '1507.07370-2-13-1'], ['1507.07370-1-11-2', '1507.07370-2-13-2'], ['1507.07370-1-11-3', '1507.07370-2-13-3'], ['1507.07370-1-37-0', '1507.07370-2-39-0'], ['1507.07370-1-37-1', '1507.07370-2-39-1'], ['1507.07370-1-37-2', '1507.07370-2-39-2'], ['1507.07370-1-8-0', '1507.07370-2-8-0'], ['1507.07370-1-20-0', '1507.07370-2-22-0'], ['1507.07370-1-20-1', '1507.07370-2-22-1'], ['1507.07370-1-20-2', '1507.07370-2-22-2'], ['1507.07370-1-14-0', '1507.07370-2-16-0'], ['1507.07370-1-14-1', '1507.07370-2-16-1'], ['1507.07370-1-30-0', '1507.07370-2-32-0'], ['1507.07370-1-17-1', '1507.07370-2-19-1'], ['1507.07370-1-17-2', '1507.07370-2-19-2'], ['1507.07370-1-19-0', '1507.07370-2-21-0'], ['1507.07370-1-19-1', '1507.07370-2-21-1'], ['1507.07370-1-19-2', '1507.07370-2-21-2'], ['1507.07370-1-23-0', '1507.07370-2-25-0'], ['1507.07370-1-23-1', '1507.07370-2-25-1'], ['1507.07370-1-23-2', '1507.07370-2-25-2'], ['1507.07370-1-23-3', '1507.07370-2-25-3'], ['1507.07370-1-29-0', '1507.07370-2-31-0'], ['1507.07370-1-0-0', '1507.07370-2-0-0'], ['1507.07370-1-0-1', '1507.07370-2-0-1'], ['1507.07370-1-0-2', '1507.07370-2-0-2'], ['1507.07370-1-7-0', '1507.07370-2-7-0'], ['1507.07370-1-7-1', '1507.07370-2-7-1'], ['1507.07370-1-10-0', '1507.07370-2-12-0'], ['1507.07370-1-10-1', '1507.07370-2-12-1'], ['1507.07370-1-10-2', '1507.07370-2-12-2'], ['1507.07370-1-34-0', '1507.07370-2-36-0'], ['1507.07370-1-34-2', '1507.07370-2-36-2'], ['1507.07370-1-34-3', '1507.07370-2-36-3'], ['1507.07370-1-34-4', '1507.07370-2-36-4']]","[['1507.07370-1-22-1', '1507.07370-2-24-1'], ['1507.07370-1-3-1', '1507.07370-2-3-1'], ['1507.07370-1-4-0', '1507.07370-2-4-0'], ['1507.07370-1-4-1', '1507.07370-2-4-1'], ['1507.07370-1-4-2', '1507.07370-2-4-2'], ['1507.07370-1-30-1', '1507.07370-2-32-1'], ['1507.07370-1-17-0', '1507.07370-2-19-0']]",[],"[['1507.07370-1-34-1', '1507.07370-2-36-1'], ['1507.07370-1-34-5', '1507.07370-2-36-5']]",[],"['1507.07370-1-1-0', '1507.07370-1-30-2', '1507.07370-1-31-0', '1507.07370-1-33-0', '1507.07370-1-36-0', '1507.07370-1-36-1', '1507.07370-2-1-0', '1507.07370-2-6-0', '1507.07370-2-32-2', '1507.07370-2-33-0', '1507.07370-2-35-0', '1507.07370-2-38-0', '1507.07370-2-38-1', '1507.07370-2-41-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1507.07370,,,
cond-mat-0301201,"{'cond-mat-0301201-1-0-0': 'In this paper we report the results of simulations of a 2D gravity driven, dissipative granular flow through a hopper system.', 'cond-mat-0301201-1-0-1': 'Measurements of impulse distributions [MATH] on the simulated system show flow-velocity-invariant behavior of the distribution for impulses larger than the average impulse [MATH].', 'cond-mat-0301201-1-0-2': 'For small impulses, however, [MATH] decreases significantly with flow velocity, a phenomenon which can be attributed exclusively to collisions between grains undergoing frequent collisions.', 'cond-mat-0301201-1-0-3': 'Visualizations of the system also show that these frequently colliding particles tend to form increasingly large linear clusters as the flow velocity decreases.', 'cond-mat-0301201-1-0-4': 'A model is proposed for the form of [MATH], given distributions of cluster size and velocity, which accurately predicts the observed form of the distribution.', 'cond-mat-0301201-1-0-5': 'Thus the impulse distribution provides some insight into the formation and properties of these ""dynamic"" force chains.', 'cond-mat-0301201-1-1-0': 'IntroductionGranular materials exhibit a wide spectrum of behavior ranging from gaseous to liquid to solid.', 'cond-mat-0301201-1-1-1': 'Remarkably, all of these phases of granular matter respond to external stimuli in a manner strikingly different from ordinary fluids and solids [CITATION].', 'cond-mat-0301201-1-1-2': 'In static granular piles, the spatially inhomogeneous manner in which stress is transmitted from the bulk of the pile to the boundary has been well documented[CITATION].', 'cond-mat-0301201-1-1-3': 'Experiments have shown that the force distribution [MATH] at the walls is exponential at large forces and exhibits a plateau at small forces[CITATION].', 'cond-mat-0301201-1-1-4': 'In addition, the highly stressed grains in static packings are organized into linear structures termed ""force chains""[CITATION].', 'cond-mat-0301201-1-1-5': 'The appearance of similar large scale structures in flowing granular matter would have significant implications for both continuum theories and descriptions of jamming in non-thermal systems.', 'cond-mat-0301201-1-1-6': 'A recent proposal for a unified picture of jamming in thermal and non-thermal systems suggests that jamming occurs due the formation of force chains whose presence is signalled by the appearance of a plateau in [MATH] [CITATION].', 'cond-mat-0301201-1-1-7': 'A continuum description of steady-state flow on an inclined plane models the system as a collection of transient 1D chains immersed in a viscous fluid[CITATION].', 'cond-mat-0301201-1-1-8': 'Indeed, transient ""clusters"" have been identified experimentally in granular surface flows [CITATION] and shear flows[CITATION], and simulations of chute flows have shown evidence for a plateau in [MATH] as the system approaches jamming[CITATION].', 'cond-mat-0301201-1-2-0': 'Recent experiments have been performed in a two dimensional hopper geometry to explore the presence of incipient force chains in purely collisional gravity-driven flow [CITATION].', 'cond-mat-0301201-1-2-1': 'Measurements of the time trace of the impulse delivered to a transducer placed at the side wall of the hopper have shown that the distribution of impulses, [MATH], displays an exponential decay at large [MATH], as for the case of static materials.', 'cond-mat-0301201-1-2-2': 'This exponential form of the distribution is maintained for all flow velocities from the largest measured ([MATH] = 60.0 cm/s) to the minimum flow velocity prior to the point at which the system no longer exhibits sustained flow ([MATH] = 9.4 cm/s).', 'cond-mat-0301201-1-2-3': 'However, at small [MATH], [MATH] develops an upward trend which becomes increasingly more evident as the flow velocity decreases.', 'cond-mat-0301201-1-3-0': 'In this letter, we report the results of event-driven simulations of a system of inelastic, monodispersed hard disks in the experimental geometry of Ref. [CITATION].', 'cond-mat-0301201-1-3-1': 'Our simulations provide clear evidence of an increasing proportion of collisions with small impulses as the flow velocity is decreased.', 'cond-mat-0301201-1-3-2': 'This results in the formation of a plateau in [MATH] as the minimum flow velocity for sustained flow is approached.', 'cond-mat-0301201-1-3-3': 'In addition, we observe the formation of clusters of disks which collide ""frequently"" and are reminiscent of the ""collapse strings"" observed in freely cooling granular matter[CITATION].', 'cond-mat-0301201-1-3-4': 'We present a model calculation which strongly suggests that the increase of small impulse events is associated with the growth of these clusters.', 'cond-mat-0301201-1-4-0': 'SimulationsThe grain dynamics used in the simulations are as in Ref. [CITATION] (momentum is conserved and at each interparticle collision the energy loss is proportional to [MATH] where [MATH] is the coefficient of restitution).', 'cond-mat-0301201-1-4-1': 'To ensure that the pressure is independent of the height the side walls must absorb some vertical momentum, therefore we impose the condition that collisions with the walls are inelastic in the tangential direction.', 'cond-mat-0301201-1-4-2': 'The flow velocity is controlled similarly to the experiments, by adjusting the width of the hopper opening.', 'cond-mat-0301201-1-4-3': 'However, as we wish to observe the system over many events, particles exiting the system at the bottom must be replaced at the top to create sustained flow.', 'cond-mat-0301201-1-4-4': 'This necessitates the introduction of a probability of reflection [MATH] at the bottom of the hopper (this would be equivalent to the presence of a sieve in experiment).', 'cond-mat-0301201-1-4-5': 'In our case, it provides another parameter with which we can tune the flow velocity.', 'cond-mat-0301201-1-4-6': 'Typically, our simulations were done on systems of 500 disks, with [MATH] = 0.9 and [MATH] = 0.4.', 'cond-mat-0301201-1-4-7': 'The simulation was run for [MATH] events for each flow velocity, with [MATH] discarded initially to allow the system time to reach steady state.', 'cond-mat-0301201-1-5-0': 'Simulation Results The physical quantity that is most closely related to [MATH] in our flowing hard-disk system is the distribution of impulses transferred at each collision, [MATH].', 'cond-mat-0301201-1-5-1': 'Unlike the experiments done in Ref. [CITATION], where the measurements were made only at the wall, we measure the magnitude of the momentum transfer at each collision event (note that the impulse and flow velocity are measured in units such that disk diameter [MATH], disk mass [MATH] and acceleration due to gravity [MATH] are all equal to one).', 'cond-mat-0301201-1-5-2': 'We observe an exponential form of [MATH] at impulses larger than the average impulse (Fig. [REF]) for the full range of flow rates.', 'cond-mat-0301201-1-5-3': 'After scaling the impulse by the average impulse, [MATH], the curves collapse onto each other for large impulses.', 'cond-mat-0301201-1-5-4': 'The small impulse behavior of the distribution, however, changes markedly with changes in the flow velocity.', 'cond-mat-0301201-1-5-5': 'As the flow velocity decreases, the height of the distribution at a minimum impulse ([MATH] for this distribution) increases and [MATH] begins to develop a plateau at small [MATH].', 'cond-mat-0301201-1-5-6': 'If the ratio between [MATH] at the peak impulse to [MATH] at the minimum impulse [MATH] is calculated and plotted against the flow velocity, a dramatic increase is observed (see inset to Fig. [REF]).', 'cond-mat-0301201-1-5-7': 'Extrapolating to small flow velocities indicates that this ratio becomes one at a non-zero flow rate.', 'cond-mat-0301201-1-6-0': 'Impulse distribution scaled to the average impulse, [MATH], with flow velocity decreasing from the bottom curve to the top curve.', 'cond-mat-0301201-1-6-1': 'The inset shows the ratio of [MATH] to [MATH] as a function of the flow velocity.', 'cond-mat-0301201-1-7-0': 'It should also be noted that the height of the distribution at [MATH] makes a sharp upturn which becomes more pronounced as the flow velocity decreases (the upturn is not present for the fastest flow velocity).', 'cond-mat-0301201-1-7-1': 'While we do not fully understand the reasons for this phenomenon, we believe it is not the same upward trend visible in the experiments.', 'cond-mat-0301201-1-7-2': 'That feature was a smooth continuation of the large impulse curve, while in our distribution a peak is seen.', 'cond-mat-0301201-1-7-3': 'In addition, our upturn occurs only at [MATH], regardless of the bin width used in constructing the distribution.', 'cond-mat-0301201-1-7-4': 'Such small impulses are not likely to be distinguishable in experiment due to the resolution limit imposed by the transducer.', 'cond-mat-0301201-1-7-5': 'For the remainder of our analysis, we will consider the impulse distribution without the [MATH] point.', 'cond-mat-0301201-1-8-0': 'The basic form of the impulse distribution, a peak at a finite value of [MATH] and an exponential tail can be understood on the basis of a flow of uncorrelated hard disks.', 'cond-mat-0301201-1-8-1': 'If the disks were uncorrelated, the impulse distribution would be a convolution of the individual momentum (velocity) distributions.', 'cond-mat-0301201-1-8-2': 'Since there is an average flow velocity, this would give rise to a peak in the distribution.', 'cond-mat-0301201-1-8-3': 'The exponential tail would be observed if the individual velocity distributions also had exponential tails.', 'cond-mat-0301201-1-8-4': 'The velocity distribution observed in our simulations can, to a first approximation, be described by Gaussians with exponential tails.', 'cond-mat-0301201-1-8-5': 'This argument would then suggest that the exponential tail arises from uncorrelated particles and is a consequence of the shape of the velocity distribution.', 'cond-mat-0301201-1-8-6': 'Viewed from this perspective, the lack of change in shape of [MATH] at large [MATH] as the flow approaches jamming is not surprising.', 'cond-mat-0301201-1-8-7': 'We will discuss this further in the context of a simple model to be presented below.', 'cond-mat-0301201-1-9-0': 'The change in [MATH] at small [MATH] is reminiscent of the universal jamming scenario[CITATION] although the direction of the change (filling up at small impulses) is different from the one observed in Lennard Jones systems and foams where the probability of finding small forces decreases as the system nears the jamming transition.', 'cond-mat-0301201-1-9-1': 'Nevertheless, the idea that changes in the distribution at small impulses or forces could be related to the appearance of structures akin to force chains is intriguing.', 'cond-mat-0301201-1-10-0': 'To explore a possible connection between the changes in [MATH] and the appearance of spatial inhomogeneities, we considered a question first asked in studies of inelastic collapse in freely cooling granular gases[CITATION], ""How many collisions does a given grain undergo in a fixed number of events?""', 'cond-mat-0301201-1-10-1': 'We define a minimum frequency of collision for a given particle as [MATH] (where [MATH] is the average time between events for a given flow velocity) and then identify those particles undergoing collisions with a frequency [MATH] as frequently colliding.', 'cond-mat-0301201-1-10-2': 'This allows us to define a minimum number of collisions a frequently colliding particle must undergo in a given number of events.', 'cond-mat-0301201-1-10-3': 'We then construct an image of our system at every 1000 events, and color all disks satisfying the criteria (see Fig. [REF]a,b).', 'cond-mat-0301201-1-10-4': 'As we decrease the flow velocity, the frequently colliding particles form increasingly larger linear clusters (compare Fig. [REF]a, where [MATH] = 2.03 in our units or 35.6 cm/s and Fig. [REF]b, where [MATH] = 0.96 or 16.9 cm/s).', 'cond-mat-0301201-1-10-5': 'These 1D structures observed in our simulation are reminiscent of the transient solid chains postulated by the hydrodynamic model of Ref. [CITATION].', 'cond-mat-0301201-1-10-6': 'Comparisons of the impulse distribution of the frequently colliding particles and the impulse distribution of the remaining ""rarely colliding"" particles (Fig. [REF] c,d) reveals that it is the contribution from the frequently colliding particles which causes the height of the total impulse distribution at [MATH] to increase in the manner seen in Fig. [REF]a.', 'cond-mat-0301201-1-10-7': 'The large impulse behavior appears to be dominated by the rarely colliding particles.', 'cond-mat-0301201-1-10-8': 'These observations are relatively insensitive to changes in [MATH] provided it is larger than a minimum value.', 'cond-mat-0301201-1-11-0': '(a) (b) Sample image of simulation for [MATH] = 2.03 (a) and for [MATH] = 0.96 (b).', 'cond-mat-0301201-1-11-1': '(c) (d) Impulse distributions for all particles, frequently colliding particles and rarely colliding particles for [MATH] = 2.03 (c) and for [MATH] = 0.96 (d).', 'cond-mat-0301201-1-11-2': 'Note that these distributions are unnormalized.', 'cond-mat-0301201-1-12-0': 'Impulse Distribution Model A possible connection between the shape of the impulse distribution and the development of linear clusters of frequently colliding particles can be drawn by investigating the following model.', 'cond-mat-0301201-1-12-1': 'Consider a one dimensonal cluster of particles which are all moving with the same velocity.', 'cond-mat-0301201-1-12-2': 'Now if another particle, travelling at some speed [MATH] relative to the cluster (see Fig. [REF]a) collides inelastically with one end of the chain, the impulse associated with that collision will be [MATH] where [MATH] is related to the coefficient of restitution by [MATH][CITATION].', 'cond-mat-0301201-1-12-3': 'If the chain was only comprised of one grain, the impulse distribution [MATH] (given an initial incoming speed [MATH]) would be a single spike of height 1 located at [MATH].', 'cond-mat-0301201-1-12-4': 'For two particles, [MATH] would be a bar of height 1 with limits at [MATH] and [MATH].', 'cond-mat-0301201-1-12-5': 'Continuing this argument for clusters containing [MATH] particles, then as [MATH] becomes very large, the leftmost limit of [MATH] approaches zero (Fig. [REF]b).', 'cond-mat-0301201-1-13-0': '(a) Particle of speed [MATH] incident on 1D cluster of [MATH] stationary particles.', 'cond-mat-0301201-1-13-1': '(b) Resulting form of [MATH].', 'cond-mat-0301201-1-14-0': 'Given a distribution of speeds [MATH] for the incident particle, and a distribution of cluster sizes [MATH], the total impulse distribution [MATH] is: [EQUATION]', 'cond-mat-0301201-1-14-1': 'If the cluster size distribution falls off sharply, then the shape of [MATH] will essentially reflect the shape of [MATH].', 'cond-mat-0301201-1-14-2': 'If, however, the cluster size distribution becomes broad, then the small impulse end of [MATH] will flatten out reflecting the nature of [MATH] for large [MATH].', 'cond-mat-0301201-1-15-0': 'As detailed previously, the criterion that we use to identify the clusters in our simulations is that of ""frequent"" collisions.', 'cond-mat-0301201-1-15-1': 'Due to the inelastic nature of the collisions, the velocities of these particles become highly correlated[CITATION] and they can be idealized as clusters of disks moving with the same velocity.', 'cond-mat-0301201-1-15-2': 'This velocity correlation within spatial clusters of particles has been directly observed in granular surface flows[CITATION].', 'cond-mat-0301201-1-15-3': 'Since we observe these clusters to be linear and growing in size with decreasing flow rates, it is plausible that the changes in [MATH] observed in our simulations is related to a change in distribution of the size of the clusters of frequently-colliding disks.', 'cond-mat-0301201-1-15-4': 'In order to put this conjecture on a firmer footing, we calculated the impulse distribution from our model using forms of [MATH] and [MATH] which provide good fits to our simulation data.', 'cond-mat-0301201-1-15-5': 'The form of [MATH] that we use is a Gaussian with an exponential tail.', 'cond-mat-0301201-1-15-6': 'This form is representative of the simulation results for [MATH] with the center of the Gaussian reflecting the average flow velocity.', 'cond-mat-0301201-1-15-7': 'The observed cluster size distribution is consistent with the form [MATH] with [MATH] increasing as the flow velocity decreases.', 'cond-mat-0301201-1-15-8': 'Fig. [REF]a shows [MATH] obtained from our simulations for three different flow rates.', 'cond-mat-0301201-1-15-9': 'Note that a change in [MATH] will produce a change in [MATH] for a given flow velocity but will not alter the shape of [MATH] nor the trend of increasing [MATH] with decreasing flow velocity.', 'cond-mat-0301201-1-16-0': '(a) Simulation results for [MATH] for varying flow rates.', 'cond-mat-0301201-1-16-1': '(b) Results of the proposed model for [MATH] with [MATH] increasing from the bottom curve to the top curve.', 'cond-mat-0301201-1-16-2': 'The inset shows the ratio of [MATH] to [MATH] as a function of [MATH].', 'cond-mat-0301201-1-17-0': 'Using these forms of [MATH] and [MATH], we find that the [MATH] obtained from the simple model has a form which is remarkably similar to the measured [MATH], as shown in Fig. [REF]b.', 'cond-mat-0301201-1-17-1': 'The control parameter in the model is [MATH] and the contribution at small impulses increases with [MATH] as is evident from the inset to Fig. [REF]b.', 'cond-mat-0301201-1-17-2': 'The correspondence between the model and our simulations strongly suggests that the observed changes in [MATH] are associated with the growth of clusters of frequently colliding particles and as [MATH] diverges, the ratio of [MATH] to [MATH] approaches unity.', 'cond-mat-0301201-1-17-3': 'The tail of the [MATH] distribution is controlled by the shape of [MATH] as we have verified within the model.', 'cond-mat-0301201-1-18-0': 'The picture which seems to be emerging from our simulations is that of increasingly larger scale spatial heterogeneities developing as the system approaches jamming.', 'cond-mat-0301201-1-18-1': 'Since the clusters are essentially the same as the ones identified in freely cooling granular matter, their origin lies in the dissipative nature of the medium.', 'cond-mat-0301201-1-18-2': 'The heterogeneities reflect strong velocity correlations of grains and leave a distinctive signature in the impulse distribution.', 'cond-mat-0301201-1-18-3': 'Whether the clusters that we have identified are indeed incipient force chains remains to be verified.', 'cond-mat-0301201-1-18-4': 'If this connection can be established, for example through the calculation of stress correlations in the flowing medium, then our analysis will provide a natural connection between [MATH] and force chains.', 'cond-mat-0301201-1-18-5': 'Moreover, it would a indicate that jamming occurs through the formation of system spanning clusters of grains whose velocities are strongly correlated.'}","{'cond-mat-0301201-2-0-0': 'In this paper we report the results of simulations of a 2D gravity driven, dissipative granular flow through a hopper system.', 'cond-mat-0301201-2-0-1': 'Measurements of impulse distributions [MATH] on the simulated system show flow-velocity-invariant behavior of the distribution for impulses larger than the average impulse [MATH].', 'cond-mat-0301201-2-0-2': 'For small impulses, however, [MATH] decreases significantly with flow velocity, a phenomenon which can be attributed exclusively to collisions between grains undergoing frequent collisions.', 'cond-mat-0301201-2-0-3': 'Visualizations of the system also show that these frequently colliding particles tend to form increasingly large linear clusters as the flow velocity decreases.', 'cond-mat-0301201-2-0-4': 'A model is proposed for the form of [MATH], given distributions of cluster size and velocity, which accurately predicts the observed form of the distribution.', 'cond-mat-0301201-2-0-5': 'Thus the impulse distribution provides some insight into the formation and properties of these ""dynamic"" force chains.', 'cond-mat-0301201-2-1-0': 'IntroductionGranular materials exhibit a wide spectrum of behavior ranging from gaseous to liquid to solid.', 'cond-mat-0301201-2-1-1': 'Remarkably, all of these phases of granular matter respond to external stimuli in a manner strikingly different from ordinary fluids and solids [CITATION].', 'cond-mat-0301201-2-1-2': 'In static granular piles, the spatially inhomogeneous manner in which stress is transmitted from the bulk of the pile to the boundary has been well documented[CITATION].', 'cond-mat-0301201-2-1-3': 'Experiments have shown that the force distribution [MATH] at the walls is exponential at large forces and exhibits a plateau at small forces[CITATION].', 'cond-mat-0301201-2-1-4': 'In addition, the highly stressed grains in static packings are organized into linear structures termed ""force chains""[CITATION].', 'cond-mat-0301201-2-1-5': 'The appearance of similar large scale structures in flowing granular matter would have significant implications for both continuum theories and descriptions of jamming in non-thermal systems.', 'cond-mat-0301201-2-1-6': 'A recent proposal for a unified picture of jamming in thermal and non-thermal systems suggests that jamming occurs due the formation of force chains whose presence is signalled by the appearance of a plateau in [MATH] [CITATION].', 'cond-mat-0301201-2-1-7': 'A continuum description of steady-state flow on an inclined plane models the system as a collection of transient 1D chains immersed in a viscous fluid[CITATION].', 'cond-mat-0301201-2-1-8': 'Indeed, transient ""clusters"" have been identified experimentally in granular surface flows [CITATION] and shear flows[CITATION], and simulations of chute flows have shown evidence for a plateau in [MATH] as the system approaches jamming[CITATION].', 'cond-mat-0301201-2-2-0': 'Recent experiments have been performed in a two dimensional hopper geometry to explore the presence of incipient force chains in purely collisional gravity-driven flow [CITATION].', 'cond-mat-0301201-2-2-1': 'Measurements of the time trace of the impulse delivered to a transducer placed at the side wall of the hopper have shown that the distribution of impulses, [MATH], displays an exponential decay at large [MATH], as for the case of static materials.', 'cond-mat-0301201-2-2-2': 'This exponential form of the distribution is maintained for all flow velocities from the largest measured ([MATH] = 60.0 cm/s) to the minimum flow velocity prior to the point at which the system no longer exhibits sustained flow ([MATH] = 9.4 cm/s).', 'cond-mat-0301201-2-2-3': 'However, at small [MATH], [MATH] develops an upward trend which becomes increasingly more evident as the flow velocity decreases.', 'cond-mat-0301201-2-3-0': 'In this letter, we report the results of event-driven simulations of a system of inelastic, monodispersed hard disks in the experimental geometry of Ref. [CITATION].', 'cond-mat-0301201-2-3-1': 'Our simulations provide clear evidence of an increasing proportion of collisions with small impulses as the flow velocity is decreased.', 'cond-mat-0301201-2-3-2': 'This results in the formation of a plateau in [MATH] as the minimum flow velocity for sustained flow is approached.', 'cond-mat-0301201-2-3-3': 'In addition, we observe the formation of clusters of disks which collide ""frequently"" and are reminiscent of the ""collapse strings"" observed in freely cooling granular matter[CITATION].', 'cond-mat-0301201-2-3-4': 'We present a model calculation which strongly suggests that the increase of small impulse events is associated with the growth of these clusters.', 'cond-mat-0301201-2-4-0': 'SimulationsThe grain dynamics used in the simulations are as in Ref. [CITATION] (momentum is conserved and at each interparticle collision the energy loss is proportional to [MATH] where [MATH] is the coefficient of restitution).', 'cond-mat-0301201-2-4-1': 'To ensure that the pressure is independent of the height the side walls must absorb some vertical momentum, therefore we impose the condition that collisions with the walls are inelastic in the tangential direction.', 'cond-mat-0301201-2-4-2': 'The flow velocity is controlled similarly to the experiments, by adjusting the width of the hopper opening.', 'cond-mat-0301201-2-4-3': 'However, as we wish to observe the system over many events, particles exiting the system at the bottom must be replaced at the top to create sustained flow.', 'cond-mat-0301201-2-4-4': 'This necessitates the introduction of a probability of reflection [MATH] at the bottom of the hopper (this would be equivalent to the presence of a sieve in experiment).', 'cond-mat-0301201-2-4-5': 'In our case, it provides another parameter with which we can tune the flow velocity.', 'cond-mat-0301201-2-4-6': 'Typically, our simulations were done on systems of 500 disks, with [MATH] = 0.9 and [MATH] = 0.4.', 'cond-mat-0301201-2-4-7': 'The simulation was run for [MATH] events for each flow velocity, with [MATH] discarded initially to allow the system time to reach steady state.', 'cond-mat-0301201-2-5-0': 'Simulation Results The physical quantity that is most closely related to [MATH] in our flowing hard-disk system is the distribution of impulses transferred at each collision, [MATH].', 'cond-mat-0301201-2-5-1': 'Unlike the experiments done in Ref. [CITATION], where the measurements were made only at the wall, we measure the magnitude of the momentum transfer at each collision event (note that the impulse and flow velocity are measured in units such that disk diameter [MATH], disk mass [MATH] and acceleration due to gravity [MATH] are all equal to one).', 'cond-mat-0301201-2-5-2': 'We observe an exponential form of [MATH] at impulses larger than the average impulse (Fig. [REF]) for the full range of flow rates.', 'cond-mat-0301201-2-5-3': 'After scaling the impulse by the average impulse, [MATH], the curves collapse onto each other for large impulses.', 'cond-mat-0301201-2-5-4': 'The small impulse behavior of the distribution, however, changes markedly with changes in the flow velocity.', 'cond-mat-0301201-2-5-5': 'As the flow velocity decreases, the height of the distribution at a minimum impulse ([MATH] for this distribution) increases and [MATH] begins to develop a plateau at small [MATH].', 'cond-mat-0301201-2-5-6': 'If the ratio between [MATH] at the peak impulse to [MATH] at the minimum impulse [MATH] is calculated and plotted against the flow velocity, a dramatic increase is observed (see inset to Fig. [REF]).', 'cond-mat-0301201-2-5-7': 'Extrapolating to small flow velocities indicates that this ratio becomes one at a non-zero flow rate.', 'cond-mat-0301201-2-6-0': 'Impulse distribution scaled to the average impulse, [MATH], with flow velocity decreasing from the bottom curve to the top curve.', 'cond-mat-0301201-2-6-1': 'The inset shows the ratio of [MATH] to [MATH] as a function of the flow velocity.', 'cond-mat-0301201-2-7-0': 'It should also be noted that the height of the distribution at [MATH] makes a sharp upturn which becomes more pronounced as the flow velocity decreases (the upturn is not present for the fastest flow velocity).', 'cond-mat-0301201-2-7-1': 'While we do not fully understand the reasons for this phenomenon, we believe it is not the same upward trend visible in the experiments.', 'cond-mat-0301201-2-7-2': 'That feature was a smooth continuation of the large impulse curve, while in our distribution a peak is seen.', 'cond-mat-0301201-2-7-3': 'In addition, our upturn occurs only at [MATH], regardless of the bin width used in constructing the distribution.', 'cond-mat-0301201-2-7-4': 'Such small impulses are not likely to be distinguishable in experiment due to the resolution limit imposed by the transducer.', 'cond-mat-0301201-2-7-5': 'For the remainder of our analysis, we will consider the impulse distribution without the [MATH] point.', 'cond-mat-0301201-2-8-0': 'The basic form of the impulse distribution, a peak at a finite value of [MATH] and an exponential tail can be understood on the basis of a flow of uncorrelated hard disks.', 'cond-mat-0301201-2-8-1': 'If the disks were uncorrelated, the impulse distribution would be a convolution of the individual momentum (velocity) distributions.', 'cond-mat-0301201-2-8-2': 'Since there is an average flow velocity, this would give rise to a peak in the distribution.', 'cond-mat-0301201-2-8-3': 'The exponential tail would be observed if the individual velocity distributions also had exponential tails.', 'cond-mat-0301201-2-8-4': 'The velocity distribution observed in our simulations can, to a first approximation, be described by Gaussians with exponential tails.', 'cond-mat-0301201-2-8-5': 'This argument would then suggest that the exponential tail arises from uncorrelated particles and is a consequence of the shape of the velocity distribution.', 'cond-mat-0301201-2-8-6': 'Viewed from this perspective, the lack of change in shape of [MATH] at large [MATH] as the flow approaches jamming is not surprising.', 'cond-mat-0301201-2-8-7': 'We will discuss this further in the context of a simple model to be presented below.', 'cond-mat-0301201-2-9-0': 'The change in [MATH] at small [MATH] is reminiscent of the universal jamming scenario[CITATION] although the direction of the change (filling up at small impulses) is different from the one observed in Lennard Jones systems and foams where the probability of finding small forces decreases as the system nears the jamming transition.', 'cond-mat-0301201-2-9-1': 'Nevertheless, the idea that changes in the distribution at small impulses or forces could be related to the appearance of structures akin to force chains is intriguing.', 'cond-mat-0301201-2-10-0': 'To explore a possible connection between the changes in [MATH] and the appearance of spatial inhomogeneities, we considered a question first asked in studies of inelastic collapse in freely cooling granular gases[CITATION], ""How many collisions does a given grain undergo in a fixed number of events?""', 'cond-mat-0301201-2-10-1': 'We define a minimum frequency of collision for a given particle as [MATH] (where [MATH] is the average time between events for a given flow velocity) and then identify those particles undergoing collisions with a frequency [MATH] as frequently colliding.', 'cond-mat-0301201-2-10-2': 'This allows us to define a minimum number of collisions a frequently colliding particle must undergo in a given number of events.', 'cond-mat-0301201-2-10-3': 'We then construct an image of our system at every 1000 events, and color all disks satisfying the criteria (see Fig. [REF]a,b).', 'cond-mat-0301201-2-10-4': 'As we decrease the flow velocity, the frequently colliding particles form increasingly larger linear clusters (compare Fig. [REF]a, where [MATH] = 2.03 in our units or 35.6 cm/s and Fig. [REF]b, where [MATH] = 0.96 or 16.9 cm/s).', 'cond-mat-0301201-2-10-5': 'These 1D structures observed in our simulation are reminiscent of the transient solid chains postulated by the hydrodynamic model of Ref. [CITATION].', 'cond-mat-0301201-2-10-6': 'Comparisons of the impulse distribution of the frequently colliding particles and the impulse distribution of the remaining ""rarely colliding"" particles (Fig. [REF] c,d) reveals that it is the contribution from the frequently colliding particles which causes the height of the total impulse distribution at [MATH] to increase in the manner seen in Fig. [REF]a.', 'cond-mat-0301201-2-10-7': 'The large impulse behavior appears to be dominated by the rarely colliding particles.', 'cond-mat-0301201-2-10-8': 'These observations are relatively insensitive to changes in [MATH] provided it is larger than a minimum value.', 'cond-mat-0301201-2-11-0': '(a) (b) Sample image of simulation for [MATH] = 2.03 (a) and for [MATH] = 0.96 (b).', 'cond-mat-0301201-2-11-1': '(c) (d) Impulse distributions for all particles, frequently colliding particles and rarely colliding particles for [MATH] = 2.03 (c) and for [MATH] = 0.96 (d).', 'cond-mat-0301201-2-11-2': 'Note that these distributions are unnormalized.', 'cond-mat-0301201-2-12-0': 'Impulse Distribution Model A possible connection between the shape of the impulse distribution and the development of linear clusters of frequently colliding particles can be drawn by investigating the following model.', 'cond-mat-0301201-2-12-1': 'Consider a one dimensonal cluster of particles which are all moving with the same velocity.', 'cond-mat-0301201-2-12-2': 'Now if another particle, travelling at some speed [MATH] relative to the cluster (see Fig. [REF]a) collides inelastically with one end of the chain, the impulse associated with that collision will be [MATH] where [MATH] is related to the coefficient of restitution by [MATH][CITATION].', 'cond-mat-0301201-2-12-3': 'If the chain was only comprised of one grain, the impulse distribution [MATH] (given an initial incoming speed [MATH]) would be a single spike of height 1 located at [MATH].', 'cond-mat-0301201-2-12-4': 'For two particles, [MATH] would be a bar of height 1 with limits at [MATH] and [MATH].', 'cond-mat-0301201-2-12-5': 'Continuing this argument for clusters containing [MATH] particles, then as [MATH] becomes very large, the leftmost limit of [MATH] approaches zero (Fig. [REF]b).', 'cond-mat-0301201-2-13-0': '(a) Particle of speed [MATH] incident on 1D cluster of [MATH] stationary particles.', 'cond-mat-0301201-2-13-1': '(b) Resulting form of [MATH].', 'cond-mat-0301201-2-14-0': 'Given a distribution of speeds [MATH] for the incident particle, and a distribution of cluster sizes [MATH], the total impulse distribution [MATH] is: [EQUATION]', 'cond-mat-0301201-2-14-1': 'If the cluster size distribution falls off sharply, then the shape of [MATH] will essentially reflect the shape of [MATH].', 'cond-mat-0301201-2-14-2': 'If, however, the cluster size distribution becomes broad, then the small impulse end of [MATH] will flatten out reflecting the nature of [MATH] for large [MATH].', 'cond-mat-0301201-2-15-0': 'As detailed previously, the criterion that we use to identify the clusters in our simulations is that of ""frequent"" collisions.', 'cond-mat-0301201-2-15-1': 'Due to the inelastic nature of the collisions, the velocities of these particles become highly correlated[CITATION] and they can be idealized as clusters of disks moving with the same velocity.', 'cond-mat-0301201-2-15-2': 'This velocity correlation within spatial clusters of particles has been directly observed in granular surface flows[CITATION].', 'cond-mat-0301201-2-15-3': 'Since we observe these clusters to be linear and growing in size with decreasing flow rates, it is plausible that the changes in [MATH] observed in our simulations is related to a change in distribution of the size of the clusters of frequently-colliding disks.', 'cond-mat-0301201-2-15-4': 'In order to put this conjecture on a firmer footing, we calculated the impulse distribution from our model using forms of [MATH] and [MATH] which provide good fits to our simulation data.', 'cond-mat-0301201-2-15-5': 'The form of [MATH] that we use is a Gaussian with an exponential tail.', 'cond-mat-0301201-2-15-6': 'This form is representative of the simulation results for [MATH] with the center of the Gaussian reflecting the average flow velocity.', 'cond-mat-0301201-2-15-7': 'The observed cluster size distribution is consistent with the form [MATH] with [MATH] increasing as the flow velocity decreases.', 'cond-mat-0301201-2-15-8': 'Fig. [REF]a shows [MATH] obtained from our simulations for three different flow rates.', 'cond-mat-0301201-2-15-9': 'Note that a change in [MATH] will produce a change in [MATH] for a given flow velocity but will not alter the shape of [MATH] nor the trend of increasing [MATH] with decreasing flow velocity.', 'cond-mat-0301201-2-16-0': '(a) Simulation results for [MATH] for varying flow rates.', 'cond-mat-0301201-2-16-1': '(b) Results of the proposed model for [MATH] with [MATH] increasing from the bottom curve to the top curve.', 'cond-mat-0301201-2-16-2': 'The inset shows the ratio of [MATH] to [MATH] as a function of [MATH].', 'cond-mat-0301201-2-17-0': 'Using these forms of [MATH] and [MATH], we find that the [MATH] obtained from the simple model has a form which is remarkably similar to the measured [MATH], as shown in Fig. [REF]b.', 'cond-mat-0301201-2-17-1': 'The control parameter in the model is [MATH] and the contribution at small impulses increases with [MATH] as is evident from the inset to Fig. [REF]b.', 'cond-mat-0301201-2-17-2': 'The correspondence between the model and our simulations strongly suggests that the observed changes in [MATH] are associated with the growth of clusters of frequently colliding particles and as [MATH] diverges, the ratio of [MATH] to [MATH] approaches unity.', 'cond-mat-0301201-2-17-3': 'The tail of the [MATH] distribution is controlled by the shape of [MATH] as we have verified within the model.', 'cond-mat-0301201-2-18-0': 'The picture which seems to be emerging from our simulations is that of increasingly larger scale spatial heterogeneities developing as the system approaches jamming.', 'cond-mat-0301201-2-18-1': 'Since the clusters are essentially the same as the ones identified in freely cooling granular matter, their origin lies in the dissipative nature of the medium.', 'cond-mat-0301201-2-18-2': 'The heterogeneities reflect strong velocity correlations of grains and leave a distinctive signature in the impulse distribution.', 'cond-mat-0301201-2-18-3': 'Whether the clusters that we have identified are indeed incipient force chains remains to be verified.', 'cond-mat-0301201-2-18-4': 'If this connection can be established, for example through the calculation of stress correlations in the flowing medium, then our analysis will provide a natural connection between [MATH] and force chains.', 'cond-mat-0301201-2-18-5': 'Moreover, it would a indicate that jamming occurs through the formation of system spanning clusters of grains whose velocities are strongly correlated.'}","[['cond-mat-0301201-1-0-0', 'cond-mat-0301201-2-0-0'], ['cond-mat-0301201-1-0-1', 'cond-mat-0301201-2-0-1'], ['cond-mat-0301201-1-0-2', 'cond-mat-0301201-2-0-2'], ['cond-mat-0301201-1-0-3', 'cond-mat-0301201-2-0-3'], ['cond-mat-0301201-1-0-4', 'cond-mat-0301201-2-0-4'], ['cond-mat-0301201-1-0-5', 'cond-mat-0301201-2-0-5'], ['cond-mat-0301201-1-1-0', 'cond-mat-0301201-2-1-0'], ['cond-mat-0301201-1-1-1', 'cond-mat-0301201-2-1-1'], ['cond-mat-0301201-1-1-2', 'cond-mat-0301201-2-1-2'], ['cond-mat-0301201-1-1-3', 'cond-mat-0301201-2-1-3'], ['cond-mat-0301201-1-1-4', 'cond-mat-0301201-2-1-4'], ['cond-mat-0301201-1-1-5', 'cond-mat-0301201-2-1-5'], ['cond-mat-0301201-1-1-6', 'cond-mat-0301201-2-1-6'], ['cond-mat-0301201-1-1-7', 'cond-mat-0301201-2-1-7'], ['cond-mat-0301201-1-1-8', 'cond-mat-0301201-2-1-8'], ['cond-mat-0301201-1-10-0', 'cond-mat-0301201-2-10-0'], ['cond-mat-0301201-1-10-1', 'cond-mat-0301201-2-10-1'], ['cond-mat-0301201-1-10-2', 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'cond-mat-0301201-2-14-2'], ['cond-mat-0301201-1-11-1', 'cond-mat-0301201-2-11-1'], ['cond-mat-0301201-1-11-2', 'cond-mat-0301201-2-11-2'], ['cond-mat-0301201-1-18-0', 'cond-mat-0301201-2-18-0'], ['cond-mat-0301201-1-18-1', 'cond-mat-0301201-2-18-1'], ['cond-mat-0301201-1-18-2', 'cond-mat-0301201-2-18-2'], ['cond-mat-0301201-1-18-3', 'cond-mat-0301201-2-18-3'], ['cond-mat-0301201-1-18-4', 'cond-mat-0301201-2-18-4'], ['cond-mat-0301201-1-18-5', 'cond-mat-0301201-2-18-5']]",[],[],[],[],"['cond-mat-0301201-1-11-0', 'cond-mat-0301201-2-11-0']","{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}",https://arxiv.org/abs/cond-mat/0301201,,,
hep-lat-0108006,"{'hep-lat-0108006-1-0-0': 'We study the topological susceptibility, [MATH], in QCD with two quark flavours using lattice field configurations that have been produced with an [MATH] improved quark action.', 'hep-lat-0108006-1-0-1': 'We find clear evidence for the expected suppression at small quark mass, and examine the variation of [MATH] with this mass.', 'hep-lat-0108006-1-0-2': 'The resulting estimate of the pion decay constant, [MATH], is consistent with the experimental value of [MATH].', 'hep-lat-0108006-1-0-3': 'We compare [MATH] to the large-[MATH] prediction and find consistency over a large range of quark masses.', 'hep-lat-0108006-1-0-4': 'We discuss the benefits of the non-perturbative action improvement scheme and of the stategy of keepng the lattice spacing (nearly) fixed as the quark mass is varied.', 'hep-lat-0108006-1-0-5': 'We compare our results with other studies and suggest why such a quark mass dependence has not always been seen.', 'hep-lat-0108006-1-1-0': '# Introduction', 'hep-lat-0108006-1-2-0': 'In gluodynamics (the pure gauge or ""quenched"" theory) lattice calculations of the continuum topological susceptibility now appear to be relatively free of the systematic errors arising from the discretisation, the finite volumes and the various measurement algorithms employed (for a recent review, see [CITATION]).', 'hep-lat-0108006-1-3-0': 'The inclusion of sea quarks in (""dynamical"") lattice simulations, even at the relatively large quark masses currently employed, is numerically extremely expensive, and can only be done for lattices with relatively few sites (typically [MATH]).', 'hep-lat-0108006-1-3-1': 'To avoid significant finite volume errors, the lattice must then be relatively coarse, with, in our case, a spacing [MATH].', 'hep-lat-0108006-1-3-2': 'This is a significant fraction of the mean instanton radius, as calculated in gluodynamics, and thus precludes a robust, detailed study of the local topological features of the vacuum in the presence of sea quarks.', 'hep-lat-0108006-1-3-3': 'The topological susceptibility, on the other hand, may be calculated with some confidence and provides one of the first opportunities to test some of the more interesting predictions of QCD.', 'hep-lat-0108006-1-3-4': 'Indeed, it is in these measurements that we find some of the most striking evidence for the presence of the sea quarks (or, alternatively, for a strong quenching effect) in the lattice simulations.', 'hep-lat-0108006-1-4-0': 'We recall that the ensembles used here have been produced with two notable features [CITATION].', 'hep-lat-0108006-1-4-1': 'The first is the use of an improved action, such that leading order lattice discretisation effects are expected to depend quadratically, rather than linearly, on the lattice spacing (just as in gluodynamics).', 'hep-lat-0108006-1-4-2': 'In addition, the action parameters have been chosen to maintain a relatively constant lattice spacing, particularly for the larger values of the quark mass.', 'hep-lat-0108006-1-5-0': 'These features have allowed us to see the first clear evidence [CITATION] for the expected suppression of the topological susceptibility in the chiral limit, despite our relatively large quark masses.', 'hep-lat-0108006-1-5-1': 'From this behaviour we can directly estimate the pion decay constant without needing to know the lattice operator renormalisation factors that arise in more conventional calculations.', 'hep-lat-0108006-1-6-0': 'The structure of this paper is as follows.', 'hep-lat-0108006-1-6-1': 'In Section [REF] we discuss the measurement of the topological susceptibility and its expected behaviour both near the chiral limit, and in the limit of a large number of colours, [MATH].', 'hep-lat-0108006-1-6-2': 'In Section [REF] we describe the UKQCD ensembles and the lattice measurements of the topological susceptibility over a range of sea quark masses.', 'hep-lat-0108006-1-6-3': 'We fit these with various ansatze motivated by the previous section.', 'hep-lat-0108006-1-6-4': 'We compare our findings with other recent studies in Section [REF].', 'hep-lat-0108006-1-6-5': 'Finally, we provide a summary in Section [REF].', 'hep-lat-0108006-1-7-0': ""These results were presented at the IOP2000 [CITATION], the Confinement IV [CITATION] and, in a much more preliminary form, the Lattice '99 [CITATION] conferences."", 'hep-lat-0108006-1-7-1': 'Since then, we have increased the size of several ensembles and included a new parameter set to try and address the issue of discretisation effects and the continuum limit in our results.', 'hep-lat-0108006-1-7-2': 'We also have more accurate results from the quenched theory with which to compare.', 'hep-lat-0108006-1-7-3': 'A brief summary of this work appears in [CITATION].', 'hep-lat-0108006-1-8-0': '# The topological susceptibility', 'hep-lat-0108006-1-9-0': 'In four-dimensional Euclidean space-time, SU(3) gauge field configurations can be separated into topological classes, and moving between different classes is not possible by a smooth deformation of the fields.', 'hep-lat-0108006-1-9-1': 'The classes are characterised by an integer-valued winding number.', 'hep-lat-0108006-1-9-2': 'This Pontryagin index, or topological charge [MATH], can be obtained by integrating the local topological charge density [EQUATION] over all space-time [EQUATION]', 'hep-lat-0108006-1-9-3': 'The topological susceptibility is the expectation value of the squared charge, normalised by the volume [EQUATION]', 'hep-lat-0108006-1-9-4': 'An isolated topological charge induces an exact zero-mode in the quark Dirac operator.', 'hep-lat-0108006-1-9-5': ""As a result sea quarks in the vacuum induce an instanton-anti-instanton attraction which becomes stronger as the quark masses, [MATH], [MATH], , decrease towards zero (the 'chiral limit'), and the topological charge and susceptibility will be suppressed to leading order in the quark mass [CITATION], [EQUATION] where [EQUATION] is the chiral condensate (see [CITATION] for a recent discussion)."", 'hep-lat-0108006-1-9-6': 'Here we have assumed [MATH] and we neglect the contribution of heavier quarks.', 'hep-lat-0108006-1-9-7': 'PCAC theory relates this to the pion decay constant [MATH] and mass [MATH] via the Gell-Mann-Oakes-Renner relation as [EQUATION] and we may combine these for [MATH] degenerate light flavours to obtain [EQUATION] in a convention where the experimental value of [MATH] .', 'hep-lat-0108006-1-9-8': 'This relation should hold in the limit [EQUATION] which is the Leutwyler-Smilga parameter to leading order in [MATH].', 'hep-lat-0108006-1-9-9': 'We anticipate our results here to say that even on our most chiral lattices the lhs of Eqn. [REF] is of order 10 and so this bound is well satisfied.', 'hep-lat-0108006-1-9-10': 'Thus a calculation of [MATH] as a function of [MATH] should allow us to obtain a value of [MATH].', 'hep-lat-0108006-1-9-11': 'This method has an advantage over more conventional calculations in that it does not require us to know lattice operator renormalisation constants which are required for matching matrix elements, but which are usually difficult to calculate.', 'hep-lat-0108006-1-9-12': 'In principle, we require instead knowledge of the renormalisation of the topological charge operators, but we see in the next Section that this problem can be readily overcome.', 'hep-lat-0108006-1-10-0': 'As [MATH] and [MATH] increase away from zero we expect higher order terms to check the rate of increase of the topological susceptibility so that, as [MATH], [MATH] approaches the quenched value, [MATH].', 'hep-lat-0108006-1-10-1': 'In fact, as we shall see below, the values of [MATH] that we obtain are not very much smaller than [MATH].', 'hep-lat-0108006-1-10-2': 'So there is the danger of a substantial systematic error in simply applying Eqn. [REF] at our smallest values of [MATH] in order to estimate [MATH].', 'hep-lat-0108006-1-10-3': 'To estimate this error it would be useful to have some understanding of how [MATH] behaves over the whole range of [MATH].', 'hep-lat-0108006-1-10-4': 'This is the question to which we now turn.', 'hep-lat-0108006-1-11-0': 'There are two quite different reasons why [MATH] might not be much smaller than [MATH].', 'hep-lat-0108006-1-11-1': 'The obvious first possibility is that [MATH] is large.', 'hep-lat-0108006-1-11-2': 'The second possibility is more subtle: [MATH] may be small but QCD may be close to its large-[MATH] limit [CITATION].', 'hep-lat-0108006-1-11-3': 'Because fermion effects are non-leading in powers of [MATH], we expect [MATH] for any fixed, non-zero value of [MATH], however small, as the number of colours [MATH].', 'hep-lat-0108006-1-11-4': ""There are phenomenological reasons [CITATION] for believing that QCD is 'close' to [MATH], and so this is not an unrealistic consideration."", 'hep-lat-0108006-1-11-5': 'Moreover in the case of [MATH] gauge theories it has been shown [CITATION] that even SU(2) is close to SU([MATH]).', 'hep-lat-0108006-1-11-6': 'Recent calculations in four dimensions [CITATION] indicate that the same is true there.', 'hep-lat-0108006-1-11-7': 'In the present simulations, the lighter quark masses straddle the strange quark mass and so it is not obvious if we should regard them as being large or small.', 'hep-lat-0108006-1-11-8': 'We shall therefore take seriously both the possibilities discussed above.', 'hep-lat-0108006-1-12-0': 'We start by assuming the quark mass is small but that we are close to the large-[MATH] limit.', 'hep-lat-0108006-1-12-1': 'In this limit, the topological susceptibility is known [CITATION] to vary as [EQUATION] where [MATH], [MATH] are the quantities at leading order in [MATH].', 'hep-lat-0108006-1-12-2': 'In the chiral limit, at fixed [MATH], this reproduces Eqn. [REF].', 'hep-lat-0108006-1-12-3': 'In the large-[MATH] limit, at fixed [MATH], it tends to the quenched susceptibility [MATH] because [MATH].', 'hep-lat-0108006-1-12-4': 'The corrections to Eqn. [REF] are of higher order in [MATH] and/or lower order in [MATH].', 'hep-lat-0108006-1-13-0': 'We now consider the alternative possibility: that [MATH] is not small, that higher-order corrections to [MATH] will be important for most of the values of [MATH] at which we perform calculations, and that we therefore need an expression for [MATH] that interpolates between [MATH] and [MATH].', 'hep-lat-0108006-1-13-1': 'Clearly one cannot hope to derive such an expression from first principles, so we will simply choose one that we can plausibly argue is approximately correct.', 'hep-lat-0108006-1-13-2': 'The form we choose is [EQUATION] where [MATH] is the pion decay constant in the chiral limit.', 'hep-lat-0108006-1-13-3': 'The coefficients have been chosen so that this reproduces Eqn. [REF] when [MATH] and [MATH] when [MATH].', 'hep-lat-0108006-1-13-4': 'Thus, this interpolation formula possesses the correct limits and it approaches those limits with power-like corrections.', 'hep-lat-0108006-1-14-0': 'We shall use the expressions in Eqns. [REF], [REF] and [REF] to analyse the [MATH] dependence of our calculated values of [MATH] and to obtain a value of [MATH] together with an estimate of the systematic error on that value.', 'hep-lat-0108006-1-14-1': 'In addition, the comparison with Eqn. [REF] can provide us with some evidence for whether QCD is close to its large-[MATH] limit or not.', 'hep-lat-0108006-1-15-0': '# Lattice measurements', 'hep-lat-0108006-1-16-0': 'We have calculated [MATH] on four complete ensembles of dynamical configurations produced by the UKQCD collaboration, as well as two which are still in progress [CITATION].', 'hep-lat-0108006-1-16-1': 'Details of these data sets are given in Table [REF].', 'hep-lat-0108006-1-16-2': 'The SU(3) gauge fields are governed by the Wilson plaquette action, with ""clover"" improved Wilson fermions.', 'hep-lat-0108006-1-16-3': 'The improvement is non-perturbative, with [MATH] chosen to render the leading order discretisation errors quadratic (rather than linear) in the lattice spacing, [MATH].', 'hep-lat-0108006-1-17-0': 'The theory has two coupling constants.', 'hep-lat-0108006-1-17-1': 'In pure gluodynamics the gauge coupling, [MATH], controls the lattice spacing, with larger values reducing [MATH] as we move towards the critical value at [MATH].', 'hep-lat-0108006-1-17-2': 'In simulations with dynamical fermions it has the same role for a fixed fermion coupling, [MATH].', 'hep-lat-0108006-1-17-3': 'The latter controls the quark mass, with [MATH] from below corresponding to the massless limit.', 'hep-lat-0108006-1-17-4': 'In dynamical simulations, however, the fermion coupling also affects the lattice spacing, which will become larger as [MATH] is reduced (and hence [MATH] increased) at fixed [MATH].', 'hep-lat-0108006-1-18-0': 'The three least chiral UKQCD ensembles (by which we mean largest [MATH]) are e[MATH], e[MATH] and e[MATH].', 'hep-lat-0108006-1-18-1': ""By appropriately decreasing [MATH] as [MATH] is increased, the couplings are 'matched' to maintain a constant lattice spacing [CITATION] (which is 'equivalent' to [MATH] in gluodynamics with a Wilson action [CITATION]) whilst approaching the chiral limit."", 'hep-lat-0108006-1-18-2': 'The physical volume and discretisation effects should thus be very similar on these lattices.', 'hep-lat-0108006-1-18-3': 'The remaining ensembles have lower quark masses, but are at a slightly reduced lattice spacing.', 'hep-lat-0108006-1-18-4': '(To have maintained a matched lattice spacing here would have required reducing [MATH] to values where the non-perturbative value [MATH] is not known.)', 'hep-lat-0108006-1-18-5': 'As the lattices are all [MATH], we believe that the minor reduction in the lattice volume should not lead to significant finite volume corrections.', 'hep-lat-0108006-1-18-6': 'We also remark that ensembles e[MATH] and e[MATH] have been matched to have approximately the same chirality, but at (mildly) different lattice spacings.', 'hep-lat-0108006-1-19-0': 'Four-dimensional lattice theories are scale free, and the dimensionless lattice quantities must be cast in physical units through the use of a known scale.', 'hep-lat-0108006-1-19-1': 'For this work, we use the Sommer scale [CITATION] both to define the lattice spacing for the matching procedure, and to set the scale.', 'hep-lat-0108006-1-19-2': 'The measured value of [MATH] on each ensemble, as listed in Table [REF], corresponds to the same physical value of [MATH].', 'hep-lat-0108006-1-19-3': '([MATH] is the dimensionless lattice value of [MATH] in lattice units i.e. [MATH].', 'hep-lat-0108006-1-19-4': 'We use the same notation for other quantities.)', 'hep-lat-0108006-1-19-5': 'As we are in the scaling window of the theory, we can then use the naive dimensions of the various operators to relate lattice and physical quantities, e.g. [MATH], where we have incorporated the expected non-perturbative removal of the corrections linear in the lattice spacing.', 'hep-lat-0108006-1-20-0': 'Further details of the parameters and the scale determination are given in [CITATION].', 'hep-lat-0108006-1-20-1': 'Measurements were made on ensembles of 400-800 configurations of size [MATH], separated by ten hybrid Monte Carlo trajectories.', 'hep-lat-0108006-1-20-2': 'Correlations in the data were managed through jack-knife binning of the data, using ten bins whose size is large enough that neighbouring bin averages may be regarded as uncorrelated.', 'hep-lat-0108006-1-21-0': 'We begin, however, with a discussion of lattice operators and results in the quenched theory.', 'hep-lat-0108006-1-22-0': '## Lattice operators and [MATH]', 'hep-lat-0108006-1-23-0': 'The simplest lattice topological charge density operator is [EQUATION] where [MATH] denotes the product of SU(3) link variables around a given plaquette.', 'hep-lat-0108006-1-23-1': 'We use a reflection-symmetrised version and form', 'hep-lat-0108006-1-24-0': 'Q & = & 132^2 _n q(n) ,', 'hep-lat-0108006-1-25-0': '& = & Q^2 L^3 T', 'hep-lat-0108006-1-26-0': 'with [MATH] the lattice volume.', 'hep-lat-0108006-1-26-1': 'In general, [MATH] will not give an integer-valued topological charge due to finite lattice spacing effects.', 'hep-lat-0108006-1-26-2': 'There are at least three sources of these.', 'hep-lat-0108006-1-26-3': 'First is the breaking of scale invariance by the lattice which leads to the smallest instantons having a suppressed action (at least with the Wilson action) and a topological charge less than unity (at least with the operator in Eqn. [REF]).', 'hep-lat-0108006-1-26-4': 'We do not address this problem in this study, although attempts can be made to correct for it [CITATION], but simply accept this as part of the overall [MATH] error.', 'hep-lat-0108006-1-26-5': 'In addition to this, the underlying topological signal on the lattice is distorted by the presence of large amounts of UV noise on the scale of the lattice spacing [CITATION], and by a multiplicative renormalisation factor [CITATION] that is unity in the continuum, but otherwise suppresses the observed charge.', 'hep-lat-0108006-1-26-6': 'Various solutions to these problems exist [CITATION].', 'hep-lat-0108006-1-26-7': ""In this study we opt for the 'cooling' approach."", 'hep-lat-0108006-1-26-8': 'Cooling explicitly erases the ultraviolet fluctuations so that the perturbative lattice renormalisation factors for the topological charge and susceptibility are driven to their trivial continuum values, leaving [MATH] corrections that may be absorbed into all the other lattice corrections of this order.', 'hep-lat-0108006-1-26-9': ""We cool by moving through the lattice in a 'staggered' fashion, cooling each link by minimising the Wilson gauge action applied to each of the three Cabibbo-Marinari SU(2) subgroups in the link element in turn."", 'hep-lat-0108006-1-26-10': '(The Wilson gauge action is the most local, and thus particularly efficient at removing short distance fluctuations whilst preserving the long range correlations in the fields.)', 'hep-lat-0108006-1-26-11': ""Carrying out this procedure once on every link constitutes a cooling sweep (or 'cool')."", 'hep-lat-0108006-1-26-12': 'The violation of the instanton scale invariance on the lattice, with a Wilson action, is such that an isolated instanton cooled in this way will slowly shrink, and will eventually disappear when its core size is of the order of a lattice spacing, leading to a corresponding jump in the topological charge.', 'hep-lat-0108006-1-26-13': 'Such events can, of course, be detected by monitoring [MATH] as a function of the number of cooling sweeps, [MATH].', 'hep-lat-0108006-1-26-14': 'Instanton-anti-instanton pairs may also annihilate, but this has no net effect on [MATH].', 'hep-lat-0108006-1-26-15': 'However, these observations do motivate us to perform the minimum number of cools necessary to obtain an estimate of [MATH] that is stable with further increasing [MATH] (subject to the above).', 'hep-lat-0108006-1-27-0': 'To estimate this point we calculate the normalised correlation function between the topological charges measured after [MATH] cooling sweeps, and a nominally asymptotic 25 cooling sweeps: [EQUATION]', 'hep-lat-0108006-1-27-1': 'In Fig. [REF] we show a typical plot for ensemble e[MATH].', 'hep-lat-0108006-1-27-2': 'As discussed before, we have opted not to attempt to round the topological charge to integer values.', 'hep-lat-0108006-1-27-3': 'We find [MATH] and [MATH] to be stable within statistical errors for [MATH], and the results presented here are for [MATH].', 'hep-lat-0108006-1-28-0': 'The topological charge of a configuration is related to the smallest eigenvalues of the Dirac matrix and as such is often believed to be one of the slowest modes to decorrelate during Monte Carlo simulations.', 'hep-lat-0108006-1-28-1': 'It is crucial for the error analysis that the bin sizes for the data are at least twice the integrated autocorrelation times.', 'hep-lat-0108006-1-28-2': 'In Fig. [REF] we plot a typical time series of the topological charge measured every ten hybrid Monte Carlo trajectories.', 'hep-lat-0108006-1-28-3': 'The rapid variation between configurations suggests that the integrated autocorrelation time is small even for the topological charge.', 'hep-lat-0108006-1-28-4': 'Estimates of this are given in Table [REF] in units of ten trajectories.', 'hep-lat-0108006-1-28-5': 'These reinforce the impression gained from the time series plots.', 'hep-lat-0108006-1-28-6': 'The bins used in the jack-knife statistical analysis are between 400 and 1000 trajectories in length and thus may be confidently assumed to be statistically independent.', 'hep-lat-0108006-1-28-7': 'It is interesting that although autocorrelation times are hard to estimate accurately, it does appear that they increase as we move away from the chiral limit (c.f. Ref. [CITATION]).', 'hep-lat-0108006-1-29-0': 'In Fig. [REF] we divide the topological charge measurements made over an ensemble into bins of unit width centred on the integers, and plot a histogram of the probability of finding a configuration with each charge, with errors from the jack-knife analysis.', 'hep-lat-0108006-1-29-1': 'We find for all our ensembles that these histograms are very close to being symmetric, centred around [MATH] and consistent with a Gaussian envelope.', 'hep-lat-0108006-1-29-2': 'The hybrid Monte Carlo appears to be sampling the topological sectors correctly, and it is legitimate to extract an estimate of the topological susceptibility.', 'hep-lat-0108006-1-29-3': 'On the histograms we show this estimate as a Gaussian curve [EQUATION]', 'hep-lat-0108006-1-29-4': 'The central line uses our estimate of [MATH], whilst the outlying curves use the central value plus or minus one standard deviation.', 'hep-lat-0108006-1-29-5': 'The agreement with the histograms is good.', 'hep-lat-0108006-1-30-0': 'We also remark that on a lattice one obviously loses instantons with sizes [MATH].', 'hep-lat-0108006-1-30-1': 'Since the (pure gauge) instanton density decreases as [MATH] when [MATH] this would appear to induce a negligible [MATH] error in the susceptibility.', 'hep-lat-0108006-1-30-2': 'However this is only true for [MATH], and the error can be substantial for the coarse lattices often used in dynamical simulations.', 'hep-lat-0108006-1-31-0': 'In general, then, we expect the topological charge and susceptibility to be suppressed at non-zero lattice spacing.', 'hep-lat-0108006-1-31-1': 'In gluodynamics with the Wilson action this can be crudely fitted by a leading order, and negative, correction term [CITATION] [EQUATION] in the window of lattice spacings that is comparable to those used in current UKQCD dynamical simulations.', 'hep-lat-0108006-1-31-2': 'We shall use this formula as a guide in Section [REF].', 'hep-lat-0108006-1-32-0': 'We would like to compare the measurements in the presence of sea quarks with an appropriate quenched limit.', 'hep-lat-0108006-1-32-1': ""In principle this is ambiguous, as at different lattice spacings the 'equivalent' quenched limit will be different."", 'hep-lat-0108006-1-32-2': 'The variation in this quenched value, [MATH], over the range in lattice spacings of the UKQCD ensembles is, however, much less than the statistical errors on the measurements themselves.', 'hep-lat-0108006-1-32-3': 'This is a consequence of our strategy of approximately matching [MATH] for heavier [MATH].', 'hep-lat-0108006-1-32-4': 'Without this, if the calculations were performed at fixed [MATH], the lattice spacing would become increasingly coarse with increasing [MATH], and the reduction in the quenched susceptibility would be much more pronounced over this range of [MATH].', 'hep-lat-0108006-1-32-5': 'We shall return to this important point when we discuss other work in section [REF].', 'hep-lat-0108006-1-32-6': 'Note that in this transition to physical units we include the quoted statistical error in our determination of [MATH], but do not attempt to include any systematic uncertainties; there are several methods for determining [MATH] which differ at finite lattice spacing.', 'hep-lat-0108006-1-32-7': 'At [MATH] the variation in [MATH] is around 20%.', 'hep-lat-0108006-1-32-8': 'For comparison purposes we must use consistent methods for determination, and if one does so then one finds [CITATION] that [MATH] at [MATH] in the quenched theory, demonstrating that this provides an appropriate quenched limit for our calculations.', 'hep-lat-0108006-1-32-9': 'The topological susceptibility has been calculated on [MATH] lattices at [MATH] in [CITATION] and that will provide our value of [MATH].', 'hep-lat-0108006-1-32-10': 'Note that in contrast to the dynamical measurements, in the quenched case the topological charge measurements were made after 20 cooling sweeps.', 'hep-lat-0108006-1-32-11': 'The links were cooled in a sequential, rather than staggered, manner, and a non-symmetrised topological charge density operator was employed.', 'hep-lat-0108006-1-32-12': 'Such details should, however, only change the estimate within statistical errors.', 'hep-lat-0108006-1-33-0': '## Sea quark effects in the topological susceptibility', 'hep-lat-0108006-1-34-0': 'In Table [REF] we give our estimates of the topological susceptibility in physical units, using [MATH] as the scale.', 'hep-lat-0108006-1-34-1': 'In Fig. [REF] we plot [MATH] versus a similarly scaled pseudoscalar meson mass (calculated, of course, with valence quarks that are degenerate with those in the sea, i.e. [MATH]).', 'hep-lat-0108006-1-34-2': 'We also plot the corresponding value of the quenched topological susceptibility, as calculated at [MATH].', 'hep-lat-0108006-1-35-0': 'Comparing the dynamical and quenched values, the effects of the sea quarks are clear.', 'hep-lat-0108006-1-35-1': 'Whilst the measurement on e[MATH] and e[MATH] are consistent with the quenched value, moving to smaller [MATH]) the topological susceptibility is increasingly suppressed.', 'hep-lat-0108006-1-36-0': 'We can make this observation more quantitative by attempting to fit our values of [MATH] with the expected functional form in Eqn. [REF], so extracting a value of [MATH].', 'hep-lat-0108006-1-36-1': 'But we must first be clear whether this fit is justified, and what exactly we are extrapolating in, bearing in mind that Eqn. [REF] is strictly a chiral expansion that describes the behaviour for small sea quark masses in the continuum limit.', 'hep-lat-0108006-1-37-0': 'An immediate concern is that our cooling technique will occasionally misidentify the value of [MATH] and, in addition, that at finite lattice spacing the exact zero modes associated with the topological charge [MATH] are shifted away from zero.', 'hep-lat-0108006-1-37-1': 'All this implies that [MATH] will not in fact vanish as [MATH].', 'hep-lat-0108006-1-37-2': 'However we expect this effect to be small for the following reasons.', 'hep-lat-0108006-1-37-3': 'First, the use of an improved fermion action should ensure that the zero-mode shift will only be significant for very small instantons, i.e. those whose sizes are [MATH].', 'hep-lat-0108006-1-37-4': 'These are unlikely to survive the cooling and should not contribute to our calculated value of [MATH].', 'hep-lat-0108006-1-37-5': 'Large instantons, on the other hand, for which any zero mode shift should be negligible, will certainly survive the cooling.', 'hep-lat-0108006-1-37-6': 'The remaining ambiguity involves the smaller, but not very small, instantons.', 'hep-lat-0108006-1-37-7': 'These might be erased by the cooling but the probability is small simply because the number of such charges is small [CITATION].', 'hep-lat-0108006-1-37-8': 'For example, we can see from Fig. 12 in [CITATION] that the cooling only appears to cut out instantons with [MATH].', 'hep-lat-0108006-1-37-9': 'Since the lattice spacing in that plot is a factor of 1.5 smaller than at our equivalent quenched [MATH] value of 5.93, we would expect that cooling at [MATH] would affect instantons that have [MATH] in that plot.', 'hep-lat-0108006-1-37-10': 'We see from the SU(3) curve in that figure (after scaling up by a factor of [MATH] to take us from a volume of [MATH] at [MATH] to a volume of [MATH] at [MATH]) that this involves less than one topological charge per field configuration.', 'hep-lat-0108006-1-37-11': 'This is a small effect in the present context.', 'hep-lat-0108006-1-37-12': 'Thus it is reasonable to assume that this effect can be neglected for values of [MATH] and [MATH] comparable to the ones that we study, and that we can then apply Eqn. [REF] to values of the susceptibility obtained by varying [MATH] at a fixed value of the lattice spacing: except that now the decay constant [MATH] will be the one appropriate to that lattice spacing.', 'hep-lat-0108006-1-38-0': 'Now, whilst most of our data points are evaluated on a trajectory of constant lattice spacing in the parameter space [CITATION], not all are.', 'hep-lat-0108006-1-38-1': 'If [MATH] varied significantly with [MATH] over this range of [MATH], it would not be clear how to perform a consistent chiral extrapolation through the data points.', 'hep-lat-0108006-1-38-2': 'The non-perturbative improvement of the action, however, removes the leading order lattice spacing dependence and the residual corrections in this range of lattice spacings appear to be small, at least in measurements of (quenched) hadron spectroscopy [CITATION].', 'hep-lat-0108006-1-38-3': 'An indication of the possible size of the effect on topological observables comes from comparing our two measurements of [MATH] at [MATH].', 'hep-lat-0108006-1-38-4': 'The range of lattice spacings here ([MATH] varies from 4.75 to 5.14) is comparable to that over our total data set.', 'hep-lat-0108006-1-38-5': 'The accompanying shift in the topological susceptiblity is, however, within our statistical errors.', 'hep-lat-0108006-1-38-6': 'Accordingly, we proceed now to attempt a common chiral extrapolation to the data, assuming throughout that lattice corrections to the relations discussed before are too small to be discernable in our limited data.', 'hep-lat-0108006-1-38-7': 'We shall at the end of this Section return to the issue of scaling violations.', 'hep-lat-0108006-1-39-0': 'For this purpose, and in the light of the discussion at the end of section [REF], it is useful to redisplay the data in Fig. [REF], where the leading order chiral behaviour would then be a horizontal line, [EQUATION] and including the first correction gives a generic straight line [EQUATION]', 'hep-lat-0108006-1-39-1': 'In each case the intercept is related to the decay constant by [MATH].', 'hep-lat-0108006-1-39-2': 'We now follow a standard fitting procedure, first using the most chiral points, then systematically adding the less chiral points until the fit becomes unacceptably bad.', 'hep-lat-0108006-1-39-3': 'The larger the number of points one can add in this way, the more evidence one has for the fitted form and the more confident one is that the systematic errors, associated with the neglected higher order corrections, are small.', 'hep-lat-0108006-1-39-4': 'The results of performing such fits are shown in Table [REF] and those using the two and four most chiral points respectively are plotted on Fig. [REF].', 'hep-lat-0108006-1-39-5': 'We see from the Table that the fits using Eqn. [REF] show much greater stability and these are the ones that will provide our eventual best estimate for [MATH].', 'hep-lat-0108006-1-40-0': 'We should comment briefly on the determination of the fitting parameter errors.', 'hep-lat-0108006-1-40-1': 'In performing all but the constant fit we must contend with the data having (small) errors on the abscissa in addition to the ordinate.', 'hep-lat-0108006-1-40-2': 'In order to estimate their affect on the fitting parameters, we first perform fits to the data assuming that the abscissa data take their central values.', 'hep-lat-0108006-1-40-3': 'Identical fits are then made using the central values plus one, and then minus one standard deviation.', 'hep-lat-0108006-1-40-4': 'The spread of the fit parameters obtained provides what is probably a crude over-estimate of this error (given there is some correlation between the ordinal and abscissal uncertainties) but is sufficient to show that it is minor.', 'hep-lat-0108006-1-40-5': 'We show this spread as a second error, and for estimates of the decay constant we add it in quadrature to the other fit parameter error.', 'hep-lat-0108006-1-41-0': 'It is remarkable that we can obtain stable fits to most of our data using just the first correction term in Eqn. [REF].', 'hep-lat-0108006-1-41-1': 'Nonetheless, as we can see in Fig. [REF], our values of the susceptibility are not very much smaller than the [MATH] quenched value and we need to have some estimate of the possible systematic errors that may arise from neglecting the higher order corrections that will eventually check the rise in [MATH].', 'hep-lat-0108006-1-41-2': 'As discussed earlier we shall do so by exploring two possibilities.', 'hep-lat-0108006-1-41-3': ""One is that the reason why [MATH] is close to [MATH] is not that [MATH] is 'large' but rather that [MATH] is large."", 'hep-lat-0108006-1-41-4': 'Then the values of [MATH] should follow the form in Eqn. [REF].', 'hep-lat-0108006-1-41-5': 'A second possibility is simply that our values of [MATH] are indeed large.', 'hep-lat-0108006-1-41-6': 'In that case we have argued that the functional form Eqn. [REF] should be a reasonable representation of the true mass dependence.', 'hep-lat-0108006-1-41-7': 'We now perform both types of fit in turn.', 'hep-lat-0108006-1-42-0': 'We begin with the first possibility, and therefore fit the data with the following ansatz [EQUATION] where we expect [MATH] up to [MATH] corrections .', 'hep-lat-0108006-1-42-1': 'To test this we fit up to seven data points.', 'hep-lat-0108006-1-42-2': 'The first six are measured in the dynamical simulations.', 'hep-lat-0108006-1-42-3': 'The final quantity is the quenched susceptibility at [MATH].', 'hep-lat-0108006-1-43-0': 'We also expect, from the Maclaurin chiral expansion of Eqn. [REF], [EQUATION] that [MATH] is related to the decay constant as before, [MATH].', 'hep-lat-0108006-1-43-1': 'We present the results of the fits in Table [REF].', 'hep-lat-0108006-1-43-2': 'We find the UKQCD data to be well fitted by this form, but the asymptotic value is higher than the number we use for the quenched limit (in contrast to earlier estimates of this value).', 'hep-lat-0108006-1-43-3': 'Incorporating this number in the fit leads to a poorer [MATH], and a less robust fit.', 'hep-lat-0108006-1-44-0': 'We turn now to fits based on the functional form in Eqn. [REF].', 'hep-lat-0108006-1-44-1': 'We therefore use the ansatz [EQUATION] where once again we expect [MATH] and from the expansion [EQUATION] we expect [MATH].', 'hep-lat-0108006-1-44-2': 'Note that in contrast to Eqn. [REF], Eqn. [REF] has no term that is cubic in [MATH] and the rise will remain approximately quadratic for a greater range in [MATH].', 'hep-lat-0108006-1-44-3': 'That this need be no bad thing is suggested by the relatively large range over which we could fit Eqn. [REF].', 'hep-lat-0108006-1-44-4': 'Indeed, we see from the fits listed in Table [REF] that this form fits our data quite well.', 'hep-lat-0108006-1-45-0': 'Typical examples of the fits from Eqn. [REF] and Eqn. [REF] are shown in Figs. [REF] and [REF].', 'hep-lat-0108006-1-45-1': 'The similarity of the two functions is apparent.', 'hep-lat-0108006-1-45-2': 'In Table [REF] we use the fit parameters to construct the first three expansion coefficients in the Maclaurin series for the various fit functions, describing the chiral behaviour of [MATH].', 'hep-lat-0108006-1-45-3': 'The fits are consistent with one another.', 'hep-lat-0108006-1-46-0': 'The fitted asymptote of the susceptibility at large [MATH] is given by [MATH].', 'hep-lat-0108006-1-46-1': 'We see from Table [REF] that these are broadly consistent with the quenched value, and our large statistical errors do not currently allow us to resolve any [MATH] deviation from this.', 'hep-lat-0108006-1-47-0': 'As an aside, we ask what happens if we cast aside some of our theoretical expectations and ask how strong is the evidence from our data that (a) the dependence is on [MATH] rather than on some other power, and (b) the susceptibility really does go to zero as [MATH]?', 'hep-lat-0108006-1-47-1': 'To answer the first question we perform fits of the kind Eqn. [REF] but replacing [MATH] by [MATH].', 'hep-lat-0108006-1-47-2': 'We find, using all seven values of [MATH], that [MATH]; a value broadly consistent with [MATH].', 'hep-lat-0108006-1-47-3': 'The [MATH]/d.o.f. is poorer, however, than for the fit with a power fixed to 2 (possible as there is one fewer d.o.f.) suggesting that the data does not warrant the use of such an extra parameter.', 'hep-lat-0108006-1-47-4': 'As for the second question, we add a constant [MATH] to Eqn. [REF] and find [MATH].', 'hep-lat-0108006-1-47-5': 'Again this is consistent with our theoretical expectation; and again the [MATH]/d.o.f. is worse.', 'hep-lat-0108006-1-47-6': 'In both cases, however, the fits are not robust, with the fit parameters ill-constrained by our data.', 'hep-lat-0108006-1-47-7': '(See Table [REF] for details of the above two fits.)', 'hep-lat-0108006-1-48-0': 'Finally, we attempt to address the issue of discretisation effects.', 'hep-lat-0108006-1-48-1': 'Our use of the non-perturbative [MATH] should have eliminated the leading [MATH] errors.', 'hep-lat-0108006-1-48-2': 'Although [MATH] is small there is, of course, no guarantee that the coefficient of this correction might not be enhanced.', 'hep-lat-0108006-1-48-3': 'We may begin to attempt to address this more quantitatively through a combined fit that includes the first order discretisation correction in [MATH].', 'hep-lat-0108006-1-48-4': 'Our motivation here is not so much to give a continuum limit (our data will not really support reliably such a long extrapolation in [MATH]) as to control the variations due to differing discretisation in our data over the relatively small range of lattice spacings in our study.', 'hep-lat-0108006-1-48-5': 'For this reason we fit the simplest combined ansatz [EQUATION] to the five most chiral of our data points.', 'hep-lat-0108006-1-48-6': 'The resultant parameters are shown in Table [REF].', 'hep-lat-0108006-1-48-7': 'The change in discretisation errors across out data set is clearly small, as we expected from the success of the previous fits.', 'hep-lat-0108006-1-48-8': 'This justifies the use of a single chiral extrapolation over this limited range in [MATH].', 'hep-lat-0108006-1-48-9': 'Whilst it does not, however, rule out deviations between the results of this and its equivalent in the continuum limit, it does give some indication that these deviations will be small.', 'hep-lat-0108006-1-48-10': 'Clearly far greater accuracy in the measurements is needed to allow a confident extrapolation to [MATH].', 'hep-lat-0108006-1-49-0': 'Given the consistency of our description of the small [MATH] regime from our measurements, it is reasonable to use the values of [MATH] to estimate the pion decay constant, [MATH].', 'hep-lat-0108006-1-49-1': 'This is done in units of [MATH] in Tables [REF] and [REF].', 'hep-lat-0108006-1-49-2': 'We use the common chiral fit of Eqn. [REF] over the largest acceptable range to provide us with our best estimate and its statistical error.', 'hep-lat-0108006-1-49-3': 'We then use the fits with other functional forms to provide us with the systematic error.', 'hep-lat-0108006-1-49-4': 'This produces an estimate [EQUATION] where the first error is statistical and the second is systematic.', 'hep-lat-0108006-1-49-5': 'This is of course no more than our best estimate of the value of [MATH] corresponding to our lattice spacing of [MATH].', 'hep-lat-0108006-1-49-6': 'This value will contain corresponding lattice spacing corrections and these must be estimated before making a serious comparison with the experimental value.', 'hep-lat-0108006-1-49-7': 'We merely note that using [MATH] we obtain from Eqn. [REF] the value [EQUATION] which is reasonably close to the experimental value [MATH].', 'hep-lat-0108006-1-50-0': '# Comparison with other studies', 'hep-lat-0108006-1-51-0': 'During the course of this work, there have appeared a number of other studies of the topological susceptibility in lattice QCD; in particular by the Pisa group [CITATION], the CP-PACS collaboration [CITATION], the SESAM-T[MATH]L collaboration [CITATION] and the Boulder group [CITATION].', 'hep-lat-0108006-1-52-0': 'The most recent studies [CITATION] are consistent with our findings, but the earlier ones found no significant decrease of the susceptibility with decreasing quark (or pion) mass when everything was expressed in physical, rather than lattice, units.', 'hep-lat-0108006-1-52-1': 'Indeed when our detailed results and analysis were first publicised [CITATION] all the other studies then available [CITATION] (and indeed [CITATION]) appeared to contradict our findings and it was therefore necessary for us to provide some reason why this might be so.', 'hep-lat-0108006-1-52-2': 'Although the situation is now different, the lessons are still useful and we will therefore briefly summarise the main point here.', 'hep-lat-0108006-1-52-3': 'For more details we refer the reader to [CITATION].', 'hep-lat-0108006-1-53-0': 'All these other calculations differ from our study in having been performed at fixed [MATH].', 'hep-lat-0108006-1-53-1': 'That implies that the lattice spacing [MATH] decreases as [MATH] is decreased.', 'hep-lat-0108006-1-53-2': 'In typical current calculations this variation in [MATH] is substantial.', 'hep-lat-0108006-1-53-3': '(See for example Fig. 4 of [CITATION].)', 'hep-lat-0108006-1-53-4': 'At the smallest values of [MATH] the lattice spacing cannot be allowed to be too fine, because the total spatial volume must remain adequately large.', 'hep-lat-0108006-1-53-5': 'This implies that at the larger values of [MATH] the lattice spacing is quite coarse.', 'hep-lat-0108006-1-53-6': 'Over such a range of lattice spacings, the topological susceptibility in the pure gauge theory typically shows a large variation (as, for example, in eqn. [REF]).', 'hep-lat-0108006-1-53-7': 'Since for coarser [MATH] more instantons (those with [MATH]) are excluded, and more of those remaining are narrow in lattice units (with a correspondingly suppressed lattice topological charge) we expect that this variation is quite general, and not a special feature of the pure gauge theory with a Wilson action.', 'hep-lat-0108006-1-53-8': 'In lattice QCD, we therefore expect two simultaneous effects in [MATH] as we decrease [MATH] at fixed [MATH].', 'hep-lat-0108006-1-53-9': 'First, because of the [MATH] lattice corrections just discussed, [MATH] will (like [MATH]) tend to increase.', 'hep-lat-0108006-1-53-10': 'Second, it will tend to decrease because of the physical quark mass dependence.', 'hep-lat-0108006-1-53-11': 'In the range of quark masses covered in current calculations this latter decrease is not very large (as we have seen in our work) and we suggest that the two effects may largely compensate each other so as to produce a susceptibility that shows very little variation with [MATH], in contrast to the ratio [MATH] which does.', 'hep-lat-0108006-1-54-0': 'To illustrate this consider the fixed-[MATH] calculation in [CITATION].', 'hep-lat-0108006-1-54-1': 'The range of quark masses covered in that work corresponds to [MATH] decreasing from about 6.5 to about 3.0.', 'hep-lat-0108006-1-54-2': 'Simultaneously [MATH] decreases from about 0.437 to about 0.274.', 'hep-lat-0108006-1-54-3': 'Over this range of [MATH] the pure gauge susceptibility increases by almost a factor of two, as we see using Eqn. [REF].', 'hep-lat-0108006-1-54-4': 'Clearly this is large enough to compensate for the expected variation of the susceptibility.', 'hep-lat-0108006-1-55-0': 'As [MATH] is increased, the [MATH] variation with [MATH] of the corresponding quenched suseptibility [MATH] will clearly diminish.', 'hep-lat-0108006-1-55-1': 'Thus we suggested in [CITATION] that if CP-PACS were to repeat their susceptibility calculations on their larger [MATH] ensembles, they would find an [MATH] variation of [MATH] consistent with ours.', 'hep-lat-0108006-1-55-2': 'This is what they have done in their most recent work [CITATION] with the result that we predicted.', 'hep-lat-0108006-1-55-3': 'All this emphasises the utility of the UKQCD strategy of decoupling the variation of lattice corrections from the physical [MATH] dependence, by performing calculations at fixed [MATH] rather than at fixed [MATH].', 'hep-lat-0108006-1-56-0': '# Summary', 'hep-lat-0108006-1-57-0': 'We have calculated the topological susceptibility in lattice QCD with two light quark flavours, using lattice field configurations in which the lattice spacing is approximately constant as the quark mass is varied.', 'hep-lat-0108006-1-57-1': 'We find that there is clear evidence for the expected suppression of [MATH] with decreasing (sea) quark mass.', 'hep-lat-0108006-1-58-0': 'We have discussed this behaviour in the context of chiral and large [MATH] expansions, and find good agreement with the functional forms expected there.', 'hep-lat-0108006-1-58-1': 'We are not able to make a stronger statement about how close QCD is to its large [MATH] limit, owing to the relatively large statistical errors on our calculated values, particularly at larger quark masses.', 'hep-lat-0108006-1-58-2': 'This situation should change in the near future and, together with the increasing availability of information on the large-[MATH] behaviour of the pure gauge (quenched) theory [CITATION], a more precise comparison will become possible.', 'hep-lat-0108006-1-59-0': 'The consistent leading order chiral behaviour from our various fitting ansatze allows us to make an estimate for the pion decay constant, [MATH], for the lattice spacing of [MATH].', 'hep-lat-0108006-1-59-1': '(Here the first error is statistical and the second has to do with the chiral extrapolation.)', 'hep-lat-0108006-1-59-2': 'We use a lattice fermion action in which the leading [MATH] discretisation errors have been removed.', 'hep-lat-0108006-1-59-3': 'Since the more accurate (quenched) hadron masses show little residual lattice spacing dependence [CITATION], we might expect that this value of [MATH] is close to its continuum limit.', 'hep-lat-0108006-1-59-4': 'In any case, we note that it is in agreement with the experimental value, [MATH].'}","{'hep-lat-0108006-2-0-0': 'We study the topological susceptibility, [MATH], in QCD with two quark flavours using lattice field configurations that have been produced with an [MATH] improved quark action.', 'hep-lat-0108006-2-0-1': 'We find clear evidence for the expected suppression at small quark mass, and examine the variation of [MATH] with this mass.', 'hep-lat-0108006-2-0-2': 'The resulting estimate of the pion decay constant, [MATH], is consistent with the experimental value of [MATH].', 'hep-lat-0108006-2-0-3': 'We compare [MATH] to the large-[MATH] prediction and find consistency over a large range of quark masses.', 'hep-lat-0108006-2-0-4': 'We discuss the benefits of the non-perturbative action improvement scheme and of the stategy of keeping the lattice spacing (nearly) fixed as the quark mass is varied.', 'hep-lat-0108006-2-0-5': 'We compare our results with other studies and suggest why such a quark mass dependence has not always been seen.', 'hep-lat-0108006-2-1-0': '# Introduction', 'hep-lat-0108006-2-2-0': 'In gluodynamics (the pure gauge or ""quenched"" theory) lattice calculations of the continuum topological susceptibility now appear to be relatively free of the systematic errors arising from the discretisation, the finite volumes and the various measurement algorithms employed (for a recent review, see [CITATION]).', 'hep-lat-0108006-2-3-0': 'The inclusion of sea quarks in (""dynamical"") lattice simulations, even at the relatively large quark masses currently employed, is numerically extremely expensive, and can only be done for lattices with relatively few sites (typically [MATH]).', 'hep-lat-0108006-2-3-1': 'To avoid significant finite volume errors, the lattice must then be relatively coarse, with, in our case, a spacing [MATH].', 'hep-lat-0108006-2-3-2': 'This is a significant fraction of the mean instanton radius, as calculated in gluodynamics, and thus precludes a robust, detailed study of the local topological features of the vacuum in the presence of sea quarks.', 'hep-lat-0108006-2-3-3': 'The topological susceptibility, on the other hand, may be calculated with some confidence and provides one of the first opportunities to test some of the more interesting predictions of QCD.', 'hep-lat-0108006-2-3-4': 'Indeed, it is in these measurements that we find some of the most striking evidence for the presence of the sea quarks (or, alternatively, for a strong quenching effect) in the lattice simulations.', 'hep-lat-0108006-2-4-0': 'We recall that the ensembles used here have been produced with two notable features [CITATION].', 'hep-lat-0108006-2-4-1': 'The first is the use of an improved action, such that leading order lattice discretisation effects are expected to depend quadratically, rather than linearly, on the lattice spacing (just as in gluodynamics).', 'hep-lat-0108006-2-4-2': 'In addition, the action parameters have been chosen to maintain a relatively constant lattice spacing, particularly for the larger values of the quark mass.', 'hep-lat-0108006-2-5-0': 'These features have allowed us to see the first clear evidence [CITATION] for the expected suppression of the topological susceptibility in the chiral limit, despite our relatively large quark masses.', 'hep-lat-0108006-2-5-1': 'From this behaviour we can directly estimate the pion decay constant without needing to know the lattice operator renormalisation factors that arise in more conventional calculations.', 'hep-lat-0108006-2-6-0': 'The structure of this paper is as follows.', 'hep-lat-0108006-2-6-1': 'In Section [REF] we discuss the measurement of the topological susceptibility and its expected behaviour both near the chiral limit, and in the limit of a large number of colours, [MATH].', 'hep-lat-0108006-2-6-2': 'In Section [REF] we describe the UKQCD ensembles and the lattice measurements of the topological susceptibility over a range of sea quark masses.', 'hep-lat-0108006-2-6-3': 'We fit these with various ansatze motivated by the previous section.', 'hep-lat-0108006-2-6-4': 'We compare our findings with other recent studies in Section [REF].', 'hep-lat-0108006-2-6-5': 'Finally, we provide a summary in Section [REF].', 'hep-lat-0108006-2-7-0': ""These results were presented at the IOP2000 [CITATION], the Confinement IV [CITATION] and, in a much more preliminary form, the Lattice '99 [CITATION] conferences."", 'hep-lat-0108006-2-7-1': 'Since then, we have increased the size of several ensembles and included a new parameter set to try and address the issue of discretisation effects in our results.', 'hep-lat-0108006-2-7-2': 'We also have more accurate results from the quenched theory with which to compare.', 'hep-lat-0108006-2-7-3': 'A brief summary of this work appears in [CITATION].', 'hep-lat-0108006-2-8-0': '# The topological susceptibility', 'hep-lat-0108006-2-9-0': 'In four-dimensional Euclidean space-time, SU(3) gauge field configurations can be separated into topological classes, and moving between different classes is not possible by a smooth deformation of the fields.', 'hep-lat-0108006-2-9-1': 'The classes are characterised by an integer-valued winding number.', 'hep-lat-0108006-2-9-2': 'This Pontryagin index, or topological charge [MATH], can be obtained by integrating the local topological charge density [EQUATION] over all space-time [EQUATION]', 'hep-lat-0108006-2-9-3': 'The topological susceptibility is the expectation value of the squared charge, normalised by the volume [EQUATION]', 'hep-lat-0108006-2-9-4': 'An isolated topological charge induces an exact zero-mode in the quark Dirac operator.', 'hep-lat-0108006-2-9-5': ""As a result sea quarks in the vacuum induce an instanton-anti-instanton attraction which becomes stronger as the quark masses, [MATH], [MATH], , decrease towards zero (the 'chiral limit'), and the topological charge and susceptibility will be suppressed to leading order in the quark mass [CITATION], [EQUATION] where [EQUATION] is the chiral condensate (see [CITATION] for a recent discussion)."", 'hep-lat-0108006-2-9-6': 'Here we have assumed [MATH] and we neglect the contribution of heavier quarks.', 'hep-lat-0108006-2-9-7': 'PCAC theory relates this to the pion decay constant [MATH] and mass [MATH] via the Gell-Mann-Oakes-Renner relation as [EQUATION] and we may combine these for [MATH] degenerate light flavours to obtain [EQUATION] in a convention where the experimental value of [MATH] .', 'hep-lat-0108006-2-9-8': 'This relation should hold in the limit [EQUATION] which is the Leutwyler-Smilga parameter to leading order in [MATH].', 'hep-lat-0108006-2-9-9': 'We anticipate our results here to say that even on our most chiral lattices the lhs of Eqn. [REF] is of order 10 and so this bound is well satisfied.', 'hep-lat-0108006-2-9-10': 'Thus a calculation of [MATH] as a function of [MATH] should allow us to obtain a value of [MATH].', 'hep-lat-0108006-2-9-11': 'This method has an advantage over more conventional calculations in that it does not require us to know lattice operator renormalisation constants which are required for matching matrix elements, but which are usually difficult to calculate.', 'hep-lat-0108006-2-9-12': 'In principle, we require instead knowledge of the renormalisation of the topological charge operators, but we see in the next Section that this problem can be readily overcome.', 'hep-lat-0108006-2-10-0': 'As [MATH] and [MATH] increase away from zero we expect higher order terms to check the rate of increase of the topological susceptibility so that, as [MATH], [MATH] approaches the quenched value, [MATH].', 'hep-lat-0108006-2-10-1': 'In fact, as we shall see below, the values of [MATH] that we obtain are not very much smaller than [MATH].', 'hep-lat-0108006-2-10-2': 'So there is the danger of a substantial systematic error in simply applying Eqn. [REF] at our smallest values of [MATH] in order to estimate [MATH].', 'hep-lat-0108006-2-10-3': 'To estimate this error it would be useful to have some understanding of how [MATH] behaves over the whole range of [MATH].', 'hep-lat-0108006-2-10-4': 'This is the question to which we now turn.', 'hep-lat-0108006-2-11-0': 'There are two quite different reasons why [MATH] might not be much smaller than [MATH].', 'hep-lat-0108006-2-11-1': 'The obvious first possibility is that [MATH] is large.', 'hep-lat-0108006-2-11-2': 'The second possibility is more subtle: [MATH] may be small but QCD may be close to its large-[MATH] limit [CITATION].', 'hep-lat-0108006-2-11-3': 'Because fermion effects are non-leading in powers of [MATH], we expect [MATH] for any fixed, non-zero value of [MATH], however small, as the number of colours [MATH].', 'hep-lat-0108006-2-11-4': ""There are phenomenological reasons [CITATION] for believing that QCD is 'close' to [MATH], and so this is not an unrealistic consideration."", 'hep-lat-0108006-2-11-5': 'Moreover in the case of [MATH] gauge theories it has been shown [CITATION] that even SU(2) is close to SU([MATH]).', 'hep-lat-0108006-2-11-6': 'Recent calculations in four dimensions [CITATION] indicate that the same is true there.', 'hep-lat-0108006-2-11-7': 'In the present simulations, the lighter quark masses straddle the strange quark mass and so it is not obvious if we should regard them as being large or small.', 'hep-lat-0108006-2-11-8': 'We shall therefore take seriously both the possibilities discussed above.', 'hep-lat-0108006-2-12-0': 'We start by assuming the quark mass is small but that we are close to the large-[MATH] limit.', 'hep-lat-0108006-2-12-1': 'In this limit, the topological susceptibility is known [CITATION] to vary as [EQUATION] where [MATH], [MATH] are the quantities at leading order in [MATH].', 'hep-lat-0108006-2-12-2': 'In the chiral limit, at fixed [MATH], this reproduces Eqn. [REF].', 'hep-lat-0108006-2-12-3': 'In the large-[MATH] limit, at fixed [MATH], it tends to the quenched susceptibility [MATH] because [MATH].', 'hep-lat-0108006-2-12-4': 'The corrections to Eqn. [REF] are of higher order in [MATH] and/or lower order in [MATH].', 'hep-lat-0108006-2-13-0': 'We now consider the alternative possibility: that [MATH] is not small, that higher-order corrections to [MATH] will be important for most of the values of [MATH] at which we perform calculations, and that we therefore need an expression for [MATH] that interpolates between [MATH] and [MATH].', 'hep-lat-0108006-2-13-1': 'Clearly one cannot hope to derive such an expression from first principles, so we will simply choose one that we can plausibly argue is approximately correct.', 'hep-lat-0108006-2-13-2': 'The form we choose is [EQUATION] where [MATH] is the pion decay constant in the chiral limit.', 'hep-lat-0108006-2-13-3': 'The coefficients have been chosen so that this reproduces Eqn. [REF] when [MATH] and [MATH] when [MATH].', 'hep-lat-0108006-2-13-4': 'Thus, this interpolation formula possesses the correct limits and it approaches those limits with power-like corrections.', 'hep-lat-0108006-2-14-0': 'We shall use the expressions in Eqns. [REF], [REF] and [REF] to analyse the [MATH] dependence of our calculated values of [MATH] and to obtain a value of [MATH] together with an estimate of the systematic error on that value.', 'hep-lat-0108006-2-14-1': 'In addition, the comparison with Eqn. [REF] can provide us with some evidence for whether QCD is close to its large-[MATH] limit or not.', 'hep-lat-0108006-2-15-0': '# Lattice measurements', 'hep-lat-0108006-2-16-0': 'We have calculated [MATH] on five complete ensembles of dynamical configurations produced by the UKQCD collaboration, as well as one which is still in progress [CITATION].', 'hep-lat-0108006-2-16-1': 'Details of these data sets are given in Table [REF].', 'hep-lat-0108006-2-16-2': 'The SU(3) gauge fields are governed by the Wilson plaquette action, with ""clover"" improved Wilson fermions.', 'hep-lat-0108006-2-16-3': 'The improvement is non-perturbative, with [MATH] chosen to render the leading order discretisation errors quadratic (rather than linear) in the lattice spacing, [MATH].', 'hep-lat-0108006-2-17-0': 'The theory has two coupling constants.', 'hep-lat-0108006-2-17-1': 'In pure gluodynamics the gauge coupling, [MATH], controls the lattice spacing, with larger values reducing [MATH] as we move towards the critical value at [MATH].', 'hep-lat-0108006-2-17-2': 'In simulations with dynamical fermions it has the same role for a fixed fermion coupling, [MATH].', 'hep-lat-0108006-2-17-3': 'The latter controls the quark mass, with [MATH] from below corresponding to the massless limit.', 'hep-lat-0108006-2-17-4': 'In dynamical simulations, however, the fermion coupling also affects the lattice spacing, which will become larger as [MATH] is reduced (and hence [MATH] increased) at fixed [MATH].', 'hep-lat-0108006-2-18-0': 'The three least chiral UKQCD ensembles (by which we mean largest [MATH]) are e[MATH], e[MATH] and e[MATH].', 'hep-lat-0108006-2-18-1': ""By appropriately decreasing [MATH] as [MATH] is increased, the couplings are 'matched' to maintain a constant lattice spacing [CITATION] (which is 'equivalent' to [MATH] in gluodynamics with a Wilson action [CITATION]) whilst approaching the chiral limit."", 'hep-lat-0108006-2-18-2': 'The physical volume and discretisation effects should thus be very similar on these lattices.', 'hep-lat-0108006-2-18-3': 'The remaining ensembles have lower quark masses, but are at a slightly reduced lattice spacing.', 'hep-lat-0108006-2-18-4': '(To have maintained a matched lattice spacing here would have required reducing [MATH] to values where the non-perturbative value [MATH] is not known.)', 'hep-lat-0108006-2-18-5': 'As the lattices are all [MATH], we believe that the minor reduction in the lattice volume should not lead to significant finite volume corrections.', 'hep-lat-0108006-2-18-6': 'We also remark that ensembles e[MATH] and e[MATH] have been matched to have approximately the same chirality, but at (mildly) different lattice spacings.', 'hep-lat-0108006-2-19-0': 'Four-dimensional lattice theories are scale free, and the dimensionless lattice quantities must be cast in physical units through the use of a known scale.', 'hep-lat-0108006-2-19-1': 'For this work, we use the Sommer scale [CITATION] both to define the lattice spacing for the matching procedure, and to set the scale.', 'hep-lat-0108006-2-19-2': 'The measured value of [MATH] on each ensemble, as listed in Table [REF], corresponds to the same physical value of [MATH].', 'hep-lat-0108006-2-19-3': '([MATH] is the dimensionless lattice value of [MATH] in lattice units i.e. [MATH].', 'hep-lat-0108006-2-19-4': 'We use the same notation for other quantities.)', 'hep-lat-0108006-2-19-5': 'As we are in the scaling window of the theory, we can then use the naive dimensions of the various operators to relate lattice and physical quantities, e.g. [MATH], where we have incorporated the expected non-perturbative removal of the corrections linear in the lattice spacing.', 'hep-lat-0108006-2-20-0': 'Further details of the parameters and the scale determination are given in [CITATION].', 'hep-lat-0108006-2-20-1': 'Measurements were made on ensembles of 400-800 configurations of size [MATH], separated by ten hybrid Monte Carlo trajectories.', 'hep-lat-0108006-2-20-2': 'Correlations in the data were managed through jack-knife binning of the data, using ten bins whose size is large enough that neighbouring bin averages may be regarded as uncorrelated.', 'hep-lat-0108006-2-21-0': 'We begin, however, with a discussion of lattice operators and results in the quenched theory.', 'hep-lat-0108006-2-22-0': '## Lattice operators and [MATH]', 'hep-lat-0108006-2-23-0': 'The simplest lattice topological charge density operator is [EQUATION] where [MATH] denotes the product of SU(3) link variables around a given plaquette.', 'hep-lat-0108006-2-23-1': 'We use a reflection-symmetrised version and form', 'hep-lat-0108006-2-24-0': 'Q & = & 132^2 _n q(n) ,', 'hep-lat-0108006-2-25-0': '& = & Q^2 L^3 T', 'hep-lat-0108006-2-26-0': 'with [MATH] the lattice volume.', 'hep-lat-0108006-2-26-1': 'In general, [MATH] will not give an integer-valued topological charge due to finite lattice spacing effects.', 'hep-lat-0108006-2-26-2': 'There are at least three sources of these.', 'hep-lat-0108006-2-26-3': 'First is the breaking of scale invariance by the lattice which leads to the smallest instantons having a suppressed action (at least with the Wilson action) and a topological charge less than unity (at least with the operator in Eqn. [REF]).', 'hep-lat-0108006-2-26-4': 'We do not address this problem in this study, although attempts can be made to correct for it [CITATION], but simply accept this as part of the overall [MATH] error.', 'hep-lat-0108006-2-26-5': 'In addition to this, the underlying topological signal on the lattice is distorted by the presence of large amounts of UV noise on the scale of the lattice spacing [CITATION], and by a multiplicative renormalisation factor [CITATION] that is unity in the continuum, but otherwise suppresses the observed charge.', 'hep-lat-0108006-2-26-6': 'Various solutions to these problems exist [CITATION].', 'hep-lat-0108006-2-26-7': ""In this study we opt for the 'cooling' approach."", 'hep-lat-0108006-2-26-8': 'Cooling explicitly erases the ultraviolet fluctuations so that the perturbative lattice renormalisation factors for the topological charge and susceptibility are driven to their trivial continuum values, leaving [MATH] corrections that may be absorbed into all the other lattice corrections of this order.', 'hep-lat-0108006-2-26-9': ""We cool by moving through the lattice in a 'staggered' fashion, cooling each link by minimising the Wilson gauge action applied to each of the three Cabibbo-Marinari SU(2) subgroups in the link element in turn."", 'hep-lat-0108006-2-26-10': '(The Wilson gauge action is the most local, and thus particularly efficient at removing short distance fluctuations whilst preserving the long range correlations in the fields.)', 'hep-lat-0108006-2-26-11': ""Carrying out this procedure once on every link constitutes a cooling sweep (or 'cool')."", 'hep-lat-0108006-2-26-12': 'The violation of the instanton scale invariance on the lattice, with a Wilson action, is such that an isolated instanton cooled in this way will slowly shrink, and will eventually disappear when its core size is of the order of a lattice spacing, leading to a corresponding jump in the topological charge.', 'hep-lat-0108006-2-26-13': 'Such events can, of course, be detected by monitoring [MATH] as a function of the number of cooling sweeps, [MATH].', 'hep-lat-0108006-2-26-14': 'Instanton-anti-instanton pairs may also annihilate, but this has no net effect on [MATH].', 'hep-lat-0108006-2-26-15': 'However, these observations do motivate us to perform the minimum number of cools necessary to obtain an estimate of [MATH] that is stable with further increasing [MATH] (subject to the above).', 'hep-lat-0108006-2-27-0': 'To estimate this point we calculate the normalised correlation function between the topological charges measured after [MATH] cooling sweeps, and a nominally asymptotic 25 cooling sweeps: [EQUATION]', 'hep-lat-0108006-2-27-1': 'In Fig. [REF] we show a typical plot for ensemble e[MATH].', 'hep-lat-0108006-2-27-2': 'As discussed before, we have opted not to attempt to round the topological charge to integer values.', 'hep-lat-0108006-2-27-3': 'We find [MATH] and [MATH] to be stable within statistical errors for [MATH], and the results presented here are for [MATH].', 'hep-lat-0108006-2-28-0': 'The topological charge of a configuration is related to the smallest eigenvalues of the Dirac matrix and as such is often believed to be one of the slowest modes to decorrelate during Monte Carlo simulations.', 'hep-lat-0108006-2-28-1': 'It is crucial for the error analysis that the bin sizes for the data are at least twice the integrated autocorrelation times.', 'hep-lat-0108006-2-28-2': 'In Fig. [REF] we plot a typical time series of the topological charge measured every ten hybrid Monte Carlo trajectories.', 'hep-lat-0108006-2-28-3': 'The rapid variation between configurations suggests that the integrated autocorrelation time is small even for the topological charge.', 'hep-lat-0108006-2-28-4': 'Estimates of this are given in Table [REF] in units of ten trajectories.', 'hep-lat-0108006-2-28-5': 'These reinforce the impression gained from the time series plots.', 'hep-lat-0108006-2-28-6': 'The bins used in the jack-knife statistical analysis are between 400 and 1000 trajectories in length and thus may be confidently assumed to be statistically independent.', 'hep-lat-0108006-2-28-7': 'It is interesting that although autocorrelation times are hard to estimate accurately, it does appear that they increase as we move away from the chiral limit (c.f. Ref. [CITATION]).', 'hep-lat-0108006-2-29-0': 'In Fig. [REF] we divide the topological charge measurements made over an ensemble into bins of unit width centred on the integers, and plot a histogram of the probability of finding a configuration with each charge, with errors from the jack-knife analysis.', 'hep-lat-0108006-2-29-1': 'We find for all our ensembles that these histograms are very close to being symmetric, centred around [MATH] and consistent with a Gaussian envelope.', 'hep-lat-0108006-2-29-2': 'The hybrid Monte Carlo appears to be sampling the topological sectors correctly, and it is legitimate to extract an estimate of the topological susceptibility.', 'hep-lat-0108006-2-29-3': 'On the histograms we show this estimate as a Gaussian curve [EQUATION]', 'hep-lat-0108006-2-29-4': 'The central line uses our estimate of [MATH], whilst the outlying curves use the central value plus or minus one standard deviation.', 'hep-lat-0108006-2-29-5': 'The agreement with the histograms is good.', 'hep-lat-0108006-2-30-0': 'We also remark that on a lattice one obviously loses instantons with sizes [MATH].', 'hep-lat-0108006-2-30-1': 'Since the (pure gauge) instanton density decreases as [MATH] when [MATH] this would appear to induce a negligible [MATH] error in the susceptibility.', 'hep-lat-0108006-2-30-2': 'However this is only true for [MATH], and the error can be substantial for the coarse lattices often used in dynamical simulations.', 'hep-lat-0108006-2-31-0': 'In general, then, we expect the topological charge and susceptibility to be suppressed at non-zero lattice spacing.', 'hep-lat-0108006-2-31-1': 'In gluodynamics with the Wilson action this suppression can, typically, be fitted by a leading order, and negative, [MATH] correction term starting from quite moderate values of [MATH].', 'hep-lat-0108006-2-31-2': 'Of course different ways of calculating the topological charge differ substantially at finite values of [MATH], even if they agree in the continuum limit.', 'hep-lat-0108006-2-31-3': '(See, for example, Table 27 in [CITATION].)', 'hep-lat-0108006-2-31-4': 'An important factor for non-zero [MATH] is whether the topological charge is rounded to the nearest integer after cooling or not, as can be seen in Table 8 in [CITATION].', 'hep-lat-0108006-2-31-5': 'A gluodynamic calculation of [MATH] that uses a method very similar to the one used in the present paper, in particular an unrounded topological charge, can be found in the last column of Table 8 in [CITATION].', 'hep-lat-0108006-2-31-6': 'The values of the susceptibility listed there can be fitted, for [MATH], by [EQUATION]', 'hep-lat-0108006-2-31-7': 'In obtaining this fit we have used the interpolation formula for [MATH] that is given in [CITATION].', 'hep-lat-0108006-2-31-8': 'We shall use this formula in Section [REF] as a guide to the typical variation of [MATH] with [MATH].', 'hep-lat-0108006-2-32-0': 'In the next Section we shall want to compare our calculations of [MATH], as obtained in the presence of sea quarks, with an appropriate quenched limit.', 'hep-lat-0108006-2-32-1': ""The 'equivalent' quenched limit will, of course, depend on the lattice spacing, i.e. on the value of [MATH]."", 'hep-lat-0108006-2-32-2': 'However, because of our strategy of varying [MATH] so as to approximately match the values of [MATH] at the different values of [MATH], the variation in this quenched value, [MATH], over the range of lattice spacings of our ensembles, is in fact much less than the statistical errors on the measurements themselves.', 'hep-lat-0108006-2-32-3': '(If the calculations had been performed at fixed [MATH], the lattice spacing would have become increasingly coarse with increasing [MATH], and the reduction in the quenched susceptibility would have been much more pronounced over this range of [MATH].', 'hep-lat-0108006-2-32-4': 'We shall return to this important point when we discuss other work in section [REF].)', 'hep-lat-0108006-2-32-5': 'One finds [CITATION] that [MATH] at [MATH] in the quenched theory, demonstrating that this provides an appropriate quenched limit for our calculations (see Table [REF]).', 'hep-lat-0108006-2-32-6': 'At [MATH] the interpolation formula for [MATH] that we used to obtain Eqn. [REF] gives [MATH] and if we insert this in Eqn. [REF] we obtain [MATH].', 'hep-lat-0108006-2-32-7': 'Different methods for calculating [MATH] differ by [MATH] terms, and part of the difference between the UKQCD value, [MATH], and the Sommer value, [MATH], might be due to this.', 'hep-lat-0108006-2-32-8': 'If we rescale the susceptibility to account for this, then the value of [MATH] drops to [MATH].', 'hep-lat-0108006-2-32-9': 'To do better than this we need to take into account the fact that the calculations of the topological charge that enter into Eqn. [REF] are obtained by methods that are not exactly the same as those used in the calculations of the present paper.', 'hep-lat-0108006-2-32-10': 'The potentially significant differences are that 20 cooling sweeps and an unsymmetrised topological charge were used in [CITATION].', 'hep-lat-0108006-2-32-11': 'while we use 10 cooling sweeps and a symmetrised charge.', 'hep-lat-0108006-2-32-12': 'To estimate the systematic shift induced by these differences we have performed calculations on 300 [MATH] lattice field configurations generated at [MATH] (separated by 50 Monte Carlo sweeps).', 'hep-lat-0108006-2-32-13': 'We find that there is no significant difference between the susceptibility as calculated by the symmetrised and unsymmetrised charges, whether after 10 or 20 cooling sweeps.', 'hep-lat-0108006-2-32-14': 'There is, on the other hand, a small but significant difference between the susceptibility as calculated after 10 and 20 cooling sweeps.', 'hep-lat-0108006-2-32-15': 'This reduces our estimate of the equivalent quenched susceptibility by [MATH].', 'hep-lat-0108006-2-32-16': 'So taking all this into account we take our equivalent quenched susceptibility to be given by [EQUATION]', 'hep-lat-0108006-2-33-0': '## Sea quark effects in the topological susceptibility', 'hep-lat-0108006-2-34-0': 'In Table [REF] we give our estimates of the topological susceptibility in physical units, using [MATH] as the scale.', 'hep-lat-0108006-2-34-1': 'In Fig. [REF] we plot [MATH] versus a similarly scaled pseudoscalar meson mass (calculated, of course, with valence quarks that are degenerate with those in the sea, i.e. [MATH]).', 'hep-lat-0108006-2-34-2': 'We also plot the corresponding value of the quenched topological susceptibility, as calculated at [MATH].', 'hep-lat-0108006-2-35-0': 'Comparing the dynamical and quenched values, the effects of the sea quarks are clear.', 'hep-lat-0108006-2-35-1': 'Whilst the measurement on e[MATH] and e[MATH] are consistent with the quenched value, moving to smaller [MATH]) the topological susceptibility is increasingly suppressed.', 'hep-lat-0108006-2-36-0': 'We can make this observation more quantitative by attempting to fit our values of [MATH] with the expected functional form in Eqn. [REF], so extracting a value of [MATH].', 'hep-lat-0108006-2-36-1': 'But we must first be clear whether this fit is justified, and what exactly we are extrapolating in, bearing in mind that Eqn. [REF] is strictly a chiral expansion that describes the behaviour for small sea quark masses in the continuum limit.', 'hep-lat-0108006-2-37-0': 'An immediate concern is that our cooling technique will occasionally misidentify the value of [MATH] and, in addition, that at finite lattice spacing the exact zero modes associated with the topological charge [MATH] are shifted away from zero.', 'hep-lat-0108006-2-37-1': 'All this implies that [MATH] will not in fact vanish as [MATH].', 'hep-lat-0108006-2-37-2': 'However we expect this effect to be small for the following reasons.', 'hep-lat-0108006-2-37-3': 'First, the use of an improved fermion action should ensure that the zero-mode shift will only be significant for very small instantons, i.e. those whose sizes are [MATH].', 'hep-lat-0108006-2-37-4': 'These are unlikely to survive the cooling and should not contribute to our calculated value of [MATH].', 'hep-lat-0108006-2-37-5': 'Large instantons, on the other hand, for which any zero mode shift should be negligible, will certainly survive the cooling.', 'hep-lat-0108006-2-37-6': 'The remaining ambiguity involves the smaller, but not very small, instantons.', 'hep-lat-0108006-2-37-7': 'These might be erased by the cooling but the probability is small simply because the number of such charges is small [CITATION].', 'hep-lat-0108006-2-37-8': 'For example, we can see from Fig. 12 in [CITATION] that the cooling only appears to cut out instantons with [MATH].', 'hep-lat-0108006-2-37-9': 'Since the lattice spacing in that plot is a factor of 1.5 smaller than at our equivalent quenched [MATH] value of 5.93, we would expect that cooling at [MATH] would affect instantons that have [MATH] in that plot.', 'hep-lat-0108006-2-37-10': 'We see from the SU(3) curve in that figure (after scaling up by a factor of [MATH] to take us from a volume of [MATH] at [MATH] to a volume of [MATH] at [MATH]) that this involves less than one topological charge per field configuration.', 'hep-lat-0108006-2-37-11': 'This is a small effect in the present context.', 'hep-lat-0108006-2-37-12': 'Thus it is reasonable to assume that this effect can be neglected for values of [MATH] and [MATH] comparable to the ones that we study, and that we can then apply Eqn. [REF] to values of the susceptibility obtained by varying [MATH] at a fixed value of the lattice spacing: except that now the decay constant [MATH] will be the one appropriate to that lattice spacing.', 'hep-lat-0108006-2-38-0': 'Now, whilst most of our data points are evaluated on a trajectory of constant lattice spacing in the parameter space [CITATION], not all are.', 'hep-lat-0108006-2-38-1': 'If [MATH] varied significantly with [MATH] over this range of [MATH], it would not be clear how to perform a consistent chiral extrapolation through the data points.', 'hep-lat-0108006-2-38-2': 'The non-perturbative improvement of the action, however, removes the leading order lattice spacing dependence and the residual corrections in this range of lattice spacings appear to be small, at least in measurements of (quenched) hadron spectroscopy [CITATION].', 'hep-lat-0108006-2-38-3': 'An indication of the possible size of the effect on topological observables comes from comparing our two measurements of [MATH] at [MATH].', 'hep-lat-0108006-2-38-4': 'The range of lattice spacings here ([MATH] varies from 4.75 to 5.14) is comparable to that over our total data set.', 'hep-lat-0108006-2-38-5': 'The accompanying shift in the topological susceptiblity is, however, within our statistical errors.', 'hep-lat-0108006-2-38-6': 'Accordingly, we proceed now to attempt a common chiral extrapolation to the data, assuming throughout that lattice corrections to the relations discussed before are too small to be discernable in our limited data.', 'hep-lat-0108006-2-38-7': 'We shall at the end of this Section return to the issue of scaling violations.', 'hep-lat-0108006-2-39-0': 'For this purpose, and in the light of the discussion at the end of section [REF], it is useful to redisplay the data in Fig. [REF], where the leading order chiral behaviour would then be a horizontal line, [EQUATION] and including the first correction gives a generic straight line [EQUATION]', 'hep-lat-0108006-2-39-1': 'In each case the intercept is related to the decay constant by [MATH].', 'hep-lat-0108006-2-39-2': 'We now follow a standard fitting procedure, first using the most chiral points, then systematically adding the less chiral points until the fit becomes unacceptably bad.', 'hep-lat-0108006-2-39-3': 'The larger the number of points one can add in this way, the more evidence one has for the fitted form and the more confident one is that the systematic errors, associated with the neglected higher order corrections, are small.', 'hep-lat-0108006-2-39-4': 'The results of performing such fits are shown in Table [REF] and those using the two and four most chiral points respectively are plotted on Fig. [REF].', 'hep-lat-0108006-2-39-5': 'We see from the Table that the fits using Eqn. [REF] show much greater stability and these are the ones that will provide our eventual best estimate for [MATH].', 'hep-lat-0108006-2-40-0': 'We should comment briefly on the determination of the fitting parameter errors.', 'hep-lat-0108006-2-40-1': 'In performing all but the constant fit we must contend with the data having (small) errors on the abscissa in addition to the ordinate.', 'hep-lat-0108006-2-40-2': 'In order to estimate their affect on the fitting parameters, we first perform fits to the data assuming that the abscissa data take their central values.', 'hep-lat-0108006-2-40-3': 'Identical fits are then made using the central values plus one, and then minus one standard deviation.', 'hep-lat-0108006-2-40-4': 'The spread of the fit parameters obtained provides what is probably a crude over-estimate of this error (given there is some correlation between the ordinal and abscissal uncertainties) but is sufficient to show that it is minor.', 'hep-lat-0108006-2-40-5': 'We show this spread as a second error, and for estimates of the decay constant we add it in quadrature to the other fit parameter error.', 'hep-lat-0108006-2-41-0': 'It is remarkable that we can obtain stable fits to most of our data using just the first correction term in Eqn. [REF].', 'hep-lat-0108006-2-41-1': 'Nonetheless, as we can see in Fig. [REF], our values of the susceptibility are not very much smaller than the [MATH] quenched value and we need to have some estimate of the possible systematic errors that may arise from neglecting the higher order corrections that will eventually check the rise in [MATH].', 'hep-lat-0108006-2-41-2': 'As discussed earlier we shall do so by exploring two possibilities.', 'hep-lat-0108006-2-41-3': ""One is that the reason why [MATH] is close to [MATH] is not that [MATH] is 'large' but rather that [MATH] is large."", 'hep-lat-0108006-2-41-4': 'Then the values of [MATH] should follow the form in Eqn. [REF].', 'hep-lat-0108006-2-41-5': 'A second possibility is simply that our values of [MATH] are indeed large.', 'hep-lat-0108006-2-41-6': 'In that case we have argued that the functional form Eqn. [REF] should be a reasonable representation of the true mass dependence.', 'hep-lat-0108006-2-41-7': 'We now perform both types of fit in turn.', 'hep-lat-0108006-2-42-0': 'We begin with the first possibility, and therefore fit the data with the following ansatz [EQUATION] where we expect [MATH] up to [MATH] corrections .', 'hep-lat-0108006-2-42-1': 'To test this we fit up to seven data points.', 'hep-lat-0108006-2-42-2': 'The first six are measured in the dynamical simulations.', 'hep-lat-0108006-2-42-3': 'The final quantity is the quenched susceptibility at [MATH].', 'hep-lat-0108006-2-43-0': 'We also expect, from the Maclaurin chiral expansion of Eqn. [REF], [EQUATION] that [MATH] is related to the decay constant as before, [MATH].', 'hep-lat-0108006-2-43-1': 'We present the results of the fits in Table [REF].', 'hep-lat-0108006-2-43-2': 'We find the UKQCD data to be well fitted by this form, but the asymptotic value is higher than the number we use for the quenched limit (in contrast to earlier estimates of this value).', 'hep-lat-0108006-2-43-3': 'Incorporating this number in the fit leads to a poorer [MATH], and a less robust fit.', 'hep-lat-0108006-2-44-0': 'We turn now to fits based on the functional form in Eqn. [REF].', 'hep-lat-0108006-2-44-1': 'We therefore use the ansatz [EQUATION] where once again we expect [MATH] and from the expansion [EQUATION] we expect [MATH].', 'hep-lat-0108006-2-44-2': 'Note that in contrast to Eqn. [REF], Eqn. [REF] has no term that is cubic in [MATH] and the rise will remain approximately quadratic for a greater range in [MATH].', 'hep-lat-0108006-2-44-3': 'That this need be no bad thing is suggested by the relatively large range over which we could fit Eqn. [REF].', 'hep-lat-0108006-2-44-4': 'Indeed, we see from the fits listed in Table [REF] that this form fits our data quite well.', 'hep-lat-0108006-2-45-0': 'Typical examples of the fits from Eqn. [REF] and Eqn. [REF] are shown in Figs. [REF] and [REF].', 'hep-lat-0108006-2-45-1': 'The similarity of the two functions is apparent.', 'hep-lat-0108006-2-45-2': 'In Table [REF] we use the fit parameters to construct the first three expansion coefficients in the Maclaurin series for the various fit functions, describing the chiral behaviour of [MATH].', 'hep-lat-0108006-2-45-3': 'The fits are consistent with one another.', 'hep-lat-0108006-2-46-0': 'The fitted asymptote of the susceptibility at large [MATH] is given by [MATH].', 'hep-lat-0108006-2-46-1': 'We see from Table [REF] that these are broadly consistent with the quenched value, and our large statistical errors do not currently allow us to resolve any [MATH] deviation from this.', 'hep-lat-0108006-2-47-0': 'As an aside, we ask what happens if we cast aside some of our theoretical expectations and ask how strong is the evidence from our data that (a) the dependence is on [MATH] rather than on some other power, and (b) the susceptibility really does go to zero as [MATH]?', 'hep-lat-0108006-2-47-1': 'To answer the first question we perform fits of the kind Eqn. [REF] but replacing [MATH] by [MATH].', 'hep-lat-0108006-2-47-2': 'We find, using all seven values of [MATH], that [MATH]; a value broadly consistent with [MATH].', 'hep-lat-0108006-2-47-3': 'The [MATH]/d.o.f. is poorer, however, than for the fit with a power fixed to 2 (possible as there is one fewer d.o.f.) suggesting that the data does not warrant the use of such an extra parameter.', 'hep-lat-0108006-2-47-4': 'As for the second question, we add a constant [MATH] to Eqn. [REF] and find [MATH].', 'hep-lat-0108006-2-47-5': 'Again this is consistent with our theoretical expectation; and again the [MATH]/d.o.f. is worse.', 'hep-lat-0108006-2-47-6': 'In both cases, however, the fits are not robust, with the fit parameters ill-constrained by our data.', 'hep-lat-0108006-2-47-7': '(See Table [REF] for details of the above two fits.)', 'hep-lat-0108006-2-48-0': 'Finally, we attempt to address the issue of discretisation effects.', 'hep-lat-0108006-2-48-1': 'Our use of the non-perturbative [MATH] should have eliminated the leading [MATH] errors.', 'hep-lat-0108006-2-48-2': 'Although [MATH] is small there is, of course, no guarantee that the coefficient of this correction might not be enhanced.', 'hep-lat-0108006-2-48-3': 'We may begin to attempt to address this more quantitatively through a combined fit that includes the first order discretisation correction in [MATH].', 'hep-lat-0108006-2-48-4': 'Our motivation here is not so much to give a continuum limit (our data will not really support reliably such a long extrapolation in [MATH]) as to control the variations due to differing discretisation in our data over the relatively small range of lattice spacings in our study.', 'hep-lat-0108006-2-48-5': 'For this reason we fit the simplest combined ansatz [EQUATION] to the five most chiral of our data points.', 'hep-lat-0108006-2-48-6': 'The resultant parameters are shown in Table [REF].', 'hep-lat-0108006-2-48-7': 'The change in discretisation errors across out data set is clearly small, as we expected from the success of the previous fits.', 'hep-lat-0108006-2-48-8': 'This justifies the use of a single chiral extrapolation over this limited range in [MATH].', 'hep-lat-0108006-2-48-9': 'Whilst it does not, however, rule out deviations between the results of this and its equivalent in the continuum limit, it does give some indication that these deviations will be small.', 'hep-lat-0108006-2-48-10': 'Clearly far greater accuracy in the measurements is needed to allow a confident extrapolation to [MATH].', 'hep-lat-0108006-2-49-0': 'Given the consistency of our description of the small [MATH] regime from our measurements, it is reasonable to use the values of [MATH] to estimate the pion decay constant, [MATH].', 'hep-lat-0108006-2-49-1': 'This is done in units of [MATH] in Tables [REF] and [REF].', 'hep-lat-0108006-2-49-2': 'We use the common chiral fit of Eqn. [REF] over the largest acceptable range to provide us with our best estimate and its statistical error.', 'hep-lat-0108006-2-49-3': 'We then use the fits with other functional forms to provide us with the systematic error.', 'hep-lat-0108006-2-49-4': 'This produces an estimate [EQUATION] where the first error is statistical and the second is systematic.', 'hep-lat-0108006-2-49-5': 'This is of course no more than our best estimate of the value of [MATH] corresponding to our lattice spacing of [MATH].', 'hep-lat-0108006-2-49-6': 'This value will contain corresponding lattice spacing corrections and these must be estimated before making a serious comparison with the experimental value.', 'hep-lat-0108006-2-49-7': 'We merely note that using [MATH] we obtain from Eqn. [REF] the value [EQUATION] which is reasonably close to the experimental value [MATH].', 'hep-lat-0108006-2-50-0': '# Comparison with other studies', 'hep-lat-0108006-2-51-0': 'During the course of this work, there have appeared a number of other studies of the topological susceptibility in lattice QCD; in particular by the Pisa group [CITATION], the CP-PACS collaboration [CITATION], the SESAM-T[MATH]L collaboration [CITATION] and the Boulder group [CITATION].', 'hep-lat-0108006-2-52-0': 'The most recent studies [CITATION] are consistent with our findings, but the earlier ones found no significant decrease of the susceptibility with decreasing quark (or pion) mass when everything was expressed in physical, rather than lattice, units.', 'hep-lat-0108006-2-52-1': 'Indeed when our detailed results and analysis were first publicised [CITATION] all the other studies then available [CITATION] (and indeed [CITATION]) appeared to contradict our findings and it was therefore necessary for us to provide some reason why this might be so.', 'hep-lat-0108006-2-52-2': 'Although the situation is now different, the lessons are still useful and we will therefore briefly summarise the main point here.', 'hep-lat-0108006-2-52-3': 'For more details we refer the reader to [CITATION].', 'hep-lat-0108006-2-53-0': 'All these other calculations differ from our study in having been performed at fixed [MATH].', 'hep-lat-0108006-2-53-1': 'That implies that the lattice spacing [MATH] decreases as [MATH] is decreased.', 'hep-lat-0108006-2-53-2': 'In typical current calculations this variation in [MATH] is substantial.', 'hep-lat-0108006-2-53-3': '(See for example Fig. 4 of [CITATION].)', 'hep-lat-0108006-2-53-4': 'At the smallest values of [MATH] the lattice spacing cannot be allowed to be too fine, because the total spatial volume must remain adequately large.', 'hep-lat-0108006-2-53-5': 'This implies that at the larger values of [MATH] the lattice spacing is quite coarse.', 'hep-lat-0108006-2-53-6': 'Over such a range of lattice spacings, the topological susceptibility in the pure gauge theory typically shows a large variation (as, for example, in eqn. [REF]).', 'hep-lat-0108006-2-53-7': 'Since for coarser [MATH] more instantons (those with [MATH]) are excluded, and more of those remaining are narrow in lattice units (with a correspondingly suppressed lattice topological charge) we expect that this variation is quite general, and not a special feature of the pure gauge theory with a Wilson action.', 'hep-lat-0108006-2-53-8': 'In lattice QCD, we therefore expect two simultaneous effects in [MATH] as we decrease [MATH] at fixed [MATH].', 'hep-lat-0108006-2-53-9': 'First, because of the [MATH] lattice corrections just discussed, [MATH] will (like [MATH]) tend to increase.', 'hep-lat-0108006-2-53-10': 'Second, it will tend to decrease because of the physical quark mass dependence.', 'hep-lat-0108006-2-53-11': 'In the range of quark masses covered in current calculations this latter decrease is not very large (as we have seen in our work) and we suggest that the two effects may largely compensate each other so as to produce a susceptibility that shows very little variation with [MATH], in contrast to the ratio [MATH] which does.', 'hep-lat-0108006-2-54-0': 'To illustrate this consider the fixed-[MATH] calculation in [CITATION].', 'hep-lat-0108006-2-54-1': 'The range of quark masses covered in that work corresponds to [MATH] decreasing from about 6.5 to about 3.0.', 'hep-lat-0108006-2-54-2': 'Simultaneously [MATH] decreases from about 0.437 to about 0.274.', 'hep-lat-0108006-2-54-3': 'Over this range of [MATH] the pure gauge susceptibility increases by almost a factor of two, as we see using Eqn. [REF].', 'hep-lat-0108006-2-54-4': 'Clearly this is large enough to compensate for the expected variation of the susceptibility.', 'hep-lat-0108006-2-55-0': 'As [MATH] is increased, the [MATH] variation with [MATH] of the corresponding quenched susceptibility [MATH] will clearly diminish.', 'hep-lat-0108006-2-55-1': 'Thus we suggested in [CITATION] that if CP-PACS were to repeat their susceptibility calculations on their larger [MATH] ensembles, they would find an [MATH] variation of [MATH] consistent with ours.', 'hep-lat-0108006-2-55-2': 'This is what they have done in their most recent work [CITATION] with the result that we predicted.', 'hep-lat-0108006-2-55-3': 'All this emphasises the utility of the UKQCD strategy of decoupling the variation of lattice corrections from the physical [MATH] dependence, by performing calculations at fixed [MATH] rather than at fixed [MATH].', 'hep-lat-0108006-2-56-0': '# Summary', 'hep-lat-0108006-2-57-0': 'We have calculated the topological susceptibility in lattice QCD with two light quark flavours, using lattice field configurations in which the lattice spacing is approximately constant as the quark mass is varied.', 'hep-lat-0108006-2-57-1': 'We find that there is clear evidence for the expected suppression of [MATH] with decreasing (sea) quark mass.', 'hep-lat-0108006-2-58-0': 'We have discussed this behaviour in the context of chiral and large [MATH] expansions, and find good agreement with the functional forms expected there.', 'hep-lat-0108006-2-58-1': 'We are not able to make a stronger statement about how close QCD is to its large [MATH] limit, owing to the relatively large statistical errors on our calculated values, particularly at larger quark masses.', 'hep-lat-0108006-2-58-2': 'This situation should change in the near future and, together with the increasing availability of information on the large-[MATH] behaviour of the pure gauge (quenched) theory [CITATION], a more precise comparison will become possible.', 'hep-lat-0108006-2-59-0': 'The consistent leading order chiral behaviour from our various fitting ansatze allows us to make an estimate for the pion decay constant, [MATH], for the lattice spacing of [MATH].', 'hep-lat-0108006-2-59-1': '(Here the first error is statistical and the second has to do with the chiral extrapolation.)', 'hep-lat-0108006-2-59-2': 'We use a lattice fermion action in which the leading [MATH] discretisation errors have been removed.', 'hep-lat-0108006-2-59-3': 'Since the more accurate (quenched) hadron masses show little residual lattice spacing dependence [CITATION], we might expect that this value of [MATH] is close to its continuum limit.', 'hep-lat-0108006-2-59-4': 'In any case, we note that it is in agreement with the experimental value, [MATH].'}","[['hep-lat-0108006-1-26-0', 'hep-lat-0108006-2-26-0'], ['hep-lat-0108006-1-26-1', 'hep-lat-0108006-2-26-1'], ['hep-lat-0108006-1-26-2', 'hep-lat-0108006-2-26-2'], ['hep-lat-0108006-1-26-3', 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1803.01639,"{'1803.01639-1-0-0': 'Determining the relative importance of environmental factors, biotic interactions and stochasticity in assembling and maintaining species-rich communities remains a major challenge in ecology.', '1803.01639-1-0-1': 'In plant communities, interactions between individuals of different species are expected to leave a spatial signature in the form of positive or negative spatial correlations over distances relating to the spatial scale of interaction.', '1803.01639-1-0-2': 'Most studies using spatial point process tools have found relatively little evidence for interactions between pairs of species.', '1803.01639-1-0-3': 'More interactions tend to be detected in communities with fewer species.', '1803.01639-1-0-4': 'However, there is currently no understanding of how the power to detect spatial interactions may change with sample size (number of individuals of each species), or the scale and intensity of interactions.', '1803.01639-1-0-5': 'Consequently, it is not known how much of the general patterns is generated by random variation and how much is due to structured behavior of biological processes.', '1803.01639-1-1-0': 'We use a simple 2-species model, developed from shot-noise models, where the scale and intensity of interactions are controlled to simulate point pattern data.', '1803.01639-1-1-1': 'In combination with an approximation to the variance of the spatial summary statistics that we sample, we investigate the power of current spatial point pattern methods to correctly reject the null model of bivariate species independence.', '1803.01639-1-2-0': 'We show that the power to detect interactions is positively related to the abundances of the species tested, and the intensity and scale of interactions.', '1803.01639-1-2-1': 'In contrast, increasing imbalance in abundances has a negative effect on the power to detect interactions.', '1803.01639-1-2-2': 'At population sizes typically found in currently available datasets for species-rich plant communities we find only a very low power to detect interactions.', '1803.01639-1-2-3': 'Differences in power may explain the increased frequency of interactions in communities with fewer species.', '1803.01639-1-2-4': 'Furthermore, the community-wide frequency of detected interactions is very sensitive to a minimum abundance criterion for including species in the analyses.', '1803.01639-1-3-0': 'Our results imply the minimum abundance thresholds for species inclusion and the differences in abundance distributions between communities could explain many of the general patterns reported in recent analyses.', '1803.01639-1-3-1': 'We suggest ecologists perform power analyses on their data before interpreting any results.', '1803.01639-1-4-0': 'Keywords: Interspecific interactions; co-associations; community ecology; neighborhood analysis; null model; spatial point patterns; statistical power', '1803.01639-1-5-0': '# Introduction', '1803.01639-1-6-0': 'Understanding the contribution of biological interactions to the assembly and regulation of natural communities remains a key goal in ecology.', '1803.01639-1-6-1': 'The continual development and refinement of methods to detect interactions from spatial, temporal and spatio-temporal data has therefore been a mainstay of the literature on the subject.', '1803.01639-1-7-0': 'A particular focus on the role of competition can be found in plant ecology, not least because plants seem to require the same few nutrients , but also because their sessile nature might permit an understanding of processes from the spatial pattern of individuals , and allow for easier experimental manipulation .', '1803.01639-1-7-1': 'Multiple methods exist to detect interspecific interactions but in non-manipulative field conditions there are often only two choices, both of which rely upon data where the location, identity and often size of every individual is recorded .', '1803.01639-1-7-2': 'The first option is to fit growth and/or survival models that take into account the identity and size of nearby neighbours .', '1803.01639-1-7-3': 'However, this requires repeated sampling over time in order to track the fate of every individual and very often such data is not available.', '1803.01639-1-7-4': 'The second option is to investigate the spatial pattern of the community to test the null hypothesis that species are independently arranged with respect to one another.', '1803.01639-1-7-5': ""Inference from a single snapshot of the community relies upon the assumption that spatial data retains the 'memory' of the birth, death and growth of the individuals and consequently the effect of interspecific interactions should show up as inter-species spatial dependence after any effect of the abiotic environment has been removed ."", '1803.01639-1-7-6': 'Dependent interactions can then be categorised as being a competitive interaction if the species are spatially segregated, and facilitation if they are aggregated in space, although confirmation via experimental manipulation is still advisable.', '1803.01639-1-7-7': ""Due to less restrictive data requirements (the test can be carried out on a single sampling of the community), the spatial snapshot option has proven more popular, and the test methods employ well-established spatial statistics such as Ripley's [MATH] or the pair correlation function to test the null model of spatial independence ."", '1803.01639-1-8-0': 'The results of previous spatial analyses of multi-species communities have found only a very low frequency of interspecific spatial interactions (aggregation/segregation) over scales relevant to plant competition, implying interspecific interactions are generally rare, or weak (as discussed by ).', '1803.01639-1-8-1': 'However, comparisons of different plant communities have also revealed a positive relationship between the frequency of spatial independence and the number of species in the community .', '1803.01639-1-8-2': 'Spatial independence between all pairs of species is one of the key assumptions of null models for biodiversity , and the low frequency of detected interactions has been put forward in support of this assertion.', '1803.01639-1-8-3': 'However, classical niche theory also predicts the strength of interspecific interactions to decline as the number of coexisting species increases , with the relative strength of interspecific interactions being proportional to [MATH] for [MATH] species.', '1803.01639-1-8-4': 'Therefore the main difference between the theories is that null models for biodiversity assume spatial independence for all communities regardless of species richness, whereas niche theory predicts spatial interactions are likely to be stronger, and therefore more frequently detected in less species-rich communities.', '1803.01639-1-8-5': 'Hence we argue the spatial analyses appear to better support the predictions of classical niche theory.', '1803.01639-1-9-0': 'However, both the low frequency of interspecies interactions and the relationship between species richness and species interactions could arise due to the ability of the statistical tests to detect significant interactions at the sample sizes being used.', '1803.01639-1-9-1': 'Because of the unequal treatment of the null and alternative hypothesis in classical testing, failure to reject the hypothesis of no interaction does not provide concrete proof of a lack of interactions.', '1803.01639-1-9-2': 'As pointed out by [CITATION], when species are rare the rate at which two species might co-occur in space is also very low and the statistical tests used might not be able to detect any interaction, even if it were very strong.', '1803.01639-1-9-3': 'If, as is often the case, species-rich communities have few common and many rare species, then we would expect to detect few significant interactions.', '1803.01639-1-9-4': 'Indeed, several investigations have found the frequency of dependent (i.e. positive or negative) spatial associations between species to be positively related to the abundance of both species being considered .', '1803.01639-1-10-0': 'For all tests a lower limit on the abundances of species to be included in the analyses must normally be set.', '1803.01639-1-10-1': 'Previous investigations have used a range of lower abundance thresholds including 100 , 70 , 30 and even 18 individuals.', '1803.01639-1-10-2': 'However, how and why is the lower threshold of individuals selected?', '1803.01639-1-10-3': 'What are the limits of our analyses to detect significant interspecific interactions, even if they might be quite strong?', '1803.01639-1-10-4': 'Currently, little is known about the statistical power of the tests for spatial independence between pairs of species that are commonly used and consequently there are no guidelines for the lower abundance threshold.', '1803.01639-1-10-5': 'As such care is required when interpreting failures to reject the null hypothesis, and we argue it is hard to make strong statements about the relative roles of stochastic- and niche-based processes across different communities until we gain a better understanding of the power of the methods to detect departures from spatial independence.', '1803.01639-1-11-0': 'Here we will elaborate on the statistical power of commonly used methods to detect significant interactions from spatial point pattern data.', '1803.01639-1-11-1': 'We shall study this problem by constructing a simple model for generating bivariate patterns where we can directly control the strength of interaction, and by utilising an approximation to the variance of the spatial summary statistic.', '1803.01639-1-11-2': ""We will show how the power to detect significant interactions is very much a function of the species' abundances, the strength of the interaction (normally the variable we are trying to infer, and therefore unknown), and the spatial scale over which the test is performed."", '1803.01639-1-11-3': 'Unfortunately, it is not possible to provide definitive sample size criteria since the power also changes with the summary statistic and test method being used.', '1803.01639-1-11-4': 'Despite this, we believe that even a rough understanding of the power of the tests to detect dependent structure is better than no understanding.', '1803.01639-1-11-5': 'With this caveat in mind, our analyses will suggest previous abundance thresholds for species inclusion are likely too low to detect even very strong interactions in the most species-rich communities being tested, thus questioning the previously derived conclusion of a lack of dependence between species.', '1803.01639-1-11-6': 'Since power can be estimated from Monte-Carlo simulations we hope our results will motivate ecologists to think more about the issue of sample size in future studies and therefore help to resolve the debate over the relative importance of biotic interactions in species-rich communities.', '1803.01639-1-12-0': '# Materials and Methods', '1803.01639-1-13-0': '## Summary statistics for bivariate interaction', '1803.01639-1-14-0': 'Consider data for two species labelled 1 and 2 given as two sets of locations of individuals [MATH] and [MATH] respectively, where the locations are observed in a well-defined area.', '1803.01639-1-14-1': ""We will call the combined set of points [MATH] a bivariate point pattern, and refer to the individuals' locations simply as points."", '1803.01639-1-14-2': 'Technical details are left to Appendix [REF], but in brief we assume that the data generating mechanisms can be described by some processes [MATH] and [MATH], and the goal of statistical analysis is to draw conclusions about the processes using the observed set [MATH].', '1803.01639-1-14-3': 'We start by assuming that the processes are second-order stationary, which means there is no underlying heterogeneity in the abiotic environment (e.g. elevation, soil chemistry) that also affects the distributions of the species, and implies that the statistics calculated from the data do not depend on any particular location in the observation window (see the Discussion for extensions).', '1803.01639-1-14-4': 'Although ecological communities are rarely well approximated by stationary models, we motivate studying the stationary case as this must be explored first, before any more complex scenarios can be understood.', '1803.01639-1-15-0': ""We will focus our attention on the second-order statistic commonly known as Ripley's [MATH] and its derivative, the pair correlation function; our rationale being these two summaries are amongst the most popular when characterizing joint dependence ."", '1803.01639-1-15-1': 'First (as is standard) we define the intensity of a point process [MATH] as the expected number of points per unit area.', '1803.01639-1-15-2': 'The cross-[MATH] or partial-[MATH], denoted here by [MATH], is a function defined as the expected number of points of species [MATH] in a circle of radius [MATH] placed on a random individual of species [MATH], scaled with intensity [MATH] to remove dimension and facilitate comparisons.', '1803.01639-1-15-3': 'Due to symmetry, it follows that [MATH].', '1803.01639-1-15-4': 'The parameter [MATH] controls for spatial scale and allows for multi-scale analysis.', '1803.01639-1-16-0': 'The derivative of [MATH] in [MATH] is denoted by [MATH], and is called the cross- or partial- pair correlation function (pcf).', '1803.01639-1-16-1': 'The pcf describes the aggregation/segregation of cross species point locations at distance [MATH] where the probability of having a species 1 individual in some small region and a species 2 individual in some small region distance [MATH] away is relative to [MATH].', '1803.01639-1-16-2': 'The quantities are scaled so that for independent processes the expectation is [MATH] and [MATH].', '1803.01639-1-16-3': 'The different statistics are used to ask subtly different questions, with [MATH] testing for species independence up-to distance [MATH], and [MATH] testing for independence at distance [MATH].', '1803.01639-1-17-0': '## Model generated data for illustration', '1803.01639-1-18-0': 'For better understanding of the power of bivariate point pattern statistics, we develop a simple two-species model for which the level of cross-species aggregation/segregation can be controlled directly and explicitly by two parameters that determine the spatial scale and the strength of the interaction.', '1803.01639-1-18-1': 'Using this model we can provide power estimates for different sample sizes and interaction scales and strengths using simulations.', '1803.01639-1-18-2': 'The details of the model are provided in Appendix [REF].', '1803.01639-1-18-3': 'Briefly, we assume species 1 is insensitive to the presence of species 2, but that the spatial distribution of species 2 is dependent on the spatial distribution of species 1.', '1803.01639-1-18-4': 'Asymmetric interactions are a reasonable starting point given they are thought to be quite common in plant communities and theory suggests competitive asymmetry may help maintain diversity in competitive communities .', '1803.01639-1-18-5': 'The locations of all [MATH] individuals are given by a Poisson process so species 1 exhibits no intraspecific spatial structure.', '1803.01639-1-18-6': 'The [MATH] individuals are placed with distribution that depends on the locations of species 1.', '1803.01639-1-18-7': 'Importantly the model has [EQUATION] where [MATH] is a decreasing function whose exponential decay is controlled by the parameter [MATH], and has a range ([MATH] is non-zero) of approximately [MATH].', '1803.01639-1-18-8': 'This function is analogous to the interaction or competition kernels used in spatially explicit birth-death models .', '1803.01639-1-18-9': ""The strength of interspecies' interaction, as summarized by [MATH], is controlled by the parameter [MATH]."", '1803.01639-1-18-10': 'If [MATH] the two species exhibit segregation [MATH], if [MATH] the two species exhibit aggregation or clustering [MATH], and when [MATH] the two species are independent.', '1803.01639-1-18-11': 'The reader should note that this model is simply a pattern generating process for illustration, rather than a mechanistic model, and we simulate patterns conditional on fixed [MATH] and [MATH] as we want full control over them (for the unconditional model the abundances are random, like in the birth and death processes, see e.g. ).', '1803.01639-1-18-12': 'Example point patterns showing inter-species aggregation and segregation can be found in Appendix [REF], Figure [REF].', '1803.01639-1-19-0': '## Testing bivariate independence + We now turn our attention to the main problem of determining if the processes [MATH] and [MATH], as observed through the bivariate point pattern [MATH], are statistically independent.', '1803.01639-1-19-1': 'If the processes were independent, then the observed pattern would be a random superposition of the two processes.', '1803.01639-1-19-2': 'We will take this as our independence or null hypothesis which now needs to be tested using the observed data.', '1803.01639-1-20-0': 'To test if the independence hypothesis is compatible with the data, observed values of a chosen test statistic are compared to the distribution of the test statistic under the independence model.', '1803.01639-1-20-1': 'We can test either a) at some specific range, which we call pointwise tests or b) simultaneously over multiple ranges.', '1803.01639-1-20-2': 'For both types of tests the idea is to compute some test statistic [MATH] from the data, and compare it to the values of [MATH] (its distribution) as if the null hypothesis were true.', '1803.01639-1-20-3': 'If the data value is sufficiently extreme, we have reason to reject the null hypothesis.', '1803.01639-1-21-0': 'The true distribution of the test statistic under independence is rarely known in point pattern applications, and needs to be approximated by an empirical distribution derived from simulations under the independence model.', '1803.01639-1-21-1': 'This approach is known as Monte Carlo testing .', '1803.01639-1-21-2': 'We consider the observation area to be rectangular, in which case the independence simulation consists of randomly shifting pattern 1 (or 2 or both) with a toroidal wrap .', '1803.01639-1-21-3': 'This keeps the intra-species statistics of the patterns intact while ""breaking"" any inter-species dependencies, and can also be used for inhomogeneous patterns .', '1803.01639-1-22-0': 'For the purposes of this discussion, we will consider only the simple pointwise testing scenario, for which we can employ an analytical approach using a Gaussian approximation to the distribution corresponding to the random shift simulations.', '1803.01639-1-22-1': 'As we will show, the approximation is very useful since it is not only computationally very efficient relative to the MC simulations, but also allows some analytical insight into what affects the power of the tests.', '1803.01639-1-22-2': 'The pointwise tests we will study are comparable to simultaneous tests when the best range to test at is known (see Table [REF] in Appendix [REF]).', '1803.01639-1-22-3': 'As detailed in Appendix [REF], we can choose an unbiased estimator [MATH] for which approximately it holds: [EQUATION] where [MATH] is the value under the correct model.', '1803.01639-1-22-4': 'Conditional on the observed point counts [MATH], the variance of [MATH] can be approximated by [EQUATION] where [MATH] are constants depending on the range [MATH] and the geometry of the observation area (see Appendix [REF]).', '1803.01639-1-22-5': 'The variance is exact when [MATH] and [MATH] are uniformly distributed, but works quite well also for internally aggregated/segregated patterns as we will see later on in Section [REF].', '1803.01639-1-22-6': 'Although we focus on [MATH], the approach is nearly identical for [MATH], only the constants are different.', '1803.01639-1-23-0': '## Power of a statistical test', '1803.01639-1-24-0': 'Denote the null hypothesis of bivariate independence by [MATH], the test statistic by [MATH], and a confidence level of the test by [MATH] where [MATH].', '1803.01639-1-24-1': ""Recall that [MATH] is the researcher's fixed accepted margin of making a false positive decision, also known as type I error, defined mathematically as [EQUATION] where [MATH] is the distribution of [MATH] when [MATH] is true, [MATH] is the corresponding threshold value for [MATH] so that if [MATH] under [MATH] we reject the null hypothesis [MATH]."", '1803.01639-1-24-2': 'The condition refers to [MATH] being tested.', '1803.01639-1-24-3': 'On the other hand, the power of a test is the probability of a true positive judgment, i.e. the probability of rejection when the hypothesis [MATH] does not hold.', '1803.01639-1-24-4': 'Consider first the margin of making a false negative judgment, [EQUATION] also known as type II error.', '1803.01639-1-24-5': 'Then the power of the test is defined as [EQUATION].', '1803.01639-1-24-6': 'Therefore, a test is powerful if it can correctly reject the wrong null model with a high probability.', '1803.01639-1-25-0': 'Consider the idealized situation of testing the cross-species independence using the pointwise summary [MATH] for some fixed spatial scale [MATH] only.', '1803.01639-1-25-1': 'For the test statistic [MATH] the null hypothesis [MATH]: ""random superposition"" implies [MATH].', '1803.01639-1-25-2': 'Let us now consider the situation that in truth [MATH].', '1803.01639-1-25-3': 'Then if we accept the approximate Gaussianity of the test statistic as shown in the previous section, it follows by elementary manipulations that [EQUATION] where [MATH] is the cumulative distribution function of the standard Normal distribution, with [MATH]-quantiles [MATH].', '1803.01639-1-25-4': 'Notice that the sign of interaction does not matter, meaning that due to symmetry of the Gaussian distribution aggregation is as easy or hard to detect as segregation of similar strength.', '1803.01639-1-25-5': 'Also notice how the power is dependent on the variance ([MATH]) of the test statistic used.', '1803.01639-1-25-6': 'The smaller the variance, the higher the power, which explains why different unbiased estimators of [MATH] have been developed and, while all being correct in the sense of bias, they can lead to different rates of detecting interactions because of different variances.', '1803.01639-1-26-0': 'We can now use the power formula and our approximation for the variance (equation [REF]) to illustrate how to', '1803.01639-1-27-0': '# Results', '1803.01639-1-28-0': 'The power formula (equation [REF]) is a good approximation to the power of the toroidal shift Monte Carlo test (Fig. [REF]).', '1803.01639-1-28-1': ""There is very little difference between the test's true power and the approximative power given by the analytical formula, with the analytical approximation slightly overestimating the power (at most [MATH])."", '1803.01639-1-28-2': 'This implies that we can discuss the power and its effect on ecological interpretations using the convenient analytical formula, acknowledging the small optimistic bias.', '1803.01639-1-29-0': 'As indicated by equation [REF] the variance of the estimator for the [MATH]-function is increased when either or both of [MATH] and [MATH] are small.', '1803.01639-1-29-1': 'This means that both the imbalance in population abundances as well as the total number of individuals affect our ability to detect bivariate interactions.', '1803.01639-1-29-2': 'We shall investigate each of these in turn, as well as the spatial range of testing.', '1803.01639-1-30-0': '## Power in balanced scenarios and the importance of the spatial scale of testing', '1803.01639-1-31-0': 'Fig. [REF] depicts the pointwise powers for different balanced ([MATH]) low-abundance scenarios when data is segregated (aggregated results are nearly identical).', '1803.01639-1-31-1': 'Visual inspection of the example point patterns (Fig. [REF], top row) already gives some indication that departures from spatial independence might be hard to detect for the lowest abundances.', '1803.01639-1-31-2': 'More formal analysis of the power quantifies the increase in ability to detect interactions with increasing abundances ([MATH], [MATH]) of the species being investigated and how this is affected by the spatial scale at which the hypothesis is tested (Fig. [REF], bottom row).', '1803.01639-1-31-3': 'In all cases the power to detect the interaction at small spatial scales ([MATH]) is low because, although the interaction is at its strongest here, the variance of [MATH] is relatively high and overwhelms the ecological signal.', '1803.01639-1-31-4': 'The trade-off between signal and noise leads to a unimodal relationship between power and the neighbourhood radius [MATH], with the peak being approximately at [MATH] for the interaction range [MATH] for all abundance sizes considered (Fig. [REF]).', '1803.01639-1-31-5': 'We will refer to this peak in power with [MATH] as the optimal range for testing, and will focus on this best case scenario for the results presented below.', '1803.01639-1-31-6': 'The unimodal relationship highlights the point that having some prior knowledge about the likely ranges of biotic interactions is going to be important for detecting interactions.', '1803.01639-1-32-0': 'Previous results based on in situ data analysis suggest detectable interactions between trees typically occur over 10-20m .', '1803.01639-1-32-1': 'Scaling our analyses accordingly, we can use the power formula to estimate the population sizes we require in order to reliably detect an interaction of a given strength and range (Fig. [REF]).', '1803.01639-1-32-2': 'If for example we wish to be 75% sure a true positive is not to be missed when the interaction strength is weak ([MATH]= -0.1), then we require species with populations of approximately 400 individuals for the 10 unit interaction neighbourhood ([MATH] = 10) and 250 individuals for 20 unit neghbourhood ([MATH] = 20).', '1803.01639-1-32-3': 'This value is surprisingly large compared to what data we commonly have available to us.', '1803.01639-1-33-0': 'In contrast, for the maximum possible negative interaction strength ([MATH]= -1), a similar level of power is reached with only around 35 individuals for [MATH] = 10 unit and 18 individuals for [MATH] = 20.', '1803.01639-1-33-1': 'Conversely, if we have a pair of species with [MATH] = 50, and we wish to be 75% sure a true positive is not missed, we must hope that the true interaction [MATH] when coupled with short interaction range ([MATH]) is at least [MATH], and if coupled with long interaction range ([MATH]) is at least [MATH].', '1803.01639-1-33-2': 'It therefore seems likely that only the very strongest interactions are detectable with the number of individuals that are typically found in the species-rich datasets.', '1803.01639-1-34-0': '## Imbalances in species abundance', '1803.01639-1-35-0': ""Since most communities exhibit a 'hollow curve' distribution of population abundances , an imbalance in population sizes is very common."", '1803.01639-1-35-1': 'From the variance formula [REF] it is clear that imbalance has a strong effect on the power because the term [MATH], and hence the variance, increases with imbalance.', '1803.01639-1-35-2': 'This relationship is confirmed when we use the power formula to quantify the effect of population imbalance for different interaction strengths and combined population sizes (Fig. [REF]).', '1803.01639-1-35-3': 'So, for example, for an interaction strength of [MATH] = 0.1 and a desired power of 80, a combined individual count of about 750 is required when the populations are perfectly balanced, but 1000 are required when one species is five times more abundant than the second species, and a surprisingly large 1500 required when one species is ten times more abundant than the other.', '1803.01639-1-35-4': 'Alternatively, consider that we require 90 power, and that the interactions are assumed to be [MATH] and of short range, [MATH].', '1803.01639-1-35-5': 'Then, to be on the safe side, we should attain samples of sizes at least [MATH] or [MATH], depending on the imbalance.', '1803.01639-1-36-0': '## Power at rainforest sample sizes', '1803.01639-1-37-0': 'We now consider how our understanding of the power to detect interactions might affect results for observed plant communities.', '1803.01639-1-37-1': 'For simplicity, let us assume interactions are of the type given by our model and that every species is interacting with every other species in an identical manner (so [MATH] and range [MATH] are the same for all pairs of species).', '1803.01639-1-37-2': 'Since the power is the probability of detecting interactions, we can get a rough estimate of the number of detected cross-species interactions by assuming the tests are independent, and summing up the powers.', '1803.01639-1-37-3': 'This then allows a coarse comparison of recently reported frequencies of detected interactions in tropical forests with the expected frequency of detected interactions as a function of power.', '1803.01639-1-38-0': 'Fig. [REF] shows the expected number of cross-species interactions detected as a function of abundance for various hypothetical interaction strengths and ranges.', '1803.01639-1-38-1': 'The abundances are taken from the Barro Colorado Island (BCI) 1995 census of woody plants with diameter at breast height at least 1cm .', '1803.01639-1-38-2': 'The abundances are highly skewed, with a large proportion of low abundance species, and we show the power in two cases, when the pool of species consists of those with abundance at least 30 and 100.', '1803.01639-1-38-3': 'Reducing the species pool by increasing the abundance threshold naturally increases the proportions of detection, and highlights the importance of using similar thresholds when comparing different communities.', '1803.01639-1-38-4': 'It is striking how little power is to be expected for most of the species even when assuming strong interaction ([MATH] = -0.75).', '1803.01639-1-38-5': 'Only when the abundance of a species reaches thousands, can we be expected to detect even 50% of the interactions present.', '1803.01639-1-38-6': 'This is a very thought-provoking result, as the lack of detection might be explained simply by a lack of power in the majority of species-pairs.', '1803.01639-1-39-0': 'Making the same assumptions about identical interactions between all pairs of species, we conducted a comparative analysis using abundances of plants in the BCI 1995 census, the Changbaishan (CBS) forest plot , and the Sinharaja 1995 census .', '1803.01639-1-39-1': 'Following [CITATION], we considered only large plants (diameter [MATH]10cm) and keep only those species with at least 70 large individuals (note that thresholded CBS at 50 individuals).', '1803.01639-1-39-2': 'The analysis shows a big difference in the expected proportion of interactions that would be detected due to differences in the species abundances of the communities (Table [REF]).', '1803.01639-1-39-3': 'Although the actual proportions differ, qualitatively, these results are in a sense similar to those reported by [CITATION] who detected that approximately [MATH] and [MATH] of interactions departed from their null hypothesis in the CBS, Sinharaja and BCI communities, respectively.', '1803.01639-1-40-0': 'These toy examples highlight how the relationships of the power to detect interactions with population sizes, strength of interaction and spatial scale of interaction can in principle lead to patterns similar to those described in previous studies.', '1803.01639-1-40-1': 'Fortunately these artificial examples can be taken as ""worst case"" scenarios.', '1803.01639-1-40-2': 'Many of the species in forest plots are highly localised to environmental niches , in which case the context of testing needs to be defined more accurately and the pool of potential interactions limited, thus regaining power.', '1803.01639-1-41-0': '# Discussion', '1803.01639-1-42-0': 'Understanding the relative strength and therefore the importance of interspecific interactions is one of the key goals of community ecology, and the null model approach has been popular for characterizing spatial point patterns of (predominantly) diverse plant communities .', '1803.01639-1-42-1': 'However, there has been little guidance on when a given test is likely able to detect species associations that are present.', '1803.01639-1-42-2': 'Here we have made a first step in closing this important gap in our understanding.', '1803.01639-1-42-3': 'Our results clarify the quantitative relationships between the strength of the underlying biological interaction, sample size (number of individuals of both species under investigation), and the spatial scale over which the test is being performed.', '1803.01639-1-42-4': 'We have also shown that statistical power may explain both the low detection rate of biological interactions in plant communities, and the negative relationship between species-richness and frequency of detected interspecific interactions in comparative studies.', '1803.01639-1-43-0': 'Ecologists have had to rely largely upon their intuition for deciding the minimum population size to include in their analyses with the result that a range of criteria up to 100 individuals have been used.', '1803.01639-1-43-1': 'For species-rich communities, where many interspecific interactions may necessarily be weak , abundances of both species may need to be in the hundreds of individuals before any interaction is detected (Fig. [REF]), and this implies previous abundance thresholds are likely too low to detect many interactions.', '1803.01639-1-43-2': 'As several authors have acknowledged, the failure to reject the null hypothesis of spatial independence in so many species-pairs does not necessarily mean interspecific interactions are not occurring, or present .', '1803.01639-1-43-3': 'We hope our study highlights how the power of the tests can be assessed and should be factored into the interpretation of the results.', '1803.01639-1-43-4': 'The power formula can also be used in estimating the area of observation necessary to increase the power to a desirable level (Appendix [REF]).', '1803.01639-1-43-5': 'We stress that there is still much to be learned about the power of the statistical tests used in earlier studies, given the assumptions we had to make, and that the reader should take our contribution as a first step that offers a rough guide to sample sizes that are required to make strong statements about the frequency and strength of interspecific interactions.', '1803.01639-1-44-0': 'Although our model is clearly mis-specified as we use tests assuming that intensity is not dependent on abiotic features of the environment, the general applicability of our results are to some extend valid in an inhomogeneous setting.', '1803.01639-1-44-1': 'In particular we would still expect a positive relationship between population size and frequency of interactions to emerge simply due to an increase in power at larger sample sizes.', '1803.01639-1-44-2': 'Such a positive relationship has already been reported in a number of empirical studies that take habitat associations into consideration .', '1803.01639-1-44-3': 'It is possible that common species are better competitors and are somehow suppressing the abundance of the weaker competitors, but without experimental manipulation, or perhaps different analyses using repeated sampling over time , it is hard to distinguish whether this pattern is a result of biological processes or the ability of the statistical methods to detect interactions at different population sizes.', '1803.01639-1-45-0': 'The spatial scale over which tests are performed is important for the ability to detect spatial dependencies (Fig. [REF]), and our results are similar to empirical studies that often find few negative interactions at the shortest distances, even though this is where the interactions are likely to be strongest .', '1803.01639-1-45-1': 'Short scales suffer from having high variability due to the relatively small number of neighbours possible in a small area, but at longer distances, the effect of neighbours is weaker.', '1803.01639-1-45-2': 'Hence there is a sweet spot where this trade-off is maximised, and the location of this is likely dependent on several factors, not least of which is the scale over which interactions are occurring.', '1803.01639-1-45-3': 'For woody plants, there have been several studies that have fitted neighbourhood growth or survival models to individual-based data that tracks the fate of trees over time , and most results seem to point to interactions being confined to 10-30m radius around an individual.', '1803.01639-1-45-4': 'However, little is known about how the spatial scale of interspecific interactions change with life history stage, environmental conditions, or even species identity even though the latter has been shown to be very important for determining coexistence .', '1803.01639-1-45-5': 'Any changes to the scales of interactions will have consequences for the hypothesis testing methods discussed here, but until more is understood about the spatial scales of interactions between species, it seems sensible to test over ranges reported in earlier studies.', '1803.01639-1-46-0': 'Our discussion up to this point has been in the context of stationary, most notably homogeneous, data.', '1803.01639-1-46-1': 'Most recent analyses have tried to factor out the effects of spatial heterogeneity in the abiotic environment by using inhomogeneous Poisson processes as the null model .', '1803.01639-1-46-2': 'Currently it is hard to predict whether the power of an inhomogeneous analogue of our scenario would be lower or higher.', '1803.01639-1-46-3': 'On the one hand we could expect higher power to detect interactions because the model better captures the underlying processes that generate the spatial distributions of the species within the community.', '1803.01639-1-46-4': 'However, we also expect variance to be increased, since extra parameters need to be estimated leaving a fewer degrees of freedom per parameter.', '1803.01639-1-46-5': 'For example, tests using the inhomogoneneous Poisson process method use a smoothing kernel to approximately remove the effects of large scale structure assumed to be caused by habitat associations (see e.g. ).', '1803.01639-1-46-6': 'Typically, the same smoothing parameter is used for all species, which is a sensible assumption when little is known about the spatial scale of habitat associations, but there is no reason to suspect a single smoothing parameter is appropriate for all species.', '1803.01639-1-46-7': 'An open challenge is to better understand how mis-specification of the smoothing parameter will bias the detection of interactions.', '1803.01639-1-46-8': 'Again, we feel that using a biologically motivated model to simulate data is a useful approach for exploring such issues.', '1803.01639-1-47-0': 'Finally, we remind the reader that the spatial statistics used in the null model approach do not say anything directly about the processes that may have created the patterns, and different processes could generate the same summary statistic.', '1803.01639-1-47-1': 'As an alternative, model-based approaches, either in the form we use here (which include the familiar Thomas Cluster models) or birth-death models could also be applied to the inference of biological interactions from point pattern data .', '1803.01639-1-47-2': 'Model fitting will normally lead to estimation of parameters that can also be estimated in the field (eg. dispersal kernels, interaction kernels), we therefore feel that their continued development will help improve understanding of the processes underpinning the results returned .', '1803.01639-1-48-0': 'In conclusion, we hope our main contribution is to encourage more users to consider explicitly the ability of the spatial point pattern tests to detect significant associations between species.', '1803.01639-1-48-1': 'We have shown that the data requirements to detect even strong interactions may be quite high, mirroring results for null model tests of species co-occurrences in community matrix data .', '1803.01639-1-48-2': 'On this basis, we suggest it is desirable to only interpret the frequency of interactions across large numbers of species once the effect of different powers to detect interactions for pairs of species of given population sizes has been (even approximately) factored out.', '1803.01639-1-48-3': 'This seems especially important in comparative analyses across different communities where the spatial scales, strengths of interactions and the species abundance distributions may differ and affect the power to detect interactions.', '1803.01639-1-49-0': ""# Authors' contributions"", '1803.01639-1-50-0': 'TR conceived the idea during discussions with DM and SO; TR derived the model and formulas, designed and executed computations, and contributed extensively to the manuscript; DM contributed to the rainforest experiment and extensively to the manuscript; SO contributed to the manuscript; All authors contributed to the intellectual core of the manuscript.'}","{'1803.01639-2-0-0': 'Determining the importance of biotic interactions in assembling and maintaining species-rich communities remains a major challenge in ecology.', '1803.01639-2-0-1': 'In plant communities, interactions between individuals of different species are expected to generate positive or negative spatial interspecific correlations over short distances.', '1803.01639-2-0-2': 'Recent studies using individual-based point pattern datasets have concluded that (i) detectable interspecific interactions are generally rare, but (ii) are most common in communities with fewer species; and (iii) the most abundant species tend to have the highest frequency of interactions.', '1803.01639-2-0-3': 'However, there is currently no understanding of how the detection of spatial interactions may change with the abundances of each species, or the scale and intensity of interactions.', '1803.01639-2-0-4': 'Here, we ask if statistical power is sufficient to explain all three key results.', '1803.01639-2-1-0': 'We use a simple 2-species model, where the scale and intensity of interactions are controlled to simulate point pattern data.', '1803.01639-2-1-1': 'In combination with an approximation to the variance of the spatial summary statistics that we sample, we investigate the power of current spatial point pattern methods to correctly reject the null model of pairwise species independence.', '1803.01639-2-2-0': 'We show the power to detect interactions is positively related to both the abundances of the species tested, and the intensity and scale of interactions, but negatively related to imbalance in abundances.', '1803.01639-2-2-1': 'Differences in detection power in combination with the abundance distributions found in natural communities are sufficient to explain all the three key empirical results, even if all species are interacting identically with all other species.', '1803.01639-2-2-2': 'Critically, many hundreds of individuals of both species may be required to detect even intense pairwise interactions, implying current abundance thresholds for including species in the analyses are too low.', '1803.01639-2-3-0': 'Synthesis: The widespread failure to reject the null model of spatial interspecific independence could be due to low power of the statistical tests rather than any key biological processes.', '1803.01639-2-3-1': 'Our analyses are a first step in quantifying how much data is required to make strong statements about the role of biotic interactions in diverse plant communities, and power should be factored into analyses and considered when designing empirical studies.', '1803.01639-2-4-0': 'Keywords: Determinants of plant community diversity and structure; Interspecific interactions; community ecology; neighborhood analysis; null model; spatial point patterns; statistical power', '1803.01639-2-5-0': '# Introduction', '1803.01639-2-6-0': 'Understanding the contribution of biological interactions to the assembly and regulation of natural communities remains a key goal in ecology.', '1803.01639-2-6-1': 'The continual development and refinement of methods to detect interactions from spatial, temporal and spatio-temporal data has therefore been a mainstay of the literature on the subject.', '1803.01639-2-7-0': 'A particular focus on the role of competition can be found in plant ecology, not least because plants seem to require the same few nutrients , but also because their sessile nature might permit an understanding of processes from the spatial pattern of individuals , and allow for easier experimental manipulation .', '1803.01639-2-7-1': 'Multiple methods exist to detect interspecific interactions but in non-manipulative field conditions there are often only two choices, both of which rely upon data where the location, identity and often size of every individual is recorded .', '1803.01639-2-7-2': 'The first option is to fit growth and/or survival models that take into account the identity and size of nearby neighbours .', '1803.01639-2-7-3': 'However, this requires repeated sampling over time in order to track the fate of every individual and very often such data is not available.', '1803.01639-2-7-4': 'Another issue is that because all interaction parameters are fitted at once, considering all pairwise interactions is very difficult due the large number of parameters.', '1803.01639-2-7-5': ""As a consequence neighbouring individuals are usually lumped into conspecifics and heterospecifics with the potential problem that interspecific interactions are 'lost' due to cancelling out of weak and strong, and/or positive and negative effects of different species."", '1803.01639-2-7-6': 'The second option is to investigate the spatial pattern of the community to test the null hypothesis that species are independently arranged with respect to one another.', '1803.01639-2-7-7': ""Inference from a single snapshot of the community relies upon the assumption that spatial data retains the 'memory' of the birth, death and growth of the individuals and consequently the effect of interspecific interactions should show up as inter-species spatial dependence after any effect of the abiotic environment has been removed ."", '1803.01639-2-7-8': 'Under the assumption that all pairwise tests are independent, each pair of species can be assessed individually, and dependent interactions are categorised as being a competitive interaction if the species are spatially segregated, and facilitation if they are aggregated in space, although confirmation via experimental manipulation is still advisable.', '1803.01639-2-7-9': ""Due to less restrictive data requirements (the test can be carried out on a single sampling of the community), the spatial snapshot option has proven to be very popular, and the test methods employ well-established spatial statistics such as Ripley's [MATH] or the pair correlation function to test the null model of spatial independence ."", '1803.01639-2-8-0': 'The results of previous spatial analyses of multi-species communities have found only a very low frequency of interspecific spatial interactions (aggregation/segregation) over scales relevant to plant competition, implying interspecific interactions are generally rare, or weak (as discussed by ).', '1803.01639-2-8-1': 'However, comparisons of different plant communities have also revealed a positive relationship between the frequency of spatial independence and the number of species in the community .', '1803.01639-2-8-2': 'Spatial independence between all pairs of species is one of the key assumptions of null models for biodiversity , and the low frequency of detected interactions has been put forward in support of this assertion .', '1803.01639-2-8-3': 'However, classical niche theory also predicts the strength of interspecific interactions to decline as the number of coexisting species increases (equation 4 in ), with the relative strength of interspecific interactions being proportional to [MATH] for [MATH] species.', '1803.01639-2-8-4': 'Therefore the main difference between the theories is that null models for biodiversity assume spatial independence for all communities regardless of species richness, whereas niche theory predicts spatial interactions are likely to be stronger, and therefore more frequently detected in less species-rich communities.', '1803.01639-2-8-5': 'Hence we argue the spatial analyses appear to better support the predictions of classical niche theory.', '1803.01639-2-9-0': 'However, both the low frequency of interspecies interactions and the relationship between species richness and species interactions could arise due to the ability of the statistical tests to detect significant interactions at the sample sizes being used.', '1803.01639-2-9-1': 'Because of the unequal treatment of the null and alternative hypothesis in classical testing, failure to reject the hypothesis of no interaction does not provide concrete proof of a lack of interactions.', '1803.01639-2-9-2': 'As pointed out by [CITATION], when species are rare the rate at which two species might co-occur in space is also very low and the statistical tests used might not be able to detect any interaction, even if it were very strong.', '1803.01639-2-9-3': 'If, as is often the case, species-rich communities have few common and many rare species, then we would expect to detect few significant interactions.', '1803.01639-2-9-4': 'Indeed, several investigations have found the frequency of significant spatial associations between species to be positively related to the abundance of both species being considered , raising the possibility that interactions can only be detected amongst the most abundant species.', '1803.01639-2-10-0': 'For all tests a lower limit on the abundances of species to be included in the analyses must normally be set, and this acknowledges there is a limit to our ability to detect even strong interactions in small sample sizes.', '1803.01639-2-10-1': 'Previous investigations have used a range of lower abundance thresholds including 100 , 70 , 30 and even 18 individuals.', '1803.01639-2-10-2': 'However, how and why is the lower threshold of individuals selected?', '1803.01639-2-10-3': 'What are the limits of our analyses to detect significant interspecific interactions?', '1803.01639-2-10-4': 'We are aware of no study that investigates the statistical power of the tests for spatial independence between pairs of species that are commonly used and consequently there are no guidelines for the lower abundance threshold.', '1803.01639-2-10-5': 'As such care is required when interpreting failures to reject the null hypothesis, and we argue it is hard to make strong statements about the relative roles of stochastic- and niche-based processes across different communities until we gain a better understanding of the power of the methods to detect departures from spatial independence.', '1803.01639-2-10-6': 'In other words, is spatial independence a good first approximation in species rich plant communities because of diffuse competition leading to weak interactions, or is it because the statistical methods lack the power to detect the interactions for the given sample sizes typically available?', '1803.01639-2-11-0': 'Here we will elaborate on the statistical power of commonly used methods to detect significant interactions from spatial point pattern data.', '1803.01639-2-11-1': 'We shall study this problem by constructing a simple model for generating bivariate patterns where we can directly control the strength of interaction, and by utilising an approximation to the variance of the spatial summary statistic.', '1803.01639-2-11-2': ""We will show how the power to detect significant interactions is very much a function of the species' abundances, the strength of the interaction (normally the variable we are trying to infer, and therefore unknown), and the spatial scale over which the test is performed."", '1803.01639-2-11-3': 'Unfortunately, it is not possible to provide definitive sample size criteria since the power also changes with the summary statistic and test method being used.', '1803.01639-2-11-4': 'Despite this, we believe that even a rough understanding of the power of the tests to detect dependent structure is better than no understanding.', '1803.01639-2-11-5': 'With this caveat in mind, our analyses will suggest previous abundance thresholds for species inclusion are likely too low to detect even very strong interactions in the most species-rich communities being tested, thus questioning the previously derived conclusion of a lack of dependence between species.', '1803.01639-2-11-6': 'Since power can be estimated from Monte-Carlo simulations we hope our results will motivate ecologists to think more about the issue of sample size in future studies and therefore help to resolve the debate over the relative importance of biotic interactions in species-rich communities.', '1803.01639-2-12-0': '# Materials and Methods', '1803.01639-2-13-0': '## Summary statistics for bivariate interaction', '1803.01639-2-14-0': 'Consider data for two species labelled 1 and 2 given as two sets of locations of individuals [MATH] and [MATH] respectively, where the locations are observed in a well-defined area.', '1803.01639-2-14-1': ""We will call the combined set of points [MATH] a bivariate point pattern, and refer to the individuals' locations simply as points."", '1803.01639-2-14-2': 'Technical details are left to Appendix [REF], but in brief we assume that the data generating mechanisms can be described by some processes [MATH] and [MATH], and the goal of statistical analysis is to draw conclusions about the processes using the observed set [MATH].', '1803.01639-2-14-3': 'We start by assuming that the processes are second-order stationary, which means there is no underlying heterogeneity in the abiotic environment (e.g. elevation, soil chemistry) that also affects the distributions of the species, and implies that the statistics calculated from the data do not depend on any particular location in the observation window (see the Discussion for extensions).', '1803.01639-2-14-4': 'Although ecological communities are rarely well approximated by stationary models, we motivate studying the stationary case as this must be explored first, before any more complex scenarios can be understood.', '1803.01639-2-15-0': ""We will focus our attention on the second-order statistic commonly known as Ripley's [MATH] and its derivative, the pair correlation function; our rationale being these two summaries are amongst the most popular when characterizing joint dependence ."", '1803.01639-2-15-1': 'First (as is standard) we define the intensity of a point process [MATH] as the expected number of points per unit area.', '1803.01639-2-15-2': 'The cross-[MATH] or partial-[MATH], denoted here by [MATH], is a function defined as the expected number of points of species [MATH] in a circle of radius [MATH] placed on a random individual of species [MATH], scaled with intensity [MATH] to remove dimension and facilitate comparisons.', '1803.01639-2-15-3': 'Due to symmetry, it follows that [MATH].', '1803.01639-2-15-4': 'The parameter [MATH] controls for spatial scale and allows for multi-scale analysis.', '1803.01639-2-16-0': 'The derivative of [MATH] in [MATH] is denoted by [MATH], and is called the cross- or partial- pair correlation function (pcf).', '1803.01639-2-16-1': 'The pcf describes the aggregation/segregation of cross species point locations at distance [MATH] where the probability of having a species 1 individual in some small region and a species 2 individual in some small region distance [MATH] away is relative to [MATH].', '1803.01639-2-16-2': 'The quantities are scaled so that for independent processes the expectation is [MATH] and [MATH].', '1803.01639-2-16-3': 'The different statistics are used to ask subtly different questions, with [MATH] testing for species independence up-to distance [MATH], and [MATH] testing for independence at distance [MATH].', '1803.01639-2-17-0': '## Model generated data for illustration', '1803.01639-2-18-0': 'For better understanding of the power of bivariate point pattern statistics, we develop a simple two-species model for which the level of cross-species aggregation/segregation can be controlled directly and explicitly by two parameters that determine the spatial scale and the strength of the interaction.', '1803.01639-2-18-1': 'Using this model we can provide power estimates for different sample sizes and interaction scales and strengths using simulations.', '1803.01639-2-18-2': 'The details of the model are provided in Appendix [REF].', '1803.01639-2-18-3': 'Briefly, we assume species 1 is insensitive to the presence of species 2, but that the spatial distribution of species 2 is dependent on the spatial distribution of species 1.', '1803.01639-2-18-4': 'Asymmetric interactions are a reasonable starting point given they are thought to be quite common in plant communities and theory suggests competitive asymmetry may help maintain diversity in competitive communities .', '1803.01639-2-18-5': 'The locations of all [MATH] individuals are given by a Poisson process so species 1 exhibits no intraspecific spatial structure.', '1803.01639-2-18-6': 'The [MATH] individuals are placed with distribution that depends on the locations of species 1.', '1803.01639-2-18-7': 'Importantly the model has [EQUATION] where [MATH] is a decreasing function whose exponential decay is controlled by the parameter [MATH], and has a range ([MATH] is non-zero) of approximately [MATH].', '1803.01639-2-18-8': 'This function is analogous to the interaction or competition kernels used in spatially explicit birth-death models .', '1803.01639-2-18-9': ""The strength of interspecies' interaction, as summarized by [MATH], is controlled by the parameter [MATH]."", '1803.01639-2-18-10': 'If [MATH] the two species exhibit segregation [MATH], if [MATH] the two species exhibit aggregation or clustering [MATH], and when [MATH] the two species are independent.', '1803.01639-2-18-11': 'The reader should note that this model is simply a pattern generating process for illustration, rather than a mechanistic model, and we simulate patterns conditional on fixed [MATH] and [MATH] as we want full control over them (for the unconditional model the abundances are random, like in the birth and death processes, see e.g. ).', '1803.01639-2-18-12': 'Example point patterns showing inter-species aggregation and segregation can be found in Appendix [REF], Fig. [REF].', '1803.01639-2-19-0': '## Testing bivariate independence + We now turn our attention to the main problem of determining if the processes [MATH] and [MATH], as observed through the bivariate point pattern [MATH], are statistically independent.', '1803.01639-2-19-1': 'If the processes were independent, then the observed pattern would be a random superposition of the two processes.', '1803.01639-2-19-2': 'We will take this as our independence or null hypothesis which now needs to be tested using the observed data.', '1803.01639-2-20-0': 'To test if the independence hypothesis is compatible with the data, observed values of a chosen test statistic are compared to the distribution of the test statistic under the independence model.', '1803.01639-2-20-1': 'We can test either a) at some specific range, which we call pointwise tests or b) simultaneously over multiple ranges.', '1803.01639-2-20-2': 'For both types of tests the idea is to compute some test statistic [MATH] from the data, and compare it to the values of [MATH] (its distribution) as if the null hypothesis were true.', '1803.01639-2-20-3': 'If the data value is sufficiently extreme, we have reason to reject the null hypothesis.', '1803.01639-2-21-0': 'The true distribution of the test statistic under independence is rarely known in point pattern applications, and needs to be approximated by an empirical distribution derived from simulations under the independence model.', '1803.01639-2-21-1': 'This approach is known as Monte Carlo testing .', '1803.01639-2-21-2': 'We consider the observation area to be rectangular, in which case the independence simulation consists of randomly shifting pattern 1 (or 2 or both) with a toroidal wrap .', '1803.01639-2-21-3': 'This keeps the intra-species statistics of the patterns intact while ""breaking"" any inter-species dependencies, and can also be used for inhomogeneous patterns .', '1803.01639-2-22-0': 'For the purposes of this discussion, we will consider only the simple pointwise testing scenario, for which we can employ an analytical approach using a Gaussian approximation to the distribution corresponding to the random shift simulations.', '1803.01639-2-22-1': 'As we will show, the approximation is very useful since it is not only computationally very efficient relative to the MC simulations, but also allows some analytical insight into what affects the power of the tests.', '1803.01639-2-22-2': 'The pointwise tests we will study are comparable to simultaneous tests when the best range to test at is known (see Table [REF] in Appendix [REF]).', '1803.01639-2-22-3': 'As detailed in Appendix [REF], we can choose an unbiased estimator [MATH] for which approximately it holds: [EQUATION] where [MATH] is the value under the correct model.', '1803.01639-2-22-4': 'Conditional on the observed point counts [MATH], the variance of [MATH] can be approximated by [EQUATION] where [MATH] are constants depending on the range [MATH] and the geometry of the observation area (see Appendix [REF]).', '1803.01639-2-22-5': 'The variance is exact when [MATH] and [MATH] are uniformly distributed, but works quite well also for internally aggregated/segregated patterns as we will see later on in Section [REF].', '1803.01639-2-22-6': 'Although we focus on [MATH], the approach is nearly identical for [MATH], only the constants are different.', '1803.01639-2-23-0': '## Power of a statistical test', '1803.01639-2-24-0': 'Denote the null hypothesis of bivariate independence by [MATH], the test statistic by [MATH], and a confidence level of the test by [MATH] where [MATH].', '1803.01639-2-24-1': ""Recall that [MATH] is the researcher's fixed accepted margin of making a false positive decision, also known as type I error, defined mathematically as [EQUATION] where [MATH] is the distribution of [MATH] when [MATH] is true, [MATH] is the corresponding threshold value for [MATH] so that if [MATH] under [MATH] we reject the null hypothesis [MATH]."", '1803.01639-2-24-2': 'The condition refers to [MATH] being tested.', '1803.01639-2-24-3': 'On the other hand, the power of a test is the probability of a true positive judgment, i.e. the probability of rejection when the hypothesis [MATH] does not hold.', '1803.01639-2-24-4': 'Consider first the margin of making a false negative judgment, [EQUATION] also known as type II error.', '1803.01639-2-24-5': 'Then the power of the test is defined as [EQUATION].', '1803.01639-2-24-6': 'Therefore, a test is powerful if it can correctly reject the wrong null model with a high probability.', '1803.01639-2-25-0': 'Consider the idealized situation of testing the cross-species independence using the pointwise summary [MATH] for some fixed spatial scale [MATH] only.', '1803.01639-2-25-1': 'For the test statistic [MATH] the null hypothesis [MATH]: ""random superposition"" implies [MATH].', '1803.01639-2-25-2': 'Let us now consider the situation that in truth [MATH].', '1803.01639-2-25-3': 'Then if we accept the approximate Gaussianity of the test statistic as shown in the previous section, it follows by elementary manipulations that [EQUATION] where [MATH] is the cumulative distribution function of the standard Normal distribution, with [MATH]-quantiles [MATH].', '1803.01639-2-25-4': 'Notice that the sign of interaction does not matter, meaning that due to symmetry of the Gaussian distribution aggregation is as easy or hard to detect as segregation of similar strength.', '1803.01639-2-25-5': 'Also notice how the power is dependent on the variance ([MATH]) of the test statistic used.', '1803.01639-2-25-6': 'The smaller the variance, the higher the power, which explains why different unbiased estimators of [MATH] have been developed and, while all being correct in the sense of bias, they can lead to different rates of detecting interactions because of different variances.', '1803.01639-2-26-0': 'We can now use the power formula and our approximation for the variance (equation [REF]) to illustrate how to', '1803.01639-2-27-0': '# Results', '1803.01639-2-28-0': 'The power formula (equation [REF]) is a good approximation to the power of the toroidal shift Monte Carlo test (Fig. [REF]).', '1803.01639-2-28-1': ""There is very little difference between the test's true power and the approximative power given by the analytical formula, with the analytical approximation slightly overestimating the power (at most [MATH])."", '1803.01639-2-28-2': 'This implies that we can discuss the power and its effect on ecological interpretations using the convenient analytical formula, acknowledging the small optimistic bias.', '1803.01639-2-29-0': 'As indicated by equation [REF] the variance of the estimator for the [MATH]-function is increased when either or both of [MATH] and [MATH] are small.', '1803.01639-2-29-1': 'This means that both the imbalance in population abundances as well as the total number of individuals affect our ability to detect bivariate interactions.', '1803.01639-2-29-2': 'We shall investigate each of these in turn, as well as the spatial range of testing.', '1803.01639-2-30-0': '## Power in balanced scenarios and the importance of the spatial scale of testing', '1803.01639-2-31-0': 'Fig. [REF] depicts the pointwise powers for different balanced ([MATH]) low-abundance scenarios when data is segregated (aggregated results are nearly identical).', '1803.01639-2-31-1': 'Visual inspection of the example point patterns (Fig. [REF], top row) already gives some indication that departures from spatial independence might be hard to detect for the lowest abundances.', '1803.01639-2-31-2': 'More formal analysis of the power quantifies the increase in ability to detect interactions with increasing abundances ([MATH], [MATH]) of the species being investigated and how this is affected by the spatial scale at which the hypothesis is tested (Fig. [REF], bottom row).', '1803.01639-2-31-3': 'In all cases the power to detect the interaction at small spatial scales ([MATH]) is low because, although the interaction is at its strongest here, the variance of [MATH] is relatively high and overwhelms the ecological signal.', '1803.01639-2-31-4': 'The trade-off between signal and noise leads to a unimodal relationship between power and the neighbourhood radius [MATH], with the peak being approximately at [MATH] for the interaction range [MATH] for all abundance sizes considered (Fig. [REF]).', '1803.01639-2-31-5': 'We will refer to this peak in power with [MATH] as the optimal range for testing, and will focus on this best case scenario for the results presented below.', '1803.01639-2-31-6': 'The unimodal relationship highlights the point that having some prior knowledge about the likely ranges of biotic interactions is going to be important for detecting interactions.', '1803.01639-2-32-0': 'Previous results based on in situ data analysis suggest detectable interactions between trees typically occur over 10-20m .', '1803.01639-2-32-1': 'Scaling our analyses accordingly, we can use the power formula to estimate the population sizes we require in order to reliably detect an interaction of a given strength and range (Fig. [REF]).', '1803.01639-2-32-2': 'If for example we wish to be 75% sure a true positive is not to be missed when the interaction strength is weak ([MATH]= -0.1), then we require species with populations of approximately 400 individuals for the 10 unit interaction neighbourhood ([MATH] = 10) and 250 individuals for 20 unit neghbourhood ([MATH] = 20).', '1803.01639-2-32-3': 'This value is surprisingly large compared to what data we commonly have available to us.', '1803.01639-2-33-0': 'In contrast, for the maximum possible negative interaction strength ([MATH]= -1), a similar level of power is reached with only around 35 individuals for [MATH] = 10 unit and 18 individuals for [MATH] = 20.', '1803.01639-2-33-1': 'Conversely, if we have a pair of species with [MATH] = 50, and we wish to be 75% sure a true positive is not missed, we must hope that the true interaction [MATH] when coupled with short interaction range ([MATH]) is at least [MATH], and if coupled with long interaction range ([MATH]) is at least [MATH].', '1803.01639-2-33-2': 'It therefore seems likely that only the very strongest interactions are detectable with the number of individuals that are typically found in the species-rich datasets.', '1803.01639-2-34-0': '## Imbalances in species abundance', '1803.01639-2-35-0': ""Since most communities exhibit a 'hollow curve' distribution of population abundances , an imbalance in population sizes is very common."", '1803.01639-2-35-1': 'From the variance formula [REF] it is clear that imbalance has a strong effect on the power because the term [MATH], and hence the variance, increases with imbalance.', '1803.01639-2-35-2': 'This relationship is confirmed when we use the power formula to quantify the effect of population imbalance for different interaction strengths and combined population sizes (Fig. [REF]).', '1803.01639-2-35-3': 'So, for example, for an interaction strength of [MATH] = 0.1 and a desired power of 80, a combined individual count of about 750 is required when the populations are perfectly balanced, but 1000 are required when one species is five times more abundant than the second species, and a surprisingly large 1500 required when one species is ten times more abundant than the other.', '1803.01639-2-35-4': 'Alternatively, consider that we require 90 power, and that the interactions are assumed to be [MATH] and of short range, [MATH].', '1803.01639-2-35-5': 'Then, to be on the safe side, we should attain samples of sizes at least [MATH] or [MATH], depending on the imbalance.', '1803.01639-2-36-0': '## Power at rainforest sample sizes', '1803.01639-2-37-0': 'We now consider how our understanding of the power to detect interactions might affect results for observed plant communities.', '1803.01639-2-37-1': 'For simplicity, let us assume interactions are of the type given by our model and that every species is interacting with every other species in an identical manner (so [MATH] and range [MATH] are the same for all pairs of species).', '1803.01639-2-37-2': 'Since the power is the probability of detecting interactions, we can get a rough estimate of the number of detected cross-species interactions by assuming the tests are independent, and summing up the powers.', '1803.01639-2-37-3': 'This then allows a coarse comparison of recently reported frequencies of detected interactions in tropical forests with the expected frequency of detected interactions as a function of power.', '1803.01639-2-38-0': 'Fig. [REF] shows the expected number of cross-species interactions detected as a function of abundance for various hypothetical interaction strengths and ranges.', '1803.01639-2-38-1': 'The abundances are taken from the Barro Colorado Island (BCI) 1995 census of woody plants with diameter at breast height at least 1cm .', '1803.01639-2-38-2': 'The abundances are highly skewed, with a large proportion of low abundance species, and we show the power in two cases, when the pool of species consists of those with abundance at least 30 and 100.', '1803.01639-2-38-3': 'Reducing the species pool by increasing the abundance threshold naturally increases the proportions of detection, and highlights the importance of using similar thresholds when comparing different communities.', '1803.01639-2-38-4': 'It is striking how little power is to be expected for most of the species even when assuming strong interaction ([MATH] = -0.75).', '1803.01639-2-38-5': 'Only when the abundance of a species reaches thousands, can we be expected to detect even 50% of the interactions present.', '1803.01639-2-38-6': 'This is a very thought-provoking result, as the lack of detection might be explained simply by a lack of power in the majority of species-pairs.', '1803.01639-2-39-0': 'Making the same assumptions about identical interactions between all pairs of species, we conducted a comparative analysis using abundances of plants in the BCI 1995 census, the Changbaishan (CBS) forest plot , and the Sinharaja 1995 census .', '1803.01639-2-39-1': 'Following [CITATION], we considered only large plants (diameter [MATH]10cm) and keep only those species with at least 70 large individuals (note that thresholded CBS at 50 individuals).', '1803.01639-2-39-2': 'The analysis shows a big difference in the expected proportion of interactions that would be detected due to differences in the species abundances of the communities (Table [REF]).', '1803.01639-2-39-3': 'Although the actual proportions differ, qualitatively, these results are in a sense similar to those reported by [CITATION] who detected that approximately [MATH] and [MATH] of interactions departed from their null hypothesis in the CBS, Sinharaja and BCI communities, respectively.', '1803.01639-2-40-0': 'These toy examples highlight how the relationships of the power to detect interactions with population sizes, strength of interaction and spatial scale of interaction can in principle lead to patterns similar to those described in previous studies.', '1803.01639-2-40-1': 'Fortunately these artificial examples can be taken as ""worst case"" scenarios.', '1803.01639-2-40-2': 'Many of the species in forest plots are highly localised to environmental niches , in which case the context of testing needs to be defined more accurately and the pool of potential interactions limited, thus regaining power.', '1803.01639-2-41-0': '# Discussion', '1803.01639-2-42-0': 'Understanding the relative strength and therefore the importance of interspecific interactions is one of the key goals of community ecology, and the null model approach has been popular for characterizing spatial point patterns of (predominantly) diverse plant communities .', '1803.01639-2-42-1': 'However, there has been little guidance on when a given test is likely able to detect species associations that are present.', '1803.01639-2-42-2': 'Here we have made a first step in closing this important gap in our understanding.', '1803.01639-2-42-3': 'Our results clarify the quantitative relationships between the strength of the underlying biological interaction, sample size (number of individuals of both species under investigation), and the spatial scale over which the test is being performed.', '1803.01639-2-42-4': 'We have also shown that statistical power may explain both the low detection rate of biological interactions in plant communities, and the negative relationship between species-richness and frequency of detected interspecific interactions in comparative studies.', '1803.01639-2-43-0': 'Ecologists have had to rely largely upon their intuition for deciding the minimum population size to include in their analyses with the result that a range of criteria up to 100 individuals have been used.', '1803.01639-2-43-1': 'For species-rich communities, where many interspecific interactions may necessarily be weak , abundances of both species may need to be in the hundreds of individuals before any interaction is detected (Fig. [REF]), and this implies previous abundance thresholds are likely too low to detect many interactions.', '1803.01639-2-43-2': 'As several authors have acknowledged, the failure to reject the null hypothesis of spatial independence in so many species-pairs does not necessarily mean interspecific interactions are not occurring, or present .', '1803.01639-2-43-3': 'We hope our study highlights how the power of the tests can be assessed and should be factored into the interpretation of the results.', '1803.01639-2-43-4': 'The power formula can also be used in estimating the area of observation necessary to increase the power to a desirable level (Appendix [REF]), so can also be used to aid study design.', '1803.01639-2-43-5': 'Despite this, we do stress that there is still much to be learned about the power of the statistical tests used in earlier studies, given the assumptions we had to make, and that the reader should take our contribution as a first step that offers a rough guide to sample sizes that are required to make strong statements about the frequency and strength of interspecific interactions.', '1803.01639-2-44-0': 'Although our model is clearly mis-specified as we use tests assuming that intensity is not dependent on abiotic features of the environment, the general applicability of our results will carry-over into the inhomogeneous setting.', '1803.01639-2-44-1': 'In particular we would still expect a positive relationship between population size and frequency of interactions to emerge simply due to an increase in power at larger sample sizes.', '1803.01639-2-44-2': 'Such a positive relationship has already been reported in a number of empirical studies that take habitat associations into consideration .', '1803.01639-2-44-3': 'It is possible that common species are better competitors and are somehow suppressing the abundance of the weaker competitors, but without experimental manipulation, or perhaps different analyses using repeated sampling over time , it is hard to distinguish whether this pattern is a result of biological processes or the ability of the statistical methods to detect interactions at different population sizes.', '1803.01639-2-45-0': 'The spatial scale over which tests are performed is important for the ability to detect spatial dependencies (Fig. [REF]), and our results are similar to empirical studies that often find few negative interactions at the shortest distances, even though this is where the interactions are likely to be strongest .', '1803.01639-2-45-1': 'Short scales suffer from having high variability due to the relatively small number of neighbours possible in a small area, but at longer distances, the effect of neighbours is weaker.', '1803.01639-2-45-2': 'Hence there is a sweet spot where this trade-off is maximised, and the location of this is likely dependent on several factors, not least of which is the scale over which interactions are occurring (e.g. Fig. 2 in [CITATION] for an empirical example).', '1803.01639-2-45-3': 'For woody plants, there have been several studies that have fitted neighbourhood growth or survival models to individual-based data that tracks the fate of trees over time , and most results seem to point to interactions being confined to 10-30m radius around an individual.', '1803.01639-2-45-4': 'However, little is known about how the spatial scales of interspecific interactions change with life history stage, environmental conditions, or even species identity even though the latter has been shown to be very important for determining coexistence .', '1803.01639-2-45-5': 'Any changes to the scales of interactions will have consequences for the hypothesis testing methods discussed here, but until more is understood about the spatial scales of interactions between species, it seems sensible to test over ranges reported in earlier studies.', '1803.01639-2-46-0': 'Our discussion up to this point has been in the context of stationary, most notably homogeneous, data.', '1803.01639-2-46-1': 'Most recent analyses have tried to factor out the effects of spatial heterogeneity in the abiotic environment by using inhomogeneous Poisson processes as the null model .', '1803.01639-2-46-2': 'Currently it is hard to predict whether the power of an inhomogeneous analogue of our scenario would be lower or higher.', '1803.01639-2-46-3': 'On the one hand we could expect higher power to detect interactions because the model better captures the underlying processes that generate the spatial distributions of the species within the community.', '1803.01639-2-46-4': 'However, we also expect variance to be increased, since extra parameters need to be estimated leaving a fewer degrees of freedom per parameter.', '1803.01639-2-46-5': 'For example, tests using the inhomogoneneous Poisson process method use a smoothing kernel to approximately remove the effects of large scale structure assumed to be caused by habitat associations (see e.g. ).', '1803.01639-2-46-6': 'Typically, the same smoothing parameter is used for all species, which is a sensible assumption when little is known about the spatial scale of habitat associations, but there is no reason to suspect a single smoothing parameter is appropriate for all species.', '1803.01639-2-46-7': 'An open challenge is to better understand how mis-specification of the smoothing parameter will bias the detection of interactions.', '1803.01639-2-46-8': 'Again, we feel that using a biologically motivated model to simulate data is a useful approach for exploring such issues.', '1803.01639-2-47-0': 'Finally, we remind the reader that the spatial statistics used in the null model approach do not say anything directly about the processes that may have created the patterns, and different processes could generate the same summary statistic.', '1803.01639-2-47-1': 'As an alternative, model-based approaches, either in the form we use here (which include the familiar Thomas Cluster models) or birth-death models could also be applied to the inference of biological interactions from point pattern data .', '1803.01639-2-47-2': 'Model fitting will normally lead to estimation of parameters that can also be estimated in the field (eg. dispersal kernels, interaction kernels), we therefore feel that their continued development will help improve understanding of the processes underpinning the results returned .', '1803.01639-2-48-0': 'In conclusion, we hope our main contribution is to encourage more users to consider explicitly the ability of the spatial point pattern tests to detect significant associations between species.', '1803.01639-2-48-1': 'We have shown that the data requirements to detect even strong interactions may be quite high, mirroring results for null model tests of species co-occurrences in community matrix data .', '1803.01639-2-48-2': 'On this basis, we suggest it is desirable to only interpret the frequency of interactions across large numbers of species once the effect of different powers to detect interactions for pairs of species of given population sizes has been (even approximately) factored out.', '1803.01639-2-48-3': 'This seems especially important in comparative analyses across different communities where the spatial scales, strengths of interactions and the species abundance distributions may differ and affect the power to detect interactions.', '1803.01639-2-49-0': ""# Authors' contributions"", '1803.01639-2-50-0': 'TR conceived the idea during discussions with DM and SO; TR derived the model and formulas, designed and executed computations, and contributed extensively to the manuscript; DM contributed to the rainforest experiment and extensively to the manuscript; SO contributed to the manuscript.', '1803.01639-2-50-1': 'All authors contributed to the intellectual core of the manuscript.'}","[['1803.01639-1-7-0', '1803.01639-2-7-0'], ['1803.01639-1-7-1', '1803.01639-2-7-1'], ['1803.01639-1-7-2', '1803.01639-2-7-2'], ['1803.01639-1-7-3', '1803.01639-2-7-3'], ['1803.01639-1-7-4', '1803.01639-2-7-6'], ['1803.01639-1-7-5', '1803.01639-2-7-7'], ['1803.01639-1-39-0', '1803.01639-2-39-0'], ['1803.01639-1-39-1', '1803.01639-2-39-1'], ['1803.01639-1-39-2', '1803.01639-2-39-2'], ['1803.01639-1-39-3', '1803.01639-2-39-3'], ['1803.01639-1-40-0', '1803.01639-2-40-0'], ['1803.01639-1-40-1', '1803.01639-2-40-1'], ['1803.01639-1-40-2', '1803.01639-2-40-2'], ['1803.01639-1-47-0', '1803.01639-2-47-0'], ['1803.01639-1-47-1', '1803.01639-2-47-1'], ['1803.01639-1-47-2', '1803.01639-2-47-2'], ['1803.01639-1-48-0', '1803.01639-2-48-0'], ['1803.01639-1-48-1', 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'1803.01639-2-45-4']]",[],"[['1803.01639-1-7-6', '1803.01639-2-7-8'], ['1803.01639-1-4-0', '1803.01639-2-4-0'], ['1803.01639-1-2-0', '1803.01639-2-2-0'], ['1803.01639-1-2-3', '1803.01639-2-2-1'], ['1803.01639-1-10-0', '1803.01639-2-10-0'], ['1803.01639-1-10-3', '1803.01639-2-10-3'], ['1803.01639-1-0-1', '1803.01639-2-0-1'], ['1803.01639-1-0-4', '1803.01639-2-0-3'], ['1803.01639-1-9-4', '1803.01639-2-9-4']]",[],[],"{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1803.01639,,,
quant-ph-0102095,"{'quant-ph-0102095-1-0-0': 'Description of scalar charged particles, based on the Feshbach - Villars formalism, is proposed.', 'quant-ph-0102095-1-0-1': 'Particles are described by an object that is Wigner function in usual coordinates and momentums and density matrix in charge variable.', 'quant-ph-0102095-1-0-2': 'It is possible to introduce the usual Wigner function for a large class of dynamical variables.', 'quant-ph-0102095-1-0-3': 'Such an approach contains explicitly a measuring device frame.', 'quant-ph-0102095-1-0-4': 'From our point of view it corresponds to the Copenhagen interpretation of quantum mechanics.', 'quant-ph-0102095-1-0-5': 'It is shown how physical properties of such particles depend on the definition of operator of coordinate.', 'quant-ph-0102095-1-0-6': 'Evolution equation for the Wigner function of a single-charge state in classical limit coincides with the Liouville equation.', 'quant-ph-0102095-1-0-7': 'Localization peculiarities manifest themselves in specific constraints on possible initial conditions.', 'quant-ph-0102095-1-1-0': '# Introduction', 'quant-ph-0102095-1-2-0': 'Question about the wave function nature has its origin in early years of quantum mechanics.', 'quant-ph-0102095-1-2-1': 'For a long time it has been considered as philosophical rather then physical question [CITATION].', 'quant-ph-0102095-1-2-2': 'However, it is very actual now because of the recent theoretical and experimental progress in quantum information [CITATION].', 'quant-ph-0102095-1-3-0': 'One of significant points in understanding the wave function nature is the Einstein - Podolsky - Rosen paradox and existence of quantum correlations related to it.', 'quant-ph-0102095-1-3-1': 'Since 1980, such specific behavior of quantum systems has been confirmed many times in experiments [CITATION].', 'quant-ph-0102095-1-3-2': 'It is very important that such correlations ""spread"" in space instantly.', 'quant-ph-0102095-1-3-3': 'Nevertheless, if one stays at position of the Copenhagen interpretation, there is no violation of casualty principle.', 'quant-ph-0102095-1-4-0': 'However, due to the fact that collapse of wave function takes place at a given instant (or maybe in a small time interval) in the whole space, one can speak of violation of casualty principle and a conflict between quantum mechanics and special relativity [CITATION] in some interpretations of quantum mechanics.', 'quant-ph-0102095-1-4-1': 'Following the Bell and Eberhard idea that consistent description should contain a preferred frame, relativistic classical and quantum mechanics is built in [CITATION] in such way that the relativity principle is generalized and the casualty is not violated.', 'quant-ph-0102095-1-4-2': 'Basic assumption of this theory is that transformation from one frame to another is realized by operators which are isomorphic to the Lorentz group and depend on certain four-vector (as parameter) in such a way that the instant time hyper-plain is invariant.', 'quant-ph-0102095-1-4-3': 'This vector is interpreted as a relative to an observer four-velocity of the preferred frame.', 'quant-ph-0102095-1-4-4': 'Under this assumption postulate about the light speed constancy is changed to the postulate about the average speed of light on closed path constancy.', 'quant-ph-0102095-1-5-0': 'One of remarkable peculiarities of this theory is in existence of well-defined position operator [CITATION] which coincides with the Newton - Wigner position operator [CITATION] in the preferred frame.', 'quant-ph-0102095-1-5-1': 'It means that in this approach measurement of position does not create a particle - antiparticle couple since the odd part responsible for appearance of a different charge state superposition is absent.', 'quant-ph-0102095-1-5-2': 'Hence, there exists a supposition that localization of relativistic particles can bear information about presence of the preferred frame in the Universe.', 'quant-ph-0102095-1-6-0': 'The above arguments show the fundamental role determination of the position operator structure can play.', 'quant-ph-0102095-1-6-1': 'Moreover, the question, if it is an one-particle, or, in principle, a many-particle operator, is of specific interest as well.', 'quant-ph-0102095-1-6-2': 'Therefore, it is very important to find situations where the odd part of the position operator could manifest itself.', 'quant-ph-0102095-1-6-3': 'It would be possible to realize such observation on strongly localized (near the Compton wavelength) states of a separate particle.', 'quant-ph-0102095-1-6-4': 'However, because of such states being a problem for laboratory experiments, it is of great interest how this peculiarity manifests itself in a many-particle system.', 'quant-ph-0102095-1-7-0': 'The Weyl - Wigner - Moyal (WWM) formalism is a convenient method to describe both one-particle and many-particle quantum systems in non-relativistic theory [CITATION].', 'quant-ph-0102095-1-7-1': 'Nevertheless, attempts to generalize it for relativistic case lead to a number of problems.', 'quant-ph-0102095-1-7-2': 'The first problem is that Weyl rule [CITATION] does not include time as a dynamical variable, and the scalar product in Hilbert space of states is formulated not for functions square integrable over the whole space-time but in the three-dimensional space or in a space-like hyper-surface only.', 'quant-ph-0102095-1-7-3': 'In [18] this problem was resolved by generalization of the spatial integration over the whole space-time without Weyl rule application.', 'quant-ph-0102095-1-7-4': 'The WWM formalism in the framework of the stochastic formulation of quantum mechanics, where scalar product is formulated for square integrable functions in whole space-time, leads to Lorenz invariant expressions as well [CITATION].', 'quant-ph-0102095-1-8-0': 'The matrix-valued Wigner function formalism has been developed for the general case of many component equations and, in particular, for [MATH] spin particles on the basis of usual Weyl rule [CITATION].', 'quant-ph-0102095-1-8-1': 'Certainly, such equations are not Lorenz invariant, though average values coincide with their analogues in usual approach.', 'quant-ph-0102095-1-9-0': 'Next essential problem in relativistic WWM formalism is in absence of a well-defined position operator.', 'quant-ph-0102095-1-9-1': 'Unlike the works mentioned above, where Wigner function is determined by means of usual position operator, in [CITATION] formalism with the Newton - Wigner position operator is developed.', 'quant-ph-0102095-1-9-2': 'This approach is not Lorenz invariant as well, and respective results differ from the standard ones.', 'quant-ph-0102095-1-9-3': 'However, they can be related to [CITATION], where the consistent definition of the position operator is possible.', 'quant-ph-0102095-1-10-0': 'The goals of this work are in formulating the WWM relativistic formalism for scalar charged particles under the approach of [CITATION] and finding a set of specific peculiarities in relativistic quantum system behavior, that are related to the non-trivial structure of the position operator.', 'quant-ph-0102095-1-10-1': 'It turns out that values of some observables depend directly on the position operator definition that, as mentioned above, can be a consequence of existence of the preferred frame in the Universe.', 'quant-ph-0102095-1-11-0': 'In Sec.[REF] we introduce the Weyl rule for matrix-valued observables in the case of scalar charged particles and discuss peculiarities of correspondence between classical and quantum dynamical algebras.', 'quant-ph-0102095-1-11-1': 'Sec.[REF] is devoted to the matrix-valued Wigner function and quantum Liouville equation.', 'quant-ph-0102095-1-11-2': 'Here we also discuss the absence of Lorenz invariance of this approach, and how it can be related to Copenhagen interpretation of quantum mechanics.', 'quant-ph-0102095-1-11-3': 'In Sec. [REF] we consider how the usual Weyl rule turns into the Feshbach - Villars representation.', 'quant-ph-0102095-1-11-4': 'Among the whole set of dynamical variables, we separate a special class of observables with the Weyl symbols independent of charge variable (charge-invariant variables).', 'quant-ph-0102095-1-11-5': 'They have several remarkable properties.', 'quant-ph-0102095-1-11-6': 'In particular, we find the relationship between their even and odd parts.', 'quant-ph-0102095-1-11-7': 'The usual (not matrix-valued) Wigner function can be introduced for such observables.', 'quant-ph-0102095-1-11-8': 'It is considered in Sec. [REF] (for a brief abstract of this approach see [CITATION]).', 'quant-ph-0102095-1-11-9': 'This object includes four components: one corresponds to particle, second one corresponds to antiparticle (even part of Wigner function), and two more are interference terms (odd part of Wigner function).', 'quant-ph-0102095-1-11-10': 'Evolution equation for odd part turns it in classical limit to zero, and equation for even part coincides with its analogue in the Newton - Wigner position operator approach [CITATION].', 'quant-ph-0102095-1-11-11': 'The difference reveals itself in peculiarities of constraints on initial conditions that are considered in Sec. [REF].', 'quant-ph-0102095-1-12-0': '# Weyl rule and specific properties of dynamical algebra for scalar charged particles', 'quant-ph-0102095-1-13-0': 'In this and the next Sections we apply the methods developed in [CITATION] to Klein - Gordon equation that is written using the Feshbach - Villars formalism [CITATION].', 'quant-ph-0102095-1-13-1': 'It makes possible to take into account the charge variable explicitly.', 'quant-ph-0102095-1-13-2': 'With account of the fact that Hilbert space of states for scalar charged particles has an indefinite metric, we have to distinguish between covariant and contravariant basis vectors and coordinates for subspace corresponding to the charge variable.', 'quant-ph-0102095-1-13-3': 'Hence, arbitrary state is expanded on basis vectors in the following way: [EQUATION] where [EQUATION]', 'quant-ph-0102095-1-13-4': 'Here [MATH] is an arbitrary dynamical variable and the symbol of summarizing can be interpreted as integration with respect to [MATH].', 'quant-ph-0102095-1-13-5': 'Greek indices take values [MATH].', 'quant-ph-0102095-1-13-6': '[MATH] plays the role of metric tensor (see [CITATION]).', 'quant-ph-0102095-1-13-7': 'In notations like [MATH], [MATH], [MATH], [MATH], [MATH],..., symbols [MATH], [MATH] do not correspond to specific operators and mean only values of the corresponding matrix elements.', 'quant-ph-0102095-1-13-8': 'There are some peculiarities in notations of operators as well.', 'quant-ph-0102095-1-13-9': '[MATH] is a full operator that acts on all dynamical variables.', 'quant-ph-0102095-1-13-10': '[MATH] is an operator that acts on the charge variables only (or, in other words, it is a [MATH]-numeric matrix).', 'quant-ph-0102095-1-14-0': 'For a consistent development of the WWM formalism we shall formulate the Weyl rule bringing into correspondence matrix-valued classical variables (Weyl symbols) to quantum-mechanical operators.', 'quant-ph-0102095-1-14-1': 'Here we should take into account that in our case classical variables are operators acting on the charge variable.', 'quant-ph-0102095-1-14-2': 'But because of operator of quasi-probability density in coordinate representation being proportional to identity matrix in charge space, succession of this operator and a matrix-valued Weyl symbol does not matter.', 'quant-ph-0102095-1-14-3': 'Choosing it in an arbitrary way, one can write the Weyl rule as follows: [EQUATION] where [MATH] is the matrix-valued Weyl symbol, [MATH] stands for corresponding quantum-mechanical operator, [MATH] is the operator of quasi-probability density that is the Fourier transform of the displacement operator.', 'quant-ph-0102095-1-14-4': 'Following [CITATION] one can obtain the expansion of [MATH] on eigen vectors of the position operator: [EQUATION] where [MATH] is dimensionality of the physical space.', 'quant-ph-0102095-1-15-0': 'Substituting ([REF]) to ([REF]) one can find the expression for an arbitrary operator expansion: [EQUATION]', 'quant-ph-0102095-1-15-1': 'Similarly such expression can be written by expending the operator of quasi-probability density on the eigen vectors of the momentum operator: [EQUATION]', 'quant-ph-0102095-1-15-2': 'Matrix elements of this operator one can find in the form: [EQUATION]', 'quant-ph-0102095-1-15-3': 'By changing variables in this expression in standard way and performing Fourier transformation one can get the expression that reconstructs matrix-valued Weyl symbol through operator: [EQUATION]', 'quant-ph-0102095-1-15-4': 'In [CITATION] the integral form of the Klein - Gordon equation with pseudo-differential symbols in the Feshbach - Villars representation (where Hamiltonian matrix has a diagonal form) is obtained.', 'quant-ph-0102095-1-15-5': 'In fact, it means that the Newton - Wigner coordinate is used there instead of usual coordinate.', 'quant-ph-0102095-1-15-6': 'Here we obtain another integral form of Klein - Gordon equation using the usual position operator: [EQUATION] where [EQUATION] is the matrix-valued Weyl symbol of the Hamiltonian in general case (for a particle in electromagnetic field), which depends on both coordinate and momentum.', 'quant-ph-0102095-1-15-7': 'Unlike [CITATION] this equation takes into account that position operator mixes the states with the different signs of charge.', 'quant-ph-0102095-1-16-0': 'Till now,consideration of the matrix-valued WWM formalism is very similar to the usual one.', 'quant-ph-0102095-1-16-1': 'Considerable differences appear at determining the matrix-valued Moyal bracket and in classical limit.', 'quant-ph-0102095-1-16-2': 'To consider this question one should find the matrix-valued Weyl symbol from product of two operators in standard way.', 'quant-ph-0102095-1-17-0': 'Let [MATH] and [MATH] be matrix-valued Weyl symbols of two operators [MATH] and [MATH] respectively.', 'quant-ph-0102095-1-17-1': 'Let us introduce operators [MATH] as follows: [EQUATION]', 'quant-ph-0102095-1-17-2': 'Then, using ([REF]) and consideration like used in [CITATION], one can obtain the matrix-valued Weyl symbol of the operator [MATH]: [EQUATION]', 'quant-ph-0102095-1-17-3': 'For the matrix-valued Weyl symbol of the operator [MATH] one can find a similar expression.', 'quant-ph-0102095-1-17-4': 'Therefore, the matrix-valued Moyal bracket can be determined in the following form: [EQUATION]', 'quant-ph-0102095-1-17-5': 'Unlike the usual case, matrix-valued Weyl symbols do not commutate, so it is not possible to represent such a Moyal bracket as sinus from the operator of Poisson bracket.', 'quant-ph-0102095-1-17-6': 'This fact results in classical limit in some peculiarities.', 'quant-ph-0102095-1-17-7': 'In general case, one can express the classical limit of the matrix-valued Moyal bracket via the matrix-valued Poisson bracket [CITATION] and commutator of the matrix-valued Weyl symbols: [EQUATION]', 'quant-ph-0102095-1-17-8': 'If the matrix-valued Weyl symbols (for example position and Hamiltonian) commutate, the matrix-valued Moyal bracket is in fact a usual one and coincides with Poisson bracket in classical limit.', 'quant-ph-0102095-1-17-9': 'If they do not commutate with each other, three cases are possible.', 'quant-ph-0102095-1-17-10': 'Let [MATH] when [MATH].', 'quant-ph-0102095-1-17-11': 'If [MATH], classical limit does not exist for such couple of the matrix-valued Weyl symbols, since the value of the matrix-valued Moyal bracket increases infinitely.', 'quant-ph-0102095-1-17-12': 'If [MATH], it turns to zero in classical limit.', 'quant-ph-0102095-1-17-13': 'If [MATH], the classical limit exists.', 'quant-ph-0102095-1-17-14': 'Here we can take as an example the Newton -Wigner position operator [CITATION].', 'quant-ph-0102095-1-17-15': 'Part of its matrix-valued Weyl symbol,not commutating with Hamiltonian, is proportional to Planck constant thus in this case there exists a well-defined classical limit.', 'quant-ph-0102095-1-18-0': '# Matrix-valued Wigner function and quantum Liouville equation', 'quant-ph-0102095-1-19-0': 'Using ([REF]), one can find the average value of an arbitrary operator [MATH] expressed via its matrix-valued Weyl symbol: [EQUATION]', 'quant-ph-0102095-1-19-1': 'Then, introducing the matrix-valued Wigner function [EQUATION] this expression can be written in a simpler form: [EQUATION]', 'quant-ph-0102095-1-19-2': 'Substitution of the operator of quasi-probability density expansion ([REF]) to ([REF]) leads to the expression for the matrix-valued Wigner function in coordinate representation.', 'quant-ph-0102095-1-19-3': 'As a result we get a formula that coincides with definition given in [CITATION]: [EQUATION]', 'quant-ph-0102095-1-19-4': 'Next, let us describe a method to obtain evolution equation for the matrix-valued Wigner function of a scalar charged particle.', 'quant-ph-0102095-1-19-5': 'To do this one shall differentiate expression ([REF]) with respect to time.', 'quant-ph-0102095-1-19-6': 'Value of the wave function derivative is taken from the integral form of the Klein - Gordon equation ([REF]).', 'quant-ph-0102095-1-19-7': 'After standard transformations, similar to the usual WWM formalism, we obtain the quantum Liouville equation [EQUATION] where the matrix-valued Moyal bracket is determined by expression ([REF]).', 'quant-ph-0102095-1-20-0': 'At first sight, this equation has a considerable defect in comparison to similar expressions in [CITATION].', 'quant-ph-0102095-1-20-1': 'Namely it is not Lorenz invariant.', 'quant-ph-0102095-1-20-2': 'For interpretation of this fact, we have to take into account arguments from the measurement theory.', 'quant-ph-0102095-1-20-3': 'The fact is that average values calculated in this approach coincide with ones in the usual (Schrodinger) representation of quantum mechanics that is Lorenz invariant.', 'quant-ph-0102095-1-20-4': 'Nevertheless, unlike the stochastic formulation of quantum mechanics [CITATION], scalar product is determined here with functions that are square integrable in certain space-like hyper-surface, rather than over the whole space-time.', 'quant-ph-0102095-1-20-5': 'Moreover, the value of scalar product does not depend on its choice [CITATION].', 'quant-ph-0102095-1-20-6': 'One can relate this hyper-surface with the measurement device frame.', 'quant-ph-0102095-1-20-7': 'In other words, the wave function collapse occurs in a frame in which the equation ([REF]) is written.', 'quant-ph-0102095-1-20-8': 'Absence of Lorenz invariance is the consequence of the fact that the Weyl rule does not include time as an independent dynamical variable.', 'quant-ph-0102095-1-21-0': 'In [CITATION] the process of relativistic measurement is considered.', 'quant-ph-0102095-1-21-1': 'In that work the point of view is expressed, that wave function (and as result the Wigner function, we notice) has no objective value and does not transform covariantly when there are classical interventions.', 'quant-ph-0102095-1-21-2': 'Here we note that in principle equation ([REF]) can be written with four-dimensional Lorenz invariant symbols only.', 'quant-ph-0102095-1-21-3': 'But to do this we have to incorporate in theory a certain time-like unit vector in a way similar to Tomonaga-Schwinger approach to quantum field theory [CITATION].', 'quant-ph-0102095-1-21-4': 'It is the four-velocity of the frame where wave function collapse occurs (the measuring device frame) relative to the second static observer (watching observer).', 'quant-ph-0102095-1-21-5': 'This frame has principally another sense to that of the preferred frame in [CITATION].', 'quant-ph-0102095-1-21-6': 'In the approach [CITATION] collapse of wave function has to take place in all frames in the whole space (because of the instant time hyper-surface being invariant).', 'quant-ph-0102095-1-21-7': 'In our approach this process obeys the relativity of simultaneity.', 'quant-ph-0102095-1-21-8': 'If it were possible to observe the wave function directly, such a hypothetical watching observer would see the wave function collapse as a certain moving front.', 'quant-ph-0102095-1-21-9': 'As a result, at a certain instant, in static frame (attached to the watching observer), a state with a part of the wave function before measurement on one hand and a part of the wave function after measurement on another hand, is realized Fig. [REF].', 'quant-ph-0102095-1-21-10': 'But perhaps it is impossible to propose even gedanken experiment without use of a superluminar signal to interpret the results in favor of one of approaches.', 'quant-ph-0102095-1-22-0': 'Hence we find to be very important and of fundamental value the fact that quantum mechanics in the Wigner representation necessarily includes the four-velocity of measuring device frame in explicit form.', 'quant-ph-0102095-1-23-0': '# Transformation to the Feshbach - Villars representation', 'quant-ph-0102095-1-24-0': 'In the Feshbach - Villars representation Hamiltonian matrix has a diagonal form and indices in the charge space correspond to the particle and antiparticle [CITATION].', 'quant-ph-0102095-1-24-1': 'Therefore, it is convenient to distinguish between solutions with different charge signs and to give them explicit physical sense.', 'quant-ph-0102095-1-24-2': 'Furthermore, in this approach an influence of the odd part of the position operator on values of some physical variables is more evident.', 'quant-ph-0102095-1-25-0': 'Operator [MATH]transforms to the Feshbach - Villars representation by formula: [EQUATION] where the transformation matrix has the form: [EQUATION]', 'quant-ph-0102095-1-25-1': 'Here and below [EQUATION] is the energy of a relativistic free particle.', 'quant-ph-0102095-1-26-0': 'Next, we apply ([REF]) to the Weyl rule in the form ([REF]).', 'quant-ph-0102095-1-26-1': 'As result, we obtain the expression that brings into correspondence the matrix-valued Weyl symbol to the operator in the Feshbach -Villars representation: [EQUATION]', 'quant-ph-0102095-1-26-2': 'In general, it is rather difficult to interpret this expression since there is a complicated dependence on integration variable under the integral sign.', 'quant-ph-0102095-1-26-3': 'Here we do not go beyond the simplest case.', 'quant-ph-0102095-1-27-0': 'Let the matrix-valued Weyl symbol be proportional to the identity matrix: [EQUATION]', 'quant-ph-0102095-1-27-1': 'In principle, one can say that such symbols do not depend on the charge variable, so that the class of dynamical variables, which corresponds to those, we denominate here, to be more brief, as a class of charge-invariant variables.', 'quant-ph-0102095-1-27-2': 'Most of dynamical variables that we consider in relativistic (non-quantum) mechanics belong to this class.', 'quant-ph-0102095-1-27-3': 'The reason for it is in absence of a dependence on the charge variable in classical mechanics.', 'quant-ph-0102095-1-27-4': 'Hence, the question about the classical limit is for such variables especially interesting.', 'quant-ph-0102095-1-28-0': 'The Weyl rule for charge-invariant variables in the Feshbach - Villars representation has the form: [EQUATION]', 'quant-ph-0102095-1-28-1': 'Unlike [CITATION] and non-relativistic case there is a matrix-valued variable here: [EQUATION]', 'quant-ph-0102095-1-28-2': 'It contains even and odd parts and is expressed via energy of a free particle ([REF]): [EQUATION]', 'quant-ph-0102095-1-28-3': 'Consequences from ([REF]) are expressions for even [MATH] and odd [MATH] parts of the operator of a charge-invariant observable in terms of its Weyl symbol: [EQUATION]', 'quant-ph-0102095-1-28-4': 'Matrix elements with eigen vectors of the momentum of the operator of a charge-invariant variable and its even and odd parts can be written in such a form: [EQUATION]', 'quant-ph-0102095-1-28-5': 'Then, like it is done in Sec. [REF], one can obtain a formula that reconstructs the Weyl symbol of operator and its even and odd parts in the Feshbach - Villars representation: [EQUATION]', 'quant-ph-0102095-1-28-6': 'Comparing ([REF]) and ([REF]) we conclude that matrix elements (integral kernels) of even and odd parts of the operator of an arbitrary charge-invariant variable are uniquely related with each other due to the Weyl rule: [EQUATION]', 'quant-ph-0102095-1-28-7': 'A consequence from this expression is the fact that odd part of an independent on position operator is zero.', 'quant-ph-0102095-1-28-8': 'If one uses as [MATH], for example, the scalar potential of electric field, expression ([REF]) establishes a quantitative relationship between effects of motion of a particle in electric field and its interaction with polarizable vacuum (trembling motion, Zitterbewegung) [CITATION].', 'quant-ph-0102095-1-29-0': 'Consider now peculiarities of time derivatives of charge-invariant variables.', 'quant-ph-0102095-1-29-1': 'In coordinate representation, the matrix-valued Weyl symbol of such operator has the form: [EQUATION]', 'quant-ph-0102095-1-29-2': 'The Weyl rule ([REF]) (in the Feshbach - Villars representation) for it can be written as follows: [EQUATION] or, in another form: [EQUATION] where we have introduced a new matrix-valued variable of three arguments: [EQUATION]', 'quant-ph-0102095-1-29-3': 'If [MATH] depends on [MATH] linearly, the formula ([REF]) can be presented in especially simple form: [EQUATION]', 'quant-ph-0102095-1-29-4': 'The time derivative of an operator is a composition of its even and odd parts where the odd part has a classical limit because of absence of interference terms.', 'quant-ph-0102095-1-29-5': 'As an example one can take the Newton - Wigner position operator that is even part of the usual position operator.', 'quant-ph-0102095-1-30-0': 'In general case this formula contains in higher orders of [MATH] non-standard terms, since [MATH] distinguishes from [MATH] .', 'quant-ph-0102095-1-30-1': 'However, in classical limit they vanish and this expression takes the form ([REF]).', 'quant-ph-0102095-1-31-0': 'Hence, in the Feshbach - Villars representation, where we distinguish between solutions with different charge signs, odd part of the position results not only in odd part of operators emergence, but leads to some peculiarities in their even parts as well.', 'quant-ph-0102095-1-31-1': 'Particularly, it appears in the specifics of the semi-classical limit.', 'quant-ph-0102095-1-32-0': '# Wigner function and quantum Liouville equation for charge-invariant variables', 'quant-ph-0102095-1-33-0': 'It is easy to see from ([REF]) that it is possible to introduce usual Wigner function for the charge invariant variables in such a way that their average values are determined by formula: [EQUATION]', 'quant-ph-0102095-1-33-1': 'To show it, we expand the operator of quasi-probability density in the Feshbach - Villars representation through the eigen vectors of momentum: [EQUATION]', 'quant-ph-0102095-1-33-2': 'Wigner function is determined as the average value of this operator over an arbitrary state that contains in general case components with both signs of charge: [EQUATION]', 'quant-ph-0102095-1-33-3': 'There are four components in this expression.', 'quant-ph-0102095-1-33-4': 'Two of them are the average value of the even part of operator of quasi-probability density, and another two are the average value of the odd part.', 'quant-ph-0102095-1-33-5': 'Let us introduce the symbols: [EQUATION]', 'quant-ph-0102095-1-33-6': 'It should be noticed that here and below the object [MATH] is not the matrix-valued Wigner function in the sense of [CITATION] and Sec. [REF] of this work.', 'quant-ph-0102095-1-34-0': 'Substituting ([REF]) into ([REF]), we obtain for the Wigner function components following expressions [EQUATION]', 'quant-ph-0102095-1-34-1': 'Even components of the Wigner function ([REF]) correspond to charge definite state.', 'quant-ph-0102095-1-34-2': 'The value of odd components ([REF]) for such a state is zero.', 'quant-ph-0102095-1-34-3': 'The expression ([REF]) differs from analogous one for non-relativistic Wigner function and relativistic one determined using the Newton - Wigner position operator [CITATION] by the function [MATH]under the integral sign (see ([REF])).', 'quant-ph-0102095-1-34-4': 'This function has a specific feature:its expansion on [MATH], [MATH] does not contain square terms.', 'quant-ph-0102095-1-34-5': 'It means that in non-relativistic limit the expression ([REF]) coincides with usual determination of the Wigner function.', 'quant-ph-0102095-1-35-0': 'We obtain the evolution equations for every component separately.', 'quant-ph-0102095-1-35-1': 'The general principle here is the same as in Sec. [REF].', 'quant-ph-0102095-1-35-2': 'However, instead of equation ([REF]) one shall use the integral form of Klein -Gordon equation in the Feshbach - Villars representation [CITATION]: [EQUATION]', 'quant-ph-0102095-1-35-3': 'Following equations can be obtained in standard way, through differentiating Wigner function components with respect to time: [EQUATION]', 'quant-ph-0102095-1-35-4': 'Nevertheless, the Wigner function components are not independent, i.e. a specific constraint is imposed on solutions of the system ([REF]),([REF]) .', 'quant-ph-0102095-1-35-5': 'To find it one should take Fourier transform and make the standard change of variables for every component ([REF]),([REF]).', 'quant-ph-0102095-1-35-6': 'As a result, we obtain following expressions: [EQUATION]', 'quant-ph-0102095-1-35-7': 'Now we divide ([REF]) by ([REF]) and ([REF]) by ([REF]), and equal resulting expressions to each other because of equality of their left-hand sides.', 'quant-ph-0102095-1-35-8': 'This gives us the constraint we are looking for: [EQUATION]', 'quant-ph-0102095-1-35-9': 'Equation ([REF]) contains explicitly the imaginary unit, so the question is actual if the Wigner function is real.', 'quant-ph-0102095-1-35-10': 'To test it we consider the complex-conjugate expressions to ([REF]),([REF]).', 'quant-ph-0102095-1-35-11': 'After some easy transformations we obtain following identities: [EQUATION]', 'quant-ph-0102095-1-35-12': 'They mean, that even components of Wigner function are real.', 'quant-ph-0102095-1-35-13': 'Odd components are complex-conjugate to each other so that their sum is real as well.', 'quant-ph-0102095-1-36-0': 'It is essential that the equation for even components of Wigner function coincides with analogous expression obtained in [CITATION] for the formalism where the Newton - Wigner position operator is used.', 'quant-ph-0102095-1-36-1': 'Hence, dynamics of quasi-distribution functions for systems of particles with charges of same sign (charge definite states) is identical in both cases.', 'quant-ph-0102095-1-37-0': '# Statistical properties of the Wigner function for charge-invariant variables', 'quant-ph-0102095-1-38-0': 'Constraint on the initial conditions of the Wigner function is the general peculiarity of the approach described here because equations are identical in both cases (for charge definite states).', 'quant-ph-0102095-1-38-1': 'In this Section we show how some theorems and properties differ from their analogues in usual WWM formalism [CITATION] and in approach where the Newton - Wigner position operator is used [CITATION].', 'quant-ph-0102095-1-39-0': 'First of all one should notice the property of normability.', 'quant-ph-0102095-1-39-1': 'Even part of Wigner function ([REF]) is normalized in whole phase space, and integral of odd part ([REF]) is zero.', 'quant-ph-0102095-1-40-0': 'Consider now the compatibility of Wigner function ([REF]) with distributions in the coordinate and momentum spaces for a single-charge state.', 'quant-ph-0102095-1-40-1': 'For this purpose we integrate ([REF]) by coordinate.', 'quant-ph-0102095-1-40-2': 'As a result, we obtain distribution in momentum space: [EQUATION]', 'quant-ph-0102095-1-40-3': 'This function always has a definite sign, so that it can be interpreted as the probability density.', 'quant-ph-0102095-1-40-4': 'One can obtain a more non-trivial result for distribution in coordinate space.', 'quant-ph-0102095-1-40-5': 'The result of integrating of ([REF]) by momentum can be given in such a form: [EQUATION] or, which is the same, [EQUATION] where [MATH] is the wave function in representation of the Newton - Wigner coordinate [CITATION].', 'quant-ph-0102095-1-40-6': 'It is obvious, that quasi-distribution ([REF]), ([REF]) is not sign-definite.', 'quant-ph-0102095-1-40-7': 'This property is typical for bosons and makes difficulties in probability interpretation [CITATION].', 'quant-ph-0102095-1-40-8': 'Nevertheless, formally use of such quasi-probability makes it possible to calculate average values of the variables dependent on coordinate.', 'quant-ph-0102095-1-41-0': 'Average values of variables depending on momentum only do not differ from similar ones in usual approach.', 'quant-ph-0102095-1-41-1': 'Let us find the peculiarities of higher moments of coordinate in our case.', 'quant-ph-0102095-1-41-2': 'For this purpose we integrate ([REF]) with [MATH] by coordinate and use the identity: [EQUATION]', 'quant-ph-0102095-1-41-3': 'After some obvious transformations the result for [MATH]-th moment of coordinate can be written as follows: [EQUATION]', 'quant-ph-0102095-1-41-4': 'First moment (average coordinate) has the value similar to one in the Newton - Wigner coordinate approach.', 'quant-ph-0102095-1-41-5': 'Differences manifest themselves in higher moments.', 'quant-ph-0102095-1-41-6': 'As an example, we apply this expression to second moment of coordinate: [EQUATION]', 'quant-ph-0102095-1-41-7': 'Except of the usual part, this formula also contains an additional term that causes the peculiarities related to determination of the position operator.', 'quant-ph-0102095-1-41-8': 'For strongly localized states, it results in formal violation of uncertainty relation.', 'quant-ph-0102095-1-42-0': 'Similar to the case of the usual WWM formalism [CITATION], we prove two criterions that make it possible to select Wigner functions for pure and mixed states out of the whole set of functions of the variables [MATH].', 'quant-ph-0102095-1-43-0': 'Criterion of pure state.', 'quant-ph-0102095-1-44-0': 'For the functions [MATH] and [MATH] to be even and odd components of the Wigner function for charge invariant variables, it is necessary and sufficient that equalities ([REF]),([REF]),([REF]) hold true, and the following conditions are satisfied: [EQUATION]', 'quant-ph-0102095-1-44-1': 'Necessity.', 'quant-ph-0102095-1-44-2': 'Following [CITATION] we start from an obvious identity: [EQUATION]', 'quant-ph-0102095-1-44-3': 'Applying it to the formulas (51) - (54) one obtains the equalities: [EQUATION]', 'quant-ph-0102095-1-44-4': 'Now, substituting the explicit form of [MATH] and [MATH]from ([REF]), we obtain ([REF]),([REF]).', 'quant-ph-0102095-1-45-0': 'Sufficiency.', 'quant-ph-0102095-1-45-1': 'Let components of the function satisfy the conditions ([REF]), ([REF]) or, that is the same, ([REF]),([REF]).', 'quant-ph-0102095-1-45-2': 'So one can write following conditions: [EQUATION] where [MATH] are certain functions.', 'quant-ph-0102095-1-45-3': 'Let us show that they can be chosen in a way to be consistent with each other and present a wave function for a scalar charged particle.', 'quant-ph-0102095-1-45-4': 'From ([REF]),([REF]) and ([REF]),([REF]) it follows: [EQUATION]', 'quant-ph-0102095-1-45-5': 'Then one obtains: [EQUATION]', 'quant-ph-0102095-1-45-6': 'Hence, one can say that our system is described by the following wave functions: [EQUATION]', 'quant-ph-0102095-1-45-7': 'Now one has to prove that wave functions presented in such a way (with one and two tildes) are consistent with each other.', 'quant-ph-0102095-1-45-8': 'For this purpose combination of expressions ([REF]) - ([REF]) and ([REF]) - ([REF]) is substituted to condition ([REF]).', 'quant-ph-0102095-1-45-9': 'As a result, we get: [EQUATION]', 'quant-ph-0102095-1-45-10': 'This leads to the following equality: [EQUATION]', 'quant-ph-0102095-1-45-11': 'If this condition is fulfilled, one can accept as the wave function [EQUATION]', 'quant-ph-0102095-1-45-12': 'Then the condition ([REF]) is satisfied and, with account of the fact that components [MATH] differ from components of [MATH] by a constant phase, condition ([REF]) is satisfied as well.', 'quant-ph-0102095-1-45-13': 'Hence, conditions of this criterion allow us to introduce the wave function ([REF]).', 'quant-ph-0102095-1-45-14': 'Then, similar to how it is done in [CITATION], one can show that it satisfies the Klein - Gordon equation in the Feshbach - Villars representation.', 'quant-ph-0102095-1-45-15': 'Which was to be proved.', 'quant-ph-0102095-1-46-0': 'This criterion differs from a similar one in usual WWM formalism [CITATION] and in the approach that uses the Newton - Wigner position operator by difference of the right parts of the conditions ([REF]),([REF]) from zero.', 'quant-ph-0102095-1-46-1': 'The obvious consequence from it is the fact that the one-particle state, where Wigner function would be a joint Gauss distribution by coordinate and momentum, is impossible in the approach described here.', 'quant-ph-0102095-1-47-0': 'Consider another property of Wigner function, true for both pure and mixed states.', 'quant-ph-0102095-1-47-1': 'For this purpose, we introduce the formula for square of the module of scalar product of state [MATH] and state [MATH]: [EQUATION]', 'quant-ph-0102095-1-47-2': 'It is easy to test it by substituting expressions ([REF]),([REF]) for the Wigner function components into it.', 'quant-ph-0102095-1-47-3': 'Then, let [MATH] presents a mixed state consisting of orthogonal states described by [MATH]: [EQUATION]', 'quant-ph-0102095-1-47-4': 'Taking into account that sum of [MATH](the probability of system to be in state [MATH]) squares is less than unity (consequence from ([REF]) ) one obtains for an arbitrary state: [EQUATION]', 'quant-ph-0102095-1-47-5': 'Moreover, for a pure state this inequality turns to an equality.', 'quant-ph-0102095-1-47-6': 'Expression ([REF]) can be used as necessary and sufficient condition for both pure and mixed states.', 'quant-ph-0102095-1-48-0': 'This condition is written for a state that is the superposition of states with different signs of charges, that is more typical for many-particle case.', 'quant-ph-0102095-1-48-1': 'It takes a more simple form for a state where only one sign of charge is realized: [EQUATION]', 'quant-ph-0102095-1-48-2': 'Though this inequality does not contain interference terms, it differs from its non-relativistic analogue by the function ([REF]) being present.', 'quant-ph-0102095-1-48-3': 'Hence, dynamical variable that contains square and higher moments of coordinate can have peculiarities in many-particle systems that are described by statistical physics.', 'quant-ph-0102095-1-49-0': 'The type of influence of the odd part of the position operator on dynamical variables can be illustrated with the one-particle state that is the Gauss distribution in momentum space with arbitrary square of dispersion [MATH]: [EQUATION] where we have chosen the natural units of measure, [MATH].', 'quant-ph-0102095-1-49-1': 'Dependence of position dispersion on momentum dispersion is shown in Fig. [REF].', 'quant-ph-0102095-1-49-2': 'Its peculiarity is the formal violation of uncertainty relation.', 'quant-ph-0102095-1-49-3': 'Under very strong localization, states even with negative square of dispersion are possible, that was shown in [CITATION].', 'quant-ph-0102095-1-49-4': 'This fact is a property of scalar charged particles, and localization peculiarities for fermions rather different.', 'quant-ph-0102095-1-50-0': '# Conclusions', 'quant-ph-0102095-1-51-0': 'The usual (not Lorenz invariant) Weyl rule makes it possible to introduce Wigner function that is not Lorenz invariant but all average values calculated with it, coincide with ones calculated with Lorenz invariant wave function.', 'quant-ph-0102095-1-51-1': 'It results in fact that quantum mechanics in the Wigner formulation contains with necessity a measuring device frame.', 'quant-ph-0102095-1-51-2': 'In principle, we can write the evolution equations using only four-dimensional Lorenz invariant symbols, but it is needed to introduce a certain time-like vector for it.', 'quant-ph-0102095-1-51-3': 'It is the four-velocity of the frame in which wave function collapse occurs, relatively to the second (static) observer.', 'quant-ph-0102095-1-51-4': 'It should be noticed, that this approach differs from [CITATION] where there is the preferred frame.', 'quant-ph-0102095-1-51-5': 'In the last cases such frame has a global sense and its introduction is related to attempts of correct tachyons description and, as a consequence, to a possible explanation of the instant quantum correlations (in a relativistic case) from position of de Broglie - Bohm quantum mechanics.', 'quant-ph-0102095-1-52-0': 'Phase space for a scalar charged particle is not limited by three couples of the momentums and coordinates only.', 'quant-ph-0102095-1-52-1': 'The charge part of it exists as well.', 'quant-ph-0102095-1-52-2': 'However, in the approach presented here we leave the operator nature of such variables without modification.', 'quant-ph-0102095-1-52-3': 'As a result, the matrix-valued Wigner function is the density matrix in charge space with standard rules of average values calculation as well.', 'quant-ph-0102095-1-53-0': 'If we limit our consideration only by such elements of dynamical algebra that do not depend on variables of the charge space, it is possible to introduce the usual Wigner function.', 'quant-ph-0102095-1-53-1': 'This object differs from Wigner function for non-relativistic particle and from Wigner function in the Newton - Wigner position operator approach as well.', 'quant-ph-0102095-1-53-2': 'First of all it should be noticed that it contains four components corresponding to particles, antiparticles and their interference with each other.', 'quant-ph-0102095-1-53-3': 'Moreover, even for one-particle case, when only one component exists, definition of Wigner function differs as the result of odd part of the position operator being present, from the usual one.', 'quant-ph-0102095-1-54-0': 'It results in non-standard behavior of some physical variables, even in absence of conditions when particles creation is possible.', 'quant-ph-0102095-1-54-1': 'However, it is observed not for all physical variables.', 'quant-ph-0102095-1-54-2': 'For example, energy and number of particles (that are usually considered in statistical physics) do not show such peculiarities.', 'quant-ph-0102095-1-54-3': 'Hence, one can expect such effects on the quadratic and higher moments of coordinate.', 'quant-ph-0102095-1-54-4': 'As an example, one can take the dispersion that can be interpreted as real (physical) size of system that is not limited by external borders.', 'quant-ph-0102095-1-55-0': 'One can separate two groups of effects that result from this approach: ones related to interference between particles and antiparticles, and ones that take place in systems with same charge signs.', 'quant-ph-0102095-1-55-1': 'Effects of the first group result from presence of the odd part of the Wigner function, second ones are due to the specific function [MATH] being present in the even component of the Wigner function.', 'quant-ph-0102095-1-55-2': 'At the one-particle level it manifests itself, for example, in violation of the uncertainty relation.', 'quant-ph-0102095-1-55-3': 'Perhaps, such effects can exist in many-particle systems as well.', 'quant-ph-0102095-1-56-0': 'Even and odd components of Wigner function in the system of quantum Liouville equations are not mixed up together.', 'quant-ph-0102095-1-56-1': 'This results from absence of particles creation from vacuum in the system.', 'quant-ph-0102095-1-56-2': 'For example, in an electric field vacuum is not stable [CITATION], and this can be interpreted as a consequence of the odd part of the position operator being present as well.', 'quant-ph-0102095-1-56-3': 'It manifests itself in mixing up of different components in the quantum Liouville equation.', 'quant-ph-0102095-1-56-4': 'There is another situation for the instant magnetic field: particles are not created and the Wigner function components are not mixed up [CITATION].', 'quant-ph-0102095-1-56-5': 'Nevertheless, even components of position and momentum of such a system satisfy not usual commutation relations but deformed Heisenberg - Weyl algebra ones [CITATION].', 'quant-ph-0102095-1-56-6': 'These facts can mean that in external electromagnetic fields odd part of the position operator reveals itself especially strong.', 'quant-ph-0102095-1-57-0': 'Speakable and Unspeakable in Quantum Mechanics.', 'quant-ph-0102095-1-57-1': '(Cambridge University Press, Cambridge, 1987).', 'quant-ph-0102095-1-58-0': 'b15 K. Imre and E.Ozizmir, M. Rosenbaum ,P.F. Zweifel, J. Math.', 'quant-ph-0102095-1-59-0': 'b18 S.R. de Groot, W.A. van Leeuwen, Ch.', 'quant-ph-0102095-1-59-1': 'G. van Weert, Relativistic kinetic theory.', 'quant-ph-0102095-1-60-0': 'b19 P.R. Holland and A. Kyprianidis, Z. Maric, J.P. Vigier, Phys.', 'quant-ph-0102095-1-61-0': 'b20 P. Gerard, P.A. Markovich, N.J. Mauser and F. Poupaud, Comm.', 'quant-ph-0102095-1-62-0': 'Possible peculiarities of synchrotron radiation in a strong magnetic field.', 'quant-ph-0102095-1-62-1': 'Submitted to Space Science and Technology (Kosmichna Nauka i Tehnologiya), Kiev.', 'quant-ph-0102095-1-62-2': 'This work was presented in VIII Ukrainian Conference on Plasma Physics and Controlled Fusion, Alushta, Crimea, 11 - 17 September, 2000.', 'quant-ph-0102095-1-63-0': 'b25 C. Dewdney, P.R. Holland, A. Kyprianidis, Z. Maric and J.P. Vigier, Phys.', 'quant-ph-0102095-1-64-0': 'b28 D.I. Blokhintsev, On the localization of micro-particles in space and time.', 'quant-ph-0102095-1-65-0': 'b30 A.A. Grib, S.G. Mamaev, B.M. Mostapenko, Vacuum quantum effects in strong fields (Energoatomizdat, Moscow, 1988) [in Russian].'}","{'quant-ph-0102095-2-0-0': 'A description of scalar charged particles, based on the Feshbach - Villars formalism, is proposed.', 'quant-ph-0102095-2-0-1': 'Particles are described by an object that is a Wigner function in usual coordinates and momenta and a density matrix in the charge variable.', 'quant-ph-0102095-2-0-2': 'It is possible to introduce the usual Wigner function for a large class of dynamical variables.', 'quant-ph-0102095-2-0-3': 'Such an approach explicitly contains a measuring device frame.', 'quant-ph-0102095-2-0-4': 'From our point of view it corresponds to the Copenhagen interpretation of quantum mechanics.', 'quant-ph-0102095-2-0-5': 'It is shown how physical properties of such particles depend on the definition of the coordinate operator.', 'quant-ph-0102095-2-0-6': 'The evolution equation for the Wigner function of a single-charge state in the classical limit coincides with the Liouville equation.', 'quant-ph-0102095-2-0-7': 'Localization peculiarities manifest themselves in specific constraints on possible initial conditions.', 'quant-ph-0102095-2-1-0': '# Introduction', 'quant-ph-0102095-2-2-0': 'The question regarding the nature of the wavefunction has its origin in the early years of quantum mechanics.', 'quant-ph-0102095-2-2-1': 'For a long time it has been considered as a philosophical rather then a physical question [CITATION].', 'quant-ph-0102095-2-2-2': 'However, it is now very real because of the recent theoretical and experimental progress in quantum information [CITATION].', 'quant-ph-0102095-2-3-0': 'One of the significant points in understanding the nature of the wave function is the Einstein - Podolsky - Rosen paradox and the existence of quantum correlations related to it.', 'quant-ph-0102095-2-3-1': 'Since 1980, such specific behavior of quantum systems has been confirmed many times in experiments [CITATION].', 'quant-ph-0102095-2-3-2': 'It is very important that such correlations ""spread"" in space instantly.', 'quant-ph-0102095-2-3-3': 'Nevertheless, if one adheres to the Copenhagen interpretation, there is no violation of causality principle.', 'quant-ph-0102095-2-4-0': 'However, due to the fact that collapse of the wavefunction takes place at a given instant (or maybe in a small time interval) in the whole space, one can speak of violation of causality principle and a conflict between quantum mechanics and special relativity [CITATION] in some interpretations of quantum mechanics.', 'quant-ph-0102095-2-4-1': 'Following the idea of Bell and Eberhard that a consistent description should contain a preferred frame, relativistic classical and quantum mechanics is built in [CITATION] in such a way that the relativity principle is generalized and the causality is not violated.', 'quant-ph-0102095-2-4-2': 'A basic assumption of this theory is that transformation from one frame to another is realized by operators which are isomorphic to the Lorentz group and depend on a certain four-vector (as parameter) in such a way that the instant time hyper-plain is invariant.', 'quant-ph-0102095-2-4-3': 'This vector is interpreted as a relative to an observer four-velocity of the preferred frame.', 'quant-ph-0102095-2-4-4': 'Under this assumption the postulate about the light speed constancy is changed to the postulate about the constancy of the average speed of light on closed path.', 'quant-ph-0102095-2-5-0': 'One of the remarkable peculiarities of this theory is in the existence of a well-defined position operator [CITATION] which coincides with the Newton - Wigner position operator [CITATION] in the preferred frame.', 'quant-ph-0102095-2-5-1': 'It means that in this approach measurement of position does not create a particle - antiparticle couple since the odd part responsible for appearance of a different charge state superposition is absent.', 'quant-ph-0102095-2-5-2': 'Hence, there exists a supposition that localization of relativistic particles can bear information about presence of the preferred frame in the Universe.', 'quant-ph-0102095-2-6-0': 'The above arguments show the fundamental role that determination of the position operator structure can play.', 'quant-ph-0102095-2-6-1': 'Moreover, the question of whether it is a one-particle, or, in principle, a many-particle operator, is of specific interest as well.', 'quant-ph-0102095-2-6-2': 'Therefore, it is very important to find situations where the odd part of the position operator could manifest itself.', 'quant-ph-0102095-2-6-3': 'It would be possible to realize such observation on strongly localized (near the Compton wavelength) states of a separate particle.', 'quant-ph-0102095-2-6-4': 'However, because such states are a problem for laboratory experiments, it is of great interest how this peculiarity manifests itself in a many-particle system.', 'quant-ph-0102095-2-7-0': 'The Weyl - Wigner - Moyal (WWM) formalism is a convenient method to describe both one-particle and many-particle quantum systems in non-relativistic theory [CITATION].', 'quant-ph-0102095-2-7-1': 'Nevertheless, attempts to generalize it for the relativistic case lead to a number of problems.', 'quant-ph-0102095-2-7-2': 'The first problem is that the Weyl rule [CITATION] does not include time as a dynamical variable, and the scalar product in the Hilbert space of states is formulated for functions square integrable not over the whole space-time but in the three-dimensional space or in a space-like hyper-surface only.', 'quant-ph-0102095-2-7-3': 'In [18] this problem was resolved by generalization of the spatial integration over the whole space-time without Weyl rule application.', 'quant-ph-0102095-2-7-4': 'The WWM formalism in the framework of the stochastic formulation of quantum mechanics, where the scalar product is formulated for square integrable functions in the whole space-time, also leads to Lorenz-invariant expressions [CITATION].', 'quant-ph-0102095-2-8-0': 'The matrix-valued Wigner function formalism has been developed for the general case of many component equations and, in particular, for [MATH] spin particles on the basis of the usual Weyl rule [CITATION].', 'quant-ph-0102095-2-8-1': 'Certainly, such equations are not Lorenz invariant, though average values coincide with their analogues in the usual approach.', 'quant-ph-0102095-2-9-0': 'The next essential problem in relativistic WWM formalism is in absence of a well-defined position operator.', 'quant-ph-0102095-2-9-1': 'Unlike the works mentioned above, where the Wigner function is determined by means of the usual position operator, in [CITATION] a formalism with the Newton - Wigner position operator is developed.', 'quant-ph-0102095-2-9-2': 'This approach is not Lorenz invariant either, and respective results differ from the standard ones.', 'quant-ph-0102095-2-9-3': 'However, they can be related to [CITATION], where the consistent definition of the position operator is possible.', 'quant-ph-0102095-2-10-0': 'The goals of this paper are in formulating the WWM relativistic formalism for scalar charged particles under the approach of [CITATION] and finding a set of specific peculiarities in relativistic quantum system behavior, that are related to the non-trivial structure of the position operator.', 'quant-ph-0102095-2-10-1': 'It turns out that values of some observables depend directly on the position operator definition, that, as mentioned above, can be a consequence of existence of the preferred frame in the Universe.', 'quant-ph-0102095-2-11-0': 'In Sec.[REF] we introduce the Weyl rule for matrix-valued observables in the case of scalar charged particles and discuss peculiarities of correspondence between classical and quantum dynamical algebras.', 'quant-ph-0102095-2-11-1': 'Sec.[REF] is devoted to the matrix-valued Wigner function and quantum Liouville equation.', 'quant-ph-0102095-2-11-2': 'Here we also discuss the absence of Lorenz invariance of this approach, and how it can be related to the Copenhagen interpretation of quantum mechanics.', 'quant-ph-0102095-2-11-3': 'In Sec. [REF] we consider how the usual Weyl rule turns into the Feshbach - Villars representation.', 'quant-ph-0102095-2-11-4': 'Among the whole set of dynamical variables, we separate a special class of observables with the Weyl symbols independent of charge variable (charge-invariant variables).', 'quant-ph-0102095-2-11-5': 'They have several remarkable properties.', 'quant-ph-0102095-2-11-6': 'In particular, we find the relationship between their even and odd parts.', 'quant-ph-0102095-2-11-7': 'The usual (not matrix-valued) Wigner function can be introduced for such observables.', 'quant-ph-0102095-2-11-8': 'It is considered in Sec. [REF] (for a brief abstract of this approach see [CITATION]).', 'quant-ph-0102095-2-11-9': 'This object includes four components: one corresponds to a particle, second one corresponds to an antiparticle (even part of Wigner function), and two more are interference terms (odd part of Wigner function).', 'quant-ph-0102095-2-11-10': 'The evolution equation for the odd part becomes zero in the classical limit, and the equation for the even part coincides with its analogue in the Newton - Wigner position operator approach [CITATION].', 'quant-ph-0102095-2-11-11': 'The difference reveals itself in peculiarities of constraints on initial conditions that are considered in Sec. [REF].', 'quant-ph-0102095-2-12-0': '# Weyl rule and specific properties of dynamical algebra for scalar charged particles', 'quant-ph-0102095-2-13-0': 'In this and the following Sections we apply the methods developed in [CITATION] to the Klein - Gordon equation that is written using the Feshbach - Villars formalism [CITATION].', 'quant-ph-0102095-2-13-1': 'This makes it possible to take the charge variable into account explicitly.', 'quant-ph-0102095-2-13-2': 'With account of the fact that the Hilbert space of states for scalar charged particles has an indefinite metric, we have to distinguish between covariant and contravariant basis vectors and coordinates for subspace corresponding to the charge variable.', 'quant-ph-0102095-2-13-3': 'Hence, arbitrary state is expanded on basis vectors in the following way: [EQUATION] where [EQUATION]', 'quant-ph-0102095-2-13-4': 'Here [MATH] is an arbitrary dynamical variable and the summarizing symbol can be interpreted as an integration with respect to [MATH].', 'quant-ph-0102095-2-13-5': 'Greek indices take values [MATH].', 'quant-ph-0102095-2-13-6': '[MATH] plays the role of metric tensor (see [CITATION]).', 'quant-ph-0102095-2-13-7': 'In notations like [MATH], [MATH], [MATH], [MATH], [MATH],..., symbols [MATH], [MATH] do not correspond to specific operators and mean only values of the corresponding matrix elements.', 'quant-ph-0102095-2-13-8': 'There are some peculiarities in notations of operators as well.', 'quant-ph-0102095-2-13-9': '[MATH] is a full operator that acts on all dynamical variables.', 'quant-ph-0102095-2-13-10': '[MATH] is an operator that acts on the charge variables only (or, in other words, it is a [MATH]-numeric matrix).', 'quant-ph-0102095-2-14-0': 'For a consistent development of the WWM formalism we shall formulate the Weyl rule bringing into correspondence matrix-valued classical variables (Weyl symbols) to quantum-mechanical operators.', 'quant-ph-0102095-2-14-1': 'Here we should take into account that in our case classical variables are operators acting on the charge variable.', 'quant-ph-0102095-2-14-2': 'However, because the operator of quasi-probability density in the coordinate representation is proportional to the identity matrix in charge space, the succession of this operator and a matrix-valued Weyl symbol does not matter.', 'quant-ph-0102095-2-14-3': 'Choosing it in an arbitrary way, one can write the Weyl rule as follows: [EQUATION] where [MATH] is the matrix-valued Weyl symbol, [MATH] stands for corresponding quantum-mechanical operator, [MATH] is the operator of quasi-probability density that is the Fourier transform of the displacement operator.', 'quant-ph-0102095-2-14-4': 'Following [CITATION] one can obtain the expansion of [MATH] on eigenvectors of the position operator: [EQUATION] where [MATH] is dimensionality of the physical space.', 'quant-ph-0102095-2-15-0': 'Substituting ([REF]) to ([REF]) one can find the expression for an arbitrary operator expansion: [EQUATION]', 'quant-ph-0102095-2-15-1': 'Similarly, such expression can be written by expending the operator of quasi-probability density on the eigenvectors of the momentum operator: [EQUATION]', 'quant-ph-0102095-2-15-2': 'Matrix elements of this operator can be found in the form: [EQUATION]', 'quant-ph-0102095-2-15-3': 'By changing variables in this expression in a standard way and performing Fourier transformation one can obtain the expression that reconstructs matrix-valued Weyl symbol through operator: [EQUATION]', 'quant-ph-0102095-2-15-4': 'In [CITATION] the integral form of the Klein - Gordon equation with pseudo-differential symbols in the Feshbach - Villars representation (where Hamiltonian matrix has a diagonal form) is obtained.', 'quant-ph-0102095-2-15-5': 'In fact, it means that the Newton - Wigner coordinate is used there instead of the usual coordinate.', 'quant-ph-0102095-2-15-6': 'Here we obtain another integral form of the Klein - Gordon equation using the usual position operator: [EQUATION] where [EQUATION] is the matrix-valued Weyl symbol of the Hamiltonian in the general case (for a particle in electromagnetic field), which depends on both coordinate and momentum.', 'quant-ph-0102095-2-15-7': 'Unlike [CITATION] this equation takes into account that the position operator mixes states with different charge signs.', 'quant-ph-0102095-2-16-0': 'Until now,consideration of the matrix-valued WWM formalism has been very similar to the usual one.', 'quant-ph-0102095-2-16-1': 'Considerable differences appear in determining the matrix-valued Moyal bracket and in the classical limit.', 'quant-ph-0102095-2-16-2': 'To consider this question one should find the matrix-valued Weyl symbol from the product of two operators in a standard way.', 'quant-ph-0102095-2-17-0': 'Let [MATH] and [MATH] be matrix-valued Weyl symbols of two operators [MATH] and [MATH] respectively.', 'quant-ph-0102095-2-17-1': 'Let us introduce operators [MATH] as follows: [EQUATION]', 'quant-ph-0102095-2-17-2': 'Then, using ([REF]) and consideration like that used in [CITATION], one can obtain the matrix-valued Weyl symbol of the operator [MATH]: [EQUATION]', 'quant-ph-0102095-2-17-3': 'For the matrix-valued Weyl symbol of the operator [MATH] one can find a similar expression.', 'quant-ph-0102095-2-17-4': 'Therefore, the matrix-valued Moyal bracket can be determined in the following form: [EQUATION]', 'quant-ph-0102095-2-17-5': 'Unlike the usual case, matrix-valued Weyl symbols do not commutate, so it is not possible to represent such a Moyal bracket as a sinus from the operator of the Poisson bracket.', 'quant-ph-0102095-2-17-6': 'This fact results in the classical limit in some peculiarities.', 'quant-ph-0102095-2-17-7': 'In the general case, one can express the classical limit of the matrix-valued Moyal bracket via the matrix-valued Poisson bracket [CITATION] and the commutator of the matrix-valued Weyl symbols: [EQUATION]', 'quant-ph-0102095-2-17-8': 'If the matrix-valued Weyl symbols (for example position and Hamiltonian) commutate, the matrix-valued Moyal bracket is in fact the usual one and coincides with the Poisson bracket in the classical limit.', 'quant-ph-0102095-2-17-9': 'If they do not commutate with each other, three cases are possible.', 'quant-ph-0102095-2-17-10': 'Let [MATH] when [MATH].', 'quant-ph-0102095-2-17-11': 'If [MATH], the classical limit does not exist for such a couple of the matrix-valued Weyl symbols, since the value of the matrix-valued Moyal bracket increases infinitely.', 'quant-ph-0102095-2-17-12': 'If [MATH], it becomes zero in the classical limit.', 'quant-ph-0102095-2-17-13': 'If [MATH], the classical limit exists.', 'quant-ph-0102095-2-17-14': 'Here we can take as an example the Newton -Wigner position operator [CITATION].', 'quant-ph-0102095-2-17-15': ""Part of its matrix-valued Weyl symbol,not commutating with the Hamiltonian, is proportional to Planck's constant, thus in this case there exists a well-defined classical limit."", 'quant-ph-0102095-2-18-0': '# Matrix-valued Wigner function and quantum Liouville equation', 'quant-ph-0102095-2-19-0': 'Using ([REF]), one can find the average value of an arbitrary operator [MATH] expressed via its matrix-valued Weyl symbol: [EQUATION]', 'quant-ph-0102095-2-19-1': 'Then, introducing the matrix-valued Wigner function [EQUATION] this expression can be written in a simpler form: [EQUATION]', 'quant-ph-0102095-2-19-2': 'Substitution of the operator of quasi-probability density expansion ([REF]) into ([REF]) leads to the expression for the matrix-valued Wigner function in the coordinate representation.', 'quant-ph-0102095-2-19-3': 'As a result we obtain a formula that coincides with the definition given in [CITATION]: [EQUATION]', 'quant-ph-0102095-2-19-4': 'Next, let us describe a method to obtain evolution equation for the matrix-valued Wigner function of a scalar charged particle.', 'quant-ph-0102095-2-19-5': 'To do this one shall differentiate expression ([REF]) with respect to time.', 'quant-ph-0102095-2-19-6': 'The value of the wavefunction derivative is taken from the integral form of the Klein - Gordon equation ([REF]).', 'quant-ph-0102095-2-19-7': 'After standard transformations, similar to the usual WWM formalism, we obtain the quantum Liouville equation [EQUATION] where the matrix-valued Moyal bracket is determined by expression ([REF]).', 'quant-ph-0102095-2-20-0': 'At first sight, this equation has a considerable defect in comparison to similar expressions in [CITATION]; namely, it is not Lorentz invariant.', 'quant-ph-0102095-2-20-1': 'For an interpretation of this fact, we have to take into account arguments from the measurement theory.', 'quant-ph-0102095-2-20-2': 'The fact is that average values calculated in this approach coincide with ones in the usual (Schrodinger) representation of quantum mechanics that is Lorentz invariant.', 'quant-ph-0102095-2-20-3': 'Nevertheless, unlike the stochastic formulation of quantum mechanics [CITATION], the scalar product is determined here with functions that are square integrable in a certain space-like hyper-surface, rather than over the whole space-time.', 'quant-ph-0102095-2-20-4': 'Moreover, the value of scalar product does not depend on its choice [CITATION].', 'quant-ph-0102095-2-20-5': 'One can relate this hyper-surface to the measurement device frame.', 'quant-ph-0102095-2-20-6': 'In other words, the wavefunction collapse occurs in a frame in which equation ([REF]) is written.', 'quant-ph-0102095-2-20-7': 'The absence of Lorentz invariance is a consequence of the fact that the Weyl rule does not include time as an independent dynamical variable.', 'quant-ph-0102095-2-21-0': 'In [CITATION] the process of relativistic measurement is considered.', 'quant-ph-0102095-2-21-1': 'In that work the point of view is expressed, that the wavefunction (and as result the Wigner function, we notice) has no objective value and does not covariantly transform when there are classical interventions.', 'quant-ph-0102095-2-21-2': 'Here we note that in principle equation ([REF]) can be written with four-dimensional Lorentz invariant symbols only, but to do this we have to incorporate in the theory a certain time-like unit vector in a way similar to Tomonaga-Schwinger approach to quantum field theory [CITATION].', 'quant-ph-0102095-2-21-3': 'It is the four-velocity of the frame where the wavefunction collapse occurs (the measuring device frame) relative to the second static observer (watching observer).', 'quant-ph-0102095-2-21-4': 'This frame has principally another sense to that of the preferred frame in [CITATION].', 'quant-ph-0102095-2-21-5': 'In that approach [CITATION] collapse of the wavefunction has to take place in all frames in the whole space (because of the instant time hyper-surface being invariant).', 'quant-ph-0102095-2-21-6': 'In our approach this process obeys the relativity of simultaneity.', 'quant-ph-0102095-2-21-7': 'If it were possible to observe the wave function directly, such a hypothetical watching observer would see the wavefunction collapse as a certain moving front.', 'quant-ph-0102095-2-21-8': 'As a result, at a certain instant, in a static frame (attached to the watching observer), a state with a part of the wavefunction before measurement on one hand and a part of the wavefunction after measurement on the other hand, is realized (Fig. [REF]).', 'quant-ph-0102095-2-21-9': 'But perhaps it is impossible to propose even gedanken experiment without use of a superluminar signal to interpret the results in favor of one of the approaches.', 'quant-ph-0102095-2-22-0': 'Hence we find to be very important and of fundamental value the fact that quantum mechanics in the Wigner representation necessarily includes the four-velocity of the measuring device frame in explicit form.', 'quant-ph-0102095-2-23-0': '# Transformation to the Feshbach - Villars representation', 'quant-ph-0102095-2-24-0': 'In the Feshbach - Villars representation, Hamiltonian matrix has a diagonal form and indices in the charge space correspond to the particle and antiparticle [CITATION].', 'quant-ph-0102095-2-24-1': 'Therefore, it is convenient to distinguish between solutions with different charge signs and to give them explicit physical sense.', 'quant-ph-0102095-2-24-2': 'Furthermore, in this approach an influence of the odd part of the position operator on values of some physical variables is more evident.', 'quant-ph-0102095-2-25-0': 'Operator [MATH]transforms to the Feshbach - Villars representation by the following formula: [EQUATION] where the transformation matrix has the form: [EQUATION]', 'quant-ph-0102095-2-25-1': 'Here and below [EQUATION] is the energy of a relativistic free particle.', 'quant-ph-0102095-2-26-0': 'Next, we apply ([REF]) to the Weyl rule in the form ([REF]).', 'quant-ph-0102095-2-26-1': 'As result, we obtain the expression that brings into correspondence the matrix-valued Weyl symbol to the operator in the Feshbach-Villars representation: [EQUATION]', 'quant-ph-0102095-2-26-2': 'In general, it is rather difficult to interpret this expression since there is a complicated dependence on the integration variable under the integral sign.', 'quant-ph-0102095-2-26-3': 'Here we do not go beyond the simplest case.', 'quant-ph-0102095-2-27-0': 'Let the matrix-valued Weyl symbol be proportional to the identity matrix: [EQUATION]', 'quant-ph-0102095-2-27-1': 'In principle, one can say that such symbols do not depend on the charge variable, so that the class of dynamical variables, which corresponds to those, we denominate here, to be more brief, as a class of charge-invariant variables.', 'quant-ph-0102095-2-27-2': 'Most of the dynamical variables that we consider in relativistic (non-quantum) mechanics belong to this class.', 'quant-ph-0102095-2-27-3': 'The reason for this is the absence of a dependence on the charge variable in classical mechanics.', 'quant-ph-0102095-2-27-4': 'Hence, the question about the classical limit is for such variables especially interesting.', 'quant-ph-0102095-2-28-0': 'The Weyl rule for charge-invariant variables in the Feshbach - Villars representation has the form: [EQUATION]', 'quant-ph-0102095-2-28-1': 'Unlike [CITATION] and non-relativistic case there is a matrix-valued variable here: [EQUATION]', 'quant-ph-0102095-2-28-2': 'It contains even and odd parts and is expressed via the energy of a free particle ([REF]): [EQUATION]', 'quant-ph-0102095-2-28-3': 'Consequences from ([REF]) are expressions for even [MATH] and odd [MATH] parts of the operator of a charge-invariant observable in terms of its Weyl symbol: [EQUATION]', 'quant-ph-0102095-2-28-4': 'Matrix elements with eigenvectors of the momentum of the operator of a charge-invariant variable and its even and odd parts can be written in the following form: [EQUATION]', 'quant-ph-0102095-2-28-5': 'Then, as was achieved in Sec. [REF], one can obtain a formula that reconstructs the Weyl symbol from the operator, with even and odd parts, in the FeshbachVillars representation: [EQUATION]', 'quant-ph-0102095-2-28-6': 'Comparing ([REF]) and ([REF]) we conclude that matrix elements (integral kernels) of even and odd parts of the operator of an arbitrary charge-invariant variable are uniquely related to each other due to the Weyl rule: [EQUATION]', 'quant-ph-0102095-2-28-7': 'A consequence of this expression is the fact that independent of position the odd part of an operator is zero.', 'quant-ph-0102095-2-28-8': 'If one uses as [MATH], for example, the scalar potential of an electric field, expression ([REF]) establishes a quantitative relationship between effects of motion of a particle in an electric field and its interaction with a polarizable vacuum (trembling motion, Zitterbewegung) [CITATION].', 'quant-ph-0102095-2-29-0': 'Consider now time derivative peculiarities of charge-invariant variables.', 'quant-ph-0102095-2-29-1': 'In the coordinate representation, the matrix-valued Weyl symbol of such an operator has the form: [EQUATION]', 'quant-ph-0102095-2-29-2': 'The Weyl rule ([REF]) (in the Feshbach - Villars representation) for this can be written as follows: [EQUATION] or, in another form: [EQUATION] where we have introduced a new matrix-valued variable of three arguments: [EQUATION]', 'quant-ph-0102095-2-29-3': 'If [MATH] on [MATH] linearly depends, formula ([REF]) can be presented in a particularly simple form: [EQUATION]', 'quant-ph-0102095-2-29-4': 'The time derivative of an operator is a composition of its even and odd parts, where the odd part has a classical limit because of the absence of interference terms.', 'quant-ph-0102095-2-29-5': 'As an example one can take the Newton - Wigner position operator that is even part of the usual position operator.', 'quant-ph-0102095-2-30-0': 'In the general case this formula contains in higher orders of [MATH] non-standard terms, since [MATH] distinguishes from [MATH] .', 'quant-ph-0102095-2-30-1': 'However, in the classical limit they vanish and this expression takes the form of ([REF]).', 'quant-ph-0102095-2-31-0': 'Hence, in the Feshbach - Villars representation, where we distinguish between solutions with different charge signs, the odd part of the position results not only in the emergence of the odd part of operators, but leads to some peculiarities in their even parts as well.', 'quant-ph-0102095-2-31-1': 'In particular, it appears in the specifics of the semi-classical limit.', 'quant-ph-0102095-2-32-0': '# Wigner function and quantum Liouville equation for charge-invariant variables', 'quant-ph-0102095-2-33-0': 'It is easy to see from ([REF]) that it is possible to introduce the usual Wigner function for the charge invariant variables in such a way that their average values are determined by the formula: [EQUATION]', 'quant-ph-0102095-2-33-1': 'To show this, we expand the operator of quasi-probability density in the Feshbach - Villars representation through the eigenvectors of momentum: [EQUATION]', 'quant-ph-0102095-2-33-2': 'The Wigner function is determined as the average value of this operator over an arbitrary state that contains in the general case components with both charge signs: [EQUATION]', 'quant-ph-0102095-2-33-3': 'There are four components in this expression.', 'quant-ph-0102095-2-33-4': 'Two of them are the average value of the even part of the operator of quasi-probability density, and the other two are the average values of the odd part.', 'quant-ph-0102095-2-33-5': 'Let us introduce the symbols: [EQUATION]', 'quant-ph-0102095-2-33-6': 'It should be noted here and below that the object [MATH] is not the matrix-valued Wigner function in the sense of [CITATION] and Sec. [REF] of this work.', 'quant-ph-0102095-2-34-0': 'Substituting ([REF]) into ([REF]), we obtain for the Wigner function components following expressions [EQUATION]', 'quant-ph-0102095-2-34-1': 'Even components of the Wigner function ([REF]) correspond to a charge definite state.', 'quant-ph-0102095-2-34-2': 'The value of odd components ([REF]) for such a state is zero.', 'quant-ph-0102095-2-34-3': 'The expression ([REF]) differs from analogous one for a non-relativistic Wigner function and relativistic one determined using the Newton - Wigner position operator [CITATION] by the function [MATH]under the integral sign (see ([REF])).', 'quant-ph-0102095-2-34-4': 'This function has a specific feature: its expansion on [MATH], [MATH] does not contain square terms.', 'quant-ph-0102095-2-34-5': 'This means that in the non-relativistic limit expression ([REF]) coincides with the usual determination of the Wigner function.', 'quant-ph-0102095-2-35-0': 'We obtain the evolution equations for every component separately.', 'quant-ph-0102095-2-35-1': 'The general principle here is the same as in Sec. [REF].', 'quant-ph-0102095-2-35-2': 'However, instead of equation ([REF]) we shall use the integral form of Klein -Gordon equation in the Feshbach - Villars representation [CITATION]: [EQUATION]', 'quant-ph-0102095-2-35-3': 'The following equations can be obtained in a standard way, through differentiating Wigner function components with respect to time: [EQUATION]', 'quant-ph-0102095-2-35-4': 'Nevertheless, the Wigner function components are not independent, i.e. a specific constraint is imposed on solutions of the system ([REF]),([REF]) .', 'quant-ph-0102095-2-35-5': 'To find this one should take the Fourier transform and make the standard change of variables for every component ([REF]),([REF]).', 'quant-ph-0102095-2-35-6': 'As a result, we obtain the following expressions: [EQUATION]', 'quant-ph-0102095-2-35-7': 'Now we divide ([REF]) by ([REF]) and ([REF]) by ([REF]), and equate the resulting expressions with each other due to the equality of their left-hand sides.', 'quant-ph-0102095-2-35-8': 'This gives us the constraint we are looking for: [EQUATION]', 'quant-ph-0102095-2-35-9': 'Equation ([REF]) explicitly contains the imaginary unit, so the actual question is whether the Wigner function is real.', 'quant-ph-0102095-2-35-10': 'To test this we consider the complex-conjugate expressions for ([REF]),([REF]).', 'quant-ph-0102095-2-35-11': 'After some easy transformations we obtain the following identities: [EQUATION]', 'quant-ph-0102095-2-35-12': 'These mean that even components of the Wigner function are real.', 'quant-ph-0102095-2-35-13': 'Odd components are complex conjugate to each other so that their sum is real as well.', 'quant-ph-0102095-2-36-0': 'It is essential that the equation for even components of the Wigner function coincides with the analogous expression obtained in [CITATION] for the formalism where the Newton - Wigner position operator is used.', 'quant-ph-0102095-2-36-1': 'Hence, the dynamics of quasi-distribution functions for systems of particles with charges of the same sign (charge definite states) is identical in both cases.', 'quant-ph-0102095-2-37-0': '# Statistical properties of the Wigner function for charge-invariant variables', 'quant-ph-0102095-2-38-0': 'Constraint on the initial conditions of the Wigner function is the general peculiarity of the approach described here because equations are identical in both cases (for charge definite states).', 'quant-ph-0102095-2-38-1': 'In this Section we show how some theorems and properties differ from their analogues in the usual WWM formalism [CITATION] and in an approach where the Newton - Wigner position operator is used [CITATION].', 'quant-ph-0102095-2-39-0': 'First of all one should note the property of normality.', 'quant-ph-0102095-2-39-1': 'Even part of the Wigner function ([REF]) is normalized in the whole phase space, and the integral of the odd part ([REF]) is zero.', 'quant-ph-0102095-2-40-0': 'Consider now the compatibility of the Wigner function ([REF]) with distributions in the coordinate and momentum spaces for a single-charge state.', 'quant-ph-0102095-2-40-1': 'For this purpose we integrate ([REF]) by coordinate.', 'quant-ph-0102095-2-40-2': 'As a result, we obtain the distribution in momentum space: [EQUATION]', 'quant-ph-0102095-2-40-3': 'This function always has a definite sign, so that it can be interpreted as the probability density.', 'quant-ph-0102095-2-40-4': 'One can obtain a more non-trivial result for distribution in coordinate space.', 'quant-ph-0102095-2-40-5': 'The result of integrating of ([REF]) by momentum can be given in the following form: [EQUATION] or, which is the same, [EQUATION] where [MATH] is the wavefunction in the representation of the Newton - Wigner coordinate [CITATION].', 'quant-ph-0102095-2-40-6': 'It is obvious, that quasi-distribution, ([REF]) and ([REF]), is not sign-definite.', 'quant-ph-0102095-2-40-7': 'This property is typical for bosons and causes difficulties in probability interpretation [CITATION].', 'quant-ph-0102095-2-40-8': 'Nevertheless, formal use of such quasi-probability makes it possible to calculate average values of the variables dependent on coordinate.', 'quant-ph-0102095-2-41-0': 'Average values of variables only depending on momentum do not differ from similar ones in usual approach.', 'quant-ph-0102095-2-41-1': 'Let us find the peculiarities of higher moments of coordinate in our case.', 'quant-ph-0102095-2-41-2': 'For this purpose we integrate ([REF]) with [MATH] by coordinate and use the identity: [EQUATION]', 'quant-ph-0102095-2-41-3': 'After some obvious transformations the result for the [MATH]-th moment of the coordinate can be written as follows: [EQUATION]', 'quant-ph-0102095-2-41-4': 'The first moment (average coordinate) has a value similar to one in the Newton - Wigner coordinate approach.', 'quant-ph-0102095-2-41-5': 'Differences manifest themselves in higher moments.', 'quant-ph-0102095-2-41-6': 'As an example, we apply this expression to second moment of the coordinate: [EQUATION]', 'quant-ph-0102095-2-41-7': 'As well as the usual part, this formula also contains an additional term that causes the peculiarities related to determination of the position operator.', 'quant-ph-0102095-2-41-8': 'For strongly localized states, it results in formal violation of the uncertainty relation.', 'quant-ph-0102095-2-42-0': 'Similar to the case of the usual WWM formalism [CITATION], we prove two criteria that make it possible to select Wigner functions for pure and mixed states out of the whole set of functions of the variables [MATH].', 'quant-ph-0102095-2-43-0': 'Criterion of pure state.', 'quant-ph-0102095-2-44-0': 'For the functions [MATH] and [MATH] to be even and odd components of the Wigner function for charge invariant variables, it is necessary and sufficient that equalities ([REF]),([REF]),([REF]) hold true, and the following conditions are satisfied: [EQUATION]', 'quant-ph-0102095-2-44-1': 'Necessity.', 'quant-ph-0102095-2-44-2': 'Following [CITATION] we start from an obvious identity: [EQUATION]', 'quant-ph-0102095-2-44-3': 'Applying it to formulas (51) - (54) one obtains the equalities [EQUATION]', 'quant-ph-0102095-2-44-4': 'Now, substituting the explicit form of [MATH] and [MATH]from ([REF]), we obtain ([REF]),([REF]).', 'quant-ph-0102095-2-45-0': 'Sufficiency.', 'quant-ph-0102095-2-45-1': 'Let components of the function satisfy conditions ([REF]), ([REF]) or, equivalently, ([REF]),([REF]).', 'quant-ph-0102095-2-45-2': 'So one can write following conditions: [EQUATION] where [MATH] are certain functions.', 'quant-ph-0102095-2-45-3': 'Let us show that they can be chosen in such a way as to be consistent with each other and present a wavefunction for a scalar charged particle.', 'quant-ph-0102095-2-45-4': 'From ([REF]),([REF]) and ([REF]),([REF]) it follows that [EQUATION]', 'quant-ph-0102095-2-45-5': 'Then one obtains [EQUATION]', 'quant-ph-0102095-2-45-6': 'Hence, one can say that our system is described by the following wavefunctions: [EQUATION]', 'quant-ph-0102095-2-45-7': 'Now one has to prove that wavefunctions presented in such a way (with one and two tildes) are consistent with each other.', 'quant-ph-0102095-2-45-8': 'For this purpose, a combination of expressions ([REF]) - ([REF]) and ([REF]) and ([REF]) is substituted into condition ([REF]).', 'quant-ph-0102095-2-45-9': 'As a result, we obtain [EQUATION]', 'quant-ph-0102095-2-45-10': 'This leads to the following equality: [EQUATION]', 'quant-ph-0102095-2-45-11': 'If this condition is fulfilled, one can accept as the wavefunction [EQUATION]', 'quant-ph-0102095-2-45-12': 'Then condition ([REF]) is satisfied and, taking account of the fact that components of [MATH] differ from components of [MATH] by a constant phase, condition ([REF]) is satisfied as well.', 'quant-ph-0102095-2-45-13': 'Hence, conditions of this criterion allow us to introduce wavefunction ([REF]).', 'quant-ph-0102095-2-45-14': 'Then, similar to how it is done in [CITATION], one can show that it satisfies the Klein - Gordon equation in the Feshbach - Villars representation.', 'quant-ph-0102095-2-45-15': 'Which was to be proved.', 'quant-ph-0102095-2-46-0': 'This criterion differs from a similar one in the usual WWM formalism [CITATION] and in the approach that uses the Newton - Wigner position operator by difference of the right-hand parts of conditions ([REF]),([REF]) from zero.', 'quant-ph-0102095-2-46-1': 'The obvious consequence from this is the fact that the one-particle state, where Wigner function would be a joint Gauss distribution by coordinate and momentum, is impossible in the approach described here.', 'quant-ph-0102095-2-47-0': 'Consider another property of Wigner function, true for both pure and mixed states.', 'quant-ph-0102095-2-47-1': 'For this purpose, we introduce the formula for the square of the module of the scalar product of state [MATH] and state [MATH]: [EQUATION]', 'quant-ph-0102095-2-47-2': 'It is easy to test this by substituting expressions ([REF]) and ([REF]) for the Wigner function components into it.', 'quant-ph-0102095-2-47-3': 'Then, let [MATH] presents a mixed state consisting of orthogonal states described by [MATH]: [EQUATION]', 'quant-ph-0102095-2-47-4': 'Taking into account that the sum of [MATH](the probability of system to be in state [MATH]) squares is less than unity (a consequence from ([REF]) ) one obtains for an arbitrary state [EQUATION]', 'quant-ph-0102095-2-47-5': 'Moreover, for a pure state this inequality turns into an equality.', 'quant-ph-0102095-2-47-6': 'Expression ([REF]) can be used as necessary and sufficient condition for both pure and mixed states.', 'quant-ph-0102095-2-48-0': 'This condition is written for a state that is the superposition of states with different charges signs, which is more typical for the many-particle case.', 'quant-ph-0102095-2-48-1': 'It takes a simpler form for a state where only one charge sign is realized: [EQUATION]', 'quant-ph-0102095-2-48-2': 'Though this inequality does not contain interference terms, it differs from its non-relativistic analogue due to function ([REF]) being present.', 'quant-ph-0102095-2-48-3': 'Hence, a dynamical variable that contains square and higher moments of coordinate can have peculiarities in many-particle systems that are described by statistical physics.', 'quant-ph-0102095-2-49-0': 'The type of influence of the odd part of the position operator on dynamical variables can be illustrated with the one-particle state that is the Gauss distribution in momentum space with arbitrary square of dispersion [MATH]: [EQUATION] where we have chosen the natural units of measure, [MATH].', 'quant-ph-0102095-2-49-1': 'The dependence of position dispersion on momentum dispersion is shown in Fig. [REF].', 'quant-ph-0102095-2-49-2': 'Its peculiarity is the formal violation of uncertainty relation.', 'quant-ph-0102095-2-49-3': 'Under very strong localization, states even with negative square of dispersion are possible, which was shown in [CITATION].', 'quant-ph-0102095-2-49-4': 'This fact is a property of scalar charged particles, and localization peculiarities for fermions rather different.', 'quant-ph-0102095-2-50-0': '# Conclusions', 'quant-ph-0102095-2-51-0': 'The usual (not Lorentz invariant) Weyl rule makes it possible to introduce the Wigner function that is not Lorentz invariant, but all average values calculated with it coincide with ones calculated with Lorentz invariant wavefunction.', 'quant-ph-0102095-2-51-1': 'This results in the fact that quantum mechanics in the Wigner formulation contains with necessity a measuring device frame.', 'quant-ph-0102095-2-51-2': 'In principle, we can write the evolution equations using only four-dimensional Lorentz invariant symbols, but it is necessary to introduce a certain time-like vector for it.', 'quant-ph-0102095-2-51-3': 'It is the four-velocity of the frame in which wavefunction collapse occurs, relative to the second (static) observer.', 'quant-ph-0102095-2-51-4': 'It should be noted, that this approach differs from [CITATION] where there is the preferred frame.', 'quant-ph-0102095-2-51-5': 'In the last cases such a frame has a global sense and its introduction is related to attempts of correct tachyons description and, as a consequence, to a possible explanation of the instant quantum correlations (in a relativistic case) from the position of de Broglie - Bohm quantum mechanics.', 'quant-ph-0102095-2-52-0': 'Phase space for a scalar charged particle is not only limited by three couples of the momenta and coordinates .', 'quant-ph-0102095-2-52-1': 'The charge part of it exists as well.', 'quant-ph-0102095-2-52-2': 'However, in the approach presented here we leave the operator nature of such variables without modification.', 'quant-ph-0102095-2-52-3': 'As a result, the matrix-valued Wigner function is the density matrix in charge space with standard rules of average values calculation as well.', 'quant-ph-0102095-2-53-0': 'If we limit our consideration only to such elements of dynamical algebra that do not depend on variables of the charge space, it is possible to introduce the usual Wigner function.', 'quant-ph-0102095-2-53-1': 'This object differs from the Wigner function for a non-relativistic particle and from the Wigner function in the Newton - Wigner position operator approach as well.', 'quant-ph-0102095-2-53-2': 'First of all it should be noted that it contains four components, corresponding to particles, antiparticles and their interference with each other.', 'quant-ph-0102095-2-53-3': 'Moreover, even for the one-particle case, when only one component exists, the definition of the Wigner function differs, as the result of odd part of the position operator being present, from the usual one.', 'quant-ph-0102095-2-54-0': 'This results in non-standard behaviour of some physical variables, even in the absence of conditions when particles creation is possible.', 'quant-ph-0102095-2-54-1': 'However, it is not observed for all physical variables.', 'quant-ph-0102095-2-54-2': 'For example, energy and number of particles (that are usually considered in statistical physics) do not show such peculiarities.', 'quant-ph-0102095-2-54-3': 'Hence, one can expect such effects on the quadratic and higher moments of coordinate.', 'quant-ph-0102095-2-54-4': 'As an example, one can take the dispersion that can be interpreted as the real (physical) size of system that is not limited by external borders.', 'quant-ph-0102095-2-55-0': 'One can separate two groups of effects that result from this approach: ones related to interference between particles and antiparticles, and ones that take place in systems with same charge signs.', 'quant-ph-0102095-2-55-1': 'Effects of the first group result from presence of the odd part of the Wigner function, the second ones are due to the specific function [MATH] being present in the even component of the Wigner function.', 'quant-ph-0102095-2-55-2': 'At the one-particle level it manifests itself, for example, in violation of the uncertainty relation.', 'quant-ph-0102095-2-55-3': 'Perhaps, such effects can exist in many-particle systems as well.', 'quant-ph-0102095-2-56-0': 'Even and odd components of Wigner function in the system of quantum Liouville equations are not mixed up together.', 'quant-ph-0102095-2-56-1': 'This results from absence of particles creation from vacuum in the system.', 'quant-ph-0102095-2-56-2': 'For example, in an electric field the vacuum is not stable [CITATION], and this can be interpreted as a consequence of the odd part of the position operator being present as well.', 'quant-ph-0102095-2-56-3': 'It manifests itself in mixing up of different components in the quantum Liouville equation.', 'quant-ph-0102095-2-56-4': 'There is another situation for the instant magnetic field: particles are not created and the Wigner function components are not mixed up [CITATION].', 'quant-ph-0102095-2-56-5': 'Nevertheless, even components of position and momentum of such a system satisfy not the usual commutation relations but the deformed Heisenberg - Weyl algebra ones [CITATION].', 'quant-ph-0102095-2-56-6': 'These facts can mean that in external electromagnetic fields the odd part of the position operator reveals itself to be especially strong.', 'quant-ph-0102095-2-57-0': 'Speakable and Unspeakable in Quantum Mechanics.', 'quant-ph-0102095-2-57-1': '(Cambridge University Press, Cambridge, 1987).', 'quant-ph-0102095-2-58-0': 'b15 K. Imre and E.Ozizmir, M. Rosenbaum ,P.F. Zweifel, J. Math.', 'quant-ph-0102095-2-59-0': 'b18 S.R. de Groot, W.A. van Leeuwen, Ch.', 'quant-ph-0102095-2-59-1': 'G. van Weert, Relativistic kinetic theory.', 'quant-ph-0102095-2-60-0': 'b19 P.R. Holland and A. Kyprianidis, Z. Maric, J.P. Vigier, Phys.', 'quant-ph-0102095-2-61-0': 'b20 P. Gerard, P.A. Markovich, N.J. Mauser and F. Poupaud, Comm.', 'quant-ph-0102095-2-62-0': 'Possible peculiarities of synchrotron radiation in a strong magnetic field.', 'quant-ph-0102095-2-62-1': 'Submitted to Space Science and Technology (Kosmichna Nauka i Tehnologiya), Kiev.', 'quant-ph-0102095-2-62-2': 'This work was presented in VIII Ukrainian Conference on Plasma Physics and Controlled Fusion, Alushta, Crimea, 11 - 17 September, 2000.', 'quant-ph-0102095-2-63-0': 'b25 C. Dewdney, P.R. Holland, A. Kyprianidis, Z. Maric and J.P. Vigier, Phys.', 'quant-ph-0102095-2-64-0': 'b28 D.I. Blokhintsev, On the localization of micro-particles in space and time.', 'quant-ph-0102095-2-65-0': 'b30 A.A. Grib, S.G. Mamaev, B.M. Mostapenko, Vacuum quantum effects in strong fields (Energoatomizdat, Moscow, 1988) [in Russian].'}","[['quant-ph-0102095-1-19-0', 'quant-ph-0102095-2-19-0'], ['quant-ph-0102095-1-19-1', 'quant-ph-0102095-2-19-1'], ['quant-ph-0102095-1-19-4', 'quant-ph-0102095-2-19-4'], ['quant-ph-0102095-1-19-5', 'quant-ph-0102095-2-19-5'], ['quant-ph-0102095-1-19-7', 'quant-ph-0102095-2-19-7'], ['quant-ph-0102095-1-59-0', 'quant-ph-0102095-2-59-0'], ['quant-ph-0102095-1-59-1', 'quant-ph-0102095-2-59-1'], ['quant-ph-0102095-1-14-0', 'quant-ph-0102095-2-14-0'], ['quant-ph-0102095-1-14-1', 'quant-ph-0102095-2-14-1'], ['quant-ph-0102095-1-14-3', 'quant-ph-0102095-2-14-3'], ['quant-ph-0102095-1-57-0', 'quant-ph-0102095-2-57-0'], ['quant-ph-0102095-1-57-1', 'quant-ph-0102095-2-57-1'], ['quant-ph-0102095-1-20-5', 'quant-ph-0102095-2-20-4'], ['quant-ph-0102095-1-25-1', 'quant-ph-0102095-2-25-1'], ['quant-ph-0102095-1-47-0', 'quant-ph-0102095-2-47-0'], ['quant-ph-0102095-1-47-3', 'quant-ph-0102095-2-47-3'], 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1111.1541,"{'1111.1541-1-0-0': 'We discover the origin of the pathologies of the disorder parameter used in previous papers to detect dual superconductivity of [MATH] vacuum, and we remove them by defining an improved disorder parameter.', '1111.1541-1-0-1': 'A check of the approach is made by numerical simulations of [MATH] gauge theory, which demonstrate that the approach is consistent and with it that deconfinement is a transition from dual superconductor to normal.', '1111.1541-1-1-0': '# Introduction', '1111.1541-1-2-0': 'A solid candidate mechanism for color confinement is dual superconductivity of the vacuum [CITATION].', '1111.1541-1-2-1': 'Here dual means interchange of electric and magnetic with respect to ordinary superconductors.', '1111.1541-1-2-2': 'The idea is that the confining vacuum is a condensate of magnetic charges, just as an ordinary superconductor is a condensate of Cooper pairs: the chromoelectric field acting between a quark-antiquark pair is thus constrained into Abrikosov flux tubes, whose energy is proportional to the length, so that the potential raises linearly at large distances, producing confinement.', '1111.1541-1-3-0': 'The role of magnetic charges in [MATH] has been actively investigated by the lattice community during the years.', '1111.1541-1-3-1': 'The most popular approach consisted in detecting monopoles in lattice configurations, and, on the basis of the empirical indications that they could be the dominant degrees of freedom [CITATION], in trying to extract from the observed monopole trajectories an effective lagrangean for them [CITATION].', '1111.1541-1-3-2': 'Monopole condensation should be readable from the parameters of this lagrangean.', '1111.1541-1-3-3': 'This program was not convincingly successful.', '1111.1541-1-3-4': 'Among others there was the intrinsic ambiguity in monopole detection, related to its dependence on the choice of the abelian projection [CITATION].', '1111.1541-1-3-5': 'Monopoles are abelian objects, and live in [MATH] subgroups of the gauge group: the number and the location of the monopoles observed in lattice configurations strongly depend on the choice of the [MATH] subgroup.', '1111.1541-1-3-6': 'This seems to contradict the guess that this choice is physically irrelevant [CITATION].', '1111.1541-1-3-7': 'This issue was recently clarified [CITATION] by showing that monopoles are gauge invariant objects, as they should for physical reasons, but their detection by the recipe of Ref. [CITATION] does indeed depend on the gauge: it was also shown that the so called ""maximal abelian projection"" is the one in which the detected monopoles correspond to the real monopoles.', '1111.1541-1-3-8': 'This legitimates the works performed in the past in the maximal abelian gauge, but maybe the analysis of the effective lagrangean could be reconsidered, keeping in mind that monopoles are expected, in principle, to be the dominant excitations in the vicinity of the deconfining transition, and not everywhere as assumed in these works.', '1111.1541-1-4-0': 'In Ref. [CITATION] it was also shown that the Higgs breaking of the magnetic [MATH] symmetry coupled to monopoles is an abelian projection independent property: if in the confined phase the system is a dual superconductor and in the deconfined phase it is normal this is true in all abelian projections.', '1111.1541-1-5-0': 'An alternative approach to the problem has been to define a disorder parameter, to detect magnetic charge condensation in the ground state [CITATION].', '1111.1541-1-5-1': 'The disorder parameter is the vacuum expectation value [MATH] of a gauge invariant operator [MATH] carrying non-zero magnetic charge: if [MATH] vacuum is a dual superconductor.', '1111.1541-1-5-2': 'In the deconfined phase [MATH].', '1111.1541-1-6-0': 'A prototype version of the operator [MATH] is that for the [MATH] lattice gauge theory [CITATION].', '1111.1541-1-6-1': 'The basic idea is to define the creation of a monopole as a shift by the classical field of a monopole of the field in the Schroedinger representation [EQUATION]', '1111.1541-1-6-2': 'In whatever gauge we quantize [MATH] is the conjugate momentum to the transverse field [MATH].', '1111.1541-1-6-3': '[MATH] is the field at [MATH] produced by a monopole sitting at [MATH], in the transverse gauge [MATH].', '1111.1541-1-6-4': 'As a consequence [EQUATION]', '1111.1541-1-6-5': 'Eq. ([REF]) is nothing but the field theory version of the elementary translation [EQUATION]', '1111.1541-1-6-6': 'A discretized version of the operator [MATH] can be constructed [CITATION], and the order parameter [MATH] with it.', '1111.1541-1-6-7': 'A theorem can be proved [CITATION] that such an order parameter is equal to the one defined and discussed analytically by completely different methods in Ref. [CITATION].', '1111.1541-1-6-8': 'One gets [EQUATION] with [MATH] the partition function for the gauge field action [MATH], [MATH] the partition function of a modified action, in which the time-space plaquettes at the time [MATH], at which the monopole is created, are modified to include the electric field term at the exponent of Eq. ([REF]).', '1111.1541-1-7-0': 'It proves convenient for numerical determinations to deal with the quantity [EQUATION] from which [MATH] is easily determined, being [MATH], [EQUATION]', '1111.1541-1-7-1': 'The numerical determination [CITATION] of [MATH] gave a spectacular demonstration of the theorem proved originally in Ref. [CITATION] for the Villain form of the action, and extended in Ref. [CITATION] to the Wilson form, and at the same time a check that the numerical simulation was correct.', '1111.1541-1-8-0': 'For [MATH], the critical coupling, [MATH] is volume independent and finite at large volumes, implying, by use of Eq. ([REF]), that [MATH] which means dual superconductivity.', '1111.1541-1-8-1': 'For [MATH], [MATH] grows negative linearly with the size [MATH] of the lattice, so that, by Eq. ([REF]) [MATH] in the thermodynamic limit.', '1111.1541-1-8-2': 'For [MATH] a sharp negative peak is observed, corresponding to a rapid decrease to zero of [MATH].', '1111.1541-1-8-3': 'The finite size scaling analysis indicates a first order transition.', '1111.1541-1-9-0': 'Notice that [MATH] is a magnetically charged Dirac-like gauge invariant operator [CITATION], so that its vacuum expectation value can be non zero.', '1111.1541-1-10-0': 'A similar game was played with the 2d Ising model [CITATION] where kinks condense in the disordered phase, with the 3d [MATH] model, where a phase transition exists related to the condensation of vortices [CITATION], and with the 3d Heisenberg model, where the well known de-magnetising transition can be related to the condensation of a kind of Weiss domains [CITATION].', '1111.1541-1-11-0': 'A lattice version of Eq. ([REF]) for non abelian [MATH] gauge theory was then constructed in Ref. [CITATION] for monopoles in specific abelian projections, and in Ref. [CITATION] with a somewhat different formulation.', '1111.1541-1-11-1': 'The construction was also extended to [MATH] gauge theory [CITATION].', '1111.1541-1-11-2': ""With the lattice sizes explored (up to [MATH]) and within the statistics reachable at that time, all the features 1), 2), 3) found in the prototype [MATH] model were reproduced, namely finite, volume independent values for [MATH] at small [MATH]'s, a negative linear increase with the spatial size of the lattice at large [MATH]'s and a negative peak at the deconfining transition as known from the Polyakov line order parameter, with scaling properties consistent with the known order and universality class of the transitions."", '1111.1541-1-11-3': 'The disorder parameter proved also to be independent of the choice of the abelian projection [CITATION] and to work also in presence of quarks [CITATION].', '1111.1541-1-11-4': 'Indications for monopole condensation were also obtained by a completely different technique in Ref. [CITATION].', '1111.1541-1-12-0': 'An attempt was then made to analyze with the same tool the deconfining transition of the gauge theory with gauge group [MATH] [CITATION].', '1111.1541-1-12-1': 'The question was specially important since the group [MATH] has no centre and no central vortices [CITATION], and therefore proving that the observed transition is indeed deconfinement as disappearance of dual superconductivity would be an indication that monopoles, as opposed to vortices, be the relevant degrees of freedom for color confinement.', '1111.1541-1-12-2': ""See also in this connection Ref. 's [CITATION]."", '1111.1541-1-12-3': 'The formal extension of the formalism to deal with monopole condensation in the case of generic gauge groups had been developed in Ref. [CITATION].', '1111.1541-1-13-0': 'To our surprise it was found [CITATION] that the expected behavior of [MATH] was there, but superimposed to a negative background increasing with the spatial size of the lattice at large volumes, in particular at low values of [MATH], so that as a consequence of Eq. ([REF]) the order parameter goes to zero everywhere in the thermodynamic limit, and can not distinguish between confined and deconfined phase.', '1111.1541-1-13-1': ""We then went back to the old data of Ref. 's [CITATION] and made a more careful analysis with much larger statistics and larger lattices, and also for [MATH] we found a similar inconvenience."", '1111.1541-1-13-2': 'Similar results were published in Ref. [CITATION], where it was argued about the very possibility of defining an order parameter for superconductivity.', '1111.1541-1-14-0': 'In this paper we track the origin of the inconvenience and define an improved order parameter which is free of it, and still is the vacuum expectation of an operator carrying non zero magnetic charge.', '1111.1541-1-14-1': 'We check the new order parameter numerically for [MATH] gauge theory and find that it satisfies all of the three features listed above.', '1111.1541-1-14-2': 'In order to understand the low [MATH] background of [MATH] we study its strong coupling expansion.', '1111.1541-1-15-0': 'As for other [MATH]-dimensional systems showing condensation of [MATH]-dimensional topological excitations which had been analyzed in the past with analogous techniques we find, by use of strong coupling expansion, that the [MATH] gauge theory [CITATION] is exempt of the problem, and so are the [MATH]-dimensional Ising model, where kinks condense in the high temperature phase [CITATION], and the [MATH]-dimensional Heisenberg magnet [CITATION].', '1111.1541-1-15-1': 'Instead a non trivial improvement is necessary for the [MATH]-dimensional [MATH] model, where vortices condense in the low temperature phase, and for a generic gauge theory, with and without matter fields, and specifically for [MATH] gauge theory.', '1111.1541-1-16-0': 'In this paper we mainly concentrate on the lattice [MATH] pure gauge theory, but the theoretical analysis is valid for any gauge group.', '1111.1541-1-17-0': 'In Section [REF] we go through the construction of the disorder parameter, we track the origin of the problem and we solve it.', '1111.1541-1-17-1': 'To do that we use the strong coupling expansion at low [MATH].', '1111.1541-1-17-2': 'Some details of the expansion are given in Appendix [REF].', '1111.1541-1-17-3': 'The origin of the problem is somehow related to the difficulties in defining the disorder parameter in presence of electric charges [CITATION].', '1111.1541-1-18-0': 'In Section [REF] we discuss the choice of the classical field configuration which is added to the quantum fluctuations to create a monopole.', '1111.1541-1-18-1': ""As a starting point we use the soliton monopole configuration of Ref. 's[CITATION]."", '1111.1541-1-18-2': 'For the sake of clarity algebraic details are presented separately in Appendix [REF].', '1111.1541-1-19-0': 'In Section [REF] we present numerical results for [MATH] pure gauge theory on the lattice, demonstrating numerically the analysis of Sections [REF] and [REF].', '1111.1541-1-20-0': 'In Section [REF] we present the numerical results of a previous attempt to solve the problem by renormalizing the disorder parameter by its value at zero temperature [CITATION] and show that it is meaningful and consistent with the approach described in Section [REF].', '1111.1541-1-21-0': 'Section [REF] contains the conclusions and the outlook.', '1111.1541-1-22-0': '# The improved disorder parameter.', '1111.1541-1-23-0': 'We start from the [MATH] gauge theory for simplicity, recalling the definitions and the notations of Ref. [CITATION] to make the argumentation self contained.', '1111.1541-1-24-0': 'In the euclidean continuum the disorder parameter Eq. ([REF]) reads [EQUATION] where [MATH], [MATH] is the usual vector potential times the electric charge [MATH], [MATH] is the action of free photons [EQUATION] and [EQUATION]', '1111.1541-1-24-1': 'On the lattice, assuming the usual Wilson form of the action [CITATION], the theory is compactified.', '1111.1541-1-24-2': 'The partition function becomes [EQUATION] and the action [CITATION], [EQUATION] [MATH] is the plaquette, the parallel transport around the elementary loop of the lattice in the plane [MATH], [MATH], [MATH] is the elementary link, [MATH].', '1111.1541-1-24-3': 'The lattice spacing has been put conventionally equal to 1.', '1111.1541-1-24-4': 'In this notation [EQUATION] [MATH] is the discretized version of the field strength tensor.', '1111.1541-1-25-0': 'The modified plaquette [MATH], which defines the action [MATH] in Eq. ([REF]), is different from [MATH] only for electric plaquettes [MATH] at [MATH], the time at which the monopole is created.', '1111.1541-1-25-1': 'In formulae [EQUATION]', '1111.1541-1-25-2': 'For all other sites and/or orientations [EQUATION]', '1111.1541-1-25-3': 'One can define [MATH] by the formula [MATH].', '1111.1541-1-25-4': ""From Eq. 's ([REF]) and ([REF]) it follows [EQUATION]"", '1111.1541-1-25-5': 'By the change of variables [CITATION] [EQUATION] in the Feynman integral [MATH], the jacobian is [MATH] and [EQUATION]', '1111.1541-1-25-6': 'For all the other components and/or sites of the lattice [MATH] .', '1111.1541-1-26-0': 'A monopole has been added at time [MATH], independent of the spatial boundary conditions (BC) used for the quantum fluctuations.', '1111.1541-1-27-0': 'By iterating the change of variables at subsequent times the result is that the effect of the operator is to add a monopole to the configurations at all times larger than [MATH].', '1111.1541-1-27-1': 'In particular, if we use periodic spatial BC all of those configurations will have the magnetic charge of the added monopole.', '1111.1541-1-27-2': 'In terms of states we have [EQUATION]', '1111.1541-1-27-3': 'By [MATH] we denote the state with one monopole, by [MATH] that with a antimonopole added to the vacuum state.', '1111.1541-1-28-0': 'Starting from Eq. ([REF]) we could have performed the change of variables [EQUATION] with the result [EQUATION]', '1111.1541-1-28-1': 'For all the other components and/or sites of the lattice [MATH].', '1111.1541-1-29-0': 'The change of variables can then be iterated with the result that an antimonopole is present at all negative times, or [EQUATION]', '1111.1541-1-29-1': 'By a similar technique we can compute the norm of the state [MATH].', '1111.1541-1-29-2': 'We can add and subtract to [MATH] the field of the classical monopole.', '1111.1541-1-29-3': 'In this case [MATH].', '1111.1541-1-29-4': 'By operating the change of variables Eq. ([REF]) at growing times and the one Eq. ([REF]) at decreasing times with the sign of the external field changed, we simply get [EQUATION]', '1111.1541-1-29-5': 'As a consequence [EQUATION]', '1111.1541-1-29-6': 'Modulo an irrelevant phase [MATH] is the properly normalized probability amplitude from the vacuum to the state with a monopole added, and therefore a correct order parameter.', '1111.1541-1-29-7': 'The discretization has preserved the unitarity of the shift operator Eq. ([REF]).', '1111.1541-1-30-0': 'At large values of [MATH], (weak coupling regime) the quantity [MATH] defined by Eq. ([REF]) can be computed in perturbation theory [CITATION] and is proportional to the lattice size [MATH] with a negative coefficient, so that [MATH] in the thermodynamic limit and the vacuum is normal.', '1111.1541-1-30-1': 'At low values of [MATH] we can compute it by a strong coupling expansion: if the coefficients of the expansion are finite in the infinite volume limit, [MATH] and the system is a dual superconductor, since the strong coupling series is known to have a finite range of convergence in [MATH].', '1111.1541-1-30-2': 'The strong coupling expansion of [MATH] Eq. ([REF]) can be computed with the standard techniques.', '1111.1541-1-30-3': 'Many terms cancel in the subtraction between the two quantities and the first non-zero term is [MATH] and corresponds to six plaquettes containing an elementary cube with an edge parallel to the time axis (see Appendix [REF]).', '1111.1541-1-30-4': 'The result is [EQUATION] where [MATH] is the field strength of the classical field of the monopole, [MATH].', '1111.1541-1-30-5': 'The sum in Eq. ([REF]) can be approximated by an integral.', '1111.1541-1-30-6': 'At large distances [MATH] is small, the cosines can be expanded and [EQUATION] which is convergent since [MATH].', '1111.1541-1-30-7': 'The sum in Eq. ([REF]) is also convergent at small distances.', '1111.1541-1-30-8': 'The argument is expected to be valid at all orders: indeed at all orders [MATH] is expected to be proportional to [MATH] where [MATH] is a combination of components of the field of the monopole [MATH] and since the figures corresponding to the non zero terms of the expansion are closed, there will be as many terms with a positive sign as with a negative sign, making the result at large distances go down as [MATH] and the result finite.', '1111.1541-1-30-9': 'Notice that the result only depends on the magnetic field, which is independent of the position of the Dirac string.', '1111.1541-1-31-0': 'We have presented the case of [MATH] in detail to be ready to do a similar analysis for non abelian gauge groups.', '1111.1541-1-31-1': 'We shall do that for the simplest case [MATH], but everything is general and applies to any group.', '1111.1541-1-31-2': 'As a result we shall understand and correct the pathologies of the existing disorder parameter quoted in Section I.', '1111.1541-1-32-0': 'The definition of the order parameter [MATH] of Ref. [CITATION] for [MATH] is the natural extension to the non-abelian case of the construction for [MATH] presented above.', '1111.1541-1-32-1': 'The original idea is to add the field of an abelian monopole to the magnetic field of the residual [MATH] of a given abelian projection.', '1111.1541-1-32-2': 'Here we shall give a more physical view of the construction, showing in particular that the choice of the abelian projection to start from is completely irrelevant.', '1111.1541-1-33-0': ""Again we have Eq. ([REF]) with [EQUATION] and Eq. 's ([REF])eqrefpp with [EQUATION] and [MATH] for all [MATH] except for the electric plaquettes at [MATH], the time at which the monopole is created, which are given by [EQUATION] [MATH] is the trace of the identity, and [EQUATION] [MATH] is a classical [MATH] monopole field configuration, e.g. the 't Hooft-Polyakov one, which we will choose with a typical length smaller than the lattice spacing so that its asymptotic form is valid."", '1111.1541-1-34-0': 'We now perform a change of variables in the Feynman integral which defines [MATH], analogous to that of Eq. ([REF]), namely [EQUATION]', '1111.1541-1-34-1': 'As a consequence [EQUATION]', '1111.1541-1-34-2': 'In Eq. ([REF]) [EQUATION]', '1111.1541-1-34-3': 'By repeated use of the Campbell-Baker-Haussdorf formula it is easy to show that, up to terms [MATH] [EQUATION]', '1111.1541-1-34-4': 'A monopole has been added at time [MATH].', '1111.1541-1-34-5': 'If we take the field [MATH] in the unitary representation, which coincides with the maximal abelian gauge [CITATION], and periodic boundary conditions for the quantum fluctuations, the magnetic monopole field at large distance will be that of the monopole added, whatever is the abelian projection we started from.', '1111.1541-1-34-6': 'The magnetic field [MATH] is directed along the color direction of the residual [MATH] and adds to the corresponding component of the quantum field.', '1111.1541-1-34-7': 'That component is in any case undefined by terms [MATH] depending on the conventions used to extract it from the configurations, again by the Campbell-Baker-Haussdorf formula.', '1111.1541-1-35-0': 'For the same reason the matrix [MATH] of Eq. ([REF]) has the same [MATH] component as [MATH] up to terms [MATH] since the terms [MATH] cancel between [MATH] and [MATH].', '1111.1541-1-35-1': 'Therefore the change of variables Eq. ([REF]) can be repeated at time [MATH] with the effect of adding a monopole at time [MATH] and so on.', '1111.1541-1-35-2': 'The monopole propagates from time [MATH] on.', '1111.1541-1-35-3': 'The same conclusion follows from the fact that the abelian projected field obeys abelian Bianchi identities [CITATION], so that the magnetic current is conserved [CITATION].', '1111.1541-1-35-4': 'Eq. ([REF]) is still valid.', '1111.1541-1-35-5': 'There is, however, a misuse of language: in the abelian case the monopole propagates unchanged from time [MATH] to [MATH], and is the ""out"" state.', '1111.1541-1-35-6': 'As we have just seen, in the non-abelian case the monopole configuration gets modified in propagating, even if it preserves its quantum numbers, so the configuration is not exactly the ""out"" state.', '1111.1541-1-36-0': 'A few remarks:', '1111.1541-1-37-0': 'All of the equations for the non abelian case also hold for the abelian one where they coincide with those described above as can easily be checked.', '1111.1541-1-38-0': ""In what follows we shall use as classical monopole field [MATH] the 't Hooft -Polyakov soliton solution in the unitary representation: as any magnetic monopole field they decrease as [MATH] at large distances."", '1111.1541-1-38-1': 'The details will be presented in the next section.', '1111.1541-1-39-0': 'As directly visible from the definition Eq. ([REF]) of [MATH], [MATH] and [MATH] are invariant under covariant transformations of [MATH], [MATH] but not under gauge transformations of the classical field.', '1111.1541-1-39-1': 'We then expect different disorder parameters in different gauges, which are however expected to be on the same footing since each of them is the transition amplitude from the ground state to a state with a monopole added.', '1111.1541-1-40-0': 'The disorder parameter of Ref. [CITATION] was [EQUATION]', '1111.1541-1-40-1': 'The correct quantum-mechanical definition of the disorder parameter, i.e. of the probability to find in the vacuum state a state with magnetic charge increased by a monopole is [EQUATION]', '1111.1541-1-40-2': 'In the case of the [MATH] gauge group we showed that [MATH] so that Eq.s ([REF]) and ([REF]) are the same thing and [MATH] as originally defined is the order parameter.', '1111.1541-1-40-3': 'For non abelian gauge groups we will show that this is not the case.', '1111.1541-1-40-4': '[MATH] can be written as [EQUATION] so that [EQUATION] which again coincides with Eq. ([REF]) in the case of [MATH], since there [MATH].', '1111.1541-1-41-0': 'We shall now compute [MATH] for the generic case and show that it is non trivial: the discrete implementation of the operator Eq. ([REF]) proposed in Ref. [CITATION] does not preserve unitarity.', '1111.1541-1-41-1': 'Not only that, but both [MATH] and [MATH] tend to zero in the large volume limit, in such a way that their ratio stays finite and different from zero at small values of [MATH].', '1111.1541-1-42-0': '[MATH] of Eq. ([REF]) is the correct order parameter, which will be computed via [MATH] of Eq. ([REF]) by use of Eq. ([REF]).', '1111.1541-1-43-0': 'We first give the expression for [MATH].', '1111.1541-1-43-1': 'It naturally derives from the definition of scalar product in the language of path integral.', '1111.1541-1-43-2': ""[EQUATION] where [MATH] everywhere except for the mixed space-time plaquettes [MATH] at the time [MATH] at which the monopole is created, where (see Eq. 's ([REF])-([REF])) [EQUATION]"", '1111.1541-1-43-3': 'Indeed by the usual change of variables in the path integral which defines [MATH] it can be shown that with this definition there is a monopole propagating in the future, and one coming from the past.', '1111.1541-1-43-4': '[MATH] and [MATH] in the equation ([REF]) cancel in the case of [MATH] because they commute with the links, everything being abelian, and therefore [MATH] everywhere.', '1111.1541-1-44-0': ""We shall now compute both [MATH] and [MATH] by the strong coupling expansion at low [MATH]'s."", '1111.1541-1-44-1': 'The details of the expansion are sketched Appendix [REF] below.', '1111.1541-1-45-0': 'Like for the case of [MATH] the first non trivial contribution to the strong coupling expansion of [MATH] is [MATH] corresponding graphically to an elementary cube, and is given by [EQUATION] and [EQUATION]', '1111.1541-1-45-1': 'By use of the definition Eq. ([REF]) we get [EQUATION]', '1111.1541-1-45-2': 'Since, as we shall see in Section [REF], in the Wu-Yang parametrization of the monopole field [MATH], it is easily seen, by expanding the cosines at large distances, that, neglecting finite terms, [EQUATION] which is linearly divergent to [MATH] with the spatial size of the lattice.', '1111.1541-1-45-3': ""This is indeed the pathology observed in lattice simulations, and what makes [MATH] at all [MATH]'s in the thermodynamic limit."", '1111.1541-1-46-0': 'If we now play the same game with the new order parameter [MATH] of Eq. ([REF]) we get instead [EQUATION] with [EQUATION]', '1111.1541-1-46-1': 'In the strong coupling again the first non trivial term corresponds to the elementary cube and gives [EQUATION] where the coefficient of proportionality is the same as in Eq. ([REF]) and [EQUATION]', '1111.1541-1-46-2': 'In computing [MATH] by use of Eq. ([REF]) the leading terms at large distances cancel and the final result is finite in the thermodynamic limit.', '1111.1541-1-47-0': 'In formulae [EQUATION]', '1111.1541-1-47-1': 'The state [MATH] has zero norm in the infinite volume limit, but its direction in Hilbert space is well defined, at least in the confined phase.', '1111.1541-1-48-0': 'This was the leading term in the strong-coupling expansion.', '1111.1541-1-48-1': 'The exact result is obtained either by studying systematically higher order terms in the strong coupling expansion, or numerically by lattice simulations.', '1111.1541-1-48-2': 'This we will do in the next section.', '1111.1541-1-48-3': 'Notice that the coefficient of the first term in the strong coupling expansion depends on the vector potential and not on the magnetic field as was the case for [MATH] gauge group.', '1111.1541-1-48-4': 'This fact is related to the remark 3) above, that the definition of [MATH] is not invariant under gauge transformations of the classical field of the monopole.', '1111.1541-1-49-0': '# The monopole configuration in non abelian gauge theories', '1111.1541-1-50-0': 'In the [MATH] gauge theory on the lattice the classical monopole configuration by which the field is shifted by the monopole creation operator has a Dirac string singularity: [MATH] is fully gauge invariant both with respect to regular gauge transformations and with respect to singular gauge transformations like those which displace the Dirac string [CITATION].', '1111.1541-1-50-1': 'There are, however, problems to satisfy in a consistent way the Dirac relation between electric and magnetic charge when electric charges are present [CITATION]: this is the case for example in non abelian gauge theories where electrically charged fields are necessarily present, at least the charged partners of the [MATH] gluon.', '1111.1541-1-51-0': 'In the non abelian case we shall view the monopole as an [MATH] configuration.', '1111.1541-1-52-0': ""Suppose we want to create a 't Hooft-Polyakov monopole in the unitary representation from the configuration with no gluons."", '1111.1541-1-53-0': 'For our purposes we may consider only the large distance form of the configuration.', '1111.1541-1-53-1': 'The notation is standard (see e.g. [CITATION]).', '1111.1541-1-54-0': 'The monopole configuration in the Wu-Yang form with discontinuity say at a polar angle [MATH] is obtained from the hedgehog gauge configuration (in polar coordinates) [EQUATION] by operating the gauge transformation [EQUATION] with [EQUATION] and [EQUATION] [MATH] is the usual Heaviside function.', '1111.1541-1-54-1': 'The requirement in the definition of [MATH] is that [MATH] is [MATH] and [MATH] independent and that [MATH].', '1111.1541-1-54-2': '[MATH] can be e.g. [MATH], or [MATH] or any fixed combination thereof.', '1111.1541-1-54-3': 'The field in the new gauge is computed by use of the formula [EQUATION]', '1111.1541-1-54-4': 'The details of the computation are given in Appendix [REF].', '1111.1541-1-54-5': 'The result is [EQUATION]', '1111.1541-1-54-6': 'The non abelian magnetic field is radial [EQUATION]', '1111.1541-1-54-7': 'As shown in Appendix [REF] the singularities cancel and the field is simply the coulombic field [MATH].', '1111.1541-1-54-8': 'Moreover the component [MATH] of the field in the Wu-Yang unitary gauge is continuous inside [MATH] as shown in Eq. ([REF]).', '1111.1541-1-54-9': 'It is instead discontinuous if viewed as an [MATH] configuration.', '1111.1541-1-54-10': 'In the limit [MATH] the singularity becomes the usual Dirac string [CITATION].', '1111.1541-1-54-11': 'To create a monopole we should in principle create, together with the abelian diagonal part of the field, a singular charged gluon field [MATH] which cancels the singularities.', '1111.1541-1-54-12': 'For the matrix element of the monopole operator which we are considering (creation from a zero gluon configuration) this procedure amounts to keep only the diagonal part and neglect its singularities (Dirac string or Wu-Yang transfer matrix).', '1111.1541-1-54-13': 'This is the recipe that will be implemented on the lattice (Section [REF]).', '1111.1541-1-55-0': 'If we act with the monopole operator on a generic configuration with spatial periodic boundary conditions, i.e. with net magnetic charge zero, the magnetic field at large distances will be the one that we add, and we are therefore automatically in the maximal abelian or unitary gauge [CITATION].', '1111.1541-1-55-1': 'The singularities cancel anyhow in the magnetic field, except possibly for the bilinear term in Eq. ([REF]) with the singular classical field [MATH] Eq. ([REF]) and the quantum fluctuation of [MATH], which should however average to zero.', '1111.1541-1-55-2': 'Again the recipe is to keep only [MATH] at [MATH].', '1111.1541-1-56-0': 'How this recipe extends to more complicated matrix elements or correlation functions is an open question which deserves further study.', '1111.1541-1-56-1': 'The emerging indication is that, contrary to a generic theory containing charged fields, in a gauge theory the problem raised in Ref. [CITATION] is structurally solved.', '1111.1541-1-56-2': 'Monopole configurations are non singular: what produces the singularity is the abelian projection.', '1111.1541-1-56-3': 'In the language of Ref. [CITATION] monopoles in a non-abelian gauge theory are violations on non-abelian Bianchi identities, which produce violations of abelian Bianchi identities only after abelian projection (Sect. 2 of Ref. [CITATION]).', '1111.1541-1-57-0': '# Numerical results', '1111.1541-1-58-0': 'In this section we present the numerical results obtained by simulating the [MATH] lattice gauge theory.', '1111.1541-1-58-1': 'Since both the actions [MATH] and [MATH] are linear in each of the link, the usual combination of heatbath [CITATION] and overrelaxation [CITATION] can be used for the update of the configuration, even when the monopole operator is present.', '1111.1541-1-58-2': 'Numerical simulations have been performed on GRID resources provided by INFN.', '1111.1541-1-59-0': 'Our aim is to show that the behaviour of the improved operator [MATH] in the [MATH] case is similar to the one observed in the [MATH] lattice gauge theory, namely we want to verify that [MATH]', '1111.1541-1-60-0': 'As previously discussed these are the properties of [MATH] which allow to interpret confinement as dual superconductivity, as they guarantee that [MATH] in the low temperature phase and [MATH] in the high temperature one.', '1111.1541-1-61-0': 'We will first of all study the behaviour of [MATH] and [MATH] for small values of [MATH] (i.e. [MATH]) for lattices with temporal extent [MATH] (for which [MATH], see e.g. [CITATION]).', '1111.1541-1-61-1': 'The numerical results for [MATH] are shown in Fig. ([REF]) and the linear divergence of [MATH] with the lattice size (as predicted by Eq. ([REF])) is clearly seen.', '1111.1541-1-61-2': 'Notice that with lower statistics and smaller lattices the points in the insert of Fig. ([REF]) would be undistinguishable within errors as was in Ref. [CITATION].', '1111.1541-1-62-0': 'This infrared divergence is instead absent in the [MATH] data, which are shown in Fig. ([REF]) and smoothly approach their thermodynamical values.', '1111.1541-1-62-1': 'Clearly this approach is slower for the data at larger [MATH] shown in the figure, which are closer to the deconfinement transition where the correlation length is larger.', '1111.1541-1-62-2': 'Note that, although the absolute values of the error bars for [MATH] and [MATH] are similar, the relative errors for [MATH] are much bigger than those for [MATH] since the value of [MATH] is about an order of magnitude smaller than [MATH].', '1111.1541-1-63-0': 'The data showing the behaviour of [MATH] in the neighbourhood of the [MATH] deconfinement transition are displayed in Fig. ([REF]) and the development of the negative peak at the transition is clearly visible.', '1111.1541-1-63-1': 'The deconfinement transition for [MATH] dimensional [MATH] lattice gauge theory is known to belong to the universality class of the 3d Ising model and from a simple scaling ansatz for [MATH] (see e.g. [CITATION]) the finite size scaling relation for the singular part of [MATH] follows [EQUATION] where [MATH] is a scaling function.', '1111.1541-1-63-2': 'From this scaling form it follows that by increasing the lattice size the peak height should increase as [MATH], while its width should shrink as [MATH].', '1111.1541-1-63-3': 'This behaviour is qualitatively visible in Fig. ([REF]) and will be now verified also quantitatively.', '1111.1541-1-64-0': 'In order to estimate the analytical background of [MATH], a fit of the form [MATH] was performed on the peak values of [MATH] and the constant background term [MATH] was used.', '1111.1541-1-64-1': 'The quality of the scaling in Eq. ([REF]) is shown in Fig. ([REF]), where the known values [MATH] and [MATH] where used for the deconfinement coupling and the 3d Ising critical index.', '1111.1541-1-64-2': 'We explicitly notice that scaling is to be expected only for [MATH], since for [MATH] should not be well defined in the thermodynamical limit and it is expected to diverge to [MATH] as will be directly tested below.', '1111.1541-1-65-0': 'To verify that the position of the peak scales correctly with the lattice spacing we simulated also lattices with different temporal extent.', '1111.1541-1-65-1': 'In Fig. ([REF]) data are shown corresponding to a lattice with [MATH] and again the negative peak develops in correspondence of the deconfinement transition, which now takes place at [MATH] (see e.g. [CITATION]).', '1111.1541-1-66-0': 'The last point that remains to be checked is the behaviour of [MATH] at temperatures higher than the deconfinement temperature.', '1111.1541-1-66-1': 'In order to have [MATH] for [MATH], [MATH] must diverge to [MATH] in the thermodynamical limit for every [MATH].', '1111.1541-1-66-2': 'This is indeed what happens, as shown in Fig. ([REF]), where two different scaling are seen: at fixed lattice size [MATH] is asymptotically linear in [MATH] while at fixed [MATH] value [MATH] is asymptotically linear in the lattice size [MATH].', '1111.1541-1-67-0': 'We have thus verified numerically that the improved monopole operator [MATH] is a well defined operator to check dual superconductivity in lattice simulations.', '1111.1541-1-67-1': 'We have thus shown that the deconfinement transition in [MATH] lattice gauge theory can be interpreted as a monopole condensation transition.', '1111.1541-1-68-0': '# Subtraction at zero temperature.', '1111.1541-1-69-0': 'For the sake of completeness we also report some numerical results of an attempt we made prior to the approach described in this paper on the way to understand the diseases of our disorder parameter [CITATION].', '1111.1541-1-69-1': 'The basic idea was that the sort of infrared divergence appearing in [MATH] would look as a short distance effect in the dual description and produce a multiplicative renormalization of [MATH].', '1111.1541-1-69-2': 'In the ratio [MATH] the constant would cancel and the deconfining transition become visible.', '1111.1541-1-69-3': 'The analogue of [MATH] becomes now [MATH].', '1111.1541-1-69-4': 'The results are shown in Figs. ([REF]) and ([REF]).', '1111.1541-1-69-5': 'We report these results because they prove to be meaningful.', '1111.1541-1-70-0': 'It is not easy to simulate the system at [MATH] and large [MATH], since the physical length grows exponentially with [MATH] and the lattice size should be increased accordingly.', '1111.1541-1-70-1': 'In the practically accessible range of values of [MATH] the results are satisfactory: the phase transition is clearly visible and the appropriate scaling is observed.', '1111.1541-1-70-2': 'In terms of the new disorder parameter this means that [MATH] is rather smooth in [MATH] and temperature independent below and around the deconfinement peak.', '1111.1541-1-70-3': ""This can indeed be directly seen in Fig. ([REF]) where the term [MATH] of Eq. ([REF]), which determines the normalization Eq. ([REF]), is shown to be [MATH] or temperature independent in a range of [MATH]'s."", '1111.1541-1-71-0': 'The behavior of [MATH] at zero temperature is also shown in Fig. ([REF]) to be volume independent and finite, meaning that there is confinement.', '1111.1541-1-72-0': '# Conclusions and outlook', '1111.1541-1-73-0': 'Understanding the mechanism of confinement of color in [MATH] is a fundamental problem in particle physics.', '1111.1541-1-73-1': 'Dual superconductivity of the confining vacuum is an appealing candidate.', '1111.1541-1-74-0': 'A disorder parameter had been introduced to detect condensation of magnetic charges, in order to demonstrate by numerical simulations that deconfinement is a transition from a dual superconductor to a normal state.', '1111.1541-1-74-1': 'The starting point was the [MATH] lattice gauge theory which has a deconfining transition well understood in terms of dual superconductivity.', '1111.1541-1-74-2': 'The generalization of the [MATH] disorder parameter to the non-abelian case proved to be affected by a pathology: in the thermodynamic limit the parameter as was constructed tends to zero both in the deconfined and in the confined phase, and is thus unable to detect superconductivity.', '1111.1541-1-75-0': 'In this paper we trace the origin of this disease and we cure it.', '1111.1541-1-75-1': 'In synthesis the original disorder parameter was not a properly normalized probability amplitude, and had limit zero at increasing volumes because the norm of the state obtained from the vacuum by addition of a monopole tends to zero.', '1111.1541-1-75-2': 'The direction of the state in the Hilbert space has instead a well defined limit, and with it the probability amplitude in the confined phase.', '1111.1541-1-75-3': 'This has been shown both by strong coupling expansion and by numerical simulations.', '1111.1541-1-76-0': 'The results support dual superconductivity as mechanism of color confinement.', '1111.1541-1-77-0': 'The disorder parameter is by construction gauge invariant if the classical field of the monopole transforms covariantly.', '1111.1541-1-77-1': 'In other words it is not invariant under gauge transformations of the classical external field.', '1111.1541-1-77-2': 'This is not a difficulty in principle for the demonstration of monopole condensation, but is unpleasant.', '1111.1541-1-78-0': 'We are trying to use alternative definitions, as could be the technique of the Schroedinger functional, which was already used in Ref. [CITATION] which could be more satisfactory.', '1111.1541-1-79-0': '# aknowledgments', '1111.1541-1-80-0': 'One of us (A. D. G.) thanks G. Paffuti for useful discussions.', '1111.1541-1-80-1': 'The numerical simulations have been performed on GRID resources provided by INFN and in particular on the CSN4 Cluster located in Pisa.', '1111.1541-1-81-0': '# Strong coupling expansion', '1111.1541-1-82-0': 'In this section we sketch the strong coupling expansion of [MATH] and [MATH] of Sec. [REF].', '1111.1541-1-82-1': 'The strong coupling expansion is a power series expansion in [MATH] (see e.g. [CITATION]).', '1111.1541-1-82-2': 'More refined methods, such as the character expansion (see [CITATION]), cannot be used here because of the monopole insertion in [MATH].', '1111.1541-1-83-0': 'For the [MATH] case we want to compute the expansion of [EQUATION] where [MATH] and [MATH] are defined in Eqs. ([REF])-([REF]).', '1111.1541-1-83-1': 'The fundamental formulae to be used are [EQUATION]', '1111.1541-1-83-2': 'It has to be noted that in the expression ([REF]) most of the terms cacels between [MATH] and [MATH].', '1111.1541-1-83-3': 'For example the lowest non-trivial contribution to both [MATH] and [MATH] are [MATH] and can be graphically represented by two superimposed plaquettes (if the two plaquettes are not superimposed the result is zero because of the first of Eqs. ([REF])).', '1111.1541-1-83-4': 'However these two contributions to [MATH] and [MATH] are equal and cancel in Eq. ([REF]).', '1111.1541-1-84-0': 'The first non-trivial contribution to [MATH] turns out to be [MATH] and is graphically represented by the six plaquettes that enclose a three-dimensional cube.', '1111.1541-1-84-1': 'If the cube has no edges along the temporal direction and/or if it is not positioned at the time at which the monopole is created, it simplifies in the difference in Eq. ([REF]).', '1111.1541-1-85-0': 'Let us denote by [MATH] the contribution to the mean value [MATH] associated to a cube sitting at [MATH] with edges along the directions [MATH] is the temporal direction).', '1111.1541-1-85-1': 'By applying 12 times (the number of the edges of the cube) the second formula of Eqs. ([REF]) we get [EQUATION] from which Eq. ([REF]) follows.', '1111.1541-1-86-0': 'In the non-abelian case the integration rules Eqs. ([REF]) become [EQUATION] and the computations goes along the same line.', '1111.1541-1-86-1': 'Also in the non-abelian case the first non-trivial contribution to [MATH] is [MATH] and correspond to the cube graph of in Fig. ([REF]).', '1111.1541-1-87-0': 'By using the second relation of Eqs. ([REF]) on the twelve edges of the cube we obtain the expressions in Eq. ([REF]) and Eq. ([REF]).', '1111.1541-1-88-0': '# Details of the gauge transformation of Section [REF].', '1111.1541-1-89-0': 'In our notation [MATH] are the three generators along the axes of the polar coordinates.', '1111.1541-1-89-1': 'In terms of the generators along the cartesian axes, [MATH],they are given by [EQUATION]', '1111.1541-1-89-2': 'By use of Eq. ([REF]) the covariant terms in Eq. ([REF]) are easily computed giving [EQUATION]', '1111.1541-1-89-3': 'The affine terms [MATH] are [EQUATION]', '1111.1541-1-89-4': ""Using the fact that [MATH], the result for [MATH] is that of Eq. 's ([REF])eqrefAteta([REF])."", '1111.1541-1-90-0': 'We now compute the magnetic field by Eq. [REF].', '1111.1541-1-90-1': 'The three terms give separately [EQUATION]', '1111.1541-1-90-2': 'The term [MATH] is continuous through [MATH] and gives the coulombic field of the monopole.', '1111.1541-1-90-3': 'Finally [EQUATION]', '1111.1541-1-90-4': 'Since [MATH], the second term in Eq. ([REF]) gives [EQUATION]', '1111.1541-1-90-5': 'Finally the bilinear term gives [EQUATION]', '1111.1541-1-90-6': 'Since [MATH] and [EQUATION] we get [EQUATION]', '1111.1541-1-90-7': 'All the singularities cancel in the sum and only the coulombic term is left.', '1111.1541-1-91-0': 'Notice that, since [MATH] is orthogonal to [MATH] [EQUATION]', '1111.1541-1-91-1': 'The discontinuity in [MATH] vanishes, and so does the second term of [MATH] which should cancel it.'}","{'1111.1541-2-0-0': 'We discover the origin of the pathologies of the disorder parameter used in previous papers to detect dual superconductivity of [MATH] vacuum, and we remove them by defining an improved disorder parameter.', '1111.1541-2-0-1': 'A check of the approach is made by numerical simulations of [MATH] gauge theory, which demonstrate that the approach is consistent and with it that deconfinement is a transition from dual superconductor to normal.', '1111.1541-2-1-0': '# Introduction', '1111.1541-2-2-0': 'A solid candidate mechanism for color confinement is dual superconductivity of the vacuum [CITATION].', '1111.1541-2-2-1': 'Here dual means interchange of electric and magnetic with respect to ordinary superconductors.', '1111.1541-2-2-2': 'The idea is that the confining vacuum is a condensate of magnetic charges, just as an ordinary superconductor is a condensate of Cooper pairs: the chromoelectric field acting between a quark-antiquark pair is thus constrained into Abrikosov flux tubes, whose energy is proportional to the length, so that the potential raises linearly at large distances, producing confinement.', '1111.1541-2-3-0': 'The role of magnetic charges in [MATH] has been actively investigated by the lattice community during the years.', '1111.1541-2-3-1': 'The most popular approach consisted in detecting monopoles in lattice configurations, and, on the basis of the empirical indications that they could be the dominant degrees of freedom [CITATION], in trying to extract from the observed monopole trajectories an effective lagrangean for them [CITATION].', '1111.1541-2-3-2': 'Monopole condensation should be readable from the parameters of this lagrangean.', '1111.1541-2-3-3': 'This program was not convincingly successful.', '1111.1541-2-3-4': 'Among others there was the intrinsic ambiguity in monopole detection, related to its dependence on the choice of the abelian projection [CITATION].', '1111.1541-2-3-5': 'Monopoles are abelian objects, and live in [MATH] subgroups of the gauge group: the number and the location of the monopoles observed in lattice configurations strongly depend on the choice of the [MATH] subgroup.', '1111.1541-2-3-6': 'This seems to contradict the guess that this choice is physically irrelevant [CITATION].', '1111.1541-2-3-7': 'This issue was recently clarified [CITATION] by showing that monopoles are gauge invariant objects, as they should for physical reasons, but their detection by the recipe of Ref. [CITATION] does indeed depend on the gauge: it was also shown that the so called ""maximal abelian projection"" is the one in which the detected monopoles correspond to the real monopoles.', '1111.1541-2-3-8': 'This legitimates the works performed in the past in the maximal abelian gauge, but maybe the analysis of the effective lagrangean could be reconsidered, keeping in mind that monopoles are expected, in principle, to be the dominant excitations in the vicinity of the deconfining transition, and not everywhere as assumed in these works.', '1111.1541-2-4-0': 'In Ref. [CITATION] it was also shown that the Higgs breaking of the magnetic [MATH] symmetry coupled to monopoles is an abelian projection independent property: if in the confined phase the system is a dual superconductor and in the deconfined phase it is normal this is true in all abelian projections.', '1111.1541-2-5-0': 'An alternative approach to the problem has been to define a disorder parameter, to detect magnetic charge condensation in the ground state [CITATION].', '1111.1541-2-5-1': 'The disorder parameter is the vacuum expectation value [MATH] of a gauge invariant operator [MATH] carrying non-zero magnetic charge: if [MATH] vacuum is a dual superconductor.', '1111.1541-2-5-2': 'In the deconfined phase [MATH].', '1111.1541-2-6-0': 'A prototype version of the operator [MATH] is that for the [MATH] lattice gauge theory [CITATION].', '1111.1541-2-6-1': 'The basic idea is to define the creation of a monopole as a shift by the classical field of a monopole of the field in the Schroedinger representation [EQUATION]', '1111.1541-2-6-2': 'In whatever gauge we quantize [MATH] is the conjugate momentum to the transverse field [MATH].', '1111.1541-2-6-3': '[MATH] is the field at [MATH] produced by a monopole sitting at [MATH], in the transverse gauge [MATH].', '1111.1541-2-6-4': 'As a consequence [EQUATION]', '1111.1541-2-6-5': 'Eq. ([REF]) is nothing but the field theory version of the elementary translation [EQUATION]', '1111.1541-2-6-6': 'A discretized version of the operator [MATH] can be constructed [CITATION], and the order parameter [MATH] with it.', '1111.1541-2-6-7': 'A theorem can be proved [CITATION] that such an order parameter is equal to the one defined and discussed analytically by completely different methods in Ref. [CITATION].', '1111.1541-2-6-8': 'One gets [EQUATION] with [MATH] the partition function for the gauge field action [MATH], [MATH] the partition function of a modified action, in which the time-space plaquettes at the time [MATH], at which the monopole is created, are modified to include the electric field term at the exponent of Eq. ([REF]).', '1111.1541-2-7-0': 'It proves convenient for numerical determinations to deal with the quantity [EQUATION] from which [MATH] is easily determined, being [MATH], [EQUATION]', '1111.1541-2-7-1': 'The numerical determination [CITATION] of [MATH] gave a spectacular demonstration of the theorem proved originally in Ref. [CITATION] for the Villain form of the action, and extended in Ref. [CITATION] to the Wilson form, and at the same time a check that the numerical simulation was correct.', '1111.1541-2-8-0': 'For [MATH], the critical coupling, [MATH] is volume independent and finite at large volumes, implying, by use of Eq. ([REF]), that [MATH] which means dual superconductivity.', '1111.1541-2-8-1': 'For [MATH], [MATH] grows negative linearly with the size [MATH] of the lattice, so that, by Eq. ([REF]) [MATH] in the thermodynamic limit.', '1111.1541-2-8-2': 'For [MATH] a sharp negative peak is observed, corresponding to a rapid decrease to zero of [MATH].', '1111.1541-2-8-3': 'The finite size scaling analysis indicates a first order transition.', '1111.1541-2-9-0': 'Notice that [MATH] is a magnetically charged Dirac-like gauge invariant operator [CITATION], so that its vacuum expectation value can be non zero.', '1111.1541-2-10-0': 'A similar game was played with the 2d Ising model [CITATION] where kinks condense in the disordered phase, with the 3d [MATH] model, where a phase transition exists related to the condensation of vortices [CITATION], and with the 3d Heisenberg model, where the well known de-magnetising transition can be related to the condensation of a kind of Weiss domains [CITATION].', '1111.1541-2-11-0': 'A lattice version of Eq. ([REF]) for non abelian [MATH] gauge theory was then constructed in Ref. [CITATION] for monopoles in specific abelian projections, and in Ref. [CITATION] with a somewhat different formulation.', '1111.1541-2-11-1': 'The construction was also extended to [MATH] gauge theory [CITATION].', '1111.1541-2-11-2': ""With the lattice sizes explored (up to [MATH]) and within the statistics reachable at that time, all the features 1), 2), 3) found in the prototype [MATH] model were reproduced, namely finite, volume independent values for [MATH] at small [MATH]'s, a negative linear increase with the spatial size of the lattice at large [MATH]'s and a negative peak at the deconfining transition as known from the Polyakov line order parameter, with scaling properties consistent with the known order and universality class of the transitions."", '1111.1541-2-11-3': 'The disorder parameter proved also to be independent of the choice of the abelian projection [CITATION] and to work also in presence of quarks [CITATION].', '1111.1541-2-11-4': 'Indications for monopole condensation were also obtained by a completely different technique in Ref. [CITATION].', '1111.1541-2-12-0': 'An attempt was then made to analyze with the same tool the deconfining transition of the gauge theory with gauge group [MATH] [CITATION].', '1111.1541-2-12-1': 'The question was specially important since the group [MATH] has no centre and no central vortices [CITATION], and therefore proving that the observed transition is indeed deconfinement as disappearance of dual superconductivity would be an indication that monopoles, as opposed to vortices, be the relevant degrees of freedom for color confinement.', '1111.1541-2-12-2': ""See also in this connection Ref. 's [CITATION]."", '1111.1541-2-12-3': 'The formal extension of the formalism to deal with monopole condensation in the case of generic gauge groups had been developed in Ref. [CITATION].', '1111.1541-2-13-0': 'To our surprise it was found [CITATION] that the expected behavior of [MATH] was there, but superimposed to a negative background increasing with the spatial size of the lattice at large volumes, in particular at low values of [MATH], so that as a consequence of Eq. ([REF]) the order parameter goes to zero everywhere in the thermodynamic limit, and can not distinguish between confined and deconfined phase.', '1111.1541-2-13-1': ""We then went back to the old data of Ref. 's [CITATION] and made a more careful analysis with much larger statistics and larger lattices, and also for [MATH] we found a similar inconvenience."", '1111.1541-2-13-2': 'Similar results were published in Ref. [CITATION], where it was argued about the very possibility of defining an order parameter for superconductivity.', '1111.1541-2-14-0': 'In this paper we track the origin of the inconvenience and define an improved order parameter which is free of it, and still is the vacuum expectation of an operator carrying non zero magnetic charge.', '1111.1541-2-14-1': 'We check the new order parameter numerically for [MATH] gauge theory and find that it satisfies all of the three features listed above.', '1111.1541-2-14-2': 'In order to understand the low [MATH] background of [MATH] we study its strong coupling expansion.', '1111.1541-2-15-0': 'As for other [MATH]-dimensional systems showing condensation of [MATH]-dimensional topological excitations which had been analyzed in the past with analogous techniques we find, by use of strong coupling expansion, that the [MATH] gauge theory [CITATION] is exempt of the problem, and so are the [MATH]-dimensional Ising model, where kinks condense in the high temperature phase [CITATION], and the [MATH]-dimensional Heisenberg magnet [CITATION].', '1111.1541-2-15-1': 'Instead a non trivial improvement is necessary for the [MATH]-dimensional [MATH] model, where vortices condense in the low temperature phase, and for a generic gauge theory, with and without matter fields, and specifically for [MATH] gauge theory.', '1111.1541-2-16-0': 'In this paper we mainly concentrate on the lattice [MATH] pure gauge theory, but the theoretical analysis is valid for any gauge group.', '1111.1541-2-17-0': 'In Section [REF] we go through the construction of the disorder parameter, we track the origin of the problem and we solve it.', '1111.1541-2-17-1': 'To do that we use the strong coupling expansion at low [MATH].', '1111.1541-2-17-2': 'Some details of the expansion are given in Appendix [REF].', '1111.1541-2-17-3': 'The origin of the problem is somehow related to the difficulties in defining the disorder parameter in presence of electric charges [CITATION].', '1111.1541-2-18-0': 'In Section [REF] we discuss the choice of the classical field configuration which is added to the quantum fluctuations to create a monopole.', '1111.1541-2-18-1': ""As a starting point we use the soliton monopole configuration of Ref. 's[CITATION]."", '1111.1541-2-18-2': 'For the sake of clarity algebraic details are presented separately in Appendix [REF].', '1111.1541-2-19-0': 'In Section [REF] we present numerical results for [MATH] pure gauge theory on the lattice, demonstrating numerically the analysis of Sections [REF] and [REF].', '1111.1541-2-20-0': 'In Section [REF] we present the numerical results of a previous attempt to solve the problem by renormalizing the disorder parameter by its value at zero temperature [CITATION] and show that it is meaningful and consistent with the approach described in Section [REF].', '1111.1541-2-21-0': 'Section [REF] contains the conclusions and the outlook.', '1111.1541-2-22-0': '# The improved disorder parameter.', '1111.1541-2-23-0': 'We start from the [MATH] gauge theory for simplicity, recalling the definitions and the notations of Ref. [CITATION] to make the argumentation self contained.', '1111.1541-2-24-0': 'In the euclidean continuum the disorder parameter Eq. ([REF]) reads [EQUATION] where [MATH], [MATH] is the usual vector potential times the electric charge [MATH], [MATH] is the action of free photons [EQUATION] and [EQUATION]', '1111.1541-2-24-1': 'On the lattice, assuming the usual Wilson form of the action [CITATION], the theory is compactified.', '1111.1541-2-24-2': 'The partition function becomes [EQUATION] and the action [CITATION], [EQUATION] [MATH] is the plaquette, the parallel transport around the elementary loop of the lattice in the plane [MATH], [MATH], [MATH] is the elementary link, [MATH].', '1111.1541-2-24-3': 'The lattice spacing has been put conventionally equal to 1.', '1111.1541-2-24-4': 'In this notation [EQUATION] [MATH] is the discretized version of the field strength tensor.', '1111.1541-2-25-0': 'The modified plaquette [MATH], which defines the action [MATH] in Eq. ([REF]), is different from [MATH] only for electric plaquettes [MATH] at [MATH], the time at which the monopole is created.', '1111.1541-2-25-1': 'In formulae [EQUATION]', '1111.1541-2-25-2': 'For all other sites and/or orientations [EQUATION]', '1111.1541-2-25-3': 'One can define [MATH] by the formula [MATH].', '1111.1541-2-25-4': ""From Eq. 's ([REF]) and ([REF]) it follows [EQUATION]"", '1111.1541-2-25-5': 'By the change of variables [CITATION] [EQUATION] in the Feynman integral [MATH], the jacobian is [MATH] and [EQUATION]', '1111.1541-2-25-6': 'For all the other components and/or sites of the lattice [MATH] .', '1111.1541-2-26-0': 'A monopole has been added at time [MATH], independent of the spatial boundary conditions (BC) used for the quantum fluctuations.', '1111.1541-2-27-0': 'By iterating the change of variables at subsequent times the result is that the effect of the operator is to add a monopole to the configurations at all times larger than [MATH].', '1111.1541-2-27-1': 'In particular, if we use periodic spatial BC all of those configurations will have the magnetic charge of the added monopole.', '1111.1541-2-27-2': 'In terms of states we have [EQUATION]', '1111.1541-2-27-3': 'By [MATH] we denote the state with one monopole, by [MATH] that with a antimonopole added to the vacuum state.', '1111.1541-2-28-0': 'Starting from Eq. ([REF]) we could have performed the change of variables [EQUATION] with the result [EQUATION]', '1111.1541-2-28-1': 'For all the other components and/or sites of the lattice [MATH].', '1111.1541-2-29-0': 'The change of variables can then be iterated with the result that an antimonopole is present at all negative times, or [EQUATION]', '1111.1541-2-29-1': 'By a similar technique we can compute the norm of the state [MATH].', '1111.1541-2-29-2': 'We can add and subtract to [MATH] the field of the classical monopole.', '1111.1541-2-29-3': 'In this case [MATH].', '1111.1541-2-29-4': 'By operating the change of variables Eq. ([REF]) at growing times and the one Eq. ([REF]) at decreasing times with the sign of the external field changed, we simply get [EQUATION]', '1111.1541-2-29-5': 'As a consequence [EQUATION]', '1111.1541-2-29-6': 'Modulo an irrelevant phase [MATH] is the properly normalized probability amplitude from the vacuum to the state with a monopole added, and therefore a correct order parameter.', '1111.1541-2-29-7': 'The discretization has preserved the unitarity of the shift operator Eq. ([REF]).', '1111.1541-2-30-0': 'At large values of [MATH], (weak coupling regime) the quantity [MATH] defined by Eq. ([REF]) can be computed in perturbation theory [CITATION] and is proportional to the lattice size [MATH] with a negative coefficient, so that [MATH] in the thermodynamic limit and the vacuum is normal.', '1111.1541-2-30-1': 'At low values of [MATH] we can compute it by a strong coupling expansion: if the coefficients of the expansion are finite in the infinite volume limit, [MATH] and the system is a dual superconductor, since the strong coupling series is known to have a finite range of convergence in [MATH].', '1111.1541-2-30-2': 'The strong coupling expansion of [MATH] Eq. ([REF]) can be computed with the standard techniques.', '1111.1541-2-30-3': 'Many terms cancel in the subtraction between the two quantities and the first non-zero term is [MATH] and corresponds to six plaquettes containing an elementary cube with an edge parallel to the time axis (see Appendix [REF]).', '1111.1541-2-30-4': 'The result is [EQUATION] where [MATH] is the field strength of the classical field of the monopole, [MATH].', '1111.1541-2-30-5': 'The sum in Eq. ([REF]) can be approximated by an integral.', '1111.1541-2-30-6': 'At large distances [MATH] is small, the cosines can be expanded and [EQUATION] which is convergent since [MATH].', '1111.1541-2-30-7': 'The sum in Eq. ([REF]) is also convergent at small distances.', '1111.1541-2-30-8': 'The argument is expected to be valid at all orders: indeed at all orders [MATH] is expected to be proportional to [MATH] where [MATH] is a combination of components of the field of the monopole [MATH] and since the figures corresponding to the non zero terms of the expansion are closed, there will be as many terms with a positive sign as with a negative sign, making the result at large distances go down as [MATH] and the result finite.', '1111.1541-2-30-9': 'Notice that the result only depends on the magnetic field, which is independent of the position of the Dirac string.', '1111.1541-2-31-0': 'We have presented the case of [MATH] in detail to be ready to do a similar analysis for non abelian gauge groups.', '1111.1541-2-31-1': 'We shall do that for the simplest case [MATH], but everything is general and applies to any group.', '1111.1541-2-31-2': 'As a result we shall understand and correct the pathologies of the existing disorder parameter quoted in Section I.', '1111.1541-2-32-0': 'The definition of the order parameter [MATH] of Ref. [CITATION] for [MATH] is the natural extension to the non-abelian case of the construction for [MATH] presented above.', '1111.1541-2-32-1': 'The original idea is to add the field of an abelian monopole to the magnetic field of the residual [MATH] of a given abelian projection.', '1111.1541-2-32-2': 'Here we shall give a more physical view of the construction, showing in particular that the choice of the abelian projection to start from is completely irrelevant.', '1111.1541-2-33-0': ""Again we have Eq. ([REF]) with [EQUATION] and Eq. 's ([REF])eqrefpp with [EQUATION] and [MATH] for all [MATH] except for the electric plaquettes at [MATH], the time at which the monopole is created, which are given by [EQUATION] [MATH] is the trace of the identity, and [EQUATION] [MATH] is a classical [MATH] monopole field configuration, e.g. the 't Hooft-Polyakov one, which we will choose with a typical length smaller than the lattice spacing so that its asymptotic form is valid."", '1111.1541-2-34-0': 'We now perform a change of variables in the Feynman integral which defines [MATH], analogous to that of Eq. ([REF]), namely [EQUATION]', '1111.1541-2-34-1': 'As a consequence [EQUATION]', '1111.1541-2-34-2': 'In Eq. ([REF]) [EQUATION]', '1111.1541-2-34-3': 'By repeated use of the Campbell-Baker-Haussdorf formula it is easy to show that, up to terms [MATH] [EQUATION]', '1111.1541-2-34-4': 'A monopole has been added at time [MATH].', '1111.1541-2-34-5': 'If we take the field [MATH] in the unitary representation, which coincides with the maximal abelian gauge [CITATION], and periodic boundary conditions for the quantum fluctuations, the magnetic monopole field at large distance will be that of the monopole added, whatever is the abelian projection we started from.', '1111.1541-2-34-6': 'The magnetic field [MATH] is directed along the color direction of the residual [MATH] and adds to the corresponding component of the quantum field.', '1111.1541-2-34-7': 'That component is in any case undefined by terms [MATH] depending on the conventions used to extract it from the configurations, again by the Campbell-Baker-Haussdorf formula.', '1111.1541-2-35-0': 'For the same reason the matrix [MATH] of Eq. ([REF]) has the same [MATH] component as [MATH] up to terms [MATH] since the terms [MATH] cancel between [MATH] and [MATH].', '1111.1541-2-35-1': 'Therefore the change of variables Eq. ([REF]) can be repeated at time [MATH] with the effect of adding a monopole at time [MATH] and so on.', '1111.1541-2-35-2': 'The monopole propagates from time [MATH] on.', '1111.1541-2-35-3': 'The same conclusion follows from the fact that the abelian projected field obeys abelian Bianchi identities [CITATION], so that the magnetic current is conserved [CITATION].', '1111.1541-2-35-4': 'Eq. ([REF]) is still valid.', '1111.1541-2-35-5': 'There is, however, a misuse of language: in the abelian case the monopole propagates unchanged from time [MATH] to [MATH], and is the ""out"" state.', '1111.1541-2-35-6': 'As we have just seen, in the non-abelian case the monopole configuration gets modified in propagating, even if it preserves its quantum numbers, so the configuration is not exactly the ""out"" state.', '1111.1541-2-36-0': 'A few remarks:', '1111.1541-2-37-0': 'All of the equations for the non abelian case also hold for the abelian one where they coincide with those described above as can easily be checked.', '1111.1541-2-38-0': ""In what follows we shall use as classical monopole field [MATH] the 't Hooft -Polyakov soliton solution in the unitary representation: as any magnetic monopole field they decrease as [MATH] at large distances."", '1111.1541-2-38-1': 'The details will be presented in the next section.', '1111.1541-2-39-0': 'As directly visible from the definition Eq. ([REF]) of [MATH], [MATH] and [MATH] are invariant under covariant transformations of [MATH], [MATH] but not under gauge transformations of the classical field.', '1111.1541-2-39-1': 'We then expect different disorder parameters in different gauges, which are however expected to be on the same footing since each of them is the transition amplitude from the ground state to a state with a monopole added.', '1111.1541-2-40-0': 'The disorder parameter of Ref. [CITATION] was [EQUATION]', '1111.1541-2-40-1': 'The correct quantum-mechanical definition of the disorder parameter, i.e. of the probability to find in the vacuum state a state with magnetic charge increased by a monopole is [EQUATION]', '1111.1541-2-40-2': 'In the case of the [MATH] gauge group we showed that [MATH] so that Eq.s ([REF]) and ([REF]) are the same thing and [MATH] as originally defined is the order parameter.', '1111.1541-2-40-3': 'For non abelian gauge groups we will show that this is not the case.', '1111.1541-2-40-4': '[MATH] can be written as [EQUATION] so that [EQUATION] which again coincides with Eq. ([REF]) in the case of [MATH], since there [MATH].', '1111.1541-2-41-0': 'We shall now compute [MATH] for the generic case and show that it is non trivial: the discrete implementation of the operator Eq. ([REF]) proposed in Ref. [CITATION] does not preserve unitarity.', '1111.1541-2-41-1': 'Not only that, but both [MATH] and [MATH] tend to zero in the large volume limit, in such a way that their ratio stays finite and different from zero at small values of [MATH].', '1111.1541-2-42-0': '[MATH] of Eq. ([REF]) is the correct order parameter, which will be computed via [MATH] of Eq. ([REF]) by use of Eq. ([REF]).', '1111.1541-2-43-0': 'We first give the expression for [MATH].', '1111.1541-2-43-1': 'It naturally derives from the definition of scalar product in the language of path integral.', '1111.1541-2-43-2': ""[EQUATION] where [MATH] everywhere except for the mixed space-time plaquettes [MATH] at the time [MATH] at which the monopole is created, where (see Eq. 's ([REF])-([REF])) [EQUATION]"", '1111.1541-2-43-3': 'Indeed by the usual change of variables in the path integral which defines [MATH] it can be shown that with this definition there is a monopole propagating in the future, and one coming from the past.', '1111.1541-2-43-4': '[MATH] and [MATH] in the equation ([REF]) cancel in the case of [MATH] because they commute with the links, everything being abelian, and therefore [MATH] everywhere.', '1111.1541-2-44-0': ""We shall now compute both [MATH] and [MATH] by the strong coupling expansion at low [MATH]'s."", '1111.1541-2-44-1': 'The details of the expansion are sketched Appendix [REF] below.', '1111.1541-2-45-0': 'Like for the case of [MATH] the first non trivial contribution to the strong coupling expansion of [MATH] is [MATH] corresponding graphically to an elementary cube, and is given by [EQUATION] and [EQUATION]', '1111.1541-2-45-1': 'By use of the definition Eq. ([REF]) we get [EQUATION]', '1111.1541-2-45-2': 'Since, as we shall see in Section [REF], in the Wu-Yang parametrization of the monopole field [MATH], it is easily seen, by expanding the cosines at large distances, that, neglecting finite terms, [EQUATION] which is linearly divergent to [MATH] with the spatial size of the lattice.', '1111.1541-2-45-3': ""This is indeed the pathology observed in lattice simulations, and what makes [MATH] at all [MATH]'s in the thermodynamic limit."", '1111.1541-2-46-0': 'If we now play the same game with the new order parameter [MATH] of Eq. ([REF]) we get instead [EQUATION] with [EQUATION]', '1111.1541-2-46-1': 'In the strong coupling again the first non trivial term corresponds to the elementary cube and gives [EQUATION] where the coefficient of proportionality is the same as in Eq. ([REF]) and [EQUATION]', '1111.1541-2-46-2': 'In computing [MATH] by use of Eq. ([REF]) the leading terms at large distances cancel and the final result is finite in the thermodynamic limit.', '1111.1541-2-47-0': 'In formulae [EQUATION]', '1111.1541-2-47-1': 'The state [MATH] has zero norm in the infinite volume limit, but its direction in Hilbert space is well defined, at least in the confined phase.', '1111.1541-2-48-0': 'This was the leading term in the strong-coupling expansion.', '1111.1541-2-48-1': 'The exact result is obtained either by studying systematically higher order terms in the strong coupling expansion, or numerically by lattice simulations.', '1111.1541-2-48-2': 'This we will do in the next section.', '1111.1541-2-48-3': 'Notice that the coefficient of the first term in the strong coupling expansion depends on the vector potential and not on the magnetic field as was the case for [MATH] gauge group.', '1111.1541-2-48-4': 'This fact is related to the remark 3) above, that the definition of [MATH] is not invariant under gauge transformations of the classical field of the monopole.', '1111.1541-2-49-0': '# The monopole configuration in non abelian gauge theories', '1111.1541-2-50-0': 'In the [MATH] gauge theory on the lattice the classical monopole configuration by which the field is shifted by the monopole creation operator has a Dirac string singularity: [MATH] is fully gauge invariant both with respect to regular gauge transformations and with respect to singular gauge transformations like those which displace the Dirac string [CITATION].', '1111.1541-2-50-1': 'There are, however, problems to satisfy in a consistent way the Dirac relation between electric and magnetic charge when electric charges are present [CITATION]: this is the case for example in non abelian gauge theories where electrically charged fields are necessarily present, at least the charged partners of the [MATH] gluon.', '1111.1541-2-51-0': 'In the non abelian case we shall view the monopole as an [MATH] configuration.', '1111.1541-2-52-0': ""Suppose we want to create a 't Hooft-Polyakov monopole in the unitary representation from the configuration with no gluons."", '1111.1541-2-53-0': 'For our purposes we may consider only the large distance form of the configuration.', '1111.1541-2-53-1': 'The notation is standard (see e.g. [CITATION]).', '1111.1541-2-54-0': 'The monopole configuration in the Wu-Yang form with discontinuity say at a polar angle [MATH] is obtained from the hedgehog gauge configuration (in polar coordinates) [EQUATION] by operating the gauge transformation [EQUATION] with [EQUATION] and [EQUATION] [MATH] is the usual Heaviside function.', '1111.1541-2-54-1': 'The requirement in the definition of [MATH] is that [MATH] is [MATH] and [MATH] independent and that [MATH].', '1111.1541-2-54-2': '[MATH] can be e.g. [MATH], or [MATH] or any fixed combination thereof.', '1111.1541-2-54-3': 'The field in the new gauge is computed by use of the formula [EQUATION]', '1111.1541-2-54-4': 'The details of the computation are given in Appendix [REF].', '1111.1541-2-54-5': 'The result is [EQUATION]', '1111.1541-2-54-6': 'The non abelian magnetic field is radial [EQUATION]', '1111.1541-2-54-7': 'As shown in Appendix [REF] the singularities cancel and the field is simply the coulombic field [MATH].', '1111.1541-2-54-8': 'Moreover the component [MATH] of the field in the Wu-Yang unitary gauge is continuous inside [MATH] as shown in Eq. ([REF]).', '1111.1541-2-54-9': 'It is instead discontinuous if viewed as an [MATH] configuration.', '1111.1541-2-54-10': 'In the limit [MATH] the singularity becomes the usual Dirac string [CITATION].', '1111.1541-2-54-11': 'To create a monopole we should in principle create, together with the abelian diagonal part of the field, a singular charged gluon field [MATH] which cancels the singularities.', '1111.1541-2-54-12': 'For the matrix element of the monopole operator which we are considering (creation from a zero gluon configuration) this procedure amounts to keep only the diagonal part and neglect its singularities (Dirac string or Wu-Yang transfer matrix).', '1111.1541-2-54-13': 'This is the recipe that will be implemented on the lattice (Section [REF]).', '1111.1541-2-55-0': 'If we act with the monopole operator on a generic configuration with spatial periodic boundary conditions, i.e. with net magnetic charge zero, the magnetic field at large distances will be the one that we add, and we are therefore automatically in the maximal abelian or unitary gauge [CITATION].', '1111.1541-2-55-1': 'The singularities cancel anyhow in the magnetic field, except possibly for the bilinear term in Eq. ([REF]) with the singular classical field [MATH] Eq. ([REF]) and the quantum fluctuation of [MATH], which should however average to zero.', '1111.1541-2-55-2': 'Again the recipe is to keep only [MATH] at [MATH].', '1111.1541-2-56-0': 'How this recipe extends to more complicated matrix elements or correlation functions is an open question which deserves further study.', '1111.1541-2-56-1': 'The emerging indication is that, contrary to a generic theory containing charged fields, in a gauge theory the problem raised in Ref. [CITATION] is structurally solved.', '1111.1541-2-56-2': 'Monopole configurations are non singular: what produces the singularity is the abelian projection.', '1111.1541-2-56-3': 'In the language of Ref. [CITATION] monopoles in a non-abelian gauge theory are violations on non-abelian Bianchi identities, which produce violations of abelian Bianchi identities only after abelian projection (Sect. 2 of Ref. [CITATION]).', '1111.1541-2-57-0': '# Numerical results', '1111.1541-2-58-0': 'In this section we present the numerical results obtained by simulating the [MATH] lattice gauge theory.', '1111.1541-2-58-1': 'Since both the actions [MATH] and [MATH] are linear in each of the link, the usual combination of heatbath [CITATION] and overrelaxation [CITATION] can be used for the update of the configuration, even when the monopole operator is present.', '1111.1541-2-58-2': 'Numerical simulations have been performed on GRID resources provided by INFN.', '1111.1541-2-59-0': 'Our aim is to show that the behaviour of the improved operator [MATH] in the [MATH] case is similar to the one observed in the [MATH] lattice gauge theory, namely we want to verify that [MATH]', '1111.1541-2-60-0': 'As previously discussed these are the properties of [MATH] which allow to interpret confinement as dual superconductivity, as they guarantee that [MATH] in the low temperature phase and [MATH] in the high temperature one.', '1111.1541-2-61-0': 'We will first of all study the behaviour of [MATH] and [MATH] for small values of [MATH] (i.e. [MATH]) for lattices with temporal extent [MATH] (for which [MATH], see e.g. [CITATION]).', '1111.1541-2-61-1': 'The numerical results for [MATH] are shown in Fig. ([REF]) and the linear divergence of [MATH] with the lattice size (as predicted by Eq. ([REF])) is clearly seen.', '1111.1541-2-61-2': 'Notice that with lower statistics and smaller lattices the points in the insert of Fig. ([REF]) would be undistinguishable within errors as was in Ref. [CITATION].', '1111.1541-2-62-0': 'This infrared divergence is instead absent in the [MATH] data, which are shown in Fig. ([REF]) and smoothly approach their thermodynamical values.', '1111.1541-2-62-1': 'Clearly this approach is slower for the data at larger [MATH] shown in the figure, which are closer to the deconfinement transition where the correlation length is larger.', '1111.1541-2-62-2': 'Note that, although the absolute values of the error bars for [MATH] and [MATH] are similar, the relative errors for [MATH] are much bigger than those for [MATH] since the value of [MATH] is about an order of magnitude smaller than [MATH].', '1111.1541-2-63-0': 'The data showing the behaviour of [MATH] in the neighbourhood of the [MATH] deconfinement transition are displayed in Fig. ([REF]) and the development of the negative peak at the transition is clearly visible.', '1111.1541-2-63-1': 'The deconfinement transition for [MATH] dimensional [MATH] lattice gauge theory is known to belong to the universality class of the 3d Ising model and from a simple scaling ansatz for [MATH] (see e.g. [CITATION]) the finite size scaling relation for the singular part of [MATH] follows [EQUATION] where [MATH] is a scaling function.', '1111.1541-2-63-2': 'From this scaling form it follows that by increasing the lattice size the peak height should increase as [MATH], while its width should shrink as [MATH].', '1111.1541-2-63-3': 'This behaviour is qualitatively visible in Fig. ([REF]) and will be now verified also quantitatively.', '1111.1541-2-64-0': 'In order to estimate the analytical background of [MATH], a fit of the form [MATH] was performed on the peak values of [MATH] and the constant background term [MATH] was used.', '1111.1541-2-64-1': 'The quality of the scaling in Eq. ([REF]) is shown in Fig. ([REF]), where the known values [MATH] and [MATH] where used for the deconfinement coupling and the 3d Ising critical index.', '1111.1541-2-64-2': 'We explicitly notice that scaling is to be expected only for [MATH], since for [MATH] should not be well defined in the thermodynamical limit and it is expected to diverge to [MATH] as will be directly tested below.', '1111.1541-2-65-0': 'To verify that the position of the peak scales correctly with the lattice spacing we simulated also lattices with different temporal extent.', '1111.1541-2-65-1': 'In Fig. ([REF]) data are shown corresponding to a lattice with [MATH] and again the negative peak develops in correspondence of the deconfinement transition, which now takes place at [MATH] (see e.g. [CITATION]).', '1111.1541-2-66-0': 'The last point that remains to be checked is the behaviour of [MATH] at temperatures higher than the deconfinement temperature.', '1111.1541-2-66-1': 'In order to have [MATH] for [MATH], [MATH] must diverge to [MATH] in the thermodynamical limit for every [MATH].', '1111.1541-2-66-2': 'This is indeed what happens, as shown in Fig. ([REF]), where two different scaling are seen: at fixed lattice size [MATH] is asymptotically linear in [MATH] while at fixed [MATH] value [MATH] is asymptotically linear in the lattice size [MATH].', '1111.1541-2-67-0': 'We have thus verified numerically that the improved monopole operator [MATH] is a well defined operator to check dual superconductivity in lattice simulations.', '1111.1541-2-67-1': 'We have thus shown that the deconfinement transition in [MATH] lattice gauge theory can be interpreted as a monopole condensation transition.', '1111.1541-2-68-0': '# Subtraction at zero temperature.', '1111.1541-2-69-0': 'For the sake of completeness we also report some numerical results of an attempt we made prior to the approach described in this paper on the way to understand the diseases of our disorder parameter [CITATION].', '1111.1541-2-69-1': 'The basic idea was that the sort of infrared divergence appearing in [MATH] would look as a short distance effect in the dual description and produce a multiplicative renormalization of [MATH].', '1111.1541-2-69-2': 'In the ratio [MATH] the constant would cancel and the deconfining transition become visible.', '1111.1541-2-69-3': 'The analogue of [MATH] becomes now [MATH].', '1111.1541-2-69-4': 'The results are shown in Figs. ([REF]) and ([REF]).', '1111.1541-2-69-5': 'We report these results because they prove to be meaningful.', '1111.1541-2-70-0': 'It is not easy to simulate the system at [MATH] and large [MATH], since the physical length grows exponentially with [MATH] and the lattice size should be increased accordingly.', '1111.1541-2-70-1': 'In the practically accessible range of values of [MATH] the results are satisfactory: the phase transition is clearly visible and the appropriate scaling is observed.', '1111.1541-2-70-2': 'In terms of the new disorder parameter this means that [MATH] is rather smooth in [MATH] and temperature independent below and around the deconfinement peak.', '1111.1541-2-70-3': ""This can indeed be directly seen in Fig. ([REF]) where the term [MATH] of Eq. ([REF]), which determines the normalization Eq. ([REF]), is shown to be [MATH] or temperature independent in a range of [MATH]'s."", '1111.1541-2-71-0': 'The behavior of [MATH] at zero temperature is also shown in Fig. ([REF]) to be volume independent and finite, meaning that there is confinement.', '1111.1541-2-72-0': '# Conclusions and outlook', '1111.1541-2-73-0': 'Understanding the mechanism of confinement of color in [MATH] is a fundamental problem in particle physics.', '1111.1541-2-73-1': 'Dual superconductivity of the confining vacuum is an appealing candidate.', '1111.1541-2-74-0': 'A disorder parameter had been introduced to detect condensation of magnetic charges, in order to demonstrate by numerical simulations that deconfinement is a transition from a dual superconductor to a normal state.', '1111.1541-2-74-1': 'The starting point was the [MATH] lattice gauge theory which has a deconfining transition well understood in terms of dual superconductivity.', '1111.1541-2-74-2': 'The generalization of the [MATH] disorder parameter to the non-abelian case proved to be affected by a pathology: in the thermodynamic limit the parameter as was constructed tends to zero both in the deconfined and in the confined phase, and is thus unable to detect superconductivity.', '1111.1541-2-75-0': 'In this paper we trace the origin of this disease and we cure it.', '1111.1541-2-75-1': 'In synthesis the original disorder parameter was not a properly normalized probability amplitude, and had limit zero at increasing volumes because the norm of the state obtained from the vacuum by addition of a monopole tends to zero.', '1111.1541-2-75-2': 'The direction of the state in the Hilbert space has instead a well defined limit, and with it the probability amplitude in the confined phase.', '1111.1541-2-75-3': 'This has been shown both by strong coupling expansion and by numerical simulations.', '1111.1541-2-76-0': 'The results support dual superconductivity as mechanism of color confinement.', '1111.1541-2-77-0': 'The disorder parameter is by construction gauge invariant if the classical field of the monopole transforms covariantly.', '1111.1541-2-77-1': 'In other words it is not invariant under gauge transformations of the classical external field.', '1111.1541-2-77-2': 'This is not a difficulty in principle for the demonstration of monopole condensation, but is unpleasant.', '1111.1541-2-78-0': 'We are trying to use alternative definitions, as could be the technique of the Schroedinger functional, which was already used in Ref. [CITATION] which could be more satisfactory.', '1111.1541-2-79-0': '# aknowledgments', '1111.1541-2-80-0': 'One of us (A. D. G.) thanks G. Paffuti for useful discussions.', '1111.1541-2-80-1': 'The numerical simulations have been performed on GRID resources provided by INFN and in particular on the CSN4 Cluster located in Pisa.', '1111.1541-2-81-0': '# Strong coupling expansion', '1111.1541-2-82-0': 'In this section we sketch the strong coupling expansion of [MATH] and [MATH] of Sec. [REF].', '1111.1541-2-82-1': 'The strong coupling expansion is a power series expansion in [MATH] (see e.g. [CITATION]).', '1111.1541-2-82-2': 'More refined methods, such as the character expansion (see [CITATION]), cannot be used here because of the monopole insertion in [MATH].', '1111.1541-2-83-0': 'For the [MATH] case we want to compute the expansion of [EQUATION] where [MATH] and [MATH] are defined in Eqs. ([REF])-([REF]).', '1111.1541-2-83-1': 'The fundamental formulae to be used are [EQUATION]', '1111.1541-2-83-2': 'It has to be noted that in the expression ([REF]) most of the terms cacels between [MATH] and [MATH].', '1111.1541-2-83-3': 'For example the lowest non-trivial contribution to both [MATH] and [MATH] are [MATH] and can be graphically represented by two superimposed plaquettes (if the two plaquettes are not superimposed the result is zero because of the first of Eqs. ([REF])).', '1111.1541-2-83-4': 'However these two contributions to [MATH] and [MATH] are equal and cancel in Eq. ([REF]).', '1111.1541-2-84-0': 'The first non-trivial contribution to [MATH] turns out to be [MATH] and is graphically represented by the six plaquettes that enclose a three-dimensional cube.', '1111.1541-2-84-1': 'If the cube has no edges along the temporal direction and/or if it is not positioned at the time at which the monopole is created, it simplifies in the difference in Eq. ([REF]).', '1111.1541-2-85-0': 'Let us denote by [MATH] the contribution to the mean value [MATH] associated to a cube sitting at [MATH] with edges along the directions [MATH] is the temporal direction).', '1111.1541-2-85-1': 'By applying 12 times (the number of the edges of the cube) the second formula of Eqs. ([REF]) we get [EQUATION] from which Eq. ([REF]) follows.', '1111.1541-2-86-0': 'In the non-abelian case the integration rules Eqs. ([REF]) become [EQUATION] and the computations goes along the same line.', '1111.1541-2-86-1': 'Also in the non-abelian case the first non-trivial contribution to [MATH] is [MATH] and correspond to the cube graph of in Fig. ([REF]).', '1111.1541-2-87-0': 'By using the second relation of Eqs. ([REF]) on the twelve edges of the cube we obtain the expressions in Eq. ([REF]) and Eq. ([REF]).', '1111.1541-2-88-0': '# Details of the gauge transformation of Section [REF].', '1111.1541-2-89-0': 'In our notation [MATH] are the three generators along the axes of the polar coordinates.', '1111.1541-2-89-1': 'In terms of the generators along the cartesian axes, [MATH],they are given by [EQUATION]', '1111.1541-2-89-2': 'By use of Eq. ([REF]) the covariant terms in Eq. ([REF]) are easily computed giving [EQUATION]', '1111.1541-2-89-3': 'The affine terms [MATH] are [EQUATION]', '1111.1541-2-89-4': ""Using the fact that [MATH], the result for [MATH] is that of Eq. 's ([REF])eqrefAteta([REF])."", '1111.1541-2-90-0': 'We now compute the magnetic field by Eq. [REF].', '1111.1541-2-90-1': 'The three terms give separately [EQUATION]', '1111.1541-2-90-2': 'The term [MATH] is continuous through [MATH] and gives the coulombic field of the monopole.', '1111.1541-2-90-3': 'Finally [EQUATION]', '1111.1541-2-90-4': 'Since [MATH], the second term in Eq. ([REF]) gives [EQUATION]', '1111.1541-2-90-5': 'Finally the bilinear term gives [EQUATION]', '1111.1541-2-90-6': 'Since [MATH] and [EQUATION] we get [EQUATION]', '1111.1541-2-90-7': 'All the singularities cancel in the sum and only the coulombic term is left.', '1111.1541-2-91-0': 'Notice that, since [MATH] is orthogonal to [MATH] [EQUATION]', '1111.1541-2-91-1': 'The discontinuity in [MATH] vanishes, and so does the second term of [MATH] which should cancel it.'}","[['1111.1541-1-23-0', '1111.1541-2-23-0'], ['1111.1541-1-12-0', '1111.1541-2-12-0'], ['1111.1541-1-12-1', '1111.1541-2-12-1'], ['1111.1541-1-12-2', '1111.1541-2-12-2'], ['1111.1541-1-12-3', '1111.1541-2-12-3'], ['1111.1541-1-67-0', '1111.1541-2-67-0'], ['1111.1541-1-67-1', '1111.1541-2-67-1'], ['1111.1541-1-65-0', '1111.1541-2-65-0'], ['1111.1541-1-65-1', '1111.1541-2-65-1'], ['1111.1541-1-10-0', '1111.1541-2-10-0'], ['1111.1541-1-19-0', '1111.1541-2-19-0'], ['1111.1541-1-4-0', '1111.1541-2-4-0'], ['1111.1541-1-9-0', '1111.1541-2-9-0'], ['1111.1541-1-34-0', '1111.1541-2-34-0'], ['1111.1541-1-34-1', '1111.1541-2-34-1'], ['1111.1541-1-34-3', '1111.1541-2-34-3'], ['1111.1541-1-34-4', '1111.1541-2-34-4'], ['1111.1541-1-34-5', '1111.1541-2-34-5'], ['1111.1541-1-34-6', '1111.1541-2-34-6'], ['1111.1541-1-34-7', '1111.1541-2-34-7'], ['1111.1541-1-84-0', '1111.1541-2-84-0'], ['1111.1541-1-84-1', '1111.1541-2-84-1'], ['1111.1541-1-55-0', '1111.1541-2-55-0'], ['1111.1541-1-55-1', '1111.1541-2-55-1'], ['1111.1541-1-55-2', '1111.1541-2-55-2'], ['1111.1541-1-90-0', '1111.1541-2-90-0'], ['1111.1541-1-90-1', '1111.1541-2-90-1'], ['1111.1541-1-90-2', '1111.1541-2-90-2'], ['1111.1541-1-90-4', '1111.1541-2-90-4'], ['1111.1541-1-90-5', '1111.1541-2-90-5'], ['1111.1541-1-90-6', '1111.1541-2-90-6'], ['1111.1541-1-90-7', '1111.1541-2-90-7'], ['1111.1541-1-77-0', '1111.1541-2-77-0'], ['1111.1541-1-77-1', '1111.1541-2-77-1'], ['1111.1541-1-77-2', '1111.1541-2-77-2'], ['1111.1541-1-42-0', '1111.1541-2-42-0'], ['1111.1541-1-69-0', '1111.1541-2-69-0'], ['1111.1541-1-69-1', '1111.1541-2-69-1'], ['1111.1541-1-69-2', '1111.1541-2-69-2'], ['1111.1541-1-69-3', '1111.1541-2-69-3'], ['1111.1541-1-69-4', '1111.1541-2-69-4'], ['1111.1541-1-69-5', '1111.1541-2-69-5'], ['1111.1541-1-17-0', '1111.1541-2-17-0'], ['1111.1541-1-17-1', '1111.1541-2-17-1'], ['1111.1541-1-17-2', '1111.1541-2-17-2'], ['1111.1541-1-17-3', '1111.1541-2-17-3'], ['1111.1541-1-31-0', '1111.1541-2-31-0'], ['1111.1541-1-31-1', '1111.1541-2-31-1'], ['1111.1541-1-31-2', '1111.1541-2-31-2'], ['1111.1541-1-64-0', '1111.1541-2-64-0'], ['1111.1541-1-64-1', '1111.1541-2-64-1'], ['1111.1541-1-64-2', '1111.1541-2-64-2'], ['1111.1541-1-6-0', '1111.1541-2-6-0'], ['1111.1541-1-6-1', '1111.1541-2-6-1'], ['1111.1541-1-6-2', '1111.1541-2-6-2'], ['1111.1541-1-6-3', '1111.1541-2-6-3'], ['1111.1541-1-6-4', '1111.1541-2-6-4'], ['1111.1541-1-6-5', '1111.1541-2-6-5'], ['1111.1541-1-6-6', '1111.1541-2-6-6'], ['1111.1541-1-6-7', '1111.1541-2-6-7'], ['1111.1541-1-6-8', '1111.1541-2-6-8'], ['1111.1541-1-86-0', '1111.1541-2-86-0'], ['1111.1541-1-86-1', '1111.1541-2-86-1'], ['1111.1541-1-8-0', '1111.1541-2-8-0'], ['1111.1541-1-8-1', '1111.1541-2-8-1'], ['1111.1541-1-8-2', '1111.1541-2-8-2'], ['1111.1541-1-8-3', '1111.1541-2-8-3'], ['1111.1541-1-59-0', '1111.1541-2-59-0'], ['1111.1541-1-29-0', '1111.1541-2-29-0'], ['1111.1541-1-29-1', '1111.1541-2-29-1'], ['1111.1541-1-29-2', '1111.1541-2-29-2'], ['1111.1541-1-29-3', '1111.1541-2-29-3'], ['1111.1541-1-29-4', '1111.1541-2-29-4'], ['1111.1541-1-29-5', '1111.1541-2-29-5'], ['1111.1541-1-29-6', '1111.1541-2-29-6'], ['1111.1541-1-29-7', '1111.1541-2-29-7'], ['1111.1541-1-14-0', '1111.1541-2-14-0'], ['1111.1541-1-14-1', '1111.1541-2-14-1'], ['1111.1541-1-14-2', '1111.1541-2-14-2'], ['1111.1541-1-35-0', '1111.1541-2-35-0'], ['1111.1541-1-35-1', '1111.1541-2-35-1'], ['1111.1541-1-35-2', '1111.1541-2-35-2'], ['1111.1541-1-35-3', '1111.1541-2-35-3'], ['1111.1541-1-35-4', '1111.1541-2-35-4'], ['1111.1541-1-35-5', '1111.1541-2-35-5'], ['1111.1541-1-35-6', '1111.1541-2-35-6'], ['1111.1541-1-30-0', '1111.1541-2-30-0'], ['1111.1541-1-30-1', '1111.1541-2-30-1'], ['1111.1541-1-30-2', '1111.1541-2-30-2'], ['1111.1541-1-30-3', '1111.1541-2-30-3'], ['1111.1541-1-30-4', '1111.1541-2-30-4'], ['1111.1541-1-30-5', '1111.1541-2-30-5'], ['1111.1541-1-30-6', '1111.1541-2-30-6'], ['1111.1541-1-30-7', '1111.1541-2-30-7'], ['1111.1541-1-30-8', '1111.1541-2-30-8'], ['1111.1541-1-30-9', '1111.1541-2-30-9'], ['1111.1541-1-18-0', '1111.1541-2-18-0'], ['1111.1541-1-18-1', '1111.1541-2-18-1'], ['1111.1541-1-18-2', '1111.1541-2-18-2'], ['1111.1541-1-37-0', '1111.1541-2-37-0'], ['1111.1541-1-91-0', '1111.1541-2-91-0'], ['1111.1541-1-91-1', '1111.1541-2-91-1'], ['1111.1541-1-38-0', '1111.1541-2-38-0'], ['1111.1541-1-38-1', '1111.1541-2-38-1'], ['1111.1541-1-46-0', '1111.1541-2-46-0'], ['1111.1541-1-46-1', '1111.1541-2-46-1'], ['1111.1541-1-46-2', '1111.1541-2-46-2'], ['1111.1541-1-20-0', '1111.1541-2-20-0'], ['1111.1541-1-27-0', '1111.1541-2-27-0'], ['1111.1541-1-27-1', '1111.1541-2-27-1'], ['1111.1541-1-27-2', '1111.1541-2-27-2'], ['1111.1541-1-27-3', '1111.1541-2-27-3'], ['1111.1541-1-82-0', '1111.1541-2-82-0'], ['1111.1541-1-82-1', '1111.1541-2-82-1'], ['1111.1541-1-82-2', '1111.1541-2-82-2'], ['1111.1541-1-41-0', '1111.1541-2-41-0'], ['1111.1541-1-41-1', '1111.1541-2-41-1'], ['1111.1541-1-54-0', '1111.1541-2-54-0'], ['1111.1541-1-54-1', '1111.1541-2-54-1'], ['1111.1541-1-54-2', '1111.1541-2-54-2'], ['1111.1541-1-54-3', '1111.1541-2-54-3'], ['1111.1541-1-54-4', '1111.1541-2-54-4'], ['1111.1541-1-54-5', '1111.1541-2-54-5'], ['1111.1541-1-54-6', '1111.1541-2-54-6'], ['1111.1541-1-54-7', '1111.1541-2-54-7'], ['1111.1541-1-54-8', '1111.1541-2-54-8'], ['1111.1541-1-54-9', '1111.1541-2-54-9'], ['1111.1541-1-54-10', '1111.1541-2-54-10'], ['1111.1541-1-54-11', '1111.1541-2-54-11'], ['1111.1541-1-54-12', '1111.1541-2-54-12'], ['1111.1541-1-54-13', '1111.1541-2-54-13'], ['1111.1541-1-63-0', '1111.1541-2-63-0'], ['1111.1541-1-63-1', '1111.1541-2-63-1'], ['1111.1541-1-63-2', '1111.1541-2-63-2'], ['1111.1541-1-63-3', '1111.1541-2-63-3'], ['1111.1541-1-28-0', '1111.1541-2-28-0'], ['1111.1541-1-28-1', '1111.1541-2-28-1'], ['1111.1541-1-26-0', '1111.1541-2-26-0'], ['1111.1541-1-3-0', '1111.1541-2-3-0'], ['1111.1541-1-3-1', '1111.1541-2-3-1'], ['1111.1541-1-3-2', '1111.1541-2-3-2'], ['1111.1541-1-3-3', '1111.1541-2-3-3'], ['1111.1541-1-3-4', '1111.1541-2-3-4'], ['1111.1541-1-3-5', '1111.1541-2-3-5'], ['1111.1541-1-3-6', '1111.1541-2-3-6'], ['1111.1541-1-3-7', '1111.1541-2-3-7'], ['1111.1541-1-3-8', '1111.1541-2-3-8'], ['1111.1541-1-45-0', '1111.1541-2-45-0'], ['1111.1541-1-45-1', '1111.1541-2-45-1'], ['1111.1541-1-45-2', '1111.1541-2-45-2'], ['1111.1541-1-45-3', '1111.1541-2-45-3'], ['1111.1541-1-5-0', '1111.1541-2-5-0'], ['1111.1541-1-5-1', '1111.1541-2-5-1'], ['1111.1541-1-5-2', '1111.1541-2-5-2'], ['1111.1541-1-50-0', '1111.1541-2-50-0'], ['1111.1541-1-50-1', '1111.1541-2-50-1'], ['1111.1541-1-58-0', '1111.1541-2-58-0'], ['1111.1541-1-58-1', '1111.1541-2-58-1'], ['1111.1541-1-58-2', '1111.1541-2-58-2'], ['1111.1541-1-39-0', '1111.1541-2-39-0'], ['1111.1541-1-39-1', '1111.1541-2-39-1'], ['1111.1541-1-73-0', '1111.1541-2-73-0'], ['1111.1541-1-73-1', '1111.1541-2-73-1'], ['1111.1541-1-61-0', '1111.1541-2-61-0'], ['1111.1541-1-61-1', '1111.1541-2-61-1'], ['1111.1541-1-61-2', '1111.1541-2-61-2'], ['1111.1541-1-51-0', '1111.1541-2-51-0'], ['1111.1541-1-24-0', '1111.1541-2-24-0'], ['1111.1541-1-24-1', '1111.1541-2-24-1'], ['1111.1541-1-24-2', '1111.1541-2-24-2'], ['1111.1541-1-24-3', '1111.1541-2-24-3'], ['1111.1541-1-24-4', '1111.1541-2-24-4'], ['1111.1541-1-83-0', '1111.1541-2-83-0'], ['1111.1541-1-83-1', '1111.1541-2-83-1'], ['1111.1541-1-83-2', '1111.1541-2-83-2'], ['1111.1541-1-83-3', '1111.1541-2-83-3'], ['1111.1541-1-83-4', '1111.1541-2-83-4'], ['1111.1541-1-53-0', '1111.1541-2-53-0'], ['1111.1541-1-53-1', 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astro-ph-0111111,"{'astro-ph-0111111-1-0-0': 'Substantial fluxes of protons and leptons with energies below the geomagnetic cutoff have been measured by the AMS experiment at altitudes of 370-390 Km, in the latitude interval [MATH]51.7[MATH].', 'astro-ph-0111111-1-0-1': 'The production mechanisms of the observed trapped fluxes are investigated in detail by means of the FLUKA Monte Carlo simulation code.', 'astro-ph-0111111-1-0-2': 'All known processes involved in the interaction of the cosmic rays with the atmosphere (detailed descriptions of the magnetic field and atmospheric density, as well as the electromagnetic and nuclear interaction processes) are included in the simulation.', 'astro-ph-0111111-1-0-3': 'The results are presented and compared with the experimental data, indicating good agreement with the observed fluxes.', 'astro-ph-0111111-1-0-4': 'The impact of secondary proton flux on particle production in atmosphere is briefly discussed.', 'astro-ph-0111111-1-1-0': '# Introduction', 'astro-ph-0111111-1-2-0': 'Cosmic rays approaching the Earth interact with the atmosphere resulting in a substantial flux of secondary particles.', 'astro-ph-0111111-1-2-1': 'A reliable estimate of the secondary flux composition and energy spectra is of considerable interest, e.g. the evaluation of background radiation for satellites and the estimate of the atmospheric neutrino production for neutrino oscillation experiments [CITATION].', 'astro-ph-0111111-1-3-0': 'The accurate AMS measurements of charged fluxes of cosmic and secondary particles in the near Earth region allow an extensive test of models describing the cosmic ray interactions with the atmosphere and the evolution of the produced secondary particles in the magnetic field.', 'astro-ph-0111111-1-4-0': ""In this work, we report results from a Monte Carlo simulation based on FLUKA 2000 [CITATION], which describes the interaction of the cosmic protons with the Earth's atmosphere, including the propagation of the secondaries in the Earth's magnetic field."", 'astro-ph-0111111-1-5-0': '# The model', 'astro-ph-0111111-1-6-0': 'An isotropic flux of protons is uniformly generated on a geocentric spherical surface with a radius of 1.07 Earth radii ([MATH] a.s.l.) in the kinetic energy range [MATH].', 'astro-ph-0111111-1-7-0': 'The energy spectrum is modeled according to a power law, modified to account for the solar modulation effects as suggested in [CITATION].', 'astro-ph-0111111-1-7-1': 'Both the spectral index and the solar modulation parameter are extracted from a fit of the AMS data [CITATION].', 'astro-ph-0111111-1-8-0': ""The magnetic field in the Earth's proximity includes two components: the Earth's magnetic field, calculated using a 10 harmonics IGRF [CITATION] implementation, and the external magnetic field, calculated using the Tsyganenko Model[CITATION]."", 'astro-ph-0111111-1-9-0': 'To account for the geomagnetic effects, for each primary proton we back-trace an antiproton of the same energy until one of the following conditions is satisfied:', 'astro-ph-0111111-1-10-0': ""the particle reaches the distance of [MATH] from the Earth's center."", 'astro-ph-0111111-1-11-0': 'the particle touches again the production sphere.', 'astro-ph-0111111-1-12-0': 'neither 1 or 2 is satisfied before a time limit is reached.', 'astro-ph-0111111-1-13-0': 'If condition 1 is satisfied the particle is on an allowed trajectory, while if condition 2 is satisfied the particle is on a forbidden one.', 'astro-ph-0111111-1-13-1': 'Condition 3 arises for only a small fraction of the events [MATH].', 'astro-ph-0111111-1-14-0': ""Particles on allowed trajectories are propagated forward and can reach the Earth's atmosphere."", 'astro-ph-0111111-1-14-1': 'The atmosphere around the Earth is simulated up to [MATH] a.s.l. using 60 concentric layers of homogeneous density and chemical composition.', 'astro-ph-0111111-1-14-2': 'Data on density and chemical composition are taken from the standard MSIS model [CITATION].', 'astro-ph-0111111-1-14-3': 'The Earth is modeled as a solid sphere which absorbs each particle reaching its surface.', 'astro-ph-0111111-1-15-0': '## The generation technique', 'astro-ph-0111111-1-16-0': 'The ideal approach in the generation of the primary cosmic rays spectra would be to start with an isotropic distribution of particles at a great distance (typically [MATH]) from the Earth where the geomagnetic field introduces negligible distortions on the interstellar flux.', 'astro-ph-0111111-1-16-1': 'However, this computational method is intrinsically inefficient since most of the particles are generated with trajectories which will not reach the Earth environment.', 'astro-ph-0111111-1-16-2': 'Kinematical cuts can be applied in order to improve the selection efficiency at generation, however they tend to introduce a bias for low rigidities particles.', 'astro-ph-0111111-1-17-0': 'A good alternative to this approach is the backtracing method [CITATION] [CITATION] adopted in the present analysis as outlined in the previous section.', 'astro-ph-0111111-1-17-1': 'In the following, we will shortly discuss the validity of the technique and report the results of a comparison of the two methods.', 'astro-ph-0111111-1-17-2': 'We recall that this method was applied for the first time in ref. [CITATION] for the generation of atmospheric neutrino fluxes.', 'astro-ph-0111111-1-18-0': 'Let us consider first the effects of the geomagnetic field on an incoming flux of charged particles in the absence of a solid Earth.', 'astro-ph-0111111-1-18-1': 'For the discussion, we start with an isotropic flux of monoenergetic', 'astro-ph-0111111-1-19-0': 'protons at large distance, i.e. at infinity, from the origin of a geocentrical reference system.', 'astro-ph-0111111-1-19-1': 'In this scenario, a negligible fraction of particles, with very particular initial kinematic parameters, will follow complicated paths and remains confined at a given distance from the origin (semi-bounded trajectories); for all practical purposes this sample can be ignored.', 'astro-ph-0111111-1-19-2': 'Most of the particles will follow unbounded trajectories, reaching again infinity after being deflected by the magnetic field.', 'astro-ph-0111111-1-20-0': 'Unbounded trajectories cross a spherical surface centered in the field source only an even number of times, as shown in fig. [REF]: we call legs the trajectory parts connecting the spherical surface to infinity and loops the parts of the trajectory starting and ending inside the spherical surface.', 'astro-ph-0111111-1-21-0': 'Since each trajectory can be followed in both directions and no source or sink of particles is contained within the surface, the incoming and outgoing fluxes are the same.', 'astro-ph-0111111-1-21-1': 'However, the presence of the magnetic field breaks the isotropy of the flux ""near"" the field source, so for a given location there is a flux dependence due to the direction.', 'astro-ph-0111111-1-22-0': 'Applying the Liouville Theorem, under the hypothesis of isotropy at infinity, it is straightforward to prove [CITATION] that the proton flux in a random point is the same as at infinity along a set of directions (allowed directions), and zero along all the others (forbidden directions).', 'astro-ph-0111111-1-23-0': 'The pattern of the allowed and forbidden directions depends on both the rigidity and the location and is known as the geomagnetic cutoff.', 'astro-ph-0111111-1-24-0': 'With the introduction of a solid Earth, all the trajectories that are crossing the Earth are broken in two or more pieces (fig. [REF]): the legs become one-way trajectories and the loops disappear.', 'astro-ph-0111111-1-25-0': 'The presence of the Earth modifies the flux which exits from the surrounding spherical surface, since particles are absorbed by the Earth, while it has only a minimal effect on the incoming flux which is modified only by the absence of certain loops.', 'astro-ph-0111111-1-26-0': ""To generate the flux of particles reaching the Earth's atmosphere, it is sufficient to follow the particles along the allowed trajectories corresponding to the legs, taking care to avoid double or multiple counting."", 'astro-ph-0111111-1-27-0': 'To respect this prescription we reject all trajectories that are back-traced to the production sphere, this allow us to correctly consider the cases like the one shown in fig. [REF].', 'astro-ph-0111111-1-28-0': ""We point out that an important difference with respect to the application in the neutrino flux calculation of [CITATION] is that for the former, the generation sphere coincided with the Earth's surface, and therefore the forbidden trajectories included those which touched again the Earth (plus those who remained trapped for a long time)."", 'astro-ph-0111111-1-28-1': 'In that case there are no problems of double counting.', 'astro-ph-0111111-1-29-0': 'To check the validity of our technique we made a test comparing the results of the ""brute force"" generation technique at 10 Earth\'s radii distance from the Earth\'s center with the backtracing technique described in this paper.', 'astro-ph-0111111-1-30-0': 'Figure [REF] shows this comparison for several cha-racteristic distributions, the agreement between the two methods is good.', 'astro-ph-0111111-1-31-0': '## The interaction model', 'astro-ph-0111111-1-32-0': ""We use the software package FLUKA 2000 [CITATION] to transport the particles and describe their interactions with Earth's atmosphere."", 'astro-ph-0111111-1-32-1': 'This package contains a tridimensional description of the interactions and should reproduce the spatial distribution of secondaries better than older models based on empirical parameterizations of accelerator data.', 'astro-ph-0111111-1-33-0': 'Interactions are treated in a theory-driven approach, and the models and their implementations are guided and checked using experimental data.', 'astro-ph-0111111-1-33-1': 'Hadron-nucleon interaction models are based on resonance production and decay below an energy of few [MATH] and on the Dual Parton Model above.', 'astro-ph-0111111-1-34-0': 'The extension to hadron-nucleus interactions is done in the framework of a generalized intra-nuclear cascade approach including the Gribov-Glauber multi-collision mechanism for higher energies followed by equilibrium processes: evaporation, fission, Fermi break-up and [MATH] de-excitation.', 'astro-ph-0111111-1-35-0': 'In fig [REF] a) we show the map of the primary proton interaction points in geographical coordinates.', 'astro-ph-0111111-1-35-1': 'The distribution reflects the influence of the geomagnetic cutoff.', 'astro-ph-0111111-1-35-2': 'Fig [REF] b) shows the interactions altitude profile, the solid histogram is for [MATH] while the dashed one is for [MATH].', 'astro-ph-0111111-1-35-3': 'The mean interaction altitude depends weakly on the energy.', 'astro-ph-0111111-1-36-0': 'In tab. [REF] we list the characteristic features of the cosmic proton interactions in the atmosphere.', 'astro-ph-0111111-1-37-0': '# Comparison with the AMS data', 'astro-ph-0111111-1-38-0': 'To compare with the AMS data, we define a detection boundary corresponding to a spherical surface at the AMS orbit altitude ([MATH] a.s.l).', 'astro-ph-0111111-1-38-1': 'We record each particle that crosses the detection boundary within the AMS field-of-view defined as a cone with a 32 degree aperture with respect to the local zenith or nadir directions.', 'astro-ph-0111111-1-39-0': 'To obtain the absolute normalization, we take into account the field-of-view, the corresponding AMS acceptance, and an equivalent time exposure (E.T.E) corresponding to the number of primary protons generated.', 'astro-ph-0111111-1-40-0': 'Our results are based on a sample of [MATH] primary protons generated in the kinetic energy range of [MATH], which corresponds to [MATH] (E.T.E).', 'astro-ph-0111111-1-41-0': '## Protons', 'astro-ph-0111111-1-42-0': 'In fig.[REF], we show the comparison between the fluxes obtained with the simulation and the measured AMS downgoing proton flux [CITATION] in nine bins of geomagnetic latitude ([MATH]).', 'astro-ph-0111111-1-42-1': 'Fig.[REF] shows the same comparison for the upgoing proton flux in four selected [MATH] bins.', 'astro-ph-0111111-1-43-0': 'As seen in fig.[REF], the simulation reproduces well at all latitudes, the high energy part of the spectrum and the falloff in the primary spectrum due to the geomagnetic cutoff, thus validating the general approach used for the generation and detection, as well as the tracing technique.', 'astro-ph-0111111-1-44-0': 'The part of the spectrum below the geomagnetic cutoff is sensitive to the interaction model.', 'astro-ph-0111111-1-44-1': 'In particular, it is populated by the low energy protons produced in primary interactions with the atmosphere and that spiral along the field lines in the vicinity of the detecting altitude.', 'astro-ph-0111111-1-45-0': 'Their number and energy distribution depend on the details of the target fragmentation model.', 'astro-ph-0111111-1-45-1': 'The results depend also on the accuracy of the particle transport algorithm and on the details of the atmosphere description.', 'astro-ph-0111111-1-45-2': 'The results shown in figs. [REF] and [REF] were obtained with FLUKA2000[CITATION] code, making use of a setup derived from the one adopted in [CITATION].', 'astro-ph-0111111-1-46-0': 'In figs. [REF] and [REF], the bin-to-bin statistical fluctuations are seen at the lower kinetic energies, particularly in the equatorial region.', 'astro-ph-0111111-1-46-1': 'This part of the spectra corresponds to the undercutoff component where the observed flux is the result of multiple detections of the same particles.', 'astro-ph-0111111-1-46-2': 'The importance of the effect on the observed fluxes is illustrated by the real proton flux, i.e. the flux obtained by counting only once each particle crossing the detector, which is indicated by the dashed distributions in figs. [REF] and [REF].', 'astro-ph-0111111-1-46-3': 'In particular, the real number of protons crossing the detector in the equatorial region is more than one order of magnitude lower than the observed flux.', 'astro-ph-0111111-1-46-4': 'As a consequence statistical fluctuations of the simulated data are amplified by this effect.', 'astro-ph-0111111-1-47-0': 'At high geomagnetic latitudes, the solid and dashed lines merge.', 'astro-ph-0111111-1-47-1': 'The effect becomes negligible for [MATH].', 'astro-ph-0111111-1-48-0': ""Within the formalism of adiabatic invariants, it is seen that trapped particles, i.e. the undercutoff protons, move along drift shells which can be associated with a characteristic residence time that depends on the fraction of the shell located inside the Earth's atmosphere."", 'astro-ph-0111111-1-48-1': 'Thus, particles moving along long-lived shells have a large probability to cross many times a geocentered spherical detector, while those moving along short-lived shells typically cross the detector one time.', 'astro-ph-0111111-1-49-0': 'The drift shells crossing the AMS orbit, at an altitude of 400 km, are in general short-lived, however in the equatorial region the long-lived are preset as well.', 'astro-ph-0111111-1-50-0': 'In consequence, the ""real"" undercutoff component of the protons fluxes is at least one order of magnitude lower than the primary CR proton component at all latitudes, even in the equatorial region where the AMS measurement indicates an important secondary proton flux.', 'astro-ph-0111111-1-50-1': 'This can be better seen in fig.[REF], where the integral primary proton flux seen by AMS is shown as a function of geomagnetic latitude.', 'astro-ph-0111111-1-50-2': 'The intensities of the ""real"" and measured undercutoff fluxes are reported in the same plot for comparison and their ratio with the primary component shown in fig.[REF].', 'astro-ph-0111111-1-50-3': 'A minor contribution from the undercutoff proton component can be therefore expected in the atmospheric shower development and neutrino production.', 'astro-ph-0111111-1-51-0': 'In figure [REF], the life time is plotted versus the kinetic energy of the trapped secondary protons for [MATH].', 'astro-ph-0111111-1-51-1': 'In the scatter plot it is possible to distinguish the populations corresponding to long-lived and short-lived shells similar to those shown in [CITATION] for leptons.', 'astro-ph-0111111-1-52-0': 'Fig. [REF] shows the distribution of trapped secondary proton end points for [MATH], fig. [REF]a is for a lifetimes smaller than 0.3 s., while fig. [REF]b is for a lifetimes greater than 0.3 s..', 'astro-ph-0111111-1-52-1': 'The end point distribution agrees with the location of the intersections of the drift shells with the atmosphere as experimentally verified by [CITATION], and discussed in [CITATION].', 'astro-ph-0111111-1-53-0': '## Electrons and positrons', 'astro-ph-0111111-1-54-0': 'In fig. [REF] we show a comparison of the simulated undercutoff electron and positron downgoing fluxes with the corresponding AMS measured fluxes [CITATION].', 'astro-ph-0111111-1-54-1': 'The AMS positron measurement is limited to energies below [MATH], corresponding to the upper limit of proton rejection of the threshold Cherenkov counter used to distinguish protons and positrons.', 'astro-ph-0111111-1-55-0': 'In fig. [REF]b-c we show the integrated positron and electron downgoing fluxes for the kinetic energy range [MATH] as a function of [MATH].', 'astro-ph-0111111-1-55-1': 'The simulation reproduces reasonably well the general behavior of the data in terms of shape and intensity; a similar agreement is observed for the upgoing lepton spectra (not shown).', 'astro-ph-0111111-1-56-0': 'As in the case of protons, statistical fluctuations affect the comparison in the equatorial region.', 'astro-ph-0111111-1-56-1': 'The real lepton fluxes, corresponding to the real proton flux described earlier, are shown as the dashed line distributions in fig. [REF]', 'astro-ph-0111111-1-57-0': 'An interesting feature of the comparison is the fact that in the equatorial region, the electrons are produced essentially by primary protons with [MATH] , while for the positrons lower energy protons contribute as well.', 'astro-ph-0111111-1-57-1': 'This distinction disappears at higher latitudes.', 'astro-ph-0111111-1-57-2': 'This behavior can be explained by the East-West asymmetry of the geomagnetic cutoff.', 'astro-ph-0111111-1-57-3': 'Westward-moving protons produce positrons which will populate the drift shells, while the produced electrons enter the atmosphere [CITATION].', 'astro-ph-0111111-1-57-4': 'The energy spectrum of eastward-moving protons is affected by the geomagnetic cutoff.', 'astro-ph-0111111-1-57-5': 'As a consequence, the interacting protons in the equatorial region have higher energies (lower flux) yielding a lower undercutoff electron flux compared to the undercutoff positron flux.', 'astro-ph-0111111-1-58-0': '# Conclusion', 'astro-ph-0111111-1-59-0': 'Our results show good agreement between simulation and the measured data.', 'astro-ph-0111111-1-59-1': 'The Monte Carlo simulation describes well the undercutoff proton and lepton fluxes.', 'astro-ph-0111111-1-59-2': 'Our results indicate that the undercutoff proton flux should have a small impact on secondary particle production in the atmosphere.', 'astro-ph-0111111-1-59-3': 'However this will be object of further and more refined study in the future.', 'astro-ph-0111111-1-60-0': 'The simulation, constrained by the high statistic measurements of AMS, can be used to assess the radiation environment in near-Earth orbit, and represent a valuable tool for more accurate calculations of particle fluxes in atmosphere.', 'astro-ph-0111111-1-61-0': 'This work has been partially supported by the Italian Space Agency (ASI) under contract ARS-98/47.'}","{'astro-ph-0111111-2-0-0': 'Substantial fluxes of protons and leptons with energies below the geomagnetic cutoff have been measured by the AMS experiment at altitudes of 370-390 Km, in the latitude interval [MATH]51.7[MATH].', 'astro-ph-0111111-2-0-1': 'The production mechanisms of the observed trapped fluxes are investigated in detail by means of the FLUKA Monte Carlo simulation code.', 'astro-ph-0111111-2-0-2': 'All known processes involved in the interaction of the cosmic protons with the atmosphere (detailed descriptions of the magnetic field and atmospheric density, as well as the electromagnetic and nuclear interaction processes) are included in the simulation.', 'astro-ph-0111111-2-0-3': 'The results are presented and compared with the experimental data, indicating good agreement with the observed fluxes.', 'astro-ph-0111111-2-0-4': 'The impact of secondary proton flux on particle production in atmosphere is briefly discussed.', 'astro-ph-0111111-2-1-0': '# Introduction', 'astro-ph-0111111-2-2-0': 'Cosmic rays approaching the Earth interact with the atmosphere resulting in a substantial flux of secondary particles.', 'astro-ph-0111111-2-2-1': 'The knowledge of composition, intensity and energy spectra of these particles is of considerable interest, e.g. for the evaluation of background radiation for satellites and the estimate of the atmospheric neutrino production for neutrino oscillation experiments [CITATION].', 'astro-ph-0111111-2-3-0': 'The AMS measurements in near earth orbit [CITATION] have allowed, for the first time, to gather accurate information on the intensity, energy spectra and geographical origin of charged particle fluxes at energies below the geomagnetic cutoff over a wide range of latitudes and at almost all longitudes.', 'astro-ph-0111111-2-3-1': 'The under cutoff component of proton fluxes at equatorial latitudes has revealed an unexpected intensity of up to [MATH] of the primary proton flux, a positron to electron flux ratio has been found in the undercutoff component which largely exceeds the cosmic one, differences in residence times and geographical origins have been reconstructed for positively and negatively charged particles.', 'astro-ph-0111111-2-4-0': 'A robust interpretation of these and many other characteristics of the undercutoff fluxes in terms of secondary particles produced in atmosphere requires an accurate description of both the interaction processes at their origin and of the geomagnetic field effects.', 'astro-ph-0111111-2-4-1': 'Recently, different interpretations of the AMS measurements have been proposed [CITATION] based on Monte Carlo simulations using different approaches on both the generation technique and the interaction model.', 'astro-ph-0111111-2-5-0': 'In this work, we report results from a fully 3D Monte Carlo simulation based on FLUKA 2000 [CITATION] for the description of cosmic ray interactions with the atmosphere.', 'astro-ph-0111111-2-5-1': 'The key features of our analysis are an efficient generation technique for the incoming proton flux and a true microscopic, theory driven treatment of the interaction processes opposite to empirical parametrization of accelerator data.', 'astro-ph-0111111-2-5-2': 'As a first attempt the contribution of [MATH] and the heavier nuclei, representing ([MATH]) [CITATION] of the all nuclei cosmic flux, is neglected.', 'astro-ph-0111111-2-6-0': 'In the following section we give a detailed description of the basic ingredients of this simulation, the generation technique and the interaction model.', 'astro-ph-0111111-2-6-1': 'In section 3 we present the results on both protons and leptons and the comparison with AMS measurements.', 'astro-ph-0111111-2-6-2': 'In section 4 we propose our conclusions.', 'astro-ph-0111111-2-7-0': '# The model', 'astro-ph-0111111-2-8-0': 'An isotropic flux of protons is uniformly generated on a geocentric spherical surface with a radius of 1.07 Earth radii ([MATH] a.s.l.) in the kinetic energy range [MATH].', 'astro-ph-0111111-2-9-0': 'We took the functional form suggested in [CITATION] to describe the proton energy spectrum, the spectral index and the solar modulation parameter are extracted from a fit to the AMS data [CITATION].', 'astro-ph-0111111-2-10-0': ""The magnetic field in the Earth's proximity includes two components: the Earth's magnetic field, calculated using a 10 harmonics IGRF [CITATION] implementation, and the external magnetic field, calculated using the Tsyganenko Model [CITATION]."", 'astro-ph-0111111-2-10-1': 'To account for the geomagnetic effects, for each primary proton we back-trace an antiproton of the same energy until one of the following conditions is satisfied:', 'astro-ph-0111111-2-11-0': ""the particle reaches the distance of [MATH] from the Earth's center."", 'astro-ph-0111111-2-11-1': 'the particle touches again the production sphere.', 'astro-ph-0111111-2-11-2': 'neither 1 or 2 is satisfied before a time limit is reached.', 'astro-ph-0111111-2-12-0': 'If condition 1 is satisfied the particle is on an allowed trajectory, while if condition 2 is satisfied the particle is on a forbidden one.', 'astro-ph-0111111-2-12-1': 'Condition 3 arises for only a small fraction of the events [MATH].', 'astro-ph-0111111-2-13-0': ""Particles on allowed trajectories are propagated forward and can reach the Earth's atmosphere."", 'astro-ph-0111111-2-13-1': 'The atmosphere around the Earth is simulated up to [MATH] a.s.l. using 60 concentric layers of homogeneous density and chemical composition.', 'astro-ph-0111111-2-13-2': 'Data on density and chemical composition are taken from the standard MSIS model [CITATION].', 'astro-ph-0111111-2-13-3': 'The Earth is modeled as a solid sphere which absorbs each particle reaching its surface.', 'astro-ph-0111111-2-14-0': '## The generation technique', 'astro-ph-0111111-2-15-0': 'The ideal approach in the generation of the primary cosmic rays spectra would be to start with an isotropic distribution of particles at a great distance (typically [MATH]) from the Earth where the geomagnetic field introduces negligible distortions on the interstellar flux.', 'astro-ph-0111111-2-15-1': 'However, this computational method is intrinsically inefficient since most of the particles are generated with trajectories which will not reach the Earth environment.', 'astro-ph-0111111-2-15-2': 'Kinematic cuts can be applied in order to improve the selection efficiency at generation, however they tend to introduce a bias particles with low rigidity.', 'astro-ph-0111111-2-16-0': 'A good alternative to this approach is the backtracing method [CITATION],[CITATION] adopted in the present analysis as outlined in the previous section.', 'astro-ph-0111111-2-16-1': 'In the following, we will shortly discuss the validity of the technique and report the results of a comparison of the two methods.', 'astro-ph-0111111-2-16-2': 'We recall that this method was applied for the first time in ref. [CITATION] for the generation of atmospheric neutrino fluxes.', 'astro-ph-0111111-2-17-0': 'Let us consider first the effects of the geomagnetic field on an incoming flux of charged particles in the absence of a solid Earth.', 'astro-ph-0111111-2-17-1': 'For the discussion, we start with an isotropic flux of monoenergetic protons at large distance, i.e. at infinity, from the origin of a geocentrical reference system.', 'astro-ph-0111111-2-17-2': 'In this scenario, a negligible fraction of particles, with very particular initial kinematic parameters, will follow complicated paths and remains confined at a given distance from the origin (semi-bounded trajectories); for all practical purposes this sample can be ignored.', 'astro-ph-0111111-2-17-3': 'Most of the particles will follow unbounded trajectories, reaching again infinity after being deflected by the magnetic field.', 'astro-ph-0111111-2-18-0': 'Unbounded trajectories cross a spherical surface centered in the field source only an even number of times, as shown in Fig. [REF]: we call legs the trajectory parts connecting the spherical surface to infinity and loops the parts of the trajectory starting and ending inside the spherical surface.', 'astro-ph-0111111-2-19-0': 'Since each trajectory can be followed in both directions and no source or sink of particles is contained within the surface, the incoming and outgoing fluxes are the same.', 'astro-ph-0111111-2-19-1': 'However, the presence of the magnetic field breaks the isotropy of the flux ""near"" the field source, so for a given location there is a flux dependence due to the direction.', 'astro-ph-0111111-2-20-0': 'Applying the Liouville Theorem, under the hypothesis of isotropy at infinity, it is straightforward to prove [CITATION] that the proton flux in a random point is the same as at infinity along a set of directions (allowed directions), and zero along all the others (forbidden directions).', 'astro-ph-0111111-2-21-0': 'The pattern of the allowed and forbidden directions depends on both the rigidity and the location and is known as the geomagnetic cutoff.', 'astro-ph-0111111-2-22-0': 'With the introduction of a solid Earth, all the trajectories that are crossing the Earth are broken in two or more pieces (Fig. [REF]): the legs become one-way trajectories and the loops disappear.', 'astro-ph-0111111-2-23-0': 'The presence of the Earth modifies the flux which exits from the surrounding spherical surface, since particles are absorbed by the Earth, while it has only a minimal effect on the incoming flux which is modified only by the absence of certain loops.', 'astro-ph-0111111-2-23-1': ""To generate the flux of particles reaching the Earth's atmosphere, it is sufficient to follow the particles along the allowed trajectories corresponding to the legs, taking care to avoid double or multiple counting."", 'astro-ph-0111111-2-24-0': 'To respect this prescription we reject all trajectories that are back-traced to the production sphere, this allow us to correctly consider the cases like the one shown in Fig. [REF].', 'astro-ph-0111111-2-25-0': ""We point out that an important difference with respect to the application in the neutrino flux calculation of [CITATION] is that for the former, the generation sphere coincided with the Earth's surface, and therefore the forbidden trajectories included those which touched again the Earth (plus those who remained trapped for a long time)."", 'astro-ph-0111111-2-25-1': 'In that case there are no problems of double counting.', 'astro-ph-0111111-2-26-0': ""To check the validity of our technique we made a test comparing the results of the inefficient generation technique at 10 Earth's radii distance from the Earth's center with the backtracing technique described in this paper."", 'astro-ph-0111111-2-27-0': 'Fig. [REF] shows this comparison for several cha-racteristic distributions, the agreement between the two methods is good.', 'astro-ph-0111111-2-28-0': '## The interaction model', 'astro-ph-0111111-2-29-0': ""We use the software package FLUKA 2000 [CITATION] to transport the particles and describe their interactions with Earth's atmosphere."", 'astro-ph-0111111-2-29-1': 'The setup of this simulation is derived from the one used in [CITATION].', 'astro-ph-0111111-2-29-2': 'This package contains a tridimensional description of both electro-magnetic and hadronic interactions.', 'astro-ph-0111111-2-29-3': 'This code is benchmarked against a wide set of data and is already used in many applications, ranging from low energy nuclear physics to high energy accelerator and cosmic ray physics.', 'astro-ph-0111111-2-29-4': 'For this reason we have preferred this model with respect to the use of ""ad hoc"" parametrizations of particle production in the energy range of our interest.', 'astro-ph-0111111-2-30-0': 'In FLUKA hadronic interactions are treated in a theory-driven approach, and the models and their implementations are guided and checked using experimental data.', 'astro-ph-0111111-2-30-1': 'Hadron-nucleon interaction models are based on resonance production and decay below an energy of few [MATH] and on the Dual Parton Model above.', 'astro-ph-0111111-2-30-2': 'The extension from hadron to hadron-nucleus interactions is done in the framework of a generalized intra-nuclear cascade approach including the Gribov-Glauber multi-collision mechanism for higher energies followed by equilibrium processes: evaporation, fission, Fermi break-up and [MATH] de-excitation.', 'astro-ph-0111111-2-30-3': 'The parameters of the models embedded in the FLUKA package are fixed only by comparing expectations with data from accelerator experiments.', 'astro-ph-0111111-2-31-0': 'In fig [REF] a) we show the map of the primary proton interaction points in geographical coordinates.', 'astro-ph-0111111-2-31-1': 'The distribution reflects the influence of the geomagnetic cutoff.', 'astro-ph-0111111-2-31-2': 'Fig [REF] b) shows the interactions altitude profile, the solid histogram is for [MATH] while the dashed one is for [MATH].', 'astro-ph-0111111-2-31-3': 'The mean interaction altitude depends weakly on the energy.', 'astro-ph-0111111-2-32-0': 'The cosmic proton impinging in the atmosphere are doing elastic scattering in the 24% of the events and inelastic interactions in the remaining 76%, in tab. [REF] we show some characteristic of the inelastic interactions as simulated by FLUKA 2000.', 'astro-ph-0111111-2-33-0': '# Comparison with the AMS data', 'astro-ph-0111111-2-34-0': 'To compare with the AMS data, we define a detection boundary corresponding to a spherical surface at the AMS orbit altitude ([MATH] a.s.l).', 'astro-ph-0111111-2-34-1': 'We record each particle that crosses the detection boundary within the AMS field-of-view, defined as a cone with a 32[MATH] aperture with respect to the local zenith or nadir directions.', 'astro-ph-0111111-2-35-0': 'To obtain the absolute normalization, we take into account the field-of-view, the corresponding AMS acceptance, and an Equivalent Time Exposure (E.T.E.) corresponding to the number of the generated primary protons.', 'astro-ph-0111111-2-36-0': 'Our results are based on a sample of [MATH] primary protons generated in the kinetic energy range of [MATH], which corresponds to [MATH] (E.T.E).', 'astro-ph-0111111-2-37-0': '## Protons', 'astro-ph-0111111-2-38-0': 'In Fig.[REF], we show the comparison between the fluxes obtained with the simulation and the measured AMS downgoing proton flux [CITATION] in nine bins of geomagnetic latitude ([MATH]) [CITATION].', 'astro-ph-0111111-2-38-1': 'Fig.[REF] shows the same comparison for the upgoing proton flux in four selected bins of [MATH].', 'astro-ph-0111111-2-39-0': 'As seen in Fig.[REF], the simulation well reproduces at all latitudes the high energy part of the spectrum and the falloff in the primary spectrum due to the geomagnetic cutoff, thus validating the general approach used for the generation and detection, as well as the tracing technique.', 'astro-ph-0111111-2-40-0': 'A good agreement among data and simulation is also found in the under-cutoff part of the spectra.', 'astro-ph-0111111-2-40-1': 'The small and systematic deficit which can be seen in the secondary component of the simulated fluxes is of the same order of the expected contribution from the interaction of cosmic He and heavier nuclei.', 'astro-ph-0111111-2-41-0': 'This flux is due to the secondaries produced in the atmosphere and that spiral along the geomagnetic field lines up to the detection altitude.', 'astro-ph-0111111-2-41-1': 'Therefore it is sensitive to specific aspects of the interaction model and to the accuracy of the particle transport algorithm.', 'astro-ph-0111111-2-42-0': 'A correct quantitative prediction of this part of the spectra depends on the quality of low energy nucleon production both in terms of yield and energy distribution.', 'astro-ph-0111111-2-42-1': 'This is in part due to the fragmentation of the target nucleus, and depends on the details of the nuclear physics algorithms describing excitation and break up.', 'astro-ph-0111111-2-43-0': 'From the analysis of the motion of the secondary protons from their production up to their detection, it can be pointed out that a fraction of the observed flux is due to a multiple counting of the same particles.', 'astro-ph-0111111-2-43-1': ""Within the formalism of adiabatic invariants [CITATION], it is seen that charged particles trapped in the geomagnetic field, i.e. the undercutoff protons, move along drift shells which can be associated with a characteristic residence time that depends on the fraction of the shell located inside the Earth's atmosphere."", 'astro-ph-0111111-2-43-2': 'Thus, particles moving along long-lived shells have a large probability to cross many times a geocentered spherical detector, while those moving along short-lived shells typically cross the detector only once.', 'astro-ph-0111111-2-44-0': 'The drift shells crossing the AMS orbit, at an altitude of 400 km, are in general short-lived, however in the equatorial region the long-lived shells are present as well [CITATION].', 'astro-ph-0111111-2-45-0': 'In the following, we will indicate as the real proton flux that one obtained by counting only once each particle crossing the detector: its intensity is indicated by the dashed distributions in figs. [REF] and [REF].', 'astro-ph-0111111-2-46-0': 'A quite relevant effect can be seen in the equatorial region: there the AMS measurement indicates an important secondary proton flux while the real number of protons crossing the detector is more than one order of magnitude lower.', 'astro-ph-0111111-2-46-1': 'At high geomagnetic latitudes, the solid and dashed lines tend to merge.', 'astro-ph-0111111-2-46-2': 'The effect becomes negligible for [MATH].', 'astro-ph-0111111-2-47-0': 'This can be better seen in Fig.[REF], where the integral primary proton flux seen by AMS is shown as a function of geomagnetic latitude.', 'astro-ph-0111111-2-47-1': 'The intensities of the real and measured undercutoff fluxes are reported in the same plot for comparison and their ratio with the primary component shown in Fig.[REF].', 'astro-ph-0111111-2-47-2': 'A minor contribution from the undercutoff proton component can be therefore expected in the atmospheric shower development and neutrino production.', 'astro-ph-0111111-2-48-0': 'In Fig. [REF], the residence time is plotted versus the kinetic energy of the trapped secondary protons for [MATH].', 'astro-ph-0111111-2-48-1': 'In the scatter plot it is possible to distinguish the populations corresponding to long-lived and short-lived shells similar to those shown in [CITATION] for leptons.', 'astro-ph-0111111-2-49-0': 'Fig. [REF] shows the distribution of trapped secondary proton end points for [MATH], Fig. [REF]a is for a lifetimes smaller than 0.3 s., while Fig. [REF]b is for a lifetimes greater than 0.3 s..', 'astro-ph-0111111-2-49-1': 'The end point distribution agrees with the location of the intersections of the drift shells with the atmosphere as experimentally verified by [CITATION], and discussed in [CITATION].', 'astro-ph-0111111-2-50-0': '## Electrons and positrons', 'astro-ph-0111111-2-51-0': 'In Fig. [REF] we show a comparison of the simulated undercutoff electron and positron downgoing fluxes with the corresponding AMS measured fluxes [CITATION].', 'astro-ph-0111111-2-52-0': 'We remind that the AMS positron measurement is restricted at energies below few GeVs, with a dependence of the maximum energy on the geomagnetic latitude which reflects the increasing proton background with [MATH].', 'astro-ph-0111111-2-53-0': 'A comparison of data and simulation in the high energy part of the electron spectra is not possible, since the cosmic electrons have not been used as an input in the current work.', 'astro-ph-0111111-2-53-1': 'However, their contribution to the cosmic rays reaching the atmosphere is [MATH] leading to a negligible effect of in the generation of the undercutoff fluxes.', 'astro-ph-0111111-2-54-0': 'The simulation well reproduces the general behavior of the undercutoff part of the spectra in terms of shape and intensity; a similar agreement is observed for the upgoing lepton spectra (not shown).', 'astro-ph-0111111-2-54-1': 'The real lepton fluxes, corresponding to the real proton flux described earlier, are shown with the dashed line distribution in Fig. [REF].', 'astro-ph-0111111-2-54-2': 'As in the case of protons, a large effect from multiple crossing is present going toward the equatorial region, more pronounced for the positron component.', 'astro-ph-0111111-2-55-0': 'As for the undercutoff protons, we would have expected a systematic deficit in the simulated electron fluxes coming from the missing contribution of helium and heavier nuclei to the CR fluxes.', 'astro-ph-0111111-2-55-1': 'Subcutoff [MATH] are mainly (97[MATH]) coming from decays of pions produced in the proton collisions with the atmospheric nuclei: charged pions contribute through the [MATH] chain, while [MATH] through [MATH] with subsequent e.m. showers.', 'astro-ph-0111111-2-55-2': 'The relative contribution of charged pions to the subcutoff electrons (positron) fluxes at AMS altitude in our simulation is found to be [MATH]), while the remaining [MATH]) appears to come from [MATH] production.', 'astro-ph-0111111-2-55-3': 'This point deserves some considerations.', 'astro-ph-0111111-2-55-4': 'The level of agreement in the comparison of data and predictions for [MATH] fluxes turns out to be an important benchmark for the interaction model in view of a discussion on particle production in atmosphere, since is strictly linked to the meson production (mostly pions at this energy).', 'astro-ph-0111111-2-55-5': 'This work complements other studies oriented to the validation of the FLUKA model in terms of particle yields.', 'astro-ph-0111111-2-55-6': 'In particular, the quality of [MATH] generation in our interaction model has been already checked in [CITATION] through the comparison with muon fluxes measurements at different depths in atmosphere.', 'astro-ph-0111111-2-55-7': 'The muons are from the charged pions decay chain and experimental data [CITATION] are well reproduced by the simulation.', 'astro-ph-0111111-2-55-8': ""In the case of [MATH] also [MATH]'s become relevant."", 'astro-ph-0111111-2-55-9': 'Usually, when parametrized interaction models are used, like in the works of ref. [CITATION], the yield of [MATH] is fixed assuming a priori an exact charge symmetry in pion production.', 'astro-ph-0111111-2-55-10': 'In practice, this is also made necessary by the large errors that affect the scarce existing experimental data on neutral pion production.', 'astro-ph-0111111-2-55-11': 'Instead, in the case of a microscopic interaction model like FLUKA, there is no constraint of this type, and the balance of [MATH] vs. [MATH] automatically emerges from the feature of the model.', 'astro-ph-0111111-2-55-12': ""Recently, it has been pointed out how in FLUKA there exists a significant violation of charge symmetry[CITATION]: [MATH]'s are in general more ([MATH]20%) of the average of [MATH] and [MATH]."", 'astro-ph-0111111-2-55-13': 'Technically this symmetry violation emerges in the hadronization of color strings and this normally occurs also in other codes, like JETSET or PITHYA[CITATION].', 'astro-ph-0111111-2-55-14': 'We cannot enter here in a detailed discussion of this point, and we limit ourselves to say that there are reasons to believe that, at least for laboratory energies below 100 GeV, the acceptable value of charge asymmetry should be lower than that resulting from the present version of the code.', 'astro-ph-0111111-2-55-15': 'However, the comparisons of predictions to data discussed in this work, combined with the mentioned work on atmospheric muons[CITATION], already tell that the predicted fraction of [MATH] cannot be significantly wrong, although no definitive quantitative conclusion can be extracted, since the nuclear component has not been yet introduced in the primary spectrum.', 'astro-ph-0111111-2-56-0': 'In Fig.[REF]b-c we show the integrated positron and electron downgoing fluxes for the kinetic energy range [MATH] as a function of [MATH].', 'astro-ph-0111111-2-56-1': 'Their ratio is shown in Fig.[REF]a.', 'astro-ph-0111111-2-56-2': 'One of the most remarkable features of the AMS measurement is the large value of this ratio, when compared to the natural cosmic value, and its latitude dependence.', 'astro-ph-0111111-2-56-3': 'In Fig. [REF], the contribution from primary protons with [MATH] to the electron and positron fluxes is illustrated by the filled area.', 'astro-ph-0111111-2-56-4': 'We can notice that in the equatorial region, the electrons are produced essentially by primary protons with [MATH] , while for the positrons lower energy protons contribute as well.', 'astro-ph-0111111-2-56-5': 'This distinction disappears at higher latitudes, where positron and electrons are produced by the protons in the same energy range.', 'astro-ph-0111111-2-57-0': 'This behavior reflects the East-West asymmetry of the geomagnetic cutoff on primary protons and the larger probabilities of escape from atmosphere for secondary electrons(positrons) generated by Westward(Eastward) moving protons [CITATION].', 'astro-ph-0111111-2-58-0': 'Positrons are preferentially injected on drift shells reaching the AMS altitude by eastward moving protons, which experience a lower rigidity cutoff than westward moving ones.', 'astro-ph-0111111-2-58-1': 'This mechanism is more effective at the equator, where the cutoff is larger and its asymmetry maximal, resulting in the excess of undercutoff positrons from low energy protons as indicated by our simulation.', 'astro-ph-0111111-2-58-2': 'The cutoff mechanism becomes irrelevant at high latitudes, where any difference in positron and electron production should be given instead by different [MATH] production.', 'astro-ph-0111111-2-58-3': 'Nor the data neither our simulation indicate, within their uncertainties, a relevant charge asymmetry from this source.', 'astro-ph-0111111-2-59-0': '# Conclusions', 'astro-ph-0111111-2-60-0': ""The interactions of cosmic ray protons with the Earth's atmosphere have been investigated by means of a fully 3D Monte Carlo program."", 'astro-ph-0111111-2-61-0': 'The proton, electron and positron undercutoff flux intensities measured by AMS, as well as their energy spectra, have been correctly reproduced by our simulation', 'astro-ph-0111111-2-62-0': 'Geomagnetic effects, and in particular the east-west asymmetry in the cosmic protons rigidity cutoff, have been confirmed as the mechanism responsible for the measured excess of the positron component.', 'astro-ph-0111111-2-63-0': 'The main features of the geographical origin and residence time distributions for both protons and leptons have been replicated and the effect of multiple crossing of the detector by spiraling secondaries in the geomagnetic field briefly discussed.', 'astro-ph-0111111-2-64-0': 'Our results indicate that the intensity of the undercutoff proton flux, when the multiple counting is taken into account, never exceeds a 10% of the cosmic proton flux, representing a negligible source for atmospheric production of secondaries.', 'astro-ph-0111111-2-64-1': 'However, this aspect will be object of further and more refined study in the future.', 'astro-ph-0111111-2-65-0': 'The analysis on the possible strategies to generate the cosmic rays incoming flux has shown the validity of a backtracing approach as an accurate and highly efficient technique.', 'astro-ph-0111111-2-66-0': 'This work provides also additional way to validate the features of the adopted particle production model.', 'astro-ph-0111111-2-66-1': 'In particular, the study of [MATH] fluxes has revealed to be to be an interesting instrument to check the meson production in primary interactions, and the results are satisfactory.', 'astro-ph-0111111-2-66-2': 'We believe that our simulation, validated by the high statistic measurements of AMS, can be used to assess the radiation environment in near Earth orbit, and represents a valuable tool for a more accurate calculation of particle fluxes in atmosphere.', 'astro-ph-0111111-2-67-0': 'This work has been partially supported by the Italian Space Agency (ASI) under contract ARS-98/47.'}","[['astro-ph-0111111-1-16-0', 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1503.04203,"{'1503.04203-1-0-0': 'We present a new model of ""Stealth Dark Matter"": a composite baryonic scalar of an [MATH] strongly-coupled theory with even [MATH].', '1503.04203-1-0-1': 'All mass scales are technically natural, and dark matter stability is automatic without imposing an additional discrete or global symmetry.', '1503.04203-1-0-2': 'Constituent fermions transform in vector-like representations of the electroweak group that permit both electroweak-breaking and electroweak-preserving mass terms.', '1503.04203-1-0-3': 'This gives a tunable coupling of stealth dark matter to the Higgs boson independent of the dark matter mass itself.', '1503.04203-1-0-4': 'We specialize to [MATH], and investigate the constraints on the model from dark meson decay, electroweak precision measurements, basic collider limits, and spin-independent direct detection scattering through Higgs exchange.', '1503.04203-1-0-5': 'We exploit our earlier lattice simulations that determined the composite spectrum as well as the effective Higgs coupling of stealth dark matter in order to place bounds from direct detection, excluding constituent fermions with dominantly electroweak-breaking masses.', '1503.04203-1-0-6': 'A lower bound on the dark baryon mass [MATH]>[MATH] GeV is obtained from the indirect requirement that the lightest dark meson not be observable at LEP II.', '1503.04203-1-0-7': 'We briefly survey some intriguing properties of stealth dark matter that are worthy of future study, including: collider studies of dark meson production and decay; indirect detection signals from annihilation; relic abundance estimates for both symmetric and asymmetric mechanisms; and direct detection through electromagnetic polarizability, a detailed study of which will appear in a companion paper.', '1503.04203-1-1-0': 'INT-PUB-15-004, LLNL-JRNL-667446', '1503.04203-1-2-0': '# Introduction', '1503.04203-1-3-0': 'Composite dark matter, made up of electroweak-charged constituents provides a straightforward mechanism for obtaining viable electrically-neutral particle dark matter that can yield the correct cosmological abundance while surviving direct and indirect detection search limits, e.g., [CITATION].', '1503.04203-1-3-1': 'In this paradigm, the dark sector consists of fermions that transform under the electroweak group and a new, strongly-coupled non-Abelian dark force.', '1503.04203-1-3-2': 'This was considered long ago in the context of technicolor theories, where the strong dynamics was doing double-duty to both break electroweak symmetry and provide a dark matter candidate [CITATION].', '1503.04203-1-4-0': 'In this paper, electroweak symmetry breaking is accomplished through the weakly-coupled Standard Model Higgs mechanism, while the new strongly-coupled sector is reserved solely for providing a viable dark matter candidate.', '1503.04203-1-4-1': 'This dark sector is not easy to detect in dark matter detection experiments or in collider experiments, and so we give it the name ""Stealth Dark Matter"".', '1503.04203-1-4-2': 'Earlier work in this direction includes [CITATION], and except for [CITATION], was often limited by the inability to perturbatively calculate the spectrum and form factors due to strong coupling.', '1503.04203-1-5-0': 'The proposed dark matter candidate is a scalar baryon of [MATH], and hence [MATH] must be even.', '1503.04203-1-5-1': 'We take the dark fermions to be in vector-like representations of the electroweak group.', '1503.04203-1-5-2': 'Hence, the constituent dark fermions can acquire bare mass terms (fermion masses that do not require electroweak symmetry breaking) while also permitting Yukawa interactions that marry dark fermion electroweak doublets with singlets.', '1503.04203-1-5-3': 'This yields a theory in which dark matter couples to the Higgs boson in a tunable way that is essentially independent of the dark matter mass itself.', '1503.04203-1-5-4': 'This is somewhat analogous to dark sector models with a dark U(1) portal (e.g., [CITATION]), where the coupling to the Standard Model is tunable through an otherwise arbitrary parameter - the kinetic mixing between the dark U(1) and hypercharge.', '1503.04203-1-6-0': 'The existence of both electroweak-preserving and electroweak-breaking masses for the dark fermions provides two main benefits.', '1503.04203-1-6-1': 'First, given that the Higgs boson couples electroweak doublets with singlets, the global flavor symmetries of the dark fermions can be completely broken to just dark baryon number.', '1503.04203-1-6-2': 'All mesons can decay through an electroweak process (e.g., electrically charged mesons through [MATH] exchange) or through the usual chiral anomaly (e.g., the lightest neutral meson).', '1503.04203-1-6-3': 'Ensuring that these particles decay before big-bang nucleosynthesis sets a weak lower bound on the Higgs interaction strength.', '1503.04203-1-6-4': '(This is in contrast with [CITATION] where additional interactions were required to ensure mesons decay, e.g., through higher dimensional operators).', '1503.04203-1-6-5': 'The second reason is related to the orientation of the chiral condensate after the dark force confines.', '1503.04203-1-6-6': 'Large vector-like masses for the dark fermions ensure that the condensate can be aligned toward the electroweak-preserving direction, and thus the dark sector leads to only small corrections to electroweak precision measurements.', '1503.04203-1-6-7': 'We estimate the size of these corrections in this paper.', '1503.04203-1-7-0': 'There are many appealing features of an electroweak-neutral composite dark matter candidate made up from fermions transforming under the electroweak group, including:', '1503.04203-1-8-0': 'We focus mainly on a confining [MATH] gauge theory dark sector with dark fermions transforming non-trivially under the electroweak group.', '1503.04203-1-8-1': 'We apply our recent results [CITATION] using lattice simulations for the spectrum and effective Higgs interaction for [MATH].', '1503.04203-1-8-2': 'As emphasized in [CITATION], this theory is well suited for lattice calculations since we are not interested in the chiral limit of vanishing dark fermion masses.', '1503.04203-1-8-3': 'Indeed, lattice simulations can efficiently simulate the parameter region where the dark fermion masses are comparable to the confinement scale, exactly where the perturbative estimates are least useful.', '1503.04203-1-9-0': 'The organization of the paper is as follows.', '1503.04203-1-9-1': 'In Sec. [REF] we discuss the assumptions and requirements to construct our stealth dark matter model.', '1503.04203-1-9-2': 'In Sec. [REF] we detail the dark fermion interactions and masses.', '1503.04203-1-9-3': 'In addition, we write the electroweak currents in terms of the dark fermion mass eigenstates of the theory, detailed in Appendix [REF].', '1503.04203-1-9-4': 'Until this point, the discussion of the model is general.', '1503.04203-1-9-5': 'In Sec. [REF], we simplify the parameter space for phenomenological and calculational purposes, applying a global custodial [MATH] symmetry and taking the approximately symmetric dark fermion mass matrix limit.', '1503.04203-1-9-6': 'Then in Sec. [REF] we discuss the light non-singlet mesons in the theory, in particular their decay rates and constraints from non-observation at LEP II.', '1503.04203-1-9-7': 'In Sec. [REF] we discuss the stealth dark matter contributions to the [MATH] parameter, and demonstrate the parametric suppression that happens in several regimes.', '1503.04203-1-9-8': 'In Sec. [REF] we obtain the Higgs boson coupling to the dark fermions.', '1503.04203-1-9-9': 'Then in Sec. [REF] we apply our previous model-independent results on the [MATH] spectrum and effective Higgs coupling to stealth dark matter.', '1503.04203-1-9-10': 'We obtain the bounds on the parameter space from the non-observation of a spin-independent direct detection signal at LUX.', '1503.04203-1-9-11': 'We briefly discuss the relic abundance of stealth dark matter in Sec. [REF].', '1503.04203-1-9-12': 'Finally we conclude with a discussion in Sec. [REF].', '1503.04203-1-10-0': '# Constructing a viable model', '1503.04203-1-11-0': '## Basic assumptions', '1503.04203-1-12-0': 'We assume that the dark matter candidate is a composite particle of a non-Abelian, confining gauge theory based on the group [MATH] with [MATH] flavors of fermions transforming in the fundamental representation.', '1503.04203-1-12-1': 'The number [MATH] is restricted by only the condition of confinement.', '1503.04203-1-12-2': 'For reasons outlined in the introduction (abundance, detectability), the dark fermions carry electroweak charges.', '1503.04203-1-12-3': 'Our model includes a tunable Higgs ""portal"" coupling between the dark sector and the Standard Model via dimension-4 Higgs couplings.', '1503.04203-1-12-4': 'We do not consider QCD-colored dark fermions since since with [MATH], dark baryons would not generally be color singlets.', '1503.04203-1-13-0': '## Requirements', '1503.04203-1-14-0': 'We require dark matter stability to be automatic, arising from a global symmetry.', '1503.04203-1-14-1': 'This motivates considering the dark baryon of the non-Abelian dark sector to be the dark matter [CITATION].', '1503.04203-1-14-2': 'In the presence of GUT-scale or Planck-scale suppressed operators, the stability of the dark baryon should be sufficient to avoid cosmological constraints.', '1503.04203-1-15-0': 'The requirement of a sufficiently preserved accidental baryon number disfavors a dark [MATH] group.', '1503.04203-1-15-1': 'First, there is no automatic baryon number in [MATH] because there is no fundamental distinction between mesons and baryons.', '1503.04203-1-15-2': 'Imposing a global [MATH] baryon number is possible (e.g. see [CITATION]) but in addition baryon number violating dimension-5 Planck-suppressed operators such as [MATH] must be absent, where [MATH] is the dark fermion.', '1503.04203-1-15-3': '(Otherwise, the dark [MATH] baryon would decay on a timescale much shorter than the age of the Universe.)', '1503.04203-1-16-0': 'For [MATH], operators involving dark baryon decay are necessarily dimension-6 or higher, and thus safe from GUT-scale or Planck-scale suppressed violations of dark baryon number.', '1503.04203-1-16-1': '[MATH] with odd [MATH] is a perfectly interesting theory, having been studied before for [MATH] by our collaboration [CITATION].', '1503.04203-1-16-2': 'There it was found that a fermionic dark baryon has a magnetic dipole interaction that leads to a significant contribution to spin-independent scattering.', '1503.04203-1-16-3': 'Constraints from the XENON100 experiment were satisfied only when the dark matter mass [MATH]>[MATH] TeV [CITATION].', '1503.04203-1-16-4': 'This strong constraint on the mass scale implies the model is difficult to test at near-future colliders.', '1503.04203-1-17-0': 'The magnetic dipole interaction (and other higher dimensional operators that require spin) are absent when the dark baryon is a scalar.', '1503.04203-1-17-1': 'We are thus naturally led to [MATH] with even [MATH], for which the otherwise strong constraints from direct detection are weakened, lowering the scales of interest into a regime that can be probed by colliders and other detection strategies.', '1503.04203-1-18-0': 'We assume the dark fermions have masses [MATH] on the order of the [MATH] confinement scale [MATH].', '1503.04203-1-18-1': 'If the masses were much smaller, the dark sector would contain light pseudo-Goldstone pions that transform under the electroweak group, which are strongly constrained by collider experiments.', '1503.04203-1-18-2': 'A dark sector with purely vector-like fermion masses has approximately stable electrically-charged mesons due to dark flavor symmetries.', '1503.04203-1-18-3': 'Conversely, a dark sector with purely electroweak breaking fermion masses has a dark matter candidate that is ruled out by spin-independent direct detection through single Higgs exchange.', '1503.04203-1-18-4': '(For example, quirky dark matter [CITATION] is now completely ruled out by Higgs exchange, given the direct detection bounds from LUX [CITATION] combined with the relatively light Higgs mass [CITATION].)', '1503.04203-1-18-5': 'Fermions with both vector-like and (small) electroweak breaking contributions to their masses can avoid both problems.', '1503.04203-1-19-0': 'We require the lightest dark baryon to be electrically neutral.', '1503.04203-1-19-1': 'We also require Higgs couplings at dimension-4 to pairs of dark fermions.', '1503.04203-1-19-2': 'These two requirements impose restrictions on the electroweak charges of the dark fermions.', '1503.04203-1-20-0': 'One solution is familiar from old technicolor theories (e.g. [CITATION]): requiring the dark fermion charges to roughly satisfy [MATH]<[MATH] where [MATH] is the [MATH] isospin.', '1503.04203-1-20-1': 'Choosing doublets ([MATH] under [MATH]) then gives a finite number of discrete possibilities.', '1503.04203-1-21-0': 'A simple model that satisfies all of these requirements is shown in Table [REF].', '1503.04203-1-21-1': 'The electric charges of the dark fermions in the broken electroweak phase are [MATH], ensuring all hadrons have integer electric charges.', '1503.04203-1-21-2': 'So long as the lightest [MATH] and [MATH] dark fermions are close in mass, the lightest baryon will be a scalar and electrically neutral.', '1503.04203-1-21-3': 'Finally, with the assignments shown in Table [REF], all gauge (and global) anomalies vanish, which is automatic with fermions that transform under vector-like representations of the [MATH] and electroweak groups.', '1503.04203-1-22-0': '# Dark fermion interactions and masses', '1503.04203-1-23-0': 'The fermions [MATH] transform under a global [MATH] flavor symmetry with [[MATH]][MATH] surviving after the weak gauging of the electroweak symmetry.', '1503.04203-1-23-1': 'From this large global symmetry, one [MATH] (diagonal) subgroup will be identified with [MATH], one [MATH] subgroup will be identified with [MATH], and one [MATH] will be identified with dark baryon number.', '1503.04203-1-23-2': 'The total fermionic content of the model is therefore 8 Weyl fermions that pair up to become 4 Dirac fermions in the fundamental or anti-fundamental representation of [MATH] with electric charges of [MATH].', '1503.04203-1-23-3': 'We use the notation where the superscript [MATH] or [MATH] (as in [MATH], [MATH] and later [MATH], [MATH], [MATH], [MATH]) denotes a fermion with electric charge of [MATH] or [MATH] respectively.', '1503.04203-1-24-0': 'The fermion kinetic terms in the Lagrangian are given by [EQUATION] where the covariant derivatives are [EQUATION] with the interactions among the electroweak group and the new [MATH].', '1503.04203-1-24-1': 'Here [MATH], [MATH] and [MATH] are the representation matrices for the fundamental of [MATH].', '1503.04203-1-25-0': 'The vector-like mass terms allowed by the gauge symmetries are [EQUATION] where [MATH] and the relative minus signs between the mass terms have been chosen for later convenience.', '1503.04203-1-25-1': 'The mass term [MATH] explicitly breaks an [[MATH]][MATH] global symmetry down to the diagonal [MATH] where the [MATH] is identified with [MATH].', '1503.04203-1-25-2': ""The mass terms [MATH] explicitly break the remaining [[MATH]][MATH] down to [MATH] where one of the [MATH]'s is identified with [MATH]."", '1503.04203-1-25-3': '(In the special case when [MATH], the global symmetry is enhanced to [MATH], where the global [MATH] acts as a custodial symmetry.)', '1503.04203-1-25-4': 'Thus, after weakly gauging the electroweak symmetry and writing arbitrary vector-like mass terms, the unbroken flavor symmetry is [MATH].', '1503.04203-1-26-0': 'Electroweak symmetry breaking mass terms arise from coupling to the Higgs field [MATH] that we take to be in the [MATH] representation.', '1503.04203-1-26-1': 'They are given by [EQUATION] where again the relative minus signs are chosen for later convenience.', '1503.04203-1-26-2': 'After electroweak symmetry breaking, [MATH], with [MATH] GeV.', '1503.04203-1-26-3': 'Replacing the Higgs field by its VEV in Eq. ([REF]), we obtain mass terms for the fermions, in 2-component notation, [EQUATION] with the mass matrices given by [EQUATION]', '1503.04203-1-26-4': 'These Yukawa couplings break the remaining [MATH] flavor symmetry to [MATH] dark baryon number.', '1503.04203-1-26-5': 'The mass matrices [MATH] and [MATH] correspond to the masses of two sets of fermions with electric charge [MATH] and [MATH] respectively, in the fundamental representation of [MATH].', '1503.04203-1-26-6': 'The two biunitary mass matrices can be diagonalized by four independent rotation angles [EQUATION] where the rotation matrices are defined by [EQUATION]', '1503.04203-1-26-7': 'The 2-component mass eigenstate spinors are thus [EQUATION] where the extra phase in Eqs. ([REF]),([REF]) ensures the [MATH] fermions will have positive mass eigenvalues.', '1503.04203-1-27-0': 'The Lagrangian for the fermion mass eigenstates becomes [EQUATION] where the mass eigenvalues are [MATH] for [MATH], and the distinction between fermions [MATH] and [MATH] allows us to write the [MATH] fermion masses as [MATH].', '1503.04203-1-27-1': 'The Dirac spinor mass eigenstates are constructed from the 2-component Weyl spinor mass eigenstates in the usual way, [EQUATION] giving the Dirac fermion masses [EQUATION]', '1503.04203-1-27-2': 'The fermion masses themselves are obtained from a straightforward diagonalization of the mass matrices, [EQUATION] with mixing angles [EQUATION] with identical expressions for [MATH] and [MATH] with the replacement [MATH] everywhere.', '1503.04203-1-28-0': 'It is important to note that the electroweak currents ([MATH], [MATH], [MATH], [MATH]) play an important role in the upcoming phenomenological discussions.', '1503.04203-1-28-1': 'Due to the extended expressions for these quantities in terms of our Dirac spinors, we have relegated a detailed derivation of the electroweak currents to Appendix [REF].', '1503.04203-1-29-0': '# Simplifications', '1503.04203-1-30-0': 'Our main interest is the more specialized case where the lightest [MATH] and [MATH] fermions are degenerate in mass to a very good approximation.', '1503.04203-1-30-1': 'This leads to a neutral scalar baryon with a vanishing charge radius.', '1503.04203-1-30-2': 'While there are several ways this could be accomplished, we can simply impose a custodial [MATH] global symmetry on the Lagrangian.', '1503.04203-1-30-3': 'In order to simplify notation, we define [MATH], [MATH] and [MATH].', '1503.04203-1-30-4': 'In the custodial [MATH] symmetric theory, [MATH] and [MATH].', '1503.04203-1-31-0': '## Custodial SU(2)', '1503.04203-1-32-0': 'An exact custodial [MATH] symmetry implies the masses and interactions are symmetric with respect to the interchange [MATH].', '1503.04203-1-32-1': 'This means the Lagrangian parameters satisfy [EQUATION]', '1503.04203-1-32-2': 'Defining the overall vector-like mass scale [MATH] and difference [MATH] to be [EQUATION] the dark fermion mass eigenvalues are [EQUATION]', '1503.04203-1-32-3': 'No [MATH] or [MATH] labels are necessary, since custodial [MATH] symmetry implies that there is one pair of Dirac fermions with electric charge [MATH] with mass [MATH] (the lightest pair), as well as a second pair of Dirac fermions with electric charge [MATH] with mass [MATH] (the heavier pair).', '1503.04203-1-32-4': 'The spectrum is illustrated in Fig. [REF].', '1503.04203-1-33-0': 'In the limit [MATH], the fermions acquire purely vector-like masses, and thus the chiral condensate of the dark force is aligned to a purely electroweak-preserving direction.', '1503.04203-1-33-1': ""In order that the chiral condensate's electroweak-preserving orientation is not significantly disrupted, we consider small electroweak breaking masses, [MATH]."", '1503.04203-1-34-0': 'This leaves two distinct regimes for the spectrum, depending on the relative sizes of [MATH] and [MATH].', '1503.04203-1-35-0': '## Approximately symmetric mass matrices', '1503.04203-1-36-0': 'A second simplification, useful to analytically and numerically evaluate our results, is to take [MATH].', '1503.04203-1-36-1': 'The mass matrices Eqs. ([REF],[REF]) are approximately symmetric.', '1503.04203-1-36-2': 'Specifically, we can write [EQUATION] and expand in powers of [MATH].', '1503.04203-1-36-3': 'For example, the dark fermion masses become simply [EQUATION] to leading order in [MATH].', '1503.04203-1-37-0': 'The distinct regimes are thus [MATH] and [MATH].', '1503.04203-1-37-1': 'In the Linear Case, electroweak symmetry breaking is (dominantly) responsible for the mass splitting between [MATH] and [MATH].', '1503.04203-1-37-2': 'In the Quadratic Case, the splitting is dominantly attributed to the vector-like mass splitting [MATH].', '1503.04203-1-37-3': 'As we shall see, the primary distinction between these two cases is in the Higgs coupling to the fermion mass eigenstates: proportional to [MATH] for the Linear Case and [MATH] for the Quadratic Case, hence the case names.', '1503.04203-1-37-4': 'A similar observation was also found in Ref. [CITATION].', '1503.04203-1-38-0': 'From this point forward unless noted otherwise, we assume the fermion mass parameters satisfy an exact custodial [MATH] and the mass matrices are approximately symmetric.', '1503.04203-1-39-0': '# Light Non-Singlet Meson Phenomenology', '1503.04203-1-40-0': 'Theories with new fermions that transform under vector-like representations of the electroweak group generically have enlarged global flavor symmetries that can prevent decay of the lightest non-singlet mesons and baryons.', '1503.04203-1-40-1': 'In the case of dark baryons, this is a feature, providing the rationale for the stability of the lightest dark baryon of the theory.', '1503.04203-1-41-0': 'In the case of the lightest non-singlet mesons, this can be problematic, since some of these mesons carry electric charge.', '1503.04203-1-41-1': 'Stable integer charged mesons are strongly constrained from collider searches as well as cosmology.', '1503.04203-1-41-2': 'One solution is to postulate additional higher dimensional operators that connect a dark fermion pair with a Standard Model fermion pair [CITATION].', '1503.04203-1-41-3': 'This must be carefully done to avoid also writing operators that violate the approximate global symmetries protecting the stability of the dark matter.', '1503.04203-1-41-4': 'In the stealth dark matter model, however, electroweak symmetry breaking can provide the source of global flavor symmetry breaking, leading to the decay of the lightest charged mesons.', '1503.04203-1-41-5': '(We will not discuss the lightest neutral mesons, but they are generically more difficult to produce in colliders, and they will decay through essentially the same mechanism as we describe for the charged mesons.)', '1503.04203-1-42-0': 'The lightest electrically charged mesons are composed dominantly of the dark fermion pairs [MATH] and [MATH].', '1503.04203-1-42-1': 'We can estimate the lightest meson lifetime by generalizing pion decay of QCD to our model.', '1503.04203-1-42-2': 'The relevant matrix element is (see, e.g., [CITATION]) [EQUATION] where [MATH] is the ""pion decay constant"" associated with the dark force in this paper.', '1503.04203-1-42-3': 'The axial part of the electroweak current can be read off from the electroweak currents given in Eqs. ([REF]),([REF]) [EQUATION] where [EQUATION] and [MATH] is identical upon [MATH].', '1503.04203-1-42-4': 'In the custodial limit, Eq. ([REF]), the axial coefficient is [EQUATION]', '1503.04203-1-42-5': 'Some insight can be gained using approximately symmetric mass matrices, Eq. ([REF]).', '1503.04203-1-42-6': 'We then obtain [EQUATION]', '1503.04203-1-42-7': 'The decay width can be obtained from pion decay of QCD by replacing [MATH] in the Standard Model with [MATH] for the dark mesons.', '1503.04203-1-42-8': 'Since the charged dark mesons of this model are much heavier than the QCD pions, there are many possible decay modes.', '1503.04203-1-42-9': 'For a general decay into a Standard Model doublet [MATH], assuming [MATH], the decay width is [EQUATION]', '1503.04203-1-42-10': 'If [MATH], the dominant decay mode is expected to be [MATH], otherwise [MATH] and [MATH], with branching ratios of roughly 70% and 30% respectively.', '1503.04203-1-42-11': 'Note that the decay width has several enhancement factors relative to the QCD pion decay width [EQUATION] where for simplicity we have neglected kinematic suppression.', '1503.04203-1-42-12': 'As an example, if [MATH], we find the lightest charged dark mesons decay faster than QCD charged pions so long as [MATH]>[MATH].', '1503.04203-1-42-13': 'This is easy to satisfy with small Yukawa couplings and dark fermion masses at or beyond the electroweak scale.', '1503.04203-1-43-0': 'We can now make some comments about existing collider constraints on non-singlet mesons.', '1503.04203-1-43-1': 'The lightest charged mesons [MATH] can be pair produced in particle colliders through the Drell-Yan process, and will decay through annihilation of the constituent fermions into a [MATH] boson.', '1503.04203-1-43-2': 'Because the Drell-Yan production is mediated by a photon and the mesons have unit electric charge, the production cross-section is substantial, leading to robust bounds from LEP-II.', '1503.04203-1-43-3': 'For charged states near the LEP-II energy threshold, the dominant decay mode is expected to be [MATH] as noted above.', '1503.04203-1-43-4': 'Reinterpreting the LEP-II bound from the pair production of supersymmetric partners to the tau (with the stau decaying into a tau and a nearly massless gravitino), we find [MATH]>[MATH] GeV [CITATION].', '1503.04203-1-43-5': 'Stronger bounds from the LHC may be possible, although existing searches do not yet give any significant constraints on the charged mesons [CITATION]; we briefly highlight the signals in the discussion.', '1503.04203-1-44-0': 'Using our lattice results from Ref. [CITATION], we can translate the experimental bound on the mass of the pseudoscalar meson into a bound on the baryon mass, [MATH] GeV when the ratio of the pseudoscalar mass to the vector meson mass is [MATH].', '1503.04203-1-45-0': '# Contributions to Electroweak Precision Observables', '1503.04203-1-46-0': 'Stealth dark matter contains dark fermions that acquire electroweak symmetry breaking contributions to their masses.', '1503.04203-1-46-1': 'Consequently, there are contributions to the electroweak precision observables of the Standard Model, generally characterized by [MATH] and [MATH] [CITATION].', '1503.04203-1-46-2': 'In the custodial [MATH] limit, Eq. ([REF]), the contribution to [MATH] vanishes.', '1503.04203-1-46-3': 'There is a contribution to [MATH], controllable through the relative size of the electroweak breaking and electroweak preserving masses of the dark fermions.', '1503.04203-1-47-0': 'The [MATH] parameter is defined in terms of momentum derivatives of current-current correlators [CITATION], [EQUATION] where the currents [MATH] and [MATH] for the stealth dark matter model are defined in Eqs. ([REF]) and ([REF]).', '1503.04203-1-47-1': 'After some algebra and identifications of symmetric contractions, these definitions of the currents in terms of 4-component fermion fields lead to the current-current correlator.', '1503.04203-1-47-2': 'In the custodial limit, we obtain [EQUATION] where the connected contributions to the correlation functions are given by [EQUATION]', '1503.04203-1-47-3': 'Here, [MATH] and the flavor indices [MATH], where it is understood that the flavors labeled [MATH] have larger fermion masses than the flavors labeled [MATH].', '1503.04203-1-47-4': 'Since the [MATH] flavors have the same mass, the [MATH] and [MATH] labels are interchangeable (i.e. everything is written in terms of the [MATH] flavors).', '1503.04203-1-48-0': 'We can obtain expressions for the mixing angle coefficients.', '1503.04203-1-48-1': 'Like the case of light meson decay, if we consider an approximately symmetric mass matrix, with Yukawa couplings given by Eq. ([REF]), all of the mixing angle coefficients are approximately equal to each other, differing only at first order in [MATH], i.e., [EQUATION]', '1503.04203-1-48-2': 'In the Linear Case, the mixing angles are approximately equal [MATH].', '1503.04203-1-48-3': 'In the Quadratic Case, all of the contributions to the [MATH] parameter are suppressed by [MATH].', '1503.04203-1-48-4': 'To calculate the [MATH] parameter in general requires lattice methods, paying close attention to the heavy-light splitting of the fermions, [MATH].', '1503.04203-1-48-5': 'To a first approximation we expect that in the limit of small mass splitting, [MATH], [EQUATION]', '1503.04203-1-48-6': 'This gives for the current-current correlator [EQUATION] where all of the [MATH] and [MATH] contributions self-cancel.', '1503.04203-1-48-7': 'Hence, we see that the contribution to the [MATH] parameter is suppressed as [MATH] or [MATH], as expected.', '1503.04203-1-49-0': '# Fermion Couplings to the Higgs Boson', '1503.04203-1-50-0': 'In terms of the gauge-eigenstate fields, the interactions of the Higgs boson with the dark-sector fermions are, in matrix notation, [EQUATION]', '1503.04203-1-50-1': 'These matrices are not simultaneously diagonalizable with the mass matrices, Eqs. ([REF]),([REF]).', '1503.04203-1-50-2': 'This means that the Higgs boson in general has off-diagonal, ""dark flavor-changing"" interactions with the mass eigenstate fields.', '1503.04203-1-50-3': 'Explicitly, we find in terms of the mixing angles [EQUATION]', '1503.04203-1-50-4': 'In the custodial [MATH] limit, we can drop the [MATH] and [MATH] labels since the Higgs coupling matrix is identical for both sets of fields.', '1503.04203-1-50-5': 'If we further take the limit of an approximately symmetric mass matrix, Eq. ([REF]), the Higgs couplings simplify to [EQUATION]', '1503.04203-1-50-6': 'We observe both diagonal and off-diagonal Higgs couplings to the fermions.', '1503.04203-1-50-7': 'The off-diagonal dark flavor-changing interactions vanish in the limit [MATH] and [MATH].', '1503.04203-1-50-8': 'In this limit an enhanced flavor symmetry among the fermions is restored, and the analogue of the GIM mechanism forbids such interactions at tree-level.', '1503.04203-1-50-9': 'The off-diagonal Higgs couplings lead to an inelastic scattering cross section when a single Higgs is exchanged.', '1503.04203-1-50-10': 'This is highly suppressed unless the mass difference [MATH] is near the (non-relativistic) kinetic energy of the dark matter in galaxy.', '1503.04203-1-50-11': 'Two off-diagonal Higgs couplings can be combined in a loop involving one heavier dark fermion and double Higgs exchange, but this is suppressed by the square of the Higgs couplings times a loop factor, as well as by the mass of the heavier fermions.', '1503.04203-1-51-0': 'The single Higgs coupling to the lightest fermions is finally [EQUATION] where [EQUATION] (Note also that the single Higgs coupling to the heaviest fermions [MATH] is identical up to an overall sign.)', '1503.04203-1-51-1': 'Depending on the relative size of [MATH] and [MATH], the Higgs boson couples linearly or quadratically proportional to the Yukawa coupling [MATH].', '1503.04203-1-51-2': 'The additional suppression of [MATH] in the Quadratic Case will imply that spin independent scattering through single Higgs exchange can be significantly weaker when the mass difference between the lightest and heaviest fermions is dominated by the electroweak preserving mass [MATH].', '1503.04203-1-52-0': '# Direct detection bounds from Higgs exchange', '1503.04203-1-53-0': 'In a previous paper [CITATION], we determined the model-independent bounds on direct detection from Higgs exchange for a scalar baryon of [MATH].', '1503.04203-1-53-1': 'The model-independent result was expressed in terms of the effective Higgs coupling to the baryon [EQUATION]', '1503.04203-1-53-2': 'The first factor, the baryon mass [MATH] (divided by the electroweak vev), as well as the third factor [EQUATION] are extracted from our lattice results [CITATION].', '1503.04203-1-53-3': 'The second factor [EQUATION] provides the effective coupling of the Higgs boson to the fermions (multiplied by [MATH]), and we have evaluated the derivative for the two cases in our model.', '1503.04203-1-54-0': 'Unfortunately, we cannot directly apply our previous results on constraints in [MATH]-[MATH] space to the parameters of the stealth dark matter model.', '1503.04203-1-54-1': 'This is because we do not know the dark fermion mass, [MATH], independent of the lattice regularization scheme.', '1503.04203-1-54-2': 'We can, however, construct a regularization-independent parameter, the effective Yukawa coupling [MATH], that is closely related to the model parameters: [EQUATION]', '1503.04203-1-54-3': 'The [MATH] parameter is therefore [EQUATION]', '1503.04203-1-54-4': 'Recasting our previous constraints in [MATH]-[MATH] space into [MATH]-[MATH] space, we can identify the region of parameter space that remains viable.', '1503.04203-1-54-5': 'The constraints for the Linear Case are shown in Fig. [REF] and the Quadratic Case in Fig. [REF].', '1503.04203-1-54-6': 'In the top two plots for the respective figures, the region above the LUX bounds represents the excluded parameter space for the model at a given dark matter mass ([MATH]) and effective Yukawa coupling ([MATH]).', '1503.04203-1-54-7': 'The figures show a clear qualitative trend in how the predictions change as a function of dark matter mass.', '1503.04203-1-54-8': 'In particular, the cross-section is independent of [MATH] for the Linear Case and inversely proportional to [MATH] in the Quadratic Case.', '1503.04203-1-54-9': 'The bottom plot in the respective figures represent the allowed [MATH] values, which are depicted as the area under the curves.', '1503.04203-1-54-10': 'By increasing the splitting [MATH] between the vector-like mass terms, significantly more [MATH] parameter space becomes available.', '1503.04203-1-55-0': '# Abundance', '1503.04203-1-56-0': 'We now provide a brief discussion of the relic abundance of stealth dark matter.', '1503.04203-1-56-1': 'In the regime where the dark fermions have masses comparable to the confinement scale of the dark force, calculating the relic abundance is an intrinsically strongly-coupled calculation.', '1503.04203-1-56-2': 'Unfortunately, this calculational difficulty is not easily overcome with lattice simulations, due to the different initial and final states.', '1503.04203-1-56-3': 'Nevertheless, it is straightforward to see that the relic abundance can match the cosmological abundance through at least two distinct mechanisms that lead to two different mass scales for stealth dark matter.', '1503.04203-1-56-4': 'In this section we discuss obtaining the abundance of stealth dark matter through thermal freezeout, leading to a symmetric abundance of dark baryons and anti-baryons.', '1503.04203-1-56-5': 'Separately, we consider the possibility of an asymmetric abundance generated through electroweak sphalerons.', '1503.04203-1-57-0': '## Symmetric Abundance', '1503.04203-1-58-0': 'In the early universe at temperatures well above the confinement scale of the [MATH] dark gauge force, the dark fermions are in thermal equilibrium with the thermal bath through their electroweak interactions.', '1503.04203-1-58-1': 'As the universe cools to temperatures below the confinement scale, the degrees of freedom change from dark fermions and gluons into the dark baryons and mesons of the low energy description.', '1503.04203-1-58-2': 'Some of the dark mesons carry electric charge, and so the dark mesons remain in thermal equilibrium with the Standard Model quarks, leptons, and gauge fields.', '1503.04203-1-58-3': 'Since the dark baryons are strongly coupled to the dark mesons, they also are kept in thermal equilibrium.', '1503.04203-1-58-4': 'As the temperature of the universe falls well below the mass of the dark baryons, they annihilate into dark mesons that subsequently thermalize and decay (or decay then thermalize) into Standard Model particles.', '1503.04203-1-58-5': 'The symmetric abundance of dark baryons is therefore determined by the annihilation rate of dark baryons into dark mesons.', '1503.04203-1-59-0': 'The annihilation rate of dark baryons to dark mesons is a strongly coupled process.', '1503.04203-1-59-1': 'We expect [MATH], [MATH], and [MATH], (and to possibly more mesons if kinematically allowed) to occur, but we do not know the dominant annihilation channel.', '1503.04203-1-59-2': 'If the 2-to-2 process [MATH] dominates, one approach is to use partial wave unitarity estimate the annihilation rate [CITATION].', '1503.04203-1-59-3': 'Using this method, we can estimate the thermally averaged annihilation rate to be [EQUATION] where [MATH] at freezeout [CITATION].', '1503.04203-1-59-4': 'Matching this cross section to the required thermal relic abundance yields [MATH] TeV.', '1503.04203-1-59-5': 'An alternative approach is to use naive dimensional analysis (NDA) [CITATION], which appears to lead to a larger dark matter mass.', '1503.04203-1-60-0': 'If the 2-to-3 or 2-to-4 processes dominate instead, the additional phase space and kinematic suppression lowers the annihilation rate, and therefore lowers the scalar baryon mass needed to obtain the cosmological abundance.', '1503.04203-1-60-1': 'For recent work that has considered the thermal relic abundance in multibody processes, see [CITATION].', '1503.04203-1-60-2': 'Suffice it to say a symmetric thermal abundance of dark baryons will match the cosmological abundance for a relatively large baryon mass that is of order tens to hundreds of TeV.', '1503.04203-1-61-0': '## Asymmetric Abundance', '1503.04203-1-62-0': 'Early work on technibaryons demonstrated that strongly-coupled dark matter could arise from an asymmetric abundance [CITATION].', '1503.04203-1-62-1': 'The main ingredient to obtain the correct cosmological abundance involved the electroweak sphaleron - the non-perturbative solution at finite temperature that allows for transitions between vacua with different [MATH] numbers.', '1503.04203-1-62-2': 'In the early universe, at temperatures much larger than the electroweak scale, electroweak sphalerons are expected to violate one accidental global symmetry, [MATH] number, leaving [MATH] and [MATH] numbers unaffected [CITATION].', '1503.04203-1-62-3': 'Here [MATH] number is proportional to the dark baryon number, with some appropriate normalization (for examples, see [CITATION]).', '1503.04203-1-63-0': 'Given a baryogenesis mechanism, the electroweak sphalerons redistribute baryon number into lepton number and dark baryon number.', '1503.04203-1-63-1': 'As the universe cools, the mass of the technibaryon becomes larger than the temperature of the Universe.', '1503.04203-1-63-2': 'Eventually, the universe cools to the point where electroweak sphalerons ""freeze out"" and can no longer continue exchanging [MATH], [MATH], and [MATH] numbers.', '1503.04203-1-63-3': 'The residual abundance of dark baryons is [MATH] where the number density is proportional to [MATH], where [MATH] is the temperature at which sphaleron interactions shut off.', '1503.04203-1-64-0': 'If the baryon and dark baryon number densities are comparable, the would-be overabundance of dark matter (from [MATH]) is compensated by the Boltzmann suppression.', '1503.04203-1-64-1': 'Very roughly, [MATH]-[MATH] TeV is the natural mass scale that matches the cosmological abundance of dark matter [CITATION].', '1503.04203-1-64-2': 'A crucial component of the early technibaryon papers [CITATION] is that the technifermions were in a purely chiral representation of the electroweak group, like the fermions of the Standard Model.', '1503.04203-1-65-0': 'In stealth dark matter, given an early baryogenesis mechanism (or other analogous mechanism to generate an asymmetry in a globally conserved quantity [CITATION]), it is possible that electroweak sphalerons could also lead to the correct relic abundance of dark baryons consistent with cosmology.', '1503.04203-1-66-0': 'There is one critical difference from the early technicolor models (as well as the quirky dark matter model): The dark fermions in stealth dark matter have both vector-like and electroweak symmetry breaking masses.', '1503.04203-1-66-1': 'This leads to a suppression of the effectiveness of the electroweak sphalerons by a factor of [MATH], c.f. Eq. ([REF]), leading to a somewhat smaller stealth baryon mass to obtain the correct relic abundance compared with a technicolor model (all other parameters equal).', '1503.04203-1-66-2': 'A more quantitative estimate is complicated by several factors:', '1503.04203-1-67-0': 'Given the exponential suppression of the asymmetric abundance as the dark baryon mass is increased, it is clear that the upper bound on the dark baryon mass is nearly the same as the technibaryon calculation (updated to the current cosmological parameters), when stealth dark fermions have vector-like masses comparable to electroweak symmetry breaking masses.', '1503.04203-1-67-1': '(This case is, however, constrained by the [MATH] parameter, see Sec. [REF]).', '1503.04203-1-67-2': 'We can therefore anticipate that a range of stealth dark matter masses will be viable, up to about a TeV.', '1503.04203-1-67-3': 'More precise predictions require further detailed investigation that is beyond the scope of this paper.', '1503.04203-1-68-0': '# Discussion', '1503.04203-1-69-0': 'We have presented a concrete model, ""stealth dark matter"", that is a composite baryonic scalar of a new [MATH] strongly-coupled confining gauge theory with dark fermions transforming under the electroweak group.', '1503.04203-1-69-1': 'Though the stealth dark matter model has a wide parameter space, we focused on dark fermion masses that respect an exact custodial [MATH].', '1503.04203-1-69-2': 'Custodial [MATH] implies the lightest bosonic baryonic composite is an electrically neutral scalar (and not a vector or spin-2) of the [MATH] dark spectrum, and in addition does not have a charge radius.', '1503.04203-1-69-3': 'This yields an exceptionally ""stealthy"" dark matter candidate, with spin-independent direct detection scattering proceeding only through Higgs exchange (studied in this paper) and the polarizability interaction (studied in our companion paper [CITATION]).', '1503.04203-1-69-4': 'Custodial [MATH] also allows for stealth dark matter to completely avoid the constraints from the [MATH] parameter.', '1503.04203-1-69-5': 'While contributions to the [MATH] parameter are present, they are suppressed by the ratio of the electroweak symmetry breaking mass-squared divided by a vector-like mass squared of the dark fermions.', '1503.04203-1-69-6': 'We also verified the lightest non-singlet mesons decay rapidly (so long as [MATH]), avoiding any cosmological issues with stable electrically-charged dark mesons.', '1503.04203-1-70-0': 'Specializing to the case of [MATH], we then applied our earlier model-independent lattice results [CITATION] to the parameters of stealth dark matter, and obtained constraints on the effective Higgs interaction.', '1503.04203-1-70-1': 'We find that the present LUX bound is able only to mildly constrain the Higgs coupling to stealth dark matter for relatively light dark baryons.', '1503.04203-1-70-2': 'Even weaker constraints arise when the effective Higgs interaction is quadratic in the Yukawa coupling, which is a natural possibility when the two pairs of dark fermions are split dominantly by vector-like masses, i.e., [MATH].', '1503.04203-1-71-0': 'While we have considered many aspects of stealth dark matter, several avenues warrant further investigation:', '1503.04203-1-72-0': 'Finally there are broader model-building questions to consider.', '1503.04203-1-72-1': 'One is the choice of scales [MATH] that has been the focus of this work.', '1503.04203-1-72-2': 'This could arise dynamically.', '1503.04203-1-72-3': 'For example, if there are sufficient flavors in the [MATH] gauge theory such that it is approximately conformal at high energies, then as the theory is run down through the dark fermion mass scale [MATH], the dark fermions integrate out, and confinement sets in at [MATH].', '1503.04203-1-72-4': 'This is well known to occur for supersymmetric [MATH] theories in the conformal window that flow to confining theories once the number of flavors drops below [MATH] [CITATION].', '1503.04203-1-72-5': 'The origin of the vector-like masses of the fermions is also an interesting model-building puzzle.', '1503.04203-1-72-6': 'However, just as SM fermion masses are vector-like below the electroweak breaking scale, we can imagine dark fermion vector-like masses could be revealed as arising from dynamics that breaks the flavor symmetries of our dark fermions at some higher scale.'}","{'1503.04203-2-0-0': 'We present a new model of ""Stealth Dark Matter"": a composite baryonic scalar of an [MATH] strongly-coupled theory with even [MATH].', '1503.04203-2-0-1': 'All mass scales are technically natural, and dark matter stability is automatic without imposing an additional discrete or global symmetry.', '1503.04203-2-0-2': 'Constituent fermions transform in vector-like representations of the electroweak group that permit both electroweak-breaking and electroweak-preserving mass terms.', '1503.04203-2-0-3': 'This gives a tunable coupling of stealth dark matter to the Higgs boson independent of the dark matter mass itself.', '1503.04203-2-0-4': 'We specialize to [MATH], and investigate the constraints on the model from dark meson decay, electroweak precision measurements, basic collider limits, and spin-independent direct detection scattering through Higgs exchange.', '1503.04203-2-0-5': 'We exploit our earlier lattice simulations that determined the composite spectrum as well as the effective Higgs coupling of stealth dark matter in order to place bounds from direct detection, excluding constituent fermions with dominantly electroweak-breaking masses.', '1503.04203-2-0-6': 'A lower bound on the dark baryon mass [MATH]>[MATH] GeV is obtained from the indirect requirement that the lightest dark meson not be observable at LEP II.', '1503.04203-2-0-7': 'We briefly survey some intriguing properties of stealth dark matter that are worthy of future study, including: collider studies of dark meson production and decay; indirect detection signals from annihilation; relic abundance estimates for both symmetric and asymmetric mechanisms; and direct detection through electromagnetic polarizability, a detailed study of which will appear in a companion paper.', '1503.04203-2-1-0': 'INT-PUB-15-004, LLNL-JRNL-667446', '1503.04203-2-2-0': '# Introduction', '1503.04203-2-3-0': 'Composite dark matter, made up of electroweak-charged constituents provides a straightforward mechanism for obtaining viable electrically-neutral particle dark matter that can yield the correct cosmological abundance while surviving direct and indirect detection search limits, e.g., [CITATION].', '1503.04203-2-3-1': 'In this paradigm, the dark sector consists of fermions that transform under the electroweak group and a new, strongly-coupled non-Abelian dark force.', '1503.04203-2-3-2': 'This was considered long ago in the context of technicolor theories, where the strong dynamics was doing double-duty to both break electroweak symmetry and provide a dark matter candidate [CITATION].', '1503.04203-2-4-0': 'In this paper, electroweak symmetry breaking is accomplished through the weakly-coupled Standard Model Higgs mechanism, while the new strongly-coupled sector is reserved solely for providing a viable dark matter candidate.', '1503.04203-2-4-1': 'This dark sector is not easy to detect in dark matter detection experiments or in collider experiments, and so we give it the name ""Stealth Dark Matter"".', '1503.04203-2-4-2': 'Earlier work in this direction includes [CITATION], and except for [CITATION], was often limited by the inability to perturbatively calculate the spectrum and form factors due to strong coupling.', '1503.04203-2-5-0': 'The proposed dark matter candidate is a scalar baryon of [MATH], and hence [MATH] must be even.', '1503.04203-2-5-1': 'We take the dark fermions to be in vector-like representations of the electroweak group.', '1503.04203-2-5-2': 'Hence, the constituent dark fermions can acquire bare mass terms (fermion masses that do not require electroweak symmetry breaking) while also permitting Yukawa interactions that marry dark fermion electroweak doublets with singlets.', '1503.04203-2-5-3': 'This yields a theory in which dark matter couples to the Higgs boson in a tunable way that is essentially independent of the dark matter mass itself.', '1503.04203-2-5-4': 'This is somewhat analogous to dark sector models with a dark U(1) portal (e.g., [CITATION]), where the coupling to the Standard Model is tunable through an otherwise arbitrary parameter - the kinetic mixing between the dark U(1) and hypercharge.', '1503.04203-2-6-0': 'The existence of both electroweak-preserving and electroweak-breaking masses for the dark fermions provides two main benefits.', '1503.04203-2-6-1': 'First, given that the Higgs boson couples electroweak doublets with singlets, the global flavor symmetries of the dark fermions can be completely broken to just dark baryon number.', '1503.04203-2-6-2': 'All mesons can decay through an electroweak process (e.g., electrically charged mesons through [MATH] exchange) or through the usual chiral anomaly (e.g., the lightest neutral meson).', '1503.04203-2-6-3': 'Ensuring that these particles decay before big-bang nucleosynthesis sets a weak lower bound on the Higgs interaction strength.', '1503.04203-2-6-4': '(This is in contrast with [CITATION] where additional interactions were required to ensure mesons decay, e.g., through higher dimensional operators).', '1503.04203-2-6-5': 'The second reason is related to the orientation of the chiral condensate after the dark force confines.', '1503.04203-2-6-6': 'Large vector-like masses for the dark fermions ensure that the condensate can be aligned toward the electroweak-preserving direction, and thus the dark sector leads to only small corrections to electroweak precision measurements.', '1503.04203-2-6-7': 'We estimate the size of these corrections in this paper.', '1503.04203-2-7-0': 'There are many appealing features of an electroweak-neutral composite dark matter candidate made up from fermions transforming under the electroweak group, including:', '1503.04203-2-8-0': 'We focus mainly on a confining [MATH] gauge theory dark sector with dark fermions transforming non-trivially under the electroweak group.', '1503.04203-2-8-1': 'We apply our recent results [CITATION] using lattice simulations for the spectrum and effective Higgs interaction for [MATH].', '1503.04203-2-8-2': 'As emphasized in [CITATION], this theory is well suited for lattice calculations since we are not interested in the chiral limit of vanishing dark fermion masses.', '1503.04203-2-8-3': 'Indeed, lattice simulations can efficiently simulate the parameter region where the dark fermion masses are comparable to the confinement scale, exactly where the perturbative estimates are least useful.', '1503.04203-2-9-0': 'The organization of the paper is as follows.', '1503.04203-2-9-1': 'In Sec. [REF] we discuss the assumptions and requirements to construct our stealth dark matter model.', '1503.04203-2-9-2': 'In Sec. [REF] we detail the dark fermion interactions and masses.', '1503.04203-2-9-3': 'In addition, we write the electroweak currents in terms of the dark fermion mass eigenstates of the theory, detailed in Appendix [REF].', '1503.04203-2-9-4': 'Until this point, the discussion of the model is general.', '1503.04203-2-9-5': 'In Sec. [REF], we simplify the parameter space for phenomenological and calculational purposes, applying a global custodial [MATH] symmetry and taking the approximately symmetric dark fermion mass matrix limit.', '1503.04203-2-9-6': 'Then in Sec. [REF] we discuss the light non-singlet mesons in the theory, in particular their decay rates and constraints from non-observation at LEP II.', '1503.04203-2-9-7': 'In Sec. [REF] we discuss the stealth dark matter contributions to the [MATH] parameter, and demonstrate the parametric suppression that happens in several regimes.', '1503.04203-2-9-8': 'In Sec. [REF] we obtain the Higgs boson coupling to the dark fermions.', '1503.04203-2-9-9': 'Then in Sec. [REF] we apply our previous model-independent results on the [MATH] spectrum and effective Higgs coupling to stealth dark matter.', '1503.04203-2-9-10': 'We obtain the bounds on the parameter space from the non-observation of a spin-independent direct detection signal at LUX.', '1503.04203-2-9-11': 'We briefly discuss the relic abundance of stealth dark matter in Sec. [REF].', '1503.04203-2-9-12': 'Finally we conclude with a discussion in Sec. [REF].', '1503.04203-2-10-0': '# Constructing a viable model', '1503.04203-2-11-0': '## Basic assumptions', '1503.04203-2-12-0': 'We assume that the dark matter candidate is a composite particle of a non-Abelian, confining gauge theory based on the group [MATH] with [MATH] flavors of fermions transforming in the fundamental representation.', '1503.04203-2-12-1': 'The number [MATH] is restricted by only the condition of confinement.', '1503.04203-2-12-2': 'For reasons outlined in the introduction (abundance, detectability), the dark fermions carry electroweak charges.', '1503.04203-2-12-3': 'Our model includes a tunable Higgs ""portal"" coupling between the dark sector and the Standard Model via dimension-4 Higgs couplings.', '1503.04203-2-12-4': 'We do not consider QCD-colored dark fermions since with [MATH], dark baryons would not generally be color singlets.', '1503.04203-2-13-0': '## Requirements', '1503.04203-2-14-0': 'We require dark matter stability to be automatic, arising from a global symmetry.', '1503.04203-2-14-1': 'This motivates considering the dark baryon of the non-Abelian dark sector to be the dark matter [CITATION].', '1503.04203-2-14-2': 'In the presence of GUT-scale or Planck-scale suppressed operators, the stability of the dark baryon should be sufficient to avoid cosmological constraints.', '1503.04203-2-15-0': 'The requirement of a sufficiently preserved accidental baryon number disfavors a dark [MATH] group.', '1503.04203-2-15-1': 'First, there is no automatic baryon number in [MATH] because there is no fundamental distinction between mesons and baryons.', '1503.04203-2-15-2': 'Imposing a global [MATH] baryon number is possible (e.g. see [CITATION]) but in addition baryon number violating dimension-5 Planck-suppressed operators such as [MATH] must be absent, where [MATH] is the dark fermion.', '1503.04203-2-15-3': '(Otherwise, the dark [MATH] baryon would decay on a timescale much shorter than the age of the Universe.)', '1503.04203-2-16-0': 'For [MATH], operators involving dark baryon decay are necessarily dimension-6 or higher, and thus safe from GUT-scale or Planck-scale suppressed violations of dark baryon number.', '1503.04203-2-16-1': '[MATH] with odd [MATH] is a perfectly interesting theory, having been studied before for [MATH] by our collaboration [CITATION].', '1503.04203-2-16-2': 'There it was found that a fermionic dark baryon has a magnetic dipole interaction that leads to a significant contribution to spin-independent scattering.', '1503.04203-2-16-3': 'Constraints from the XENON100 experiment were satisfied only when the dark matter mass [MATH]>[MATH] TeV [CITATION].', '1503.04203-2-16-4': 'This strong constraint on the mass scale implies the model is difficult to test at near-future colliders.', '1503.04203-2-17-0': 'The magnetic dipole interaction (and other higher dimensional operators that require spin) are absent when the dark baryon is a scalar.', '1503.04203-2-17-1': 'We are thus naturally led to [MATH] with even [MATH], for which the otherwise strong constraints from direct detection are weakened, lowering the scales of interest into a regime that can be probed by colliders and other detection strategies.', '1503.04203-2-18-0': 'We assume the dark fermions have masses [MATH] on the order of the [MATH] confinement scale [MATH].', '1503.04203-2-18-1': 'If the masses were much smaller, the dark sector would contain light pseudo-Goldstone pions that transform under the electroweak group, which are strongly constrained by collider experiments.', '1503.04203-2-18-2': 'A dark sector with purely vector-like fermion masses has approximately stable electrically-charged mesons due to dark flavor symmetries.', '1503.04203-2-18-3': 'Conversely, a dark sector with purely electroweak breaking fermion masses has a dark matter candidate that is ruled out by spin-independent direct detection through single Higgs exchange.', '1503.04203-2-18-4': '(For example, quirky dark matter [CITATION] is now completely ruled out by Higgs exchange, given the direct detection bounds from LUX [CITATION] combined with the relatively light Higgs mass [CITATION].)', '1503.04203-2-18-5': 'Fermions with both vector-like and (small) electroweak breaking contributions to their masses can avoid both problems.', '1503.04203-2-19-0': 'We require the lightest dark baryon to be electrically neutral.', '1503.04203-2-19-1': 'We also require Higgs couplings at dimension-4 to pairs of dark fermions.', '1503.04203-2-19-2': 'These two requirements impose restrictions on the electroweak charges of the dark fermions.', '1503.04203-2-20-0': 'One solution is familiar from old technicolor theories (e.g. [CITATION]): requiring the dark fermion charges to roughly satisfy [MATH]<[MATH] where [MATH] is the [MATH] isospin.', '1503.04203-2-20-1': 'Choosing doublets ([MATH] under [MATH]) then gives a finite number of discrete possibilities.', '1503.04203-2-21-0': 'A simple model that satisfies all of these requirements is shown in Table [REF].', '1503.04203-2-21-1': 'The electric charges of the dark fermions in the broken electroweak phase are [MATH], ensuring all hadrons have integer electric charges.', '1503.04203-2-21-2': 'So long as the lightest [MATH] and [MATH] dark fermions are close in mass, the lightest baryon will be a scalar and electrically neutral.', '1503.04203-2-21-3': 'Finally, with the assignments shown in Table [REF], all gauge (and global) anomalies vanish, which is automatic with fermions that transform under vector-like representations of the [MATH] and electroweak groups.', '1503.04203-2-22-0': '# Dark fermion interactions and masses', '1503.04203-2-23-0': 'The fermions [MATH] transform under a global [MATH] flavor symmetry with [[MATH]][MATH] surviving after the weak gauging of the electroweak symmetry.', '1503.04203-2-23-1': 'From this large global symmetry, one [MATH] (diagonal) subgroup will be identified with [MATH], one [MATH] subgroup will be identified with [MATH], and one [MATH] will be identified with dark baryon number.', '1503.04203-2-23-2': 'The total fermionic content of the model is therefore 8 Weyl fermions that pair up to become 4 Dirac fermions in the fundamental or anti-fundamental representation of [MATH] with electric charges of [MATH].', '1503.04203-2-23-3': 'We use the notation where the superscript [MATH] or [MATH] (as in [MATH], [MATH] and later [MATH], [MATH], [MATH], [MATH]) denotes a fermion with electric charge of [MATH] or [MATH] respectively.', '1503.04203-2-24-0': 'The fermion kinetic terms in the Lagrangian are given by [EQUATION] where the covariant derivatives are [EQUATION] with the interactions among the electroweak group and the new [MATH].', '1503.04203-2-24-1': 'Here [MATH], [MATH] and [MATH] are the representation matrices for the fundamental of [MATH].', '1503.04203-2-25-0': 'The vector-like mass terms allowed by the gauge symmetries are [EQUATION] where [MATH] and the relative minus signs between the mass terms have been chosen for later convenience.', '1503.04203-2-25-1': 'The mass term [MATH] explicitly breaks an [[MATH]][MATH] global symmetry down to the diagonal [MATH] where the [MATH] is identified with [MATH].', '1503.04203-2-25-2': ""The mass terms [MATH] explicitly break the remaining [[MATH]][MATH] down to [MATH] where one of the [MATH]'s is identified with [MATH]."", '1503.04203-2-25-3': '(In the special case when [MATH], the global symmetry is enhanced to [MATH], where the global [MATH] acts as a custodial symmetry.)', '1503.04203-2-25-4': 'Thus, after weakly gauging the electroweak symmetry and writing arbitrary vector-like mass terms, the unbroken flavor symmetry is [MATH].', '1503.04203-2-26-0': 'Electroweak symmetry breaking mass terms arise from coupling to the Higgs field [MATH] that we take to be in the [MATH] representation.', '1503.04203-2-26-1': 'They are given by [EQUATION] where again the relative minus signs are chosen for later convenience.', '1503.04203-2-26-2': 'After electroweak symmetry breaking, [MATH], with [MATH] GeV.', '1503.04203-2-26-3': 'Replacing the Higgs field by its VEV in Eq. ([REF]), we obtain mass terms for the fermions, in 2-component notation, [EQUATION] with the mass matrices given by [EQUATION]', '1503.04203-2-26-4': 'These Yukawa couplings break the remaining [MATH] flavor symmetry to [MATH] dark baryon number.', '1503.04203-2-26-5': 'The mass matrices [MATH] and [MATH] correspond to the masses of two sets of fermions with electric charge [MATH] and [MATH] respectively, in the fundamental representation of [MATH].', '1503.04203-2-26-6': 'The two biunitary mass matrices can be diagonalized by four independent rotation angles [EQUATION] where the rotation matrices are defined by [EQUATION]', '1503.04203-2-26-7': 'The 2-component mass eigenstate spinors are thus [EQUATION] where the extra phase in Eqs. ([REF]),([REF]) ensures the [MATH] fermions will have positive mass eigenvalues.', '1503.04203-2-27-0': 'The Lagrangian for the fermion mass eigenstates becomes [EQUATION] where the mass eigenvalues are [MATH] for [MATH], and the distinction between fermions [MATH] and [MATH] allows us to write the [MATH] fermion masses as [MATH].', '1503.04203-2-27-1': 'The Dirac spinor mass eigenstates are constructed from the 2-component Weyl spinor mass eigenstates in the usual way, [EQUATION] giving the Dirac fermion masses [EQUATION]', '1503.04203-2-27-2': 'The fermion masses themselves are obtained from a straightforward diagonalization of the mass matrices, [EQUATION] with mixing angles [EQUATION] with identical expressions for [MATH] and [MATH] with the replacement [MATH] everywhere.', '1503.04203-2-28-0': 'It is important to note that the electroweak currents ([MATH], [MATH], [MATH], [MATH]) play an important role in the upcoming phenomenological discussions.', '1503.04203-2-28-1': 'Due to the extended expressions for these quantities in terms of our Dirac spinors, we have relegated a detailed derivation of the electroweak currents to Appendix [REF].', '1503.04203-2-29-0': '# Simplifications', '1503.04203-2-30-0': 'Our main interest is the more specialized case where the lightest [MATH] and [MATH] fermions are degenerate in mass to a very good approximation.', '1503.04203-2-30-1': 'This leads to a neutral scalar baryon with a vanishing charge radius.', '1503.04203-2-30-2': 'While there are several ways this could be accomplished, we can simply impose a custodial [MATH] global symmetry on the Lagrangian.', '1503.04203-2-30-3': 'In order to simplify notation, we define [MATH], [MATH] and [MATH].', '1503.04203-2-30-4': 'In the custodial [MATH] symmetric theory, [MATH] and [MATH].', '1503.04203-2-31-0': '## Custodial SU(2)', '1503.04203-2-32-0': 'An exact custodial [MATH] symmetry implies the masses and interactions are symmetric with respect to the interchange [MATH].', '1503.04203-2-32-1': 'This means the Lagrangian parameters satisfy [EQUATION]', '1503.04203-2-32-2': 'Defining the overall vector-like mass scale [MATH] and difference [MATH] to be [EQUATION] the dark fermion mass eigenvalues are [EQUATION]', '1503.04203-2-32-3': 'No [MATH] or [MATH] labels are necessary, since custodial [MATH] symmetry implies that there is one pair of Dirac fermions with electric charge [MATH] with mass [MATH] (the lightest pair), as well as a second pair of Dirac fermions with electric charge [MATH] with mass [MATH] (the heavier pair).', '1503.04203-2-32-4': 'The spectrum is illustrated in Fig. [REF].', '1503.04203-2-33-0': 'In the limit [MATH], the fermions acquire purely vector-like masses, and thus the chiral condensate of the dark force is aligned to a purely electroweak-preserving direction.', '1503.04203-2-33-1': ""In order that the chiral condensate's electroweak-preserving orientation is not significantly disrupted, we consider small electroweak breaking masses, [MATH]."", '1503.04203-2-34-0': 'This leaves two distinct regimes for the spectrum, depending on the relative sizes of [MATH] and [MATH].', '1503.04203-2-35-0': '## Approximately symmetric mass matrices', '1503.04203-2-36-0': 'A second simplification, useful to analytically and numerically evaluate our results, is to take [MATH].', '1503.04203-2-36-1': 'The mass matrices Eqs. ([REF],[REF]) are approximately symmetric.', '1503.04203-2-36-2': 'Specifically, we can write [EQUATION] and expand in powers of [MATH].', '1503.04203-2-36-3': 'For example, the dark fermion masses become simply [EQUATION] to leading order in [MATH].', '1503.04203-2-37-0': 'The distinct regimes are thus [MATH] and [MATH].', '1503.04203-2-37-1': 'In the Linear Case [MATH], electroweak symmetry breaking is (dominantly) responsible for the mass splitting between [MATH] and [MATH].', '1503.04203-2-37-2': 'In the Quadratic Case [MATH], the splitting is dominantly attributed to the vector-like mass splitting [MATH].', '1503.04203-2-37-3': 'As we shall see, the primary distinction between these two cases is in the Higgs coupling to the fermion mass eigenstates: proportional to [MATH] for the Linear Case and [MATH] for the Quadratic Case, hence the case names.', '1503.04203-2-37-4': 'A similar observation was also found in Ref. [CITATION].', '1503.04203-2-38-0': 'From this point forward unless noted otherwise, we assume the fermion mass parameters satisfy an exact custodial [MATH] and the mass matrices are approximately symmetric.', '1503.04203-2-39-0': '# Light Non-Singlet Meson Phenomenology', '1503.04203-2-40-0': 'Theories with new fermions that transform under vector-like representations of the electroweak group generically have enlarged global flavor symmetries that can prevent decay of the lightest non-singlet mesons and baryons.', '1503.04203-2-40-1': 'In the case of dark baryons, this is a feature, providing the rationale for the stability of the lightest dark baryon of the theory.', '1503.04203-2-41-0': 'In the case of the lightest non-singlet mesons, this can be problematic, since some of these mesons carry electric charge.', '1503.04203-2-41-1': 'Stable integer charged mesons are strongly constrained from collider searches as well as cosmology.', '1503.04203-2-41-2': 'One solution is to postulate additional higher dimensional operators that connect a dark fermion pair with a Standard Model fermion pair [CITATION].', '1503.04203-2-41-3': 'This must be carefully done to avoid also writing operators that violate the approximate global symmetries protecting the stability of the dark matter.', '1503.04203-2-41-4': 'In the stealth dark matter model, however, electroweak symmetry breaking can provide the source of global flavor symmetry breaking, leading to the decay of the lightest charged mesons.', '1503.04203-2-41-5': '(We will not discuss the lightest neutral mesons, but they are generically more difficult to produce in colliders, and they will decay through essentially the same mechanism as we describe for the charged mesons.)', '1503.04203-2-42-0': 'The lightest electrically charged mesons are composed dominantly of the dark fermion pairs [MATH] and [MATH].', '1503.04203-2-42-1': 'We can estimate the lightest meson lifetime by generalizing pion decay of QCD to our model.', '1503.04203-2-42-2': 'The relevant matrix element is (see, e.g., [CITATION]) [EQUATION] where [MATH] is the ""pion decay constant"" associated with the dark force in this paper.', '1503.04203-2-42-3': 'The axial part of the electroweak current can be read off from the electroweak currents given in Eqs. ([REF]),([REF]) [EQUATION] where [EQUATION] and [MATH] is identical upon [MATH].', '1503.04203-2-42-4': 'In the custodial limit, Eq. ([REF]), the axial coefficient is [EQUATION]', '1503.04203-2-42-5': 'Some insight can be gained using approximately symmetric mass matrices, Eq. ([REF]).', '1503.04203-2-42-6': 'We then obtain [EQUATION]', '1503.04203-2-42-7': 'The decay width can be obtained from pion decay of QCD by replacing [MATH] in the Standard Model with [MATH] for the dark mesons.', '1503.04203-2-42-8': 'Since the charged dark mesons of this model are much heavier than the QCD pions, there are many possible decay modes.', '1503.04203-2-42-9': 'For a general decay into a Standard Model doublet [MATH], assuming [MATH], the decay width is [EQUATION]', '1503.04203-2-42-10': 'If [MATH], the dominant decay mode is expected to be [MATH], otherwise [MATH] and [MATH], with branching ratios of roughly 70% and 30% respectively.', '1503.04203-2-42-11': 'Note that the decay width has several enhancement factors relative to the QCD pion decay width [EQUATION] where for simplicity we have neglected kinematic suppression.', '1503.04203-2-42-12': 'As an example, if [MATH], we find the lightest charged dark mesons decay faster than QCD charged pions so long as [MATH]>[MATH].', '1503.04203-2-42-13': 'This is easy to satisfy with small Yukawa couplings and dark fermion masses at or beyond the electroweak scale.', '1503.04203-2-43-0': 'We can now make some comments about existing collider constraints on non-singlet mesons.', '1503.04203-2-43-1': 'The lightest charged mesons [MATH] can be pair produced in particle colliders through the Drell-Yan process, and will decay through annihilation of the constituent fermions into a [MATH] boson.', '1503.04203-2-43-2': 'Because the Drell-Yan production is mediated by a photon and the mesons have unit electric charge, the production cross-section is substantial, leading to robust bounds from LEP-II.', '1503.04203-2-43-3': 'For charged states near the LEP-II energy threshold, the dominant decay mode is expected to be [MATH] as noted above.', '1503.04203-2-43-4': 'Reinterpreting the LEP-II bound from the pair production of supersymmetric partners to the tau (with the stau decaying into a tau and a nearly massless gravitino), we find [MATH]>[MATH] GeV [CITATION].', '1503.04203-2-43-5': 'Stronger bounds from the LHC may be possible, although existing searches do not yet give any significant constraints on the charged mesons [CITATION]; we briefly highlight the signals in the discussion.', '1503.04203-2-44-0': 'Using our lattice results from Ref. [CITATION], we can translate the experimental bound on the mass of the pseudoscalar meson into a bound on the baryon mass, [MATH] GeV when the ratio of the pseudoscalar mass to the vector meson mass is [MATH].', '1503.04203-2-45-0': '# Contributions to Electroweak Precision Observables', '1503.04203-2-46-0': 'Stealth dark matter contains dark fermions that acquire electroweak symmetry breaking contributions to their masses.', '1503.04203-2-46-1': 'Consequently, there are contributions to the electroweak precision observables of the Standard Model, generally characterized by [MATH] and [MATH] [CITATION].', '1503.04203-2-46-2': 'In the custodial [MATH] limit, Eq. ([REF]), the contribution to [MATH] vanishes.', '1503.04203-2-46-3': 'There is a contribution to [MATH], controllable through the relative size of the electroweak breaking and electroweak preserving masses of the dark fermions.', '1503.04203-2-47-0': 'The [MATH] parameter is defined in terms of momentum derivatives of current-current correlators [CITATION], [EQUATION] where the currents [MATH] and [MATH] for the stealth dark matter model are defined in Eqs. ([REF]) and ([REF]).', '1503.04203-2-47-1': 'After some algebra and identifications of symmetric contractions, these definitions of the currents in terms of 4-component fermion fields lead to the current-current correlator.', '1503.04203-2-47-2': 'In the custodial limit, we obtain [EQUATION] where the connected contributions to the correlation functions are given by [EQUATION]', '1503.04203-2-47-3': 'Here, [MATH] and the flavor indices [MATH], where it is understood that the flavors labeled [MATH] have larger fermion masses than the flavors labeled [MATH].', '1503.04203-2-47-4': 'Since the [MATH] flavors have the same mass, the [MATH] and [MATH] labels are interchangeable (i.e. everything is written in terms of the [MATH] flavors).', '1503.04203-2-48-0': 'We can obtain expressions for the mixing angle coefficients.', '1503.04203-2-48-1': 'Like the case of light meson decay, if we consider an approximately symmetric mass matrix, with Yukawa couplings given by Eq. ([REF]), all of the mixing angle coefficients are approximately equal to each other, differing only at first order in [MATH], i.e., [EQUATION]', '1503.04203-2-48-2': 'In the Linear Case, the mixing angles are approximately equal [MATH].', '1503.04203-2-48-3': 'In the Quadratic Case, all of the contributions to the [MATH] parameter are suppressed by [MATH].', '1503.04203-2-48-4': 'To calculate the [MATH] parameter in general requires lattice methods, paying close attention to the heavy-light splitting of the fermions, [MATH].', '1503.04203-2-48-5': 'To a first approximation we expect that in the limit of small mass splitting, [MATH], [EQUATION]', '1503.04203-2-48-6': 'This gives for the current-current correlator [EQUATION] where all of the [MATH] and [MATH] contributions self-cancel.', '1503.04203-2-48-7': 'Hence, we see that the contribution to the [MATH] parameter is suppressed as [MATH] or [MATH], as expected.', '1503.04203-2-49-0': '# Fermion Couplings to the Higgs Boson', '1503.04203-2-50-0': 'In terms of the gauge-eigenstate fields, the interactions of the Higgs boson with the dark-sector fermions are, in matrix notation, [EQUATION]', '1503.04203-2-50-1': 'These matrices are not simultaneously diagonalizable with the mass matrices, Eqs. ([REF]),([REF]).', '1503.04203-2-50-2': 'This means that the Higgs boson in general has off-diagonal, ""dark flavor-changing"" interactions with the mass eigenstate fields.', '1503.04203-2-50-3': 'Explicitly, we find in terms of the mixing angles [EQUATION]', '1503.04203-2-50-4': 'In the custodial [MATH] limit, we can drop the [MATH] and [MATH] labels since the Higgs coupling matrix is identical for both sets of fields.', '1503.04203-2-50-5': 'If we further take the limit of an approximately symmetric mass matrix, Eq. ([REF]), the Higgs couplings simplify to [EQUATION]', '1503.04203-2-50-6': 'We observe both diagonal and off-diagonal Higgs couplings to the fermions.', '1503.04203-2-50-7': 'The off-diagonal dark flavor-changing interactions vanish in the limit [MATH] and [MATH].', '1503.04203-2-50-8': 'In this limit an enhanced flavor symmetry among the fermions is restored, and the analogue of the GIM mechanism forbids such interactions at tree-level.', '1503.04203-2-50-9': 'The off-diagonal Higgs couplings lead to an inelastic scattering cross section when a single Higgs is exchanged.', '1503.04203-2-50-10': 'This is highly suppressed unless the mass difference [MATH] is near the (non-relativistic) kinetic energy of the dark matter in galaxy.', '1503.04203-2-50-11': 'Two off-diagonal Higgs couplings can be combined in a loop involving one heavier dark fermion and double Higgs exchange, but this is suppressed by the square of the Higgs couplings times a loop factor, as well as by the mass of the heavier fermions.', '1503.04203-2-51-0': 'The single Higgs coupling to the lightest fermions is finally [EQUATION] where [EQUATION] (Note also that the single Higgs coupling to the heaviest fermions [MATH] is identical up to an overall sign.)', '1503.04203-2-51-1': 'Depending on the relative size of [MATH] and [MATH], the Higgs boson couples linearly or quadratically proportional to the Yukawa coupling [MATH].', '1503.04203-2-51-2': 'The additional suppression of [MATH] in the Quadratic Case will imply that spin independent scattering through single Higgs exchange can be significantly weaker when the mass difference between the lightest and heaviest fermions is dominated by the electroweak preserving mass [MATH].', '1503.04203-2-52-0': '# Direct detection bounds from Higgs exchange', '1503.04203-2-53-0': 'In a previous paper [CITATION], we determined the model-independent bounds on direct detection from Higgs exchange for a scalar baryon of [MATH].', '1503.04203-2-53-1': 'The model-independent result was expressed in terms of the effective Higgs coupling to the baryon [EQUATION]', '1503.04203-2-53-2': 'The first factor, the baryon mass [MATH] (divided by the electroweak VEV), as well as the third factor [EQUATION] are extracted from our lattice results [CITATION].', '1503.04203-2-53-3': 'The second factor [EQUATION] provides the effective coupling of the Higgs boson to the fermions (multiplied by [MATH]), and we have evaluated the derivative for the two cases in our model.', '1503.04203-2-54-0': 'Unfortunately, we cannot directly apply our previous results on constraints in [MATH]-[MATH] space to the parameters of the stealth dark matter model.', '1503.04203-2-54-1': 'This is because we do not know the dark fermion mass, [MATH], independent of the lattice regularization scheme.', '1503.04203-2-54-2': 'We can, however, construct a regularization-independent parameter, the effective Yukawa coupling [MATH], that is closely related to the model parameters: [EQUATION]', '1503.04203-2-54-3': 'The [MATH] parameter is therefore [EQUATION]', '1503.04203-2-54-4': 'Recasting our previous constraints in [MATH]-[MATH] space into [MATH]-[MATH] space, we can identify the region of parameter space that remains viable.', '1503.04203-2-54-5': 'The constraints for the Linear Case are shown in Fig. [REF] and the Quadratic Case in Fig. [REF].', '1503.04203-2-54-6': 'In the top two plots for the respective figures, the region above the LUX bounds represents the excluded parameter space for the model at a given dark matter mass ([MATH]) and effective Yukawa coupling ([MATH]).', '1503.04203-2-54-7': 'The figures show a clear qualitative trend in how the predictions change as a function of dark matter mass.', '1503.04203-2-54-8': 'In particular, the cross-section is independent of [MATH] for the Linear Case and inversely proportional to [MATH] in the Quadratic Case.', '1503.04203-2-54-9': 'The bottom plots in Figs. [REF],[REF] shows the maximum [MATH] allowed for a given dark matter mass.', '1503.04203-2-54-10': 'By increasing the splitting [MATH] between the vector-like mass terms, significantly more [MATH] parameter space becomes available.', '1503.04203-2-55-0': '# Abundance', '1503.04203-2-56-0': 'We now provide a brief discussion of the relic abundance of stealth dark matter.', '1503.04203-2-56-1': 'In the regime where the dark fermions have masses comparable to the confinement scale of the dark force, calculating the relic abundance is an intrinsically strongly-coupled calculation.', '1503.04203-2-56-2': 'Unfortunately, this calculational difficulty is not easily overcome with lattice simulations, due to the different initial and final states.', '1503.04203-2-56-3': 'Nevertheless, it is straightforward to see that the relic abundance can match the cosmological abundance through at least two distinct mechanisms that lead to two different mass scales for stealth dark matter.', '1503.04203-2-56-4': 'In this section we discuss obtaining the abundance of stealth dark matter through thermal freezeout, leading to a symmetric abundance of dark baryons and anti-baryons.', '1503.04203-2-56-5': 'Separately, we consider the possibility of an asymmetric abundance generated through electroweak sphalerons.', '1503.04203-2-57-0': '## Symmetric Abundance', '1503.04203-2-58-0': 'In the early universe at temperatures well above the confinement scale of the [MATH] dark gauge force, the dark fermions are in thermal equilibrium with the thermal bath through their electroweak interactions.', '1503.04203-2-58-1': 'As the universe cools to temperatures below the confinement scale, the degrees of freedom change from dark fermions and gluons into the dark baryons and mesons of the low energy description.', '1503.04203-2-58-2': 'Some of the dark mesons carry electric charge, and so the dark mesons remain in thermal equilibrium with the Standard Model quarks, leptons, and gauge fields.', '1503.04203-2-58-3': 'Since the dark baryons are strongly coupled to the dark mesons, they also are kept in thermal equilibrium.', '1503.04203-2-58-4': 'As the temperature of the universe falls well below the mass of the dark baryons, they annihilate into dark mesons that subsequently thermalize and decay (or decay then thermalize) into Standard Model particles.', '1503.04203-2-58-5': 'The symmetric abundance of dark baryons is therefore determined by the annihilation rate of dark baryons into dark mesons.', '1503.04203-2-59-0': 'The annihilation of dark baryons to dark mesons is a strongly coupled process.', '1503.04203-2-59-1': 'We expect [MATH], [MATH], and [MATH], (and to possibly more mesons if kinematically allowed) to occur, but we do not know the dominant annihilation channel.', '1503.04203-2-59-2': 'If the 2-to-2 process [MATH] dominates, one approach is to use partial wave unitarity to estimate the thermally averaged annihilation rate [CITATION], [EQUATION] where [MATH] at freezeout [CITATION].', '1503.04203-2-59-3': 'Matching this cross section to the required thermal relic abundance yields [MATH] TeV.', '1503.04203-2-59-4': 'An alternative approach is to use naive dimensional analysis [CITATION], which appears to lead to a larger dark matter mass.', '1503.04203-2-60-0': 'If the 2-to-3 or 2-to-4 processes dominate instead, the additional phase space and kinematic suppression lowers the annihilation rate, and therefore lowers the scalar baryon mass needed to obtain the cosmological abundance.', '1503.04203-2-60-1': 'For recent work that has considered the thermal relic abundance in multibody processes, see [CITATION].', '1503.04203-2-60-2': 'Suffice it to say a symmetric thermal abundance of dark baryons will match the cosmological abundance for a relatively large baryon mass that is of order tens to hundreds of TeV.', '1503.04203-2-61-0': '## Asymmetric Abundance', '1503.04203-2-62-0': 'Early work on technibaryons demonstrated that strongly-coupled dark matter could arise from an asymmetric abundance [CITATION].', '1503.04203-2-62-1': 'The main ingredient to obtain the correct cosmological abundance involved the electroweak sphaleron - the non-perturbative solution at finite temperature that allows for transitions between vacua with different [MATH] numbers.', '1503.04203-2-62-2': 'In the early universe, at temperatures much larger than the electroweak scale, electroweak sphalerons are expected to violate one accidental global symmetry, [MATH] number, leaving [MATH] and [MATH] numbers unaffected [CITATION].', '1503.04203-2-62-3': 'Here [MATH] number is proportional to the dark baryon number, with some appropriate normalization (for examples, see [CITATION]).', '1503.04203-2-63-0': 'Given a baryogenesis mechanism, the electroweak sphalerons redistribute baryon number into lepton number and dark baryon number.', '1503.04203-2-63-1': 'As the universe cools, the mass of the technibaryon becomes larger than the temperature of the Universe.', '1503.04203-2-63-2': 'Eventually, the universe cools to the point where electroweak sphalerons ""freeze out"" and can no longer continue exchanging [MATH], [MATH], and [MATH] numbers.', '1503.04203-2-63-3': 'The residual abundance of dark baryons is [MATH] where the number density is proportional to [MATH], where [MATH] is the temperature at which sphaleron interactions shut off.', '1503.04203-2-64-0': 'If the baryon and dark baryon number densities are comparable, the would-be overabundance of dark matter (from [MATH]) is compensated by the Boltzmann suppression.', '1503.04203-2-64-1': 'Very roughly, [MATH]-[MATH] TeV is the natural mass scale that matches the cosmological abundance of dark matter [CITATION].', '1503.04203-2-64-2': 'A crucial component of the early technibaryon papers [CITATION] is that the technifermions were in a purely chiral representation of the electroweak group, like the fermions of the Standard Model.', '1503.04203-2-65-0': 'In stealth dark matter, given an early baryogenesis mechanism (or other analogous mechanism to generate an asymmetry in a globally conserved quantity [CITATION]), it is possible that electroweak sphalerons could also lead to the correct relic abundance of dark baryons consistent with cosmology.', '1503.04203-2-66-0': 'There is one critical difference from the early technicolor models (as well as the quirky dark matter model): The dark fermions in stealth dark matter have both vector-like and electroweak symmetry breaking masses.', '1503.04203-2-66-1': 'This leads to a suppression of the effectiveness of the electroweak sphalerons by a factor of [MATH], c.f. Eq. ([REF]), leading to a somewhat smaller stealth baryon mass to obtain the correct relic abundance compared with a technicolor model (all other parameters equal).', '1503.04203-2-66-2': 'A more quantitative estimate is complicated by several factors:', '1503.04203-2-67-0': 'Given the exponential suppression of the asymmetric abundance as the dark baryon mass is increased, it is clear that the upper bound on the dark baryon mass is nearly the same as the technibaryon calculation (updated to the current cosmological parameters), when stealth dark fermions have vector-like masses comparable to electroweak symmetry breaking masses.', '1503.04203-2-67-1': '(This case is, however, constrained by the [MATH] parameter, see Sec. [REF]).', '1503.04203-2-67-2': 'We can therefore anticipate that a range of stealth dark matter masses will be viable, up to about a TeV.', '1503.04203-2-67-3': 'More precise predictions require further detailed investigation that is beyond the scope of this paper.', '1503.04203-2-68-0': '# Discussion', '1503.04203-2-69-0': 'We have presented a concrete model, ""stealth dark matter"", that is a composite baryonic scalar of a new [MATH] strongly-coupled confining gauge theory with dark fermions transforming under the electroweak group.', '1503.04203-2-69-1': 'Though the stealth dark matter model has a wide parameter space, we focused on dark fermion masses that respect an exact custodial [MATH].', '1503.04203-2-69-2': 'Custodial [MATH] implies the lightest bosonic baryonic composite is an electrically neutral scalar (and not a vector or spin-2) of the [MATH] dark spectrum, and in addition does not have a charge radius.', '1503.04203-2-69-3': 'This yields an exceptionally ""stealthy"" dark matter candidate, with spin-independent direct detection scattering proceeding only through Higgs exchange (studied in this paper) and the polarizability interaction (studied in our companion paper [CITATION]).', '1503.04203-2-69-4': 'Custodial [MATH] also allows for stealth dark matter to completely avoid the constraints from the [MATH] parameter.', '1503.04203-2-69-5': 'While contributions to the [MATH] parameter are present, they are suppressed by the ratio of the electroweak symmetry breaking mass-squared divided by a vector-like mass squared of the dark fermions.', '1503.04203-2-69-6': 'We also verified the lightest non-singlet mesons decay rapidly (so long as [MATH]), avoiding any cosmological issues with stable electrically-charged dark mesons.', '1503.04203-2-70-0': 'Specializing to the case of [MATH], we then applied our earlier model-independent lattice results [CITATION] to the parameters of stealth dark matter, and obtained constraints on the effective Higgs interaction.', '1503.04203-2-70-1': 'We find that the present LUX bound is able only to mildly constrain the Higgs coupling to stealth dark matter for relatively light dark baryons.', '1503.04203-2-70-2': 'Even weaker constraints arise when the effective Higgs interaction is quadratic in the Yukawa coupling, which is a natural possibility when the two pairs of dark fermions are split dominantly by vector-like masses, i.e., [MATH].', '1503.04203-2-71-0': 'While we have considered many aspects of stealth dark matter, several avenues warrant further investigation:', '1503.04203-2-72-0': 'Finally there are broader model-building questions to consider.', '1503.04203-2-72-1': 'One is the choice of scales [MATH] that has been the focus of this work.', '1503.04203-2-72-2': 'This could arise dynamically.', '1503.04203-2-72-3': 'For example, if there are sufficient flavors in the [MATH] gauge theory such that it is approximately conformal at high energies, then as the theory is run down through the dark fermion mass scale [MATH], the dark fermions integrate out, and confinement sets in at [MATH].', '1503.04203-2-72-4': 'This is well known to occur for supersymmetric [MATH] theories in the conformal window that flow to confining theories once the number of flavors drops below [MATH] [CITATION].', '1503.04203-2-72-5': 'The origin of the vector-like masses of the fermions is also an interesting model-building puzzle.', '1503.04203-2-72-6': 'However, just as SM fermion masses are vector-like below the electroweak breaking scale, we can imagine dark fermion vector-like masses could be revealed as arising from dynamics that breaks the flavor symmetries of our dark fermions at some higher scale.'}","[['1503.04203-1-37-0', '1503.04203-2-37-0'], ['1503.04203-1-37-3', '1503.04203-2-37-3'], ['1503.04203-1-37-4', '1503.04203-2-37-4'], ['1503.04203-1-50-0', '1503.04203-2-50-0'], ['1503.04203-1-50-1', '1503.04203-2-50-1'], ['1503.04203-1-50-2', '1503.04203-2-50-2'], ['1503.04203-1-50-3', 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'1503.04203-2-37-2'], ['1503.04203-1-12-4', '1503.04203-2-12-4'], ['1503.04203-1-53-2', '1503.04203-2-53-2'], ['1503.04203-1-59-0', '1503.04203-2-59-0'], ['1503.04203-1-59-2', '1503.04203-2-59-2'], ['1503.04203-1-59-5', '1503.04203-2-59-4']]",[],[],[],"['1503.04203-1-1-0', '1503.04203-1-7-0', '1503.04203-1-66-2', '1503.04203-1-71-0', '1503.04203-2-1-0', '1503.04203-2-7-0', '1503.04203-2-66-2', '1503.04203-2-71-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1503.04203,,,
cond-mat-0310043,"{'cond-mat-0310043-1-0-0': 'The ground-state phase diagram of two-dimensional Falicov-Kimball model with nearest-neighbour and next-nearest-neighbour hoppings (characterized by [MATH] constants, respectively) has been studied in perturbative regime, i.e. in the case when on-site Coulomb interaction constant [MATH] is much larger than [MATH] ones.', 'cond-mat-0310043-1-0-1': 'The fourth-order phase diagram exhibits a rich structure, more complicated than for the ordinary Falicov-Kimball model.', 'cond-mat-0310043-1-0-2': 'Possible experimental implications of the presence of nnn term are shortly discussed.', 'cond-mat-0310043-1-1-0': '# Introduction', 'cond-mat-0310043-1-2-0': 'The Falicov-Kimball model has been proposed in 1969 to description the metal-insulator transition [CITATION].', 'cond-mat-0310043-1-2-1': 'Later on, it has been applied in another important problems: mixed valence phenomena [CITATION], crystallization and alloy formation [CITATION] and others.', 'cond-mat-0310043-1-2-2': 'In this model, we are dealing with two types of particles defined on a [MATH]-dimensional simple cubic lattice [MATH]: immobile ""ions"" and itinerant spinless ""electrons"".', 'cond-mat-0310043-1-2-3': 'There exist also other interpretations of the model [CITATION], [CITATION].', 'cond-mat-0310043-1-3-0': 'The Hamiltonian defined on a finite subset [MATH] of [MATH] has the form [EQUATION] where [EQUATION]', 'cond-mat-0310043-1-3-1': 'Here [MATH] and [MATH] are creation and annihilation operators of an electron at lattice site [MATH], satisfying ordinary anticommutation relations.', 'cond-mat-0310043-1-3-2': 'The corresponding number particle operator is [MATH].', 'cond-mat-0310043-1-3-3': '[MATH] is a classical variable taking values [MATH] or [MATH]; it measures the number of ions at lattice site [MATH].', 'cond-mat-0310043-1-3-4': 'The chemical potentials of the ions and electrons are [MATH] and [MATH], respectively.', 'cond-mat-0310043-1-4-0': 'The Falicov-Kimball model in its basic, ""backbone"" form given by ([REF]), ([REF]) is too oversimplified to give quantitative predictions in real experiments.', 'cond-mat-0310043-1-4-1': 'However, it is nontrivial lattice model of correlated electrons and captures many aspects of behaviour of such systems.', 'cond-mat-0310043-1-4-2': 'It allows rigorous analysis in many situations; for a review, see [CITATION].', 'cond-mat-0310043-1-4-3': 'One can hope that a good understanding of this simpler model might lead to better insight into the Hubbard model, where rigorous results are rare [CITATION].', 'cond-mat-0310043-1-5-0': 'One can try to make the FK model more realistic by adding various terms to the ""backbone"" hamiltonian ([REF]), ([REF]) in the manner analogous to that in the original Hubbard paper [CITATION].', 'cond-mat-0310043-1-5-1': '(Other possibility is enlargement of the space of internal degrees of freedom [CITATION], but we will not consider it here.)', 'cond-mat-0310043-1-5-2': ""The most important among them are: consideration of another types of lattice, particle statistics and presence of magnetic field [CITATION]; correlated hopping (analysed in [CITATION], [CITATION]); taking into account the Coulomb interactions between nearest neighbours, as well as (small) hopping of heavy particles [CITATION], [CITATION]; consideration of the next-nearest-neighbour hoppings (let's name this modification as the [MATH] model in analogy with the corresponding version of the Hubbard model [CITATION])."", 'cond-mat-0310043-1-5-3': 'This last effect has been analysed in only few papers.', 'cond-mat-0310043-1-5-4': 'In [CITATION], authors established that if [MATH], then the phase diagram of the [MATH] FKM does not differ too much from the diagram of the pure FK model.', 'cond-mat-0310043-1-5-5': 'A remarkable paper is [CITATION], devoted to analysis of three-dimensional strongly asymmetric Hubbard model (i.e. generalized FK one) with three hopping parameters, for large Coulomb interaction constant [MATH], in the neighbourhood of the symmetry point.', 'cond-mat-0310043-1-5-6': 'Authors have determined rigorously the structure of ground states and proved their thermal stability up to terms proportional to [MATH] (square of the hopping constant).', 'cond-mat-0310043-1-6-0': 'In this paper, author examined influence of further terms of perturbation expansion (3-rd an 4-th ones) on the ground-state phase diagram in two-dimensional situation in the half-filling case, i.e. when the average value of the total particle number [MATH] is equal to the number of sites [MATH].', 'cond-mat-0310043-1-6-1': 'Effects of higher-order-terms turned out to be very interesting in the ordinary FK model [CITATION], [CITATION], [CITATION].', 'cond-mat-0310043-1-7-0': 'As a first step of the study, the effective Hamiltonian has been derived; it can be written as the Hamiltonian for the Ising model with complicated interactions, leading to strong frustration.', 'cond-mat-0310043-1-7-1': 'After that, ground states of the Hamiltonian have been looked for, and the phase diagram has been constructed.', 'cond-mat-0310043-1-7-2': 'In the orders 2 and 3 it was possible to determine it rigorously, whereas in the fourth order use of the restricted phase diagram method was necessary.', 'cond-mat-0310043-1-7-3': 'This method proved their utility in the analysis of another versions of the FKM [CITATION], [CITATION], [CITATION].', 'cond-mat-0310043-1-7-4': 'In this method, one constructs collection of all periodic arrangements of ions up to certain values of lattice sites [MATH] per elementary cell, and then one looks for the configuration of minimal energy among members of this set.', 'cond-mat-0310043-1-8-0': 'As one could expect, the fourth-order phase diagram turned out to be more complicated than in the case of the ordinary FKM.', 'cond-mat-0310043-1-8-1': 'In this last case, one observes five phases with period not exceeding [MATH] [CITATION], [CITATION], [CITATION]; in the case of the correlated-hopping FKM, six phases are present [CITATION], [CITATION].', 'cond-mat-0310043-1-8-2': 'In the ground state phase diagram of [MATH]-FKM, thirteen phases has been found (three of them are degenerate); the period of non-degenerate phases does not exceed 12 sites per elementary cell.', 'cond-mat-0310043-1-8-3': 'One observes also that the region occupied by one phase (FK-like one, of density [MATH]) is anomally large (one can expect that this region should occupy region of the size of the order [MATH], whereas the actual size is of the order [MATH]).', 'cond-mat-0310043-1-8-4': 'This phenomenon can be explained by the lifting of the macroscopic degeneracy present in the second order by higher-order perturbation.', 'cond-mat-0310043-1-9-0': 'Comparing phase diagrams of the FKM and [MATH] FKM, it turned out that the influence of the nnn hopping is surprisingly large.', 'cond-mat-0310043-1-10-0': 'The outline of the paper is as follows.', 'cond-mat-0310043-1-10-1': 'In the Sec. [REF], the effective Hamiltonians up to fourth order perturbation theory have been derived.', 'cond-mat-0310043-1-10-2': 'In the Sec. [REF], ground states and phase diagram of the effective Hamiltonians in subsequent orders have been determined.', 'cond-mat-0310043-1-10-3': 'Moreover, effects of neglected higher-order-terms as well as temperature have been discussed.', 'cond-mat-0310043-1-10-4': 'The last section [REF] contains summary and conclusions.', 'cond-mat-0310043-1-11-0': '# Perturbation theory and effective Hamiltonian', 'cond-mat-0310043-1-12-0': '## Nonperturbed Hamiltonian, their ground states and phase diagram', 'cond-mat-0310043-1-13-0': 'In this paper we examine the model in the range of parameters [MATH].', 'cond-mat-0310043-1-13-1': 'The value of [MATH] is usually smaller than that of [MATH], however both these quantities are of the same order.', 'cond-mat-0310043-1-14-0': 'For derivation of the effective Hamiltonian, the method worked out in the paper [CITATION] has been applied.', 'cond-mat-0310043-1-14-1': 'It has this advantage that it can serve (provided certain conditions are fulfilled) as a first step to application of the quantum Pirogov-Sinai method and proving thermal and quantum stability of ground states.', 'cond-mat-0310043-1-14-2': 'Detailed description of all these procedures can be found in [CITATION]- [CITATION].', 'cond-mat-0310043-1-14-3': 'Here, only the application of the method and results will be given, as the general scheme is identical as in the paper [CITATION].', 'cond-mat-0310043-1-15-0': 'To obtain the final expression, we must divide states of the system onto ground and excited ones, and to find corresponding projections onto both groups.', 'cond-mat-0310043-1-15-1': 'These collection of states are identical as in [CITATION].', 'cond-mat-0310043-1-16-0': 'Let us begin our analysis starting from the classical part of the Hamiltonian ([REF]); it is well known, see [CITATION].', 'cond-mat-0310043-1-16-1': 'The Hilbert space [MATH] on the [MATH]-th site is spanned by the states: [MATH] or, explicitely, [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0310043-1-16-2': 'The corresponding energies are: [MATH].', 'cond-mat-0310043-1-16-3': 'The phase diagram consist of the following four regions.', 'cond-mat-0310043-1-16-4': 'In region [MATH], defined by [MATH], all sites are empty.', 'cond-mat-0310043-1-16-5': 'In two twin regions [MATH] given by conditions: [MATH]: [MATH], [MATH], [MATH] (for [MATH], one should interchange the subscripts [MATH] and [MATH]) all sites are in the [MATH] (corresp. [MATH]) state.', 'cond-mat-0310043-1-16-6': 'In the region [MATH], given by: [MATH], all sites are doubly occupied.', 'cond-mat-0310043-1-16-7': 'This situation is illustrated on Fig. [REF].', 'cond-mat-0310043-1-17-0': 'We choose the states [MATH] and [MATH] as ground states.', 'cond-mat-0310043-1-17-1': 'They are divided from excited ones by energy gap [MATH].', 'cond-mat-0310043-1-17-2': 'It means that we analyse the phase diagram in some subset of the region [MATH] (the shaded region on the Fig. 1).', 'cond-mat-0310043-1-17-3': 'The most interesting situation takes place in the neighbourhood of the [MATH] line between regions [MATH] and [MATH]; on this line, we observe a macroscopic degeneracy.', 'cond-mat-0310043-1-18-0': 'The projection operator on ground states at [MATH]-th site is [EQUATION]', 'cond-mat-0310043-1-19-0': '## Effective Hamiltonians up to 4-th order of perturbation theory', 'cond-mat-0310043-1-20-0': 'Expression for effective Hamiltonian in fourth-order perturbation theory for the ordinary FK model can be found in [CITATION], Table 2.', 'cond-mat-0310043-1-20-1': 'The 4-th order effective Hamiltonian for [MATH] FKM can be derived using the same methodology, described in [CITATION], Sec. 3.', 'cond-mat-0310043-1-20-2': '(It should be stressed that expressions up to 4-th order have been derived, for the 3d model, in the paper [CITATION].', 'cond-mat-0310043-1-20-3': ""Unfortunately, authors didn't analyse effects of orders 3 and 4)."", 'cond-mat-0310043-1-21-0': 'As a final result of calculations, one obtains, after specialization to the half-filled case (i.e. the situation where the total number of particles is equal to the number of lattice sites):', 'cond-mat-0310043-1-22-0': '[MATH] Second-order correction: [EQUATION] where: [MATH]; [MATH] is the classical one-half spin on the lattice site [MATH]; it is related to the variable [MATH] by the formula: [MATH].', 'cond-mat-0310043-1-23-0': '[MATH] Third-order correction: [EQUATION] where summation is performed over all triples [MATH] on the lattice such that [MATH] and [MATH] are nearest neighbour bonds forming the angle [MATH].', 'cond-mat-0310043-1-24-0': '[MATH] Fourth-order correction is the most complicated one and is a sum of two-body (2b) and four-body (4b) interactions: [EQUATION]', 'cond-mat-0310043-1-24-1': 'In formulas above, we have: [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH], where: [MATH]; [MATH]; [MATH].', 'cond-mat-0310043-1-24-2': 'Sets [MATH] are defined in the following way: [MATH] (""square"") is formed by spins occupying vertices [MATH], [MATH], [MATH] and [MATH]; [MATH] (""diagonal square"") is formed by: [MATH], [MATH], [MATH] and [MATH]; [MATH] (""big triangle""): [MATH], [MATH], [MATH] and [MATH]; [MATH]: (""rhomb"") [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0310043-1-24-3': 'The summation over four-body interactions in ([REF]) is performed over all sets obtained from plaquettes [MATH] above by operations compatible with lattice symmetries (translations, rotations by multiple of [MATH], reflections, inversions); the plaquette [MATH] occupies sites [MATH] and [MATH].', 'cond-mat-0310043-1-25-0': '# Ground state phase diagrams in order 2 and 3', 'cond-mat-0310043-1-26-0': '## Order 2', 'cond-mat-0310043-1-27-0': 'The ground-state phase diagram of the system described by the Hamiltonian ([REF]) can be obtained by rewriting the Hamiltonian in the following equivalent form: [EQUATION] where [EQUATION] (lattice sites [MATH] are arranged anticlockwise on the plaquette).', 'cond-mat-0310043-1-28-0': 'It is easy to check that the Hamiltonian rewritten in the form ([REF]) is an m-potential ([CITATION], [CITATION], [CITATION]; this definition is also reminded in the Appendix).', 'cond-mat-0310043-1-28-1': 'In such a case, we can replace the process of minimization of energy over the whole lattice by the problem much simpler: the minimization of energy over the set of plaquette configurations.', 'cond-mat-0310043-1-28-2': 'These configurations are presented on Fig. [REF].', 'cond-mat-0310043-1-28-3': 'It leads to the picture of the phase diagram as illustrated on Fig. [REF].', 'cond-mat-0310043-1-28-4': 'This diagram possess two obvious symmetries.', 'cond-mat-0310043-1-28-5': 'One of them is due to symmetry of the hamiltonian ([REF]) with respect to the change of sign [MATH]; the phase diagram is also symmetric with respect to such a change of sign.', 'cond-mat-0310043-1-28-6': 'The second one is the symmetry of phase diagram with respect to the change [MATH]; however, in this case, one should also replace configurations by their mirror images (i. e. [MATH]).', 'cond-mat-0310043-1-29-0': 'Phases [MATH] (full) and [MATH] (empty; for illustration, see configuration 0 on the Fig. [REF]) are build from plaquettes [MATH] and [MATH], respectively.', 'cond-mat-0310043-1-29-1': 'These phases are unique.', 'cond-mat-0310043-1-29-2': 'We have similar situation for regions [MATH] (Neel phase; Fig. [REF], configuration 10) and [MATH] ( Fig. [REF], configuration 11 (it is an analogon of the ""planar"" phase in [CITATION]).', 'cond-mat-0310043-1-29-3': 'They are build from plaquettes [MATH] and [MATH], respectively.', 'cond-mat-0310043-1-29-4': 'Again, these phases are unique (modulo translations).', 'cond-mat-0310043-1-29-5': 'The situation for phases [MATH], [MATH] differs from previous ones.', 'cond-mat-0310043-1-29-6': 'These phases are build from plaquettes [MATH], [MATH], respectively (see Fig. [REF], configuration No. 5 as an example); however, they are non-unique and possess macroscopic degeneration.', 'cond-mat-0310043-1-29-7': 'One can easily check that there is a large dose of freedom in building of lattice configurations from these plaquettes.', 'cond-mat-0310043-1-29-8': 'We encounter here situation similar to this which happens for the antiferromagnetic Ising model on triangular lattice.', 'cond-mat-0310043-1-30-0': '## Order 3', 'cond-mat-0310043-1-31-0': 'Now, let us check how the phase diagram will change under switching third-order terms on.', 'cond-mat-0310043-1-31-1': 'Let us rewrite the third-order correction ([REF]) in the equivalent ""plaquette"" form: [EQUATION] where [EQUATION] (again spins [MATH] are arranged anticlockwise on the plaquette).', 'cond-mat-0310043-1-32-0': 'The full Hamiltonian, up to third-order terms, is the sum of terms ([REF]) and ([REF]).', 'cond-mat-0310043-1-32-1': 'As in previous Subsection, one can check that it is an m-potential.', 'cond-mat-0310043-1-32-2': 'Moreover, it turns out that the presence of third-order terms does not modify ground states of plaquettes.', 'cond-mat-0310043-1-32-3': 'In the other words, plaquette configurations which were ground-states in the second order, remain ground states also in third order!', 'cond-mat-0310043-1-32-4': 'This implies that degeneracy of phases [MATH] i [MATH] is not lifted and they still are degenerate.', 'cond-mat-0310043-1-32-5': 'What does change, it is location of the boundary between phases.', 'cond-mat-0310043-1-32-6': 'The difference in location of phase boundaries in orders 2 and 3 is of the order [MATH].', 'cond-mat-0310043-1-33-0': 'The phase diagram in third order possess certain kind of symmetry.', 'cond-mat-0310043-1-33-1': 'It is discussed in more details in the next Subsection.', 'cond-mat-0310043-1-33-2': 'At this moment, we only conclude that the phase diagram in 3.', 'cond-mat-0310043-1-33-3': 'order is a small deformation of the second-order phase diagram.', 'cond-mat-0310043-1-34-0': '## Phase diagram in fourth order', 'cond-mat-0310043-1-35-0': 'Regions occupied by phases [MATH] in both second and third order exhibit macroscopic degeneracy.', 'cond-mat-0310043-1-35-1': 'One can expect that they will be sensitive against perturbations and that in some of next orders this degeneracy will be lifted.', 'cond-mat-0310043-1-35-2': 'This happens yet in fourth order; we describe the situation below.', 'cond-mat-0310043-1-36-0': 'This picture has been obtained by the restricted phase diagram method.', 'cond-mat-0310043-1-36-1': 'Recall that in this method, one takes into account all periodic configurations up to certain values of lattice sites [MATH] per elementary cell, and then one minimizes the energy over this set of configurations.', 'cond-mat-0310043-1-37-0': 'The ground-state phase diagram is much more complicated than for the ordinary FKM - see Figs. [REF] and [REF] but it is still manageable (in some respects it is similar to the phase diagram of the FKM on triangular lattice, studied in [CITATION]).', 'cond-mat-0310043-1-37-1': 'Thirteen phases have been detected on phase diagram; it seems that three of them are degenerate.', 'cond-mat-0310043-1-37-2': 'Moreover, for each phase present for [MATH], there corresponds their ""mirror"" for [MATH] (this mirror is obtained by the change of occupied sites onto unoccupied ones and vice versa).', 'cond-mat-0310043-1-37-3': 'Periods of these phases do not exceed 12 sites per elementary cell.', 'cond-mat-0310043-1-37-4': ""Such a picture emerged at [MATH] and hasn't changed up to [MATH] (which corresponds to more than [MATH] trial configurations)."", 'cond-mat-0310043-1-37-5': 'For this reason, author claims that this phase diagram is ""non-rigorous but exact"".', 'cond-mat-0310043-1-38-0': 'The phase diagram in fourth order possess certain (pseudo)symmetries.', 'cond-mat-0310043-1-38-1': 'Let us describe the situation order-by-order.', 'cond-mat-0310043-1-39-0': 'In the second order, we have the Hamiltonian symmetric with respect to the change [MATH] and [MATH].', 'cond-mat-0310043-1-39-1': 'It is obvious that the phase diagram is symmetric with respect to the change [MATH], if one change also configuration to its mirror image.', 'cond-mat-0310043-1-40-0': 'In the third order, the Hamiltonian is no longer symmetric.', 'cond-mat-0310043-1-40-1': 'However, their ground states are easily determined, because the Hamiltonian is expressible as a sum of m-potentials, defined on [MATH] plaquettes.', 'cond-mat-0310043-1-40-2': 'It turns out that on the phase diagram, there are present the same phases as in the second order.', 'cond-mat-0310043-1-40-3': 'The difference between phase diagrams in the second and third order is apparent in location of phase boundaries; corresponding lines are shifted by a factor proportional to [MATH].', 'cond-mat-0310043-1-40-4': 'This shift is symmetric with respect to the change [MATH].', 'cond-mat-0310043-1-40-5': 'In the other words: If some boundary between phases [MATH] and [MATH] is shifted by [MATH] for [MATH], then, for [MATH], the boundary between mirrors of phases [MATH] and [MATH] is shifted by [MATH].', 'cond-mat-0310043-1-41-0': 'Let us analyse (pseudo)symmetries of the fourth order phase diagram.', 'cond-mat-0310043-1-41-1': 'Consider first some third-order configuration of minimal energy for [MATH] (call it CME[MATH]); for every such a configuration, we have its mirror, which is also the configuration of minimal energy for some [MATH] (call it CME[MATH]).', 'cond-mat-0310043-1-41-2': 'Now, let us add the fourth order contribution to the Hamiltonian.', 'cond-mat-0310043-1-41-3': 'This term is symmetric with respect to reversing of spins; so, if for [MATH] we have some configuration of minimal energy CME[MATH], then for [MATH] the configuration of minimal energy CME[MATH] will be the mirror of CME[MATH].', 'cond-mat-0310043-1-42-0': 'Let us summarize the above by the statement that the topological structure of phase diagram is the same for [MATH] and [MATH]; for every phase [MATH] appearing for [MATH], we have corresponding mirror [MATH] for [MATH].', 'cond-mat-0310043-1-42-1': 'Phase boundaries between phases [MATH] and [MATH] and their mirrors [MATH] and [MATH] are related by: [EQUATION]', 'cond-mat-0310043-1-42-2': 'Let us stress that this situation (i.e. the same structure of phase diagram for [MATH] and [MATH] is a very peculiar property of the fourth-order Hamiltonian ([REF]), ([REF]), ([REF]); it is due to degeneracy and (pseudo)symmetries of lower-order Hamiltonians.', 'cond-mat-0310043-1-42-3': 'For more general Hamiltonian, we have no such similarity.', 'cond-mat-0310043-1-43-0': 'Perhaps, the most interesting effect of the presence of the term with next-nearest-neighbour hopping is the appearance of the anomally large region occupied by the phase 5 (FK-like phase with density [MATH]).', 'cond-mat-0310043-1-43-1': 'At first sight, phases appearing in the fourth-order perturbation theory should occupy region of the width [MATH] (where [MATH] is some homogeneous fourth-order polynom in [MATH] and [MATH]).', 'cond-mat-0310043-1-43-2': 'However, it turns out that phase 5 occupies a region of width proportional to [MATH], i.e. of the same order as the Neel phase, appearing in the second order!', 'cond-mat-0310043-1-43-3': 'One can explain this phenomenon in the following way: Regions occupied by phase [MATH] (the situation with phase [MATH] is analogous) has width of the order [MATH] proportional to exhibit macroscopic degeneracy both in 2-nd and in 3-rd order.', 'cond-mat-0310043-1-43-4': ""The fourth-order perturbation lifts this degeneracy and as a result the phase [MATH] (an 'ancestor') transforms into non-degenerate phase 5 (the 'descendant') of the same density, and occupies region approximately as large as an 'ancestor' [MATH]."", 'cond-mat-0310043-1-44-0': 'Most of phases is unique, but there are also phases which remain degenerate even in the 4-th order (phases 1, 3, 12).', 'cond-mat-0310043-1-44-1': ""Strictly speaking, the restricted phase diagram method detects here only finite degeneracy, i.e. finitely many ground states with identical energy and density but different orderings (we don't count trivial degeneracy due to symmetry operations, i.e. translations, rotations and reflections)."", 'cond-mat-0310043-1-44-2': ""Number of these ground states grows with [MATH]; for [MATH] we observed: eight phases of equal energy and density [MATH], (the first two such configurations are phases 1 and 1'); eight phases of density [MATH] (the first two such configurations are phases 3 and 3'); five phases of density [MATH] (the first two of them are 12 and 12')."", 'cond-mat-0310043-1-44-3': 'Moreover, it has been observed that every member of such collection of phases with equal density is build up from identical plaquette configurations [MATH].', 'cond-mat-0310043-1-44-4': ""They can be 'glued' together in various arrangements and there is no uniqueness in such a procedure, i.e. resulting lattice configuration is non-unique."", 'cond-mat-0310043-1-44-5': 'Situation here is similar to that which happens in the second order for phases [MATH].', 'cond-mat-0310043-1-44-6': 'It is natural to conjecture that we encounter here the macroscopic degeneracy, i.e. presence of infinite number of configurations of identical density and energy.', 'cond-mat-0310043-1-44-7': ""However author (so far) can't prove this."", 'cond-mat-0310043-1-45-0': 'Phase diagrams in orders 2 and 3 are rigorous (by writing out Hamiltonians as sums of m-potentials).', 'cond-mat-0310043-1-45-1': 'Author tried to do analogous thing in the fourth-order by an attempt to construct m-potentials in a manner analogous as in [CITATION], [CITATION], [CITATION], however, it succeeded only for some phases, but not for a whole phase diagram.', 'cond-mat-0310043-1-46-0': ""An analysis above concerned the 'truncated' phase diagram, i.e. the phase diagram of the fourth-order effective Hamiltonian ([REF])."", 'cond-mat-0310043-1-46-1': 'Which changes can result as an effect coming from next (neglected) orders and temperature?', 'cond-mat-0310043-1-46-2': 'Author expects (and conjectures) that most phases (i.e. all with exception of degenerate ones: i.e. 1, 3, 11 and their descendants) are stable ones [CITATION], [CITATION].', 'cond-mat-0310043-1-46-3': 'For such phases, Peierls conditions (both classical and quantum ones - see [CITATION], [CITATION]) are fulfilled, and regions occupied by these phases deform in only small manner upon thermal and quantum perturbations.', 'cond-mat-0310043-1-46-4': 'This assertion concerns regions of phase diagram sufficiently far from phase boundaries.', 'cond-mat-0310043-1-46-5': 'For regions of width of the order of [MATH] around phase boundaries, one cannot formulate any statements without going into next orders of perturbation theory.', 'cond-mat-0310043-1-46-6': 'However, at present author cannot prove stability and Peierls conditions.', 'cond-mat-0310043-1-46-7': '(It could be proved by construction of m-potentials, which failed so far).', 'cond-mat-0310043-1-46-8': 'Regions occupied by degenerate phases constitute different problem.', 'cond-mat-0310043-1-46-9': 'Here, changes caused by thermal and quantum perturbations can be very significant; they are ""terra incognita"" and we skip this subject, leaving it as an open problem.', 'cond-mat-0310043-1-47-0': '# Summary and conclusions', 'cond-mat-0310043-1-48-0': 'The effective Hamiltonian and phase diagram for ground states of the [MATH] FKM have been determined up to fourth order of perturbation theory.', 'cond-mat-0310043-1-48-1': 'In the second and third order, phase diagram was constructed by rewritting the Hamiltonian as a sum of m-potentials.', 'cond-mat-0310043-1-48-2': 'The phase diagram in the fourth order has been determined by the method of restricted phase diagrams; author claims that such a picture is ""nonrigorous but exact"".', 'cond-mat-0310043-1-48-3': 'The phase diagram is considerably more complicated than for the ordinary FKM, but still it is manageable.', 'cond-mat-0310043-1-48-4': 'Thirteen phases are present (plus their ""mirrors""); three of these phases are degenerate (i.e. possess identical density and energy but different ordering).', 'cond-mat-0310043-1-48-5': 'Author conjectures that this degeneracy is macroscopic (the method of restricted phase diagrams handles only finite number of configurations).', 'cond-mat-0310043-1-49-0': 'Let us list some of the features of the phase diagram in fourth order:', 'cond-mat-0310043-1-50-0': 'For small [MATH], the phase diagram is similar to this for the ordinary FKM; however, for larger [MATH], these phase diagrams are quite different.', 'cond-mat-0310043-1-50-1': 'One observes anomally large region occupied by one of the phases appearing in 4-th order (phase number 5).', 'cond-mat-0310043-1-50-2': 'It could have experimental implications as a possibility of appearance of ""charge density waves"" more exotic than Neel ordering.', 'cond-mat-0310043-1-50-3': 'The phase diagram of the full model (i.e. with inclusion of neglected terms perturbation theory, as well as temperature) is more difficult to examine than for the ordinary FKM.', 'cond-mat-0310043-1-50-4': 'At this stage of investigation, almost nothing rigorous can be said.', 'cond-mat-0310043-1-50-5': 'However, author expects that after switching the quantum and thermal perturbations on, the phase diagram will change in only small manner inside regions occupied by nondegenerate phases (0,2,4-10,12).', 'cond-mat-0310043-1-50-6': 'But regions of width of the order [MATH] around the phase boundaries, as well as regions occupied by degenerate phases 1,3,11, are out of possibilities of present analysis.', 'cond-mat-0310043-1-50-7': 'The Falicov-Kimball model is sensitive to perturbations.', 'cond-mat-0310043-1-50-8': 'Modifications of the Hamiltonian such as introduction of correlated hopping, nnn-hopping or consideration of the FKM in non-perturbative regime (i.e. for values of [MATH] not very small) can significantly or even drastically modify the phase diagram.', 'cond-mat-0310043-1-51-0': '# Ground states, m-potentials', 'cond-mat-0310043-1-52-0': 'The Hamiltonian [MATH] , where [MATH], is a function defined on [MATH] - the space of all configurations of the system.', 'cond-mat-0310043-1-52-1': 'Usually Hamiltonian is defined as a sum of potentials, i.e. functions defined on subsets of [MATH]: [MATH].', 'cond-mat-0310043-1-52-2': 'Usually one imposes restrictions such that potentials are finite-range ones, i.e. such that [MATH], [MATH] finite.', 'cond-mat-0310043-1-52-3': 'It is also assumed that potentials are translation invariant.', 'cond-mat-0310043-1-53-0': 'm-potential.', 'cond-mat-0310043-1-53-1': 'Now, consider the system on an infinite lattice [MATH].', 'cond-mat-0310043-1-53-2': 'Assume that sets [MATH] (i.e. potential supports) are translation of a fixed plaquette [MATH] by a lattice vector [MATH]: [MATH], where [MATH] is an operator of such translation.', 'cond-mat-0310043-1-53-3': 'We say that the function [MATH] is an m-potential, if there exist configuration (perhaps, non-unique) [MATH] with the following properties: i) For every plaquette [MATH], the ""plaquette energies"", i.e. values of the Hamiltonian calculated on the plaquette [MATH]: [MATH] are all equal; ii) For every another configuration [MATH], the condition: [MATH] is fulfilled.', 'cond-mat-0310043-1-54-0': 'Ground states of the classical Hamiltonian.', 'cond-mat-0310043-1-54-1': 'If there exist such configuration [MATH] as above, then we call it the ground state of the Hamiltonian.', 'cond-mat-0310043-1-55-0': 'The property of the potential to be an m-potential can be reformulated as follows.', 'cond-mat-0310043-1-55-1': 'If a given potential is an m-potential, then the local minimality of energy (i.e. minimality on a plaquette) implies the global minimality on the whole lattice.', 'cond-mat-0310043-1-55-2': 'The property that a potential is an m-potential is very important one, as it replaces searching of ground states of the infinite lattice by looking for the minima on a finite set.', 'cond-mat-0310043-1-55-3': 'Unfortunately, some given potential possess this property only exceptionally.', 'cond-mat-0310043-1-55-4': '(But fortunately, in Secs. [REF] and [REF] they share such property!)', 'cond-mat-0310043-1-55-5': 'A method to avoid this obstacle is to find - for a given potential [MATH] - an equivalent potential [MATH], such that [MATH] is an m-potential.', 'cond-mat-0310043-1-55-6': 'However, in general it is difficult task.'}","{'cond-mat-0310043-2-0-0': 'The ground-state phase diagram of two-dimensional Falicov-Kimball model with nearest-neighbour and next-nearest-neighbour hoppings (characterized by [MATH] constants, respectively) has been studied in perturbative regime, i.e. in the case when on-site Coulomb interaction constant [MATH] is much larger than [MATH] ones.', 'cond-mat-0310043-2-0-1': 'The fourth-order phase diagram exhibits a rich structure, more complicated than for the ordinary Falicov-Kimball model.', 'cond-mat-0310043-2-0-2': 'Possible experimental implications of the presence of nnn term are shortly discussed.', 'cond-mat-0310043-2-1-0': '# Introduction', 'cond-mat-0310043-2-2-0': 'The Falicov-Kimball model has been proposed in 1969 to description the metal-insulator transition [CITATION].', 'cond-mat-0310043-2-2-1': 'Later on, it has been applied in another important problems: mixed valence phenomena [CITATION], crystallization and alloy formation [CITATION] and others.', 'cond-mat-0310043-2-2-2': 'In this model, we are dealing with two types of particles defined on a [MATH]-dimensional simple cubic lattice [MATH]: immobile ""ions"" and itinerant spinless ""electrons"".', 'cond-mat-0310043-2-2-3': 'There exist also other interpretations of the model [CITATION], [CITATION].', 'cond-mat-0310043-2-3-0': 'The Hamiltonian defined on a finite subset [MATH] of [MATH] has the form [EQUATION] where [EQUATION]', 'cond-mat-0310043-2-3-1': 'Here [MATH] and [MATH] are creation and annihilation operators of an electron at lattice site [MATH], satisfying ordinary anticommutation relations.', 'cond-mat-0310043-2-3-2': 'The corresponding number particle operator is [MATH].', 'cond-mat-0310043-2-3-3': '[MATH] is a classical variable taking values [MATH] or [MATH]; it measures the number of ions at lattice site [MATH].', 'cond-mat-0310043-2-3-4': 'The chemical potentials of the ions and electrons are [MATH] and [MATH], respectively.', 'cond-mat-0310043-2-4-0': 'The Falicov-Kimball model in its basic, ""backbone"" form given by ([REF]), ([REF]) is too oversimplified to give quantitative predictions in real experiments.', 'cond-mat-0310043-2-4-1': 'However, it is nontrivial lattice model of correlated electrons and captures many aspects of behaviour of such systems.', 'cond-mat-0310043-2-4-2': 'It allows rigorous analysis in many situations; for a review, see [CITATION].', 'cond-mat-0310043-2-4-3': 'One can hope that a good understanding of this simpler model might lead to better insight into the Hubbard model, where rigorous results are rare [CITATION].', 'cond-mat-0310043-2-5-0': 'One can try to make the FK model more realistic by adding various terms to the ""backbone"" hamiltonian ([REF]), ([REF]) in the manner analogous to that in the original Hubbard paper [CITATION].', 'cond-mat-0310043-2-5-1': '(Other possibility is enlargement of the space of internal degrees of freedom [CITATION], but we will not consider it here.)', 'cond-mat-0310043-2-5-2': ""The most important among them are: consideration of another types of lattice, particle statistics and presence of magnetic field [CITATION]; correlated hopping (analysed in [CITATION], [CITATION], [CITATION]); taking into account the Coulomb interactions between nearest neighbours, as well as (small) hopping of heavy particles [CITATION], [CITATION]; consideration of the next-nearest-neighbour hoppings (let's name this modification as the [MATH] model in analogy with the corresponding version of the Hubbard model [CITATION])."", 'cond-mat-0310043-2-5-3': 'This last effect has been analysed in only few papers.', 'cond-mat-0310043-2-5-4': 'In [CITATION], authors established that if [MATH], then the phase diagram of the [MATH] FKM does not differ too much from the diagram of the pure FK model.', 'cond-mat-0310043-2-5-5': 'A remarkable paper is [CITATION], devoted to analysis of three-dimensional strongly asymmetric Hubbard model (i.e. generalized FK one) with three hopping parameters, for large Coulomb interaction constant [MATH], in the neighbourhood of the symmetry point.', 'cond-mat-0310043-2-5-6': 'Authors have determined rigorously the structure of ground states and proved their thermal stability up to terms proportional to [MATH] (square of the hopping constant).', 'cond-mat-0310043-2-6-0': 'In this paper, author examined influence of further terms of perturbation expansion (3-rd an 4-th ones) on the ground-state phase diagram in two-dimensional situation in the half-filling case, i.e. when the average value of the total particle number [MATH] is equal to the number of sites [MATH].', 'cond-mat-0310043-2-6-1': 'Effects of higher-order-terms turned out to be very interesting in the ordinary FK model [CITATION], [CITATION], [CITATION].', 'cond-mat-0310043-2-7-0': 'As a first step of the study, the effective Hamiltonian has been derived; it can be written as the Hamiltonian for the Ising model with complicated interactions, leading to strong frustration.', 'cond-mat-0310043-2-7-1': 'After that, ground states of the Hamiltonian have been looked for, and the phase diagram has been constructed.', 'cond-mat-0310043-2-7-2': 'In the orders 2 and 3 it was possible to determine it rigorously, whereas in the fourth order use of the restricted phase diagram method was necessary.', 'cond-mat-0310043-2-7-3': 'This method proved their utility in the analysis of another versions of the FKM [CITATION], [CITATION], [CITATION].', 'cond-mat-0310043-2-7-4': 'In this method, one constructs collection of all periodic arrangements of ions up to certain values of lattice sites [MATH] per elementary cell, and then one looks for the configuration of minimal energy among members of this set.', 'cond-mat-0310043-2-8-0': 'As one could expect, the fourth-order phase diagram turned out to be more complicated than in the case of the ordinary FKM.', 'cond-mat-0310043-2-8-1': 'In this last case, one observes five phases with period not exceeding [MATH] [CITATION], [CITATION], [CITATION]; in the case of the correlated-hopping FKM, six phases are present [CITATION], [CITATION].', 'cond-mat-0310043-2-8-2': 'In the ground state phase diagram of [MATH]-FKM, thirteen phases has been found (three of them are degenerate); the period of non-degenerate phases does not exceed 12 sites per elementary cell.', 'cond-mat-0310043-2-8-3': 'One observes also that the region occupied by one phase (FK-like one, of density [MATH]) is anomally large (one can expect that this region should occupy region of the size of the order [MATH], whereas the actual size is of the order [MATH]).', 'cond-mat-0310043-2-8-4': 'This phenomenon can be explained by the lifting of the macroscopic degeneracy present in the second order by higher-order perturbation.', 'cond-mat-0310043-2-9-0': 'Comparing phase diagrams of the FKM and [MATH] FKM, it turned out that the influence of the nnn hopping is surprisingly large.', 'cond-mat-0310043-2-10-0': 'The outline of the paper is as follows.', 'cond-mat-0310043-2-10-1': 'In the Sec. [REF], the effective Hamiltonians up to fourth order perturbation theory have been derived.', 'cond-mat-0310043-2-10-2': 'In the Sec. [REF], ground states and phase diagram of the effective Hamiltonians in subsequent orders have been determined.', 'cond-mat-0310043-2-10-3': 'Moreover, effects of neglected higher-order-terms as well as temperature have been discussed.', 'cond-mat-0310043-2-10-4': 'The last section [REF] contains summary and conclusions.', 'cond-mat-0310043-2-11-0': '# Perturbation theory and effective Hamiltonian', 'cond-mat-0310043-2-12-0': '## Nonperturbed Hamiltonian, their ground states and phase diagram', 'cond-mat-0310043-2-13-0': 'In this paper we examine the model in the range of parameters [MATH].', 'cond-mat-0310043-2-13-1': 'The value of [MATH] is usually smaller than that of [MATH], however both these quantities are of the same order.', 'cond-mat-0310043-2-14-0': 'For derivation of the effective Hamiltonian, the method worked out in the paper [CITATION] has been applied.', 'cond-mat-0310043-2-14-1': 'It has this advantage that it can serve (provided certain conditions are fulfilled) as a first step to application of the quantum Pirogov-Sinai method and proving thermal and quantum stability of ground states.', 'cond-mat-0310043-2-14-2': 'Detailed description of all these procedures can be found in [CITATION] - [CITATION].', 'cond-mat-0310043-2-14-3': 'Here, only the application of the method and results will be given, as the general scheme is identical as in the paper [CITATION].', 'cond-mat-0310043-2-15-0': 'To obtain the final expression, we must divide states of the system onto ground and excited ones, and to find corresponding projections onto both groups.', 'cond-mat-0310043-2-15-1': 'These collection of states are identical as in [CITATION].', 'cond-mat-0310043-2-16-0': 'Let us begin our analysis starting from the classical part of the Hamiltonian ([REF]); it is well known, see [CITATION].', 'cond-mat-0310043-2-16-1': 'The Hilbert space [MATH] on the [MATH]-th site is spanned by the states: [MATH] or, explicitely, [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0310043-2-16-2': 'The corresponding energies are: [MATH].', 'cond-mat-0310043-2-16-3': 'The phase diagram consist of the following four regions.', 'cond-mat-0310043-2-16-4': 'In region [MATH], defined by [MATH], all sites are empty.', 'cond-mat-0310043-2-16-5': 'In two twin regions [MATH] given by conditions: [MATH]: [MATH], [MATH], [MATH] (for [MATH], one should interchange the subscripts [MATH] and [MATH]) all sites are in the [MATH] (corresp. [MATH]) state.', 'cond-mat-0310043-2-16-6': 'In the region [MATH], given by: [MATH], all sites are doubly occupied.', 'cond-mat-0310043-2-16-7': 'This situation is illustrated on Fig. [REF].', 'cond-mat-0310043-2-17-0': 'We choose the states [MATH] and [MATH] as ground states.', 'cond-mat-0310043-2-17-1': 'They are divided from excited ones by energy gap [MATH].', 'cond-mat-0310043-2-17-2': 'It means that we analyse the phase diagram in some subset of the region [MATH] (the shaded region on the Fig. 1).', 'cond-mat-0310043-2-17-3': 'The most interesting situation takes place in the neighbourhood of the [MATH] line between regions [MATH] and [MATH]; on this line, we observe a macroscopic degeneracy.', 'cond-mat-0310043-2-18-0': 'The projection operator on ground states at [MATH]-th site is [EQUATION]', 'cond-mat-0310043-2-19-0': '## Effective Hamiltonians up to 4-th order of perturbation theory', 'cond-mat-0310043-2-20-0': 'Expression for effective Hamiltonian in fourth-order perturbation theory for the ordinary FK model can be found in [CITATION], Table 2.', 'cond-mat-0310043-2-20-1': 'The 4-th order effective Hamiltonian for [MATH] FKM can be derived using the same methodology, described in [CITATION], Sec. 3.', 'cond-mat-0310043-2-20-2': '(It should be stressed that expressions up to 4-th order have been derived, for the 3d model, in the paper [CITATION].', 'cond-mat-0310043-2-20-3': ""Unfortunately, authors didn't analyse effects of orders 3 and 4)."", 'cond-mat-0310043-2-21-0': 'As a final result of calculations, one obtains, after specialization to the half-filled case (i.e. the situation where the total number of particles is equal to the number of lattice sites):', 'cond-mat-0310043-2-22-0': '[MATH] Second-order correction: [EQUATION] where: [MATH]; [MATH] is the classical one-half spin on the lattice site [MATH]; it is related to the variable [MATH] by the formula: [MATH].', 'cond-mat-0310043-2-23-0': '[MATH] Third-order correction: [EQUATION] where summation is performed over all triples [MATH] on the lattice such that [MATH] and [MATH] are nearest neighbour bonds forming the angle [MATH].', 'cond-mat-0310043-2-24-0': '[MATH] Fourth-order correction is the most complicated one and is a sum of two-body (2b) and four-body (4b) interactions: [EQUATION]', 'cond-mat-0310043-2-24-1': 'In formulas above, we have: [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH]; [MATH], where: [MATH]; [MATH]; [MATH].', 'cond-mat-0310043-2-24-2': 'Sets [MATH] are defined in the following way: [MATH] (""square"") is formed by spins occupying vertices [MATH], [MATH], [MATH] and [MATH]; [MATH] (""diagonal square"") is formed by: [MATH], [MATH], [MATH] and [MATH]; [MATH] (""big triangle""): [MATH], [MATH], [MATH] and [MATH]; [MATH]: (""rhomb"") [MATH], [MATH], [MATH] and [MATH].', 'cond-mat-0310043-2-24-3': 'The summation over four-body interactions in ([REF]) is performed over all sets obtained from plaquettes [MATH] above by operations compatible with lattice symmetries (translations, rotations by multiple of [MATH], reflections, inversions); the plaquette [MATH] occupies sites [MATH] and [MATH].', 'cond-mat-0310043-2-25-0': '# Ground state phase diagrams in order 2 and 3', 'cond-mat-0310043-2-26-0': '## Order 2', 'cond-mat-0310043-2-27-0': 'The ground-state phase diagram of the system described by the Hamiltonian ([REF]) can be obtained by rewriting the Hamiltonian in the following equivalent form: [EQUATION] where [EQUATION] (lattice sites [MATH] are arranged anticlockwise on the plaquette).', 'cond-mat-0310043-2-28-0': 'It is easy to check that the Hamiltonian rewritten in the form ([REF]) is an m-potential ([CITATION], [CITATION], [CITATION]; this definition is also reminded in the Appendix).', 'cond-mat-0310043-2-28-1': 'In such a case, we can replace the process of minimization of energy over the whole lattice by the problem much simpler: the minimization of energy over the set of plaquette configurations.', 'cond-mat-0310043-2-28-2': 'These configurations are presented on Fig. [REF].', 'cond-mat-0310043-2-28-3': 'It leads to the picture of the phase diagram as illustrated on Fig. [REF].', 'cond-mat-0310043-2-28-4': 'This diagram possess two obvious symmetries.', 'cond-mat-0310043-2-28-5': 'One of them is due to symmetry of the hamiltonian ([REF]) with respect to the change of sign [MATH]; the phase diagram is also symmetric with respect to such a change of sign.', 'cond-mat-0310043-2-28-6': 'The second one is the symmetry of phase diagram with respect to the change [MATH]; however, in this case, one should also replace configurations by their mirror images (i. e. [MATH]).', 'cond-mat-0310043-2-29-0': 'Phases [MATH] (full) and [MATH] (empty; for illustration, see configuration 0 on the Fig. [REF]) are build from plaquettes [MATH] and [MATH], respectively.', 'cond-mat-0310043-2-29-1': 'These phases are unique.', 'cond-mat-0310043-2-29-2': 'We have similar situation for regions [MATH] (Neel phase; Fig. [REF], configuration 10) and [MATH] ( Fig. [REF], configuration 11 (it is an analogon of the ""planar"" phase in [CITATION]).', 'cond-mat-0310043-2-29-3': 'They are build from plaquettes [MATH] and [MATH], respectively.', 'cond-mat-0310043-2-29-4': 'Again, these phases are unique (modulo translations).', 'cond-mat-0310043-2-29-5': 'The situation for phases [MATH], [MATH] differs from previous ones.', 'cond-mat-0310043-2-29-6': 'These phases are build from plaquettes [MATH], [MATH], respectively (see Fig. [REF], configuration No. 5 as an example); however, they are non-unique and possess macroscopic degeneration.', 'cond-mat-0310043-2-29-7': 'One can easily check that there is a large dose of freedom in building of lattice configurations from these plaquettes.', 'cond-mat-0310043-2-29-8': 'We encounter here situation similar to this which happens for the antiferromagnetic Ising model on triangular lattice.', 'cond-mat-0310043-2-30-0': '## Order 3', 'cond-mat-0310043-2-31-0': 'Now, let us check how the phase diagram will change under switching third-order terms on.', 'cond-mat-0310043-2-31-1': 'Let us rewrite the third-order correction ([REF]) in the equivalent ""plaquette"" form: [EQUATION] where [EQUATION] (again spins [MATH] are arranged anticlockwise on the plaquette).', 'cond-mat-0310043-2-32-0': 'The full Hamiltonian, up to third-order terms, is the sum of terms ([REF]) and ([REF]).', 'cond-mat-0310043-2-32-1': 'As in previous Subsection, one can check that it is an m-potential.', 'cond-mat-0310043-2-32-2': 'Moreover, it turns out that the presence of third-order terms does not modify ground states of plaquettes.', 'cond-mat-0310043-2-32-3': 'In the other words, plaquette configurations which were ground-states in the second order, remain ground states also in third order!', 'cond-mat-0310043-2-32-4': 'This implies that degeneracy of phases [MATH] i [MATH] is not lifted and they still are degenerate.', 'cond-mat-0310043-2-32-5': 'What does change, it is location of the boundary between phases.', 'cond-mat-0310043-2-32-6': 'The difference in location of phase boundaries in orders 2 and 3 is of the order [MATH].', 'cond-mat-0310043-2-33-0': 'The phase diagram in third order possess certain kind of symmetry.', 'cond-mat-0310043-2-33-1': 'It is discussed in more details in the next Subsection.', 'cond-mat-0310043-2-33-2': 'At this moment, we only conclude that the phase diagram in 3.', 'cond-mat-0310043-2-33-3': 'order is a small deformation of the second-order phase diagram.', 'cond-mat-0310043-2-34-0': '## Phase diagram in fourth order', 'cond-mat-0310043-2-35-0': 'Regions occupied by phases [MATH] in both second and third order exhibit macroscopic degeneracy.', 'cond-mat-0310043-2-35-1': 'One can expect that they will be sensitive against perturbations and that in some of next orders this degeneracy will be lifted.', 'cond-mat-0310043-2-35-2': 'This happens yet in fourth order; we describe the situation below.', 'cond-mat-0310043-2-36-0': 'This picture has been obtained by the restricted phase diagram method.', 'cond-mat-0310043-2-36-1': 'Recall that in this method, one takes into account all periodic configurations up to certain values of lattice sites [MATH] per elementary cell, and then one minimizes the energy over this set of configurations.', 'cond-mat-0310043-2-37-0': 'The ground-state phase diagram is much more complicated than for the ordinary FKM - see Figs. [REF] and [REF] but it is still manageable (in some respects it is similar to the phase diagram of the FKM on triangular lattice, studied in [CITATION]).', 'cond-mat-0310043-2-37-1': 'Thirteen phases have been detected on phase diagram; it seems that three of them are degenerate.', 'cond-mat-0310043-2-37-2': 'Moreover, for each phase present for [MATH], there corresponds their ""mirror"" for [MATH] (this mirror is obtained by the change of occupied sites onto unoccupied ones and vice versa).', 'cond-mat-0310043-2-37-3': 'Periods of these phases do not exceed 12 sites per elementary cell.', 'cond-mat-0310043-2-37-4': ""Such a picture emerged at [MATH] and hasn't changed up to [MATH] (which corresponds to more than [MATH] trial configurations)."", 'cond-mat-0310043-2-37-5': 'For this reason, author claims that this phase diagram is ""exact but non-rigorous"".', 'cond-mat-0310043-2-38-0': 'The phase diagram in fourth order possess certain (pseudo)symmetries.', 'cond-mat-0310043-2-38-1': 'Let us describe the situation order-by-order.', 'cond-mat-0310043-2-39-0': 'In the second order, we have the Hamiltonian symmetric with respect to the change [MATH] and [MATH].', 'cond-mat-0310043-2-39-1': 'It is obvious that the phase diagram is symmetric with respect to the change [MATH], if one change also configuration to its mirror image.', 'cond-mat-0310043-2-40-0': 'In the third order, the Hamiltonian is no longer symmetric.', 'cond-mat-0310043-2-40-1': 'However, their ground states are easily determined, because the Hamiltonian is expressible as a sum of m-potentials, defined on [MATH] plaquettes.', 'cond-mat-0310043-2-40-2': 'It turns out that on the phase diagram, there are present the same phases as in the second order.', 'cond-mat-0310043-2-40-3': 'The difference between phase diagrams in the second and third order is apparent in location of phase boundaries; corresponding lines are shifted by a factor proportional to [MATH].', 'cond-mat-0310043-2-40-4': 'This shift is symmetric with respect to the change [MATH].', 'cond-mat-0310043-2-40-5': 'In the other words: If some boundary between phases [MATH] and [MATH] is shifted by [MATH] for [MATH], then, for [MATH], the boundary between mirrors of phases [MATH] and [MATH] is shifted by [MATH].', 'cond-mat-0310043-2-41-0': 'Let us analyse (pseudo)symmetries of the fourth order phase diagram.', 'cond-mat-0310043-2-41-1': 'Consider first some third-order configuration of minimal energy for [MATH] (call it CME[MATH]); for every such a configuration, we have its mirror, which is also the configuration of minimal energy for some [MATH] (call it CME[MATH]).', 'cond-mat-0310043-2-41-2': 'Now, let us add the fourth order contribution to the Hamiltonian.', 'cond-mat-0310043-2-41-3': 'This term is symmetric with respect to reversing of spins; so, if for [MATH] we have some configuration of minimal energy CME[MATH], then for [MATH] the configuration of minimal energy CME[MATH] will be the mirror of CME[MATH].', 'cond-mat-0310043-2-42-0': 'Let us summarize the above by the statement that the topological structure of phase diagram is the same for [MATH] and [MATH]; for every phase [MATH] appearing for [MATH], we have corresponding mirror [MATH] for [MATH].', 'cond-mat-0310043-2-42-1': 'Phase boundaries between phases [MATH] and [MATH] and their mirrors [MATH] and [MATH] are related by: [EQUATION]', 'cond-mat-0310043-2-42-2': 'Let us stress that this situation (i.e. the same structure of phase diagram for [MATH] and [MATH] is a very peculiar property of the fourth-order Hamiltonian ([REF]), ([REF]), ([REF]); it is due to degeneracy and (pseudo)symmetries of lower-order Hamiltonians.', 'cond-mat-0310043-2-42-3': 'For more general Hamiltonian, we have no such similarity.', 'cond-mat-0310043-2-43-0': 'Perhaps, the most interesting effect of the presence of the term with next-nearest-neighbour hopping is the appearance of the anomally large region occupied by the phase 5 (FK-like phase with density [MATH]).', 'cond-mat-0310043-2-43-1': 'At first sight, phases appearing in the fourth-order perturbation theory should occupy region of the width [MATH] (where [MATH] is some homogeneous fourth-order polynom in [MATH] and [MATH]).', 'cond-mat-0310043-2-43-2': 'However, it turns out that phase 5 occupies a region of width proportional to [MATH], i.e. of the same order as the Neel phase, appearing in the second order!', 'cond-mat-0310043-2-43-3': 'One can explain this phenomenon in the following way: Regions occupied by phase [MATH] (the situation with phase [MATH] is analogous) has width of the order [MATH] proportional to exhibit macroscopic degeneracy both in 2-nd and in 3-rd order.', 'cond-mat-0310043-2-43-4': ""The fourth-order perturbation lifts this degeneracy and as a result the phase [MATH] (an 'ancestor') transforms into non-degenerate phase 5 (the 'descendant') of the same density, and occupies region approximately as large as an 'ancestor' [MATH]."", 'cond-mat-0310043-2-44-0': 'Most of phases is unique, but there are also phases which remain degenerate even in the 4-th order (phases 1, 3, 12).', 'cond-mat-0310043-2-44-1': ""Strictly speaking, the restricted phase diagram method detects here only finite degeneracy, i.e. finitely many ground states with identical energy and density but different orderings (we don't count trivial degeneracy due to symmetry operations, i.e. translations, rotations and reflections)."", 'cond-mat-0310043-2-44-2': ""Number of these ground states grows with [MATH]; for [MATH] we observed: eight phases of equal energy and density [MATH], (the first two such configurations are phases 1 and 1'); eight phases of density [MATH] (the first two such configurations are phases 3 and 3'); five phases of density [MATH] (the first two of them are 12 and 12')."", 'cond-mat-0310043-2-44-3': 'Moreover, it has been observed that every member of such collection of phases with equal density is build up from identical plaquette configurations [MATH].', 'cond-mat-0310043-2-44-4': ""They can be 'glued' together in various arrangements and there is no uniqueness in such a procedure, i.e. resulting lattice configuration is non-unique."", 'cond-mat-0310043-2-44-5': 'Situation here is similar to that which happens in the second order for phases [MATH].', 'cond-mat-0310043-2-44-6': 'It is natural to conjecture that we encounter here the macroscopic degeneracy, i.e. presence of infinite number of configurations of identical density and energy.', 'cond-mat-0310043-2-44-7': ""However author (so far) can't prove this."", 'cond-mat-0310043-2-45-0': 'Phase diagrams in orders 2 and 3 are rigorous (by writing out Hamiltonians as sums of m-potentials).', 'cond-mat-0310043-2-45-1': 'Author tried to do analogous thing in the fourth-order by an attempt to construct m-potentials in a manner analogous as in [CITATION], [CITATION], [CITATION], however, it succeeded only for some phases, but not for a whole phase diagram.', 'cond-mat-0310043-2-46-0': ""An analysis above concerned the 'truncated' phase diagram, i.e. the phase diagram of the fourth-order effective Hamiltonian ([REF])."", 'cond-mat-0310043-2-46-1': 'Which changes can result as an effect coming from next (neglected) orders and temperature?', 'cond-mat-0310043-2-46-2': 'Author expects (and conjectures) that most phases (i.e. all with exception of degenerate ones: i.e. 1, 3, 11 and their descendants) are stable ones [CITATION], [CITATION].', 'cond-mat-0310043-2-46-3': 'For such phases, Peierls conditions (both classical and quantum ones - see [CITATION], [CITATION]) are fulfilled, and regions occupied by these phases deform in only small manner upon thermal and quantum perturbations.', 'cond-mat-0310043-2-46-4': 'This assertion concerns regions of phase diagram sufficiently far from phase boundaries.', 'cond-mat-0310043-2-46-5': 'For regions of width of the order of [MATH] around phase boundaries, one cannot formulate any statements without going into next orders of perturbation theory.', 'cond-mat-0310043-2-46-6': 'However, at present author cannot prove stability and Peierls conditions.', 'cond-mat-0310043-2-46-7': '(It could be proved by construction of m-potentials, which failed so far).', 'cond-mat-0310043-2-46-8': 'Regions occupied by degenerate phases constitute different problem.', 'cond-mat-0310043-2-46-9': 'Here, changes caused by thermal and quantum perturbations can be very significant; they are ""terra incognita"" and we skip this subject, leaving it as an open problem.', 'cond-mat-0310043-2-47-0': '# Summary and conclusions', 'cond-mat-0310043-2-48-0': 'The effective Hamiltonian and phase diagram for ground states of the [MATH] FKM have been determined up to fourth order of perturbation theory.', 'cond-mat-0310043-2-48-1': 'In the second and third order, phase diagram was constructed by rewritting the Hamiltonian as a sum of m-potentials.', 'cond-mat-0310043-2-48-2': 'The phase diagram in the fourth order has been determined by the method of restricted phase diagrams; author claims that such a picture is ""exact but non-rigorous"".', 'cond-mat-0310043-2-48-3': 'The phase diagram is considerably more complicated than for the ordinary FKM, but still it is manageable.', 'cond-mat-0310043-2-48-4': 'Thirteen phases are present (plus their ""mirrors""); three of these phases are degenerate (i.e. possess identical density and energy but different ordering).', 'cond-mat-0310043-2-48-5': 'Author conjectures that this degeneracy is macroscopic (the method of restricted phase diagrams handles only finite number of configurations).', 'cond-mat-0310043-2-49-0': 'Let us list some of the features of the phase diagram in fourth order:', 'cond-mat-0310043-2-50-0': 'For small [MATH], the phase diagram is similar to this for the ordinary FKM; however, for larger [MATH], these phase diagrams are quite different.', 'cond-mat-0310043-2-50-1': 'One observes anomally large region occupied by one of the phases appearing in 4-th order (phase number 5).', 'cond-mat-0310043-2-50-2': 'It could have experimental implications as a possibility of appearance of ""charge density waves"" more exotic than Neel ordering.', 'cond-mat-0310043-2-50-3': 'The phase diagram of the full model (i.e. with inclusion of neglected terms perturbation theory, as well as temperature) is more difficult to examine than for the ordinary FKM.', 'cond-mat-0310043-2-50-4': 'At this stage of investigation, almost nothing rigorous can be said.', 'cond-mat-0310043-2-50-5': 'However, author expects that after switching the quantum and thermal perturbations on, the phase diagram will change in only small manner inside regions occupied by nondegenerate phases (0,2,4-10,12).', 'cond-mat-0310043-2-50-6': 'But regions of width of the order [MATH] around the phase boundaries, as well as regions occupied by degenerate phases 1,3,11, are out of possibilities of present analysis.', 'cond-mat-0310043-2-50-7': 'The Falicov-Kimball model is sensitive to perturbations.', 'cond-mat-0310043-2-50-8': 'Modifications of the Hamiltonian such as introduction of correlated hopping, nnn-hopping or consideration of the FKM in non-perturbative regime (i.e. for values of [MATH] not very small) can significantly or even drastically modify the phase diagram.', 'cond-mat-0310043-2-51-0': '# Ground states, m-potentials', 'cond-mat-0310043-2-52-0': 'The Hamiltonian [MATH] , where [MATH], is a function defined on [MATH] - the space of all configurations of the system.', 'cond-mat-0310043-2-52-1': 'Usually Hamiltonian is defined as a sum of potentials, i.e. functions defined on subsets of [MATH]: [MATH].', 'cond-mat-0310043-2-52-2': 'Usually one imposes restrictions such that potentials are finite-range ones, i.e. such that [MATH], [MATH] finite.', 'cond-mat-0310043-2-52-3': 'It is also assumed that potentials are translation invariant.', 'cond-mat-0310043-2-53-0': 'm-potential.', 'cond-mat-0310043-2-53-1': 'Now, consider the system on an infinite lattice [MATH].', 'cond-mat-0310043-2-53-2': 'Assume that sets [MATH] (i.e. potential supports) are translation of a fixed plaquette [MATH] by a lattice vector [MATH]: [MATH], where [MATH] is an operator of such translation.', 'cond-mat-0310043-2-53-3': 'We say that the function [MATH] is an m-potential, if there exist configuration (perhaps, non-unique) [MATH] with the following properties: i) For every plaquette [MATH], the ""plaquette energies"", i.e. values of the Hamiltonian calculated on the plaquette [MATH]: [MATH] are all equal; ii) For every another configuration [MATH], the condition: [MATH] is fulfilled.', 'cond-mat-0310043-2-54-0': 'Ground states of the classical Hamiltonian.', 'cond-mat-0310043-2-54-1': 'If there exist such configuration [MATH] as above, then we call it the ground state of the Hamiltonian.', 'cond-mat-0310043-2-55-0': 'The property of the potential to be an m-potential can be reformulated as follows.', 'cond-mat-0310043-2-55-1': 'If a given potential is an m-potential, then the local minimality of energy (i.e. minimality on a plaquette) implies the global minimality on the whole lattice.', 'cond-mat-0310043-2-55-2': 'The property that a potential is an m-potential is very important one, as it replaces searching of ground states of the infinite lattice by looking for the minima on a finite set.', 'cond-mat-0310043-2-55-3': 'Unfortunately, some given potential possess this property only exceptionally.', 'cond-mat-0310043-2-55-4': '(But fortunately, in Secs. [REF] and [REF] they share such property!)', 'cond-mat-0310043-2-55-5': 'A method to avoid this obstacle is to find - for a given potential [MATH] - an equivalent potential [MATH], such that [MATH] is an m-potential.', 'cond-mat-0310043-2-55-6': 'However, in general it is difficult task.'}","[['cond-mat-0310043-1-55-0', 'cond-mat-0310043-2-55-0'], ['cond-mat-0310043-1-55-1', 'cond-mat-0310043-2-55-1'], ['cond-mat-0310043-1-55-2', 'cond-mat-0310043-2-55-2'], ['cond-mat-0310043-1-55-3', 'cond-mat-0310043-2-55-3'], ['cond-mat-0310043-1-55-4', 'cond-mat-0310043-2-55-4'], ['cond-mat-0310043-1-55-5', 'cond-mat-0310043-2-55-5'], ['cond-mat-0310043-1-55-6', 'cond-mat-0310043-2-55-6'], ['cond-mat-0310043-1-27-0', 'cond-mat-0310043-2-27-0'], ['cond-mat-0310043-1-9-0', 'cond-mat-0310043-2-9-0'], ['cond-mat-0310043-1-40-0', 'cond-mat-0310043-2-40-0'], ['cond-mat-0310043-1-40-1', 'cond-mat-0310043-2-40-1'], ['cond-mat-0310043-1-40-2', 'cond-mat-0310043-2-40-2'], ['cond-mat-0310043-1-40-3', 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1811.11712,"{'1811.11712-1-0-0': '[1]ELI-ALPS, ELI-Hu NKft, Dugonics ter 13, H-6720 Szeged, Hungary [2]Institut fur Theoretische Physik, Universitat zu Koln', '1811.11712-1-1-0': 'Zulpicher Strasse 77, D-50937 Koln, Germany [3]Saint-Petersburg State University', '1811.11712-1-2-0': 'Ulyanovskaya str.', '1811.11712-1-2-1': '1, Petrodvorets, Saint-Petersburg 198504, Russia', '1811.11712-1-3-0': 'At strong coupling, the scalar Schwinger effect is studied by the field-theoretic method of worldline instantons for dynamic fields of single-pulse and sinusoidal types.', '1811.11712-1-3-1': 'By examining the Wilson loop along the closed instanton path, corrections to the results obtained from weak-coupling approximations are discovered.', '1811.11712-1-3-2': 'It shows that: [label=*)] since the additional terms depend on strength of the background field, the Wilson loop becomes non-negligible even in the extremely weak field; a similar breaking of weak-field condition happens beyond the critical field as well.', '1811.11712-1-4-0': 'In other words, considerations at strong coupling turn out to be indispensable for strongly dynamic field.', '1811.11712-1-4-1': ""Therefore, following Semenoff and Zarembo's proposal, the Schwinger effect at strong coupling is studied in the context of [MATH] supersymmetric Yang-Mills theory for the Coulomb phase."", '1811.11712-1-4-2': 'With the help of gauge/gravity duality the vacuum decay rate is evaluated by the string action with instanton worldline as boundary, which is located on a probe D3-brane.', '1811.11712-1-4-3': 'The corresponding classical worldsheets are estimated by perturbing the integrable case of the constant field.', '1811.11712-1-5-0': '# Introduction', '1811.11712-1-6-0': 'The Extreme Light Infrastructure (ELI) is designed to produce the highest power and intense laser worldwide [CITATION].', '1811.11712-1-6-1': 'It has the potential of reaching ultra-relativistic intensities, which will challenge the Schwinger limit [MATH].', '1811.11712-1-6-2': 'As laser intensity approaches this value, the vacuum becomes unstable, and charged particles produce in pairs, so that laser loses the energy, and its intensity stays within the upper limit.', '1811.11712-1-6-3': 'However, Schwinger effect is not only a phenomenon in electromagnetism, but a universal aspect of quantum vacuum in the presence of a [MATH] gauge field with a classical background, see e.g. [CITATION].', '1811.11712-1-7-0': 'The pair production rate in the constant electric field was pioneered by Sauter, Heisenberg and Euler [CITATION], and the corresponding effect was named after Schwinger [CITATION] who did the calculation based on field theoretical approaches, see e.g. [CITATION] for a current review.', '1811.11712-1-7-1': 'A semiclassical approach called worldline instantons has been introduced into the study of the cases with constant and inhomogeneous fields in the small-coupling and weak-field approximations [CITATION], in which the production rate, in Wick-rotated Euclidean space, is represented by a worldline path integral.', '1811.11712-1-7-2': 'The so-called worldline instantons are the periodic saddle points relevant for a calculation of integral by the steepest descent method.', '1811.11712-1-8-0': 'The case at strong coupling in constant field has been studied with the help of gauge/gravity duality [CITATION], also known as Semenoff-Zarembo construction, where the production rate has been obtained by calculating the classical action of bosonic string, which attaches on the probe D3-brane and couples to a Kalb-Ramond field.', '1811.11712-1-8-1': 'For an arbitrary external field, the problem is related to integrating the classical equations of motion for bosonic string, because the instanton action in the production rate is replaced by the string action in the given external field.', '1811.11712-1-8-2': 'In other words, duality converts the problem to computing the area of minimal surface [CITATION] in Euclidean [MATH], whose boundary is assumed to be the trajectory of the worldline on the probe D3-brane.', '1811.11712-1-8-3': ""In mathematics, such a Dirichlet problem is known as Plateau's problem [CITATION]."", '1811.11712-1-9-0': 'In this work, we consider the scalar pair production for dynamic external field of single-pulse and sinusoidal types at strong coupling by the method of worldline instantons, which is explained in sections [REF] and [REF].', '1811.11712-1-9-1': 'We first show in a field-theoretical approach, that besides the enhancement due to the dynamics of electric fields, a further contribution to the production rate arises from the Wilson loop.', '1811.11712-1-9-2': 'However, such a correction diverges as [MATH] and leads a contradiction on the weak-field condition, so that near the critical field, the method itself breaks down, and vacuum cascade therefore becomes vaguely.', '1811.11712-1-9-3': ""To clear the problem, we follow Semenoff and Zarembo's proposal, applying the gauge/gravity duality to study the Schwinger effect in the Coulomb phase of a [MATH] supersymmetric Yang-Mills theory in sec. [REF]."", '1811.11712-1-9-4': 'The classical solution of the corresponding string worldsheets are estimated by perturbation around the solvable case of a constant electric field.', '1811.11712-1-10-0': '# Worldline instantons at strong coupling', '1811.11712-1-11-0': 'The worldline instanton approach is a semiclassical calculation realised by the worldline path integral representations [CITATION].', '1811.11712-1-11-1': 'The so-called worldline instanton is a periodic solution of the stationary phase in the path integral.', '1811.11712-1-11-2': 'Based on this method, the pair production rate [MATH] for massive scalar QED in the small-coupling and weak-field approximation is given by [CITATION] [EQUATION] where [MATH] is 4-volume, [MATH] is a constant [EQUATION] and [MATH] represents the action of worldline instanton, i.e. [EQUATION] here [MATH] is classical background field different with fluctuation part of [MATH] gauge field [MATH], which is also included in the initial setup of path integral but cancelled due to consideration in small coupling.', '1811.11712-1-12-0': 'The functional integral in eq. [REF] can be computed by the stationary phase approximation, for more general cases with 1D temporal inhomogeneous fields [MATH], the worldline instanton is obtained by solving the instanton equation with periodic boundary condition [EQUATION]', '1811.11712-1-12-1': 'Thus the creation rate can be represented by [EQUATION] where [MATH] is a monotone decreasing function of Keldysh parameter [MATH] and calculated by substituting the solution of eq. [REF] into the worldline action with the corresponding background field [MATH].', '1811.11712-1-13-0': 'For arbitrary coupling constant, a modification is deduced as average of [MATH] Wilson loop [MATH], i.e. [EQUATION] with [EQUATION]', '1811.11712-1-13-1': 'The integral path [MATH] is along the 2D trajectory of worldline instanton.', '1811.11712-1-13-2': 'In the Feynman gauge it becomes [CITATION] [EQUATION]', '1811.11712-1-13-3': 'For the constant electric field, worldline instanton is a circle, and the production rate of single scalar pair is given by [EQUATION] where [MATH] is Schwinger limit in the natural unit.', '1811.11712-1-13-4': 'The consideration on arbitrary coupling [CITATION] leads to a correction in eq. [REF], see also sec. [REF] [EQUATION]', '1811.11712-1-13-5': 'Our aim in the next section is to calculate the Wilson loops in eq. [REF] along two specific worldlines separately.', '1811.11712-1-13-6': 'The fist one is the worldline instanton in the single-pulse field, while the second is in the sinusoidal field.', '1811.11712-1-13-7': 'We show that the average of Wilson loop diverges in the extremely weak strength of field and the weak-field condition is also broken in these two cases, as noted e.g. in [CITATION].', '1811.11712-1-14-0': '# Wilson loops along worldline instanton paths', '1811.11712-1-15-0': 'The central problem in the case with arbitrary coupling is Wilson loop.', '1811.11712-1-15-1': 'The integral in eq. [REF] generally diverges as [MATH] approaches [MATH] [CITATION].', '1811.11712-1-15-2': 'To obtain a finite result, one can introduce a small parameter [MATH] in order to perform the regularisation [EQUATION] here [MATH] and [MATH] can be understood as angular variables of coordinates for closed instanton path.', '1811.11712-1-15-3': 'The divergence of Wilson loop will be removed by subtracting the following term [EQUATION] which is called perimeter law depicted the behaviour of Wilson loop in Euclidean space [CITATION].', '1811.11712-1-15-4': 'Because all worldline instantons satisfy the condition [MATH], here [MATH] is defined in the same way with [CITATION], thus the divergent term can be reduced [EQUATION]', '1811.11712-1-15-5': 'After subtraction of this term from eq. [REF] , let the parameter [MATH] tends to zero, one obtains the regularised [MATH] [EQUATION]', '1811.11712-1-15-6': 'Therefore the regularised Wilson loop is given by [EQUATION] which appears as a factor in the final production rate.', '1811.11712-1-16-0': '## Constant electric field', '1811.11712-1-17-0': 'For constant electric field, the worldline instanton path is a circle of radius [MATH] [CITATION] [EQUATION]', '1811.11712-1-17-1': 'The zero component [MATH] denotes the Euclidean time.', '1811.11712-1-17-2': 'The instanton action for single pair production is obtained by substituting this circle solution into eq. [REF] [EQUATION]', '1811.11712-1-17-3': 'The weak field condition [MATH] implies [EQUATION]', '1811.11712-1-17-4': 'On the other side, the integral in eq. [REF] can be calculated explicitly, so is the perimeter law in eq. [REF] [EQUATION]', '1811.11712-1-17-5': 'Thus the regularised Wilson loop for single scalar pair can be recovered [CITATION] [EQUATION]', '1811.11712-1-17-6': 'Then the production rate is given by [EQUATION] from which the critical field for vacuum cascade can be estimated [EQUATION] namely [EQUATION] which is much greater than the Schwinger limit and breaks the weak field condition in eq. [REF], as noted in e.g. [CITATION].', '1811.11712-1-17-7': 'Therefore the obtained results are not valid when the field approaches to the range close to the Schwinger limit and can not answer whether the cascade of vacuum-decay happens.', '1811.11712-1-18-0': '[Single-pulse field [MATH]]Single-pulse field [MATH]', '1811.11712-1-19-0': 'For more general 1D dynamical field, we suppose that the Wilson loop is of following form [EQUATION] where [MATH] is enhancement factor with respect to the case of constant external field in eq. [REF].', '1811.11712-1-20-0': 'The first example is pair production in the single pulse field [CITATION].', '1811.11712-1-20-1': 'The 2D instanton path can be calculated analytically [EQUATION] so the worldline action for this path is given by [EQUATION]', '1811.11712-1-20-2': 'The weak-field condition leads to [EQUATION]', '1811.11712-1-20-3': 'The approximation [MATH] is reasonable here, then weak-field condition becomes [MATH].', '1811.11712-1-20-4': 'To estimate the Wilson loop, one can expand the numerator and denominator in the integrand of eq. [REF] at second order separately and find [EQUATION] while the perimeter law has a closed form [EQUATION]', '1811.11712-1-20-5': 'The enhancement factor in the week-field [MATH] condition is then [EQUATION]', '1811.11712-1-20-6': 'Since [MATH], it tends to enhance the contribution to Wilson loop in eq. [REF], moreover the pair production rate is no longer exponentially suppressed, see fig. [REF].', '1811.11712-1-21-0': 'The production rate reads [EQUATION]', '1811.11712-1-21-1': 'Note that for small [MATH], one has critical value of the field [EQUATION] it also breaks the weak field condition as the case in the constant field, even for [MATH], [MATH].', '1811.11712-1-21-2': 'In contrast with the case in constant field, the exponential factor of Wilson loop is no longer constant.', '1811.11712-1-21-3': 'It diverges as [MATH] approaches to [MATH], which leads a divergent term in the decay rate [MATH] around [MATH], even though the pre-exponential factor has to be taken account of [CITATION], because the exact form of pre-exponential factor tends to a constant as [MATH].', '1811.11712-1-21-4': 'In other wolds, production rate in eq. [REF] is not valid in the extremely weak field.', '1811.11712-1-22-0': '[Sinusoidal field [MATH]]Sinusoidal field [MATH]', '1811.11712-1-23-0': 'The second example is in sinusoidal field [CITATION].', '1811.11712-1-23-1': 'The components of worldline instanton are represented through the special functions [EQUATION]', '1811.11712-1-23-2': 'Here [MATH] is complete elliptic integral, [MATH] and [MATH] are Jacobi elliptic functions.', '1811.11712-1-23-3': 'The instanton action is given by [EQUATION]', '1811.11712-1-23-4': 'Repeating the procedure above, one gets [EQUATION] and the perimeter law [EQUATION]', '1811.11712-1-23-5': 'Thus at the week-field approximation [MATH], one has [EQUATION] [MATH] is also greater than unit and not a trivial function depending on external field, see fig. [REF].', '1811.11712-1-23-6': 'In addition, [MATH] is also divergent as [MATH] under the consideration of pre-exponential factor [CITATION].', '1811.11712-1-24-0': 'The production rate is then [EQUATION]', '1811.11712-1-24-1': 'For small [MATH], one gets [EQUATION] which breaks the weak field condition as well.', '1811.11712-1-24-2': 'From these three examples above one may note that, first the connection of coupling constant with the strength of the dynamical field is an universal aspect, i.e. the weak field condition is inevitably broken by the increasing coupling; second, the production rate diverges in the extremely weak field.', '1811.11712-1-25-0': '# Holographic Schwinger effect with dynamic field', '1811.11712-1-26-0': 'In order to answer if the vacuum decay for strong coupling happens as the strength of dynamical field close to the Schwinger limit, we consider the similar effect in the gauge/gravity duality, the gauge field and quantum gravity theory here refer in particular to the [MATH] SYM on 4D boundary of [MATH] and the type IIB superstring theory in the bulk of [MATH].', '1811.11712-1-27-0': ""According to Semenoff-Zarembo's holographic setup [CITATION], the production rate is obtained from the string worldsheet attaching to the boundary loop on probe D3-brane, i.e. [EQUATION] where [MATH] is Nambu-Goto (NG) action [EQUATION] and [MATH] is Kalb-Ramond (or NS-NS, where NS is the abbreviation of Neveu-Schwarz) 2-form as interaction term of string, [EQUATION]"", '1811.11712-1-27-1': 'Here [MATH] are the coordinates of string worldsheet and [MATH] describe the 10D [MATH] space-time with metric [MATH], the [MATH] is embedding metric of worldsheet.', '1811.11712-1-28-0': 'The shape of boundary loop on the D3-brane is equal to worldline instanton according to the assumption, thus the central problem is converted to compute the minimal surface in Euclidean [MATH] given the equation of boundary.', '1811.11712-1-29-0': '## Constant electric field', '1811.11712-1-30-0': 'The trajectory of worldline instanton in the constant field is circle, thus one can parametrise the worldsheet by [EQUATION] which is different with [CITATION], and the corresponding Jacobian between two parameterisations is of sign, i.e. [MATH]; in other words, the transformation from one to another reverses the orientation of Kalb-Ramond coupling.', '1811.11712-1-30-1': 'The NG action in this parameterisation becomes [EQUATION] where [MATH], [MATH] is location of D3-brane.', '1811.11712-1-30-2': 'Then one can solve [MATH] analytically [CITATION] [EQUATION]', '1811.11712-1-30-3': 'Here [MATH] is the trajectory of worldline instanton on the D3-brane.', '1811.11712-1-30-4': 'Substituting it into the NG-action, one obtains [EQUATION]', '1811.11712-1-30-5': 'The NS-NS term is then [EQUATION] where [MATH] can be fixed by the extremising the total action, [EQUATION]', '1811.11712-1-30-6': 'The production rate is [EQUATION] here we have replaced the parameters of string and spacetime by the one of gauge field via [MATH].', '1811.11712-1-31-0': '[Estimation for Single-pulse field [MATH]]Estimation for Single-pulse field [MATH]', '1811.11712-1-32-0': 'The worldline instanton as boundary on the D3-brane in the case with single-pulse field is shown in eq. [REF].', '1811.11712-1-32-1': 'Thus one can parametrise the worldsheet by using [MATH] an [MATH], i.e. [EQUATION] where [MATH] is regarded as initial information and not relevant to the scale [MATH].', '1811.11712-1-32-2': 'Since as [MATH] [EQUATION] the zero order of [MATH] is circle boundary, we will treat the complete worldline boundary as perturbation around the circle.', '1811.11712-1-32-3': 'The [MATH] component can be estimated by noting that [EQUATION] with the trajectory of worldline instanton [MATH].', '1811.11712-1-32-4': 'In single-pulse field, we replace [MATH] by the corresponding trajectory of worldline, i.e. [EQUATION]', '1811.11712-1-32-5': 'In the limit [MATH], [MATH] reduces to circle [EQUATION] thus the NG-action is formulated [EQUATION] where [MATH] is given in eq. [REF], while the NS-NS part right now becomes [EQUATION]', '1811.11712-1-32-6': 'The [MATH] is fixed as the stationary point of the total action, [EQUATION]', '1811.11712-1-32-7': 'The rate with correction up to the second order of [MATH] is [EQUATION]', '1811.11712-1-32-8': 'The first part standing in the exponential function comes from the circle boundary, while the second term arises from the deformation, see fig. [REF].', '1811.11712-1-33-0': 'The correction due to dynamics of background field increases the critical field, which is fixed by [EQUATION] beyond which the decay rate decreases as the similar as eq. [REF].', '1811.11712-1-34-0': '[Estimation for sinusoidal field [MATH]]Estimation for sinusoidal field [MATH]', '1811.11712-1-35-0': 'The worldline instanton in this case can be parametrised as [EQUATION] where [MATH] is defined in eq. [REF].', '1811.11712-1-35-1': 'The Maclaurin series of Jacobian elliptic functions in [MATH] provide [EQUATION]', '1811.11712-1-35-2': 'The trajectory of worldline instanton on D3-brane in this case [EQUATION] and the expansion with respect to small [MATH] is [EQUATION]', '1811.11712-1-35-3': 'The second order of NG-action and NS-NS term are separately [EQUATION] and [EQUATION] [MATH] up to second order of [MATH] is given by [EQUATION]', '1811.11712-1-35-4': 'The final rate, see fig. [REF] [EQUATION]', '1811.11712-1-35-5': 'The critical field should be greater than [MATH], i.e. [EQUATION] and the dynamic characteristic is of the similar tendency of moving critical field, namely it enhances the value of critical field.', '1811.11712-1-36-0': '# Conclusion and Discussion', '1811.11712-1-37-0': 'In this paper, the scalar Schwinger effect at strong coupling has been studied, using the field-theoretic method of worldline instantons.', '1811.11712-1-37-1': 'A non-trivial contribution to the production rate is discovered by evaluating the Wilson loop along the path of worldline instanton.', '1811.11712-1-37-2': 'This correction term depends on the field strength [MATH], and diverges as [MATH] approaches to zero even under the consideration on exact form of pre-exponential factor.', '1811.11712-1-38-0': 'However one can introduce a cutoff [MATH] by minimising eq. [REF], i.e. [MATH] is fixed by computing the following algebraic equation', '1811.11712-1-39-0': '[EQUATION] where [MATH] is fine-structure constant, [MATH] and [MATH].', '1811.11712-1-39-1': 'For the cases with small [MATH] and [MATH], one can give the estimation [EQUATION]', '1811.11712-1-39-2': 'The vacuum decay rate [REF] is valid beyond the [MATH].', '1811.11712-1-39-3': 'This provide a solution to possible problem of divergence.', '1811.11712-1-39-4': 'In addition, the introduction of the correction term leads a contradiction on the weak-field condition near critical field strength, which also makes it indispensable to further investigate the strong coupling case with dynamic field.', '1811.11712-1-40-0': 'To solve the problem of breaking-weak condition, in the context of [MATH] supersymmetric Yang-Mills theory the production rate is calculated by gauge/gravity duality, according to which the instanton action is replaced by the area of a classical string worldsheet in Euclidean [MATH], where the boundary on the probe D3-brane is given by the trajectory of worldline instantons.', '1811.11712-1-40-1': 'Thus the problem is converted to solving the classical bosonic strings with Dirichlet boundary conditions.', '1811.11712-1-40-2': 'To provide an explicit estimation, we treat the worldsheet in the particular problems as perturbations around the one with circle boundary, which has been solved exactly.', '1811.11712-1-40-3': 'Such an expansion is practical and realistic, because only low-frequency laser (comparing with electron mass) is currently operational.', '1811.11712-1-40-4': 'The obtained decay rates in two considered examples of dynamic fields are similar concave functions as the case in constant field, but the critical fields are comparably increased.'}","{'1811.11712-2-0-0': 'At strong coupling, the scalar Schwinger effect is studied by the field-theoretical method of worldline instantons for dynamic fields of single-pulse and sinusoidal types.', '1811.11712-2-0-1': 'By examining the Wilson loop along the closed instanton path, corrections to the results obtained from weak-coupling approximations are discovered.', '1811.11712-2-0-2': 'They show that: [label=*)] since the additional terms depend on strength of the background field, the Wilson loop becomes non-negligible even in the extreme weak-field limit; and a breaking of weak-field condition similar to constant field also happens beyond the critical field.', '1811.11712-2-1-0': 'In other words, considerations at strong coupling turn out to be indispensable for strongly dynamic field.', '1811.11712-2-1-1': ""Therefore, following Semenoff and Zarembo's proposal, the Schwinger effect at strong coupling is studied with an [MATH] supersymmetric Yang-Mills theory in the Coulomb phase."", '1811.11712-2-1-2': 'With the help of the gauge/gravity duality, the vacuum decay rate is evaluated by the string action with instanton worldline as boundary, which is located on a probe D3-brane.', '1811.11712-2-1-3': 'The corresponding classical worldsheets are estimated by perturbing the integrable case of a constant field.', '1811.11712-2-2-0': '# Introduction', '1811.11712-2-3-0': 'The Extreme Light Infrastructure (ELI) is designed to produce the highest power and intense laser worldwide [CITATION].', '1811.11712-2-3-1': 'It has the potential of reaching ultra-relativistic intensities, challenging the Schwinger limit [MATH].', '1811.11712-2-3-2': 'As laser intensity approaches this value, vacuum becomes unstable, and charged particles produce in pairs, so that laser loses the energy, and its intensity stays within the upper limit.', '1811.11712-2-3-3': 'However, Schwinger effect is not only a phenomenon in electromagnetism, but a universal aspect of quantum vacuum in the presence of a [MATH] gauge field with a classical background, see e.g. [CITATION].', '1811.11712-2-4-0': 'The pair-production rate in the constant electric field had been pioneered by Sauter, Heisenberg and Euler [CITATION], and the corresponding Effect was named after Schwinger [CITATION], who did the calculation based on field-theoretical approaches, see e.g. [CITATION] for a current review.', '1811.11712-2-4-1': 'A semiclassical approach called worldline instantons has been introduced more recently to the study of constant and inhomogeneous fields in the small-coupling and weak-field approximations [CITATION], where the production rate, in Wick-rotated Euclidean space, is represented by a worldline path integral.', '1811.11712-2-4-2': 'The so-called worldline instantons are the periodic saddle points relevant for a calculation of integral by the steepest descent method.', '1811.11712-2-5-0': 'The pair-production rate at strong coupling in constant field has been studied with the help of gauge/gravity duality [CITATION], also known as the Semenoff-Zarembo construction, where the production rate has been obtained by calculating the classical action of a bosonic string, which is attached to a probe D3-brane and coupled to a Kalb-Ramond field.', '1811.11712-2-5-1': 'Since the instanton action in the production rate is equivalent to the string action, which is proportional to the area, calculation of the rate is related to integrating the classical equations of motion for bosonic string in the given external field.', '1811.11712-2-5-2': 'In other words, duality converts the problem to evaluating the area of a minimal surface [CITATION] in Euclidean [MATH], the boundary of which is assumed to be the trajectory of the worldline on the probe D3-brane.', '1811.11712-2-5-3': ""In mathematics, such a Dirichlet problem is known as the Plateau's problem [CITATION]."", '1811.11712-2-6-0': 'In this work, we consider the scalar pair production in a dynamic external field of single-pulse and sinusoidal types at strong coupling by the method of worldline instantons, which is explained in sections [REF] and [REF].', '1811.11712-2-6-1': 'We first show by a field-theoretical approach, that besides the enhancement due to the dynamics of electric fields, a further contribution to the production rate arises from the Wilson loop.', '1811.11712-2-6-2': 'However, such a correction diverges as [MATH] and leads to a contradiction to the weak-field condition, so that near the critical field, the method itself breaks down, and the prediction of a vacuum cascade becomes unclear.', '1811.11712-2-6-3': ""To overcome the problem, we then follow Semenoff and Zarembo's proposal in sec. [REF], applying the gauge/gravity duality to the Schwinger effect in the Coulomb phase of an [MATH] supersymmetric Yang-Mills theory."", '1811.11712-2-6-4': 'The classical solution of the corresponding string worldsheets are estimated by perturbing the solvable case of a constant external electric field.', '1811.11712-2-7-0': '# Worldline instantons at strong coupling', '1811.11712-2-8-0': 'The worldline instanton approach is a semiclassical calculation realised by the worldline path integral representations [CITATION], in which the so-called worldline instanton is a periodic solution of the stationary phase in the path integral.', '1811.11712-2-8-1': 'Based on this method, the pair-production rate [MATH] for a massive scalar QED in the small-coupling and weak-field approximation is given by [CITATION] [EQUATION] where [MATH] is the 4-volume, [MATH] is a constant, given by [EQUATION] and [MATH] represents the action of worldline instanton, i.e. [EQUATION]', '1811.11712-2-8-2': 'In eq. [REF], [MATH] is a classical background field, not to be confused with the fluctuation part of the [MATH] gauge field [MATH], which is also included in the initial setup of path integral, but cancelled due to considerations at small coupling.', '1811.11712-2-9-0': 'The path integral in eq. [REF] can be computed by the stationary phase approximation.', '1811.11712-2-9-1': 'for more general cases with 1D temporally inhomogeneous fields [MATH], the worldline instanton is obtained by solving the instanton equation with periodic boundary condition [EQUATION]', '1811.11712-2-9-2': 'Thus the pair-production rate can be written as [EQUATION] where [MATH] is a monotonically decreasing function of Keldysh parameter [MATH] and calculated by substituting the solution of eq. [REF] into the worldline action with the corresponding background field [MATH].', '1811.11712-2-10-0': 'At arbitrary coupling constant, the production rate is changed by a factor of the averaged [MATH] Wilson loop [MATH], [EQUATION]', '1811.11712-2-10-1': 'The integral path [MATH] is along the 2D trajectory of worldline instanton.', '1811.11712-2-10-2': 'In the Feynman gauge of the [MATH] field, the Wilson loop becomes [CITATION] [EQUATION]', '1811.11712-2-10-3': 'For a constant electric field, the worldline instanton is a circle, and the production rate of a single scalar pair is given by [EQUATION] where [MATH] is the Schwinger limit in natural units.', '1811.11712-2-10-4': 'Consideration at arbitrary coupling [CITATION] leads to a correction to eq. [REF] [EQUATION] see also sec. [REF] Our aim in the next section is to calculate the Wilson loops in eq. [REF] along two specific worldlines separately.', '1811.11712-2-10-5': 'The fist one is the worldline instanton in a single-pulse field, whereas the second is in a sinusoidal field.', '1811.11712-2-10-6': 'We show that in extremely weak field, the average of regularised Wilson loop diverges; the weak-field condition is also broken in these two cases, as noted in e.g. [CITATION]; see also sec. [REF].', '1811.11712-2-11-0': '# Wilson loops along worldline instanton paths', '1811.11712-2-12-0': 'The Wilson loop in eq. [REF] plays a central role in the case of arbitrary coupling, which typically diverges as [MATH] approaches [MATH] [CITATION].', '1811.11712-2-12-1': 'To obtain a finite result, one can introduce a small regularisation parameter [MATH], such that [EQUATION] where [MATH] and [MATH] can be understood as angular coordinates for the instanton.', '1811.11712-2-12-2': 'The divergence of Wilson loop can now be removed by subtracting [EQUATION] which is an example of the perimeter law, depicting the behaviour of the Wilson loop in Euclidean space [CITATION].', '1811.11712-2-12-3': 'Because all worldline instantons satisfy [MATH], where [MATH] is defined in the same way as in [CITATION], the divergent term can be reduced to [EQUATION]', '1811.11712-2-12-4': 'After subtracting this term from eq. [REF], one obtains the regularised [MATH] by taking the limit [EQUATION]', '1811.11712-2-12-5': 'The regularised Wilson loop is then given by [EQUATION] which appears as a factor in the final expression of the production rate.', '1811.11712-2-13-0': '## Constant electric field', '1811.11712-2-14-0': 'For a constant electric field, the worldline instanton is a circle of radius [MATH], given by [CITATION] [EQUATION] where the zeroth component [MATH] denotes the Euclidean time.', '1811.11712-2-14-1': 'The instanton action for single pair production is obtained by substituting this solution into eq. [REF], yielding [EQUATION]', '1811.11712-2-14-2': 'The weak field condition [MATH] [CITATION] implies [EQUATION] where [MATH] is defined in eq. [REF].', '1811.11712-2-15-0': 'On the other hand, the integral in eq. [REF] and the perimeter law in eq. [REF] can be evaluated explicitly as [EQUATION]', '1811.11712-2-15-1': 'Hence the regularised Wilson loop for a single scalar pair can be recovered as [CITATION] [EQUATION] and the production rate is given by [EQUATION] from which the critical field for vacuum cascade can be estimated by [EQUATION]', '1811.11712-2-15-2': 'One sees that [MATH] is much greater than the Schwinger limit and breaks the weak field condition in eq. [REF], as have been noted in e.g. [CITATION].', '1811.11712-2-15-3': 'Therefore, the obtained results are not valid when the field goes close to the Schwinger limit, and cannot answer the question whether a vacuum cascade happens near the critical field strength.', '1811.11712-2-16-0': '[Single-pulse field [MATH]] Single-pulse field [MATH]', '1811.11712-2-17-0': 'For a generic 1D dynamic field, we assume that the Wilson loop is of the following form [EQUATION] where [MATH] is an enhancement factor with respect to the case of a constant external field in eq. [REF].', '1811.11712-2-18-0': 'The first example is pair production in the single-pulse field [CITATION].', '1811.11712-2-18-1': 'The instanton can be evaluated explicitly as [EQUATION] and the worldline action for this path reads [EQUATION]', '1811.11712-2-18-2': 'The weak-field condition leads to [EQUATION]', '1811.11712-2-18-3': 'The approximation [MATH] is reasonable here, so that the weak-field condition becomes [MATH].', '1811.11712-2-19-0': 'To estimate the Wilson loop, one can expand the numerator and denominator separately in the integrand of eq. [REF] up to second order, yielding [EQUATION]', '1811.11712-2-19-1': 'Meanwhile, the corresponding perimeter law has a closed form [EQUATION]', '1811.11712-2-19-2': 'The enhancement factor is then in the weak-field condition [MATH] [EQUATION]', '1811.11712-2-19-3': 'Since [MATH], the factor amplifies the contribution from the Wilson loop in eq. [REF], so that the pair-production rate is no longer exponentially suppressed, see fig. [REF].', '1811.11712-2-20-0': 'Now the exponential factor in the production rate is evaluated as [EQUATION]', '1811.11712-2-20-1': 'Note that for small [MATH], one has the critical field [EQUATION]', '1811.11712-2-20-2': 'On the one hand, as in the case with constant field, eq. [REF] also breaks the weak-field condition, even for [MATH] and [MATH].', '1811.11712-2-20-3': 'On the other hand, in contrast with the case in constant field, the exponential factor of Wilson loop is no longer constant.', '1811.11712-2-20-4': 'It diverges as [MATH] approaches [MATH], which leads to a divergent term in the decay rate [MATH] for [MATH], even if the pre-exponential factor were taken into account [CITATION], which approaches a constant as [MATH].', '1811.11712-2-20-5': 'In other words, the production rate in eq. [REF] is not valid in the extremely weak field.', '1811.11712-2-21-0': '[Sinusoidal field [MATH]] Sinusoidal field [MATH]', '1811.11712-2-22-0': 'The second example is a sinusoidal field [CITATION].', '1811.11712-2-22-1': 'The coordinates of the instanton can be represented by special functions as [EQUATION] where [MATH] is the complete elliptic integral, [MATH] and [MATH] are Jacobi elliptic functions.', '1811.11712-2-22-2': 'The instanton action is given by [EQUATION]', '1811.11712-2-22-3': 'Repeating the procedure above, one obtains the exponent [EQUATION] and the corresponding perimeter law [EQUATION]', '1811.11712-2-22-4': 'Thus at the week-field approximation [MATH], one has [EQUATION]', '1811.11712-2-22-5': 'In this case, [MATH] is also greater than unity and a non-trivial function depending on external field, see fig. [REF].', '1811.11712-2-23-0': 'In addition, [MATH] is also divergent as [MATH], even if the pre-exponential factor is considered [CITATION].', '1811.11712-2-23-1': 'The exponent factor in the production rate is then given by [EQUATION]', '1811.11712-2-23-2': 'For small [MATH], one gets [EQUATION] which breaks the weak field condition as well.', '1811.11712-2-23-3': 'From these three examples above, one may note that, first, the relation of coupling constant with the strength of the dynamic field is universal, i.e. the weak field condition is inevitably broken by the increasing coupling; and second, the production rate diverges in the extremely weak field.', '1811.11712-2-24-0': '# Holographic Schwinger effect with dynamic field', '1811.11712-2-25-0': 'In order to answer the question, if the vacuum decay for strong coupling happens as the strength of time-dependent field goes close to the Schwinger limit, we consider a similar effect in the gauge/gravity duality, where the gauge field theory refers to an [MATH] supersymmetric Yang-Mills theory on the 4D boundary of an [MATH] space, and the quantum gravity is a type 2B superstring theory in the bulk of the [MATH].', '1811.11712-2-26-0': ""According to Semenoff and Zarembo's holographic setup [CITATION],"", '1811.11712-2-27-0': 'the exponential factor in the production rate of the gauge field is obtained from the superstring counterpart by the area of the string worldsheet attached to a probe D3-brane, i.e. [EQUATION] where [MATH] is the Nambu-Goto (NG) action [CITATION] [EQUATION] and [MATH] is the Kalb-Ramond [CITATION] 2-form (or NS-NS, where NS is the abbreviation of Neveu-Schwarz [CITATION] ) as an string interaction term, [EQUATION]', '1811.11712-2-27-1': 'In eqs. [REF] and [REF], [MATH] are the coordinates on the string worldsheet, [MATH] are the coordinates of the 10D [MATH] space with metric [MATH], and [MATH] is the induced metric.', '1811.11712-2-28-0': 'In the Semenoff-Zarembo construction, the worldsheet ends on the probe D3-brane with a boundary, taking the same shape as the worldline instanton.', '1811.11712-2-28-1': 'Hence the essential problem is converted to compute the on-shell action of string in Euclidean [MATH] with the given boundary.', '1811.11712-2-28-2': 'Note that the Nambu-Goto action is proportional to the worldsheet area, and extremising the area leads to a minimal surface.', '1811.11712-2-28-3': ""In other words, calculation of the exponential factor now corresponds to a Plateau's problem in the framework of gauge/gravity duality."", '1811.11712-2-29-0': '## Constant electric field', '1811.11712-2-30-0': 'The worldline instanton in a constant field is a circle, thus the worldsheet can be parametrised by [EQUATION] which is different than the choice in [CITATION], and the two parameterisations have different chirality, i.e. [MATH], where [MATH] is the Jacobian.', '1811.11712-2-30-1': 'Therefore, the orientation of the Kalb-Ramond coupling is also reversed.', '1811.11712-2-30-2': 'The Nambu-Goto action in our parameterisation becomes [EQUATION] where [MATH], [MATH] is the location of the probe D3-brane.', '1811.11712-2-30-3': '[MATH] can be analytically as [CITATION] [EQUATION] where [MATH] is the worldline instanton on the D3-brane.', '1811.11712-2-30-4': 'Substituting it into the Nambu-Goto action, one obtains [EQUATION]', '1811.11712-2-30-5': 'Furthermore, the NS-NS term reads [EQUATION] where [MATH] can be fixed by extremising the total action, yielding [EQUATION]', '1811.11712-2-30-6': 'The exponential factor in the production rate can now be solved as [EQUATION] in which the string and spacetime parameters have been replaced by the ones of the gauge field via [MATH].', '1811.11712-2-31-0': '[Estimation of Single-pulse field [MATH]] Estimation of Single-pulse field [MATH]', '1811.11712-2-32-0': 'The instanton path as the worldsheet boundary on the D3-brane in a single-pulse field has been shown in eq. [REF].', '1811.11712-2-32-1': 'Thus one can parametrise the worldsheet by using [MATH] and [MATH], i.e. [EQUATION] where [MATH] is regarded as initial information and not relevant to the scale [MATH].', '1811.11712-2-32-2': 'As [MATH], [EQUATION] namely the zeroth order of [MATH] is just the circle boundary.', '1811.11712-2-32-3': 'Therefore, we will treat the complete worldline boundary as perturbation around the circle.', '1811.11712-2-32-4': 'The [MATH] component can be estimated by noting that [EQUATION] where [MATH] is the instanton path.', '1811.11712-2-32-5': 'In a single-pulse field, it reads [EQUATION]', '1811.11712-2-32-6': 'In the limit [MATH], [MATH] reduces to a circle [EQUATION]', '1811.11712-2-32-7': 'Hence the Nambu-Goto action is formulated as [EQUATION] where [MATH] is given in eq. [REF].', '1811.11712-2-32-8': 'On the other hand, the NS-NS part right now becomes [EQUATION]', '1811.11712-2-32-9': 'The [MATH] is fixed as the stationary point of the total action, [EQUATION]', '1811.11712-2-32-10': 'The exponential factor in the production rate with correction up to the second order of [MATH] is [EQUATION]', '1811.11712-2-32-11': 'In eq. [REF], the first part in the exponent comes from the circular boundary, while the second term arises from the deformation, see fig. [REF].', '1811.11712-2-33-0': 'The correction due to the time-dependence of the background field enhances the critical field, which is fixed by [EQUATION] beyond which the decay rate decreases as the similar as eq. [REF].', '1811.11712-2-34-0': '[Estimation of sinusoidal field [MATH]] Estimation of sinusoidal field [MATH]', '1811.11712-2-35-0': 'The worldline instanton in this case can be parametrised as [EQUATION] where [MATH] has been defined in eq. [REF].', '1811.11712-2-35-1': 'The Maclaurin series of Jacobian elliptic functions in [MATH] gives [EQUATION]', '1811.11712-2-35-2': 'The instanton path on the D3-brane in this case is [EQUATION] and its expansion for small [MATH] reads [EQUATION]', '1811.11712-2-35-3': 'The second order terms for the Nambu-Goto action and the NS-NS term are respectively [EQUATION]', '1811.11712-2-35-4': 'Up to the second order of [MATH], [MATH] is given by [EQUATION]', '1811.11712-2-35-5': 'Finally, the exponential factor in the production rate is given by [EQUATION] see fig. [REF].', '1811.11712-2-36-0': 'The critical field should be greater than [MATH], i.e. [EQUATION] and one sees that dynamics of the external field also enhances the critical field as before.', '1811.11712-2-37-0': '# Conclusion and Discussion', '1811.11712-2-38-0': 'In this paper, the scalar Schwinger effect at strong coupling has been studied, by first using the field-theoretical method of worldline instantons.', '1811.11712-2-38-1': 'A non-trivial contribution to the production rate is discovered by evaluating the Wilson loop along the instanton path, which depends on the field strength [MATH], and diverges as [MATH] approaches zero even if the pre-exponential factor is considered.', '1811.11712-2-39-0': 'However, one can introduce a cutoff [MATH] by minimising eq. [REF], i.e. [MATH] is fixed by computing the following algebraic equation', '1811.11712-2-40-0': '[EQUATION] where [MATH] is the fine-structure constant, [MATH] and [MATH].', '1811.11712-2-40-1': 'For the cases with small [MATH] and [MATH], one can give the estimation [EQUATION]', '1811.11712-2-40-2': 'The vacuum decay rate [REF] is valid beyond [MATH].', '1811.11712-2-40-3': 'This suggests a solution to the potential problem of low-field divergence.', '1811.11712-2-40-4': 'In addition, the introduction of the correction term leads to a contradiction to the weak-field condition near the critical field strength, which also makes it indispensable to further investigate the strong-coupling case in a dynamic field.', '1811.11712-2-41-0': 'In order to solve the problem of broken weak-field condition, in the context of an [MATH] supersymmetric Yang-Mills theory, the production rate is calculated by the gauge/gravity duality, according to which the instanton action has a string counterpart of the classical string action in Euclidean [MATH], where the boundary on the probe D3-brane is given by the instanton path.', '1811.11712-2-41-1': 'Thus the problem is converted to solving the classical motion of string with Dirichlet boundary conditions.', '1811.11712-2-41-2': 'To provide an explicit estimation, we treat the specific worldsheets as perturbations around the one with circle boundary, which had been solved exactly.', '1811.11712-2-41-3': 'Such an expansion is practical and realistic, because only low-frequency laser (comparing with electron mass) is currently operational.', '1811.11712-2-41-4': 'The obtained decay rates in the two examples with dynamic fields are similar concave functions as in the case with constant field, but the critical fields increase considerably.'}","[['1811.11712-1-3-1', '1811.11712-2-0-1'], ['1811.11712-1-7-2', '1811.11712-2-4-2'], ['1811.11712-1-4-0', '1811.11712-2-1-0'], ['1811.11712-1-23-3', '1811.11712-2-22-2'], ['1811.11712-1-6-0', 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'1811.11712-4-38-0'], ['1811.11712-3-33-1', '1811.11712-4-38-1'], ['1811.11712-3-33-2', '1811.11712-4-38-2'], ['1811.11712-3-33-3', '1811.11712-4-38-3'], ['1811.11712-3-33-4', '1811.11712-4-38-4'], ['1811.11712-3-33-5', '1811.11712-4-38-5'], ['1811.11712-3-33-7', '1811.11712-4-38-8'], ['1811.11712-3-13-1', '1811.11712-4-20-1'], ['1811.11712-3-13-2', '1811.11712-4-20-2'], ['1811.11712-3-8-0', '1811.11712-4-9-0'], ['1811.11712-3-8-1', '1811.11712-4-9-1'], ['1811.11712-3-8-3', '1811.11712-4-9-3'], ['1811.11712-3-30-0', '1811.11712-4-35-0'], ['1811.11712-3-30-1', '1811.11712-4-35-1'], ['1811.11712-3-35-0', '1811.11712-4-40-0'], ['1811.11712-3-35-1', '1811.11712-4-40-1'], ['1811.11712-3-35-2', '1811.11712-4-40-2'], ['1811.11712-3-35-3', '1811.11712-4-40-3'], ['1811.11712-3-35-4', '1811.11712-4-40-4'], ['1811.11712-3-35-5', '1811.11712-4-40-5'], ['1811.11712-3-35-6', '1811.11712-4-40-6'], ['1811.11712-3-35-7', '1811.11712-4-40-7'], ['1811.11712-3-35-8', '1811.11712-4-40-8'], ['1811.11712-3-35-9', '1811.11712-4-40-9'], ['1811.11712-3-35-10', '1811.11712-4-40-10'], ['1811.11712-3-35-11', '1811.11712-4-40-11'], ['1811.11712-3-35-12', '1811.11712-4-40-12'], ['1811.11712-3-35-13', '1811.11712-4-40-13'], ['1811.11712-3-31-0', '1811.11712-4-36-0'], ['1811.11712-3-31-1', '1811.11712-4-36-1'], ['1811.11712-3-31-2', '1811.11712-4-36-2'], ['1811.11712-3-31-3', '1811.11712-4-36-3'], ['1811.11712-3-16-0', '1811.11712-4-23-0'], ['1811.11712-3-16-1', '1811.11712-4-23-1'], ['1811.11712-3-4-0', '1811.11712-4-5-0'], ['1811.11712-3-4-1', '1811.11712-4-5-1'], ['1811.11712-3-4-2', '1811.11712-4-5-2'], ['1811.11712-3-7-0', '1811.11712-4-8-0'], ['1811.11712-3-7-1', '1811.11712-4-8-1'], ['1811.11712-3-7-2', '1811.11712-4-8-2'], ['1811.11712-3-41-0', '1811.11712-4-45-0'], ['1811.11712-3-41-1', '1811.11712-4-45-1'], ['1811.11712-3-41-2', '1811.11712-4-45-2'], ['1811.11712-3-14-0', '1811.11712-4-21-0'], ['1811.11712-3-14-1', '1811.11712-4-21-1'], ['1811.11712-3-14-2', '1811.11712-4-21-2'], ['1811.11712-3-14-3', '1811.11712-4-21-3'], ['1811.11712-3-1-0', '1811.11712-4-2-0'], ['1811.11712-3-1-1', '1811.11712-4-2-1'], ['1811.11712-3-1-3', '1811.11712-4-2-3'], ['1811.11712-3-9-0', '1811.11712-4-10-0'], ['1811.11712-3-9-1', '1811.11712-4-10-1'], ['1811.11712-3-9-2', '1811.11712-4-10-2'], ['1811.11712-3-9-3', '1811.11712-4-10-3'], ['1811.11712-3-9-4', '1811.11712-4-10-4'], ['1811.11712-3-9-6', '1811.11712-4-10-6'], ['1811.11712-3-9-7', '1811.11712-4-10-7'], ['1811.11712-3-19-0', '1811.11712-4-26-0'], ['1811.11712-3-19-2', '1811.11712-4-26-2'], ['1811.11712-3-19-4', '1811.11712-4-26-4'], ['1811.11712-3-43-0', '1811.11712-4-47-0'], ['1811.11712-3-43-1', '1811.11712-4-47-1'], ['1811.11712-3-43-2', '1811.11712-4-47-2'], ['1811.11712-3-43-3', '1811.11712-4-47-3'], ['1811.11712-3-43-5', '1811.11712-4-47-5'], ['1811.11712-3-43-6', '1811.11712-4-47-6'], ['1811.11712-3-2-0', '1811.11712-4-3-0'], ['1811.11712-3-2-1', '1811.11712-4-3-1'], ['1811.11712-3-2-2', '1811.11712-4-3-2'], ['1811.11712-3-2-3', '1811.11712-4-3-3'], ['1811.11712-3-3-0', '1811.11712-4-4-0'], ['1811.11712-3-3-3', '1811.11712-4-4-3'], ['1811.11712-3-24-0', '1811.11712-4-30-0'], ['1811.11712-3-24-1', '1811.11712-4-30-1'], ['1811.11712-3-24-2', '1811.11712-4-30-2'], ['1811.11712-3-24-3', '1811.11712-4-30-3'], ['1811.11712-3-24-4', '1811.11712-4-30-4'], ['1811.11712-3-24-5', '1811.11712-4-30-5'], ['1811.11712-3-24-6', '1811.11712-4-30-6'], ['1811.11712-3-22-0', '1811.11712-4-28-2'], ['1811.11712-3-22-2', '1811.11712-4-28-4'], ['1811.11712-3-22-3', '1811.11712-4-28-5'], ['1811.11712-1-20-2', '1811.11712-2-18-2'], ['1811.11712-2-5-1', '1811.11712-3-4-0'], ['1811.11712-2-5-2', '1811.11712-3-4-1'], ['1811.11712-2-5-3', '1811.11712-3-4-2'], ['1811.11712-2-23-0', '1811.11712-3-25-0'], ['1811.11712-2-23-1', '1811.11712-3-26-0'], ['1811.11712-3-5-0', '1811.11712-4-6-0'], ['1811.11712-3-5-2', '1811.11712-4-6-2'], ['1811.11712-3-5-6', '1811.11712-4-6-4'], ['1811.11712-2-6-1', 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'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1811.11712,"{'1811.11712-3-0-0': '# Introduction', '1811.11712-3-1-0': 'The Extreme Light Infrastructure (ELI) is designed to produce the highest power and intense laser worldwide [CITATION].', '1811.11712-3-1-1': 'It has a potential of reaching ultra-relativistic intensities, challenging the Schwinger limit [MATH].', '1811.11712-3-1-2': 'As laser field approaches this value, vacuum becomes unstable, and charged particles produce in pairs, so that laser loses the energy, and its intensity stays within the upper limit.', '1811.11712-3-1-3': 'However, Schwinger effect is not only a phenomenon in electromagnetism, but a universal aspect of quantum vacuum in the presence of a [MATH] gauge field with a classical background, see e.g. [CITATION].', '1811.11712-3-2-0': 'The pair-production rate in the constant electric field had been pioneered by Sauter, Heisenberg and Euler [CITATION], and the corresponding Effect was named after Schwinger [CITATION], who did the calculation based on field-theoretical approaches, see e.g. [CITATION] for a current review.', '1811.11712-3-2-1': 'A semiclassical approach called worldline instantons has been introduced more recently to the study of constant and inhomogeneous fields in the small-coupling and weak-field approximations [CITATION], where the production rate, in Wick-rotated Euclidean space, is represented by a worldline path integral.', '1811.11712-3-2-2': 'The so-called worldline instantons are the periodic saddle points relevant for a calculation of integral by the steepest descent method.', '1811.11712-3-2-3': 'The extension of inhomogeneous fields originates from more practical purpose.', '1811.11712-3-2-4': 'As analyzed in [CITATION], [MATH]-D dynamic electric fields reduce the critical value of Schwinger effect, such that pair production from vacuum is more close to experimentally observable conditions.', '1811.11712-3-3-0': ""The Schwinger effect at arbitrary coupling in constant field has also been studied in [CITATION] at the weak-field condition, which is considered originally in order to overcome some obstacles from direct application of Schwinger's approach."", '1811.11712-3-3-1': 'However the later observation (e.g. [CITATION]) found that the weak-field condition is broken around the critical field, defined as a point of vacuum decay, such that the mechanism in weak-field condition loses its prediction for this vacuum phenomenon.', '1811.11712-3-3-2': 'Inspired by the similar existence of critical value of electric field in string theory, it is of a possibility to clarify the vacuum decay in the Coulomb phase beyond the weak-field condition with the help of gauge/gravity duality [CITATION].', '1811.11712-3-3-3': 'This is also known as the Semenoff-Zarembo construction, where the production rate has been obtained by calculating the classical action of a bosonic string, which is attached to a probe D3-brane and coupled to a Kalb-Ramond field.', '1811.11712-3-4-0': 'Since the instanton action in the production rate is equivalent to the string action, which is proportional to the area, calculation of the rate is related to integrating the classical equations of motion for bosonic string in the given external field.', '1811.11712-3-4-1': 'In other words, duality converts the problem to evaluating the area of a minimal surface [CITATION] in Euclidean [MATH], the boundary of which is assumed to be the trajectory of the worldline on the probe D3-brane.', '1811.11712-3-4-2': ""In mathematics, such a Dirichlet problem is known as the Plateau's problem [CITATION]."", '1811.11712-3-5-0': 'In this work, we consider the scalar pair production in a dynamic external field of single-pulse and sinusoidal types at strong coupling and weak-field limit based on the method of worldline instantons, which is explained in sections [REF] and [REF].', '1811.11712-3-5-1': 'We first show by a field-theoretical approach, that besides the enhancement due to the dynamics of electric fields, a further contribution to the production rate arises from the Wilson loop.', '1811.11712-3-5-2': 'However, such a correction seems to diverge as Keldysh adiabaticity parameter [MATH] based on our estimated formula; and it leads to a contradiction to the weak-field condition, so that near the critical field, the method itself breaks down, and the prediction of a vacuum cascade becomes unclear.', '1811.11712-3-5-3': 'The first problem arises because our estimation for Wilson loop is equivalent to adiabatic expansion (i.e. expansion around [MATH]) and does not provide information for all region.', '1811.11712-3-5-4': 'This also reflects the non-perturbative aspect of Schwinger effect from another angle.', '1811.11712-3-5-5': ""To overcome the second problem, we then follow Semenoff and Zarembo's proposal in sec. [REF], applying the gauge/gravity duality to the Schwinger effect in the Coulomb phase of an [MATH] supersymmetric Yang-Mills theory."", '1811.11712-3-5-6': 'The classical solution of the corresponding string worldsheets are estimated by perturbing the solvable case of a constant external electric field.', '1811.11712-3-6-0': '# Worldline instantons at strong coupling', '1811.11712-3-7-0': 'The worldline instanton approach is a semiclassical calculation realised by the worldline path integral representations [CITATION], in which the so-called worldline instanton is a periodic solution of the stationary phase in the path integral.', '1811.11712-3-7-1': 'Based on this method, the pair-production rate [MATH] for a massive scalar QED in the small-coupling and weak-field approximation is computed by [CITATION] [EQUATION] where [MATH] is the 4-volume, [MATH] is a constant, given by [EQUATION] and [MATH] represents the action of worldline instanton, i.e. [EQUATION]', '1811.11712-3-7-2': 'In eq. [REF], [MATH] is a classical background field, not to be confused with the fluctuation part of the [MATH] gauge field [MATH], which is also included in the initial setup of path integral, but cancelled due to considerations at small coupling.', '1811.11712-3-8-0': 'The path integral in eq. [REF] can be computed by the stationary phase approximation in weak-field condition [EQUATION]', '1811.11712-3-8-1': 'For more general cases with 1D temporally inhomogeneous fields [MATH], the worldline instanton is obtained by solving the instanton equation with periodic boundary condition [EQUATION]', '1811.11712-3-8-2': 'Thus the pair-production rate can be written as [EQUATION] where [MATH] is a monotonically decreasing function of Keldysh parameter [MATH] and calculated by substituting the solution of eq. [REF] into the worldline action with the corresponding background field [MATH].', '1811.11712-3-8-3': 'In addition, the weak-field condition eq. [REF] is of various forms according to different dynamic fields, because it depends on specific instanton solution.', '1811.11712-3-9-0': 'At arbitrary coupling constant but with weak-field condition [CITATION], the production rate is modified by a factor, [EQUATION] where [MATH] is the average of [MATH] Wilson loop [MATH] [EQUATION]', '1811.11712-3-9-1': 'The integral path [MATH] is along the 2D trajectory of worldline instanton.', '1811.11712-3-9-2': 'In the Feynman gauge of the [MATH] field, the Wilson loop becomes [CITATION] [EQUATION]', '1811.11712-3-9-3': 'After simplification, [MATH] can be represented via a double contour integrals [EQUATION]', '1811.11712-3-9-4': 'For a constant electric field, the worldline instanton is a circle, and the production rate of a single scalar pair is given by [EQUATION] where [MATH] is the Schwinger limit in natural units.', '1811.11712-3-9-5': 'Consideration at arbitrary coupling [CITATION] leads to a correction to eq. [REF] [EQUATION] see also sec. [REF].', '1811.11712-3-9-6': 'Our aim in the next section is to calculate the Wilson loops in eq. [REF] along two specific worldlines separately.', '1811.11712-3-9-7': 'The first one is the worldline instanton in a single-pulse field, whereas the second is in a sinusoidal field.', '1811.11712-3-9-8': 'We show that the weak-field condition is also broken in these two cases, as noted in e.g. [CITATION].', '1811.11712-3-10-0': '# Wilson loops along worldline instanton paths', '1811.11712-3-11-0': 'The Wilson loop in eq. [REF] plays a central role in the case of arbitrary coupling, which typically diverges as [MATH] approaches [MATH] [CITATION].', '1811.11712-3-11-1': 'To obtain a finite result, one can introduce a small regularisation parameter [MATH], such that [EQUATION] where [MATH] and [MATH] can be understood as angular coordinates for the instanton.', '1811.11712-3-11-2': 'The divergence of Wilson loop can now be removed by subtracting [EQUATION] which is an example of the perimeter law, depicting the behaviour of the Wilson loop in Euclidean space [CITATION].', '1811.11712-3-11-3': 'Because all worldline instantons satisfy [MATH], where [MATH] is defined in the same way as in [CITATION], the divergent term can be reduced to [EQUATION]', '1811.11712-3-11-4': 'After subtracting this term from eq. [REF], one obtains the regularised [MATH] by taking the limit [EQUATION]', '1811.11712-3-11-5': 'The regularised Wilson loop is then given by [EQUATION] which appears as a factor in the final expression of the production rate.', '1811.11712-3-11-6': 'Practically eq. [REF] is hard to be computed analytically, thus the central problem is to find an effective estimation.', '1811.11712-3-12-0': '## Constant electric field', '1811.11712-3-13-0': 'For a constant electric field, the worldline instanton is a circle of radius [MATH], given by [CITATION] [EQUATION] [MATH] where the zeroth component [MATH] denotes the Euclidean time.', '1811.11712-3-13-1': 'The instanton action for single pair production is obtained by substituting this solution into eq. [REF], yielding [EQUATION]', '1811.11712-3-13-2': 'The weak-field condition [MATH] [CITATION] implies [EQUATION] where [MATH] is defined in eq. [REF].', '1811.11712-3-14-0': 'On the other hand, the integral in eq. [REF] and the perimeter law in eq. [REF] can be evaluated explicitly as [EQUATION]', '1811.11712-3-14-1': 'Hence the regularised Wilson loop for a single scalar pair can be recovered as [CITATION] [EQUATION] and the production rate is given by [EQUATION] from which the critical field for vacuum cascade can be estimated by [EQUATION]', '1811.11712-3-14-2': 'One sees that [MATH] is much greater than the Schwinger limit and breaks the weak field condition in eq. [REF], as have been noted in e.g. [CITATION].', '1811.11712-3-14-3': 'Therefore, the obtained results are not valid when the field goes close to the critical limit, and cannot answer the question whether a vacuum cascade happens near the critical field strength.', '1811.11712-3-15-0': '[Single-pulse field [MATH]] Single-pulse field [MATH]', '1811.11712-3-16-0': 'For a generic 1D dynamic field, we assume that the Wilson loop is of the following form [EQUATION] where [MATH] is an enhancement factor with respect to the case of a constant external field in eq. [REF].', '1811.11712-3-16-1': 'This name of [MATH] is seen to be appropriate from the fact that [MATH] is a monotonically decreasing function and tends to unit at adiabatic limit [MATH].', '1811.11712-3-17-0': 'In practices, [MATH] can not be evaluated in a closed form, and will be estimated in two approximation schemes.', '1811.11712-3-17-1': 'We can expand the numerator and denominator separately with respect to [MATH] in the integrand of eq. [REF] up to second order, which will be called the [MATH]-expansion.', '1811.11712-3-17-2': 'Before applying this approximation to single-pulse field, we apply it to the case of a constant field.', '1811.11712-3-17-3': 'The integrand after expansion becomes [EQUATION] while the subtracting term eq. [REF] will not change.', '1811.11712-3-17-4': 'Then repeating the similar procedure, one gets [EQUATION] which leads a [MATH] derivation comparing with eq. [REF].', '1811.11712-3-17-5': ""Later we will see that this [MATH] derivation happens in all considered cases, it's caused by estimation method."", '1811.11712-3-17-6': 'Hence there must be a [MATH] derivation has to be calibrated in the final results, this is also equivalent to modify the boundaries of integration.', '1811.11712-3-18-0': 'Alternatively, one can also expand the integrand of eq. [REF] at [MATH], i.e. an [MATH]-expansion.', '1811.11712-3-18-1': 'This approach not only provides an efficient estimation for some cases, but also is of a physical meaning, emphasising the nonperturbative nature of Schwinger effect.', '1811.11712-3-18-2': 'Moreover, it also gives a direct comparison of dynamic fields at adiabatic limit with constant field.', '1811.11712-3-19-0': 'Let us see the first nontrivial example, pair production in the single-pulse field [CITATION].', '1811.11712-3-19-1': 'The instanton has be evaluated explicitly to be [EQUATION] and the worldline action for this path reads [EQUATION]', '1811.11712-3-19-2': 'The weak-field condition leads to [EQUATION]', '1811.11712-3-19-3': 'At the nonperturbative region [MATH], the weak-field condition reduces to [MATH].', '1811.11712-3-19-4': 'In other words, the dynamics of field also decreases the upper limit of weak-field condition.', '1811.11712-3-20-0': 'Now we turn to the the Wilson loop.', '1811.11712-3-20-1': 'The corresponding perimeter law has a closed form [EQUATION]', '1811.11712-3-20-2': 'With the [MATH]-expansion up to [MATH], eq. [REF] reads [EQUATION]', '1811.11712-3-20-3': 'With the [MATH]-expansion up to [MATH], eq. [REF] reads [EQUATION]', '1811.11712-3-20-4': 'The enhancement factor is then in the weak-field condition [EQUATION]', '1811.11712-3-20-5': 'Since [MATH], the factor amplifies the contribution from the Wilson loop in eq. [REF], so that the pair-production rate is no longer exponentially suppressed, see fig. [REF].', '1811.11712-3-21-0': 'One may note that [MATH] as [MATH] approaches zero.', '1811.11712-3-21-1': 'The [MATH] derivation is predicted and has to be subtracted.', '1811.11712-3-21-2': 'Moreover the series expansion of closed form eq. [REF] is comparable with adiabatic [MATH]-expansion [EQUATION]', '1811.11712-3-21-3': 'From this point of view, [MATH]-expansion can be regarded as a kind of expansion around adiabatic limit.', '1811.11712-3-22-0': 'The exponential factor in the production rate is evaluated as [EQUATION]', '1811.11712-3-22-1': 'The first part is the main contribution from instanton action, the second the part arises due to the Wilson loop correction from [MATH]-expansion.', '1811.11712-3-22-2': 'If the critical field is defined as saddle point, at which the exponential suppression is precisely zero, then one could have [EQUATION]', '1811.11712-3-22-3': 'On the one hand, as in the case with constant field, eq. [REF] also breaks the weak-field condition at nonperturbative region [MATH].', '1811.11712-3-22-4': 'On the other hand, in contrast with the case in constant field, the exponential factor of Wilson loop is no longer constant.', '1811.11712-3-22-5': 'In addition, the estimation result eq. [REF] (or eq. [REF]) seems to be divergent as [MATH] approaches [MATH], even if the pre-exponential factor of Feynman integral were taken into account [CITATION].', '1811.11712-3-22-6': 'It simply implies both estimations ([MATH]-expansion and [MATH]-expansion) are not valid in the perturbative region [MATH].', '1811.11712-3-23-0': '[Sinusoidal field [MATH]] Sinusoidal field [MATH]', '1811.11712-3-24-0': 'The second example is a sinusoidal field [CITATION].', '1811.11712-3-24-1': 'The coordinates of the instanton can be represented by special functions as [EQUATION] where [MATH] is the complete elliptic integral, [MATH] and [MATH] are Jacobi elliptic functions.', '1811.11712-3-24-2': 'The instanton action is given by [EQUATION]', '1811.11712-3-24-3': 'Repeating the procedure above, one obtains the corresponding perimeter law in a closed form [EQUATION]', '1811.11712-3-24-4': 'The exponent in the [MATH]-expansion reads [EQUATION] in which [MATH] is the Jacobi amplitude function.', '1811.11712-3-24-5': 'Thus at the week-field approximation [MATH] [CITATION], one has [EQUATION]', '1811.11712-3-24-6': 'Where the deviation [MATH] has been subtracted.', '1811.11712-3-24-7': 'In this case, [MATH] is also greater than unity and a non-trivial function depending on external field, and it tends to [MATH] as [MATH] approaches zero, which is comparable with [MATH]-expansion result [EQUATION] see fig. [REF].', '1811.11712-3-25-0': 'In addition, [MATH] is also divergent as [MATH], even if the pre-exponential factor is considered [CITATION].', '1811.11712-3-25-1': 'As we pointed in the last subsection, this divergence happens because of the inappropriate consideration on the region.', '1811.11712-3-26-0': 'The exponent factor in the production rate is then given by [EQUATION]', '1811.11712-3-26-1': 'For small [MATH], one gets [EQUATION] which breaks the weak-field condition at nonperturbative region as well.', '1811.11712-3-27-0': 'From above three examples, one may note that, first, the relation of coupling constant with the strength of the dynamic field is universal, i.e. the weak-field condition is inevitably broken by the increasing coupling; and second, the production rate diverges as [MATH].', '1811.11712-3-27-1': 'The later one simply means that estimation results of Wilson loop are out of valid domain, while the former makes the vacuum decay around the critical field ambiguous [CITATION].', '1811.11712-3-28-0': '# Holographic Schwinger effect with dynamic field', '1811.11712-3-29-0': 'In order to answer the question, if the vacuum decay for strong coupling happens as the strength of time-dependent field goes close to the critical limit [CITATION], we consider a similar effect in the context of gauge/gravity duality, where the gauge field theory refers to an [MATH] supersymmetric Yang-Mills theory on the 4D boundary of an [MATH] space, and the quantum gravity is a type 2B superstring theory in the bulk of the [MATH].', '1811.11712-3-29-1': 'The same as the case with constant field, we expect that the string theory could shed some light on the catastrophic vacuum decay through the duality principle.', '1811.11712-3-30-0': ""According to the Semenoff and Zarembo's holographic setup [CITATION], the exponential factor in the production rate of the gauge field is obtained from the superstring counterpart by the area of the string worldsheet attached to a probe D3-brane, i.e. [EQUATION] where [MATH] is the Nambu-Goto (NG) action [CITATION] [EQUATION] depending on the induced metric [EQUATION] and [MATH] is the Kalb-Ramond [CITATION] 2-form (or NS-NS, where NS is the abbreviation of Neveu-Schwarz [CITATION] ) as an string interaction term, [EQUATION]"", '1811.11712-3-30-1': 'In eqs. [REF] and [REF], [MATH] are the coordinates on the string worldsheet, [MATH] are the coordinates of the [MATH]-D [MATH] space with metric [MATH], and [MATH] is the induced metric.', '1811.11712-3-31-0': 'In the Semenoff-Zarembo construction, the worldsheet ends on the probe D3-brane with a boundary, taking the same shape as the worldline instanton.', '1811.11712-3-31-1': 'Hence the essential problem is converted to compute the on-shell action of string in Euclidean [MATH] with the given boundary.', '1811.11712-3-31-2': 'Note that the Nambu-Goto action is proportional to the worldsheet area, and extremising the area leads to a minimal surface.', '1811.11712-3-31-3': ""In other words, calculation of the exponential factor now corresponds to a Plateau's problem in the framework of gauge/gravity duality."", '1811.11712-3-32-0': '## Constant electric field', '1811.11712-3-33-0': 'The worldline instanton in a constant field is a circle, thus the worldsheet can be parametrised by [EQUATION] which is different than the choice in [CITATION], and the two parameterisations have different chirality, i.e. [MATH], where [MATH] is the Jacobian.', '1811.11712-3-33-1': 'Therefore, the orientation of the Kalb-Ramond coupling is also reversed.', '1811.11712-3-33-2': 'The Nambu-Goto action in our parameterisation becomes [EQUATION] where [MATH], [MATH] is the location of the probe D3-brane.', '1811.11712-3-33-3': '[MATH] can be analytically as [CITATION] [EQUATION] where [MATH] is the worldline instanton on the D3-brane.', '1811.11712-3-33-4': 'Substituting it into the Nambu-Goto action, one obtains [EQUATION]', '1811.11712-3-33-5': 'Furthermore, the NS-NS term reads [EQUATION] where [MATH] can be fixed by extremising the total action, yielding [EQUATION]', '1811.11712-3-33-6': 'Where [MATH] is defined as critical values The exponential factor in the production rate can now be solved as [EQUATION] in which the string and spacetime parameters have been replaced by the ones of the gauge field via [MATH].', '1811.11712-3-33-7': ""This is the result obtained in [CITATION] and agrees with the Schwinger's formula in the weak-field limit."", '1811.11712-3-34-0': '[Estimation of Single-pulse field [MATH]] Estimation of Single-pulse field [MATH]', '1811.11712-3-35-0': 'The instanton path as the worldsheet boundary on the D3-brane in a single-pulse field has been shown in eq. [REF].', '1811.11712-3-35-1': 'Thus one can parametrise the worldsheet by using [MATH] and [MATH], i.e. [EQUATION] where [MATH] is regarded as initial information and not relevant to the scale [MATH].', '1811.11712-3-35-2': 'The simplicity of Semenoff-Zarembo construction for constant field led us to speculate that the similar production rates could have been obtained by repeating above procedure.', '1811.11712-3-35-3': 'However, it is not anything like worldline instanton, the string worldsheets are not exactly integrable for dynamic fields in our cases.', '1811.11712-3-35-4': 'Thus to make an effect estimation, we expand the instanton at adiabatic limit, i.e. [MATH], [EQUATION] namely the zeroth order of [MATH] is just the circle boundary eq. [REF].', '1811.11712-3-35-5': 'Therefore, we will treat the complete worldline boundary as perturbation around the circle.', '1811.11712-3-35-6': 'The [MATH] component can be estimated by noting that [EQUATION] where [MATH] is the instanton path.', '1811.11712-3-35-7': 'In a single-pulse field, it reads [EQUATION]', '1811.11712-3-35-8': 'In the limit [MATH], [MATH] reduces to a circle [EQUATION]', '1811.11712-3-35-9': 'Hence the Nambu-Goto action is formulated as [EQUATION] where [MATH] is given in eq. [REF].', '1811.11712-3-35-10': 'On the other hand, the NS-NS part right now becomes [EQUATION]', '1811.11712-3-35-11': 'The [MATH] is fixed as the stationary point of the total action, [EQUATION]', '1811.11712-3-35-12': ""The exponential factor in the production rate with correction up to the second order of [MATH] is [EQUATION] here [MATH] denotes the effective coupling at large 't Hooft limit."", '1811.11712-3-35-13': 'In eq. [REF], the first part in the exponent comes from the circular boundary, while the second term arises from the deformation, see fig. [REF].', '1811.11712-3-36-0': 'If one defines the critical field as stationary point of exponent, the correction (up to [MATH]) due to the time-dependence of the background field leads a lager critical value comparing with constant field, which is fixed by [EQUATION] beyond which the decay rate decreases as the similar as eq. [REF].', '1811.11712-3-36-1': 'The enhancement up to [MATH] can also be noted from the negative sign of [MATH] in eq. [REF].', '1811.11712-3-36-2': 'In other words, the correction plays a role of suppression of pair production.', '1811.11712-3-37-0': '[Estimation of sinusoidal field [MATH]] Estimation of sinusoidal field [MATH]', '1811.11712-3-38-0': 'The worldline instanton in this case can be parametrised as [EQUATION] where [MATH] has been defined in eq. [REF].', '1811.11712-3-38-1': 'The Maclaurin series of Jacobian elliptic functions in [MATH] gives [EQUATION]', '1811.11712-3-38-2': 'The instanton path on the D3-brane in this case is [EQUATION] and its expansion for small [MATH] reads [EQUATION]', '1811.11712-3-38-3': 'The second order terms for the Nambu-Goto action and the NS-NS term are respectively [EQUATION]', '1811.11712-3-38-4': 'Up to the second order of [MATH], [MATH] is given by [EQUATION]', '1811.11712-3-38-5': 'Finally, the exponential factor in the production rate is given by [EQUATION] see fig. [REF].', '1811.11712-3-39-0': 'The critical field should be greater than [MATH], i.e. [EQUATION] and one sees that dynamics of the external field also enhances the critical field as before.', '1811.11712-3-40-0': '# Conclusion and Discussion', '1811.11712-3-41-0': 'In this paper, the scalar Schwinger effect for dynamic fields at strong coupling and weak-field limit has been studied, by first using the field-theoretical method of worldline instantons.', '1811.11712-3-41-1': 'A non-trivial contribution to the production rate is discovered by evaluating the Wilson loop along the instanton path, which depends on the Keldysh adiabaticity parameter [MATH].', '1811.11712-3-41-2': 'Thus one may expect that such correction may save the weak-field condition in strong coupling.', '1811.11712-3-41-3': 'However after computations, we find that the introduction of the correction term also leads to a contradiction to the weak-field condition near the critical field strength, which makes it indispensable to further investigate the strong-coupling case in a dynamic field.', '1811.11712-3-42-0': 'In addition, we note that the correction based on our estimation diverges as [MATH] approaches zero, even if the pre-exponential factor of Feynman integral is considered.', '1811.11712-3-42-1': 'This divergence arises because obtained results are valid only in the nonperturbative region [MATH], beyond which the estimation of Wilson loop loses its meaning.', '1811.11712-3-42-2': 'In other words, the method for estimating Wilson loop is a kind of local analysis around adiabatic limit.', '1811.11712-3-42-3': 'Nevertheless a numerical calculation suggests a convergent value of Wilson loop at [MATH], combining the main distribution of instanton action, it provides a full interpolation picture between the nonperturbative and the pertubative regions as the adiabaticity parameter [MATH] goes from [MATH] to [MATH].', '1811.11712-3-43-0': 'In order to clarify the vacuum cascade beyond the weak-field condition, in the context of an [MATH] supersymmetric Yang-Mills theory, the production rate is calculated by the gauge/gravity duality, according to which the instanton action has a string counterpart of the classical string action in Euclidean [MATH], where the boundary on the probe D3-brane is given by the instanton path.', '1811.11712-3-43-1': 'Thus the problem is converted to solving the classical motion of string with Dirichlet boundary conditions.', '1811.11712-3-43-2': 'However the string worldsheets for dynamic fields are not integrable as in the worldline instantons.', '1811.11712-3-43-3': 'To provide an explicit estimation, we treat the specific worldsheets as perturbations around the one with circle boundary, which had been solved exactly.', '1811.11712-3-43-4': ""Such an expansion is an adiabatic approximation, it's practical and realistic, because only low-frequency laser (comparing with electron mass) is currently operational."", '1811.11712-3-43-5': 'The obtained decay rates in the two examples with dynamic fields are similar concave functions as in the case with constant field, but the critical fields increase considerably.', '1811.11712-3-43-6': 'In other words, up to [MATH] the correction due to the dynamics of electric field suppresses the pair production, which is opposite to cases of worldline instantons.'}","{'1811.11712-4-0-0': 'At strong-coupling and weak-field limit, the scalar Schwinger effect is studied by the field-theoretical method of worldline instantons for dynamic fields of single-pulse and sinusoidal types.', '1811.11712-4-0-1': 'By examining the Wilson loop along the closed instanton path, corrections to the results obtained from weak-coupling approximations are discovered.', '1811.11712-4-0-2': 'They show that this part of contribution for production rate becomes dominant as Keldysh parameter increases, it makes the consideration at strong coupling turn out to be indispensable for dynamic fields.', '1811.11712-4-0-3': 'Moreover a breaking of weak-field condition similar to constant field also happens around the critical field, defined as a point of vacuum cascade.', '1811.11712-4-0-4': ""In order to make certain whether the vacuum cascade occurs beyond the weak-field condition, following Semenoff and Zarembo's proposal, the Schwinger effects of dynamic fields are studied with an [MATH] supersymmetric Yang-Mills theory in the Coulomb phase."", '1811.11712-4-0-5': 'With the help of the gauge/gravity duality, the vacuum decay rate is evaluated by the string action with instanton worldline as boundary, which is located on a probe D3-brane.', '1811.11712-4-0-6': 'The corresponding classical worldsheets are estimated by perturbing the integrable case of a constant field.', '1811.11712-4-1-0': '# Introduction', '1811.11712-4-2-0': 'The Extreme Light Infrastructure (ELI) is designed to produce the highest power and intense laser worldwide [CITATION].', '1811.11712-4-2-1': 'It has a potential of reaching ultra-relativistic intensities, challenging the Schwinger limit [MATH].', '1811.11712-4-2-2': 'As laser field approaches this value, vacuum becomes unstable, and a large amount of charged particles produces in pairs, so that laser loses the energy, and its intensity stays within the upper limit.', '1811.11712-4-2-3': 'However, Schwinger effect is not only a phenomenon in electromagnetism, but a universal aspect of quantum vacuum in the presence of a [MATH] gauge field with a classical background, see e.g. [CITATION].', '1811.11712-4-3-0': 'The pair-production rate in the constant electric field had been pioneered by Sauter, Heisenberg and Euler [CITATION], and the corresponding Effect was named after Schwinger [CITATION], who did the calculation based on field-theoretical approaches, see e.g. [CITATION] for a current review.', '1811.11712-4-3-1': 'A semiclassical approach called worldline instantons has been introduced more recently to the study of constant and inhomogeneous fields in the small-coupling and weak-field approximations [CITATION], where the production rate, in Wick-rotated Euclidean space, is represented by a worldline path integral.', '1811.11712-4-3-2': 'The so-called worldline instantons are the periodic saddle points relevant for a calculation of integral by the steepest descent method.', '1811.11712-4-3-3': 'The extension of inhomogeneous fields originates from more practical purpose.', '1811.11712-4-3-4': 'As analysed in [CITATION], [MATH]-D dynamic electric fields reduce the critical value of Schwinger effect, such that pair production from vacuum is more close to experimentally observable conditions.', '1811.11712-4-4-0': ""The Schwinger effect at arbitrary coupling in constant field has also been studied in [CITATION] at the weak-field condition, which is considered originally in order to overcome some obstacles from direct application of Schwinger's approach."", '1811.11712-4-4-1': 'However the later observation (e.g. [CITATION]) found that the weak-field condition is broken around the critical field, defined as a point of vacuum cascade, such that the mechanism in weak-field condition loses its prediction for this vacuum phenomenon.', '1811.11712-4-4-2': 'Inspired by the similar existence of critical value of electric field in string theory, it is of a possibility to clarify the vacuum cascade in the Coulomb phase beyond the weak-field condition with the help of gauge/gravity duality [CITATION].', '1811.11712-4-4-3': 'This is also known as the Semenoff-Zarembo construction, where the production rate has been obtained by calculating the classical action of a bosonic string, which is attached to a probe D3-brane and coupled to a Kalb-Ramond field.', '1811.11712-4-5-0': 'Since the instanton action in the production rate is equivalent to the string action, which is proportional to the area, calculation of the rate is related to integrating the classical equations of motion for bosonic string in the given external field.', '1811.11712-4-5-1': 'In other words, duality converts the problem to evaluating the area of a minimal surface [CITATION] in Euclidean [MATH], the boundary of which is assumed to be the trajectory of the worldline on the probe D3-brane.', '1811.11712-4-5-2': ""In mathematics, such a Dirichlet problem is known as the Plateau's problem [CITATION]."", '1811.11712-4-6-0': 'In this work, we consider the scalar pair production in a dynamic external field of single-pulse and sinusoidal types at strong coupling and weak-field limit based on the method of worldline instantons, which is explained in sections [REF] and [REF].', '1811.11712-4-6-1': 'We first show by a field-theoretical approach, that besides the enhancement due to the dynamics of electric fields, a further contribution to the production rate arises from the Wilson loop, and it becomes dominant in production rate as Keldysh parameter increases.', '1811.11712-4-6-2': 'However, such a correction seems to diverge as Keldysh adiabaticity parameter [MATH] based on our estimated formula; and it leads to a contradiction to the weak-field condition, so that near the critical field, the method itself breaks down, and the prediction of a vacuum cascade becomes unclear.', '1811.11712-4-6-3': ""To overcome the problem of breaking weak-field condition, we then follow Semenoff and Zarembo's proposal in sec. [REF], applying the gauge/gravity duality to the Schwinger effect in the Coulomb phase of an [MATH] supersymmetric Yang-Mills theory."", '1811.11712-4-6-4': 'The classical solution of the corresponding string worldsheets are estimated by perturbing the solvable case of a constant external electric field.', '1811.11712-4-7-0': '# Worldline instantons at strong coupling', '1811.11712-4-8-0': 'The worldline instanton approach is a semiclassical calculation realised by the worldline path integral representations [CITATION], in which the so-called worldline instanton is a periodic solution of the stationary phase in the path integral.', '1811.11712-4-8-1': 'Based on this method, the pair-production rate [MATH] for a massive scalar QED in the small-coupling and weak-field approximation is computed by [CITATION] [EQUATION] where [MATH] is the 4-volume, [MATH] is a constant, given by [EQUATION] and [MATH] represents the action of worldline instanton, i.e. [EQUATION]', '1811.11712-4-8-2': 'In eq. [REF], [MATH] is a classical background field, not to be confused with the fluctuation part of the [MATH] gauge field [MATH], which is also included in the initial setup of path integral, but cancelled due to considerations at small coupling.', '1811.11712-4-9-0': 'The path integral in eq. [REF] can be computed by the stationary phase approximation in weak-field condition [EQUATION]', '1811.11712-4-9-1': 'For more general cases with 1D temporally inhomogeneous fields [MATH], the worldline instanton is obtained by solving the instanton equation with periodic boundary condition [EQUATION]', '1811.11712-4-9-2': 'Thus the pair-production rate can be written as [EQUATION] where [MATH] is a monotonically decreasing function of Keldysh parameter [CITATION] [MATH] and calculated by substituting the solution of eq. [REF] into the worldline action with the corresponding background field [MATH].', '1811.11712-4-9-3': 'In addition, the weak-field condition eq. [REF] is of various forms according to different dynamic fields, because it depends on specific instanton solution.', '1811.11712-4-10-0': 'At arbitrary coupling constant but with weak-field condition [CITATION], the production rate is modified by a factor, [EQUATION] where [MATH] is the average of [MATH] Wilson loop [MATH] [EQUATION]', '1811.11712-4-10-1': 'The integral path [MATH] is along the 2D trajectory of worldline instanton.', '1811.11712-4-10-2': 'In the Feynman gauge of the [MATH] field, the Wilson loop becomes [CITATION] [EQUATION]', '1811.11712-4-10-3': 'After simplification, [MATH] can be represented via a double contour integrals [EQUATION]', '1811.11712-4-10-4': 'For a constant electric field, the worldline instanton is a circle, and the production rate of a single scalar pair is given by [EQUATION] where [MATH] is the Schwinger limit in natural units.', '1811.11712-4-10-5': 'Consideration at arbitrary coupling [CITATION] leads to a correction from eq. [REF] [EQUATION] see also sec. [REF].', '1811.11712-4-10-6': 'Our aim in the next section is to calculate the Wilson loops in eq. [REF] along two specific worldlines separately.', '1811.11712-4-10-7': 'The first one is the worldline instanton in a single-pulse field, whereas the second is in a sinusoidal field.', '1811.11712-4-10-8': 'We show that the corrections due to the dynamic fields depend on Keldysh parameter, and the weak-field condition is also broken in these two cases, as noted in e.g. [CITATION].', '1811.11712-4-11-0': '# Wilson loops along worldline instanton paths', '1811.11712-4-12-0': 'The Wilson loop in eq. [REF] plays a central role in the case of arbitrary coupling, the integral eq. [REF] standing on the exponent diverges as [MATH] approaches [MATH].', '1811.11712-4-12-1': 'A regulator [MATH] has been introduced in [CITATION], such that eq. [REF] becomes [EQUATION] where prime indicates derivative with respect to the instanton parameter, and [MATH] and [MATH] can be understood as angular coordinates for the instanton.', '1811.11712-4-13-0': 'One sees that the integrand in [MATH] behaves like [MATH] as [MATH], rendering the integral divergent.', '1811.11712-4-13-1': 'Introducing [MATH] makes the integral regular, and the divergence can now seen explicitly by expanding the numerator and denominator of the integrand separately.', '1811.11712-4-13-2': 'Up to [MATH], the numerator reads [EQUATION] whereas the denominator becomes [EQUATION]', '1811.11712-4-13-3': 'The condition [MATH] has also been used, because all worldline instantons satisfy [MATH], where [MATH] is defined in the same way as in [CITATION].', '1811.11712-4-13-4': 'Hence up to second order of [MATH], one has [EQUATION] [MATH], and [EQUATION]', '1811.11712-4-13-5': 'One sees that the divergence of [MATH] now is removed by introducing a subtraction term [EQUATION] which is an example of the perimeter law, depicting the behaviour of the Wilson loop in Euclidean space [CITATION].', '1811.11712-4-13-6': 'Since [MATH] is independent of integration variable, the subtraction term can be worked out as [EQUATION]', '1811.11712-4-13-7': 'After subtracting this term from eq. [REF], one obtains the physical [MATH] by taking the limit of regularised [MATH] [EQUATION]', '1811.11712-4-13-8': 'The physical Wilson loop is then given by [EQUATION] which appears as a factor in the final expression of the production rate.', '1811.11712-4-14-0': 'In our practice with the dynamic fields, [MATH] has yet to be worked out in a closed form, and its estimation is to be discussed in sec. [REF].', '1811.11712-4-15-0': '## Estimation of [MATH] and [MATH]', '1811.11712-4-16-0': 'To derive an approximation of [MATH], one may keep the finite terms in eq. [REF] and [REF], i.e. [EQUATION]', '1811.11712-4-16-1': 'This will be called a [MATH]-expansion up to second order of [MATH], which follows straightforwardly from the separation of divergent term.', '1811.11712-4-16-2': 'This method belongs to rational approximation.', '1811.11712-4-16-3': 'Furthermore in our application in dynamic fields, this expansion can be worked out in a closed form easily.', '1811.11712-4-16-4': 'Also note the [MATH] term in eq. [REF], which will be mentioned again later with example.', '1811.11712-4-16-5': 'The validity of [MATH]-expansion is closely related to the uniform convergence of integrand, and demonstrated in [REF].', '1811.11712-4-16-6': 'If one uses diagonal Pade approximant for integrand rather than [MATH]-expansion, the convergence is obvious [CITATION].', '1811.11712-4-17-0': 'Alternatively, one may also expand the integrand of [MATH] with respect to [MATH] for the well-defined point [MATH], i.e. [EQUATION] where [MATH].', '1811.11712-4-17-1': 'This will be called a [MATH]-expansion.', '1811.11712-4-17-2': 'If the sequence [MATH] is integrable term by term, such that the interchanging summation and integration is valid, then comparing with the similar expansion of [MATH], one obtains the finite terms of each order by [EQUATION] where [MATH] are coefficients of expansion for [MATH] with respect to [MATH] [EQUATION] while [MATH] are coefficients of expansion for [MATH] with respect to [MATH].', '1811.11712-4-17-3': 'Since [MATH] is the pole of first order for [MATH] as a function of [MATH], [MATH] having the same pole of [MATH] is obviouse.', '1811.11712-4-17-4': 'For those points, where interchanging operation not long stands, this approach fails, e.g. at [MATH].', '1811.11712-4-17-5': 'In our application in dynamic fields, this expansion can also be worked out in a closed form at each order.', '1811.11712-4-17-6': 'However, the number of terms in the expansion increases exponentially, and the result are obtained by computer algebra system.', '1811.11712-4-18-0': 'Yet another way of estimating [MATH] is numerical integration, in which the regulator [MATH] is still needed.', '1811.11712-4-18-1': 'There are polynomial contributions of [MATH] in the bare term [MATH], and one might think taking a small [MATH] would give a good result.', '1811.11712-4-18-2': 'However, [MATH] and the counter term [MATH] both diverges like [MATH] as [MATH].', '1811.11712-4-18-3': 'A small [MATH] leads to a numerically dissatisfying operation, in which two big numbers cancels, yielding a small result and a great loss of significance.', '1811.11712-4-18-4': 'This problem becomes catastrophic for the dynamic fields when [MATH].', '1811.11712-4-18-5': 'In order to overcome the potentially catastrophic cancellation, we use linear extrapolation near [MATH], in which for each [MATH], [MATH] is numerically calculated for several different values of [MATH].', '1811.11712-4-18-6': 'The limit of [MATH] is then obtained by linearly extrapolate the series of results with respect to [MATH].', '1811.11712-4-18-7': 'In this approach, the error due to extrapolation can also be obtained by estimation of the parameters in linear regression.', '1811.11712-4-18-8': 'Furthermore, in our application the numerator and denominator in eq. [REF] scales as [MATH] when [MATH], so that for a fixed [MATH], at large [MATH] the regularised integrand is dominated by the regulator on the denominator.', '1811.11712-4-18-9': 'This is overcome by scaling [MATH] accordingly, such that the subtraction term in eq. [REF] remains constant with respect to [MATH].', '1811.11712-4-19-0': '## Constant electric field', '1811.11712-4-20-0': 'For a constant electric field, the worldline instanton is a circle of radius [MATH], given by [CITATION] [EQUATION] where the zeroth component [MATH] denotes the Euclidean time.', '1811.11712-4-20-1': 'The instanton action for single pair production is obtained by substituting this solution into eq. [REF], yielding [EQUATION]', '1811.11712-4-20-2': 'The weak-field condition [MATH] [CITATION] implies [EQUATION] where [MATH] is defined in eq. [REF].', '1811.11712-4-21-0': 'On the other hand, the integral in eq. [REF] and the perimeter law in eq. [REF] can be evaluated explicitly as [EQUATION]', '1811.11712-4-21-1': 'Hence the regularised Wilson loop for a single scalar pair can be recovered as [CITATION] [EQUATION] and the production rate is given by [EQUATION] from which the critical field for vacuum cascade can be estimated by [EQUATION]', '1811.11712-4-21-2': 'One sees that [MATH] is much greater than the Schwinger limit and breaks the weak field condition in eq. [REF], as have been noted in e.g. [CITATION].', '1811.11712-4-21-3': 'Therefore, the obtained results are not valid when the field goes close to the critical limit, and cannot answer the question whether a vacuum cascade happens near the critical field strength.', '1811.11712-4-22-0': '## Single-pulse field [MATH]', '1811.11712-4-23-0': 'For a generic 1D dynamic field, we assume that the Wilson loop is of the following form [EQUATION] where [MATH] is an enhancement factor with respect to the case of a constant external field in eq. [REF].', '1811.11712-4-23-1': 'This name of [MATH] is seen to be appropriate from the fact that [MATH] is a monotonically decreasing function and tends to unit at adiabatic limit [MATH].', '1811.11712-4-24-0': 'Before applying nonlinear approximation schemes as we discussed in sec. [REF] to single-pulse field, we apply it to the case of a constant field.', '1811.11712-4-24-1': 'The integrand after [MATH]-expansion becomes [EQUATION] while the subtracting term eq. [REF] will not change.', '1811.11712-4-24-2': 'Then repeating the similar procedure, one gets [EQUATION] which leads to a [MATH] deviation compared to eq. [REF].', '1811.11712-4-24-3': 'This term comes from [MATH] in [MATH] and does not depend on the specific form of worldline path.', '1811.11712-4-24-4': 'Later we will see that this [MATH] deviation happens in both cases considered in 2nd-order [MATH]-expansion, which is caused by accuracy of estimation method, thus it approaches to zero as the approximation order increases, see fig. [REF].', '1811.11712-4-25-0': 'In other words, the emergence of finite terms are expected for each orders, and the higher-order contribution should cancel [MATH] from the 2nd-order.', '1811.11712-4-25-1': 'Based on this consideration we remove [MATH] derivation directly in the final results, which should not be confused with the counter term eq. [REF].', '1811.11712-4-26-0': 'Let us see the first nontrivial example, pair production in the single-pulse field [CITATION].', '1811.11712-4-26-1': 'The instanton has been evaluated explicitly to be [EQUATION] and the worldline action for this path reads [EQUATION]', '1811.11712-4-26-2': 'The weak-field condition leads to [EQUATION]', '1811.11712-4-26-3': 'At the nonperturbative region [CITATION] [MATH], the weak-field condition reduces to [MATH].', '1811.11712-4-26-4': 'In other words, the dynamics of field also decreases the upper limit of weak-field condition.', '1811.11712-4-27-0': 'Now we turn to the the Wilson loop.', '1811.11712-4-27-1': 'The corresponding perimeter law has a closed form [EQUATION]', '1811.11712-4-27-2': 'With the [MATH]-expansion up to [MATH], eq. [REF] reads [EQUATION] which leads to [EQUATION]', '1811.11712-4-27-3': 'One may note that [MATH] as [MATH] approaches zero.', '1811.11712-4-27-4': 'The [MATH] derivation is predicted and has to be subtracted, i.e. the enhancement factor is then in the weak-field condition [EQUATION]', '1811.11712-4-27-5': 'This operation guarantees the condition: [MATH] as [MATH].', '1811.11712-4-27-6': 'Since [MATH], the factor amplifies the contribution from the Wilson loop in eq. [REF], so that the pair-production rate is no longer exponentially suppressed, see fig. [REF].', '1811.11712-4-28-0': 'Alternatively, the [MATH]-expansion can be worked out as a power series of [MATH], where eq. [REF] has been used by replacing [MATH].', '1811.11712-4-28-1': 'After removing the poles at each order of [MATH], one obtains [EQUATION]', '1811.11712-4-28-2': 'The exponential factor in the production rate is evaluated as [EQUATION]', '1811.11712-4-28-3': 'The first part is the main contribution from instanton action, the second part arises due to the Wilson loop correction from [MATH]-expansion.', '1811.11712-4-28-4': 'If the critical field is defined as saddle point, at which the exponential suppression is precisely zero, then one could have [EQUATION]', '1811.11712-4-28-5': 'On the one hand, as in the case with constant field, eq. [REF] also breaks the weak-field condition at nonperturbative region [MATH].', '1811.11712-4-28-6': 'On the other hand, in contrast with the case in constant field, the exponential factor of Wilson loop is no longer constant, such that for given [MATH] its contribution for production rate becomes dominant as [MATH] increases.', '1811.11712-4-28-7': 'In addition, both estimation eq. [REF] (or eq. [REF]) and numerical result seem to be divergent as [MATH] approaches [MATH], even if the pre-exponential factor of Feynman integral were taken into account [CITATION].', '1811.11712-4-28-8': 'It implies that Wilson loop ought to be of a pole at [MATH], and loses its meaning at this point, where the instanton trajectory collapses to a singular point.', '1811.11712-4-29-0': '## Sinusoidal field [MATH]', '1811.11712-4-30-0': 'The second example is a sinusoidal field [CITATION].', '1811.11712-4-30-1': 'The coordinates of the instanton can be represented by special functions as [EQUATION] where [MATH] is the complete elliptic integral, [MATH] and [MATH] are Jacobi elliptic functions.', '1811.11712-4-30-2': 'The instanton action is given by [EQUATION]', '1811.11712-4-30-3': 'Repeating the procedure above, one obtains the corresponding perimeter law in a closed form [EQUATION]', '1811.11712-4-30-4': 'The exponent in the [MATH]-expansion reads [EQUATION] in which [MATH] is the Jacobi amplitude function.', '1811.11712-4-30-5': 'Thus at the week-field approximation [MATH] [CITATION], one has [EQUATION]', '1811.11712-4-30-6': 'Where the deviation [MATH] has been subtracted.', '1811.11712-4-30-7': 'In this case, [MATH] is also greater than unity and a non-trivial function depending on external field, and it tends to [MATH] as [MATH] approaches zero.', '1811.11712-4-30-8': 'Alternativeszaly, the [MATH]-expansion is implemented by expanding in [MATH] first.', '1811.11712-4-30-9': 'Removing the divergences in [MATH], one obtains [EQUATION]', '1811.11712-4-30-10': 'The results are shown in fig. [REF].', '1811.11712-4-31-0': 'In addition, similar to the example in the last subsection [MATH] is also divergent as [MATH], even if the pre-exponential factor is considered [CITATION].', '1811.11712-4-31-1': 'The exponent factor in the production rate is then given by [EQUATION]', '1811.11712-4-31-2': 'For small [MATH], one gets [EQUATION] which breaks the weak-field condition at nonperturbative region as well.', '1811.11712-4-32-0': 'From above three examples, one may note that, first, the weak-field condition in the non-perturbative ranges [MATH] is inevitably broken at strong coupling, which makes the vacuum cascade around the critical field ambiguous [CITATION]; and second, the correction due to the Wilson loop in dynamic fields is a monotonically increasing function with respect to [MATH] and diverges as [MATH].', '1811.11712-4-33-0': '# Holographic Schwinger effect with dynamic field', '1811.11712-4-34-0': 'In order to answer the question, if the vacuum cascade for strong coupling happens as the strength of time-dependent field goes close to the critical limit [CITATION], we consider a similar effect in the context of gauge/gravity duality, where the gauge field theory refers to an [MATH] supersymmetric Yang-Mills theory on the 4D boundary of an [MATH] space, and the quantum gravity is a type 2@B superstring theory in the bulk of the [MATH].', '1811.11712-4-34-1': 'The same as the case with constant field, we expect that the string theory could shed some light on the catastrophic vacuum cascade through the duality principle.', '1811.11712-4-35-0': ""According to the Semenoff and Zarembo's holographic setup [CITATION], the exponential factor in the production rate of the gauge field is obtained from the superstring counterpart by the area of the string worldsheet attached to a probe D3-brane, i.e. [EQUATION] where [MATH] is the Nambu-Goto (NG) action [CITATION] [EQUATION] depending on the induced metric [EQUATION] and [MATH] is the Kalb-Ramond [CITATION] 2-form (or NS-NS, where NS is the abbreviation of Neveu-Schwarz [CITATION] ) as an string interaction term, [EQUATION]"", '1811.11712-4-35-1': 'In eqs. [REF] and [REF], [MATH] are the coordinates on the string worldsheet, [MATH] are the coordinates of the [MATH]-D [MATH] space with metric [MATH], and [MATH] is the induced metric.', '1811.11712-4-36-0': 'In the Semenoff-Zarembo construction, the worldsheet ends on the probe D3-brane with a boundary, taking the same shape as the worldline instanton.', '1811.11712-4-36-1': 'Hence the essential problem is converted to compute the on-shell action of string in Euclidean [MATH] with the given boundary.', '1811.11712-4-36-2': 'Note that the Nambu-Goto action is proportional to the worldsheet area, and extremising the area leads to a minimal surface.', '1811.11712-4-36-3': ""In other words, calculation of the exponential factor now corresponds to a Plateau's problem in the framework of gauge/gravity duality."", '1811.11712-4-37-0': '## Constant electric field', '1811.11712-4-38-0': 'The worldline instanton in a constant field is a circle, thus the worldsheet can be parametrised by [EQUATION] which is different than the choice in [CITATION], and the two parameterisations have different chirality, i.e. [MATH], where [MATH] is the Jacobian.', '1811.11712-4-38-1': 'Therefore, the orientation of the Kalb-Ramond coupling is also reversed.', '1811.11712-4-38-2': 'The Nambu-Goto action in our parameterisation becomes [EQUATION] where [MATH], [MATH] is the location of the probe D3-brane.', '1811.11712-4-38-3': '[MATH] can be analytically as [CITATION] [EQUATION] where [MATH] is the worldline instanton on the D3-brane.', '1811.11712-4-38-4': 'Substituting it into the Nambu-Goto action, one obtains [EQUATION]', '1811.11712-4-38-5': 'Furthermore, the NS-NS term reads [EQUATION] where [MATH] can be fixed by extremising the total action, yielding [EQUATION]', '1811.11712-4-38-6': 'Where [MATH] is defined as critical values.', '1811.11712-4-38-7': 'The exponential factor in the production rate can now be solved as [EQUATION] in which the string and spacetime parameters have been replaced by the ones of the gauge field via [MATH].', '1811.11712-4-38-8': ""This is the result obtained in [CITATION] and agrees with the Schwinger's formula in the weak-field limit."", '1811.11712-4-39-0': '## Estimation of Single-pulse field [MATH]', '1811.11712-4-40-0': 'The instanton path as the worldsheet boundary on the D3-brane in a single-pulse field has been shown in eq. [REF].', '1811.11712-4-40-1': 'Thus one can parametrise the worldsheet by using [MATH] and [MATH], i.e. [EQUATION] where [MATH] is regarded as initial information and not relevant to the scale [MATH].', '1811.11712-4-40-2': 'The simplicity of Semenoff-Zarembo construction for constant field led us to speculate that the similar production rates could have been obtained by repeating above procedure.', '1811.11712-4-40-3': 'However, it is not anything like worldline instanton, the string worldsheets are not exactly integrable for dynamic fields in our cases.', '1811.11712-4-40-4': 'Thus to make an effect estimation, we expand the instanton at adiabatic limit, i.e. [MATH], [EQUATION] namely the zeroth order of [MATH] is just the circle boundary eq. [REF].', '1811.11712-4-40-5': 'Therefore, we will treat the complete worldline boundary as perturbation around the circle.', '1811.11712-4-40-6': 'The [MATH] component can be estimated by noting that [EQUATION] where [MATH] is the instanton path.', '1811.11712-4-40-7': 'In a single-pulse field, it reads [EQUATION]', '1811.11712-4-40-8': 'In the limit [MATH], [MATH] reduces to a circle [EQUATION]', '1811.11712-4-40-9': 'Hence the Nambu-Goto action is formulated as [EQUATION] where [MATH] is given in eq. [REF].', '1811.11712-4-40-10': 'On the other hand, the NS-NS part right now becomes [EQUATION]', '1811.11712-4-40-11': 'The [MATH] is fixed as the stationary point of the total action, [EQUATION]', '1811.11712-4-40-12': ""The exponential factor in the production rate with correction up to the second order of [MATH] is [EQUATION] here [MATH] denotes the effective coupling at large 't Hooft limit."", '1811.11712-4-40-13': 'In eq. [REF], the first part in the exponent comes from the circular boundary, while the second term arises from the deformation, see fig. [REF].', '1811.11712-4-41-0': 'If one defines the critical field as stationary point of exponent, the correction (up to [MATH]) due to the time-dependence of the background field leads a lager critical value comparing with constant field, which is fixed by [EQUATION] beyond which the decay rate decreases as the similar as eq. [REF].', '1811.11712-4-41-1': 'The enhancement up to [MATH] can also be noted from the negative sign of [MATH] in eq. [REF].', '1811.11712-4-41-2': 'In other words, the correction plays a role of suppression of pair production.', '1811.11712-4-42-0': '## Estimation of sinusoidal field [MATH]', '1811.11712-4-43-0': 'The worldline instanton in this case can be parametrised as [EQUATION] where [MATH] has been defined in eq. [REF].', '1811.11712-4-43-1': 'The Maclaurin series of Jacobian elliptic functions in [MATH] gives [EQUATION]', '1811.11712-4-43-2': 'The instanton path on the D3-brane in this case is [EQUATION] and its expansion for small [MATH] reads [EQUATION]', '1811.11712-4-43-3': 'The second order terms for the Nambu-Goto action [EQUATION] and the NS-NS term are [EQUATION]', '1811.11712-4-43-4': 'Up to the second order of [MATH], [MATH] is given by [EQUATION]', '1811.11712-4-43-5': 'Finally, the exponential factor in the production rate is given by [EQUATION] see fig. [REF].', '1811.11712-4-43-6': 'The critical field should be greater than [MATH], i.e. [EQUATION] and one sees that dynamics of the external field also enhances the critical field as before.', '1811.11712-4-44-0': '# Conclusion and Discussion', '1811.11712-4-45-0': 'In this paper, the scalar Schwinger effect for dynamic fields at strong coupling and weak-field limit has been studied, by first using the field-theoretical method of worldline instantons.', '1811.11712-4-45-1': 'A non-trivial contribution to the production rate is discovered by evaluating the Wilson loop along the instanton path, which depends on the Keldysh adiabaticity parameter [MATH].', '1811.11712-4-45-2': 'Thus one may expect that such correction may save the weak-field condition in strong coupling.', '1811.11712-4-45-3': 'However after computations, we find that the introduction of the correction term also leads to a contradiction to the weak-field condition near the critical field strength.', '1811.11712-4-46-0': 'We note also that the correction from Wilson loop is a monotonically increasing function with respect to [MATH], which makes the contribution for production rate from Wilson loop become dominant as [MATH] increases.', '1811.11712-4-46-1': 'Moreover both [MATH]-expansion and numerical calculation suggest a divergent value as [MATH] approaches infinity, even if the pre-exponential factor of Feynman integral is considered.', '1811.11712-4-46-2': 'One possible explanation is that the Wilson loop loses its meaning at [MATH], because the instanton trajectory collapses to a singular point.', '1811.11712-4-47-0': 'In order to clarify the vacuum cascade beyond the weak-field condition, in the context of an [MATH] supersymmetric Yang-Mills theory, the production rate is calculated by the gauge/gravity duality, according to which the instanton action has a string counterpart of the classical string action in Euclidean [MATH], where the boundary on the probe D3-brane is given by the instanton path.', '1811.11712-4-47-1': 'Thus the problem is converted to solving the classical motion of string with Dirichlet boundary conditions.', '1811.11712-4-47-2': 'However the string worldsheets for dynamic fields are not integrable as in the worldline instantons.', '1811.11712-4-47-3': 'To provide an explicit estimation, we treat the specific worldsheets as perturbations around the one with circle boundary, which had been solved exactly.', '1811.11712-4-47-4': 'Such an expansion is an adiabatic approximation, it is practical and realistic, because only low-frequency laser (comparing with electron mass) is currently operational.', '1811.11712-4-47-5': 'The obtained decay rates in the two examples with dynamic fields are similar concave functions as in the case with constant field, but the critical fields increase considerably.', '1811.11712-4-47-6': 'In other words, up to [MATH] the correction due to the dynamics of electric field suppresses the pair production, which is opposite to cases of worldline instantons.', '1811.11712-4-48-0': '# The validity of [MATH]-expansion', '1811.11712-4-49-0': 'For an approximation theory [MATH] of full function [MATH], we say it is valid if [MATH] in some domain.', '1811.11712-4-49-1': 'The purpose of [MATH]-expansion is to provide a nonlinear approximation for [MATH] by establishing a sequence of functions [MATH], which satisfies [MATH].', '1811.11712-4-50-0': 'In order to establish nonlinear approximation, we note that the integrand of [MATH] can be expanded as [EQUATION] where the coefficients in the expansion are separately [EQUATION] and for given any closed and smooth instanton trajectories, both Taylor expansions in the denominator and numerator are of infinite radius of convergence separately.', '1811.11712-4-50-1': 'Therefore both expansions are uniformly convergent separately, see 8.1 theorem in [CITATION].', '1811.11712-4-51-0': 'Then one can define the [MATH]-th order of [MATH]-expansion by the integral of rational series with type [MATH] [EQUATION] where the integrand is uniformly convergent to eq. [REF] due to the quotient law of convergent sequences and absence of poles for [MATH].', '1811.11712-4-51-1': 'The uniform convergence can be shown by using theorem 7.9 in [CITATION].', '1811.11712-4-51-2': 'Generally for given two uniformly convergent sequences [MATH], [MATH] and [MATH] does not have zeroes in considering domain, then the quotient [MATH] is uniformly convergent to [MATH].', '1811.11712-4-51-3': 'Namely the supremum of [MATH] [EQUATION] approaches zero as [MATH], because both sequences are uniform convergent.', '1811.11712-4-51-4': 'Thus the interchange of limit and integral operations is valid in the corresponding domain (see 7.16 theorem in [CITATION]), i.e. [EQUATION] which is exactly what we expect [MATH].', '1811.11712-4-52-0': 'Y.-F.W. is grateful to Claus Kiefer and Nick Kwidzinski (Cologne), Chao Li (Princeton) and Ziping Rao (Vienna).', '1811.11712-4-52-1': 'H.G. is supported by ELI-ALPS project, co-financed by the European Union and the European Regional Development Fund No. GINOP-2.3.6-15-2015-00001.', '1811.11712-4-52-2': 'Y.-F.W. is supported by the Bonn-Cologne Graduate School for Physics and Astronomy (BCGS).'}",
0809.4488,"{'0809.4488-1-0-0': 'In a universe with a cosmological constant, the time variation of the gravitational potential is, in principle, observable.', '0809.4488-1-0-1': 'Using an N-body simulation of a [MATH]CDM universe, we show that linear theory is not sufficiently accurate to predict the power spectrum of the time derivative, [MATH], needed to compute the imprint of large-scale structure on the cosmic microwave background (CMB).', '0809.4488-1-0-2': 'The linear part of the [MATH] power spectrum (the integrated Sachs-Wolfe effect or ISW) drops quickly as the relative importance of [MATH] diminishes at high redshift, while the non-linear part (the Rees-Sciama effect) evolves more slowly with redshift.', '0809.4488-1-0-3': 'Therefore, the deviation of the total power spectrum from linear theory occurs at larger scales at higher redshifts.', '0809.4488-1-0-4': 'For the cross-correlation power spectrum of galaxy samples with the CMB, deviation from linear theory by about [MATH] can occur at [MATH] to [MATH], or equivalently at scales of two to four degrees.', '0809.4488-1-0-5': 'The non-linear contribution completely dominates at smaller scales.', '0809.4488-1-0-6': 'Ignoring the contribution of the Rees-Sciama effect in analyses of the cross-correlation of large-scale structure and the CMB leads to an overestimate of [MATH], though small compared to current measurement errors.', '0809.4488-1-0-7': 'Tests using the cross-correlation of SDSS galaxy sample and SDSS quasar sample with the CMB indicate that the Rees-Sciama contribution cannot be disentangled from the ISW effect in these samples because the statistical errors are too large.', '0809.4488-1-0-8': 'However, this may no longer be the case in upcoming surveys such as Pan-STARRS.', '0809.4488-1-0-9': ""We also show that on arc-minute scales the Rees-Sciama effect will give the dominant contaminating contribution to the expected cross-correlation signal induced by the Sunyaev-Zel'dovich effect."", '0809.4488-1-1-0': '# Introduction', '0809.4488-1-2-0': 'The most intriguing topic in contemporary cosmology is the nature of the dark energy which appears to dominate the energy density of the Universe at late times.', '0809.4488-1-2-1': 'Strong evidence for the existence of dark energy comes from both the combined analysis of the cosmic microwave background radiation (CMB) and the galaxy large-scale structure (LSS) , and from high redshift type Ia supernovae .', '0809.4488-1-2-2': 'Both of these techniques infer the presence of dark energy from geometrical measures.', '0809.4488-1-2-3': 'A complementary probe of dark energy is provided by techniques that measure the dynamical effect of dark energy through its influence on the rate of growth of structure.', '0809.4488-1-2-4': 'Large deep galaxy redshift surveys (like the EUCLID (the ESA Mission to Map the Dark Universe) and the ADEPT (Advanced Dark Energy Physics Telescope)) are being planned.', '0809.4488-1-2-5': 'They will exploit the redshift space anisotropy of galaxy clustering, caused by coherent flows into overdense regions and outflows from underdense regions, to measure directly the growth rate as a function of redshift.', '0809.4488-1-2-6': 'The Integrated Sachs-Wolfe (ISW) effect in which the decay of the large-scale potential fluctuations induces CMB temperature perturbations provides another measure of the dynamical effect of dark energy.', '0809.4488-1-3-0': 'In principle, the ISW effect could be detected directly in the CMB power spectrum at very low multiples.', '0809.4488-1-3-1': 'In the [MATH]CDM cosmology, it would boost the plateau in the power spectrum at [MATH].', '0809.4488-1-3-2': 'However, as the increase of the power is not large in comparison to the cosmic variance, it cannot be unambiguously detected even by the WMAP data .', '0809.4488-1-3-3': 'A more sensitive technique is to search for the ISW signal in the cross-correlation of the LSS with the CMB.', '0809.4488-1-3-4': 'As the expected signal is weak and occurs on large scales, a very large galaxy survey is needed to trace the LSS.', '0809.4488-1-3-5': 'Currently individual detections based on surveys such as APM, 2MASS, NVSS and SDSS are not of very high statistical significance and precise measurements await the construction of new, larger surveys (BOSS, Pan-STARRS1 ...).', '0809.4488-1-3-6': 'If such surveys are to place robust, meaningful constraints on the properties of the dark energy it is important fully to take account of other processes beyond the (linear) ISW effect that may contribute to the cross-correlation signal.', '0809.4488-1-3-7': 'Here, we focus on deviations caused by non-linear gravitational evolution, the Rees-Sciama effect .', '0809.4488-1-4-0': 'Other processes are known to contribute to the cross-correlation signal.', '0809.4488-1-4-1': ""First, the thermal Sunyaev-Zel'dovich (SZ) effect caused by hot ionized gas in galaxy clusters induces an anti-cross-correlation signal which can cancel the ISW effect on small scales."", '0809.4488-1-4-2': 'Its statistical contribution can be modelled and subtracted given the value of [MATH] (the [MATH] linear mass fluctuations within a sphere of 8 [MATH] Mpc) which determines the abundance of galaxy clusters .', '0809.4488-1-4-3': 'Also, since the SZ effect is frequency dependent, it can be subtracted in frequency space given sufficient spectral coverage.', '0809.4488-1-4-4': 'Second, the redshift dependence of galaxy bias, if not properly taken into account, can introduce systematics effects in the determination of dark energy parameters.', '0809.4488-1-4-5': 'Other effects such as lensing magnification and the Doppler redshift effect can also boost the cross-correlation signal, but are only important at high redshift .', '0809.4488-1-4-6': 'These effects are well documented and can be calibrated and removed.', '0809.4488-1-5-0': 'In this paper, we will solely explore the contribution of the non-linear terms, or the Rees-Sciama (RS) effect, on the cross-correlation signal.', '0809.4488-1-5-1': 'The RS effect arises from the non-linear evolution of the potential .', '0809.4488-1-5-2': 'It is believed to be much smaller than the CMB signal at all scales .', '0809.4488-1-5-3': 'Indeed, compared with the CMB power spectrum, the RS effect is orders of magnitude lower.', '0809.4488-1-5-4': 'Also, compared with the complete integrated ISW power spectrum, the RS effect has been shown, using the halo model approach , to be unimportant at [MATH].', '0809.4488-1-5-5': 'However, the RS effect has not been taken into account in cross-correlation analyses and it is important to assess its importance ahead of the completion of the next generation of large deep galaxy surveys.', '0809.4488-1-6-0': 'We use a large N-body simulation to investigate the effect of the non-linear contribution on the interpretation of the ISW cross-correlation signal.', '0809.4488-1-6-1': 'We use the [MATH]-particle L-BASICC simulation described by [CITATION] which, with a box size of 1340 [MATH] Mpc, is ideal for this purpose because not only does it enable us to extrapolate our analysis to non-linear scales at different redshifts, but it includes the very large scale power necessary to check the agreement with linear theory.', '0809.4488-1-6-2': 'The cosmology adopted in the L-BASICC simulation is [MATH]CDM, with [MATH], [MATH], [MATH], [MATH] and [MATH] km s[MATH] Mpc[MATH].', '0809.4488-1-7-0': 'The paper is organised as follows.', '0809.4488-1-7-1': 'In 2, we compute the power spectrum of the ISW plus RS effects from our simulation and compare them with linear theory.', '0809.4488-1-7-2': 'In 3, we analyse these two effects in terms of the cross-correlation of the LSS with the CMB.', '0809.4488-1-7-3': 'Finally, in 3, we discuss our results and present our conclusions.', '0809.4488-1-8-0': '# Time derivative of the potential', '0809.4488-1-9-0': 'The integrated Sachs-Wolfe effect results from the late time decay of gravitational potential fluctuations.', '0809.4488-1-9-1': 'The net blueshift or redshift of the CMB photons, caused by the change in the potential during the passage of the photons, induces net temperature fluctuations of the black body spectrum, [EQUATION] where [MATH] is the time derivative of the gravitational potential, [MATH] is the lookback time, with [MATH] at the present and [MATH] at the last scattering surface.', '0809.4488-1-9-2': 'The angular power spectrum of these temperature fluctuations (see the Appendix) is given by [EQUATION] where [MATH] is the comoving distance to lookback time, [MATH], [MATH] is the spherical Bessel function and [MATH] is the 3-D power spectrum of [MATH] fluctuations.', '0809.4488-1-9-3': ""To derive the the final expression we have used Limber's approximation by assuming [MATH] ."", '0809.4488-1-10-0': 'The ISW effect consists of the temperature fluctuations described by these equations when linear theory is used to compute [MATH] and its fluctuation power spectrum [MATH].', '0809.4488-1-10-1': 'Using a simulation to determine the non-linear contributions we can quantify the full ISW plus Rees-Sciama effect.', '0809.4488-1-10-2': 'In Fourier space, the time derivative of the gravitational potential can be expressed as: [EQUATION] where [MATH] is the expansion factor, [MATH] is the Hubble constant, [MATH] is the present mass density parameter and [MATH] is the time derivative of the density fluctuation.', '0809.4488-1-10-3': 'Combining this with the Fourier space form of the continuity equation, [MATH] gives: [EQUATION] where [MATH] is the momentum density field in Fourier space divided by the mean mass density.', '0809.4488-1-10-4': 'This enables us to estimate the Fourier transform of the [MATH] field of the simulation from the Fourier transforms of the density and momentum fields.', '0809.4488-1-10-5': 'Using equation ([REF]), the resulting power spectrum, [MATH], can be written as [EQUATION]', '0809.4488-1-10-6': 'In linear theory, [MATH] and [MATH], where [MATH] is the linear density power spectrum at the present time and [MATH] is the growth factor normalised to be unity at present.', '0809.4488-1-10-7': 'Therefore, the power spectrum of the linear ISW effect is [EQUATION] where [MATH].', '0809.4488-1-10-8': 'For easy comparison at different redshifts in the simulation, we defined a scaled [MATH] power spectrum, [MATH] which from Eq. (5) is simply [EQUATION]', '0809.4488-1-10-9': 'Our measurements of the [MATH] power spectrum are shown in Fig. [REF].', '0809.4488-1-10-10': 'The results from linear theory are also plotted.', '0809.4488-1-10-11': 'We find the total scaled [MATH] power spectrum can be well fitted by a broken power law plus the linear scaled [MATH] power spectrum [MATH], where [EQUATION]', '0809.4488-1-10-12': 'Here A and B are two free parameters that we use to fit the model to the simulation results at each redshift up to [MATH].', '0809.4488-1-10-13': 'To interpolate the model to intermediate redshifts we linearly interpolate the values of A and B from the nearest two simulation outputs.', '0809.4488-1-10-14': 'Our model is compared to the simulation results in Fig. [REF].', '0809.4488-1-11-0': 'We see in Fig. [REF] that the linear theory reproduces [MATH] at [MATH] only at [MATH] Mpc[MATH].', '0809.4488-1-11-1': 'It fails at progressively larger scales as the redshift increases.', '0809.4488-1-11-2': 'At [MATH], the ISW+RS power spectrum deviates from linear theory at [MATH] Mpc[MATH]; by [MATH], linear theory agrees with the simulation results only at [MATH] Mpc[MATH].', '0809.4488-1-11-3': 'The reason for this surprising behaviour is that the linear part of the [MATH] drops quickly to zero as the relative importance of [MATH] diminishes at high redshift, while the non-linear part evolves more slowly with redshift.', '0809.4488-1-11-4': 'Therefore, the deviation of the total power spectrum from linear theory happens at larger scales at higher redshifts.', '0809.4488-1-11-5': 'We find that the momentum power spectrum, [MATH], and the correlation power spectrum of the density and momentum, [MATH], behave similarly to the [MATH] power spectrum, namely, their deviation from linear theory occurs at larger scales at higher redshift.', '0809.4488-1-11-6': 'This is in contrast with the power spectrum of the density field which only deviates from linear theory at low redshift and on small scales.', '0809.4488-1-11-7': 'In another words, at the same redshift, the deviation from linear theory occurs at smaller scales for the density field than for the other fields.', '0809.4488-1-12-0': 'The sharp increase of [MATH] measured from the simulation at small scales ([MATH] Mpc[MATH]) is due to discreteness in the [MATH] particle L-BASICC simulation.', '0809.4488-1-12-1': 'We used the much higher resolution [MATH] particle Millennium simulation to verify that our model remains accurate at smaller scales and is robust to shot noise corrections.', '0809.4488-1-13-0': 'We can now compute the induced angular power spectrum of CMB temperature fluctuations by performing the integral in equation (2) over the redshift range [MATH] using our model for the 3-D power spectrum, [MATH].', '0809.4488-1-13-1': 'The overall result is shown in Fig. [REF] along with the contributions coming from different redshift intervals.', '0809.4488-1-13-2': 'For the overall angular power spectrum the deviation of the model from the linear theory happens at [MATH]100.', '0809.4488-1-13-3': 'This result confirms the prediction of [CITATION] based on the halo model.', '0809.4488-1-13-4': 'However, we also see that the failure of linear theory, as judged by our simulation results, occurs at smaller and smaller [MATH] as redshift increases.', '0809.4488-1-13-5': 'For example, above [MATH], the deviation occurs at [MATH] and, for larger values of [MATH] than this, linear theory becomes extremely inaccurate.', '0809.4488-1-14-0': 'In order to evaluate how the breakdown of linear theory depends on redshift, we plot the evolution of the [MATH] power at a given scale as a function of redshift in Fig. [REF].', '0809.4488-1-14-1': 'Generally, the deviations of linear theory from the simulation results decrease with scale and increase with redshift.', '0809.4488-1-14-2': 'At [MATH] Mpc[MATH], deviations start to be seen at [MATH] and, at [MATH] Mpc[MATH], linear theory has become inaccurate at all redshifts.', '0809.4488-1-14-3': 'In the right-hand panel, which shows results in [MATH] space, we find no deviations up to [MATH], but for [MATH], linear theory has clearly broken down at all redshifts.', '0809.4488-1-14-4': 'Interestingly, at high redshift, the [MATH] power in the simulation appears to be independent of [MATH] while, in linear theory, this quantity drops monotonically with [MATH].', '0809.4488-1-15-0': '# the LSS-CMB cross-correlation', '0809.4488-1-16-0': 'We now use the accurate information about the 3-D matter power spectrum provided by our simulation at different redshifts to quantify the non-linear contribution to the power spectrum of the cross-correlation between CMB fluctuations and LSS.', '0809.4488-1-17-0': 'The cross-correlation between LSS and CMB maps has been shown to be a powerful tool for verifying the existence of dark energy and constraining its properties.', '0809.4488-1-17-1': 'Current measurements of the cross-correlation have low statistical significance because the volumes probed by LSS surveys are relatively small, but this situation will improve greatly with upcoming surveys.', '0809.4488-1-17-2': 'For example, Pan-STARRS1 will survey three quarters of the sky, obtaining photometry for galaxies up to [MATH] mag in the [MATH]-band.', '0809.4488-1-17-3': 'The mean galaxy redshift in this ""[MATH] survey"" will be [MATH].', '0809.4488-1-17-4': 'Pan-STARRS1 will also carry out a deeper but smaller ""MDS"" survey covering 84 sq deg of the sky to [MATH] mag in [MATH] for which [MATH] .', '0809.4488-1-17-5': 'Cross-correlating such photometric redshift galaxy samples with a CMB map (from WMAP or Planck) will make it possible for the first time to perform ISW tomography.', '0809.4488-1-17-6': 'Galaxy samples would be divided into different redshift slices and each one cross-correlated with the CMB map.', '0809.4488-1-17-7': 'Values of the dark energy equation of state parameter, [MATH], could then be measured using the results from the different redshift slices, effectively constraining the evolution of [MATH].', '0809.4488-1-18-0': 'To illustrate how ISW tomography may work, we follow [CITATION] and model the redshift distribution of galaxies tracing the LSS as [EQUATION] but then choose the parameters [MATH] and [MATH] to emulate plausible photometric redshift slices.', '0809.4488-1-18-1': '(The same functional form was also taken by [CITATION] to model the SDSS LRG sample.)', '0809.4488-1-18-2': 'We assume [MATH] so that the width of [MATH] is much greater than the photometric redshift errors.', '0809.4488-1-18-3': 'We shift the function into different redshift intervals by using [MATH] and [MATH].', '0809.4488-1-18-4': 'The median redshift of these samples is [MATH] and [MATH] respectively.', '0809.4488-1-18-5': 'The cross-correlation power spectrum (derived in an analogous way to the auto-correlation function detailed in the appendix) is given as: [EQUATION] where [MATH] is the cross power spectrum of the potential field and the galaxy density field, [MATH] is the galaxy bias parameter at redshift [MATH], and [MATH] is the normalised galaxy selection function, where [MATH].', '0809.4488-1-18-6': 'We adopt the small angle approximation in which [MATH], where [MATH] is the comoving distance.', '0809.4488-1-18-7': 'For simplicity, we assume the galaxy bias parameter to be unity.', '0809.4488-1-18-8': 'In angular space, the cross-correlation becomes: [EQUATION] where [MATH] are Legendre polynomials.', '0809.4488-1-18-9': 'In actual measurements of CMB fluctuations, the monopole and dipole are subtracted.', '0809.4488-1-18-10': 'Therefore, we set the power at [MATH] and [MATH] to zero before converting the signal into real space.', '0809.4488-1-18-11': 'To ensure that the results at smaller angles ([MATH] degree) converge accurately, we sum the power up to [MATH].', '0809.4488-1-19-0': 'The cross-correlation results are shown in Fig. [REF].', '0809.4488-1-19-1': 'The contribution from the non-linear RS effect can be seen to become increasingly important as the redshift of the sample increases.', '0809.4488-1-19-2': 'Even at the lowest redshift shown in the left-hand panel, the total cross-correlation signal begins to deviate from the linear theory predictions at [MATH] less than 20.', '0809.4488-1-19-3': 'At higher redshifts, the difference between linear theory and the full calculation becomes as much as a factor of two.', '0809.4488-1-19-4': 'In angular space, on the right-hand panel, the RS effect is seen to dominates at scales smaller than one degree and it is still significant even at a few degrees.', '0809.4488-1-19-5': 'At high redshifts, the RS effect can be the dominant contribution on intermediate scales.', '0809.4488-1-20-0': 'The statistical significance of current measurements of the CMB-LSS cross-correlation is not yet high enough to detect the effects we are discussing.', '0809.4488-1-20-1': 'This is illustrated in Fig. [REF] where we compare actual data with predictions of the ISW effect on its own (red lines) and the combined ISW+RS effects (blue lines).', '0809.4488-1-20-2': 'The left-hand panel shows the cross-correlation of the power spectra of WMAP3 and the SDSS4 LRG sample of [CITATION].', '0809.4488-1-20-3': 'The mean redshift of the SDSS4 LRG sample is [MATH].', '0809.4488-1-20-4': 'According to [CITATION], the best fit value of [MATH] for these data is 0.83.', '0809.4488-1-20-5': 'The red lines show our reproduction of their best fit model generated from linear theory assuming their best fit cosmological parameters.', '0809.4488-1-20-6': 'The right-hand panel shows the cross-correlation of WMAP3 and the SDSS4 quasar sample given by [CITATION].', '0809.4488-1-20-7': 'Since our simulation assumes [MATH], we do not have a precise estimate of the ISW+RS for the best fit model of [CITATION].', '0809.4488-1-20-8': 'However, for illustration purposes, the blue lines in the figure show the sum of the RS contribution derived from our simulation with the ISW amplitude appropriate for [MATH] in the left-hand panel and for [MATH] in the right-hand panel.', '0809.4488-1-20-9': 'In the examples of Fig. [REF], we see that the result of the full calculation begins to deviate from the linear theory prediction at a few degrees.', '0809.4488-1-20-10': 'However, these data are too noisy to detect this effect.', '0809.4488-1-21-0': '# Conclusions', '0809.4488-1-22-0': 'We have used an N-body simulation to calculate the non-linear (Rees-Sciama) contribution to the Integrated Sachs Wolfe effect.', '0809.4488-1-22-1': 'The comparison of the 3-D and 2-D power spectra measured from the simulation with those given by linear theory reveals a strong discrepancy whose magnitude and physical scale increase with redshift.', '0809.4488-1-22-2': 'We investigated the strength of this effect on the cross-correlation of the CMB map with galaxy samples in terms of angular power spectra and in angular coordinates at different redshifts.', '0809.4488-1-22-3': 'We conclude that the non-linear contribution dominates the cross-correlation signal at scales smaller than one degree, and makes a significant contribution at a scale of a few degrees.', '0809.4488-1-22-4': 'The non-linear effect alters not only the amplitude, but also the shape of the cross-correlation power spectrum.', '0809.4488-1-22-5': 'Since it is also a positive signal, ignoring this effect leads to an overestimation of the dark energy content of the universe.', '0809.4488-1-23-0': 'With current galaxy samples which cover relatively small volumes, it is not yet possible to disentangle the contribution of the Rees-Sciama effect from that of the ISW effect within the noise.', '0809.4488-1-23-1': 'We illustrated this by calculating the cross-correlation of the WMAP3 fluctuations with those in the SDSS4 LRG and in the SDSS4 quasar samples.', '0809.4488-1-23-2': 'However, in future surveys like Pan-STARRS and LSST, for which the number of galaxies and the sky coverage will increase dramatically, the error bars of the cross-correlation will be much smaller.', '0809.4488-1-23-3': 'In this case, the importance of the Rees-Sciama effect will become significant, especially for high redshift samples.', '0809.4488-1-23-4': 'The dominant contribution of the non-linear cross-correlation at scales of arc-minutes would tend to cancel the negative SZ-CMB cross-correlation signal .', '0809.4488-1-24-0': 'Our analysis is based on a simulation that assumes a [MATH]CDM cosmology.', '0809.4488-1-24-1': 'The non-linear contribution depends on the values of the cosmological parameters.', '0809.4488-1-24-2': 'In a flat universe with a cosmological constant, the RS effect will become increasingly dominant relative to the ISW effect as the value of [MATH] decreases.', '0809.4488-1-24-3': 'In the most extreme case, if [MATH], the ISW effect will vanish, leaving only the RS effect .', '0809.4488-1-24-4': 'The analysis of this paper could be generalized either using re-normalised perturbation theory , or simulations with different dark energy models.', '0809.4488-1-24-5': 'In any case, more general modelling of the non-linear effect will be required for an accurate interpretation of future measurements of the LSS-CMB cross-correlation.', '0809.4488-1-25-0': '# ACKNOWLEDGEMENT', '0809.4488-1-26-0': 'The Millennium Run simulation used in this paper was carried out by the Virgo Consortium at the Computing Centre of the Max-Planck Society in Garching.', '0809.4488-1-26-1': 'YC is supported by the Marie Curie Early Stage Training Host Fellowship ICCIPPP, which is funded by the European Commission.', '0809.4488-1-26-2': 'We thank Raul Angulo for providing the L-BASICC simulation, which was carried out on the Cosmology Machine at Durham, and for useful discussions.', '0809.4488-1-26-3': 'CSF acknowledges a Royal-Society Wolfson Research Merit Award.', '0809.4488-1-27-0': '# Angular Power Spectra', '0809.4488-1-28-0': 'Here we derive the relationship between the 3-D power spectrum of gravitational potential fluctuations, [MATH] and the resulting angular power spectrum of the induced CMB temperature fluctuations.', '0809.4488-1-28-1': 'Expanding the pattern of temperature fluctuations, [MATH], in terms of spherical harmonics we have [EQUATION] which using equation ([REF]) becomes [EQUATION]', '0809.4488-1-28-2': 'Writing [MATH] in terms of a Fourier expansion and using the spherical harmonic expansion of a plane wave, [MATH] this becomes [EQUATION]', '0809.4488-1-28-3': 'Hence the angular power spectrum, [MATH], is given by [EQUATION]', '0809.4488-1-28-4': 'Using the identity [MATH] and the orthogonality relationship of spherical harmonics [EQUATION] this becomes [EQUATION]', '0809.4488-1-28-5': ""This exact relationship can be simplified by using Limber's approximation."", '0809.4488-1-28-6': 'For small angular separations, [MATH], at comoving distance, [MATH], the wave number, [MATH], can expressed in terms its components parallel and perpendicular to the line of sight and approximated by [MATH], where [MATH], namely, the power is dominated by that perpendicular to the line of sight and there is not any correlation between different shells of [MATH] along the line of sight.', '0809.4488-1-28-7': 'Combining this with the orthogonality relation for spherical Bessel functions, [EQUATION] we arrive at [EQUATION]'}","{'0809.4488-2-0-0': 'In a universe with a cosmological constant, the large-scale gravitational potential varies in time and this is, in principle, observable.', '0809.4488-2-0-1': 'Using an N-body simulation of a [MATH]CDM universe, we show that linear theory is not sufficiently accurate to predict the power spectrum of the time derivative, [MATH], needed to compute the imprint of large-scale structure on the cosmic microwave background (CMB).', '0809.4488-2-0-2': 'The linear part of the [MATH] power spectrum (the integrated Sachs-Wolfe effect or ISW) drops quickly as the relative importance of [MATH] diminishes at high redshift, while the non-linear part (the Rees-Sciama effect) evolves more slowly with redshift.', '0809.4488-2-0-3': 'Therefore, the deviation of the total power spectrum from linear theory occurs at larger scales at higher redshifts.', '0809.4488-2-0-4': 'The deviation occurs at [MATH] Mpc[MATH] at [MATH].', '0809.4488-2-0-5': 'The cross-correlation power spectrum of the density [MATH] and [MATH] behaves differently to the power spectrum of [MATH].', '0809.4488-2-0-6': 'Firstly, the deviation from linear theory occurs at smaller scales ([MATH] Mpc[MATH] at [MATH]).', '0809.4488-2-0-7': 'Secondly, the correlation becomes negative when the non-linear effect dominates.', '0809.4488-2-0-8': 'For the cross-correlation power spectrum of galaxy samples with the CMB, the non-linear effect becomes significant at [MATH] and rapidly makes the cross power spectrum negative.', '0809.4488-2-0-9': 'For high redshift samples, the cross-correlation is expected to be suppressed by [MATH] on arc-minute scales.', '0809.4488-2-0-10': 'The Rees-Sciama effect makes a negligible contribution to the large-scale ISW cross-correlation measurement.', '0809.4488-2-0-11': ""However, on arc-minute scales it will contaminate the expected cross-correlation signal induced by the Sunyaev-Zel'dovich effect."", '0809.4488-2-1-0': '# Introduction', '0809.4488-2-2-0': 'The most intriguing topic in contemporary cosmology is the nature of the dark energy which appears to dominate the energy density of the Universe at late times.', '0809.4488-2-2-1': 'Strong evidence for the existence of dark energy comes from both the combined analysis of the cosmic microwave background radiation (CMB) and the galaxy large-scale structure (LSS) , and from high redshift type Ia supernovae .', '0809.4488-2-2-2': 'Both of these techniques infer the presence of dark energy from geometrical measures.', '0809.4488-2-2-3': 'A complementary probe of dark energy is provided by techniques that measure the dynamical effect of dark energy through its influence on the rate of growth of structure.', '0809.4488-2-2-4': 'Large deep galaxy redshift surveys (like the EUCLID, the ESA Mission to Map the Dark Universe, and the JDEM, the Joint Dark Energy Mission) is being planned.', '0809.4488-2-2-5': 'They will exploit the redshift space anisotropy of galaxy clustering, caused by coherent flows into overdense regions and outflows from underdense regions, to measure directly the growth rate as a function of redshift.', '0809.4488-2-2-6': 'The Integrated Sachs-Wolfe (ISW) effect in which the decay of the large-scale potential fluctuations induces CMB temperature perturbations provides another measure of the dynamical effect of dark energy.', '0809.4488-2-3-0': 'In principle, the ISW effect could be detected directly in the CMB power spectrum at very low multiples.', '0809.4488-2-3-1': 'In the [MATH]CDM cosmology, it would boost the plateau in the power spectrum at [MATH].', '0809.4488-2-3-2': 'However, as the increase of the power is not large in comparison to the cosmic variance, it cannot be unambiguously detected even by the WMAP data .', '0809.4488-2-3-3': 'A more sensitive technique is to search for the ISW signal in the cross-correlation of the LSS with the CMB.', '0809.4488-2-3-4': 'As the expected signal is weak and occurs on large scales, a very large galaxy survey is needed to trace the LSS.', '0809.4488-2-3-5': 'Currently individual detections based on surveys such as APM, 2MASS, NVSS and SDSS are not of very high statistical significance and precise measurements await the construction of new, larger surveys (BOSS, Pan-STARRS1 ).', '0809.4488-2-3-6': 'If such surveys are to place robust, meaningful constraints on the properties of the dark energy it is important to take full account of other processes beyond the (linear) ISW effect that may contribute to the cross-correlation signal.', '0809.4488-2-3-7': 'Here, we focus on deviations caused by non-linear gravitational evolution, the Rees-Sciama effect .', '0809.4488-2-4-0': 'Other processes are known to contribute to the cross-correlation signal.', '0809.4488-2-4-1': ""First, the thermal Sunyaev-Zel'dovich (SZ) effect caused by hot ionized gas in galaxy clusters induces an anti-cross-correlation signal which can cancel the ISW effect on small scales."", '0809.4488-2-4-2': 'Its statistical contribution can be modelled and subtracted given the value of [MATH] (the [MATH] linear mass fluctuations within a sphere of 8 [MATH] Mpc) which determines the abundance of galaxy clusters .', '0809.4488-2-4-3': 'Also, since the thermal SZ effect is frequency dependent, it can be subtracted in frequency space given sufficient spectral coverage.', '0809.4488-2-4-4': 'Second, the redshift dependence of galaxy bias, if not properly taken into account, can introduce systematic effects in the determination of dark energy parameters.', '0809.4488-2-4-5': 'Other effects such as lensing magnification and the Doppler redshift effect can also boost the cross-correlation signal, but are only important at high redshift .', '0809.4488-2-4-6': 'These effects are well documented and can be calibrated and removed.', '0809.4488-2-5-0': 'In this paper, we will solely explore the contribution of the non-linear terms, or the Rees-Sciama (RS) effect, on the cross-correlation signal.', '0809.4488-2-5-1': 'The RS effect arises from the non-linear evolution of the potential .', '0809.4488-2-5-2': 'It is believed to be much smaller than the CMB signal at all scales .', '0809.4488-2-5-3': 'Indeed, compared with the CMB power spectrum, the RS effect is orders of magnitude lower.', '0809.4488-2-5-4': 'Also, compared with the complete integrated ISW power spectrum, the RS effect has been shown, using the halo model approach , to be unimportant at [MATH].', '0809.4488-2-5-5': 'However, the RS effect has not been taken into account in cross-correlation analyses and it is important to assess its importance ahead of the completion of the next generation of large deep galaxy surveys.', '0809.4488-2-6-0': 'We use a large N-body simulation to investigate the effect of the non-linear contribution on the interpretation of the ISW cross-correlation signal.', '0809.4488-2-6-1': 'We use the [MATH]-particle L-BASICC simulation described by [CITATION] which, with a box size of 1340 [MATH] Mpc, is ideal for this purpose because not only does it enable us to extrapolate our analysis to non-linear scales at different redshifts, but it includes the very large scale power necessary to check the agreement with linear theory.', '0809.4488-2-6-2': 'The cosmology adopted in the L-BASICC simulation is [MATH]CDM, with [MATH], [MATH], [MATH], [MATH] and [MATH] km s[MATH] Mpc[MATH].', '0809.4488-2-7-0': 'The paper is organised as follows.', '0809.4488-2-7-1': 'In 2, we compute the power spectrum of the ISW plus RS effects from our simulation and compare them with linear theory.', '0809.4488-2-7-2': 'In 3, we analyse these two effects in terms of the cross-correlation of the LSS with the CMB.', '0809.4488-2-7-3': 'Finally, in 3, we discuss our results and present our conclusions.', '0809.4488-2-8-0': '# Time derivative of the potential', '0809.4488-2-9-0': 'The integrated Sachs-Wolfe effect results from the late time decay of gravitational potential fluctuations.', '0809.4488-2-9-1': 'The net blueshift or redshift of the CMB photons, caused by the change in the potential during the passage of the photons, induces net temperature fluctuations of the black body spectrum, [EQUATION] where [MATH] is the time derivative of the gravitational potential, [MATH] is the lookback time, with [MATH] at the present and [MATH] at the last scattering surface.', '0809.4488-2-9-2': 'The angular power spectrum of these temperature fluctuations (see the Appendix) is given by [EQUATION] where [MATH] is the comoving distance to lookback time, [MATH], [MATH] is the spherical Bessel function and [MATH] is the 3-D power spectrum of [MATH] fluctuations.', '0809.4488-2-9-3': ""To derive the the final expression we have used Limber's approximation by assuming [MATH] ."", '0809.4488-2-10-0': 'The ISW effect consists of the temperature fluctuations described by these equations when linear theory is used to compute [MATH] and its fluctuation power spectrum [MATH].', '0809.4488-2-10-1': 'Using a simulation to determine the non-linear contributions we can quantify the full ISW plus Rees-Sciama effect.', '0809.4488-2-10-2': 'In Fourier space, the time derivative of the gravitational potential can be expressed as: [EQUATION] where [MATH] is the expansion factor, [MATH] is the Hubble constant, [MATH] is the present mass density parameter and [MATH] is the time derivative of the density fluctuation.', '0809.4488-2-10-3': 'Combining this with the Fourier space form of the continuity equation, [MATH] gives: [EQUATION] where [MATH] is the momentum density field in Fourier space divided by the mean mass density.', '0809.4488-2-10-4': 'This enables us to estimate the Fourier transform of the [MATH] field of the simulation from the Fourier transforms of the density and momentum fields.', '0809.4488-2-10-5': 'Using equation ([REF]), the resulting power spectrum, [MATH], can be written as [EQUATION]', '0809.4488-2-10-6': 'In linear theory, [MATH] and [MATH], where [MATH] is the linear density power spectrum at the present time and [MATH] is the growth factor normalised to be unity at present.', '0809.4488-2-10-7': 'Therefore, the power spectrum of the linear ISW effect is [EQUATION] where [MATH].', '0809.4488-2-10-8': 'For easy comparison at different redshifts in the simulation, we defined a scaled [MATH] power spectrum, [MATH] which from Eq. (5) is simply [EQUATION]', '0809.4488-2-10-9': 'Our measurements of the [MATH] power spectrum are shown in Fig. [REF].', '0809.4488-2-10-10': 'The results from linear theory are also plotted.', '0809.4488-2-10-11': 'We find the total scaled [MATH] power spectrum can be well fitted by a broken power law plus the linear scaled [MATH] power spectrum [MATH], where [EQUATION]', '0809.4488-2-10-12': 'Here A and B are two free parameters that we use to fit the model to the simulation results at each redshift up to [MATH].', '0809.4488-2-10-13': 'To interpolate the model to intermediate redshifts we linearly interpolate the values of A and B from the nearest two simulation outputs.', '0809.4488-2-10-14': 'Our model is compared to the simulation results in Fig. [REF].', '0809.4488-2-11-0': 'We see in Fig. [REF] that the linear theory reproduces [MATH] at [MATH] only at [MATH] Mpc[MATH].', '0809.4488-2-11-1': 'It fails at progressively larger scales as the redshift increases.', '0809.4488-2-11-2': 'At [MATH], the ISW+RS power spectrum deviates from linear theory at [MATH] Mpc[MATH]; by [MATH], linear theory agrees with the simulation results only at [MATH] Mpc[MATH].', '0809.4488-2-11-3': 'The reason for this surprising behaviour is that the linear part of the [MATH] drops quickly to zero as the relative importance of [MATH] diminishes at high redshift, while the non-linear part evolves more slowly with redshift.', '0809.4488-2-11-4': 'Therefore, the deviation of the total power spectrum from linear theory happens at larger scales at higher redshifts.', '0809.4488-2-11-5': 'We find that the momentum power spectrum, [MATH], and the correlation power spectrum of the density and momentum, [MATH], behave similarly to the [MATH] power spectrum, namely, their deviation from linear theory occurs at larger scales at higher redshift.', '0809.4488-2-11-6': 'This is in contrast with the power spectrum of the density field which only deviates from linear theory at low redshift and on small scales.', '0809.4488-2-11-7': 'In another words, at the same redshift, the deviation from linear theory occurs at smaller scales for the density field than for the other fields.', '0809.4488-2-12-0': 'The sharp increase of [MATH] measured from the simulation at small scales ([MATH] Mpc[MATH]) is due to discreteness in the [MATH] particle L-BASICC simulation.', '0809.4488-2-12-1': 'We used the much higher resolution [MATH] particle Millennium simulation to verify that our model remains accurate at smaller scales and is robust to shot noise corrections.', '0809.4488-2-13-0': 'We can now compute the induced angular power spectrum of CMB temperature fluctuations by performing the integral in equation (2) over the redshift range [MATH] using our model for the 3-D power spectrum, [MATH].', '0809.4488-2-13-1': 'The overall result is shown in Fig. [REF] along with the contributions coming from different redshift intervals.', '0809.4488-2-13-2': 'For the overall angular power spectrum the deviation of the model from the linear theory happens at [MATH]100.', '0809.4488-2-13-3': 'This result confirms the prediction of [CITATION] based on the halo model.', '0809.4488-2-13-4': 'However, we also see that the failure of linear theory, as judged by our simulation results, occurs at smaller and smaller [MATH] as redshift increases.', '0809.4488-2-13-5': 'For example, above [MATH], the deviation occurs at [MATH] and, for larger values of [MATH] than this, linear theory becomes extremely inaccurate.', '0809.4488-2-14-0': 'In order to evaluate how the breakdown of linear theory depends on redshift, we plot the evolution of the [MATH] power at a given scale as a function of redshift in Fig. [REF].', '0809.4488-2-14-1': 'Generally, the deviations of linear theory from the simulation results decrease with scale and increase with redshift.', '0809.4488-2-14-2': 'At [MATH] Mpc[MATH], deviations start to be seen at [MATH] and, at [MATH] Mpc[MATH], linear theory has become inaccurate at all redshifts.', '0809.4488-2-14-3': 'In the right-hand panel, which shows results in [MATH] space, we find no deviations up to [MATH], but for [MATH], linear theory has clearly broken down at all redshifts.', '0809.4488-2-14-4': 'Interestingly, at high redshift, the [MATH] power in the simulation appears to be independent of [MATH] while, in linear theory, this quantity drops monotonically with [MATH].', '0809.4488-2-15-0': '# the LSS-CMB cross-correlation', '0809.4488-2-16-0': 'To illustrate the contribution of the Rees-Sciama effect to the cross-correlation of the density field [MATH] with [MATH], we can compute the 3-D cross-correlation power spectrum [MATH] from our simulations: [EQUATION]', '0809.4488-2-16-1': 'In linear theory, [MATH], where [MATH] is the linear density power spectrum at present time.', '0809.4488-2-16-2': 'Results from simulations are shown in Fig. [REF].', '0809.4488-2-16-3': 'Comparing with linear theory, we find the non-linear contribution appears at somewhat smaller scales than that of the auto correlation power spectrum of [MATH].', '0809.4488-2-16-4': 'At [MATH], the deviation occurs at [MATH] Mpc[MATH].', '0809.4488-2-16-5': 'However, it then dominates rapidly, making the cross-correlation power spectrum negative.', '0809.4488-2-16-6': 'This indicates that once the non-linear effect dominates, the potential of overdense regions evolves faster than the expansion of the universe and gets deeper, bringing a net redshift to the CMB photon passing through it.', '0809.4488-2-16-7': 'This induces a negative cross-correlation for the CMB and LSS.', '0809.4488-2-16-8': 'The effect is the same for not fully-developed voids.', '0809.4488-2-16-9': 'To quantify the effect on the large scale ISW cross-correlation measurements, we model the 3-D cross power spectra from simulations at each redshift outputs.', '0809.4488-2-16-10': 'Interpolating to intermediate redshifts we are then able to calculate the projected 2-D power spectra.', '0809.4488-2-17-0': 'The cross-correlation between LSS and CMB maps has been shown to be a powerful tool for verifying the existence of dark energy and constraining its properties.', '0809.4488-2-17-1': 'Current measurements of the cross-correlation have low statistical significance because the volumes probed by LSS surveys are relatively small, but this situation will improve greatly with upcoming surveys.', '0809.4488-2-17-2': 'For example, Pan-STARRS1 will survey three quarters of the sky, obtaining photometry for galaxies up to [MATH] mag in the [MATH]-band.', '0809.4488-2-17-3': 'The mean galaxy redshift in this ""[MATH] survey"" will be [MATH].', '0809.4488-2-17-4': 'Pan-STARRS1 will also carry out a deeper but smaller ""MDS"" survey covering 84 sq deg of the sky to [MATH] mag in [MATH] for which [MATH] .', '0809.4488-2-17-5': 'Cross-correlating such photometric redshift galaxy samples with a CMB map (from WMAP or Planck) will make it possible for the first time to perform ISW tomography.', '0809.4488-2-17-6': 'Galaxy samples would be divided into different redshift slices and each one cross-correlated with the CMB map.', '0809.4488-2-17-7': 'Values of the dark energy equation of state parameter, [MATH], could then be measured using the results from the different redshift slices, effectively constraining the evolution of [MATH].', '0809.4488-2-18-0': 'To illustrate how ISW tomography may work, we follow [CITATION] and model the redshift distribution of galaxies tracing the LSS as [EQUATION] but then choose the parameters [MATH] and [MATH] to emulate plausible photometric redshift slices.', '0809.4488-2-18-1': '(The same functional form was also taken by [CITATION] to model the SDSS LRG sample.)', '0809.4488-2-18-2': 'We assume [MATH] so that the width of [MATH] is much greater than the expected photometric redshift errors.', '0809.4488-2-18-3': 'We shift the function into different redshift intervals by using [MATH] and [MATH].', '0809.4488-2-18-4': 'The median redshift of these samples is [MATH] and [MATH] respectively.', '0809.4488-2-18-5': 'The cross-correlation power spectrum (derived in an analogous way to the auto-correlation function detailed in the appendix) is given as: [EQUATION] where [MATH] is the cross power spectrum of the potential field and the galaxy density field, [MATH] is the galaxy bias parameter at redshift [MATH], and [MATH] is the normalised galaxy selection function, where [MATH].', '0809.4488-2-18-6': 'We adopt the small angle approximation in which [MATH], where [MATH] is the comoving distance.', '0809.4488-2-18-7': 'For simplicity, in this illustrations, we assume the galaxy bias parameter to be unity.', '0809.4488-2-18-8': 'In angular space, the cross-correlation becomes: [EQUATION] where [MATH] are Legendre polynomials.', '0809.4488-2-18-9': 'In actual measurements of CMB fluctuations, the monopole and dipole are subtracted.', '0809.4488-2-18-10': 'Therefore, we set the power at [MATH] and [MATH] to zero before converting the signal into real space.', '0809.4488-2-18-11': 'To ensure that the results at smaller angles ([MATH] degree) converge accurately, we sum the power up to [MATH].', '0809.4488-2-19-0': 'The cross-correlation results are shown in Fig. [REF].', '0809.4488-2-19-1': 'The contribution from the non-linear RS effect can be seen to become increasingly important as the redshift of the sample increases.', '0809.4488-2-19-2': 'The cross-correlation power spectra decrease and deviate from linear theory rapidly at [MATH] due to the non-linear effect.', '0809.4488-2-19-3': 'They turn negative at [MATH].', '0809.4488-2-19-4': 'This results is consistent with that given by [CITATION], who used similar method for illustrating the ISW and Rees-Sciama effect on the CMB-Week lensing cross-correlation.', '0809.4488-2-19-5': 'In angular space, on the right-hand panel, the RS effect is negligible at [MATH] degree for all redshift samples.', '0809.4488-2-19-6': 'It appears at sub-degree scales at the high samples.', '0809.4488-2-19-7': 'It suppresses the cross-correlation power spectrum by about [MATH] at acrminute scales.', '0809.4488-2-20-0': 'The statistical significance of current measurements of the CMB-LSS cross-correlation is not yet high enough to detect the effects we are discussing.', '0809.4488-2-20-1': 'Most current measurements can only measure the cross-correlation at degree scales or above .', '0809.4488-2-20-2': 'Future CMB or SZ surveys with high resolution might be able to resolve this contribution.', '0809.4488-2-21-0': '# Conclusions', '0809.4488-2-22-0': 'We have used an N-body simulation to calculate the non-linear (Rees-Sciama) contribution to the Integrated Sachs Wolfe effect.', '0809.4488-2-22-1': 'The comparison of the 3-D and 2-D power spectra measured from the simulation with those given by linear theory reveals a strong nonlinear contribution whose magnitude and physical scale increase with redshift.', '0809.4488-2-22-2': 'We investigated the strength of this effect on the cross-correlation of the CMB map with galaxy samples in terms of angular power spectra and in angular coordinates at different redshifts.', '0809.4488-2-22-3': 'We find that the non-linear contribution to the cross-correlation signal at sub-degree scales.', '0809.4488-2-22-4': 'The non-linear effect alters not only the amplitude, but also the shape of the cross-correlation power spectrum.', '0809.4488-2-22-5': 'With current galaxy samples which cover relatively small volumes, it is not yet possible to disentangle the contribution of the Rees-Sciama effect from that of the ISW effect within the noise.', '0809.4488-2-22-6': 'However, in future surveys like Pan-STARRS and LSST, for which the number of galaxies and the sky coverage will increase dramatically, the error bars of the cross-correlation will be much smaller.', '0809.4488-2-22-7': 'In this case, the importance of the Rees-Sciama effect may become significant for high redshift samples.', '0809.4488-2-22-8': 'The effect of the non-linear cross-correlation at scales of arc-minutes would lead a higher apparent SZ-CMB cross-correlation signal .', '0809.4488-2-23-0': 'Our analysis is based on a simulation that assumes a [MATH]CDM cosmology.', '0809.4488-2-23-1': 'The non-linear contribution depends on the values of the cosmological parameters.', '0809.4488-2-23-2': 'In a flat universe with a cosmological constant, the RS effect will become increasingly dominant relative to the ISW effect as the value of [MATH] decreases.', '0809.4488-2-23-3': 'In the most extreme case, if [MATH], the ISW effect will vanish, leaving only the RS effect .', '0809.4488-2-23-4': 'The analysis of this paper could be generalized either using re-normalised perturbation theory , or simulations with different dark energy models.', '0809.4488-2-23-5': 'In any case, more general modelling of the non-linear effect will be required for an accurate interpretation of future measurements of the LSS-CMB cross-correlation.', '0809.4488-2-24-0': '# ACKNOWLEDGEMENT', '0809.4488-2-25-0': 'The Millennium Run simulation used in this paper was carried out by the Virgo Consortium at the Computing Centre of the Max-Planck Society in Garching.', '0809.4488-2-25-1': 'YC is supported by the Marie Curie Early Stage Training Host Fellowship ICCIPPP, which is funded by the European Commission.', '0809.4488-2-25-2': 'We thank Raul Angulo for providing the L-BASICC simulation, which was carried out on the Cosmology Machine at Durham, and for useful discussions.', '0809.4488-2-25-3': 'We also thank Anthony Challinor, Enrique Gaztanaga and Uros Seljak for comments that allowed us to identify an error in an earlier version of this paper.', '0809.4488-2-25-4': 'CSF acknowledges a Royal-Society Wolfson Research Merit Award.', '0809.4488-2-26-0': '# Angular Power Spectra', '0809.4488-2-27-0': 'Here we derive the relationship between the 3-D power spectrum of gravitational potential fluctuations, [MATH] and the resulting angular power spectrum of the induced CMB temperature fluctuations.', '0809.4488-2-27-1': 'Expanding the pattern of temperature fluctuations, [MATH], in terms of spherical harmonics we have [EQUATION] which using equation ([REF]) becomes [EQUATION]', '0809.4488-2-27-2': 'Writing [MATH] in terms of a Fourier expansion and using the spherical harmonic expansion of a plane wave, [MATH] this becomes [EQUATION]', '0809.4488-2-27-3': 'Hence the angular power spectrum, [MATH], is given by [EQUATION]', '0809.4488-2-27-4': 'Using the identity [MATH] and the orthogonality relationship of spherical harmonics [EQUATION] this becomes [EQUATION]', '0809.4488-2-27-5': ""This exact relationship can be simplified by using Limber's approximation."", '0809.4488-2-27-6': 'For small angular separations, [MATH], at comoving distance, [MATH], the wave number, [MATH], can expressed in terms its components parallel and perpendicular to the line of sight and approximated by [MATH], where [MATH], namely, the power is dominated by that perpendicular to the line of sight and there is not any correlation between different shells of [MATH] along the line of sight.', '0809.4488-2-27-7': 'Combining this with the orthogonality relation for spherical Bessel functions, [EQUATION] we arrive at [EQUATION]'}","[['0809.4488-1-12-0', '0809.4488-2-12-0'], ['0809.4488-1-12-1', '0809.4488-2-12-1'], ['0809.4488-1-7-0', '0809.4488-2-7-0'], ['0809.4488-1-7-1', '0809.4488-2-7-1'], ['0809.4488-1-7-2', '0809.4488-2-7-2'], ['0809.4488-1-7-3', '0809.4488-2-7-3'], ['0809.4488-1-28-0', '0809.4488-2-27-0'], ['0809.4488-1-28-1', '0809.4488-2-27-1'], ['0809.4488-1-28-2', '0809.4488-2-27-2'], ['0809.4488-1-28-3', '0809.4488-2-27-3'], ['0809.4488-1-28-4', '0809.4488-2-27-4'], ['0809.4488-1-28-5', '0809.4488-2-27-5'], ['0809.4488-1-28-6', '0809.4488-2-27-6'], ['0809.4488-1-28-7', '0809.4488-2-27-7'], ['0809.4488-1-19-0', '0809.4488-2-19-0'], ['0809.4488-1-19-1', 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'0809.4488-2-10-3'], ['0809.4488-1-10-4', '0809.4488-2-10-4'], ['0809.4488-1-10-5', '0809.4488-2-10-5'], ['0809.4488-1-10-6', '0809.4488-2-10-6'], ['0809.4488-1-10-7', '0809.4488-2-10-7'], ['0809.4488-1-10-8', '0809.4488-2-10-8'], ['0809.4488-1-10-9', '0809.4488-2-10-9'], ['0809.4488-1-10-10', '0809.4488-2-10-10'], ['0809.4488-1-10-11', '0809.4488-2-10-11'], ['0809.4488-1-10-12', '0809.4488-2-10-12'], ['0809.4488-1-10-13', '0809.4488-2-10-13'], ['0809.4488-1-10-14', '0809.4488-2-10-14'], ['0809.4488-1-17-0', '0809.4488-2-17-0'], ['0809.4488-1-17-1', '0809.4488-2-17-1'], ['0809.4488-1-17-2', '0809.4488-2-17-2'], ['0809.4488-1-17-3', '0809.4488-2-17-3'], ['0809.4488-1-17-4', '0809.4488-2-17-4'], ['0809.4488-1-17-5', '0809.4488-2-17-5'], ['0809.4488-1-17-6', '0809.4488-2-17-6'], ['0809.4488-1-17-7', '0809.4488-2-17-7'], ['0809.4488-1-24-0', '0809.4488-2-23-0'], ['0809.4488-1-24-1', '0809.4488-2-23-1'], ['0809.4488-1-24-2', '0809.4488-2-23-2'], ['0809.4488-1-24-3', '0809.4488-2-23-3'], 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'0809.4488-3-11-2'], ['0809.4488-2-11-2', '0809.4488-3-11-6'], ['0809.4488-2-11-6', '0809.4488-3-11-6'], ['0809.4488-2-3-5', '0809.4488-3-3-6'], ['0809.4488-1-22-3', '0809.4488-2-22-3'], ['0809.4488-1-23-4', '0809.4488-2-22-3'], ['0809.4488-1-23-4', '0809.4488-2-22-8']]",[],[],"{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/0809.4488,"{'0809.4488-3-0-0': 'In a universe with a cosmological constant, the large-scale gravitational potential varies in time and this is, in principle, observable.', '0809.4488-3-0-1': 'Using an N-body simulation of a [MATH]CDM universe, we show that linear theory is not sufficiently accurate to predict the power spectrum of the time derivative, [MATH], needed to compute the imprint of large-scale structure on the cosmic microwave background (CMB).', '0809.4488-3-0-2': 'The linear part of the [MATH] power spectrum (the integrated Sachs-Wolfe effect or ISW) drops quickly as the relative importance of [MATH] diminishes at high redshift, while the non-linear part (the Rees-Sciama effect or RS) evolves more slowly with redshift.', '0809.4488-3-0-3': 'Therefore, the deviation of the total power spectrum from linear theory occurs at larger scales at higher redshifts.', '0809.4488-3-0-4': 'The deviation occurs at [MATH] Mpc[MATH] at [MATH].', '0809.4488-3-0-5': 'The cross-correlation power spectrum of the density [MATH] with [MATH] behaves differently to the power spectrum of [MATH].', '0809.4488-3-0-6': 'Firstly, the deviation from linear theory occurs at smaller scales ([MATH] Mpc[MATH] at [MATH]).', '0809.4488-3-0-7': 'Secondly, the correlation becomes negative when the non-linear effect dominates.', '0809.4488-3-0-8': 'For the cross-correlation power spectrum of galaxy samples with the CMB, the non-linear effect becomes significant at [MATH] and rapidly makes the cross power spectrum negative.', '0809.4488-3-0-9': 'For high redshift samples, the cross-correlation is expected to be suppressed by [MATH] on arcminute scales.', '0809.4488-3-0-10': 'The RS effect makes a negligible contribution to the large-scale ISW cross-correlation measurement.', '0809.4488-3-0-11': ""However, on arc-minute scales it will contaminate the expected cross-correlation signal induced by the Sunyaev-Zel'dovich effect."", '0809.4488-3-1-0': '# Introduction', '0809.4488-3-2-0': 'The most intriguing topic in contemporary cosmology is the nature of the dark energy which appears to dominate the energy density of the Universe at late times.', '0809.4488-3-2-1': 'Strong evidence for the existence of dark energy comes from both the combined analysis of the cosmic microwave background radiation (CMB) and the galaxy large-scale structure (LSS) , and from high redshift type Ia supernovae .', '0809.4488-3-2-2': 'Both of these techniques infer the presence of dark energy from geometrical measures.', '0809.4488-3-2-3': 'A complementary probe of dark energy is provided by techniques that measure the dynamical effect of dark energy through its influence on the rate of growth of structure.', '0809.4488-3-2-4': 'Large deep galaxy redshift surveys (like the EUCLID, the ESA Mission to Map the Dark Universe, and the JDEM, the Joint Dark Energy Mission) are being planned that will exploit the redshift space anisotropy of galaxy clustering, caused by coherent flows into overdense regions and outflows from underdense regions, to measure directly the growth rate as a function of redshift.', '0809.4488-3-3-0': 'The Integrated Sachs-Wolfe (ISW) effect , in which the decay of the large-scale potential fluctuations induces CMB temperature perturbations, provides another measure of the dynamical effect of dark energy.', '0809.4488-3-3-1': 'In principle, the ISW effect could be detected directly in the CMB power spectrum at very low multiples.', '0809.4488-3-3-2': 'In the [MATH]CDM cosmology, it would boost the plateau in the power spectrum at [MATH].', '0809.4488-3-3-3': 'However, as the increase of the power is not large in comparison to the cosmic variance, it cannot be unambiguously detected even in the WMAP data .', '0809.4488-3-3-4': 'A more sensitive technique is to search for the ISW signal in the cross-correlation of the LSS with the CMB.', '0809.4488-3-3-5': 'As the expected signal is weak and occurs on large scales, a very large galaxy survey is needed to trace the LSS.', '0809.4488-3-3-6': 'Currently individual detections based on surveys such as APM, 2MASS, NVSS and SDSS are not of very high statistical significance .', '0809.4488-3-3-7': 'There are also analyses of the ISW cross-correlation, which combine multiple galaxy survey samples and achieve [MATH] detection of the ISW effect (Ho et al. 2008; Giannantonio et al. 2008).', '0809.4488-3-3-8': 'These measurements may be the best that can be obtained before the next generation of surveys (BOSS, Pan-STARRS1) come to fruition and make redshift tomography possible.', '0809.4488-3-3-9': 'If such surveys are to place robust, meaningful constraints on the properties of the dark energy it is important to take full account of other processes beyond the (linear) ISW effect that may contribute to the cross-correlation signal.', '0809.4488-3-3-10': 'Here, we focus on deviations caused by non-linear gravitational evolution, the Rees-Sciama effect .', '0809.4488-3-4-0': 'Other processes are known to contribute to the cross-correlation signal.', '0809.4488-3-4-1': ""First, the thermal Sunyaev-Zel'dovich (SZ) effect caused by hot ionized gas in galaxy clusters induces an anti-cross-correlation signal which can cancel the ISW effect on small scales."", '0809.4488-3-4-2': 'Its statistical contribution can be modelled and subtracted given the value of [MATH] (the [MATH] linear mass fluctuations within a sphere of 8 [MATH] Mpc) which determines the abundance of galaxy clusters .', '0809.4488-3-4-3': 'Also, since the thermal SZ effect is frequency dependent, it can be subtracted in frequency space given sufficient spectral coverage.', '0809.4488-3-4-4': 'Second, the redshift dependence of galaxy bias, if not properly taken into account, can introduce systematic effects in the determination of dark energy parameters.', '0809.4488-3-4-5': 'Other effects such as lensing magnification and the Doppler redshift effect can also boost the cross-correlation signal, but are only important at high redshift .', '0809.4488-3-4-6': 'These effects are well documented and can be calibrated and removed.', '0809.4488-3-5-0': 'In this paper we will solely explore the contribution of the non-linear terms, or the Rees-Sciama (RS) effect, on the cross-correlation signal.', '0809.4488-3-5-1': 'The RS effect arises from the non-linear evolution of the potential .', '0809.4488-3-5-2': 'It is believed to be much smaller than the CMB signal at all scales .', '0809.4488-3-5-3': 'Indeed, compared with the CMB power spectrum, the RS effect is orders of magnitude lower.', '0809.4488-3-5-4': 'Also, compared with the complete integrated ISW power spectrum, the RS effect has been shown, using the halo model approach , to be unimportant at [MATH].', '0809.4488-3-5-5': 'However, the RS effect has not been taken into account in cross-correlation analyses and it is important to assess its importance ahead of the completion of the next generation of large deep galaxy surveys.', '0809.4488-3-6-0': 'We use a large N-body simulation to investigate the effect of the non-linear contribution on the interpretation of the ISW cross-correlation signal.', '0809.4488-3-6-1': 'We use the [MATH]-particle L-BASICC simulation described by [CITATION] which, with a box size of 1340 [MATH] Mpc, is ideal for this purpose because not only does it enable us to extrapolate our analysis to non-linear scales at different redshifts, but it includes the very large scale power necessary to check the agreement with linear theory.', '0809.4488-3-6-2': 'The cosmology adopted in the L-BASICC simulation is [MATH]CDM, with [MATH], [MATH], [MATH], [MATH] and [MATH] km s[MATH] Mpc[MATH].', '0809.4488-3-7-0': 'The paper is organised as follows.', '0809.4488-3-7-1': 'In 2, we compute the power spectrum of the ISW plus RS effects from our simulation and compare them with linear theory.', '0809.4488-3-7-2': 'In 3, we analyse these two effects in terms of the cross-correlation of the LSS with the CMB.', '0809.4488-3-7-3': 'Finally, in 4, we discuss our results and present our conclusions.', '0809.4488-3-8-0': '# Time derivative of the potential', '0809.4488-3-9-0': 'The integrated Sachs-Wolfe effect results from the late time decay of gravitational potential fluctuations.', '0809.4488-3-9-1': 'The net blueshift or redshift of the CMB photons caused by the change in the potential during the passage of the photons induces net temperature fluctuations of the black body spectrum, [EQUATION] where [MATH] is the time derivative of the gravitational potential, [MATH] is the lookback time, with [MATH] at the present and [MATH] at the last scattering surface.', '0809.4488-3-9-2': 'The angular power spectrum of these temperature fluctuations (see Appendix A) is given by [EQUATION] where [MATH] is the comoving distance to lookback time, [MATH], [MATH] is the spherical Bessel function and [MATH] is the 3-D power spectrum of [MATH] fluctuations.', '0809.4488-3-9-3': ""To derive the final expression we have used Limber's approximation by assuming [MATH] ."", '0809.4488-3-10-0': 'The ISW effect consists of the temperature fluctuations described by these equations when linear theory is used to compute [MATH] and its fluctuation power spectrum [MATH].', '0809.4488-3-10-1': 'Using a simulation to determine the non-linear contributions we can quantify the full ISW plus Rees-Sciama effect.', '0809.4488-3-10-2': 'In Fourier space, the time derivative of the gravitational potential can be expressed as: [EQUATION] where [MATH] is the expansion factor, [MATH] is the Hubble constant, [MATH] is the present mass density parameter and [MATH] is the time derivative of the density fluctuation.', '0809.4488-3-10-3': 'Combining this with the Fourier space form of the continuity equation, [MATH] gives: [EQUATION] where [MATH] is the momentum density field in Fourier space divided by the mean mass density.', '0809.4488-3-10-4': 'This enables us to estimate the Fourier transform of the [MATH] field of the simulation from the Fourier transforms of the density and momentum fields.', '0809.4488-3-10-5': 'Using equation ([REF]), the resulting power spectrum, [MATH], can be written as [EQUATION]', '0809.4488-3-10-6': 'In linear theory, [MATH] and [MATH], where [MATH] is the linear density power spectrum at the present time and [MATH] is the growth factor normalised to be unity at present.', '0809.4488-3-10-7': 'Therefore, the power spectrum of the linear ISW effect is [EQUATION] where [MATH].', '0809.4488-3-10-8': 'For easy comparison at different redshifts in the simulation, we defined a scaled [MATH] power spectrum, [MATH] which from Eq. (5) is simply [EQUATION]', '0809.4488-3-10-9': 'Our measurements of the [MATH] power spectrum are shown in Fig. [REF].', '0809.4488-3-10-10': 'The results from linear theory are also plotted.', '0809.4488-3-10-11': 'We find the total scaled [MATH] power spectrum can be well fitted by a broken power law plus the linear scaled [MATH] power spectrum [MATH], where [EQUATION]', '0809.4488-3-10-12': 'Here A and B are two free parameters that we use to fit the model to the simulation results at each redshift up to [MATH].', '0809.4488-3-10-13': 'To interpolate the model to intermediate redshifts we linearly interpolate the values of A and B from the nearest two simulation outputs.', '0809.4488-3-10-14': 'Our model is compared to the simulation results in Fig. [REF].', '0809.4488-3-11-0': 'We see in Fig. [REF] that the linear theory reproduces the ISW+RS [MATH] at [MATH] only at [MATH] Mpc[MATH].', '0809.4488-3-11-1': 'It fails at progressively larger scales as the redshift increases.', '0809.4488-3-11-2': 'By [MATH], linear theory agrees with the simulation results only at [MATH] Mpc[MATH].', '0809.4488-3-11-3': 'The reason for this surprising behaviour is that the linear part of the [MATH] drops quickly to zero as the relative importance of [MATH] diminishes at high redshift, while the non-linear part evolves more slowly with redshift.', '0809.4488-3-11-4': 'Therefore, the deviation of the total power spectrum from linear theory happens at larger scales at higher redshifts.', '0809.4488-3-11-5': 'We find that the momentum power spectrum, [MATH], and the correlation power spectrum of the density and momentum, [MATH], behave similarly to the [MATH] power spectrum, namely, their deviation from linear theory occurs at larger scales at higher redshift.', '0809.4488-3-11-6': 'This is in contrast with the power spectrum of the density field which deviates from linear theory on progressively larger scales at lower and lower redshift.', '0809.4488-3-11-7': 'In another words, at the same redshift, the deviation from linear theory occurs at smaller scales for the density field than for the other fields.', '0809.4488-3-12-0': 'The sharp increase of [MATH] measured from the simulation at small scales ([MATH] Mpc[MATH]) is due to discreteness in the [MATH] particle L-BASICC simulation.', '0809.4488-3-12-1': 'We used the much higher resolution [MATH] particle Millennium simulation to verify that our model remains accurate at smaller scales and is robust to shot noise corrections.', '0809.4488-3-13-0': 'We can now compute the induced angular power spectrum of CMB temperature fluctuations by performing the integral in equation (2) over the redshift range [MATH] using our model for the 3-D power spectrum, [MATH].', '0809.4488-3-13-1': 'The overall result is shown in Fig. [REF] along with the contributions coming from different redshift intervals.', '0809.4488-3-13-2': 'For the overall angular power spectrum the deviation of the model from the linear theory happens at [MATH]100.', '0809.4488-3-13-3': 'This result confirms the prediction of [CITATION] based on the halo model.', '0809.4488-3-13-4': 'However, we also see that the failure of linear theory, as judged by our simulation results, occurs at smaller and smaller [MATH] as redshift increases.', '0809.4488-3-13-5': 'For example, above [MATH], the deviation occurs at [MATH] and, for larger values of [MATH] than this, linear theory becomes extremely inaccurate.', '0809.4488-3-14-0': 'In order to evaluate how the breakdown of linear theory depends on redshift, we plot the evolution of the [MATH] power at a given scale as a function of redshift in Fig. [REF].', '0809.4488-3-14-1': 'Generally, the deviations of linear theory from the simulation results decrease with scale and increase with redshift.', '0809.4488-3-14-2': 'At [MATH] Mpc[MATH], deviations start to be seen at [MATH] and, at [MATH] Mpc[MATH], linear theory has become inaccurate at all redshifts.', '0809.4488-3-14-3': 'In the right-hand panel, which shows results in [MATH] space, we find no deviations up to [MATH], but for [MATH], linear theory has clearly broken down at all redshifts.', '0809.4488-3-14-4': 'Interestingly, at high redshift, the [MATH] power in the simulation appears to be independent of [MATH] while, in linear theory, this quantity drops monotonically with [MATH].', '0809.4488-3-15-0': '# the LSS-CMB cross-correlation', '0809.4488-3-16-0': 'To illustrate the contribution of the Rees-Sciama effect to the cross-correlation of the density field [MATH] with [MATH], we can compute the 3-D cross-correlation power spectrum [MATH] from our simulations: [EQUATION]', '0809.4488-3-16-1': 'In linear theory, [MATH], where [MATH] is the linear density power spectrum at the present time.', '0809.4488-3-16-2': 'Results from our simulations are shown in Fig. [REF].', '0809.4488-3-16-3': 'Comparing with linear theory, we find that the non-linear contribution appears at somewhat smaller scales than that of the autocorrelation power spectrum of [MATH].', '0809.4488-3-16-4': 'At [MATH], the deviation occurs at [MATH] Mpc[MATH].', '0809.4488-3-16-5': 'However, it then dominates rapidly, making the cross-correlation power spectrum negative.', '0809.4488-3-16-6': 'This indicates that once the non-linear effect dominates, the potential of overdense regions evolves faster than the expansion of the universe, becoming deeper and thus imparting a net redshift to CMB photons passing through them.', '0809.4488-3-16-7': 'The sense of the effect from underdense regions is reversed, but the effect generated by the overdense regions is dominant and induces a negative cross-correlation between the CMB and the LSS.', '0809.4488-3-17-0': 'To quantify the effect on the large-scale ISW cross-correlation measurements, we model the 3-D cross power spectrum from the simulations at each redshift output.', '0809.4488-3-17-1': 'We use linear theory to model the linear regime but once the cross power spectrum starts to deviate from linear theory, we fit it with a function of the form [MATH], where [MATH] and [MATH] are free parameters.', '0809.4488-3-17-2': 'Interpolating to intermediate redshifts, we are then able to calculate the projected 2-D power spectrum.', '0809.4488-3-18-0': 'The cross-correlation between LSS and CMB maps has been shown to be a powerful tool for verifying the existence of dark energy and constraining its properties.', '0809.4488-3-18-1': 'Current measurements of the cross-correlation have low statistical significance because the volumes probed by LSS surveys are relatively small, but this situation will improve greatly with upcoming surveys.', '0809.4488-3-18-2': 'For example, Pan-STARRS1 will survey three quarters of the sky, obtaining photometry for galaxies up to [MATH] mag in the [MATH]-band.', '0809.4488-3-18-3': 'The mean galaxy redshift in this ""[MATH] survey"" will be [MATH].', '0809.4488-3-18-4': 'Pan-STARRS1 will also carry out a deeper but smaller ""MDS"" survey covering 84 sq deg of the sky to [MATH] mag in [MATH] for which [MATH] .', '0809.4488-3-18-5': 'Cross-correlating such photometric redshift galaxy samples with a CMB map (from WMAP or Planck) will make it possible for the first time to perform ISW tomography.', '0809.4488-3-18-6': 'Galaxy samples would be divided into different redshift slices and each one cross-correlated with the CMB map.', '0809.4488-3-18-7': 'Values of the dark energy equation of state parameter, [MATH], could then be measured using the results from the different redshift slices, effectively constraining the evolution of [MATH].', '0809.4488-3-19-0': 'To illustrate how ISW tomography may work, we follow [CITATION] and model the redshift distribution of galaxies tracing the LSS as [EQUATION] but then choose the parameters [MATH] and [MATH] to emulate plausible photometric redshift slices.', '0809.4488-3-19-1': '(The same functional form was also taken by [CITATION] to model the SDSS LRG sample.)', '0809.4488-3-19-2': 'We assume [MATH] so that the width of [MATH] is much greater than the expected photometric redshift errors.', '0809.4488-3-19-3': 'We shift the function into different redshift intervals by using [MATH] and [MATH].', '0809.4488-3-19-4': 'The median redshift of these samples is [MATH] and [MATH] respectively.', '0809.4488-3-19-5': 'The cross-correlation power spectrum (derived in an analogous way to the auto-correlation function detailed in the appendix) is given as: [EQUATION] where [MATH] is the cross power spectrum of the potential field and the galaxy density field, [MATH] is the galaxy bias parameter at redshift [MATH], and [MATH] is the normalised galaxy selection function, where [MATH].', '0809.4488-3-19-6': 'We adopt the small angle approximation in which [MATH], where [MATH] is the comoving distance.', '0809.4488-3-19-7': 'For simplicity, in this illustration, we assume the galaxy bias parameter to be unity.', '0809.4488-3-19-8': 'In angular space, the cross-correlation becomes: [EQUATION] where [MATH] are Legendre polynomials.', '0809.4488-3-19-9': 'In actual measurements of CMB fluctuations, the monopole and dipole are subtracted.', '0809.4488-3-19-10': 'Therefore, we set the power at [MATH] and [MATH] to zero before converting the signal into real space.', '0809.4488-3-19-11': 'To ensure that the results at smaller angles ([MATH]) converge accurately, we sum the power up to [MATH].', '0809.4488-3-20-0': 'The cross-correlation results are shown in Fig. [REF].', '0809.4488-3-20-1': 'The contribution from the non-linear RS effect can be seen to become increasingly important as the redshift of the sample increases.', '0809.4488-3-20-2': 'The cross-correlation power spectrum decreases and deviates from linear theory rapidly at [MATH] due to the non-linear effect.', '0809.4488-3-20-3': 'It turns negative at [MATH].', '0809.4488-3-20-4': 'This result is consistent with that obtained by [CITATION] who used a similar method to illustrate the impact of the ISW and RS effects on the CMB-weak lensing cross-correlation.', '0809.4488-3-20-5': 'In angular coordinates, shown on the right-hand panel, the RS effect is negligible at [MATH] at all redshifts.', '0809.4488-3-20-6': 'For the high redshift samples, it becomes important at sub-degree scales for the high redshift samples where it suppresses the cross-correlation power spectrum by about [MATH] at acrminute scales.', '0809.4488-3-21-0': 'The statistical significance of current measurements of the CMB-LSS cross-correlation is not yet high enough to detect the effects we are discussing.', '0809.4488-3-21-1': 'Most current measurements can only determine the cross-correlation at degree scales or above .', '0809.4488-3-21-2': 'Future CMB or SZ surveys with high resolution might be able to resolve this contribution.', '0809.4488-3-22-0': '# Conclusions', '0809.4488-3-23-0': 'We have used an N-body simulation to calculate the non-linear (Rees-Sciama) contribution to the Integrated Sachs Wolfe effect.', '0809.4488-3-23-1': 'The comparison of the 3-D and 2-D power spectra measured from the simulation with those given by linear theory reveals a strong nonlinear contribution whose physical scale increase with redshift.', '0809.4488-3-23-2': 'We investigated the strength of this effect on the cross-correlation of the CMB with galaxy samples in terms of angular power spectra and in angular coordinates at different redshifts.', '0809.4488-3-23-3': 'We find that there is a non-linear contribution to the cross-correlation signal at sub-degree scales.', '0809.4488-3-23-4': 'The non-linear effect alters not only the amplitude, but also the shape of the cross-correlation power spectrum.', '0809.4488-3-23-5': 'With current galaxy samples which cover relatively small volumes, it is not yet possible to disentangle the contribution of the RS effect from that of the ISW effect within the noise.', '0809.4488-3-23-6': 'However, in future surveys like Pan-STARRS and LSST, for which the number of galaxies and the sky coverage will increase dramatically, the error bars on the cross-correlation will be much smaller.', '0809.4488-3-23-7': 'In this case, the importance of the Rees-Sciama effect may become significant for high redshift samples.', '0809.4488-3-23-8': 'The effect of the non-linear cross-correlation at scales of arcminutes would contaminate the SZ signal in the CMB and this could confuse its interpretation .', '0809.4488-3-24-0': 'Our analysis is based on a simulation that assumes a [MATH]CDM cosmology.', '0809.4488-3-24-1': 'The non-linear contribution depends on the values of the cosmological parameters.', '0809.4488-3-24-2': 'In a flat universe with a cosmological constant, the RS effect will become increasingly dominant relative to the ISW effect as the value of [MATH] decreases.', '0809.4488-3-24-3': 'In the most extreme case, if [MATH], the ISW effect will vanish, leaving only the RS effect .', '0809.4488-3-24-4': 'The analysis of this paper could be generalized either using re-normalised perturbation theory , or simulations with different dark energy models.', '0809.4488-3-24-5': 'In any case, more general modelling of the non-linear effect will be required for an accurate interpretation of future measurements of the LSS-CMB cross-correlation.', '0809.4488-3-25-0': '# ACKNOWLEDGEMENT', '0809.4488-3-26-0': 'The Millennium simulation used in this paper was carried out by the Virgo Consortium at the Computing Centre of the Max-Planck Society in Garching.', '0809.4488-3-26-1': 'YC is supported by the Marie Curie Early Stage Training Host Fellowship ICCIPPP, which is funded by the European Commission.', '0809.4488-3-26-2': 'We thank Raul Angulo for providing the L-BASICC simulation, which was carried out on the Cosmology Machine at Durham, and for useful discussions.', '0809.4488-3-26-3': 'We also thank Anthony Challinor, Enrique Gaztanaga and Uros Seljak for comments that allowed us to identify an error in an earlier version of this paper.', '0809.4488-3-26-4': 'This work was supported in part by an STFC rolling grant.', '0809.4488-3-26-5': 'CSF acknowledges a Royal-Society Wolfson Research Merit Award.', '0809.4488-3-27-0': '# Angular Power Spectra', '0809.4488-3-28-0': 'Here we derive the relationship between the 3-D power spectrum of gravitational potential fluctuations, [MATH] and the resulting angular power spectrum of the induced CMB temperature fluctuations.', '0809.4488-3-28-1': 'Expanding the pattern of temperature fluctuations, [MATH], in terms of spherical harmonics we have [EQUATION] which using equation ([REF]) becomes [EQUATION]', '0809.4488-3-28-2': 'Writing [MATH] in terms of a Fourier expansion and using the spherical harmonic expansion of a plane wave, [MATH] this becomes [EQUATION]', '0809.4488-3-28-3': 'Hence the angular power spectrum, [MATH], is given by [EQUATION]', '0809.4488-3-28-4': 'Using the identity [MATH] and the orthogonality relationship of spherical harmonics [EQUATION] this becomes [EQUATION]', '0809.4488-3-28-5': ""This exact relationship can be simplified by using Limber's approximation."", '0809.4488-3-28-6': 'For small angular separations, [MATH], at comoving distance, [MATH], the wave number, [MATH], can be expressed in terms of its components parallel and perpendicular to the line of sight and approximated by [MATH], where [MATH], namely, the power is dominated by that perpendicular to the line of sight and there is no correlation between different shells of [MATH] along the line of sight.', '0809.4488-3-28-7': 'Combining this with the orthogonality relation for spherical Bessel functions, [EQUATION] we arrive at [EQUATION] .', '0809.4488-3-28-8': '[CITATION] find that the difference between this approximation and the full calculation is less than [MATH] at [MATH].', '0809.4488-3-28-9': ""In this paper we are mainly concerned with even smaller scales ([MATH]) and so we are justified in using Limber's approximation."", '0809.4488-3-28-10': 'We also tested the difference between using [MATH] and the more accurate [MATH] and find very little difference to our results.'}",,
1906.06378,"{'1906.06378-1-0-0': 'The lattice thermal conductivity ([MATH]) is a critical property of thermoelectrics, thermal barrier coating materials and semiconductors.', '1906.06378-1-0-1': 'While accurate empirical measurements of [MATH] are extremely challenging, it is usually approximated through computational approaches, such as semi-empirical models, Green-Kubo formalism coupled with molecular dynamics simulations, and first-principles based methods.', '1906.06378-1-0-2': 'However, these theoretical methods are not only limited in terms of their accuracy, but sometimes become computationally intractable owing to their cost.', '1906.06378-1-0-3': 'Thus, in this work, we build a machine learning (ML)-based model to accurately and instantly predict [MATH] of inorganic materials, using a benchmark data set of experimentally measured [MATH] of about 100 inorganic solids.', '1906.06378-1-0-4': 'We use advanced and universal feature engineering techniques along with the Gaussian process regression algorithm, and compare the performance of our ML model with past theoretical works.', '1906.06378-1-0-5': 'The trained ML model is not only helpful for rational design and screening of novel materials, but we also identify key features governing the thermal transport behavior in non-metals.', '1906.06378-1-1-0': 'The lattice thermal conductivity ([MATH]) dictates the ability of a non-metal to conduct heat, and serves as a critical design parameter for a wide range of applications, including thermoelectrics for power generation,[CITATION] thermal barrier coatings for integrated circuits, [CITATION] and semiconductors for microelectronic devices.', '1906.06378-1-1-1': '[CITATION] Depending on the specific application, materials with different ranges of [MATH] values are desired.', '1906.06378-1-1-2': 'For example, low [MATH] is preferred as thermoelectrics (e.g., PbTe and Bi[MATH]Te[MATH]) to maximize the thermoelectric figure of merit, while for semiconductors (e.g., SiC and BP), high [MATH] is required to avoid overheating in electronic devices.', '1906.06378-1-1-3': 'Motivated by their practical and technological significance, extensive theoretical and empirical efforts have been made to compute [MATH], aimed at discovering materials with targeted thermal conductivity for specific applications.', '1906.06378-1-2-0': 'In one of the early and famous theoretical works, [MATH] of inorganic materials was estimated using semi-empirical Slack model,[CITATION] which relies on the Debye temperature ([MATH]) and the Gruneisen parameter ([MATH]) as inputs, obtained from either experimental measurements or first-principles calculations.', '1906.06378-1-2-1': '[CITATION] Although the Slack model can provide a quick [MATH] estimate, the uncertainty in its input parameters ([MATH], [MATH]) severely impacts its prediction accuracy.', '1906.06378-1-2-2': 'Slight modifications in the functional form of the Slack model (or its closely related Debye-Callaway model [CITATION]) have also been attempted by treating certain power coefficients as fitting parameters, which are determined using experimentally measured [MATH] values.', '1906.06378-1-2-3': 'However, the underlying problem of [MATH] and [MATH] uncertainty and their unavailability for new materials persists.', '1906.06378-1-2-4': 'Alternatively, the Green-Kubo formalism, combined with non-equilibrium molecular dynamics simulations, has been employed to predict [MATH] in semiconductors (e.g., Si).', '1906.06378-1-2-5': '[CITATION] However, this method can only be used for materials for which reliable atomistic force fields are available.', '1906.06378-1-2-6': 'With the recent developments of computing power and first-principles implementations, the ab initio Green-Kubo approach has been proposed to compute the [MATH] of Si and ZrO[MATH], but it is limited by the high computational cost to achieve the heat flux and system size convergences.[', '1906.06378-1-2-7': '[CITATION] Additionally, the phonon Boltzmann transport equation can now be solved numerically within the relaxation time approximation.[', '1906.06378-1-2-8': '[CITATION] In this approach, [MATH] is computed from the group velocity, the mode-dependent heat capacity, and the single-mode relaxation time (approximated by the phonon lifetime), all of which rely on the anharmonic force constants computed at the first-principles level.', '1906.06378-1-2-9': 'Even if only the first anharmonic (third-order) terms are considered, the computations involved are already so expensive that such methods are restricted to simple systems with small unit cells.', '1906.06378-1-2-10': 'Convergence issues related to the anharmonic terms truncation and dealing with large or complex systems are ubiquitous and generally hard to control.', '1906.06378-1-3-0': 'Machine learning (ML) based methods, which are emerging in Materials Science and Engineering,[CITATION] provide yet another approach to build surrogate models to rapidly predict the thermal conductivity of materials.', '1906.06378-1-3-1': 'Seko et al. developed ML models based on [MATH] computed for 110 materials (by solving the phonon Boltzmann transport equation as mentioned above) and a set of descriptors characterizing elemental and structural properties.', '1906.06378-1-3-2': '[CITATION] The main concern with such ML model is the discrepancy between the DFT computed training data and the actual experimental values (especially for solids with very high [MATH]) which directly impacts the accuracy of these models.', '1906.06378-1-3-3': 'Furthermore, the identification of key features in determining the [MATH] is far from trivial.', '1906.06378-1-4-0': 'To fill the above-mentioned gaps, we have built an ML model for [MATH], starting from a benchmark empirical data set of 100 solid materials.', '1906.06378-1-4-1': 'The scheme adopted in this work is illustrated in Figure [REF].', '1906.06378-1-4-2': 'First, the recently released Matminer package [CITATION] was used to generate a comprehensive list of 63 features to numerically represent the materials.', '1906.06378-1-4-3': 'This step was followed by the recursive feature elimination algorithm, down selecting the relevant features.', '1906.06378-1-4-4': 'The Gaussian process regression (GPR) algorithm, with 5-fold cross-validation (CV), was then utilized to build predictive models.', '1906.06378-1-4-5': 'The performance of the [MATH] models was compared with past studies and validated by 5 unseen materials.', '1906.06378-1-4-6': 'The developed ML model, which is trained on the [MATH] dataset spanning across 3 orders of magnitude, can be used to instantly predict [MATH] of new inorganic materials while the associated GPR uncertainty could indicate whether the new materials are within the training domain or not.', '1906.06378-1-4-7': 'It is hoped that the model developed in this work can be used to screen new inorganic materials with targeted [MATH], and it can be systematically improved when new materials are identified and added to the initial dataset.', '1906.06378-1-5-0': 'Figure [REF] and Table [REF] summarize the dataset of empirically measured [MATH] values (at 300 K) for 100 inorganic materials collected from the literature, [CITATION] including 81 binary and 19 ternary compounds.', '1906.06378-1-5-1': 'The dataset is significantly diverse in chemical compositions (35 cations and 22 anions), crystal structures (with space group 225, 216, 122, 186, etc.), and the range of [MATH], which spans over 3 orders of magnitude ([MATH]).', '1906.06378-1-5-2': 'The entire [MATH] data set-along with the bulk modulus feature values-is provided in Table S1 of the Supporting Information (SI).', '1906.06378-1-6-0': 'Given the wide range of [MATH], our learning problem was framed in the logarithmic scale, i.e., log([MATH]) was set as the target property, to allow better generalization of the ML models across the entire range.', '1906.06378-1-6-1': 'Furthermore, 95 out of 100 cases were used to train (with CV) the ML models, while the remaining 5 data points were held-out separately (completely unseen to the entire training process) to further validate the performance of the learned [MATH] model.', '1906.06378-1-6-2': 'For cases where multiple [MATH] values were reported in the literature, their average was used to train the ML model.', '1906.06378-1-7-0': 'To build accurate and reliable ML models, it is important to include relevant features that collectively capture the trends in the [MATH] values across the different materials.', '1906.06378-1-7-1': 'The features should not only uniquely represent each material, but also be readily available to allow instant predictions for new cases.', '1906.06378-1-7-2': 'In this regard, Matminer is a good resource to easily and quickly generate features, applicable specifically to the field of materials science.', '1906.06378-1-7-3': '[CITATION] In total, 61 features, belonging to three distinct categories, i.e., elemental, structural and pertaining to valence electrons, were obtained using the Matminer package [CITATION] by providing the chemical formula and the atomic configuration of all compounds.', '1906.06378-1-7-4': 'A total of 18 elemental properties were derived, including atomic radius, atomic mass, atom number, periodic table group and row, block, Mendeleev number, covalent radius, volume per atom from ground state, molar volume, coordination number (cn), Pauling electronegativity, first ionization energy, melting point, boiling point, thermal conductivity, average bond length and angle of a specific site with all its nearest neighbors.', '1906.06378-1-7-5': 'Since our dataset consists of binaries and ternaries, each of these elemental feature values was obtained by taking the minimum, maximum, and weighted average over the constituting chemical species, resulting in a total of 54 elemental features.', '1906.06378-1-7-6': 'For the structural features, volume per atom, packing fraction and density were considered.', '1906.06378-1-7-7': 'These quantities were computed for the crystal structure obtained from the Materials Project database.', '1906.06378-1-7-8': '[CITATION] Moreover, 4 features that capture the average number of valence electrons in the [MATH], [MATH], [MATH], and [MATH] shells of the constituting elements were also included.', '1906.06378-1-7-9': 'Finally, two additional features, DFT computed bulk modulus and the space group number, were also incorporated, resulting in a 63-dimensional feature vector.', '1906.06378-1-7-10': 'The values for bulk modulus of all compounds were obtained from the Material Project database.', '1906.06378-1-7-11': '[CITATION] As per standard ML practices, all features were scaled from 0 to 1 during model training.', '1906.06378-1-8-0': 'To retain only the relevant features, recursive feature elimination (RFE) using linear support vector regression algorithm (with 5-fold CV) was performed on the initial 63-dimensional feature vector and the dataset of 95 training points.', '1906.06378-1-8-1': 'RFE eliminates the irrelevant features by recursively ranking feature importance and pruning the least important ones.', '1906.06378-1-8-2': 'In our case, it reduced the dimensionality from 63 to 29 (see Table S2 of the SI).', '1906.06378-1-8-3': 'We also used random forest algorithm for feature dimensionality reduction.', '1906.06378-1-8-4': 'In particular, we trained the data set of 95 points using 100 trees, and used the feature importance/weight to determine the relevance of the features.', '1906.06378-1-8-5': 'As discussed in Section 2 of the SI, nearly 40 features were identified to be important using the random forest method, most of which were found to be consistent to those retained from the RFE scheme discussed earlier.', '1906.06378-1-8-6': 'This provides more confidence to the RFE based dimensionality reduction step performed in this work.', '1906.06378-1-8-7': 'Overall, the 29-dimensional feature vector obtained after RFE resulted in more accurate models than the original 63-dimensional feature, as will be discussed in detail next, while a detailed comparison of the RFE and random forest methods is provided in SI.', '1906.06378-1-9-0': 'The Gaussian process regression (GPR) with the radial basis function (RBF) kernel was utilized to train the ML models.', '1906.06378-1-9-1': 'In this case, the co-variance function between two materials with features [MATH] and [MATH] is given by [EQUATION]', '1906.06378-1-9-2': 'Here, three hyper parameters [MATH], [MATH] and [MATH] signify the variance, the length-scale parameter and the expected noise in the data, respectively.', '1906.06378-1-9-3': 'These hyper parameters were determined during the training of the models by maximizing the log-likelihood estimate.', '1906.06378-1-9-4': 'Further, 5-fold CV was adopted to avoid overfitting.', '1906.06378-1-9-5': 'Two error metrics, namely, the root mean square error (RMSE) and the coefficient of determination ([MATH]), were used to evaluate the performance of the ML models.', '1906.06378-1-9-6': 'To estimate the prediction errors on unseen data, learning curves were generated by varying the size of the training and the test sets.', '1906.06378-1-9-7': 'We note that the test sets were obtained by excluding the training points from the data set of 95 points.', '1906.06378-1-9-8': 'The left-out set of 5 points was completely separated from the learning process, and was used for just evaluation purposes on a few ""extrapolative"" material cases.', '1906.06378-1-9-9': 'Additionally, for each case, statistically meaningful results were obtained by averaging RMSE results over 100 runs with random training and test splits.', '1906.06378-1-10-0': 'It is worth analyzing the correlation between these 29 features and the empirically measured [MATH] to see how much trend is captured by these elemental, structural and chemical attributes.', '1906.06378-1-10-1': 'While in Figure [REF] we plot the [MATH] vs four important features, the corresponding plots for the remaining cases are provided in Figure S3 of SI.', '1906.06378-1-10-2': 'A strong positive correlation between [MATH]) and bulk modulus, and a strong inverse relation between [MATH]) and the mean average bond length are evident from the figure.', '1906.06378-1-10-3': 'While density alone does not show a strong correlation with [MATH]), the combined feature [MATH] does indeed show a very strong linear relation.', '1906.06378-1-10-4': 'This is in-line with the physical understanding that group velocity, which is an integral part of the semi-empirical models discussed earlier, can be approximated as [MATH].', '1906.06378-1-10-5': 'Thus, bulk modulus can be considered to play a critical role in influencing the [MATH] of different inorganic non-metals.', '1906.06378-1-10-6': 'Similarly, the inverse relationship between [MATH]) and the mean average bond length is also physically meaningful as when the bonds are shorter, the force constant is larger, and the resulting [MATH] is larger.', '1906.06378-1-10-7': 'For the case of mean atomic mass (a common feature used in the past ML model works), a slightly dispersed relationship is observed, indicating that it may be less important in governing [MATH], as was the case with the rest of the 25 features illustrated in Figure S3 of the SI.', '1906.06378-1-11-0': 'Next, the performance of the ML models can be evaluated from the learning curves presented in Figure [REF](a), wherein average RMSE on the training and the test sets as a function of training set size are included.', '1906.06378-1-11-1': 'The error bars denote the 1[MATH] deviation in the reported RMSE values over 100 runs.', '1906.06378-1-11-2': 'Results using both the initial set of 63 features (GPR), and those for the reduced 29 features (GPR-RFE) are included.', '1906.06378-1-11-3': 'Clearly, the RFE dimensionality reduction step leads to improved model performance with lower test errors, which signify better generalization of these models for unseen data.', '1906.06378-1-11-4': 'As expected, the test RMSE of both the GPR and the GPR-RFE models decreases with increase in training set size, reaching a convergence of [MATH] in test error and of [MATH] in train error for GPR-RFE models when the training set is about 80 % of the data (i.e. 76 points).', '1906.06378-1-11-5': 'Figure [REF](b) and (c) show the performance of GPR-RFE models via the example parity plots (i.e., ML predicted vs experimental log([MATH])), using 76 and 95 train points, respectively.', '1906.06378-1-11-6': 'The error bars in these cases represent the GPR uncertainty.', '1906.06378-1-11-7': 'Pretty high [MATH] coefficient ([MATH]) on the test set in both these cases suggest a good [MATH] model has indeed been developed.', '1906.06378-1-12-0': 'We compared the performance of our ML model with other semi-empirical models by computing the average factor difference (AFD),[CITATION] using the definition [MATH], where [MATH], with [MATH] being the number of data points.', '1906.06378-1-12-1': 'As shown in Table [REF], the computed AFD of GPR-RFE models using the entire set of 95 points is 1.36 [MATH]0.03, which is comparable to the reported values of 1.38 and 1.46, respectively, obtained using the Slack[CITATION] and Debye-Callaway [CITATION] models.', '1906.06378-1-12-2': 'More importantly, the latter two ML models rely on the features that are much more difficult to obtain owing to their dependence on the use of the Slack or Debye-Callaway models, while the ML model presented here uses easily and rapidly accessible chemical and structural features, making it more inexpensive and flexible.', '1906.06378-1-13-0': 'In order to further validate the generality and the accuracy of our ML models, we used the GPR-RFE models trained on the entire set of 95 points (see Figure [REF](a)) to predict the log([MATH]) of 5 unseen inorganic solids present in the hold-out set.', '1906.06378-1-13-1': 'These include Sc[MATH]O[MATH], Ga[MATH]O[MATH], MnO, AlCuO[MATH], and Ca[MATH]Al[MATH]Sb[MATH].', '1906.06378-1-13-2': 'Figure [REF](c) shows the comparison between the predicted and the experimental log([MATH]), with error bars capturing the GPR uncertainty.', '1906.06378-1-13-3': 'A good performance for these 5 unseen data points is clearly evident.', '1906.06378-1-13-4': 'The high GPR uncertainty in the case of Ca[MATH]Al[MATH]Sb[MATH] correctly signals its differences from the training data, and the application of the ML model in the ""extrapolative"" regime.', '1906.06378-1-13-5': 'Overall, the results presented here strongly advocate the good performance of the GPR-RFE models developed, which can be used to provide an inexpensive and accurate [MATH] prediction for other inorganic non-metals.', '1906.06378-1-14-0': 'In conclusion, we have developed a simple and general ML model to predict [MATH] of inorganic solid materials.', '1906.06378-1-14-1': 'This model is faster and more accurate than traditional computational methods.', '1906.06378-1-14-2': 'This work involves curating a benchmark dataset of experimental values of [MATH] of 100 inorganic solids, generating and optimizing a comprehensive set of features (using the Matminer package), and training the Gaussian Process Regression model on the data prepared.', '1906.06378-1-14-3': 'The accuracy of the developed ML models was found to be comparable to past semi-empirical models.', '1906.06378-1-14-4': 'Additionally, key features in determining [MATH] were identified.', '1906.06378-1-14-5': 'Overall, this present work would be useful for rational design and screening of new materials with desired [MATH] for specific applications, and fundamentally understanding the heat transport in inorganic solid materials.', '1906.06378-1-15-0': 'This work is supported by the Office of Naval Research through N0014-17-1-2656, a Multi-University Research Initiative (MURI) grant.', '1906.06378-1-16-0': '# Conflict of Interest: The authors declare no competing financial interest.', '1906.06378-1-17-0': 'The following files are available free of charge.'}","{'1906.06378-2-0-0': 'The lattice thermal conductivity ([MATH]) is a critical property of thermoelectrics, thermal barrier coating materials and semiconductors.', '1906.06378-2-0-1': 'While accurate empirical measurements of [MATH] are extremely challenging, it is usually approximated through computational approaches, such as semi-empirical models, Green-Kubo formalism coupled with molecular dynamics simulations, and first-principles based methods.', '1906.06378-2-0-2': 'However, these theoretical methods are not only limited in terms of their accuracy, but sometimes become computationally intractable owing to their cost.', '1906.06378-2-0-3': 'Thus, in this work, we build a machine learning (ML)-based model to accurately and instantly predict [MATH] of inorganic materials, using a benchmark data set of experimentally measured [MATH] of about 100 inorganic solids.', '1906.06378-2-0-4': 'We use advanced and universal feature engineering techniques along with the Gaussian process regression algorithm, and compare the performance of our ML model with past theoretical works.', '1906.06378-2-0-5': 'The trained ML model is not only helpful for rational design and screening of novel materials, but we also identify key features governing the thermal transport behavior in non-metals.', '1906.06378-2-1-0': '# Introduction', '1906.06378-2-2-0': 'The lattice thermal conductivity ([MATH]) dictates the ability of a non-metal to conduct heat, and serves as a critical design parameter for a wide range of applications, including thermoelectrics for power generation [CITATION], thermal barrier coatings for integrated circuits [CITATION], and semiconductors for microelectronic devices [CITATION].', '1906.06378-2-2-1': 'Depending on the specific application, materials with different ranges of [MATH] values are desired.', '1906.06378-2-2-2': 'For example, low [MATH] is preferred as thermoelectrics (e.g., PbTe and Bi[MATH]Te[MATH]) to maximize the thermoelectric figure of merit, while for semiconductors (e.g., SiC and BP), high [MATH] is required to avoid overheating in electronic devices.', '1906.06378-2-2-3': 'Motivated by their practical and technological significance, extensive theoretical and empirical efforts have been made to compute [MATH], aimed at discovering materials with targeted thermal conductivity for specific applications.', '1906.06378-2-3-0': 'In one of the early and famous theoretical works, [MATH] of inorganic materials was estimated using semi-empirical Slack model [CITATION], which relies on the Debye temperature ([MATH]) and the Gruneisen parameter ([MATH]) as inputs, obtained from either experimental measurements or first-principles calculations [CITATION].', '1906.06378-2-3-1': 'Although the Slack model can provide a quick [MATH] estimate, the uncertainty in its input parameters ([MATH], [MATH]) severely impacts its prediction accuracy.', '1906.06378-2-3-2': 'Slight modifications in the functional form of the Slack model (or its closely related Debye-Callaway model [CITATION]) have also been attempted by treating certain power coefficients as fitting parameters, which are determined using experimentally measured [MATH] values.', '1906.06378-2-3-3': 'However, the underlying problem of [MATH] and [MATH] uncertainty and their unavailability for new materials persists.', '1906.06378-2-3-4': 'Alternatively, the Green-Kubo formalism, combined with non-equilibrium molecular dynamics simulations, has been employed to predict [MATH] in semiconductors (e.g., Si) [CITATION].', '1906.06378-2-3-5': 'However, this method can only be used for materials for which reliable atomistic force fields are available.', '1906.06378-2-3-6': 'With the recent developments of computing power and first-principles implementations, the ab initio Green-Kubo approach has been proposed to compute the [MATH] of Si and ZrO[MATH], but it is limited by the high computational cost to achieve the heat flux and system size convergences [CITATION].', '1906.06378-2-3-7': 'Additionally, the phonon Boltzmann transport equation (BTE) can now be solved numerically within the relaxation time approximation [CITATION].', '1906.06378-2-3-8': 'In this approach, [MATH] is computed from the group velocity, the mode-dependent heat capacity, and the single-mode relaxation time (approximated by the phonon lifetime), all of which rely on either the harmonic or the anharmonic force constants computed at the first-principles level.', '1906.06378-2-3-9': 'While BTE calculations could in principle be done for large systems [CITATION], they are generally restricted to small unit cells owing to high computational costs.', '1906.06378-2-4-0': 'Machine learning (ML) based methods, which are emerging in Materials Science and Engineering [CITATION] provide yet another approach to build surrogate models to rapidly predict the thermal conductivity of materials.', '1906.06378-2-4-1': 'Seko et al. developed ML models based on [MATH] computed for 110 materials (by solving the phonon Boltzmann transport equation as mentioned above) and a set of descriptors characterizing elemental and structural properties [CITATION].', '1906.06378-2-4-2': 'The main concern with such ML model is the discrepancy between the DFT computed training data and the actual experimental values (especially for solids with very high [MATH]) which directly impacts the accuracy of these models.', '1906.06378-2-4-3': 'Furthermore, the identification of key features in determining the [MATH] is far from trivial.', '1906.06378-2-5-0': 'To fill the above-mentioned gaps, we have built an ML model for [MATH], starting from a benchmark empirical data set of 100 inorganic compounds.', '1906.06378-2-5-1': 'The scheme adopted in this work is illustrated in Figure [REF].', '1906.06378-2-5-2': 'First, the recently released Matminer package [CITATION] was used to generate a comprehensive list of 63 features to numerically represent the materials.', '1906.06378-2-5-3': 'This step was followed by the recursive feature elimination algorithm, down selecting the relevant features.', '1906.06378-2-5-4': 'The Gaussian process regression (GPR) algorithm, with 5-fold cross-validation (CV), was then utilized to build predictive models.', '1906.06378-2-5-5': 'The performance of the [MATH] models was compared with past studies and validated by 5 unseen materials.', '1906.06378-2-5-6': 'The developed ML model, which is trained on the [MATH] dataset spanning across 3 orders of magnitude, can be used to instantly predict [MATH] of new inorganic materials while the associated GPR uncertainty could indicate whether the new materials are within the training domain or not.', '1906.06378-2-5-7': 'It is hoped that the model developed in this work can be used to screen new inorganic materials with targeted [MATH], and it can be systematically improved when new materials are identified and added to the initial dataset.', '1906.06378-2-6-0': '# Technical Details', '1906.06378-2-7-0': '## Data set', '1906.06378-2-8-0': 'Figure [REF] and Table [REF] summarize the dataset of empirically measured [MATH] values (at room temperature) for 100 single crystal inorganic materials collected from the literature [CITATION], including 81 binary and 19 ternary compounds.', '1906.06378-2-8-1': '[MATH] of single-element materials are excluded since thermal conductivity of individual elements within a compound were used as features.', '1906.06378-2-8-2': 'The dataset is significantly diverse in chemical compositions (35 cations and 22 anions), crystal structures (with space group 225, 216, 122, 186, etc.), and the range of [MATH], which spans over 3 orders of magnitude ([MATH]).', '1906.06378-2-8-3': 'The entire [MATH] data set-along with the bulk modulus feature values-is provided in Table S1 of the Supporting Information (SI).', '1906.06378-2-9-0': 'Given the wide range of [MATH], our learning problem was framed in the logarithmic scale, i.e., log([MATH]) was set as the target property, to allow better generalization of the ML models across the entire range.', '1906.06378-2-9-1': 'Furthermore, 95 out of 100 cases were used to train (with CV) the ML models, while the remaining 5 data points were held-out separately (completely unseen to the entire training process) to further validate the performance of the learned [MATH] model.', '1906.06378-2-9-2': 'For cases where multiple [MATH] values were reported in the literature, their average was used to train the ML model.', '1906.06378-2-10-0': '## Feature set and dimensionality reduction', '1906.06378-2-11-0': 'To build accurate and reliable ML models, it is important to include relevant features that collectively capture the trends in the [MATH] values across the different materials.', '1906.06378-2-11-1': 'The features should not only uniquely represent each material, but also be readily available to allow instant predictions for new cases.', '1906.06378-2-11-2': 'In this regard, Matminer is a good resource to easily and quickly generate features, applicable specifically to the field of materials science [CITATION].', '1906.06378-2-11-3': 'In total, 61 features, belonging to three distinct categories, i.e., elemental, structural and pertaining to valence electrons, were obtained using the Matminer package [CITATION] by providing the chemical formula and the atomic configuration of all compounds.', '1906.06378-2-11-4': 'A total of 18 elemental properties were derived, including atomic radius, atomic mass, atom number, periodic table group and row, block, Mendeleev number, covalent radius, volume per atom from ground state, molar volume, coordination number (cn), Pauling electronegativity, first ionization energy, melting point, boiling point, thermal conductivity, average bond length and angle of a specific site with all its nearest neighbors.', '1906.06378-2-11-5': 'Since our dataset consists of binaries and ternaries, each of these elemental feature values was obtained by taking the minimum, maximum, and weighted average over the constituting chemical species, resulting in a total of 54 elemental features.', '1906.06378-2-11-6': 'For the structural features, volume per atom, packing fraction and density were considered.', '1906.06378-2-11-7': 'These quantities were computed for the crystal structure obtained from the Materials Project database [CITATION].', '1906.06378-2-11-8': 'Moreover, 4 features that capture the average number of valence electrons in the [MATH], [MATH], [MATH], and [MATH] shells of the constituting elements were also included.', '1906.06378-2-11-9': 'Finally, two additional features, DFT computed bulk modulus and the space group number, were also incorporated, resulting in a 63-dimensional feature vector.', '1906.06378-2-11-10': 'The values for bulk modulus of all compounds were obtained from the Material Project database [CITATION].', '1906.06378-2-11-11': 'As per standard ML practices, all features were scaled from 0 to 1 during model training.', '1906.06378-2-12-0': 'To retain only the relevant features, recursive feature elimination (RFE) using linear support vector regression algorithm (with 5-fold CV) was performed on the initial 63-dimensional feature vector and the dataset of 95 training points.', '1906.06378-2-12-1': 'RFE eliminates the irrelevant features by recursively ranking feature importance and pruning the least important ones.', '1906.06378-2-12-2': 'In our case, it reduced the dimensionality from 63 to 29 (see Table S2 of the SI).', '1906.06378-2-12-3': 'We also used random forest algorithm for feature dimensionality reduction.', '1906.06378-2-12-4': 'In particular, we trained the data set of 95 points using 100 trees, and used the feature importance/weight to determine the relevance of the features.', '1906.06378-2-12-5': 'As discussed in Section 2 of the SI, nearly 40 features were identified to be important using the random forest method, most of which were found to be consistent to those retained from the RFE scheme discussed earlier.', '1906.06378-2-12-6': 'This provides more confidence to the RFE based dimensionality reduction step performed in this work.', '1906.06378-2-12-7': 'Overall, the 29-dimensional feature vector obtained after RFE resulted in more accurate models than the original 63-dimensional feature, as will be discussed in detail next, while a detailed comparison of the RFE and random forest methods is provided in SI.', '1906.06378-2-13-0': '## Gaussian Process Regression', '1906.06378-2-14-0': 'The Gaussian process regression (GPR) with the radial basis function (RBF) kernel was utilized to train the ML models.', '1906.06378-2-14-1': 'In this case, the co-variance function between two materials with features [MATH] and [MATH] is given by [EQUATION]', '1906.06378-2-14-2': 'Here, three hyper parameters [MATH], [MATH] and [MATH] signify the variance, the length-scale parameter and the expected noise in the data, respectively.', '1906.06378-2-14-3': 'These hyper parameters were determined during the training of the models by maximizing the log-likelihood estimate.', '1906.06378-2-14-4': 'Further, 5-fold CV was adopted to avoid overfitting.', '1906.06378-2-14-5': 'Two error metrics, namely, the root mean square error (RMSE) and the coefficient of determination ([MATH]), were used to evaluate the performance of the ML models.', '1906.06378-2-14-6': 'To estimate the prediction errors on unseen data, learning curves were generated by varying the size of the training and the test sets.', '1906.06378-2-14-7': 'We note that the test sets were obtained by excluding the training points from the data set of 95 points.', '1906.06378-2-14-8': 'The left-out set of 5 points was completely separated from the learning process, and was used for just evaluation purposes on a few ""extrapolative"" material cases.', '1906.06378-2-14-9': 'Additionally, for each case, statistically meaningful results were obtained by averaging RMSE results over 100 runs with random training and test splits.', '1906.06378-2-15-0': '# Results and Discussion', '1906.06378-2-16-0': 'It is worth analyzing the correlation between these 29 features and the empirically measured [MATH] to see how much trend is captured by these elemental, structural and chemical attributes.', '1906.06378-2-16-1': 'While in Figure [REF] we plot the [MATH] vs four important features, the corresponding plots for the remaining cases are provided in Figure S3 of SI.', '1906.06378-2-16-2': 'A strong positive correlation between [MATH]) and bulk modulus, and a strong inverse relation between [MATH]) and the mean average bond length are evident from the figure.', '1906.06378-2-16-3': 'While density alone does not show a strong correlation with [MATH]), the combined feature [MATH] does indeed show a very strong linear relation.', '1906.06378-2-16-4': 'This is in-line with the physical understanding that group velocity, which is an integral part of the semi-empirical models discussed earlier, can be approximated as [MATH].', '1906.06378-2-16-5': 'Thus, bulk modulus can be considered to play a critical role in influencing the [MATH] of different inorganic non-metals.', '1906.06378-2-16-6': 'Similarly, the inverse relationship between [MATH]) and the mean average bond length is also physically meaningful as when the bonds are shorter, the force constant is larger, and the resulting [MATH] is larger.', '1906.06378-2-16-7': 'For the case of mean atomic mass (a common feature used in the past ML model works), a slightly dispersed relationship is observed, indicating that it may be less important in governing [MATH], as was the case with the rest of the 25 features illustrated in Figure S3 of the SI.', '1906.06378-2-17-0': 'Next, the performance of the ML models can be evaluated from the learning curves presented in Figure [REF](a), wherein average RMSE on the training and the test sets as a function of training set size are included.', '1906.06378-2-17-1': 'The error bars denote the 1[MATH] deviation in the reported RMSE values over 100 runs.', '1906.06378-2-17-2': 'Results using both the initial set of 63 features (GPR), and those for the reduced 29 features (GPR-RFE) are included.', '1906.06378-2-17-3': 'Clearly, the RFE dimensionality reduction step leads to improved model performance with lower test errors, which signify better generalization of these models for unseen data.', '1906.06378-2-17-4': 'As expected, the test RMSE of both the GPR and the GPR-RFE models decreases with increase in training set size, reaching a convergence of [MATH] in test error and of [MATH] in train error for GPR-RFE models when the training set is about 80 % of the data (i.e. 76 points).', '1906.06378-2-17-5': 'Figure [REF](b) and (c) show the performance of GPR-RFE models via the example parity plots (i.e., ML predicted vs experimental log([MATH])), using 76 and 95 train points, respectively.', '1906.06378-2-17-6': 'The error bars in these cases represent the GPR uncertainty.', '1906.06378-2-17-7': 'Pretty high [MATH] coefficient ([MATH]) on the test set in both these cases suggest a good [MATH] model has indeed been developed.', '1906.06378-2-18-0': 'We compared the performance of our ML model with other semi-empirical models by computing the average factor difference (AFD) [CITATION], using the definition [MATH], where [MATH], with [MATH] being the number of data points.', '1906.06378-2-18-1': 'As shown in Table [REF], the computed AFD of GPR-RFE models using the entire set of 95 points is 1.36 [MATH]0.03, which is comparable to the reported values of 1.38 and 1.46, respectively, obtained using the Slack [CITATION] and Debye-Callaway [CITATION] models.', '1906.06378-2-18-2': 'More importantly, the latter two ML models rely on the features that are much more difficult to obtain owing to their dependence on the use of the Slack or Debye-Callaway models, while the ML model presented here uses easily and rapidly accessible chemical and structural features, making it more inexpensive and flexible.', '1906.06378-2-19-0': 'In order to further validate the generality and the accuracy of our ML models, we used the GPR-RFE models trained on the entire set of 95 points (see Figure [REF](a)) to predict the log([MATH]) of 5 unseen inorganic solids with various space group numbers present in the hold-out set.', '1906.06378-2-19-1': 'These include Sc[MATH]O[MATH] (206), Ga[MATH]O[MATH] (12), MnO (225), AlCuO[MATH] (166), and Ca[MATH]Al[MATH]Sb[MATH] (55), where the number within brackets is the space group number.', '1906.06378-2-19-2': 'Figure [REF](c) shows the comparison between the predicted and the experimental log([MATH]), with error bars capturing the GPR uncertainty.', '1906.06378-2-19-3': 'A good performance for these 5 unseen data points is clearly evident.', '1906.06378-2-19-4': 'The high GPR uncertainty in the case of Ca[MATH]Al[MATH]Sb[MATH] correctly signals its space group number differences from that of the majority training data, and the application of the ML model in the ""extrapolative"" regime.', '1906.06378-2-19-5': 'Overall, the results presented here strongly advocate the good performance of the GPR-RFE models developed, which can be used to provide an inexpensive and accurate [MATH] prediction for other inorganic materials, especially for materials with rock-salt or zincblende structures.', '1906.06378-2-19-6': 'Moreover, the predictive ML model can be easily improved by actively learning on more diverse training data sets, while even extending it to predict temperature-dependence by training on temperature-dependent thermal conductivity data.', '1906.06378-2-20-0': '# Conclusion', '1906.06378-2-21-0': 'In conclusion, we have developed a simple and general ML model to predict [MATH] of inorganic solid materials.', '1906.06378-2-21-1': 'This model is faster, and at par or more accurate than traditional physics-based computational methods.', '1906.06378-2-21-2': 'This work involves curating a benchmark dataset of experimental values of [MATH] of 100 inorganic compounds, generating and optimizing a comprehensive set of features (using the Matminer package), and training the Gaussian Process Regression model on the data prepared.', '1906.06378-2-21-3': 'The accuracy of the developed ML models was found to be comparable to past semi-empirical models.', '1906.06378-2-21-4': 'Additionally, key features in determining [MATH] were identified.', '1906.06378-2-21-5': 'Overall, this present work would be useful for rational design and screening of new materials with desired [MATH] for specific applications, and fundamentally understanding the heat transport in inorganic solid materials.', '1906.06378-2-22-0': '# Data Availability', '1906.06378-2-23-0': 'The entire experimental [MATH] data set and DFT computed bulk modulus are available in Table S1 of the Supporting Information.', '1906.06378-2-24-0': 'This work is supported by the Office of Naval Research through N0014-17-1-2656, a Multi-University Research Initiative (MURI) grant.', '1906.06378-2-25-0': '# Conflict of Interest: The authors declare no competing financial interest.', '1906.06378-2-26-0': 'The following files are available free of charge.'}","[['1906.06378-1-14-0', '1906.06378-2-21-0'], ['1906.06378-1-14-3', '1906.06378-2-21-3'], ['1906.06378-1-14-4', '1906.06378-2-21-4'], ['1906.06378-1-14-5', '1906.06378-2-21-5'], ['1906.06378-1-12-2', '1906.06378-2-18-2'], ['1906.06378-1-15-0', '1906.06378-2-24-0'], ['1906.06378-1-5-1', '1906.06378-2-8-2'], ['1906.06378-1-5-2', '1906.06378-2-8-3'], ['1906.06378-1-11-0', '1906.06378-2-17-0'], ['1906.06378-1-11-1', '1906.06378-2-17-1'], ['1906.06378-1-11-2', '1906.06378-2-17-2'], ['1906.06378-1-11-3', '1906.06378-2-17-3'], ['1906.06378-1-11-4', '1906.06378-2-17-4'], 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'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1906.06378,"{'1906.06378-3-0-0': 'The lattice thermal conductivity ([MATH]) is a critical property of thermoelectrics, thermal barrier coating materials and semiconductors.', '1906.06378-3-0-1': 'While accurate empirical measurements of [MATH] are extremely challenging, it is usually approximated through computational approaches, such as semi-empirical models, Green-Kubo formalism coupled with molecular dynamics simulations, and first-principles based methods.', '1906.06378-3-0-2': 'However, these theoretical methods are not only limited in terms of their accuracy, but sometimes become computationally intractable owing to their cost.', '1906.06378-3-0-3': 'Thus, in this work, we build a machine learning (ML)-based model to accurately and instantly predict [MATH] of inorganic materials, using a benchmark data set of experimentally measured [MATH] of about 100 inorganic materials.', '1906.06378-3-0-4': 'We use advanced and universal feature engineering techniques along with the Gaussian process regression algorithm, and compare the performance of our ML model with past theoretical works.', '1906.06378-3-0-5': 'The trained ML model is not only helpful for rational design and screening of novel materials, but we also identify key features governing the thermal transport behavior in non-metals.', '1906.06378-3-1-0': '# Introduction', '1906.06378-3-2-0': 'The lattice thermal conductivity ([MATH]) dictates the ability of a non-metal to conduct heat, and serves as a critical design parameter for a wide range of applications, including thermoelectrics for power generation [CITATION], thermal barrier coatings for integrated circuits [CITATION], and semiconductors for microelectronic devices [CITATION].', '1906.06378-3-2-1': 'Depending on the specific application, materials with different ranges of [MATH] values are desired.', '1906.06378-3-2-2': 'For example, low [MATH] is preferred as thermoelectrics (e.g., PbTe and Bi[MATH]Te[MATH]) to maximize the thermoelectric figure of merit, while for semiconductors (e.g., SiC and BP), high [MATH] is required to avoid overheating in electronic devices.', '1906.06378-3-2-3': 'Motivated by their practical and technological significance, extensive theoretical and empirical efforts have been made to compute [MATH], aimed at discovering materials with targeted thermal conductivity for specific applications.', '1906.06378-3-3-0': 'In one of the early and famous theoretical works, [MATH] of inorganic materials was estimated using semi-empirical Slack model [CITATION], which relies on the Debye temperature ([MATH]) and the Gruneisen parameter ([MATH]) as inputs, obtained from either experimental measurements or first-principles calculations [CITATION].', '1906.06378-3-3-1': 'Although the Slack model can provide a quick [MATH] estimate, the uncertainty in its input parameters ([MATH], [MATH]) severely impacts its prediction accuracy.', '1906.06378-3-3-2': 'Slight modifications in the functional form of the Slack model (or its closely related Debye-Callaway model [CITATION]) have also been attempted by treating certain power coefficients as fitting parameters, which are determined using experimentally measured [MATH] values.', '1906.06378-3-4-0': 'However, the underlying problem of [MATH] and [MATH] uncertainty and their unavailability for new materials persists.', '1906.06378-3-4-1': 'Alternatively, the Green-Kubo formalism, combined with non-equilibrium molecular dynamics simulations, has been employed to predict [MATH] in semiconductors (e.g., Si) [CITATION].', '1906.06378-3-4-2': 'However, this method can only be used for materials for which reliable atomistic force fields are available.', '1906.06378-3-4-3': 'With the recent developments of computing power and first-principles implementations, the ab initio Green-Kubo approach has been proposed to compute the [MATH] of Si and ZrO[MATH], but it is limited by the high computational cost to achieve the heat flux and system size convergences [CITATION].', '1906.06378-3-4-4': 'Additionally, the phonon Boltzmann transport equation (BTE) can now be solved numerically within the relaxation time approximation [CITATION].', '1906.06378-3-4-5': 'In this approach, [MATH] is computed from the group velocity, the mode-dependent heat capacity, and the single-mode relaxation time (approximated by the phonon lifetime), all of which rely on either the harmonic or the anharmonic force constants computed at the first-principles level.', '1906.06378-3-4-6': 'While BTE calculations could in principle be done for large systems [CITATION], they are generally restricted to small unit cells owing to high computational costs.', '1906.06378-3-5-0': 'Machine learning (ML) based methods, which are emerging in Materials Science and Engineering [CITATION] provide yet another approach to build surrogate models to rapidly predict the thermal conductivity of materials.', '1906.06378-3-5-1': 'Seko et al. developed ML models based on [MATH] computed for 110 materials (by solving the phonon Boltzmann transport equation as mentioned above) and a set of descriptors characterizing elemental and structural properties [CITATION].', '1906.06378-3-5-2': 'The main concern with such ML model is the discrepancy between the DFT computed training data and the actual experimental values (especially for solids with very high [MATH]) which directly impacts the accuracy of these models.', '1906.06378-3-5-3': 'Furthermore, the identification of key features in determining the [MATH] is far from trivial.', '1906.06378-3-6-0': 'To fill the above-mentioned gaps, we have built an ML model for [MATH], starting from a benchmark empirical data set of 100 inorganic compounds.', '1906.06378-3-6-1': 'The scheme adopted in this work is illustrated in Figure [REF].', '1906.06378-3-6-2': 'First, the recently released Matminer package [CITATION] was used to generate a comprehensive list of 63 features to numerically represent the materials.', '1906.06378-3-6-3': 'This step was followed by the recursive feature elimination algorithm, down selecting the relevant features.', '1906.06378-3-6-4': 'The Gaussian process regression (GPR) algorithm, with 5-fold cross-validation (CV), was then utilized to build predictive models.', '1906.06378-3-6-5': 'The performance of the [MATH] models was compared with past studies and validated by 5 unseen materials.', '1906.06378-3-6-6': 'The developed ML model, which is trained on the [MATH] dataset spanning across 3 orders of magnitude, can be used to instantly predict [MATH] of new inorganic materials while the associated GPR uncertainty could indicate whether the new materials are within the training domain or not.', '1906.06378-3-6-7': 'It is hoped that the model developed in this work can be used to screen new inorganic materials with targeted [MATH], and it can be systematically improved when new materials are identified and added to the initial dataset.', '1906.06378-3-7-0': '# Technical Details', '1906.06378-3-8-0': '## Data set', '1906.06378-3-9-0': 'Figure [REF] and Table [REF] summarize the dataset of empirically measured [MATH] values (at room temperature) for 100 single crystal inorganic materials collected from the literature [CITATION], including 81 binary and 19 ternary compounds.', '1906.06378-3-9-1': '[MATH] of single-element materials are excluded since thermal conductivity of individual elements within a compound were used as features.', '1906.06378-3-9-2': 'The dataset is significantly diverse in chemical compositions (35 cations and 22 anions), crystal structures (with space group 225, 216, 122, 186, etc.), and the range of [MATH], which spans over 3 orders of magnitude ([MATH]).', '1906.06378-3-9-3': 'The entire [MATH] data set-along with the bulk modulus feature values-is provided in Table S1 of the Supporting Information (SI).', '1906.06378-3-10-0': 'Given the wide range of [MATH], our learning problem was framed in the logarithmic scale, i.e., log([MATH]) was set as the target property, to allow better generalization of the ML models across the entire range.', '1906.06378-3-10-1': 'Furthermore, 95 out of 100 cases were used to train (with CV) the ML models, while the remaining 5 data points were held-out separately (completely unseen to the entire training process) to further validate the performance of the learned [MATH] model.', '1906.06378-3-10-2': 'For cases where multiple [MATH] values were reported in the literature, their average was used to train the ML model.', '1906.06378-3-11-0': '## Feature set and dimensionality reduction', '1906.06378-3-12-0': 'To build accurate and reliable ML models, it is important to include relevant features that collectively capture the trends in the [MATH] values across the different materials.', '1906.06378-3-12-1': 'The features should not only uniquely represent each material, but also be readily available to allow instant predictions for new cases.', '1906.06378-3-12-2': 'In this regard, Matminer is a good resource to easily and quickly generate features, applicable specifically to the field of materials science [CITATION].', '1906.06378-3-12-3': 'In total, 61 features, belonging to three distinct categories, i.e., elemental, structural and pertaining to valence electrons, were obtained using the Matminer package [CITATION] by providing the chemical formula and the atomic configuration of all compounds.', '1906.06378-3-12-4': 'A total of 18 elemental properties were derived, including atomic radius, atomic mass, atom number, periodic table group and row, block, Mendeleev number, covalent radius, volume per atom from ground state, molar volume, coordination number (cn), Pauling electronegativity, first ionization energy, melting point, boiling point, thermal conductivity, average bond length and angle of a specific site with all its nearest neighbors.', '1906.06378-3-12-5': 'Since our dataset consists of binaries and ternaries, each of these elemental feature values was obtained by taking the minimum, maximum, and weighted average over the constituting chemical species, resulting in a total of 54 elemental features.', '1906.06378-3-12-6': 'For the structural features, volume per atom, packing fraction and density were considered.', '1906.06378-3-12-7': 'These quantities were computed for the crystal structure obtained from the Materials Project database [CITATION].', '1906.06378-3-12-8': 'Moreover, 4 features that capture the average number of valence electrons in the [MATH], [MATH], [MATH], and [MATH] shells of the constituting elements were also included.', '1906.06378-3-12-9': 'Finally, two additional features, DFT computed bulk modulus and the space group number, were also incorporated, resulting in a 63-dimensional feature vector.', '1906.06378-3-12-10': 'The values for bulk modulus of all compounds were obtained from the Material Project database [CITATION].', '1906.06378-3-12-11': 'As per standard ML practices, all features were scaled from 0 to 1 during model training.', '1906.06378-3-13-0': 'To retain only the relevant features, recursive feature elimination (RFE) using linear support vector regression algorithm (with 5-fold CV) was performed on the initial 63-dimensional feature vector and the dataset of 95 training points.', '1906.06378-3-13-1': 'RFE eliminates the irrelevant features by recursively ranking feature importance and pruning the least important ones.', '1906.06378-3-13-2': 'In our case, it reduced the dimensionality from 63 to 29 (see Table S2 of the SI).', '1906.06378-3-13-3': 'We also used random forest algorithm for feature dimensionality reduction.', '1906.06378-3-13-4': 'In particular, we trained the data set of 95 points using 100 trees, and used the feature importance/weight to determine the relevance of the features.', '1906.06378-3-13-5': 'As discussed in Section 2 of the SI, nearly 40 features were identified to be important using the random forest method, most of which were found to be consistent to those retained from the RFE scheme discussed earlier.', '1906.06378-3-13-6': 'This provides more confidence to the RFE based dimensionality reduction step performed in this work.', '1906.06378-3-13-7': 'Overall, the 29-dimensional feature vector obtained after RFE resulted in more accurate models than the original 63-dimensional feature, as will be discussed in detail next, while a detailed comparison of the RFE and random forest methods is provided in SI.', '1906.06378-3-14-0': '## Gaussian Process Regression', '1906.06378-3-15-0': 'The Gaussian process regression (GPR) with the radial basis function (RBF) kernel was utilized to train the ML models.', '1906.06378-3-15-1': 'In this case, the co-variance function between two materials with features [MATH] and [MATH] is given by [EQUATION]', '1906.06378-3-15-2': 'Here, three hyper parameters [MATH], [MATH] and [MATH] signify the variance, the length-scale parameter and the expected noise in the data, respectively.', '1906.06378-3-15-3': 'These hyper parameters were determined during the training of the models by maximizing the log-likelihood estimate.', '1906.06378-3-15-4': 'Further, 5-fold CV was adopted to avoid overfitting.', '1906.06378-3-15-5': 'Two error metrics, namely, the root mean square error (RMSE) and the coefficient of determination ([MATH]), were used to evaluate the performance of the ML models.', '1906.06378-3-15-6': 'To estimate the prediction errors on unseen data, learning curves were generated by varying the size of the training and the test sets.', '1906.06378-3-15-7': 'We note that the test sets were obtained by excluding the training points from the data set of 95 points.', '1906.06378-3-15-8': 'The left-out set of 5 points was completely separated from the learning process, and was used for just evaluation purposes on a few ""extrapolative"" material cases.', '1906.06378-3-15-9': 'Additionally, for each case, statistically meaningful results were obtained by averaging RMSE results over 100 runs with random training and test splits.', '1906.06378-3-16-0': '# Results and Discussion', '1906.06378-3-17-0': 'It is worth analyzing the correlation between these 29 features and the empirically measured [MATH] to see how much trend is captured by these elemental, structural and chemical attributes.', '1906.06378-3-17-1': 'While in Figure [REF] we plot the [MATH] vs four important features, the corresponding plots for the remaining cases are provided in Figure S3 of SI.', '1906.06378-3-17-2': 'A strong positive correlation between [MATH]) and bulk modulus, and a strong inverse relation between [MATH]) and the mean average bond length are evident from the figure.', '1906.06378-3-17-3': 'While density alone does not show a strong correlation with [MATH]), the combined feature [MATH] does indeed show a very strong linear relation.', '1906.06378-3-17-4': 'This is in-line with the physical understanding that group velocity, which is an integral part of the semi-empirical models discussed earlier, is related to the lattice anharmonic force constants, and can be approximated as [MATH].', '1906.06378-3-17-5': 'Thus, bulk modulus can be considered to play a critical role in influencing the [MATH] of different inorganic non-metals.', '1906.06378-3-17-6': 'Similarly, the inverse relationship between [MATH]) and the mean average bond length is also physically meaningful as when the bonds are shorter, the bond-strength anharmonicity are stronger, and the resulting [MATH] is larger.', '1906.06378-3-17-7': 'For the case of mean atomic mass (a common feature used in the past ML model works), a slightly dispersed relationship is observed, indicating that it may be less important in governing [MATH], as was the case with the rest of the 25 features illustrated in Figure S3 of the SI.', '1906.06378-3-18-0': 'Next, the performance of the ML models can be evaluated from the learning curves presented in Figure [REF](a), wherein average RMSE on the training and the test sets as a function of training set size are included.', '1906.06378-3-18-1': 'The error bars denote the 1[MATH] deviation in the reported RMSE values over 100 runs.', '1906.06378-3-18-2': 'Results using both the initial set of 63 features (GPR), and those for the reduced 29 features (GPR-RFE) are included.', '1906.06378-3-18-3': 'Clearly, the RFE dimensionality reduction step leads to improved model performance with lower test errors, which signify better generalization of these models for unseen data.', '1906.06378-3-18-4': 'As expected, the test RMSE of both the GPR and the GPR-RFE models decreases with increase in training set size, reaching a convergence of [MATH] in test error and of [MATH] in train error for GPR-RFE models when the training set is about 80 % of the data (i.e. 76 points).', '1906.06378-3-18-5': 'Figure [REF](b) and (c) show the performance of GPR-RFE models via the example parity plots (i.e., ML predicted vs experimental log([MATH])), using 76 and 95 train points, respectively.', '1906.06378-3-18-6': 'The error bars in these cases represent the GPR uncertainty.', '1906.06378-3-18-7': 'Pretty high [MATH] coefficient ([MATH]) on the test set in both these cases suggest a good [MATH] model has indeed been developed.', '1906.06378-3-19-0': 'We compared the performance of our ML model with other semi-empirical models by computing the average factor difference (AFD) [CITATION], using the definition [MATH], where [MATH], with [MATH] being the number of data points.', '1906.06378-3-19-1': 'As shown in Table [REF], the computed AFD of GPR-RFE models using the entire set of 95 points is 1.36 [MATH]0.03, which is comparable to the reported values of 1.38 and 1.46, respectively, obtained using the Slack [CITATION] and Debye-Callaway [CITATION] models.', '1906.06378-3-19-2': 'More importantly, the latter two ML models rely on the experimental/computed features that are much more difficult to obtain owing to their dependence on the use of the Slack or Debye-Callaway models.', '1906.06378-3-19-3': 'The ML model presented here uses easily and rapidly accessible chemical and structural features derived directly from the identity of the material, making it more inexpensive and flexible.', '1906.06378-3-19-4': 'In addition, the predicted GPR uncertainty can be used to guide the next experiments via active learning [CITATION].', '1906.06378-3-19-5': 'Further, we note that the possibility of further diversifying our ML model with data from first-principles or semi-empirical methods using multi-fidelity fusion approaches also exists [CITATION].', '1906.06378-3-20-0': 'In order to further validate the generality and the accuracy of our ML models, we used the GPR-RFE models trained on the entire set of 95 points (see Figure [REF](a)) to predict the log([MATH]) of 5 unseen inorganic solids with various space group numbers present in the hold-out set.', '1906.06378-3-20-1': 'These include Sc[MATH]O[MATH] (206), Ga[MATH]O[MATH] (12), MnO (225), AlCuO[MATH] (166), and Ca[MATH]Al[MATH]Sb[MATH] (55), where the number within brackets is the space group number.', '1906.06378-3-20-2': 'Figure [REF](c) shows the comparison between the predicted and the experimental log([MATH]), with error bars capturing the GPR uncertainty.', '1906.06378-3-20-3': 'A good performance for these 5 unseen data points is clearly evident.', '1906.06378-3-20-4': 'The high GPR uncertainty in the case of Ca[MATH]Al[MATH]Sb[MATH] correctly signals its space group number differences from that of the majority training data, and the application of the ML model in the ""extrapolative"" regime.', '1906.06378-3-20-5': 'Overall, the results presented here strongly advocate the good performance of the GPR-RFE models developed, which can be used to provide an inexpensive and accurate [MATH] prediction for other inorganic materials, especially for materials with rock-salt or zincblende structures.', '1906.06378-3-21-0': 'Additionally, it is noteworthy that the space group number is one of the important features in our ML model, although it has little physical meaning beyond allowing the model to distinguish between different structure types.', '1906.06378-3-21-1': 'If we intentionally eliminated it from the 29 features, the test RMSE of 5 unseen materials increases from 0.12 to 0.39, as shown in Figure [REF](d).', '1906.06378-3-21-2': 'This issue is due to the limitation of our present dataset, most of which belong to space groups 225 and 216.', '1906.06378-3-21-3': 'As a result, the space group number is required to distinguish materials in terms of their structures in the ML model.', '1906.06378-3-21-4': 'However, this problem can be solved when more data with more diverse space groups are included in the training dataset.', '1906.06378-3-21-5': 'Furthermore, our present ML model is more suitable for defect-free inorganic materials.', '1906.06378-3-21-6': 'There are some accuracy limitations of our model to predict [MATH] of materials with defects, allotropic materials and intermetallic compounds.', '1906.06378-3-21-7': 'However, the predictive ML model can be more easily improved by actively learning on more diverse training (even temperature-dependent) data sets compared with previous semi-empirical models, due to the easily accessible features.', '1906.06378-3-22-0': '# Conclusion', '1906.06378-3-23-0': 'In conclusion, we have developed a simple and general ML model to predict [MATH] of inorganic solid materials.', '1906.06378-3-23-1': 'This model is faster, and at par or more accurate than traditional physics-based computational methods.', '1906.06378-3-23-2': 'This work involves curating a benchmark dataset of experimental values of [MATH] of 100 inorganic compounds, generating and optimizing a comprehensive set of features (using the Matminer package), and training the Gaussian Process Regression model on the data prepared.', '1906.06378-3-23-3': 'The accuracy of the developed ML models was found to be comparable to past semi-empirical models.', '1906.06378-3-23-4': 'Additionally, key features in determining [MATH] were identified.', '1906.06378-3-23-5': 'Overall, this present work would be useful for rational design and screening of new materials with desired [MATH] for specific applications, and fundamentally understanding the heat transport in inorganic solid materials.', '1906.06378-3-24-0': '# Data Availability', '1906.06378-3-25-0': 'The entire experimental [MATH] data set and DFT computed bulk modulus are available in Table S1 of the Supporting Information.'}",,
0911.3954,"{'0911.3954-1-0-0': 'We address the problem of two interacting atoms of different species inside a cavity and find the explicit solutions of the corresponding eigenvalue problem.', '0911.3954-1-0-1': 'Closed expressions for concurrence and purity as a function of time when the cavity is prepared in a number state are found.', '0911.3954-1-0-2': 'The behavior in the concurrence-purity plane is discussed.', '0911.3954-1-1-0': 'The system of two two-level atoms (TLA) inside a cavity has attracted considerable attention, both because it has become experimentally feasible and because it is the paradigm to study the evolution of entanglement under decoherence.', '0911.3954-1-1-1': 'This combination is remarkable, because entanglement is a central resource and decoherence the major impediment for quantum information processing [CITATION].', '0911.3954-1-1-2': 'The relation between concurrence and purity of the central system yields the simplest access to the problem.', '0911.3954-1-2-0': 'Different models of two identical TLA as a central system coupled to a cavity mode in resonance with the atomic transition as environment have been studied [CITATION].', '0911.3954-1-2-1': 'In this letter, we show that one can define a wider class of such systems that remains solvable in closed form and includes the above mentioned cases.', '0911.3954-1-2-2': 'Specifically we consider atoms with different coupling to the cavity mode, different detuning and include dipole-dipole as well as Ising interactions between the atoms.', '0911.3954-1-3-0': 'We show that the total number of excitations is still a conserved quantity.', '0911.3954-1-3-1': 'Using the basis in which the corresponding operator is diagonal, the Hamiltonian will be transformed to block diagonal form, with maximally [MATH] blocks.', '0911.3954-1-3-2': 'Interestingly we could use a special case of this solution to construct an exactly solvable relativistic model [CITATION] with three degrees of freedom, namely a Dirac oscillator [CITATION] coupled to an isospin field.', '0911.3954-1-4-0': 'Finally we shall proceed to apply the closed solution to study the evolution of concurrence and purity of two interacting TLA with equal coupling and zero detuning but arbitrary dipole-dipole and Ising interactions.', '0911.3954-1-4-1': 'The interaction free case basically provides the borders of the evolution of the interacting problem in a concurrence-purity (CP) diagram, a third boundary being provided by the relative strength of the two interactions.', '0911.3954-1-5-0': 'The case analyzed should be experimentally feasible in cavity QED [CITATION].', '0911.3954-1-5-1': 'While dipole-dipole interactions commonly appear in QED, an Ising interaction might be simulated as proposed in references [CITATION].', '0911.3954-1-5-2': 'To relate experiment and theory it will be important that the actual solutions available are more general, as the exact conditions of the example might not be met.', '0911.3954-1-6-0': 'We consider the following Hamiltonian for two TLA coupled to a cavity mode in the interaction picture and rotating-wave approximation ([MATH]):', '0911.3954-1-7-0': 'where [MATH] is the detuning of the corresponding atomic transition frequency from the cavity mode, [MATH] is the coupling to the mode, [MATH] and [MATH] are the strengths of the dipole-dipole and Ising interaction respectively.', '0911.3954-1-7-1': 'We use the standard definitions of creation and annihilation operators for the harmonic oscillator ([MATH]) and for the raising and lowering operators [MATH], with the Pauli matrices ([MATH], [MATH], [MATH]).', '0911.3954-1-8-0': 'The operator [MATH] provides an additional constant of motion and it can be interpreted as the number of excitations in the system.', '0911.3954-1-8-1': 'Clearly, [MATH] and in general this is the only commuting observable of this problem.', '0911.3954-1-8-2': 'Therefore we choose the following basis for which [MATH] is diagonal', '0911.3954-1-9-0': 'Here [MATH] describes a state of [MATH] photons in the cavity, [MATH] and [MATH] describe the ground and excited states of a TLA respectively.', '0911.3954-1-9-1': 'For any given [MATH] they satisfy the relation [MATH].', '0911.3954-1-9-2': 'In this basis [MATH] is a block-diagonal matrix and each block [MATH] is a [MATH] matrix with elements [MATH].', '0911.3954-1-9-3': 'Explicitly, one has', '0911.3954-1-10-0': 'For [MATH], the basis is reduced to the triplet [MATH], [MATH] and [MATH].', '0911.3954-1-10-1': 'For [MATH] it is reduced to the singlet [MATH].', '0911.3954-1-10-2': 'The singlet is stationary and represents the situation where both atoms are in the ground state and there are no photons in the cavity.', '0911.3954-1-11-0': 'Solving the resulting eigenvalue problem implies diagonalizing each block of the Hamiltonian.', '0911.3954-1-11-1': 'In general the characteristic polynomial for the eigenvalues leads to a depressed quartic equation with eigenvalues: [EQUATION] where we used the following definitions:', '0911.3954-1-12-0': 'The eigenvectors before normalization read', '0911.3954-1-13-0': 'and the orthogonal transformation which diagonalizes the Hamiltonian is given by [MATH].', '0911.3954-1-14-0': ""Since our aim is to calculate the entanglement between the two atoms, we treat the cavity's degree of freedom as an environment."", '0911.3954-1-14-1': 'For this purpose it is convenient to start with product states of cavity and central system functions.', '0911.3954-1-14-2': 'We restrict ourselves to a definite value of the observable [MATH] and choose a number state for the cavity i.e. [EQUATION]', '0911.3954-1-14-3': 'In the same subspace of fixed eigenvalue of [MATH], one could also use the state [MATH] or [MATH] as initial product states.', '0911.3954-1-14-4': 'This type of initial state guarantees that the evolution stays confined in a four dimensional subspace.', '0911.3954-1-14-5': 'The time evolution of the state vector under the Hamiltonian ([REF]), can be written as [EQUATION] with the following coefficients [EQUATION]', '0911.3954-1-14-6': 'For readability, we shall omit the time dependence in the coefficients, [MATH].', '0911.3954-1-15-0': 'Starting from the density matrix of the whole system [MATH], we take a partial trace over the cavity degree of freedom to compute the reduced density matrix of the two TLA, given by [EQUATION]', '0911.3954-1-15-1': 'The purity [MATH] measures the entanglement between the central system and the environment, i.e. the decoherence of the two TLA and we find', '0911.3954-1-16-0': 'The concurrence [CITATION] is used to measure the entanglement between the atoms.', '0911.3954-1-16-1': 'It is defined as [MATH], where [MATH] are the eigenvalues of [MATH] in non-increasing order.', '0911.3954-1-16-2': 'In our case the concurrence is given by', '0911.3954-1-17-0': 'Some interesting features can already be inferred by inspecting ([REF]) and ([REF]).', '0911.3954-1-17-1': 'For [MATH] we have [MATH] and the purity has a minimum value of [MATH].', '0911.3954-1-17-2': 'As for the concurrence one can note the absence of entanglement sudden death [CITATION] in that particular case.', '0911.3954-1-18-0': 'Now we specialize in the symmetric case with equal couplings to the cavity, zero detunings and no interaction between the atoms.', '0911.3954-1-18-1': 'We find explicit solutions in time (not shown here) which in turn can be inverted to find an explicit relation of concurrence in terms of purity.', '0911.3954-1-18-2': 'Concurrence is represented by up to two different Curves in the CP-Plane', '0911.3954-1-19-0': 'and with', '0911.3954-1-20-0': 'We find two separate cases:', '0911.3954-1-21-0': 'For [MATH], the concurrence in the CP-plane is determined by the two curves:', '0911.3954-1-22-0': 'Otherwise, the concurrence is determined only by the curve: [EQUATION]', '0911.3954-1-22-1': 'In figure [REF] we show these solutions in the CP-Plane for [MATH] and different values of [MATH].', '0911.3954-1-22-2': 'The frontier of the gray zone corresponds to the so-called maximally entangled mixed states (MEMS).', '0911.3954-1-22-3': 'The gray dashed thick line corresponds to the Werner states [CITATION] .', '0911.3954-1-22-4': 'The red curve shows the case when the starting state is the symmetric Bell state, [MATH].', '0911.3954-1-22-5': 'This solution has the explicit form [EQUATION] and it can be seen that a certain region coincides with the curve of the MEMS.', '0911.3954-1-22-6': 'The dashed blue curve represents the situation with an initial state determined by [MATH] and [MATH].', '0911.3954-1-23-0': 'The behaviour for [MATH] is qualitatively the same and when one takes as initial state the symmetric Bell state, [MATH],the curves converge [MATH] to', '0911.3954-1-24-0': 'This convergence is quite fast.', '0911.3954-1-24-1': 'Actually in figure [REF] we took [MATH] and the red curve is a very good approximation to [MATH].', '0911.3954-1-25-0': 'Some interesting features are found if one takes into account the interaction between the atoms.', '0911.3954-1-25-1': 'We now give explicit solutions of this case in time domain.', '0911.3954-1-25-2': 'A closed solution in the [MATH]Plane is now more complicated due to the inclusion of a new frequency.', '0911.3954-1-25-3': 'Using the frequency [EQUATION] which is closely related with the eigenvalues of the Hamiltonian ([REF]), and the time-dependent functions', '0911.3954-1-26-0': 'we find the following solutions for purity and concurrence', '0911.3954-1-27-0': 'Figures [REF] and [REF] show in black the curves for the interacting case for the same initial states as the blue dashed curves.', '0911.3954-1-27-1': 'One can note that in the [MATH]-plane the curves now form a Lissajous-like figures with their frontier defined by the curves [MATH] for the starting value of [MATH] (lower frontier) and for [MATH] (upper frontier).', '0911.3954-1-27-2': 'Note that for increasing values of either of the interactions the curve in the [MATH]-Plane does not fill the entire region enclosed by the curves [MATH].', '0911.3954-1-27-3': 'If both interactions have the same strength the curve will fill again the entire area, except in the case when there are commensurable frequencies in [MATH] and [MATH], eq. ([REF]).', '0911.3954-1-27-4': 'That is the case of figure [REF] [MATH] where the black curve is closed.', '0911.3954-1-27-5': 'The minimum value of the purity can be calculated as [MATH].', '0911.3954-1-27-6': 'A separate case arises when [MATH] and [EQUATION] the minimum value is [MATH].', '0911.3954-1-28-0': 'We have shown, that many solvable models discussed for two TLA in a cavity belong to a wider class of exactly solvable models with applications well beyond quantum optics.', '0911.3954-1-28-1': 'The usefulness of such models was displayed by calculating the evolution of concurrence and purity in a particular, but interacting case where interesting evolutions in a CP diagram where shown.', '0911.3954-1-28-2': 'The new results are experimentally accessible, particularly as the general solution is given and thus the corresponding results are implicitly available if parameters of the experiment are slightly different.', '0911.3954-1-28-3': 'The parameter space will be further explored in a full length paper.', '0911.3954-1-28-4': 'Interesting situations include placing one TLA outside the cavity or at a node of the mode and using different detuning or more complicated initial states.', '0911.3954-1-29-0': 'An interesting perspective would be to extend this technique to situations, where the couplings are chosen such that -rather than four dimensional spaces where exact solutions are available- we would have larger but finite spaces known in molecular physics as polyades, which are accessible to treatments with Lie algebraic [CITATION] and semi-classical [CITATION] techniques.'}","{'0911.3954-2-0-0': 'We address the problem of two interacting atoms of different species inside a cavity and find the explicit solutions of the corresponding eigenvalues and eigenfunctions using a new invariant.', '0911.3954-2-0-1': 'This model encompasses various commonly used models.', '0911.3954-2-0-2': 'By way of example we obtain closed expressions for concurrence and purity as a function of time for the case where the cavity is prepared in a number state.', '0911.3954-2-0-3': 'We discuss the behaviour of these quantities and and their relative behaviour in the concurrence-purity plane.', '0911.3954-2-1-0': 'The system of two two-level atoms (TLA) inside a cavity has attracted considerable attention, both because it has become experimentally feasible and because it is the paradigm to study the evolution of entanglement under decoherence.', '0911.3954-2-1-1': 'This combination is remarkable, because entanglement is a central resource and decoherence the major impediment for quantum information processing [CITATION].', '0911.3954-2-1-2': 'The relation between concurrence and purity of the central system yields the simplest access to the problem.', '0911.3954-2-2-0': 'Different models of two identical TLA as a central system coupled to a cavity mode in resonance with the atomic transition as environment have been studied [CITATION].', '0911.3954-2-2-1': 'In this paper, we show that one can define a wider class of such systems that remains solvable in closed form and includes the above mentioned cases.', '0911.3954-2-2-2': 'Specifically we consider atoms with different coupling to the cavity mode, different detuning and include dipole-dipole as well as Ising interactions between the atoms.', '0911.3954-2-3-0': 'We show that the total number of excitations is a conserved quantity.', '0911.3954-2-3-1': 'Using the basis in which the corresponding operator is diagonal, the Hamiltonian will be transformed to block diagonal form, with maximally [MATH] blocks.', '0911.3954-2-3-2': 'Interestingly we could use a special case of this solution to construct an exactly solvable relativistic model [CITATION] with three degrees of freedom, namely a Dirac oscillator [CITATION] coupled to an isospin field.', '0911.3954-2-4-0': 'Note that models with different interacting TLA and a single excitation on a continuum of modes have been solved [CITATION] using the pseudomode approach [CITATION] which in those cases results in a single mode with losses.', '0911.3954-2-4-1': 'While quite similar, the loss term violates the conservation law we use and thus these are not particular cases of our model.', '0911.3954-2-5-0': 'In order to focus on a particular new aspect, namely the interplay of Ising and dipole-dipole interaction, we shall choose an example where other features of our model are simplified.', '0911.3954-2-5-1': 'Thus we shall apply the closed solution to study the time evolution of concurrence and purity of two interacting TLA with equal coupling and zero detuning but arbitrary dipole-dipole and Ising interactions.', '0911.3954-2-5-2': 'The interaction free case basically provides the borders of the evolution if we look at the interacting problem in a concurrence-purity ([MATH]) diagram, a third boundary being provided by the relative strength of the two interactions.', '0911.3954-2-6-0': 'Particular cases of the general model, for which solutions are available, should be experimentally feasible in cavity QED [CITATION].', '0911.3954-2-6-1': 'While dipole-dipole interactions commonly appear in QED, an Ising interaction might be simulated as proposed in references [CITATION].', '0911.3954-2-6-2': 'Wether a particular case, such as the one we discuss, will actually be measured depends on specific difficulties in forming the initial state, as well as the amount of interest such a case may arouse.', '0911.3954-2-6-3': 'Some such cases, including initial coherent states, will be studied in a forthcoming paper [CITATION].', '0911.3954-2-7-0': 'Consider the Hamiltonian for two TLA coupled to a cavity mode and set [MATH], we use the rotating wave approximation and work in the interaction picture so we end up with [EQUATION] where [MATH] is the detuning of the corresponding atomic transition frequency from the frequency of the cavity mode which does not appear due to our choice of the interaction picture.', '0911.3954-2-8-0': '[MATH] is the coupling to the mode, [MATH] and [MATH] are the strengths of the dipole-dipole and Ising interactions respectively.', '0911.3954-2-8-1': 'We use the standard definitions of creation and annihilation operators for the harmonic oscillator ([MATH]) and for the raising and lowering operators [MATH], with the Pauli matrices ([MATH], [MATH], [MATH]).', '0911.3954-2-9-0': 'The operator [MATH] provides an additional constant of motion and it can be interpreted as the number of excitations in the system.', '0911.3954-2-9-1': 'Clearly, [MATH] and in general this is the only commuting observable of this problem.', '0911.3954-2-9-2': 'Therefore we choose the following basis for which [MATH] is diagonal [EQUATION]', '0911.3954-2-9-3': 'Here [MATH] describes a state of [MATH] photons in the cavity, [MATH] and [MATH] describe the ground and excited states of a TLA respectively.', '0911.3954-2-9-4': 'For any given [MATH] they satisfy the relation [MATH].', '0911.3954-2-9-5': 'In this basis [MATH] is a block-diagonal matrix and each block [MATH] is a [MATH] matrix with elements [MATH].', '0911.3954-2-9-6': 'Explicitly, one has [EQUATION]', '0911.3954-2-9-7': 'For [MATH], the basis is reduced to the three states [MATH], [MATH] and [MATH].', '0911.3954-2-9-8': 'For [MATH] it is reduced to one single [MATH].', '0911.3954-2-9-9': 'This single state is stationary and represents the situation where both atoms are in the ground state and there are no photons in the cavity.', '0911.3954-2-10-0': 'Solving the resulting eigenvalue problem implies diagonalizing each block of the Hamiltonian.', '0911.3954-2-10-1': 'In general the characteristic polynomial for the eigenvalues leads to a depressed quartic equation with eigenvalues: [EQUATION] where we used the following definitions: [EQUATION]', '0911.3954-2-10-2': 'The eigenvectors before normalization read as [EQUATION] and the orthogonal transformation which diagonalizes the Hamiltonian is given by [MATH].', '0911.3954-2-11-0': 'By way of example we now treat a special case where we calculate the entanglement and purity of the pair of atoms considering the cavity mode as environment.', '0911.3954-2-11-1': 'For this purpose it is convenient to start with product states of cavity and central system functions.', '0911.3954-2-11-2': 'We restrict ourselves to a definite value of the observable [MATH] and choose a number state for the cavity i.e. [EQUATION]', '0911.3954-2-11-3': 'In the same subspace of fixed eigenvalue of [MATH], one could also use the state [MATH] or [MATH] as initial product states.', '0911.3954-2-11-4': 'This type of initial state guarantees that the evolution stays confined in a four dimensional subspace.', '0911.3954-2-11-5': 'The time evolution of the state vector under the Hamiltonian ([REF]), can be written as [EQUATION] with the following coefficients [EQUATION]', '0911.3954-2-11-6': 'For readability, we shall omit the time dependence in the coefficients, [MATH].', '0911.3954-2-12-0': 'Starting from the density matrix of the whole system [MATH], we take a partial trace over the cavity degree of freedom to compute the reduced density matrix of the two TLA, given by [EQUATION]', '0911.3954-2-12-1': 'The purity [MATH] measures the entanglement between the central system and the environment, i.e. the decoherence of the two TLA and we find [EQUATION]', '0911.3954-2-12-2': 'The concurrence [CITATION] is used to measure the entanglement between the atoms.', '0911.3954-2-12-3': 'It is defined as [MATH], where [MATH] are the eigenvalues of [MATH] in non-increasing order.', '0911.3954-2-12-4': 'In our case the concurrence is given by [EQUATION]', '0911.3954-2-12-5': 'Some interesting features can already be inferred by inspecting ([REF]) and ([REF]).', '0911.3954-2-12-6': 'For [MATH] we have [MATH] and the purity has a minimum value of [MATH].', '0911.3954-2-12-7': 'As for the concurrence one can note the absence of entanglement sudden death [CITATION] in that particular case.', '0911.3954-2-13-0': 'Now we specialize in the symmetric case with equal couplings to the cavity, zero detunings, but allow both types of interactions between the atoms.', '0911.3954-2-13-1': 'With these restrictions we are able to find explicit solutions in the time domain.', '0911.3954-2-13-2': 'Using the definitions [EQUATION] where [MATH] is a frequency closely related with the eigenvalues of the Hamiltonian ([REF]), and the time-dependent functions [EQUATION] we find the following solutions for the purity and the concurrence as functions of time [EQUATION]', '0911.3954-2-13-3': 'In figure [REF] we present these solutions for the situation where there are no initial photons in the cavity, namely [MATH].', '0911.3954-2-13-4': 'We present three cases: in red and blue for noninteracting atoms with initial states defined by [MATH] and [MATH] respectively and an interacting case with [MATH], [MATH] and [MATH] shown in black.', '0911.3954-2-13-5': 'Both quantities, concurrence and purity, display oscillatory behaviour, with one frequency in the noninteracting case and two frequencies in the interacting case as can be verified in eqs. ([REF]) and ([REF]).', '0911.3954-2-13-6': 'One can also note that with interaction (black curve) the concurrence increases while the minimum value of purity is greater in contrast to the corresponding noninteracting case (blue-dashed curve).', '0911.3954-2-13-7': 'Similar behaviour in time domain has already been studied in other references like [CITATION].', '0911.3954-2-14-0': 'The graphs in the time domain look pretty standard and this does not change if both interactions are present.', '0911.3954-2-14-1': 'It is therefore convenient to visualize the joint dynamics in a concurrence vs purity plane, the [MATH]-Plane.', '0911.3954-2-14-2': 'Figure [REF] [MATH] shows the corresponding plane for the curves in figure [REF] with the same colour code, but now the black curve is parametrized up to [MATH].', '0911.3954-2-14-3': 'In this plane we have plotted to guide the eye, a gray zone corresponding to the concurrence and purity combinations that can not be obtained in physical states and its lower frontier corresponds to the maximally entangled mixed states (MEMS), which for a given value of the purity maximize the concurrence [CITATION].', '0911.3954-2-14-4': 'The gray dashed line is defined by the Werner states [MATH], [MATH] [CITATION].', '0911.3954-2-15-0': 'One can note as well, that the dynamic of the interacting (black) case is enclosed by the noninteracting curves, the lower bound given by the blue curve with the same initial state as the black one, while the upper bound given by an initial state given by [MATH].', '0911.3954-2-15-1': 'Perhaps the most important feature here, that one can not easily visualize in the time domain, is that for an initial bell state with [MATH] and no interaction between the atoms, the curve (red) follows precisely, as we shall prove below, the one that determines the mentioned MEMS.', '0911.3954-2-15-2': 'For this we need first to obtain the analytic solutions in the [MATH]-plane.', '0911.3954-2-16-0': 'We take the explicit solutions in time in eqs. ([REF]), with [MATH], and invert them to find an explicit relation of the concurrence in terms of the purity.', '0911.3954-2-16-1': 'In this non-interacting case, concurrence is represented by up to two different curves in the [MATH]-Plane [EQUATION] with [MATH] and [MATH] as given in eq. ([REF])and with [EQUATION]', '0911.3954-2-16-2': 'We find two separate cases:', '0911.3954-2-17-0': 'For [MATH], the concurrence in the [MATH]-plane is determined by the two curves: [EQUATION]', '0911.3954-2-17-1': 'Otherwise, the concurrence is determined only by the curve: [EQUATION]', '0911.3954-2-17-2': 'In figure [REF] we show these solutions in the CP-Plane for [MATH] and different values of [MATH].', '0911.3954-2-17-3': 'The red curve shows the case when the starting state is the symmetric Bell state, [MATH].', '0911.3954-2-17-4': 'This solution has the explicit form [EQUATION] and it can be seen that in a certain region it coincides with the curve for the MEMS.', '0911.3954-2-17-5': 'In fact [MATH] coincides precisely with the curve of the MEMS for [MATH].', '0911.3954-2-17-6': 'The dashed blue curve represents the situation with an initial state determined by a pure but not fully entangled state.', '0911.3954-2-18-0': 'The behaviour for [MATH] is qualitatively the same and when one takes as initial state the symmetric Bell state, [MATH],the curves converges [MATH] to [EQUATION]', '0911.3954-2-18-1': 'Actually in figure [REF] we took [MATH] and the red curve is a very good approximation to [MATH].', '0911.3954-2-18-2': 'We note however that in the limit [MATH] this curve, which is actually an upper bound, lies below the werner curve.', '0911.3954-2-18-3': 'For finite [MATH] the curves [MATH] intersects the werner curve in an additional point apart from [MATH].', '0911.3954-2-18-4': 'This means that there is a small region (hardly visible in figure [REF]) above the werner curve that can be reached by the dynamics.', '0911.3954-2-18-5': 'We do not write here explicit expressions for the dashed blue curves, as they can be obtained from equations ([REF]) and ([REF]).', '0911.3954-2-19-0': 'Figures [REF] and [REF] show in black the curves for the interacting case for the same initial states as the blue dashed curves.', '0911.3954-2-19-1': 'One can note that in the [MATH]-plane the curves now form a Lissajous-like figures with their frontier defined by the curves [MATH] for the starting value of [MATH] (lower frontier) and for [MATH] (upper frontier).', '0911.3954-2-19-2': 'Note that for increasing values of the difference of the interactions the curve in the [MATH]-Plane does not fill the entire region enclosed by the curves [MATH].', '0911.3954-2-19-3': 'The region filled by the black curve in figure [REF] [MATH] is smaller than in [REF] [MATH], because in [REF] [MATH] we use a larger value of [MATH].', '0911.3954-2-19-4': 'The minimum value of the purity can be calculated as [MATH], this is the lower bound for [MATH] for the black curves in the figures [REF][MATH] and [REF][MATH].', '0911.3954-2-19-5': 'A separate case arises when [MATH] and [EQUATION] the minimum value is [MATH], figure [REF][MATH].', '0911.3954-2-19-6': 'If both interactions have the same strength the curve will fill again the entire area, except in the case when there are commensurable frequencies in [MATH] and [MATH], equation ([REF]).', '0911.3954-2-19-7': 'That is the case of figure [REF] [MATH] where the black curve is closed.', '0911.3954-2-20-0': 'We have given closed solutions for the dynamics of two different TLA in a cavity interacting by dipole-dipole and Ising interaction.', '0911.3954-2-20-1': 'Many solvable models discussed for two TLA in a cavity belong to this wider class of exactly solvable models including a model for a Dirac Oscillator outside the realm of quantum optics [CITATION].', '0911.3954-2-20-2': 'The effectiveness of the general solution presented was displayed by calculating the evolution of concurrence and purity and fully determining the region of its evolution in a [MATH] diagram in a particular, but interacting case.', '0911.3954-2-20-3': 'Interesting features appear when including both types of interactions.', '0911.3954-2-20-4': 'Intuitively one might think of less decoherence and a more robust entanglement with increasing interaction between the atoms.', '0911.3954-2-20-5': 'This is true if we have either of the interactions, but not necessarily if one takes interactions of similar strength.', '0911.3954-2-21-0': 'The parameter space of the model will be further explored in a full length paper.', '0911.3954-2-21-1': 'Interesting situations include placing one TLA outside the cavity or at a node of the mode and using different detuning as well as coherent or more complicated initial states.', '0911.3954-2-22-0': 'An interesting perspective would be to extend this technique to situations, where the couplings are chosen such that -rather than four dimensional spaces where exact solutions are available- we would have larger but finite spaces known in molecular physics as polyades, which are accessible to treatments with Lie algebraic [CITATION] and semi-classical [CITATION] techniques.', '0911.3954-2-23-0': 'Another worthwhile line of research may be to find an even more general class of solvable models including the one presented here and the ones using a pseudomode approach [CITATION].', '0911.3954-2-24-0': 'Financial support under project IN114310 by PAPIIT, Universidad Nacional Autonoma de Mexico is aknowledged.'}","[['0911.3954-1-5-1', '0911.3954-2-6-1'], ['0911.3954-1-3-1', '0911.3954-2-3-1'], ['0911.3954-1-3-2', '0911.3954-2-3-2'], ['0911.3954-1-22-0', '0911.3954-2-17-1'], ['0911.3954-1-22-1', '0911.3954-2-17-2'], ['0911.3954-1-22-4', '0911.3954-2-17-3'], ['0911.3954-1-14-1', '0911.3954-2-11-1'], ['0911.3954-1-14-2', '0911.3954-2-11-2'], ['0911.3954-1-14-3', '0911.3954-2-11-3'], ['0911.3954-1-14-4', '0911.3954-2-11-4'], ['0911.3954-1-14-5', '0911.3954-2-11-5'], ['0911.3954-1-14-6', '0911.3954-2-11-6'], ['0911.3954-1-27-0', '0911.3954-2-19-0'], ['0911.3954-1-27-1', '0911.3954-2-19-1'], ['0911.3954-1-27-4', '0911.3954-2-19-7'], ['0911.3954-1-29-0', '0911.3954-2-22-0'], ['0911.3954-1-1-0', '0911.3954-2-1-0'], ['0911.3954-1-1-1', '0911.3954-2-1-1'], ['0911.3954-1-1-2', '0911.3954-2-1-2'], ['0911.3954-1-2-0', '0911.3954-2-2-0'], ['0911.3954-1-2-2', '0911.3954-2-2-2'], ['0911.3954-1-7-1', '0911.3954-2-8-1'], ['0911.3954-1-15-0', '0911.3954-2-12-0'], ['0911.3954-1-16-0', '0911.3954-2-12-2'], ['0911.3954-1-16-1', '0911.3954-2-12-3'], ['0911.3954-1-17-0', '0911.3954-2-12-5'], ['0911.3954-1-17-1', '0911.3954-2-12-6'], ['0911.3954-1-17-2', '0911.3954-2-12-7'], ['0911.3954-1-11-0', '0911.3954-2-10-0'], ['0911.3954-1-8-0', '0911.3954-2-9-0'], ['0911.3954-1-8-1', '0911.3954-2-9-1'], ['0911.3954-1-9-0', '0911.3954-2-9-3'], ['0911.3954-1-9-1', '0911.3954-2-9-4'], ['0911.3954-1-9-2', '0911.3954-2-9-5'], ['0911.3954-1-24-1', '0911.3954-2-18-1'], ['0911.3954-1-28-3', '0911.3954-2-21-0'], ['0911.3954-1-28-4', '0911.3954-2-21-1'], ['0911.3954-1-4-0', '0911.3954-2-5-1'], ['0911.3954-1-4-1', '0911.3954-2-5-2'], ['0911.3954-1-3-0', '0911.3954-2-3-0'], ['0911.3954-1-22-5', '0911.3954-2-17-4'], ['0911.3954-1-27-2', '0911.3954-2-19-2'], ['0911.3954-1-27-3', '0911.3954-2-19-6'], ['0911.3954-1-27-6', '0911.3954-2-19-5'], ['0911.3954-1-0-0', '0911.3954-2-0-0'], ['0911.3954-1-2-1', '0911.3954-2-2-1'], ['0911.3954-1-15-1', '0911.3954-2-12-1'], ['0911.3954-1-16-2', '0911.3954-2-12-4'], ['0911.3954-1-8-2', '0911.3954-2-9-2'], ['0911.3954-1-10-0', '0911.3954-2-9-7'], ['0911.3954-1-10-2', '0911.3954-2-9-9'], ['0911.3954-1-23-0', '0911.3954-2-18-0'], ['0911.3954-1-5-0', '0911.3954-2-6-0'], ['0911.3954-1-22-6', '0911.3954-2-17-6'], ['0911.3954-1-14-0', '0911.3954-2-11-0'], ['0911.3954-1-27-5', '0911.3954-2-19-4'], ['0911.3954-1-0-1', '0911.3954-2-0-2'], ['0911.3954-1-0-2', '0911.3954-2-0-3'], ['0911.3954-1-18-1', '0911.3954-2-16-0'], ['0911.3954-1-18-2', '0911.3954-2-16-1'], ['0911.3954-1-7-0', '0911.3954-2-7-0'], ['0911.3954-1-7-0', '0911.3954-2-8-0'], ['0911.3954-1-13-0', '0911.3954-2-10-2'], ['0911.3954-1-10-1', '0911.3954-2-9-8'], ['0911.3954-1-25-3', '0911.3954-2-13-2']]","[['0911.3954-1-5-1', '0911.3954-2-6-1'], ['0911.3954-1-3-1', 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'0911.3954-2-12-5'], ['0911.3954-1-17-1', '0911.3954-2-12-6'], ['0911.3954-1-17-2', '0911.3954-2-12-7'], ['0911.3954-1-11-0', '0911.3954-2-10-0'], ['0911.3954-1-8-0', '0911.3954-2-9-0'], ['0911.3954-1-8-1', '0911.3954-2-9-1'], ['0911.3954-1-9-0', '0911.3954-2-9-3'], ['0911.3954-1-9-1', '0911.3954-2-9-4'], ['0911.3954-1-9-2', '0911.3954-2-9-5'], ['0911.3954-1-24-1', '0911.3954-2-18-1']]","[['0911.3954-1-28-3', '0911.3954-2-21-0'], ['0911.3954-1-28-4', '0911.3954-2-21-1'], ['0911.3954-1-4-0', '0911.3954-2-5-1'], ['0911.3954-1-4-1', '0911.3954-2-5-2'], ['0911.3954-1-3-0', '0911.3954-2-3-0'], ['0911.3954-1-22-5', '0911.3954-2-17-4'], ['0911.3954-1-27-2', '0911.3954-2-19-2'], ['0911.3954-1-27-3', '0911.3954-2-19-6'], ['0911.3954-1-27-6', '0911.3954-2-19-5'], ['0911.3954-1-0-0', '0911.3954-2-0-0'], ['0911.3954-1-2-1', '0911.3954-2-2-1'], ['0911.3954-1-15-1', '0911.3954-2-12-1'], ['0911.3954-1-16-2', '0911.3954-2-12-4'], ['0911.3954-1-8-2', '0911.3954-2-9-2'], ['0911.3954-1-10-0', '0911.3954-2-9-7'], ['0911.3954-1-10-2', '0911.3954-2-9-9'], ['0911.3954-1-23-0', '0911.3954-2-18-0']]",[],"[['0911.3954-1-5-0', '0911.3954-2-6-0'], ['0911.3954-1-22-6', '0911.3954-2-17-6'], ['0911.3954-1-14-0', '0911.3954-2-11-0'], ['0911.3954-1-27-5', '0911.3954-2-19-4'], ['0911.3954-1-0-1', '0911.3954-2-0-2'], ['0911.3954-1-0-2', '0911.3954-2-0-3'], ['0911.3954-1-18-1', '0911.3954-2-16-0'], ['0911.3954-1-18-2', '0911.3954-2-16-1'], ['0911.3954-1-7-0', '0911.3954-2-7-0'], ['0911.3954-1-7-0', '0911.3954-2-8-0'], ['0911.3954-1-13-0', '0911.3954-2-10-2'], ['0911.3954-1-10-1', '0911.3954-2-9-8']]","[['0911.3954-1-25-3', '0911.3954-2-13-2']]","['0911.3954-1-6-0', '0911.3954-1-9-3', '0911.3954-1-11-1', '0911.3954-1-12-0', '0911.3954-1-19-0', '0911.3954-1-20-0', '0911.3954-1-21-0', '0911.3954-1-26-0', '0911.3954-2-16-2']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/0911.3954,,,
1205.5217,"{'1205.5217-1-0-0': 'In the classical setting, the modular equation of level [MATH] for the modular curve [MATH] is the polynomial relation satisfied by [MATH] and [MATH], where [MATH] is the standard elliptic [MATH]-function.', '1205.5217-1-0-1': 'In this paper, we will describe a method to compute modular equations in the setting of Shimura curves.', '1205.5217-1-0-2': 'The main ingredient is the explicit method for computing Hecke operators on the spaces of modular forms on Shimura curves developed in [CITATION].', '1205.5217-1-1-0': 'The author was partially supported by Grant 99-2115-M-009-011-MY3 of the National Science Council, Taiwan (R.O.C.).', '1205.5217-1-2-0': 'Introduction Let [MATH] be the elliptic [MATH]-function.', '1205.5217-1-2-1': 'For a positive integer [MATH], consider the set [EQUATION]', '1205.5217-1-2-2': 'The group [MATH] acts on [MATH] by multiplication on the left and it can be easily checked that the cardinality of [MATH] is finite.', '1205.5217-1-2-3': 'Moreover, the multiplication of any element in [MATH] on the right of the cosets in [MATH] simply permutes the cosets.', '1205.5217-1-2-4': 'Thus, any symmetric sum of [MATH], where [MATH] runs over a complete set of representatives for [MATH], will be a modular function on [MATH] and therefore can be written as a rational function of [MATH].', '1205.5217-1-2-5': 'In fact, because the only possible pole occurs at the cusp, this rational function of [MATH] is actually a polynomial.', '1205.5217-1-2-6': 'Then the modular polynomial [MATH] of level [MATH] is defined to be the polynomial in [MATH] such that [EQUATION].', '1205.5217-1-2-7': 'In fact, it can be easily proved that this polynomial [MATH] is in [MATH].', '1205.5217-1-2-8': 'Since [MATH] is the moduli space of isomorphism classes of elliptic curves over [MATH], the equation [MATH] is the modular equation of level [MATH] for moduli of elliptic curves over [MATH].', '1205.5217-1-2-9': 'That is, if [MATH] and [MATH] are two elliptic curves over [MATH] admitting a cyclic [MATH]-isogeny between them, then their [MATH]-invariants [MATH] and [MATH] satisfy [MATH].', '1205.5217-1-3-0': 'Observe that if the Fourier expansion of [MATH] is [MATH], [MATH], then the Fourier expansion of [MATH] is simply [MATH].', '1205.5217-1-3-1': 'Thus, in principle, to determine [MATH], one just has to compute enough Fourier coefficients for [MATH] and solve a system of linear equations.', '1205.5217-1-3-2': 'Of course, the main difficulty in practice is that the coefficients are gigantic.', '1205.5217-1-3-3': 'On the other hand, because the coefficients of [MATH] are all integers, one can compute the reduction of [MATH] modulo [MATH] for a suitable number of primes [MATH] and then use the Chinese remainder theorem to recover the coefficients.', '1205.5217-1-3-4': 'See [CITATION] for the current state of the art in the computation of [MATH].', '1205.5217-1-4-0': 'In this paper, we shall consider modular equations in the settings of Shimura curves.', '1205.5217-1-4-1': 'Let [MATH] be an indefinite quaternion algebra of discriminant [MATH] over [MATH] and [MATH] be a maximal order in [MATH].', '1205.5217-1-4-2': 'Choose an embedding [MATH] and let [EQUATION] be the image of the norm-one group of [MATH] under [MATH], where [MATH] denotes the reduced norm of [MATH].', '1205.5217-1-4-3': 'Then the Shimura curve [MATH] is defined to be the quotient space [MATH].', '1205.5217-1-4-4': 'As shown in [CITATION], the Shimura curve [MATH] is the coarse moduli space for isomorphism classes of abelian surfaces with quaternionic multiplication (QM) by [MATH].', '1205.5217-1-4-5': 'Let [MATH] denote the group of Atkin-Lehner involutions on [MATH].', '1205.5217-1-4-6': 'For our purpose, we will also consider the quotient curves of [MATH] by subgroups [MATH] of [MATH].', '1205.5217-1-5-0': 'Now assume that [MATH] has genus [MATH] and choose a Hauptmodul [MATH] for [MATH] so that [MATH] generates the function field on [MATH].', '1205.5217-1-5-1': 'For a positive integer [MATH] relatively prime to [MATH], pick an element [MATH] of norm [MATH] in [MATH] such that [MATH] is an Eichler order of level [MATH].', '1205.5217-1-5-2': 'Then the modular polynomial of level [MATH] for [MATH] is defined to be the polynomial [MATH] of minimal degree, up to scalars, such that [MATH], which is essentially the rational function [MATH] such that [EQUATION].', '1205.5217-1-5-3': 'Here unlike the case of the modular curve [MATH], a symmetric sum of [MATH] as [MATH] runs through representatives of [MATH] is not equal to a polynomial of [MATH] in general.', '1205.5217-1-5-4': 'Since [MATH] is the moduli space of abelian surfaces over [MATH] with quaternionic multiplication, the modular equation of level [MATH] relates the moduli of two abelian surfaces with QM that have a certain type of isogenies between them.', '1205.5217-1-5-5': '(The precise description of the isogeny is a little complicated to be given here.', '1205.5217-1-5-6': 'See [CITATION] for details.)', '1205.5217-1-5-7': 'Modular equations for Atkin-Lehner quotients [MATH] are similarly defined.', '1205.5217-1-6-0': 'When [MATH], the problem of explicitly determining modular equations for Shimura curves is significantly more complicated than its classical counterpart.', '1205.5217-1-6-1': 'The reasons are that Shimura curves do not have cusps and it is difficult to find the Taylor expansion of an automorphic function with respect to a local parameter at any given point on the Shimura curve.', '1205.5217-1-6-2': 'When [MATH] and [MATH] are both small such that [MATH] is also of genus [MATH], it is possible to work out an explicit cover [MATH] from the ramification data alone.', '1205.5217-1-6-3': 'Then from the explicit cover, one can compute the modular equation of level [MATH].', '1205.5217-1-6-4': 'This has been done in [CITATION] for a limited number of cases.', '1205.5217-1-7-0': 'Another possible method to compute modular equations for Shimura curves uses the Schwarzian differential equation associated to a Hauptmodul [MATH].', '1205.5217-1-7-1': '(See Section [REF] for a review on the notation and properties of Schwarzian differential equation.)', '1205.5217-1-7-2': 'The idea is that with a properly chosen pair of solutions [MATH] and [MATH] of the Schwarzian differential equation and a correct positive integer [MATH], the expression [MATH] can be taken to be a local parameter at the point [MATH] of the Shimura curve with [MATH].', '1205.5217-1-7-3': 'Inverting the expression, one gets the Taylor expansion of [MATH] with respect to the local parameter.', '1205.5217-1-7-4': 'If somehow one manages to find the Taylor expansion of [MATH] (this can always be done when [MATH], then by computing enough terms and solving a system of linear equations, one gets the modular equation.', '1205.5217-1-7-5': 'However, as far as we know, there does not seem to be any paper in the literature that employs this idea to obtain modular equations for Shimura curves.', '1205.5217-1-7-6': '(The paper [CITATION] did obtain the Taylor expansion for the Hauptmodul in the case [MATH].)', '1205.5217-1-8-0': 'In this paper, we shall present a new method to compute modular equations for Shimura curves.', '1205.5217-1-8-1': 'Our method also uses Schwarzian differential equations, but relies more heavily on the arithmetic side of the theory.', '1205.5217-1-8-2': 'Namely, in an earlier work [CITATION], we showed that the spaces of automorphic forms of any given weight on a Shimura curve [MATH] of genus [MATH] can be completely characterized in terms of solutions of the Schwarzian differential equation.', '1205.5217-1-8-3': 'We [CITATION] then devised a method to compute Hecke operators with respect to our basis.', '1205.5217-1-8-4': 'The Jacquet-Langlands correspondence plays a crucial role in our approach.', '1205.5217-1-8-5': 'It turns out that our method and results in [CITATION] can also be used to compute modular equations for Shimura curves.', '1205.5217-1-8-6': 'We will describe the procedure in Section [REF] and give a detailed example in Section [REF].', '1205.5217-1-8-7': 'Using the computer algebra system MAGMA [CITATION], we have succeeded in determining modular equations of prime level up to [MATH] for [MATH] and those of prime level up to [MATH] for [MATH].', '1205.5217-1-9-0': 'The main difficulty in generalizing our method to general Shimura curves lies at the fact that our method requires that a Schwarzian differential equation is known beforehand.', '1205.5217-1-9-1': 'Because of the problem of the existence of accessory parameters, the determination of Schwarzian differential equations usually requires that an explicit cover [MATH] is known, which can be problematic when [MATH] is large.', '1205.5217-1-9-2': '(In some sense, what we do here is to deduce modular equations of higher levels from that of a given small level.)', '1205.5217-1-9-3': 'Nonetheless, once an explicit cover of Shimura curves is determined, the combination of the methods in [CITATION] and in this paper will yield modular equations for the Shimura curve.', '1205.5217-1-10-0': 'The rest of the paper is organized as follows.', '1205.5217-1-10-1': 'In Section [REF], we review the definition and properties of Schwarzian differential equations.', '1205.5217-1-10-2': 'In Section [REF], we describe our method to compute modular equations for Shimura curves [MATH], assuming that an explicit cover [MATH] is known.', '1205.5217-1-10-3': 'In Section [REF], we give a detailed example illustrating our method.', '1205.5217-1-10-4': 'In Sections [REF] and [REF], we list part of our computational results for the Shimura curves [MATH] and [MATH].', '1205.5217-1-10-5': 'Files in the MAGMA-readable format containing all our computational results are available upon request.', '1205.5217-1-11-0': 'Schwarzian differential equations', '1205.5217-1-12-0': 'In this section, we will review the definition and properties of Schwarzian differential equations.', '1205.5217-1-12-1': 'In particular, assuming the Shimura curve has genus [MATH], we will recall the characterization of the spaces of automorphic forms in terms of solutions of the associated Schwarzian differential equation.', '1205.5217-1-12-2': 'The method for computing Hecke operators on these spaces is too complicated to describe here.', '1205.5217-1-12-3': 'We refer the reader to [CITATION] for details.', '1205.5217-1-12-4': 'Most materials in this section are taken from [CITATION].', '1205.5217-1-13-0': 'Let [MATH] be a Shimura curve.', '1205.5217-1-13-1': 'We assume that the associated quaternion algebra is not [MATH] so that [MATH] has no cusps.', '1205.5217-1-13-2': 'Let [MATH] be a (meromorphic) automorphic form of weight [MATH] and [MATH] be a non-constant automorphic function on [MATH].', '1205.5217-1-13-3': 'It is known since the nineteen century that the functions [MATH], as functions of [MATH], are solutions of a certain [MATH]-st linear ordinary differential equations with algebraic functions as coefficients.', '1205.5217-1-13-4': '(See [CITATION].)', '1205.5217-1-13-5': 'In particular, since [MATH] is a meromorphic automorphic form of weight [MATH], the function [MATH] as a function of [MATH], satisfies a second-order linear ordinary differential equation.', '1205.5217-1-13-6': 'We call this differential equation the Schwarzian differential equation associated to [MATH], which has the following properties.', '1205.5217-1-14-0': '[[CITATION]]', '1205.5217-1-15-0': 'Let [MATH] be a Shimura curve of genus zero with elliptic points [MATH] of order [MATH], respectively.', '1205.5217-1-15-1': 'Let [MATH] be a Hauptmodul of [MATH] and set [MATH], [MATH].', '1205.5217-1-15-2': 'Then [MATH], as a function of [MATH], satisfies the differential equation [EQUATION] where [EQUATION] for some constants [MATH].', '1205.5217-1-15-3': 'Moreover, if [MATH] for all [MATH], then the constants [MATH] satisfy [EQUATION].', '1205.5217-1-15-4': 'Also, if [MATH], then [MATH] satisfy [EQUATION].', '1205.5217-1-16-0': 'We remark that when the Shimura curve has genus [MATH] and precisely [MATH] elliptic points, the relations among [MATH] are enough to determine the constants [MATH].', '1205.5217-1-16-1': 'This reflects the fact in classical analysis that a second-order Fuchsian differential equation with exactly three singularities is completely determined by the local exponents.', '1205.5217-1-16-2': 'When the Shimura curve has more than [MATH] elliptic points, the relations are not enough to determine [MATH].', '1205.5217-1-16-3': 'In literature, we refer to this kind of situations by saying that accessory parameters exist.', '1205.5217-1-16-4': 'In order to determine the accessory parameters, one usually tries to find an explicit cover of Shimura curves and use it to determine the Schwarzian differential equations associated to the two curves simultaneously.', '1205.5217-1-17-0': 'Now one of the key observations in [CITATION] is that the analytic behavior of the Hauptmodul [MATH] is very easy to determine and from this, one can work out a basis for the space of automorphic forms of even weight [MATH] in terms of [MATH].', '1205.5217-1-18-0': '[[CITATION]]', '1205.5217-1-19-0': 'Assume that a Shimura curve [MATH] has genus zero with elliptic points [MATH] of order [MATH], respectively.', '1205.5217-1-19-1': 'Let [MATH] be a Hauptmodul of [MATH] and set [MATH], [MATH].', '1205.5217-1-19-2': 'For a positive even integer [MATH], let [EQUATION].', '1205.5217-1-19-3': 'Then a basis for the space of automorphic forms of weight [MATH] on [MATH] is [EQUATION].', '1205.5217-1-20-0': 'The combination of two propositions shows that all automorphic forms of a given even weight [MATH] can be expressed in terms of the solutions of the Schwarzian differential equation.', '1205.5217-1-20-1': 'In [CITATION], the author developed a method for computing Hecke operators relative to the basis in Proposition [REF].', '1205.5217-1-20-2': 'The key ingredients are the Jacquet-Langlands correspondence and explicit covers of Shimura curves.', '1205.5217-1-20-3': 'We refer the reader to [CITATION] for details.', '1205.5217-1-21-0': 'Computing modular equations for Shimura curves', '1205.5217-1-22-0': 'In this section, we will present a method for computing modular equations for Shimura curves, under the working assumptions that the Schwarzian differential equations have been determined and, for a certain prime fixed [MATH], the matrices for the Hecke operator [MATH] with respect to the bases in Proposition [REF] have already been computed for sufficiently many [MATH] according to the recipe in [CITATION].', '1205.5217-1-23-0': 'Let [MATH] be a Shimura curve of genus [MATH] of the form [MATH] for some subgroup [MATH] of the group of Atkin-Lehner involutions and [MATH] be the discrete subgroup of [MATH] corresponding to [MATH].', '1205.5217-1-23-1': 'For a prime [MATH] not dividing [MATH], let [MATH], [MATH], be representatives for the cosets defining the Hecke operator [MATH], that is, [EQUATION]', '1205.5217-1-23-2': 'Then the modular equation [MATH] of level [MATH] is the polynomial relation between a Hauptmodul [MATH] for [MATH] and [MATH] for arbitrary [MATH].', '1205.5217-1-24-0': 'Pick any nonzero automorphic form [MATH] on [MATH] with the smallest possible weight [MATH].', '1205.5217-1-24-1': 'For convenience, we let [EQUATION] the summands in [REF].', '1205.5217-1-24-2': 'Then any symmetric sum of [MATH], [MATH], will be an automorphic function on [MATH] and hence can be expressed as a rational function of [MATH].', '1205.5217-1-24-3': 'In other words, there is a rational function [MATH] such that [EQUATION]', '1205.5217-1-24-4': 'That is, we have [EQUATION] for all [MATH].', '1205.5217-1-24-5': 'Now let [MATH].', '1205.5217-1-24-6': 'There is an integer [MATH], [MATH], such that [MATH].', '1205.5217-1-24-7': 'We apply the action of [MATH] to [REF] and get [EQUATION].', '1205.5217-1-24-8': 'Thus, if we let [MATH] be the rational function such that [EQUATION] which is a polynomial in [MATH], and compute the resultant [MATH] of [MATH] and [MATH] with respect to [MATH], then we have [MATH], i.e., the modular equation [MATH] appears as one of the irreducible factors in the numerator of the rational function [MATH].', '1205.5217-1-24-9': 'In general, there will be more than one irreducible factors in the numerator of [MATH], but it is not difficult to determine which one corresponds to [MATH].', '1205.5217-1-24-10': 'For example, we can use the fact that the roots of [MATH] are the coordinates of certain CM-points (also known as singular moduli) to test which factor of [MATH] is [MATH].', '1205.5217-1-24-11': 'Thus, from the above discussion, we see that the most critical part of the calculation is the determination of the rational function [MATH], which we address now.', '1205.5217-1-25-0': ""From Newton's identity, we know that the problem of determining the symmetric sums of [MATH] in [REF] is equivalent to that of determining [MATH] for [MATH]."", '1205.5217-1-25-1': 'Observe that the sum of [MATH] is equal to [MATH], by the definition of the Hecke operator [MATH].', '1205.5217-1-25-2': 'Thus, to determine [MATH], one just needs to know how the Hecke operator [MATH] acts on the basis given in Proposition [REF].', '1205.5217-1-25-3': 'This is where the work of [CITATION] comes into play.', '1205.5217-1-26-0': 'In [CITATION], we developed a method to compute [MATH], [MATH] a prime not dividing [MATH], for arbitrary weight [MATH] with respect to the basis given in Proposition [REF], assuming an explicit cover [MATH] is known.', '1205.5217-1-26-1': 'Now by the Jacquet-Langlands correspondence [CITATION], we know that [EQUATION] as Hecke modules.', '1205.5217-1-26-2': 'Here the left-hand side denotes the subspace of the newform subspace of cusp forms of weight [MATH] on [MATH] that has eigenvalue [MATH] for all [MATH], [MATH] primes, and the right-hand side denotes the space of automorphic forms of weight [MATH] on [MATH].', '1205.5217-1-26-3': 'Thus, assuming the Hecke operator [MATH] on the space [MATH] has no repeated eigenvalues, one can obtain the matrices for [MATH] with respect to the bases in Proposition [REF] from those for [MATH] and the Fourier coefficients of Hecke eigenforms in [MATH].', '1205.5217-1-27-0': 'In summary, to compute the modular equation [MATH], we follow the following steps, assuming that the Schwarzian differential equation associated to [MATH] and an explicit cover [MATH] are known for some prime [MATH] not dividing [MATH].', '1205.5217-1-28-0': '[(a)] Pick a nonzero automorphic form [MATH] on [MATH] of the smallest possible weight [MATH], expressed in the form given in Proposition [REF].', '1205.5217-1-28-1': '[(b)] Compute the matrices for [MATH] with respect to the basis in Proposition [REF] for weights [MATH] using the method in [CITATION].', '1205.5217-1-28-2': '[(c)] Compute Fourier coefficients of Hecke eigenforms in the space [MATH] in [REF] to the precision of [MATH] terms (using MAGMA [CITATION] or SAGE [CITATION]).', '1205.5217-1-28-3': '[(d)] Compute the matrices for [MATH] with respect to the basis in Proposition [REF] using informations from Steps (b) and (c).', '1205.5217-1-28-4': 'This gives us the expressions of [MATH] in terms of [MATH], where [MATH] are defined by [REF].', '1205.5217-1-28-5': ""[(e)] Use Newton's identity to convert expressions for [MATH] to those for symmetric sums of [MATH] and hence determine the rational function [MATH] in [REF]."", '1205.5217-1-28-6': '[(f)] Set [MATH].', '1205.5217-1-28-7': 'Compute and factorize the resultant [MATH] of [MATH] and [MATH] with respect to the variable [MATH].', '1205.5217-1-28-8': '[(h)] Determine which irreducible factor of the numerator of [MATH] is the modular polynomial [MATH] by using the fact that the roots of [MATH] are coordinates of some CM-points (also known as singular moduli) on [MATH].', '1205.5217-1-29-0': 'We now work out an example in details.', '1205.5217-1-30-0': 'An example', '1205.5217-1-31-0': 'In this section, we will work out the modular equation of level [MATH] for the Shimura curve [MATH].', '1205.5217-1-31-1': '(Note that this case was not covered in [CITATION].)', '1205.5217-1-32-0': 'The curve [MATH] has [MATH] elliptic points of orders [MATH], which we denote by [MATH], respectively.', '1205.5217-1-32-1': 'These are CM-points of discriminants [MATH], [MATH], [MATH], and [MATH], respectively.', '1205.5217-1-32-2': 'According to [CITATION], there is a Hauptmodul [MATH] that takes values [MATH], [MATH], and [MATH], and [MATH], respectively.', '1205.5217-1-32-3': 'Using the covering [MATH], we [CITATION] found that the Schwarzian differential equation associated to [MATH] is [EQUATION].', '1205.5217-1-32-4': 'Thus, by Proposition [REF], near the point [MATH], the [MATH]-expansion of [MATH] is the square of a linear combination of two solutions [EQUATION] of the differential equation above.', '1205.5217-1-32-5': 'Thus, by Proposition [REF], a basis for the space of automorphic forms of even weight [MATH] on [MATH] is [EQUATION] for some complex number [MATH].', '1205.5217-1-32-6': 'In particular, the one-dimensional space of automorphic forms of weight [MATH] on [MATH] is spanned by [EQUATION].', '1205.5217-1-32-7': 'Let [MATH], [MATH], be defined by [REF] with [MATH].', '1205.5217-1-32-8': 'According to the recipe described in Section [REF], we first need to compute the matrices for [MATH] with respect to the basis in [REF] for weights [MATH], [MATH].', '1205.5217-1-32-9': 'This has already been done in [CITATION].', '1205.5217-1-32-10': 'We found that the matrices of the Hecke operators [MATH] are [EQUATION]', '1205.5217-1-32-11': 'That is, for the integer [MATH] in the table, we have [EQUATION].', '1205.5217-1-33-0': 'The next informations we need are the Fourier expansions of Hecke eigenforms in the space [MATH] for [MATH].', '1205.5217-1-33-1': 'By MAGMA [CITATION], they are [EQUATION] respectively.', '1205.5217-1-33-2': 'Here each [MATH] is a root of the characteristic polynomial of [MATH] for the corresponding weight, and is different at each occurrence.', '1205.5217-1-33-3': 'From these, we deduce that the matrices for the Hecke operator [MATH] with respect to the bases in [REF] are [EQUATION]', '1205.5217-1-33-4': 'Noticing that [MATH] for any positive integer [MATH], we read from the matrices above that [EQUATION]', '1205.5217-1-33-5': 'Then the rational function [MATH] in [REF] is equal to [EQUATION]', '1205.5217-1-33-6': 'We then set [MATH] and compute the resultant [MATH] of [MATH] and [MATH] with respect to the variable [MATH].', '1205.5217-1-33-7': 'The numerator of [MATH] has two irreducible factors.', '1205.5217-1-33-8': 'To determine which one corresponds to the modular equation [MATH] of level [MATH], we use the fact that the roots of [MATH] should be the coordinates of CM-points of discriminants [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH], and these coordinates are all rational numbers.', '1205.5217-1-33-9': 'In fact, the coordinates of these CM-points were given in Table 3 of [CITATION].', '1205.5217-1-33-10': 'Those obtained numerically in [CITATION] were later verified by Errthum [CITATION] using Borcherds forms.', '1205.5217-1-33-11': '(In general, if [MATH] is the modular equation of level [MATH] for [MATH], then the zeros of the polynomial [MATH] should be coordinates of CM-points of discriminants of the form [MATH], [MATH], [MATH], and [MATH], subject to the condition that optimal embeddings of imaginary quadratic order of given discriminant into the maximal order in the quaternion algebra over [MATH] of discriminant [MATH] exist.)', '1205.5217-1-33-12': 'That is, [MATH] should factor into a product of linear factors over [MATH].', '1205.5217-1-33-13': 'Indeed, exactly one of the two irreducible factors has this property.', '1205.5217-1-33-14': 'This determines [MATH].', '1205.5217-1-33-15': 'The equation of [MATH] is given in Section [REF].', '1205.5217-1-34-0': 'Modular equations for [MATH]', '1205.5217-1-35-0': 'In this section, we consider the Shimura curve [MATH].', '1205.5217-1-35-1': 'The Hauptmodul [MATH] is chosen such that it takes values [MATH], [MATH], and [MATH] at the CM-points of discriminant [MATH], [MATH], and [MATH], respectively.', '1205.5217-1-35-2': 'For a prime [MATH], we let [MATH] denote the modular equation of level [MATH] for the Shimura curve [MATH].', '1205.5217-1-35-3': 'We have computed [MATH] for primes up to [MATH], but because the coefficients are very big, here we only list the equation for [MATH].', '1205.5217-1-35-4': '(The equation for [MATH], with a slight change of variables, is contained in Appendix A of [CITATION].)', '1205.5217-1-35-5': 'Files containing equations of other levels are available upon request.', '1205.5217-1-36-0': 'We note that for [MATH], an explicit cover [MATH] have already been determined by Elkies [CITATION].', '1205.5217-1-36-1': 'It is easier to use this explicit cover to obtain the modular equation.', '1205.5217-1-37-0': 'Write [MATH] as [MATH].', '1205.5217-1-37-1': 'Then [EQUATION]', '1205.5217-1-37-2': 'We next give a short list of irrational singular moduli obtained by factorizing [MATH] for [MATH].', '1205.5217-1-37-3': 'Note that the norms of these coordinates were already computed in [CITATION].', '1205.5217-1-37-4': 'Our computation yields their exact values, not just their norms.', '1205.5217-1-37-5': '[EQUATION]', '1205.5217-1-37-6': 'Modular equations for [MATH]', '1205.5217-1-38-0': 'In this section, we consider the Shimura curve [MATH].', '1205.5217-1-38-1': 'The Hauptmodul [MATH] is chosen such that it takes values [MATH], [MATH], [MATH], and [MATH] at the CM-points of discriminant [MATH], [MATH], [MATH], and [MATH], respectively.', '1205.5217-1-38-2': 'For a prime [MATH], we let [MATH] denote the modular equation of level [MATH] for the Shimura curve [MATH].', '1205.5217-1-38-3': 'We have computed [MATH] for primes up to [MATH], but here we only list the equations for [MATH].', '1205.5217-1-38-4': 'Files containing equations of other levels are available upon request.', '1205.5217-1-38-5': '(The equation for [MATH] is given in Section 5 of [CITATION].)', '1205.5217-1-39-0': 'Write [MATH] as [MATH].', '1205.5217-1-39-1': 'Then [EQUATION]', '1205.5217-1-39-2': 'Here we give a short table of irrational singular moduli obtained by factorizing [MATH] for [MATH].', '1205.5217-1-39-3': 'Note that the norms of these singular moduli were already computed in [CITATION] using Borcherds forms.', '1205.5217-1-39-4': '[EQUATION]'}","{'1205.5217-2-0-0': 'In the classical setting, the modular equation of level [MATH] for the modular curve [MATH] is the polynomial relation satisfied by [MATH] and [MATH], where [MATH] is the standard elliptic [MATH]-function.', '1205.5217-2-0-1': 'In this paper, we will describe a method to compute modular equations in the setting of Shimura curves.', '1205.5217-2-0-2': 'The main ingredient is the explicit method for computing Hecke operators on the spaces of modular forms on Shimura curves developed in [CITATION].', '1205.5217-2-1-0': 'The author would like to thank Professor John McKay and Professor John Voight for providing valuable comments on the paper.', '1205.5217-2-1-1': 'The author was partially supported by Grant 99-2115-M-009-011-MY3 of the National Science Council, Taiwan (R.O.C.).', '1205.5217-2-2-0': 'Introduction Let [MATH] be the elliptic [MATH]-function.', '1205.5217-2-2-1': 'For a positive integer [MATH], consider the set [EQUATION]', '1205.5217-2-2-2': 'The group [MATH] acts on [MATH] by multiplication on the left and it can be easily checked that the cardinality of [MATH] is finite.', '1205.5217-2-2-3': 'Moreover, the multiplication of any element in [MATH] on the right of the cosets in [MATH] simply permutes the cosets.', '1205.5217-2-2-4': 'Thus, any symmetric sum of [MATH], where [MATH] runs over a complete set of representatives for [MATH], will be a modular function on [MATH] and therefore can be written as a rational function of [MATH].', '1205.5217-2-2-5': 'In fact, because the only possible pole occurs at the cusp, this rational function of [MATH] is actually a polynomial.', '1205.5217-2-2-6': 'Then the modular polynomial [MATH] of level [MATH] is defined to be the polynomial in [MATH] such that [EQUATION].', '1205.5217-2-2-7': 'In fact, it can be easily proved that this polynomial [MATH] is in [MATH].', '1205.5217-2-2-8': 'Since [MATH] is the moduli space of isomorphism classes of elliptic curves over [MATH], the equation [MATH] is the modular equation of level [MATH] for moduli of elliptic curves over [MATH].', '1205.5217-2-2-9': 'That is, if [MATH] and [MATH] are two elliptic curves over [MATH] admitting a cyclic [MATH]-isogeny between them, then their [MATH]-invariants [MATH] and [MATH] satisfy [MATH].', '1205.5217-2-3-0': 'Observe that if the Fourier expansion of [MATH] is [MATH], [MATH], then the Fourier expansion of [MATH] is simply [MATH].', '1205.5217-2-3-1': 'Thus, in principle, to determine [MATH], one just has to compute enough Fourier coefficients for [MATH] and solve a system of linear equations.', '1205.5217-2-3-2': 'Of course, the main difficulty in practice is that the coefficients are gigantic.', '1205.5217-2-3-3': 'On the other hand, because the coefficients of [MATH] are all integers, one can compute the reduction of [MATH] modulo [MATH] for a suitable number of primes [MATH] and then use the Chinese remainder theorem to recover the coefficients.', '1205.5217-2-3-4': 'See [CITATION] for the current state of the art in the computation of [MATH].', '1205.5217-2-4-0': 'In this paper, we shall consider modular equations in the settings of Shimura curves.', '1205.5217-2-4-1': 'Let [MATH] be an indefinite quaternion algebra of discriminant [MATH] over [MATH] and [MATH] be a maximal order in [MATH].', '1205.5217-2-4-2': 'Choose an embedding [MATH] and let [EQUATION] be the image of the norm-one group of [MATH] under [MATH], where [MATH] denotes the reduced norm of [MATH].', '1205.5217-2-4-3': 'Then the Shimura curve [MATH] is defined to be the quotient space [MATH].', '1205.5217-2-4-4': 'As shown in [CITATION], the Shimura curve [MATH] is the coarse moduli space for isomorphism classes of abelian surfaces with quaternionic multiplication (QM) by [MATH].', '1205.5217-2-4-5': 'Let [MATH] denote the group of Atkin-Lehner involutions on [MATH].', '1205.5217-2-4-6': 'For our purpose, we will also consider the quotient curves of [MATH] by subgroups [MATH] of [MATH].', '1205.5217-2-5-0': 'Now assume that [MATH] has genus [MATH] and choose a Hauptmodul [MATH] for [MATH] so that [MATH] generates the function field on [MATH].', '1205.5217-2-5-1': 'For a positive integer [MATH] relatively prime to [MATH], pick an element [MATH] of norm [MATH] in [MATH] such that [MATH] is an Eichler order of level [MATH].', '1205.5217-2-5-2': 'Then the modular polynomial of level [MATH] for [MATH] is defined to be the polynomial [MATH] of minimal degree, up to scalars, such that [MATH], which is essentially the rational function [MATH] such that [EQUATION].', '1205.5217-2-5-3': 'Here unlike the case of the modular curve [MATH], a symmetric sum of [MATH] as [MATH] runs through representatives of [MATH] is not equal to a polynomial of [MATH] in general.', '1205.5217-2-5-4': 'Since [MATH] is the moduli space of abelian surfaces over [MATH] with quaternionic multiplication, the modular equation of level [MATH] relates the moduli of two abelian surfaces with QM that have a certain type of isogenies between them.', '1205.5217-2-5-5': '(The precise description of the isogeny is a little complicated to be given here.', '1205.5217-2-5-6': 'See [CITATION] for details.)', '1205.5217-2-5-7': 'Modular equations for Atkin-Lehner quotients [MATH] are similarly defined.', '1205.5217-2-6-0': 'When [MATH], the problem of explicitly determining modular equations for Shimura curves is significantly more complicated than its classical counterpart.', '1205.5217-2-6-1': 'The reasons are that Shimura curves do not have cusps and it is difficult to find the Taylor expansion of an automorphic function with respect to a local parameter at any given point on the Shimura curve.', '1205.5217-2-6-2': 'When [MATH] and [MATH] are both small such that [MATH] is also of genus [MATH], it is possible to work out an explicit cover [MATH] from the ramification data alone.', '1205.5217-2-6-3': 'Then from the explicit cover, one can compute the modular equation of level [MATH].', '1205.5217-2-6-4': 'This has been done in [CITATION] for a limited number of cases.', '1205.5217-2-7-0': 'Another possible method to compute modular equations for Shimura curves uses the Schwarzian differential equation associated to a Hauptmodul [MATH].', '1205.5217-2-7-1': '(See Section [REF] for a review on the notion and properties of Schwarzian differential equation.)', '1205.5217-2-7-2': 'The idea is that with a properly chosen pair of solutions [MATH] and [MATH] of the Schwarzian differential equation and a correct positive integer [MATH], the expression [MATH] can be taken to be a local parameter at the point [MATH] of the Shimura curve with [MATH].', '1205.5217-2-7-3': 'Inverting the expression, one gets the Taylor expansion of [MATH] with respect to the local parameter.', '1205.5217-2-7-4': 'If somehow one manages to find the Taylor expansion of [MATH] (this can always be done when [MATH], then by computing enough terms and solving a system of linear equations, one gets the modular equation.', '1205.5217-2-7-5': 'However, as far as we know, there does not seem to be any paper in the literature that employs this idea to obtain modular equations for Shimura curves.', '1205.5217-2-7-6': '(The paper [CITATION] did obtain the Taylor expansion for the Hauptmodul in the case [MATH].', '1205.5217-2-7-7': 'More recently, Voight and Willis [CITATION] developed a method for numerically computing Taylor expansions of automorphic forms on Shimura curves.)', '1205.5217-2-8-0': 'In his Ph.D. thesis [CITATION], Voight developed a method to compute modular equations for Shimura curves associated to quaternion algebras over totally real number fields in the cases when the Shimura curves have genus zero and precisely three elliptic points, i.e., Shimura curves associated to arithmetic triangle groups.', '1205.5217-2-8-1': 'The idea is to use hypergeometric functions, which are essentially solutions of the Schwarzian differential equations mentioned above, to numerically determine coordinates of CM-points and then use the existence of canonical models and explicit Shimura reciprocity laws to determine modular equations.', '1205.5217-2-8-2': 'Even though the equations are obtained numerically, they can be verified rigorously by showing that the monodromy group of the branched cover is correct.', '1205.5217-2-9-0': 'In this paper, we shall present a new method to compute modular equations for Shimura curves.', '1205.5217-2-9-1': 'Our method also uses Schwarzian differential equations, but relies more heavily on the arithmetic side of the theory.', '1205.5217-2-9-2': 'Namely, in an earlier work [CITATION], we showed that the spaces of automorphic forms of any given weight on a Shimura curve [MATH] of genus [MATH] can be completely characterized in terms of solutions of the Schwarzian differential equation.', '1205.5217-2-9-3': 'We [CITATION] then devised a method to compute Hecke operators with respect to our basis.', '1205.5217-2-9-4': 'The Jacquet-Langlands correspondence plays a crucial role in our approach.', '1205.5217-2-9-5': 'It turns out that our method and results in [CITATION] can also be used to compute modular equations for Shimura curves.', '1205.5217-2-9-6': 'We will describe the procedure in Section [REF] and give a detailed example in Section [REF].', '1205.5217-2-9-7': 'Using the computer algebra system MAGMA [CITATION], we have succeeded in determining modular equations of prime level up to [MATH] for [MATH] and those of prime level up to [MATH] for [MATH].', '1205.5217-2-10-0': 'The main difficulty in generalizing our method to general Shimura curves lies at the fact that our method requires that a Schwarzian differential equation is known beforehand.', '1205.5217-2-10-1': 'Because of the problem of the existence of accessory parameters, the determination of Schwarzian differential equations usually requires that an explicit cover [MATH] is known, which can be problematic when [MATH] is large.', '1205.5217-2-10-2': '(In some sense, what we do here is to deduce modular equations of higher levels from that of a given small level.)', '1205.5217-2-10-3': 'Nonetheless, once an explicit cover of Shimura curves is determined, the combination of the methods in [CITATION] and in this paper will yield modular equations for the Shimura curve.', '1205.5217-2-11-0': 'The rest of the paper is organized as follows.', '1205.5217-2-11-1': 'In Section [REF], we review the definition and properties of Schwarzian differential equations.', '1205.5217-2-11-2': 'In Section [REF], we describe our method to compute modular equations for Shimura curves [MATH], assuming that an explicit cover [MATH] is known.', '1205.5217-2-11-3': 'In Section [REF], we give a detailed example illustrating our method.', '1205.5217-2-11-4': 'In Sections [REF] and [REF], we list part of our computational results for the Shimura curves [MATH] and [MATH].', '1205.5217-2-11-5': 'Files in the MAGMA-readable format containing all our computational results are available upon request.', '1205.5217-2-12-0': 'Schwarzian differential equations', '1205.5217-2-13-0': 'In this section, we will review the definition and properties of Schwarzian differential equations.', '1205.5217-2-13-1': 'In particular, assuming the Shimura curve has genus [MATH], we will recall the characterization of the spaces of automorphic forms in terms of solutions of the associated Schwarzian differential equation.', '1205.5217-2-13-2': 'The method for computing Hecke operators on these spaces is too complicated to describe here.', '1205.5217-2-13-3': 'We refer the reader to [CITATION] for details.', '1205.5217-2-13-4': 'Most materials in this section are taken from [CITATION].', '1205.5217-2-14-0': 'Let [MATH] be a Shimura curve.', '1205.5217-2-14-1': 'We assume that the associated quaternion algebra is not [MATH] so that [MATH] has no cusps.', '1205.5217-2-14-2': 'Let [MATH] be a (meromorphic) automorphic form of weight [MATH] and [MATH] be a non-constant automorphic function on [MATH].', '1205.5217-2-14-3': 'It is known since the nineteen century that the functions [MATH], as functions of [MATH], are solutions of a certain [MATH]-st linear ordinary differential equations with algebraic functions as coefficients.', '1205.5217-2-14-4': '(See [CITATION].)', '1205.5217-2-14-5': 'In particular, since [MATH] is a meromorphic automorphic form of weight [MATH], the function [MATH] as a function of [MATH], satisfies a second-order linear ordinary differential equation.', '1205.5217-2-14-6': 'We call this differential equation the Schwarzian differential equation associated to [MATH], which has the following properties.', '1205.5217-2-15-0': '[[CITATION]]', '1205.5217-2-16-0': 'Let [MATH] be a Shimura curve of genus zero with elliptic points [MATH] of order [MATH], respectively.', '1205.5217-2-16-1': 'Let [MATH] be a Hauptmodul of [MATH] and set [MATH], [MATH].', '1205.5217-2-16-2': 'Then [MATH], as a function of [MATH], satisfies the differential equation [EQUATION] where [EQUATION] for some constants [MATH].', '1205.5217-2-16-3': 'Moreover, if [MATH] for all [MATH], then the constants [MATH] satisfy [EQUATION].', '1205.5217-2-16-4': 'Also, if [MATH], then [MATH] satisfy [EQUATION].', '1205.5217-2-17-0': 'We remark that when the Shimura curve has genus [MATH] and precisely [MATH] elliptic points, the relations among [MATH] are enough to determine the constants [MATH].', '1205.5217-2-17-1': 'This reflects the fact in classical analysis that a second-order Fuchsian differential equation with exactly three singularities is completely determined by the local exponents.', '1205.5217-2-17-2': 'When the Shimura curve has more than [MATH] elliptic points, the relations are not enough to determine [MATH].', '1205.5217-2-17-3': 'In literature, we refer to this kind of situations by saying that accessory parameters exist.', '1205.5217-2-17-4': 'In order to determine the accessory parameters, one usually tries to find an explicit cover of Shimura curves and use it to determine the Schwarzian differential equations associated to the two curves simultaneously.', '1205.5217-2-18-0': 'Now one of the key observations in [CITATION] is that the analytic behavior of the Hauptmodul [MATH] is very easy to determine and from this, one can work out a basis for the space of automorphic forms of even weight [MATH] in terms of [MATH].', '1205.5217-2-19-0': '[[CITATION]]', '1205.5217-2-20-0': 'Assume that a Shimura curve [MATH] has genus zero with elliptic points [MATH] of order [MATH], respectively.', '1205.5217-2-20-1': 'Let [MATH] be a Hauptmodul of [MATH] and set [MATH], [MATH].', '1205.5217-2-20-2': 'For a positive even integer [MATH], let [EQUATION].', '1205.5217-2-20-3': 'Then a basis for the space of automorphic forms of weight [MATH] on [MATH] is [EQUATION].', '1205.5217-2-21-0': 'The combination of two propositions shows that all automorphic forms of a given even weight [MATH] can be expressed in terms of the solutions of the Schwarzian differential equation.', '1205.5217-2-21-1': 'In [CITATION], the author developed a method for computing Hecke operators relative to the basis in Proposition [REF].', '1205.5217-2-21-2': 'The key ingredients are the Jacquet-Langlands correspondence and explicit covers of Shimura curves.', '1205.5217-2-21-3': 'We refer the reader to [CITATION] for details.', '1205.5217-2-22-0': 'Computing modular equations for Shimura curves', '1205.5217-2-23-0': 'In this section, we will present a method for computing modular equations for Shimura curves, under the working assumptions that the Schwarzian differential equations have been determined and, for a certain prime fixed [MATH], the matrices for the Hecke operator [MATH] with respect to the bases in Proposition [REF] have already been computed for sufficiently many [MATH] according to the recipe in [CITATION].', '1205.5217-2-24-0': 'Let [MATH] be a Shimura curve of genus [MATH] of the form [MATH] for some subgroup [MATH] of the group of Atkin-Lehner involutions and [MATH] be the discrete subgroup of [MATH] corresponding to [MATH].', '1205.5217-2-24-1': 'For a prime [MATH] not dividing [MATH], let [MATH], [MATH], be representatives for the cosets defining the Hecke operator [MATH], that is, [EQUATION]', '1205.5217-2-24-2': 'Then the modular equation [MATH] of level [MATH] is the polynomial relation between a Hauptmodul [MATH] for [MATH] and [MATH] for arbitrary [MATH].', '1205.5217-2-25-0': 'Pick any nonzero automorphic form [MATH] on [MATH] with the smallest possible weight [MATH].', '1205.5217-2-25-1': 'For convenience, we let [EQUATION] the summands in [REF].', '1205.5217-2-25-2': 'Then any symmetric sum of [MATH], [MATH], will be an automorphic function on [MATH] and hence can be expressed as a rational function of [MATH].', '1205.5217-2-25-3': 'In other words, there is a rational function [MATH] such that [EQUATION]', '1205.5217-2-25-4': 'That is, we have [EQUATION] for all [MATH].', '1205.5217-2-26-0': 'Let [MATH] be the rational function defined by [REF].', '1205.5217-2-26-1': 'Let [MATH] be the rational function such that [EQUATION] which is a polynomial in [MATH].', '1205.5217-2-26-2': 'Let [MATH] be the resultant of [MATH] and [MATH] with respect to the variable [MATH].', '1205.5217-2-26-3': 'Then the modular polynomial [MATH] appears as one of the irreducible factors over [MATH] in the numerator of the rational function [MATH].', '1205.5217-2-27-0': 'Let [MATH].', '1205.5217-2-27-1': 'There is an integer [MATH], [MATH], such that [MATH].', '1205.5217-2-27-2': 'We apply the action of [MATH] to [REF] and get [EQUATION].', '1205.5217-2-27-3': 'In other words, we have [EQUATION]', '1205.5217-2-27-4': 'From this equality and [REF], we see that if we let [MATH] be the resultant of [MATH] and [MATH] with respect to the variable [MATH], then [MATH].', '1205.5217-2-27-5': 'In particular, the modular polynomial [MATH] appears as an irreducible factor of [MATH] over [MATH].', '1205.5217-2-27-6': 'This proves the lemma.', '1205.5217-2-28-0': 'We remark that, in general, there will be more than one irreducible factors in the numerator of the resultant [MATH], but it is not difficult to determine which one corresponds to [MATH].', '1205.5217-2-28-1': 'For example, we can use the fact that the roots of [MATH] are the coordinates of certain CM-points (also known as singular moduli) to test which factor of [MATH] is [MATH].', '1205.5217-2-28-2': 'Thus, from the above discussion, we see that the most critical part of the calculation is the determination of the rational function [MATH], which we address now.', '1205.5217-2-29-0': ""From Newton's identity, we know that the problem of determining the symmetric sums of [MATH] in [REF] is equivalent to that of determining [MATH] for [MATH]."", '1205.5217-2-29-1': 'Observe that the sum of [MATH] is equal to [MATH], by the definition of the Hecke operator [MATH].', '1205.5217-2-29-2': 'Thus, to determine [MATH], one just needs to know how the Hecke operator [MATH] acts on the basis given in Proposition [REF].', '1205.5217-2-29-3': 'This is where the work of [CITATION] comes into play.', '1205.5217-2-30-0': 'In [CITATION], we developed a method to compute [MATH], [MATH] a prime not dividing [MATH], for arbitrary weight [MATH] with respect to the basis given in Proposition [REF], assuming an explicit cover [MATH] is known.', '1205.5217-2-30-1': 'Now recall the following explicit version of the Jacquet-Langlands correspondence.', '1205.5217-2-31-0': '[[CITATION]]', '1205.5217-2-32-0': 'Let [MATH] be discriminant of an indefinite quaternion algebra over [MATH].', '1205.5217-2-32-1': 'Let [MATH] be a positive integer relatively prime to [MATH].', '1205.5217-2-32-2': 'For an Eichler order [MATH] of level [MATH] and a positive even integer, let [MATH] denote the space of automorphic forms on the Shimura curve [MATH].', '1205.5217-2-32-3': 'Then [EQUATION] as Hecke modules.', '1205.5217-2-32-4': 'Here [EQUATION] and [MATH] denotes the newform subspace of cusp forms of weight [MATH] on [MATH].', '1205.5217-2-32-5': 'In other words, for each Hecke eigenform [MATH] in [MATH], there corresponds a Hecke eigenform [MATH] in [MATH] that shares the same Hecke eigenvalues.', '1205.5217-2-32-6': 'Moreover, for a prime divisor [MATH] of [MATH], if the Atkin-Lehner involution [MATH] acts on [MATH] by [MATH], then [EQUATION].', '1205.5217-2-33-0': 'For the situation under our consideration, we have [EQUATION] where the left-hand side denotes the subspace of the newform subspace of cusp forms of weight [MATH] on [MATH] that has eigenvalue [MATH] for all [MATH], [MATH] primes, and the right-hand side denotes the space of automorphic forms of weight [MATH] on [MATH].', '1205.5217-2-33-1': 'Thus, assuming the Hecke operator [MATH] on the space [MATH] has no repeated eigenvalues, one can obtain the matrices for [MATH] with respect to the bases in Proposition [REF] from those for [MATH] and the Fourier coefficients of Hecke eigenforms in [MATH].', '1205.5217-2-34-0': 'In summary, to compute the modular equation [MATH], we follow the following steps, assuming that the Schwarzian differential equation associated to [MATH] and an explicit cover [MATH] are known for some prime [MATH] not dividing [MATH].', '1205.5217-2-35-0': '[(a)] Pick a nonzero automorphic form [MATH] on [MATH] of the smallest possible weight [MATH], expressed in the form given in Proposition [REF].', '1205.5217-2-35-1': '[(b)] Compute the matrices for [MATH] with respect to the basis in Proposition [REF] for weights [MATH] using the method in [CITATION].', '1205.5217-2-35-2': '[(c)] Compute Fourier coefficients of Hecke eigenforms in the space [MATH] in [REF] to the precision of [MATH] terms (using MAGMA [CITATION] or SAGE [CITATION]).', '1205.5217-2-35-3': '[(d)] Compute the matrices for [MATH] with respect to the basis in Proposition [REF] using informations from Steps (b) and (c).', '1205.5217-2-35-4': 'This gives us the expressions of [MATH] in terms of [MATH], where [MATH] are defined by [REF].', '1205.5217-2-35-5': ""[(e)] Use Newton's identity to convert expressions for [MATH] to those for symmetric sums of [MATH] and hence determine the rational function [MATH] in [REF]."", '1205.5217-2-35-6': '[(f)] Set [MATH].', '1205.5217-2-35-7': 'Compute and factorize the resultant [MATH] of [MATH] and [MATH] with respect to the variable [MATH].', '1205.5217-2-35-8': '[(h)] Determine which irreducible factor of the numerator of [MATH] is the modular polynomial [MATH] by using the fact that the roots of [MATH] are coordinates of some CM-points (also known as singular moduli) on [MATH].', '1205.5217-2-36-0': 'We now work out an example in details.', '1205.5217-2-37-0': 'An example', '1205.5217-2-38-0': 'In this section, we will work out the modular equation of level [MATH] for the Shimura curve [MATH].', '1205.5217-2-38-1': '(Note that this case was not covered in [CITATION].)', '1205.5217-2-39-0': 'The curve [MATH] has [MATH] elliptic points of orders [MATH], which we denote by [MATH], respectively.', '1205.5217-2-39-1': 'These are CM-points of discriminants [MATH], [MATH], [MATH], and [MATH], respectively.', '1205.5217-2-39-2': 'According to [CITATION], there is a Hauptmodul [MATH] that takes values [MATH], [MATH], and [MATH], and [MATH], respectively.', '1205.5217-2-39-3': 'Using the covering [MATH], we [CITATION] found that the Schwarzian differential equation associated to [MATH] is [EQUATION].', '1205.5217-2-39-4': 'Thus, by Proposition [REF], near the point [MATH], the [MATH]-expansion of [MATH] is the square of a linear combination of two solutions [EQUATION] of the differential equation above.', '1205.5217-2-39-5': 'Thus, by Proposition [REF], a basis for the space of automorphic forms of even weight [MATH] on [MATH] is [EQUATION] for some complex number [MATH].', '1205.5217-2-39-6': 'In particular, the one-dimensional space of automorphic forms of weight [MATH] on [MATH] is spanned by [EQUATION].', '1205.5217-2-39-7': 'Let [MATH], [MATH], be defined by [REF] with [MATH].', '1205.5217-2-39-8': 'According to the recipe described in Section [REF], we first need to compute the matrices for [MATH] with respect to the basis in [REF] for weights [MATH], [MATH].', '1205.5217-2-39-9': 'This has already been done in [CITATION].', '1205.5217-2-39-10': 'We found that the matrices of the Hecke operators [MATH] are [EQUATION]', '1205.5217-2-39-11': 'That is, for the integer [MATH] in the table, we have [EQUATION].', '1205.5217-2-40-0': 'The next informations we need are the Fourier expansions of Hecke eigenforms in the space [MATH] for [MATH].', '1205.5217-2-40-1': 'By MAGMA [CITATION], they are [EQUATION] respectively.', '1205.5217-2-40-2': 'Here each [MATH] is a root of the characteristic polynomial of [MATH] for the corresponding weight, and is different at each occurrence.', '1205.5217-2-40-3': 'From these, we deduce that the matrices for the Hecke operator [MATH] with respect to the bases in [REF] are [EQUATION]', '1205.5217-2-40-4': 'Noticing that [MATH] for any positive integer [MATH], we read from the matrices above that [EQUATION]', '1205.5217-2-40-5': 'Then the rational function [MATH] in [REF] is equal to [EQUATION]', '1205.5217-2-40-6': 'We then set [MATH] and compute the resultant [MATH] of [MATH] and [MATH] with respect to the variable [MATH].', '1205.5217-2-40-7': 'The numerator of [MATH] has two irreducible factors.', '1205.5217-2-40-8': 'To determine which one corresponds to the modular equation [MATH] of level [MATH], we use the fact that the roots of [MATH] should be the coordinates of CM-points of discriminants [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], [MATH], and [MATH], and these coordinates are all rational numbers.', '1205.5217-2-40-9': 'In fact, the coordinates of these CM-points were given in Table 3 of [CITATION].', '1205.5217-2-40-10': 'Those obtained numerically in [CITATION] were later verified by Errthum [CITATION] using Borcherds forms.', '1205.5217-2-40-11': '(In general, if [MATH] is the modular equation of level [MATH] for [MATH], then the zeros of the polynomial [MATH] should be coordinates of CM-points of discriminants of the form [MATH], [MATH], [MATH], and [MATH], subject to the condition that optimal embeddings of imaginary quadratic order of given discriminant into the maximal order in the quaternion algebra over [MATH] of discriminant [MATH] exist.)', '1205.5217-2-40-12': 'That is, [MATH] should factor into a product of linear factors over [MATH].', '1205.5217-2-40-13': 'Indeed, exactly one of the two irreducible factors has this property.', '1205.5217-2-40-14': 'This determines [MATH].', '1205.5217-2-40-15': 'The equation of [MATH] is given in Section [REF].', '1205.5217-2-41-0': 'Modular equations for [MATH]', '1205.5217-2-42-0': 'In this section, we consider the Shimura curve [MATH].', '1205.5217-2-42-1': 'The Hauptmodul [MATH] is chosen such that it takes values [MATH], [MATH], and [MATH] at the CM-points of discriminant [MATH], [MATH], and [MATH], respectively.', '1205.5217-2-42-2': 'For a prime [MATH], we let [MATH] denote the modular equation of level [MATH] for the Shimura curve [MATH].', '1205.5217-2-42-3': 'We have computed [MATH] for primes up to [MATH], but because the coefficients are very big, here we only list the equation for [MATH].', '1205.5217-2-42-4': '(The equation for [MATH], with a slight change of variables, is contained in Appendix A of [CITATION].)', '1205.5217-2-42-5': 'Files containing equations of other levels are available upon request.', '1205.5217-2-43-0': 'We note that for [MATH], an explicit cover [MATH] have already been determined by Elkies [CITATION].', '1205.5217-2-43-1': 'It is easier to use this explicit cover to obtain the modular equation.', '1205.5217-2-44-0': 'Write [MATH] as [MATH].', '1205.5217-2-44-1': 'Then [EQUATION]', '1205.5217-2-44-2': 'We next give a short list of irrational singular moduli obtained by factorizing [MATH] for [MATH].', '1205.5217-2-44-3': 'Note that the norms of these coordinates were already computed in [CITATION].', '1205.5217-2-44-4': 'Our computation yields their exact values, not just their norms.', '1205.5217-2-44-5': '(Note that all the rational singular moduli for [MATH] were numerically determined in [CITATION] and later verified in [CITATION].)', '1205.5217-2-44-6': '[EQUATION]', '1205.5217-2-44-7': 'Modular equations for [MATH]', '1205.5217-2-45-0': 'In this section, we consider the Shimura curve [MATH].', '1205.5217-2-45-1': 'The Hauptmodul [MATH] is chosen such that it takes values [MATH], [MATH], [MATH], and [MATH] at the CM-points of discriminant [MATH], [MATH], [MATH], and [MATH], respectively.', '1205.5217-2-45-2': 'For a prime [MATH], we let [MATH] denote the modular equation of level [MATH] for the Shimura curve [MATH].', '1205.5217-2-45-3': 'We have computed [MATH] for primes up to [MATH], but here we only list the equations for [MATH].', '1205.5217-2-45-4': 'Files containing equations of other levels are available upon request.', '1205.5217-2-45-5': '(The equation for [MATH] is given in Section 5 of [CITATION].)', '1205.5217-2-46-0': 'Write [MATH] as [MATH].', '1205.5217-2-46-1': 'Then [EQUATION]', '1205.5217-2-46-2': 'Here we give a short table of irrational singular moduli obtained by factorizing [MATH] for [MATH].', '1205.5217-2-46-3': 'Note that the norms of these singular moduli were already computed in [CITATION] using Borcherds forms.', '1205.5217-2-46-4': '(All the rational singular moduli for [MATH] were numerically determined in [CITATION] and later verified in [CITATION].)', '1205.5217-2-46-5': '[EQUATION]'}","[['1205.5217-1-16-0', '1205.5217-2-17-0'], ['1205.5217-1-16-1', '1205.5217-2-17-1'], ['1205.5217-1-16-2', '1205.5217-2-17-2'], ['1205.5217-1-16-3', '1205.5217-2-17-3'], ['1205.5217-1-16-4', '1205.5217-2-17-4'], ['1205.5217-1-0-0', '1205.5217-2-0-0'], ['1205.5217-1-0-1', '1205.5217-2-0-1'], ['1205.5217-1-0-2', '1205.5217-2-0-2'], ['1205.5217-1-10-0', '1205.5217-2-11-0'], ['1205.5217-1-10-1', '1205.5217-2-11-1'], ['1205.5217-1-10-2', '1205.5217-2-11-2'], ['1205.5217-1-10-3', '1205.5217-2-11-3'], ['1205.5217-1-10-4', '1205.5217-2-11-4'], ['1205.5217-1-10-5', '1205.5217-2-11-5'], ['1205.5217-1-3-0', '1205.5217-2-3-0'], ['1205.5217-1-3-1', '1205.5217-2-3-1'], ['1205.5217-1-3-2', '1205.5217-2-3-2'], ['1205.5217-1-3-3', '1205.5217-2-3-3'], ['1205.5217-1-3-4', '1205.5217-2-3-4'], ['1205.5217-1-23-0', '1205.5217-2-24-0'], ['1205.5217-1-23-1', '1205.5217-2-24-1'], ['1205.5217-1-23-2', '1205.5217-2-24-2'], ['1205.5217-1-32-0', '1205.5217-2-39-0'], ['1205.5217-1-32-1', '1205.5217-2-39-1'], ['1205.5217-1-32-2', '1205.5217-2-39-2'], ['1205.5217-1-32-3', '1205.5217-2-39-3'], ['1205.5217-1-32-4', '1205.5217-2-39-4'], ['1205.5217-1-32-5', '1205.5217-2-39-5'], ['1205.5217-1-32-6', '1205.5217-2-39-6'], ['1205.5217-1-32-7', '1205.5217-2-39-7'], ['1205.5217-1-32-8', '1205.5217-2-39-8'], ['1205.5217-1-32-9', '1205.5217-2-39-9'], ['1205.5217-1-32-10', '1205.5217-2-39-10'], ['1205.5217-1-32-11', '1205.5217-2-39-11'], ['1205.5217-1-1-0', '1205.5217-2-1-1'], ['1205.5217-1-33-0', '1205.5217-2-40-0'], ['1205.5217-1-33-1', '1205.5217-2-40-1'], ['1205.5217-1-33-2', '1205.5217-2-40-2'], ['1205.5217-1-33-3', '1205.5217-2-40-3'], ['1205.5217-1-33-4', '1205.5217-2-40-4'], ['1205.5217-1-33-5', '1205.5217-2-40-5'], ['1205.5217-1-33-6', '1205.5217-2-40-6'], ['1205.5217-1-33-7', '1205.5217-2-40-7'], ['1205.5217-1-33-8', '1205.5217-2-40-8'], ['1205.5217-1-33-9', '1205.5217-2-40-9'], ['1205.5217-1-33-10', '1205.5217-2-40-10'], ['1205.5217-1-33-11', '1205.5217-2-40-11'], ['1205.5217-1-33-12', '1205.5217-2-40-12'], ['1205.5217-1-33-13', '1205.5217-2-40-13'], ['1205.5217-1-33-15', '1205.5217-2-40-15'], ['1205.5217-1-5-0', '1205.5217-2-5-0'], ['1205.5217-1-5-1', '1205.5217-2-5-1'], ['1205.5217-1-5-2', '1205.5217-2-5-2'], ['1205.5217-1-5-3', '1205.5217-2-5-3'], ['1205.5217-1-5-4', '1205.5217-2-5-4'], ['1205.5217-1-5-5', '1205.5217-2-5-5'], ['1205.5217-1-5-6', '1205.5217-2-5-6'], ['1205.5217-1-5-7', '1205.5217-2-5-7'], ['1205.5217-1-20-0', '1205.5217-2-21-0'], ['1205.5217-1-20-1', '1205.5217-2-21-1'], ['1205.5217-1-20-2', '1205.5217-2-21-2'], ['1205.5217-1-20-3', '1205.5217-2-21-3'], ['1205.5217-1-39-2', '1205.5217-2-46-2'], ['1205.5217-1-39-3', '1205.5217-2-46-3'], ['1205.5217-1-36-0', '1205.5217-2-43-0'], ['1205.5217-1-36-1', '1205.5217-2-43-1'], ['1205.5217-1-4-0', '1205.5217-2-4-0'], ['1205.5217-1-4-1', '1205.5217-2-4-1'], ['1205.5217-1-4-2', '1205.5217-2-4-2'], ['1205.5217-1-4-3', '1205.5217-2-4-3'], ['1205.5217-1-4-4', '1205.5217-2-4-4'], ['1205.5217-1-4-5', '1205.5217-2-4-5'], ['1205.5217-1-4-6', '1205.5217-2-4-6'], ['1205.5217-1-37-2', '1205.5217-2-44-2'], ['1205.5217-1-37-3', '1205.5217-2-44-3'], ['1205.5217-1-37-4', '1205.5217-2-44-4'], ['1205.5217-1-37-6', '1205.5217-2-44-7'], ['1205.5217-1-9-0', '1205.5217-2-10-0'], ['1205.5217-1-9-1', '1205.5217-2-10-1'], ['1205.5217-1-9-2', '1205.5217-2-10-2'], ['1205.5217-1-9-3', '1205.5217-2-10-3'], ['1205.5217-1-19-0', '1205.5217-2-20-0'], ['1205.5217-1-19-1', '1205.5217-2-20-1'], ['1205.5217-1-19-2', '1205.5217-2-20-2'], ['1205.5217-1-19-3', '1205.5217-2-20-3'], ['1205.5217-1-35-0', '1205.5217-2-42-0'], ['1205.5217-1-35-1', '1205.5217-2-42-1'], ['1205.5217-1-35-2', '1205.5217-2-42-2'], ['1205.5217-1-35-3', '1205.5217-2-42-3'], ['1205.5217-1-35-4', '1205.5217-2-42-4'], ['1205.5217-1-35-5', '1205.5217-2-42-5'], ['1205.5217-1-28-0', '1205.5217-2-35-0'], ['1205.5217-1-28-1', '1205.5217-2-35-1'], ['1205.5217-1-28-2', '1205.5217-2-35-2'], ['1205.5217-1-28-3', '1205.5217-2-35-3'], ['1205.5217-1-28-4', '1205.5217-2-35-4'], ['1205.5217-1-28-5', '1205.5217-2-35-5'], ['1205.5217-1-28-7', '1205.5217-2-35-7'], ['1205.5217-1-28-8', '1205.5217-2-35-8'], ['1205.5217-1-6-0', '1205.5217-2-6-0'], ['1205.5217-1-6-1', '1205.5217-2-6-1'], ['1205.5217-1-6-2', '1205.5217-2-6-2'], ['1205.5217-1-6-3', '1205.5217-2-6-3'], ['1205.5217-1-6-4', '1205.5217-2-6-4'], ['1205.5217-1-27-0', '1205.5217-2-34-0'], ['1205.5217-1-31-0', '1205.5217-2-38-0'], ['1205.5217-1-31-1', '1205.5217-2-38-1'], ['1205.5217-1-38-0', '1205.5217-2-45-0'], ['1205.5217-1-38-1', '1205.5217-2-45-1'], ['1205.5217-1-38-2', '1205.5217-2-45-2'], ['1205.5217-1-38-3', '1205.5217-2-45-3'], ['1205.5217-1-38-4', '1205.5217-2-45-4'], ['1205.5217-1-38-5', '1205.5217-2-45-5'], ['1205.5217-1-13-0', '1205.5217-2-14-0'], ['1205.5217-1-13-1', '1205.5217-2-14-1'], ['1205.5217-1-13-2', '1205.5217-2-14-2'], ['1205.5217-1-13-3', '1205.5217-2-14-3'], ['1205.5217-1-13-5', '1205.5217-2-14-5'], ['1205.5217-1-13-6', '1205.5217-2-14-6'], ['1205.5217-1-17-0', '1205.5217-2-18-0'], ['1205.5217-1-25-0', '1205.5217-2-29-0'], ['1205.5217-1-25-1', '1205.5217-2-29-1'], ['1205.5217-1-25-2', '1205.5217-2-29-2'], ['1205.5217-1-25-3', '1205.5217-2-29-3'], ['1205.5217-1-2-0', '1205.5217-2-2-0'], ['1205.5217-1-2-1', '1205.5217-2-2-1'], ['1205.5217-1-2-2', '1205.5217-2-2-2'], ['1205.5217-1-2-3', '1205.5217-2-2-3'], ['1205.5217-1-2-4', '1205.5217-2-2-4'], ['1205.5217-1-2-5', '1205.5217-2-2-5'], ['1205.5217-1-2-6', '1205.5217-2-2-6'], ['1205.5217-1-2-7', '1205.5217-2-2-7'], ['1205.5217-1-2-8', '1205.5217-2-2-8'], ['1205.5217-1-2-9', '1205.5217-2-2-9'], ['1205.5217-1-15-0', '1205.5217-2-16-0'], ['1205.5217-1-15-1', '1205.5217-2-16-1'], ['1205.5217-1-15-2', '1205.5217-2-16-2'], ['1205.5217-1-15-3', '1205.5217-2-16-3'], ['1205.5217-1-15-4', '1205.5217-2-16-4'], ['1205.5217-1-12-0', '1205.5217-2-13-0'], ['1205.5217-1-12-1', '1205.5217-2-13-1'], ['1205.5217-1-12-2', '1205.5217-2-13-2'], ['1205.5217-1-12-3', '1205.5217-2-13-3'], ['1205.5217-1-12-4', '1205.5217-2-13-4'], ['1205.5217-1-8-0', '1205.5217-2-9-0'], ['1205.5217-1-8-1', '1205.5217-2-9-1'], ['1205.5217-1-8-2', '1205.5217-2-9-2'], ['1205.5217-1-8-3', '1205.5217-2-9-3'], ['1205.5217-1-8-4', '1205.5217-2-9-4'], ['1205.5217-1-8-5', '1205.5217-2-9-5'], ['1205.5217-1-8-6', '1205.5217-2-9-6'], ['1205.5217-1-8-7', '1205.5217-2-9-7'], ['1205.5217-1-7-0', '1205.5217-2-7-0'], ['1205.5217-1-7-2', '1205.5217-2-7-2'], ['1205.5217-1-7-3', '1205.5217-2-7-3'], ['1205.5217-1-7-4', '1205.5217-2-7-4'], ['1205.5217-1-7-5', '1205.5217-2-7-5'], ['1205.5217-1-22-0', '1205.5217-2-23-0'], ['1205.5217-1-24-0', '1205.5217-2-25-0'], ['1205.5217-1-24-1', '1205.5217-2-25-1'], ['1205.5217-1-24-2', '1205.5217-2-25-2'], ['1205.5217-1-24-3', '1205.5217-2-25-3'], ['1205.5217-1-24-4', '1205.5217-2-25-4'], ['1205.5217-1-24-6', '1205.5217-2-27-1'], ['1205.5217-1-24-7', '1205.5217-2-27-2'], ['1205.5217-1-24-10', '1205.5217-2-28-1'], ['1205.5217-1-24-11', '1205.5217-2-28-2'], ['1205.5217-1-26-0', '1205.5217-2-30-0'], ['1205.5217-1-26-3', '1205.5217-2-33-1'], ['1205.5217-1-7-1', '1205.5217-2-7-1'], ['1205.5217-1-7-6', '1205.5217-2-7-6'], ['1205.5217-1-24-9', '1205.5217-2-28-0'], ['1205.5217-1-26-2', '1205.5217-2-33-0']]","[['1205.5217-1-16-0', '1205.5217-2-17-0'], ['1205.5217-1-16-1', '1205.5217-2-17-1'], ['1205.5217-1-16-2', '1205.5217-2-17-2'], ['1205.5217-1-16-3', '1205.5217-2-17-3'], ['1205.5217-1-16-4', '1205.5217-2-17-4'], ['1205.5217-1-0-0', 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'1205.5217-2-33-0']]",[],[],[],"['1205.5217-1-11-0', '1205.5217-1-13-4', '1205.5217-1-14-0', '1205.5217-1-18-0', '1205.5217-1-21-0', '1205.5217-1-24-5', '1205.5217-1-28-6', '1205.5217-1-29-0', '1205.5217-1-30-0', '1205.5217-1-33-14', '1205.5217-1-34-0', '1205.5217-1-37-0', '1205.5217-1-37-1', '1205.5217-1-37-5', '1205.5217-1-39-0', '1205.5217-1-39-1', '1205.5217-1-39-4', '1205.5217-2-12-0', '1205.5217-2-14-4', '1205.5217-2-15-0', '1205.5217-2-19-0', '1205.5217-2-22-0', '1205.5217-2-27-0', '1205.5217-2-31-0', '1205.5217-2-35-6', '1205.5217-2-36-0', '1205.5217-2-37-0', '1205.5217-2-40-14', '1205.5217-2-41-0', '1205.5217-2-44-0', '1205.5217-2-44-1', '1205.5217-2-44-6', '1205.5217-2-46-0', '1205.5217-2-46-1', '1205.5217-2-46-5']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1205.5217,,,
1311.6982,"{'1311.6982-1-0-0': 'We employ the adaptive resolution approach AdResS, in its recently developed Grand Canonical-like version (GC-AdResS) [Wang et al. Phys.Rev.X 3, 011018 (2013)], to calculate the excess chemical potential, [MATH], of various liquids and mixtures.', '1311.6982-1-0-1': 'We compare our results with those obtained from full atomistic simulations using the technique of thermodynamic integration and show a satisfactory agreement.', '1311.6982-1-0-2': 'Since in GC-AdResS the procedure to calculate [MATH] corresponds to the standard initial equilibration of the system, the advantage of the proposed approach is that [MATH] is automatically calculated every time a simulation is performed.', '1311.6982-1-1-0': '# Introduction', '1311.6982-1-2-0': 'The chemical potential represents an important thermodynamic information for any system, in particular for liquids, where the possibility of combining different substances for forming optimal mixtures is strictly related to knowledge of the chemical potential of each component in the mixture environment.', '1311.6982-1-2-1': 'In this perspective, molecular simulation represents a powerful tool for predicting the chemical potential of complex molecular systems.', '1311.6982-1-2-2': 'However current popular methodologies such as Widom particle insertion (IPM) [CITATION] are computationally very demanding often beyond a reasonable limit even in presence of large computational resources .', '1311.6982-1-2-3': 'Instead, other techniques, such as thermodynamic integration (TI) [CITATION], are computationally convenient but specifically designed to calculate the chemical potential and thus they may not be optimal for studying other properties, expecially when the technique requires artificial modification of the interactions.', '1311.6982-1-2-4': 'Recently we have suggested an alternative route to the calculation of the chemical potential by employing the Adaptive Resolution Simulation method in its Grand Canonical-like formulation (GC-AdResS) [CITATION].', '1311.6982-1-2-5': 'AdResS was originally designed to interface regions of space at different levels of molecular resolution within one simulation set up.', '1311.6982-1-2-6': 'This allows for large and efficient multiscale simulations where the high resolution region is restricted to a small portion of space and the rest of the system is at coarser level.', '1311.6982-1-2-7': 'The recent version of the method, GC-AdResS, given its theoretical framework, should automatically calculate the chemical potential during the process of initial equilibration: in this work we prove that this is indeed the case and report results for the chemical potential for various liquids and mixtures of particular relevance in (bio)-chemistry and material science.', '1311.6982-1-2-8': 'We compare our results with those from full atomistic TI and find a satisfactory agreement.', '1311.6982-1-2-9': 'This agreement allows us to conclude that compared to TI, GC-AdResS is a complementary tool to calculate the chemical potential, however, differently from TI and IPM, GC-AdResS simulations can be always used for standard (and multiscale) studies of generic properties of the system.', '1311.6982-1-2-10': 'Below we provide the basic technical ingredients of GC-AdResS which are relevant for the calculation of the chemical potential, more specific details can be found in [CITATION].', '1311.6982-1-3-0': '# From AdResS to GC-AdResS', '1311.6982-1-4-0': 'The original idea of AdResS is based on a simple intuitive physical principle:', '1311.6982-1-5-0': 'In [CITATION] we have defined necessary conditions in [MATH] such that the spatial probability distribution of the full-atomistic reference system was reproduced up to a certain (desired) order in the atomistic region of the adaptive system.', '1311.6982-1-5-1': 'Moreover we have shown that, because of such conditions, the accuracy in the atomistic region is independent of the accuracy of the coarse-grained model, thus, in the coarse-grained region, one can use a generic liquid of spheres whose only requirement is that it has the same molecular density of the reference system.', '1311.6982-1-5-2': 'In the simulation set up, [MATH] is calculated via an iterative procedure as a function of the molecular density in [MATH].', '1311.6982-1-5-3': 'The iterative scheme consists of calculating [MATH], ([MATH] the isothermal compressibility) and the force is considered converged when in [MATH] the target density [MATH] is reached.', '1311.6982-1-5-4': 'As a result, [MATH], acting in [MATH], assures that there are no artificial density variations across the system, thus it allows to accurately reproduce the first order of the probability distribution in the atomistic region.', '1311.6982-1-5-5': 'Higher orders can be systematically achieved by imposing in [MATH] a corrective force proportional to the gradient of the corresponding higher order of the probability distribution; for example, the gradient of the CM-CM radial distribution function for the second order.', '1311.6982-1-5-6': 'Next it was proved that indeed the target Grand Canonical distribution, that is the probability distribution of a subsystem (of the size of the atomistic region in GC-AdResS) in a large full atomistic simulation is accurately reproduced.', '1311.6982-1-5-7': 'A large number of tests were performed and the reproduction by GC-AdResS of the probability distribution was numerically proved up to (at least) the third order, more than sufficient in MD simulations.', '1311.6982-1-5-8': 'Within this framework it was finally shown that the sum of work of [MATH] and that of the thermostat corresponds to the difference in chemical potential between the atomistic and coarse-grained resolution; this subject is treated in the next section.', '1311.6982-1-6-0': '# Calculation of [MATH]', '1311.6982-1-7-0': 'In Ref. [CITATION] it has been shown that the chemical potential of the atomistic and coarse-grained resolution are related by the following formula: [EQUATION] with [MATH] the chemical potential of the coarse-grained system (in GC-AdResS this corresponds to a liquid of generic spheres), [MATH] the chemical potential of the atomistic system, [MATH] the work of the thermodynamic force in the transition region, [MATH] the work/heat provided by the thermostat in order to slowly equilibrate the inserted/removed degrees of freedom in the transition region.', '1311.6982-1-7-1': '[MATH] is composed by two parts, one, called [MATH], which compensates the dissipation of energy due to the change of interactions in [MATH], and another, [MATH], which is related to the equilibration of the reinserted/removed degrees of freedom (rotational and vibrational).', '1311.6982-1-7-2': 'According to the equipartition theorem [MATH] is equal to [MATH] per degree of freedom.', '1311.6982-1-7-3': '[MATH] can be calculated in a straightforward way.', '1311.6982-1-7-4': 'The calculation of [MATH] is instead not straightforward and we have proposed to introduce an auxiliary Hamiltonian approach where the coarse-grained and atomistic potential are interpolated, and not the forces as in AdResS.', '1311.6982-1-7-5': 'Next, we impose that the Hamiltonian system must have the same thermodynamic equilibrium of the original force-based AdResS system; this is done by introducing a thermodynamic force in the auxiliary Hamiltonian approach, which, at the target temperature, keeps the density of particles across the system as in AdResS.', '1311.6982-1-7-6': 'Since in the Hamiltonian approach we have the same equilibrium of the original adaptive (and full atomistic) system and in addition we do not need to use a thermostat, the difference between the work of the original thermodynamic force and the work of the thermodynamic force calculated in the Hamiltonian approach gives [MATH].', '1311.6982-1-7-7': 'Moreover we have proven numerically, for the case of liquid water, that [MATH], where [MATH] and [MATH] are the atomistic and coarse-grained potential.', '1311.6982-1-7-8': 'The result above implies that [MATH] can be calculated in a straightforward way during the initial equilibration within in the standard GC-AdResS code.', '1311.6982-1-7-9': 'It must be noticed that the relation we obtained for the chemical potential is the similar to that obtained in a complementary work by Potestio et al. [CITATION].', '1311.6982-1-7-10': 'In [CITATION] an Hamiltonian approach (H-AdResS) was explored and the expression of the chemical potential was derived via an elegant thermodynamic procedure.', '1311.6982-1-7-11': 'However, H-AdResS and GC-AdResS are essentially the same up to the first order in reproducing the probability distribution in the atomistic region.', '1311.6982-1-7-12': 'It is not a surprise that the calculation of the chemical potential can be done in a very similar way; in fact the chemical potential of a system in equilibrium requires (essentially) that the density of molecules is the same across the simulation box.', '1311.6982-1-7-13': 'The differences between the two methods emerge when higher orders of the probability distribution in the atomistic region are required.', '1311.6982-1-7-14': 'However the intention of this paper is not to discuss the difference between the two methods but rather to show their capability in accurately calculating the chemical potential; a discussion about the difference between the two approaches of AdResS can be found in [CITATION].', '1311.6982-1-7-15': 'At this point according to [REF], if one knows [MATH], then GC-AdResS can automatically provide [MATH].', '1311.6982-1-7-16': 'However we need to do one step more, in fact the quantity of interest is not the total chemical potential, but the excess chemical potential [MATH] which corresponds to the expression of [REF] where the kinetic (ideal gas) part is subtracted.', '1311.6982-1-7-17': 'Regarding the kinetic part, one can notice that the contribution coming from the center of mass is the same for the coarse-grained and for the atomistic molecules, thus it is automatically removed in the calculation of [REF].', '1311.6982-1-7-18': 'The kinetic part of [MATH] due to the rotational and vibrational degrees of freedom corresponds in our case to [MATH] and can be calculated by hand removing [MATH] per degree of freedom.', '1311.6982-1-7-19': 'Actually this calculation is not required, in fact the technical set up of AdResS considers the removed degrees of freedom as phantom variables but thermally equilibrate them anyway [CITATION].', '1311.6982-1-7-20': 'Thus the heat provided by the thermostat for the rotational and vibrational part is the same in the atomistic and coarse-graining molecules and is automatically removed in the difference.', '1311.6982-1-7-21': 'Finally, the calculation of [MATH] can be done with standard methods, TI or IPM, which for simple spherical molecules, like those of the coarse-grained system, requires a negligible computational cost.', '1311.6982-1-7-22': 'In conclusion, we have the final expression: [EQUATION]', '1311.6982-1-7-23': 'In the next section we apply this procedure to several liquids and mixtures.', '1311.6982-1-8-0': '# Results and Discussion', '1311.6982-1-9-0': 'We have calculated [MATH] for different liquids and mixtures, choosing cases which are representative of a large class of systems.', '1311.6982-1-9-1': 'Hydrophobic solvation in methane/water and in ethane/water mixtures, hydrophilic solvation in urea/water and acetic acid/water mixtures, a balance of both in water/tert-Butyl alcohol (TBA) mixture, other liquids, e.g. pure methanol and its role as a solvent in methanol/methane mixture.', '1311.6982-1-9-2': 'Moreover, systems as water/urea are commonly used as cosolvent of biological molecules [CITATION] while systems as tert-Butyl alcohol/water play a key role in modern technology [CITATION], thus they are of high interest per se.', '1311.6982-1-9-3': 'Results are reported in Table [REF] where the comparison with values obtained using full atomistic TI is made; in our previous work we have already shown in a clear way the convenience of GC-AdResS compared to the IPM and thus we do not consider calculations done with IPM in this paper.', '1311.6982-1-9-4': 'The agreement with full atomistic TI simulations is satisfactory in all cases, and thus it proves the accuracy of GC-AdResS in describing the essential thermodynamics of a large class of systems.', '1311.6982-1-9-5': 'We also compare the obtained values with those available in literature [CITATION].', '1311.6982-1-9-6': 'Although the concentration of the minor component in the mixtures that we consider, is higher than the concentrations considered in Refs. [CITATION], we are anyway in the very dilute regime and thus the chemical potential should not change in a significant way; we have verified such a supposed consistency.', '1311.6982-1-9-7': 'The reason why we did not treat the same concentration as those of the systems treated in literature actually illustrates the merits and limitations of GC-AdResS compared to TI.', '1311.6982-1-9-8': 'In fact TI has the advantage that one can use one single molecule in the simulation box to mimic the minor component in a mixture; in our case, instead, we must treat, technically speaking, a true mixture with a certain number of molecules of the minor component immersed in the liquid of the major component.', '1311.6982-1-9-9': 'Thus, at low concentrations, GC-AdResS simulations require larger systems than those required by TI, moreover, because of the low density of the minor component, the convergence of the corresponding thermodynamic force requires long simulations.', '1311.6982-1-9-10': 'Thus, for very dilute systems, if one is interested only in the chemical potential, then, at the current state of the art, TI shall be preferred to GC-AdResS, however if the interest goes beyond the calculation of the chemical potential, (e.g. radial distribution functions) then GC-AdResS would still be more convenient.', '1311.6982-1-9-11': 'When the concentration becomes higher, GC-AdResS becomes preferable for both tasks: general properties of the mixture and chemical potential, not only because in this case one requires larger systems, but also because the convergence of the thermodynamic force of the minor component is much faster.', '1311.6982-1-9-12': 'Moreover, we would have the flexibility of calculating the chemical potential of both components in one simulation run, whereas in TI, one needs to run two separate simulations in order to get the chemical potential of both components.', '1311.6982-1-9-13': 'In essence, GC-AdResS allows an on-the-fly determination of [MATH] of each component of a liquid, whenever a simulation is performed, without extra computational costs.', '1311.6982-1-9-14': 'Moreover, under certain conditions, e.g. high concentration of the minor component, it can become more convenient than TI, even if employed only for the calculation of [MATH].', '1311.6982-1-9-15': 'All technical details of each simulation are presented in the supporting material.', '1311.6982-1-9-16': ""The chemical potential of [MATH]-th liquid's component in a mixture is calculated as: [EQUATION] where [MATH] is the thermodynamic force applied to the molecules of the [MATH]-th component; this assures that, at the given concentration, the density of molecules of species [MATH], in the transition region, is equivalent to the density of the same liquid's component in a reference full atomistic simulation."", '1311.6982-1-9-17': '[MATH] is the gradient along [MATH] (in a rectangular box) applied to the switching function [MATH] in the center of mass of molecules of component [MATH].', '1311.6982-1-10-0': 'Fig.[REF] shows the action of the thermodynamic force and of the thermostat in the transition region [MATH] for TBA-water; the molecular density is sufficiently close to that of reference (the largest difference is below [MATH] and the average difference is below [MATH]), and thus it assures that in the atomistic region there are no (significant) artificial effects on the molecular density due to the perturbation represented by the interpolation of forces in [MATH].', '1311.6982-1-10-1': 'In Fig.[REF] we report various radial distribution functions for TBA-water in the atomistic region of the adaptive set up.', '1311.6982-1-10-2': 'The agreement with data from a full atomistic simulation is highly satisfactory.', '1311.6982-1-10-3': 'Moreover, it must be underlined that, on purpose, we have chosen extreme technical conditions, that is, a very small atomistic and coarse-grained region ([MATH]) and a relatively large transition region ([MATH]).', '1311.6982-1-10-4': 'Even in these conditions we prove that local properties as those of Fig. [REF] and Fig. [REF], together with a relevant thermodynamics quantity as [MATH] are well reproduced.', '1311.6982-1-10-5': 'This example shows the advantage of GC-AdResS, that is, a multiscale simulation where the chemical potential of each component is obtained without extra computational costs and with high accuracy in a simulation where other properties are also calculated with high accuracy.', '1311.6982-1-10-6': 'It must be also noticed that the system corresponding to the figures is, among all the system considered, the case where the action of the thermodynamic force and of the thermostat produces the less accurate agreement with the reference data.', '1311.6982-1-11-0': '# Conclusion', '1311.6982-1-12-0': 'We have shown the accuracy of GC-AdResS in calculating the excess chemical potential for a representative class of complex liquids and mixtures.', '1311.6982-1-12-1': 'The computational resources required are not prohibitive as those required by IPM and the simulations are more efficient that those done with TI when the quantities of interest are not restricted to the chemical potential only; in fact, for any system, the initial equilibration process, that is the determination of the thermodynamic force, automatically delivers the chemical potential.', '1311.6982-1-12-2': 'The only additional calculation required is that of [MATH] which implies the use of IPM or TI, but for a liquid of simple spheres, thus computationally negligible.', '1311.6982-1-12-3': 'Moreover, there exists cases where, just for the calculation of [MATH], GC-AdResS may become even more convenient than TI.', '1311.6982-1-12-4': 'Finally, the essential message is that GC-AdResS would be, per se, an efficient technique to calculate the chemical potential, however one can go much beyond and perform multiscale simulations for generic properties, and thus optimize the use of trajectories within a larger perspective of study.'}","{'1311.6982-2-0-0': 'We employ the adaptive resolution approach AdResS, in its recently developed Grand Canonical-like version (GC-AdResS) [Wang et al. Phys.Rev.X 3, 011018 (2013)], to calculate the excess chemical potential, [MATH], of various liquids and mixtures.', '1311.6982-2-0-1': 'We compare our results with those obtained from full atomistic simulations using the technique of thermodynamic integration and show a satisfactory agreement.', '1311.6982-2-0-2': 'In GC-AdResS the procedure to calculate [MATH] corresponds to the process of standard initial equilibration of the system; this implies that, independently of the specific aim of the study, [MATH], for each molecular species, is automatically calculated every time a GC-AdResS simulation is performed.', '1311.6982-2-1-0': '# Introduction', '1311.6982-2-2-0': 'The chemical potential represents an important thermodynamic information for any system, in particular for liquids, where the possibility of combining different substances for forming optimal mixtures is strictly related to knowledge of the chemical potential of each component in the mixture environment.', '1311.6982-2-2-1': 'In this perspective, molecular simulation represents a powerful tool for predicting the chemical potential of complex molecular systems.', '1311.6982-2-2-2': 'Popular, well established methodologies in Molecular Dynamics (MD) are Widom particle insertion (IPM) [CITATION] and thermodynamic integration (TI) [CITATION].', '1311.6982-2-2-3': 'IPM is computationally very demanding often beyond a reasonable limit even in presence of large computational resources, but upon convergence, is rather accurate.', '1311.6982-2-2-4': 'TI is computationally convenient but specifically designed to calculate the chemical potential and thus it may not be optimal for employing MD for studying other properties.', '1311.6982-2-2-5': 'In fact TI requires artificial modification of the atomistic interactions (see Appendix).', '1311.6982-2-2-6': 'Recently we have suggested that the chemical potential could be calculated by employing the Adaptive Resolution Simulation method in its Grand Canonical-like formulation (GC-AdResS) [CITATION].', '1311.6982-2-2-7': 'AdResS was originally designed to interface regions of space at different levels of molecular resolution within one simulation set up.', '1311.6982-2-2-8': 'This allows for large and efficient multiscale simulations where the high resolution region is restricted to a small portion of space and the rest of the system is at coarser level.', '1311.6982-2-2-9': 'The recent version of the method, GC-AdResS, given its theoretical framework, should automatically calculate the chemical potential during the process of initial equilibration: in this work we prove that this is indeed the case and report results for the chemical potential for various liquids and mixtures of particular relevance in (bio)-chemistry and material science.', '1311.6982-2-2-10': 'We compare our results with those from full atomistic TI and find a satisfactory agreement.', '1311.6982-2-2-11': 'This agreement allows us to conclude that every time a multiscale GC-AdResS is performed, [MATH] is automatically calculated for each liquid component and implicitly confirm that the basic thermodynamics of the system is well described by the method.', '1311.6982-2-2-12': 'Below we provide the basic technical ingredients of GC-AdResS which are relevant for the calculation of the chemical potential, more specific details can be found in [CITATION].', '1311.6982-2-3-0': '# From AdResS to GC-AdResS', '1311.6982-2-4-0': 'The original idea of AdResS is based on a simple intuitive physical principle:', '1311.6982-2-5-0': 'In [CITATION] we have defined necessary conditions in [MATH] such that the spatial probability distribution of the full-atomistic reference system was reproduced up to a certain (desired) order in the atomistic region of the adaptive system.', '1311.6982-2-5-1': 'Moreover we have shown that, because of such conditions, the accuracy in the atomistic region is independent of the accuracy of the coarse-grained model, thus, in the coarse-grained region, one can use a generic liquid of spheres whose only requirement is that it has the same molecular density of the reference system.', '1311.6982-2-5-2': 'In the simulation set up, [MATH] is calculated via an iterative procedure as a function of the molecular density in [MATH].', '1311.6982-2-5-3': 'The iterative scheme consists of calculating [MATH], ([MATH] the isothermal compressibility) and the force is considered converged when in [MATH] the target density [MATH] is reached.', '1311.6982-2-5-4': 'As a result, [MATH], acting in [MATH], assures that there are no artificial density variations across the system, thus it allows to accurately reproduce the first order of the probability distribution in the atomistic region.', '1311.6982-2-5-5': 'Higher orders can be systematically achieved by imposing in [MATH] a corrective force proportional to the gradient of the corresponding higher order of the probability distribution; for example, the gradient of the CM-CM radial distribution function for the second order.', '1311.6982-2-5-6': 'Next it was proved that indeed the target Grand Canonical distribution, that is the probability distribution of a subsystem (of the size of the atomistic region in GC-AdResS) in a large full atomistic simulation is accurately reproduced.', '1311.6982-2-5-7': 'A large number of tests were performed and the reproduction by GC-AdResS of the probability distribution was numerically proved up to (at least) the third order, more than sufficient in MD simulations.', '1311.6982-2-5-8': 'Within this framework it was finally shown that the sum of work of [MATH] and that of the thermostat corresponds to the difference in chemical potential between the atomistic and coarse-grained resolution; this subject is treated in the next section.', '1311.6982-2-6-0': '# Calculation of [MATH]', '1311.6982-2-7-0': 'In Ref. [CITATION] it has been shown that the chemical potential of the atomistic and coarse-grained resolution are related by the following formula: [EQUATION] with [MATH] the chemical potential of the coarse-grained system (in GC-AdResS this corresponds to a liquid of generic spheres), [MATH] the chemical potential of the atomistic system, [MATH] the work of the thermodynamic force in the transition region, [MATH] the work/heat provided by the thermostat in order to slowly equilibrate the inserted/removed degrees of freedom in the transition region.', '1311.6982-2-7-1': '[MATH] is composed by two parts, one, called [MATH], which compensates the dissipation of energy due to the change of interactions in [MATH], and another, [MATH], which is related to the equilibration of the reinserted/removed degrees of freedom (rotational and vibrational).', '1311.6982-2-7-2': 'According to the equipartition theorem [MATH] is equal to [MATH] per degree of freedom.', '1311.6982-2-7-3': '[MATH] can be calculated in a straightforward way.', '1311.6982-2-7-4': 'The calculation of [MATH] is instead not straightforward and we have proposed to introduce an auxiliary Hamiltonian approach where the coarse-grained and atomistic potential are interpolated, and not the forces as in AdResS.', '1311.6982-2-7-5': 'Next, we impose that the Hamiltonian system must have the same thermodynamic equilibrium of the original force-based AdResS system; this is done by introducing a thermodynamic force in the auxiliary Hamiltonian approach, which, at the target temperature, keeps the density of particles across the system as in AdResS.', '1311.6982-2-7-6': 'Since in the Hamiltonian approach we have the same equilibrium of the original adaptive (and full atomistic) system and in addition we do not need to use a thermostat, the difference between the work of the original thermodynamic force and the work of the thermodynamic force calculated in the Hamiltonian approach gives [MATH].', '1311.6982-2-7-7': 'Moreover we have proven numerically, for the case of liquid water, that [MATH], where [MATH] and [MATH] are the atomistic and coarse-grained potential.', '1311.6982-2-7-8': 'The result above implies that [MATH] can be calculated in a straightforward way during the initial equilibration within in the standard GC-AdResS code.', '1311.6982-2-7-9': 'It must be noticed that the relation we obtained for the chemical potential is the similar to that obtained in a complementary work by Potestio et al. [CITATION].', '1311.6982-2-7-10': 'In [CITATION] an Hamiltonian approach (H-AdResS) was explored and the expression of the chemical potential was derived via an elegant thermodynamic procedure.', '1311.6982-2-7-11': 'However, H-AdResS and GC-AdResS are essentially the same up to the first order in reproducing the probability distribution in the atomistic region.', '1311.6982-2-7-12': 'It is not a surprise that the calculation of the chemical potential can be done in a very similar way; in fact the chemical potential of a system in equilibrium requires (essentially) that the density of molecules is the same across the simulation box.', '1311.6982-2-7-13': 'The differences between the two methods emerge when higher orders of the probability distribution in the atomistic region are required.', '1311.6982-2-7-14': 'However the intention of this paper is not to discuss the difference between the two methods but rather to show their capability in accurately calculating the chemical potential; a discussion about the difference between the two approaches of AdResS can be found in [CITATION].', '1311.6982-2-7-15': 'At this point according to [REF], if one knows [MATH], then GC-AdResS can automatically provide [MATH].', '1311.6982-2-7-16': 'However we need to do one step more, in fact the quantity of interest is not the total chemical potential, but the excess chemical potential [MATH] which corresponds to the expression of [REF] where the kinetic (ideal gas) part is subtracted.', '1311.6982-2-7-17': 'Regarding the kinetic part, one can notice that the contribution coming from the center of mass is the same for the coarse-grained and for the atomistic molecules, thus it is automatically removed in the calculation of [REF].', '1311.6982-2-7-18': 'The kinetic part of [MATH] due to the rotational and vibrational degrees of freedom corresponds in our case to [MATH] and can be calculated by hand removing [MATH] per degree of freedom.', '1311.6982-2-7-19': 'Actually this calculation is not required, in fact the technical set up of AdResS considers the removed degrees of freedom as phantom variables but thermally equilibrate them anyway [CITATION].', '1311.6982-2-7-20': 'Thus the heat provided by the thermostat for the rotational and vibrational part is the same in the atomistic and coarse-graining molecules and is automatically removed in the difference.', '1311.6982-2-7-21': 'Finally, the calculation of [MATH] can be done with standard methods, TI or IPM, which for simple spherical molecules, like those of the coarse-grained system, requires a negligible computational cost.', '1311.6982-2-7-22': 'In conclusion, we have the final expression: [EQUATION]', '1311.6982-2-7-23': 'In the next section we apply this procedure to several liquids and mixtures.', '1311.6982-2-8-0': '# Results and Discussion', '1311.6982-2-9-0': 'We have calculated [MATH] for different liquids and mixtures, choosing cases which are representative of a large class of systems.', '1311.6982-2-9-1': 'Hydrophobic solvation in methane/water and in ethane/water mixtures, hydrophilic solvation in urea/water, a balance of both in water/tert-Butyl alcohol (TBA) mixture, other liquids, e.g. pure methanol and DMSO (and their mixtures with water), non aqueous mixtures in TBA/DMSO and alkane liquids such as methane, ethane and propane.', '1311.6982-2-9-2': 'Moreover, systems as water/urea are commonly used as cosolvent of biological molecules [CITATION] while systems as tert-Butyl alcohol/water play a key role in modern technology [CITATION], thus they are of high interest per se.', '1311.6982-2-9-3': 'All technical details of each simulation are presented in the Appendix.', '1311.6982-2-9-4': 'Results are reported in Table [REF] where the comparison with values obtained using full atomistic TI and available experiments, at the same concentrations, of our calculation is made; in our previous work we have already shown that value of the chemical potential of liquid water obtained with IPM is well reproduced by GC-AdResS, however the computational cost of IMP was very large, thus we do not consider calculations done with IPM in this paper.', '1311.6982-2-9-5': 'The agreement with full atomistic TI simulations is satisfactory in all cases, and thus it proves the solidity of GC-AdResS in describing the essential thermodynamics of a large class of systems.', '1311.6982-2-9-6': 'We also compare the obtained values with those available in literature [CITATION].', '1311.6982-2-9-7': 'Although the concentration of the minor component in the mixtures that we consider, is higher than the concentrations considered in Refs. [CITATION], we are anyway in the very dilute regime and thus the chemical potential should not change in a significant way; we have verified such a supposed consistency.', '1311.6982-2-9-8': ""The chemical potential of [MATH]-th liquid's component in a mixture is calculated as: [EQUATION] where [MATH] is the thermodynamic force applied to the molecules of the [MATH]-th component; this assures that, at the given concentration, the density of molecules of species [MATH], in the transition region, is equivalent to the density of the same liquid's component in a reference full atomistic simulation."", '1311.6982-2-9-9': '[MATH] is the gradient along [MATH] (in a rectangular box) applied to the switching function [MATH] in the center of mass of molecules of component [MATH].', '1311.6982-2-10-0': 'In essence, according to the results obtained, GC-AdResS allows an on-the-fly determination of [MATH] of each component of a liquid, whenever a simulation is performed, without extra computational costs.', '1311.6982-2-10-1': 'Moreover, Fig.[REF] shows the action of the thermodynamic force and of the thermostat in the transition region [MATH] for TBA-water; the molecular density is sufficiently close to that of reference (the largest difference is below [MATH] and the average difference is below [MATH]), and thus it assures that in the atomistic region there are no (significant) artificial effects on the molecular density due to the perturbation represented by the interpolation of forces in [MATH].', '1311.6982-2-10-2': 'In Fig.[REF] we report various radial distribution functions for TBA-water in the atomistic region of the adaptive set up.', '1311.6982-2-10-3': 'The agreement with data from a full atomistic simulation is highly satisfactory.', '1311.6982-2-10-4': 'Moreover, it must be underlined that, on purpose, we have chosen extreme technical conditions, that is, a very small atomistic and coarse-grained region ([MATH]) and a relatively large transition region ([MATH]).', '1311.6982-2-10-5': 'Even in these conditions we prove that local properties as those of Fig. [REF] and Fig. [REF], together with a relevant thermodynamics quantity as [MATH] are well reproduced.', '1311.6982-2-10-6': 'This example shows the key features of GC-AdResS, that is, a multiscale simulation where the chemical potential of each component is obtained without extra computational costs and with high accuracy in a simulation where other properties are also calculated with high accuracy.', '1311.6982-2-10-7': 'It must be also noticed that the system corresponding to the figures is, among all the system considered, the case where the action of the thermodynamic force and of the thermostat produces the less accurate agreement with the reference data.', '1311.6982-2-11-0': '# Current computational convenience: a critical appraisal', '1311.6982-2-12-0': 'The natural question arising from the discussion above is whether or not GC-AdResS is a more convenient technical tool for calculating [MATH] compared to TI.', '1311.6982-2-12-1': 'Currently the answer is negative although the current work is the first step towards a potentially positive answer for the future.', '1311.6982-2-12-2': 'In fact, the fastest version of AdResS is implemented in the GROMACS code [CITATION]; using the Gromacs version 4.5.1 a speedup of a factor four with respected to full atomistic simulations has been reported for aqueous mixtures [CITATION].', '1311.6982-2-12-3': 'In this case GC-AdResS was more convenient than TI because in one simulation one could obtain the chemical potential of each liquid component and at the same time calculate structural properties (e.g. radial distribution functions).', '1311.6982-2-12-4': 'However in the successive version of GROMACS 4.6.1 the performance of atomistic simulations (above all of SPC/E water) has been highly improved while the corresponding implementation of AdResS is not optimized yet.', '1311.6982-2-12-5': 'At the current state, AdResS can only assure a speed up factor between 1.5 and 1.9 for large systems (30000 molecules) compared to full atomistic simulations without SPC/E water.', '1311.6982-2-12-6': 'As a consequence for the calculation of [MATH], TI is computationally less demanding than AdResS.', '1311.6982-2-12-7': 'Another point that must be considered (in perspective) for a fair comparison between TI and GC-AdResS, is the following: even if AdResS is optimized, in any code, TI has the advantage that one can use one single molecule in the simulation box to mimic the minor component of a mixture.', '1311.6982-2-12-8': 'In our case, instead, we must treat, technically speaking, a true mixture with a certain number of molecules of the minor component immersed in the liquid of the major component.', '1311.6982-2-12-9': 'Thus, at low concentrations, GC-AdResS simulations require larger systems than those required by TI, moreover, because of the low density of the minor component, the convergence of the corresponding thermodynamic force requires long simulations.', '1311.6982-2-12-10': 'Thus, for very dilute systems, if one is interested only in the chemical potential, TI shall be preferred to GC-AdResS, however if the interest goes beyond the calculation of the chemical potential, (e.g. radial distribution functions) then (optimized) GC-AdResS would still be more convenient.', '1311.6982-2-12-11': 'When the concentration becomes higher, GC-AdResS may become preferable for both tasks: general properties of the mixture and chemical potential, not only because in this case one requires larger systems, but also because the convergence of the thermodynamic force of the minor component is much faster.', '1311.6982-2-12-12': 'Moreover, we would have the flexibility of calculating the chemical potential of both components in one simulation run, whereas in TI, one needs to run two separate simulations in order to get the chemical potential of both components.', '1311.6982-2-12-13': 'However, at the current state, the technical aspects of code optimization must be reported and we must make clear that the aim of this work is to show that the automatic calculation of [MATH], independently from the simulation code in which is implemented and its computational cost, is a ""conceptual"" feature of GC-AdResS.', '1311.6982-2-13-0': '# Conclusion', '1311.6982-2-14-0': 'We have shown the accuracy of GC-AdResS in calculating the excess chemical potential for a representative class of complex liquids and mixtures.', '1311.6982-2-14-1': 'For any system, the initial equilibration process, that is the determination of the thermodynamic force, automatically delivers the chemical potential.', '1311.6982-2-14-2': 'The only additional calculation required is that of [MATH] which implies the use of IPM or TI, but for a liquid of simple spheres, thus computationally negligible.', '1311.6982-2-14-3': 'The essential message is that GC-AdResS would be, per se, a reliable multiscale technique to calculate the chemical potential and, in perspective, upon computational/technical optimization it may become an efficient tool for calculating [MATH] compared to current techniques in MD such as TI.'}","[['1311.6982-1-10-1', '1311.6982-2-10-2'], ['1311.6982-1-10-2', '1311.6982-2-10-3'], ['1311.6982-1-10-3', '1311.6982-2-10-4'], ['1311.6982-1-10-4', '1311.6982-2-10-5'], ['1311.6982-1-10-6', '1311.6982-2-10-7'], ['1311.6982-1-5-0', '1311.6982-2-5-0'], ['1311.6982-1-5-1', '1311.6982-2-5-1'], ['1311.6982-1-5-2', '1311.6982-2-5-2'], ['1311.6982-1-5-3', '1311.6982-2-5-3'], ['1311.6982-1-5-4', '1311.6982-2-5-4'], ['1311.6982-1-5-5', '1311.6982-2-5-5'], ['1311.6982-1-5-6', '1311.6982-2-5-6'], ['1311.6982-1-5-7', '1311.6982-2-5-7'], ['1311.6982-1-5-8', '1311.6982-2-5-8'], ['1311.6982-1-0-0', '1311.6982-2-0-0'], ['1311.6982-1-0-1', '1311.6982-2-0-1'], ['1311.6982-1-7-0', '1311.6982-2-7-0'], ['1311.6982-1-7-1', '1311.6982-2-7-1'], ['1311.6982-1-7-2', '1311.6982-2-7-2'], ['1311.6982-1-7-3', 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'1311.6982-4-20-4'], ['1311.6982-3-19-5', '1311.6982-4-20-5'], ['1311.6982-3-19-6', '1311.6982-4-20-6'], ['1311.6982-3-19-7', '1311.6982-4-20-7'], ['1311.6982-3-19-8', '1311.6982-4-20-8'], ['1311.6982-3-19-9', '1311.6982-4-20-9'], ['1311.6982-3-19-10', '1311.6982-4-20-10'], ['1311.6982-3-19-11', '1311.6982-4-20-11'], ['1311.6982-3-19-12', '1311.6982-4-20-12'], ['1311.6982-3-10-0', '1311.6982-4-10-0'], ['1311.6982-3-10-1', '1311.6982-4-10-1'], ['1311.6982-3-10-2', '1311.6982-4-10-2'], ['1311.6982-3-10-3', '1311.6982-4-10-3'], ['1311.6982-3-10-4', '1311.6982-4-10-4'], ['1311.6982-3-10-5', '1311.6982-4-10-5'], ['1311.6982-3-10-6', '1311.6982-4-10-6'], ['1311.6982-3-10-7', '1311.6982-4-10-7'], ['1311.6982-3-10-8', '1311.6982-4-10-8'], ['1311.6982-3-10-9', '1311.6982-4-10-9'], ['1311.6982-3-10-10', '1311.6982-4-11-3'], ['1311.6982-3-10-11', '1311.6982-4-11-4'], ['1311.6982-3-10-12', '1311.6982-4-11-5'], ['1311.6982-3-10-13', '1311.6982-4-11-6'], ['1311.6982-3-10-14', '1311.6982-4-11-7'], 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'1311.6982-2-9-4'], ['1311.6982-1-2-2', '1311.6982-2-2-3'], ['1311.6982-1-2-4', '1311.6982-2-2-6'], ['1311.6982-1-2-9', '1311.6982-2-2-11'], ['1311.6982-1-12-1', '1311.6982-2-14-1'], ['1311.6982-1-12-4', '1311.6982-2-14-3'], ['1311.6982-2-5-3', '1311.6982-3-5-5'], ['1311.6982-2-5-5', '1311.6982-3-5-7'], ['1311.6982-2-7-6', '1311.6982-3-7-5'], ['1311.6982-2-7-9', '1311.6982-3-7-11'], ['1311.6982-2-7-18', '1311.6982-3-7-15'], ['1311.6982-2-7-19', '1311.6982-3-7-17'], ['1311.6982-2-12-13', '1311.6982-3-21-13']]",[],"['1311.6982-1-4-0', '1311.6982-2-4-0', '1311.6982-3-4-0', '1311.6982-4-4-0', '1311.6982-5-4-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '5': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1311.6982,"{'1311.6982-3-0-0': 'We employ the adaptive resolution approach AdResS, in its recently developed Grand Canonical-like version (GC-AdResS) [Wang et al. Phys.Rev.X 3, 011018 (2013)], to calculate the excess chemical potential, [MATH], of various liquids and mixtures.', '1311.6982-3-0-1': 'We compare our results with those obtained from full atomistic simulations using the technique of thermodynamic integration and show a satisfactory agreement.', '1311.6982-3-0-2': 'In GC-AdResS the procedure to calculate [MATH] corresponds to the process of standard initial equilibration of the system; this implies that, independently of the specific aim of the study, [MATH], for each molecular species, is automatically calculated every time a GC-AdResS simulation is performed.', '1311.6982-3-1-0': '# Introduction', '1311.6982-3-2-0': 'The chemical potential represents an important thermodynamic information for any system, in particular for liquids, where the possibility of combining different substances for forming optimal mixtures is strictly related to knowledge of the chemical potential of each component in the mixture environment.', '1311.6982-3-2-1': 'In this perspective, molecular simulation represents a powerful tool for predicting the chemical potential of complex molecular systems.', '1311.6982-3-2-2': 'Popular, well established methodologies in Molecular Dynamics (MD) are Widom particle insertion (IPM) [CITATION] and thermodynamic integration (TI) [CITATION].', '1311.6982-3-2-3': 'IPM is computationally very demanding often beyond a reasonable limit even in presence of large computational resources, but upon convergence, is rather accurate.', '1311.6982-3-2-4': 'TI is computationally convenient but specifically designed to calculate the chemical potential and thus it may not be optimal for employing MD for studying other properties.', '1311.6982-3-2-5': 'In fact TI requires artificial modification of the atomistic interactions (see Appendix).', '1311.6982-3-2-6': 'Recently we have suggested that the chemical potential could be calculated by employing the Adaptive Resolution Simulation method in its Grand Canonical-like formulation (GC-AdResS) [CITATION].', '1311.6982-3-2-7': 'AdResS was originally designed to interface regions of space at different levels of molecular resolution within one simulation set up.', '1311.6982-3-2-8': 'This allows for large and efficient multiscale simulations where the high resolution region is restricted to a small portion of space and the rest of the system is at coarser level.', '1311.6982-3-2-9': 'The recent version of the method, GC-AdResS, given its theoretical framework, should automatically calculate the chemical potential during the process of initial equilibration: in this work we prove that this is indeed the case and report results for the chemical potential for various liquids and mixtures of particular relevance in (bio)-chemistry and material science.', '1311.6982-3-2-10': 'We compare our results with those from full atomistic TI and find a satisfactory agreement.', '1311.6982-3-2-11': 'This agreement allows us to conclude that every time a multiscale GC-AdResS is performed, [MATH] is automatically calculated for each liquid component and implicitly confirm that the basic thermodynamics of the system is well described by the method.', '1311.6982-3-2-12': 'Moreover, in recent work AdResS has been merged with the MARTINI force field [CITATION].', '1311.6982-3-2-13': 'In this context, the possibility of checking the consistency of a quantity like the chemical potential can be used as a further argument for the validity of the method in applications to large systems of biological interest.', '1311.6982-3-2-14': 'Below we provide the basic technical ingredients of GC-AdResS which are relevant for the calculation of the chemical potential, more specific details can be found in [CITATION].', '1311.6982-3-3-0': '# From AdResS to GC-AdResS', '1311.6982-3-4-0': 'The original idea of AdResS is based on a simple intuitive physical principle:', '1311.6982-3-5-0': 'In [CITATION] we have defined necessary conditions in [MATH] such that the spatial probability distribution of the full-atomistic reference system was reproduced up to a certain (desired) order in the atomistic region of the adaptive system.', '1311.6982-3-5-1': 'We have defined the [MATH]th order of a spatial (configurational) probability distribution of [MATH] molecules, [MATH], as: [EQUATION]', '1311.6982-3-5-2': 'The first order, often mentioned in this work corresponds to the molecular number density [MATH].', '1311.6982-3-5-3': 'Moreover we have shown that, because of the necessary conditions, the accuracy in the atomistic region is independent of the accuracy of the coarse-grained model, thus, in the coarse-grained region, one can use a generic liquid of spheres whose only requirement is that it has the same molecular density of the reference system.', '1311.6982-3-5-4': 'In the simulation set up, [MATH] is calculated via an iterative procedure black using the molecular number density in [MATH].', '1311.6982-3-5-5': 'The iterative scheme consists of calculating black [MATH] the isothermal compressibility), and the thermodynamic force is considered converged when the target density [MATH] is reached in [MATH].', '1311.6982-3-5-6': 'As a result, [MATH], acting in [MATH], assures that there are no artificial density variations across the system, thus it allows to accurately reproduce the first order of the probability distribution in the atomistic region.', '1311.6982-3-5-7': 'Higher orders can be systematically achieved by imposing in [MATH] a corrective force.', '1311.6982-3-5-8': 'For example, the COM-COM radial distribution function correction for the second order [CITATION].', '1311.6982-3-5-9': 'Next it was proved that indeed the target Grand Canonical distribution, that is the probability distribution of a subsystem (of the size of the atomistic region in GC-AdResS) in a large full atomistic simulation is accurately reproduced.', '1311.6982-3-5-10': 'A large number of tests were performed and the reproduction by GC-AdResS of the probability distribution was numerically proved up to (at least) the third order, more than sufficient in MD simulations.', '1311.6982-3-5-11': 'Within this framework it was finally shown that the sum of work of [MATH] and that of the thermostat corresponds to the difference in chemical potential between the atomistic and coarse-grained resolution; this subject is treated in the next section.', '1311.6982-3-6-0': '# Calculation of Excess Chemical Potential', '1311.6982-3-7-0': 'In Ref. [CITATION] it has been shown that the chemical potential of the atomistic and coarse-grained resolution are related by the following formula: [EQUATION] with [MATH] the chemical potential of the coarse-grained system (in GC-AdResS this corresponds to a liquid of generic spheres), [MATH] the chemical potential of the atomistic system, [MATH] the work of the thermodynamic force in the transition region, [MATH] the work/heat provided by the thermostat in order to slowly equilibrate the inserted/removed degrees of freedom in the transition region.', '1311.6982-3-7-1': '[MATH] is composed by two parts, one, called [MATH], which compensates the dissipation of energy due to the non-conservative interactions in [MATH], and another, [MATH], which is related to the equilibration of the reinserted/removed degrees of freedom (rotational and vibrational).', '1311.6982-3-7-2': 'black While the determination of [MATH] is not required for our final aim (that is the calculation of the excess chemical potential, as explained later on), the calculation of [MATH] is very relevant.', '1311.6982-3-7-3': 'However this calculation is not straightforward and we have proposed to introduce an auxiliary Hamiltonian approach where the coarse-grained and atomistic potential are interpolated, and not the forces as in black the original AdResS.', '1311.6982-3-7-4': 'Next, we impose that the Hamiltonian system must have the same thermodynamic equilibrium of the original force-based GC-AdResS system; this is done by introducing a thermodynamic force in the auxiliary Hamiltonian approach, which, at the target temperature, keeps the density of particles across the system as in GC-AdResS.', '1311.6982-3-7-5': 'In the auxiliary Hamiltonian approach we have the same equilibrium as the original adaptive (and full atomistic) system and the difference between the work of the original thermodynamic force and the work of the thermodynamic force calculated in the Hamiltonian approach gives [MATH] (further details about this point are given in the Appendix [REF]).', '1311.6982-3-7-6': 'Moreover we have proven numerically, for the case of liquid water, that [MATH], where [MATH] and [MATH] are the atomistic and coarse-grained potential.', '1311.6982-3-7-7': 'black It must be noticed that the auxiliary Hamiltonian approach shall not be considered a Hamiltonian approach to adaptive resolution simulation.', '1311.6982-3-7-8': 'In fact, as discussed in Ref.[CITATION] the equilibrium is imposed artificially and per se does not have any physical meaning (for more details, see discussion in the Appendix [REF]).', '1311.6982-3-7-9': 'black In the next section of this work we show analytically that the formula above is exact (at least) at the first order w.r.t. the probability distribution of the system as defined in Eq.[REF].', '1311.6982-3-7-10': 'The result above implies that [MATH] can be calculated in a straightforward way during the initial equilibration within in the standard GC-AdResS code.', '1311.6982-3-7-11': 'It must be noticed that, within the AdResS scheme, an approach similar to the auxiliary Hamiltonian has been recently proposed and applied to liquids and mixtures (of toy models so far) by Potestio et al. [CITATION] (see also [CITATION] where such an approach is commented).', '1311.6982-3-7-12': 'At this point according to [REF], if one knows [MATH], then GC-AdResS can automatically provide [MATH].', '1311.6982-3-7-13': 'However we need to do one step more, in fact the quantity of interest is not the total chemical potential, but the excess chemical potential [MATH] which corresponds to the expression of [REF] where the kinetic (ideal gas) part is subtracted.', '1311.6982-3-7-14': 'Regarding the kinetic part, one can notice that the contribution coming from the COM is the same for the coarse-grained and for the atomistic molecules, thus it is automatically removed in the calculation of [REF].', '1311.6982-3-7-15': 'The kinetic part of [MATH] due to the rotational and vibrational degrees of freedom corresponds in our case to [MATH] and in principle can be calculated by hand (chemical potential of an isolated molecules).', '1311.6982-3-7-16': 'However such a calculation may become rather tedious for large and/or complex molecules but in our case it is actually not required.', '1311.6982-3-7-17': 'In fact the Gromacs implementation of AdResS considers the removed degrees of freedom as phantom variables but thermally equilibrate them anyway [CITATION].', '1311.6982-3-7-18': 'Thus the heat provided by the thermostat for the rotational and vibrational part is the same in the atomistic and coarse-graining molecules and is automatically removed in the difference.', '1311.6982-3-7-19': 'Finally, the calculation of [MATH] can be done with standard methods, TI or IPM, which for simple spherical molecules, like those of the coarse-grained system, requires a negligible computational cost.', '1311.6982-3-7-20': 'In conclusion, we have the final expression: [EQUATION]', '1311.6982-3-8-0': '# Analytic Derivation of [MATH]', '1311.6982-3-9-0': 'In this section we derive analytically the equivalence: [MATH] and define its conceptual limitations.', '1311.6982-3-9-1': 'We consider a potential coupling between the atomistic and coarse-grained resolution as the spatial interpolation of the atomistic and coarse-grained potential, as done instead for the forces in the standard AdResS: [EQUATION] where [MATH] and [MATH] are the atomistic and coarse-grained interaction potential between molecule [MATH] and [MATH], respectively, defined by [EQUATION] where [MATH] and [MATH] denotes the atom indexes of the corresponding molecule.', '1311.6982-3-9-2': 'The COM of the molecule is defined as: [EQUATION] where [MATH] is the mass of atom [MATH] of molecule [MATH].', '1311.6982-3-9-3': 'The potential interpolation [REF] provides an auxiliary Hamiltonian to the AdResS system, and the corresponding intermolecular force is given by: [EQUATION]', '1311.6982-3-9-4': 'We refer to the AdResS simulation using force scheme [REF] as auxiliary Hamiltonian AdResS, and all properties of this approach will be added a superscript ""H"".', '1311.6982-3-9-5': 'We define the force of changing representation by [EQUATION]', '1311.6982-3-9-6': 'We use the same notation as in our previous work [CITATION].', '1311.6982-3-9-7': 'The thermodynamic variables for the atomistic and coarse-grained regions are denoted by [MATH] and [MATH], respectively.', '1311.6982-3-9-8': 'We assume that the transition region is an infinitely thin filter (that is a much smaller region than the atomistic and coarse-grained region) that allows molecules to change resolution as they cross it.', '1311.6982-3-9-9': 'Therefore, it is reasonable to assume that: [EQUATION] where [MATH] and [MATH] are the total volume and total number of molecules of the system.', '1311.6982-3-9-10': 'In this work, we adopt the same assumptions as those listed in Sec.III.C of Ref. [CITATION], i.e. we assume the system to be in the thermodynamic limit, and molecules are short-range correlated (short-ranged must be intended as a range comparable to the size of the transition region).', '1311.6982-3-9-11': 'The thermodynamic force for GC-AdResS ([MATH]) and for the auxiliary Hamiltonian AdResS ([MATH]), enforce the system to have a flat density: [EQUATION]', '1311.6982-3-9-12': 'Where [MATH] is the equilibrium number density of the system defined by [MATH].', '1311.6982-3-9-13': 'As shown in Refs. [CITATION], [MATH] provides the balance of the grand potential or equivalently [EQUATION] where [MATH] denotes the integral of the thermodynamic force [MATH].', '1311.6982-3-10-0': ""Instead when we consider the auxiliary Hamiltonian approach, the third term on the R.H.S. of Eq. [REF] is not symmetric w.r.t molecule [MATH] and [MATH], therefore, the Newton's action-reaction law (momentum conservation) does not hold anymore."", '1311.6982-3-10-1': 'As a consequence, the pressure relation between the AT and CG resolution [REF] does not hold and should be derived again.', '1311.6982-3-10-2': 'Now assume, for simplicity and without lost of generality, that the system changes resolution only along the [MATH] direction.', '1311.6982-3-10-3': 'We impose an infinitesimal increment of the volume [MATH] to the AT region, and apply the same decrement of the volume [MATH] to the CG region.', '1311.6982-3-10-4': 'The volume of the transition region is kept constant as if it is an ideal ""piston"" that moves toward the CG region by an amount [MATH].', '1311.6982-3-10-5': 'We assume [MATH], where [MATH] is the cutting surface area.', '1311.6982-3-10-6': 'The displacement [MATH] should be infinitesimal, i.e. much smaller than the size of the transition region.', '1311.6982-3-10-7': 'This is achievable by taking the limit of [MATH], while keeping the system size fixed.', '1311.6982-3-10-8': 'It must be noticed that also the displacements of the molecules are infinitesimal, so it can be reasonably assumed that the resolution of the molecules remains the same under a displacement of [MATH].', '1311.6982-3-10-9': 'Therefore, the change of the free energy of the system is approximately: [EQUATION] where [MATH] and [MATH] are the free energies of the AT and CG region, respectively.', '1311.6982-3-10-10': 'The expression of Eq. [REF] as a sum of different terms is justified by the hypothesis of treating the system in the thermodynamic limit, and by the hypothesis that the interactions are short-ranged compared to the size of the transition region.', '1311.6982-3-10-11': '[MATH] and [MATH] is the numbers of molecules in the AT and CG region, and [MATH] and [MATH] is the volume in the AT and CG region, respectively; [MATH] is the temperature of the system.', '1311.6982-3-10-12': 'The last term is originated by the work done of the ideal piston.', '1311.6982-3-10-13': ""This latter is composed by two parts, the first corresponding to the work done by the thermodynamic force, and the second corresponding to the work done by the force of changing representation (which does not vanish due to the violation of the Newton's action-reaction law)."", '1311.6982-3-10-14': 'The first and second term of Eq. [REF] being forces based on pairwise interactions only, do not contribute to the difference of energy; in fact their total work is zero (as long as the transition region move infinitesimally along [MATH]).', '1311.6982-3-10-15': 'The notation [MATH] in Eq. [REF] denotes the ensemble average, which will be specified soon.', '1311.6982-3-10-16': 'It is straightforward to show that [EQUATION] where [MATH] is the integral of the thermodynamic force [MATH], and [MATH] is the work of changing representation, which can be explicitly written down as: [EQUATION]', '1311.6982-3-10-17': 'The average is performed over all possible positions of the second molecule (i.e. [MATH]), at fixed position of the first molecule (i.e. [MATH]) in the pairwise interaction.', '1311.6982-3-10-18': 'In case of molecules containing more than one atom, the average is also made over all possible conformations in the atomistic resolution.', '1311.6982-3-10-19': 'In the thermodynamic limit, the equilibrium volume of the AT region maximize the free energy, i.e. [MATH], which yields [EQUATION]', '1311.6982-3-10-20': 'Comparing the expression above with that obtained for GC-AdResS (Eq. [REF]), we have: [EQUATION] which relates the thermodynamic force of the auxiliary Hamiltonian AdResS and the GC-AdResS.', '1311.6982-3-11-0': 'In Ref. [CITATION] we proved that under proper assumptions, when the flat density profile is enforced by the thermodynamic force, the chemical potential difference between the different resolutions is given by [EQUATION]', '1311.6982-3-11-1': 'The same argument can be applied to the auxiliary Hamiltonian approach, and yields the chemical potential difference between the AT and CG resolutions [EQUATION]', '1311.6982-3-11-2': 'In the auxiliary Hamiltonian, we do not have the term [MATH] in the above formula (being the term [MATH] in GC-AdResS, generated by the non-conservative effect of the force interpolation).', '1311.6982-3-11-3': 'By comparing [REF] with [REF], we have the relation [EQUATION] which also relates the thermodynamic force of the auxiliary Hamiltonian AdResS and GC-AdResS.', '1311.6982-3-12-0': 'From Eq. [REF] and [REF], we find the extra work of the thermostat in GC-AdResS being identical to the work of changing representation of the auxiliary Hamiltonian approach: [EQUATION] which basically proves the statement at the beginning of this section.', '1311.6982-3-12-1': 'The ensemble average on the R.H.S. of Eq. [REF] is performed in the ensemble of the system treated with the potential interpolation approach, and the question is if the ensemble average is equivalent if it is performed in the simulation where the force interpolation approach is used.', '1311.6982-3-12-2': 'It is obvious that the spatial probability distribution corresponding to the system treated with the potential interpolation is consistent with the force interpolation at least up to the first order.', '1311.6982-3-12-3': 'It is also possible to systematically obtain equivalence in the ensemble average operation at higher orders of accuracy of the probability distribution, as, for example, it is done for the radial distribution function in Ref. [CITATION].', '1311.6982-3-12-4': 'However, here we do not consider higher order corrections, because it has been numerically shown that actually the ensemble average of [MATH] dose not depend on in which ensemble it is calculated [CITATION].', '1311.6982-3-12-5': 'Therefore, we use Eq. [REF] to calculate [MATH], and measure the ensemble average by the standard AdResS.', '1311.6982-3-12-6': 'As previously discussed, in the Gromacs implementation, the CG molecules also keep the atomistic degrees of freedom even though they are in the CG region, therefore, the kinetic part of [MATH] and [MATH] are identical, and [MATH] vanishes.', '1311.6982-3-12-7': 'Therefore, by inserting Eq. [REF] into [REF], we have [EQUATION]', '1311.6982-3-12-8': 'The extension of Eq. [REF] to multicomponent systems is reported in the Appendix [REF], while in the next section we apply the method to the calculation of [MATH] to liquids and mixtures.', '1311.6982-3-13-0': '# Results and Discussion', '1311.6982-3-14-0': 'We have calculated [MATH] for different liquids and mixtures, choosing cases which are representative of a large class of systems.', '1311.6982-3-14-1': 'Hydrophobic solvation in methane/water and in ethane/water mixtures, hydrophilic solvation in urea/water, a balance of both in water/tert-Butyl alcohol (TBA) mixture, other liquids, e.g. pure methanol and DMSO (and their mixtures with water), non aqueous mixtures in TBA/DMSO and alkane liquids such as methane, ethane and propane.', '1311.6982-3-14-2': 'Moreover, systems as water/urea are commonly used as cosolvent of biological molecules [CITATION] while systems as tert-Butyl alcohol/water play a key role in modern technology [CITATION], thus they are of high interest per se.', '1311.6982-3-14-3': 'All technical details of each simulation are presented in the Appendix [REF].', '1311.6982-3-15-0': 'Results are reported in Table [REF], where the comparison with values obtained using full atomistic TI and available experiments, at the same concentrations, of our calculation is made; in our previous work we have already shown that value of the chemical potential of liquid water obtained with IPM is well reproduced by GC-AdResS, however the computational cost of IMP was very large, thus we do not consider calculations done with IPM in this paper.', '1311.6982-3-15-1': 'The agreement with full atomistic TI simulations is satisfactory in all cases, and thus it proves the solidity of GC-AdResS in describing the essential thermodynamics of a large class of systems.', '1311.6982-3-15-2': 'We also compare the obtained values with those available in literature [CITATION].', '1311.6982-3-15-3': 'Although the concentration of the minor component in the mixtures that we consider, is higher than the concentrations considered in Refs.[CITATION], we are anyway in the very dilute regime and thus the chemical potential should not change in a significant way; black we have verified such a supposed consistency for one relevant system (see discussion about Fig.[REF]).', '1311.6982-3-15-4': ""The chemical potential of [MATH]-th liquid's component in a mixture is calculated as black (see Appendix [REF]): [EQUATION] where [MATH] is the thermodynamic force applied to the molecules of the [MATH]-th component; this assures that, at the given concentration, the density of molecules of species [MATH], in the transition region, is equivalent to the density of the same liquid's component in a reference full atomistic simulation."", '1311.6982-3-15-5': 'black The ensemble average is taken over the position of the second molecule, provided that the first molecule is of species [MATH], and located at position [MATH].', '1311.6982-3-16-0': 'black A complementary information to Table [REF] are Fig. [REF] and [REF].', '1311.6982-3-16-1': 'In Fig. [REF] we have studied the behavior of [MATH] as a function of the interaction cut-off.', '1311.6982-3-16-2': 'In fact the current version of GC-AdResS, employs the reaction field method for treating electrostatic interactions in the atomistic region, and the cut off is likely to play a role in some of the systems investigated.', '1311.6982-3-16-3': 'Fig. [REF], for the case of DMSO/water mixture, confirms our intuition and suggests that we could systematically improve the accuracy by increasing the cut-off, and at a value of about [MATH] nm, [MATH] converges.', '1311.6982-3-16-4': 'In any case, at a values of [MATH] nm, which is the one routinely used in full atomistic simulations and used by us, the value obtained with GC-AdResS is already satisfactory.', '1311.6982-3-16-5': 'The cut-off radii used for other systems are reported in Appendix [REF].', '1311.6982-3-16-6': 'A further question that may arise is the capability of our method to predict the behavior of [MATH] as a function of the concentration, above all in the very dilute regime.', '1311.6982-3-16-7': 'In Fig. [REF] we have performed such a study for the case of TBA/water mixture, we show a good agreement between GC-AdResS and TI and show that at a very dilute concentration our calculated value is close to that of experiments, moreover the trend, regarding the TI calculations, is consistent with that reported in Ref. [CITATION].', '1311.6982-3-17-0': 'In essence, according to the results obtained, GC-AdResS allows an on-the-fly determination of [MATH] of each component of a liquid, whenever a simulation is performed, without extra computational costs.', '1311.6982-3-17-1': 'Moreover, Fig.[REF] shows the action of the thermodynamic force and of the thermostat in the transition region [MATH] for TBA-water; the molecular density is sufficiently close to that of reference (the largest difference is below [MATH] and the average difference is below [MATH]), and thus it assures that in the atomistic region there are no (significant) artificial effects on the molecular density due to the perturbation represented by the interpolation of forces in [MATH].', '1311.6982-3-17-2': 'In Fig.[REF] we report various radial distribution functions for TBA-water in the atomistic region of the adaptive set up.', '1311.6982-3-17-3': 'The agreement with data from a full atomistic simulation is highly satisfactory.', '1311.6982-3-17-4': 'Moreover, it must be underlined that, on purpose, we have chosen extreme technical conditions, that is, a very small atomistic and coarse-grained region ([MATH] nm) and a relatively large transition region ([MATH] nm).', '1311.6982-3-17-5': 'Even in these conditions we prove that local properties as those of Fig. [REF] and Fig. [REF], together with a relevant thermodynamics quantity as [MATH] are well reproduced.', '1311.6982-3-17-6': 'This example shows the key features of GC-AdResS, that is, a multiscale simulation where the chemical potential of each component is obtained without extra computational costs and with high accuracy in a simulation where other properties are also calculated with high accuracy.', '1311.6982-3-17-7': 'It must be also noticed that the system corresponding to the figures is, among all the system considered, the case where the action of the thermodynamic force and of the thermostat produces the less accurate agreement with the reference data.', '1311.6982-3-18-0': '# Efficiency', '1311.6982-3-19-0': 'In order to show the numerical efficiency of our approach, we compare the time taken to do a full GC-AdResS simulation and the time for a thermodynamic integration calculation for different systems with varied concentration of TBA in water.', '1311.6982-3-19-1': 'The total time required for an GC-AdResS simulation consists of the time taken to obtain a converged thermodynamic force and the time taken to obtain the coarse-grained chemical potential.', '1311.6982-3-19-2': 'The time taken to complete TI procedure at each value of [MATH] is summed up to obtain the total time.', '1311.6982-3-19-3': 'In this work, the TI is done in two stages, first the van der Waals interactions are coupled followed by the electrostatic coupling.', '1311.6982-3-19-4': 'At each stage, 21 equally distributed values of [MATH] are used, therefore, in total 42 simulations were performed to calculate each TI chemical potential value.', '1311.6982-3-19-5': 'In an AdResS simulation, the initial guess of the thermodynamic force largely determines the time for convergence.', '1311.6982-3-19-6': 'We started with a randomly chosen initial guess ([MATH] kJ/mol, we picked a small value because the TBA molecule is hydrophilic) for system with the highest mole fraction of TBA.', '1311.6982-3-19-7': 'For all the other systems, we used the converged thermodynamic force obtained from the first system as an initial guess.', '1311.6982-3-19-8': 'The convergence was much faster in all the other cases using this approach.', '1311.6982-3-19-9': 'Table [REF] shows the number of iterations required for the thermodynamic force convergence in GC-AdResS and total time required for GC-AdResS and TI calculation.', '1311.6982-3-19-10': 'The advantage of GC-AdResS over TI is that we get two values of excess chemical potential for both solute and solvent in a single calculation, while in TI, the whole process has to be repeated to get the excess chemical potential of other component.', '1311.6982-3-19-11': 'For very dilute systems ([MATH]), however, one has to take a very large systems in GC-AdResS (see the Appendix [REF] for system size).', '1311.6982-3-19-12': 'It takes a large amount of time for the thermodynamic force to reach convergence, and hence TI is always a better option at such low concentrations with a much smaller system size at the same mole fraction.', '1311.6982-3-20-0': '# Current computational convenience: a critical appraisal', '1311.6982-3-21-0': 'The natural question arising from the discussion above is whether or not GC-AdResS is a more convenient technical tool for calculating [MATH] compared to TI.', '1311.6982-3-21-1': 'Currently the answer is neither negative nor positive, although the current work is the first step towards a potentially positive answer for the future.', '1311.6982-3-21-2': 'In fact, the fastest version of AdResS is implemented in the GROMACS code [CITATION]; using the Gromacs version 4.5.1 a speedup of a factor four with respected to full atomistic simulations has been reported for aqueous mixtures [CITATION].', '1311.6982-3-21-3': 'In this case GC-AdResS was more convenient than TI because in one simulation one could obtain the chemical potential of each liquid component and at the same time calculate structural properties (e.g. radial distribution functions).', '1311.6982-3-21-4': 'However in the successive version of GROMACS 4.6.1 the performance of atomistic simulations (above all of SPC/E water) has been highly improved while the corresponding implementation of AdResS is not optimized yet.', '1311.6982-3-21-5': 'At the current state, AdResS can only assure a speed up factor between 2 and 3 for large systems (30000 molecules) compared to full atomistic simulations (except for pure SPC/E water systems).', '1311.6982-3-21-6': 'As a consequence for the calculation of [MATH], TI is in general computationally less demanding than AdResS .', '1311.6982-3-21-7': 'Another point that must be considered (in perspective) for a fair comparison between TI and GC-AdResS, is the following: even if AdResS is optimized, in any code, TI has the advantage that one can use one single molecule in the simulation box to mimic the minor component of a mixture.', '1311.6982-3-21-8': 'In our case, instead, we must treat, technically speaking, a true mixture with a certain number of molecules of the minor component immersed in the liquid of the major component.', '1311.6982-3-21-9': 'Thus, at low concentrations, GC-AdResS simulations require larger systems than those required by TI, moreover, because of the low density of the minor component, the convergence of the corresponding thermodynamic force requires long simulations.', '1311.6982-3-21-10': 'Thus, for very dilute systems, if one is interested only in the chemical potential, TI shall be preferred to GC-AdResS, however if the interest goes beyond the calculation of the chemical potential, (e.g. radial distribution functions) then (optimized) GC-AdResS would still be more convenient.', '1311.6982-3-21-11': 'When the concentration becomes higher, GC-AdResS may become preferable for both tasks: general properties of the mixture and chemical potential, not only because in this case one requires larger systems, but also because the convergence of the thermodynamic force of the minor component is much faster.', '1311.6982-3-21-12': 'Moreover, we would have the flexibility of calculating the chemical potential of both components in one simulation run, whereas in TI, one needs to run two separate simulations in order to get the chemical potential of both components.', '1311.6982-3-21-13': 'The results reported in the previous section about the current efficiency of GC-AdResS are rather encouraging, however currently there is not a clear convenience in using GC-AdResS instead of TI for calculating [MATH]; in any case the technical aspects of code optimization must be reported and we must make clear that the aim of this work is to show that the automatic calculation of [MATH], independently from the simulation code in which is implemented and its computational cost, is a ""conceptual"" feature of GC-AdResS.', '1311.6982-3-22-0': '# Conclusion', '1311.6982-3-23-0': 'We have shown the accuracy of GC-AdResS in calculating the excess chemical potential for a representative class of complex liquids and mixtures.', '1311.6982-3-23-1': 'For any system, the initial equilibration process, that is the determination of the thermodynamic force, automatically delivers the chemical potential.', '1311.6982-3-23-2': 'The only additional calculation required is that of [MATH] which implies the use of IPM or TI, but for a liquid of simple spheres, thus computationally negligible.', '1311.6982-3-23-3': 'The essential message is that GC-AdResS would be, per se, a reliable multiscale technique to calculate the chemical potential and, in perspective, upon computational/technical optimization it may become an efficient tool for calculating [MATH] compared to current techniques in MD such as TI.'}","{'1311.6982-4-0-0': 'We employ the adaptive resolution approach AdResS, in its recently developed Grand Canonical-like version (GC-AdResS) [Wang et al. Phys.Rev.X 3, 011018 (2013)], to calculate the excess chemical potential, [MATH], of various liquids and mixtures.', '1311.6982-4-0-1': 'We compare our results with those obtained from full atomistic simulations using the technique of thermodynamic integration and show a satisfactory agreement.', '1311.6982-4-0-2': 'In GC-AdResS the procedure to calculate [MATH] corresponds to the process of standard initial equilibration of the system; this implies that, independently of the specific aim of the study, [MATH], for each molecular species, is automatically calculated every time a GC-AdResS simulation is performed.', '1311.6982-4-1-0': '# Introduction', '1311.6982-4-2-0': 'The chemical potential represents an important thermodynamic information for any system, in particular for liquids, where the possibility of combining different substances for forming optimal mixtures is strictly related to knowledge of the chemical potential of each component in the mixture environment.', '1311.6982-4-2-1': 'In this perspective, molecular simulation represents a powerful tool for predicting the chemical potential of complex molecular systems.', '1311.6982-4-2-2': 'Popular, well established methodologies in Molecular Dynamics (MD) are Widom particle insertion (IPM) [CITATION] and thermodynamic integration (TI) [CITATION].', '1311.6982-4-2-3': 'IPM is computationally very demanding often beyond a reasonable limit even in presence of large computational resources, but upon convergence, is rather accurate.', '1311.6982-4-2-4': 'TI is computationally convenient but specifically designed to calculate the chemical potential and thus it may not be optimal for employing MD for studying other properties.', '1311.6982-4-2-5': 'In fact TI requires artificial modification of the atomistic interactions (see Appendix).', '1311.6982-4-2-6': 'Recently we have suggested that the chemical potential could be calculated by employing the Adaptive Resolution Simulation method in its Grand Canonical-like formulation (GC-AdResS) [CITATION].', '1311.6982-4-2-7': 'AdResS was originally designed to interface regions of space at different levels of molecular resolution within one simulation set up.', '1311.6982-4-2-8': 'This allows for large and efficient multiscale simulations where the high resolution region is restricted to a small portion of space and the rest of the system is at coarser level.', '1311.6982-4-2-9': 'The recent version of the method, GC-AdResS, given its theoretical framework, should automatically calculate the chemical potential during the process of initial equilibration: in this work we prove that this is indeed the case and report results for the chemical potential for various liquids and mixtures of particular relevance in (bio)-chemistry and material science.', '1311.6982-4-2-10': 'We compare our results with those from full atomistic TI and find a satisfactory agreement.', '1311.6982-4-2-11': 'This agreement allows us to conclude that every time a multiscale GC-AdResS is performed, [MATH] is automatically calculated for each liquid component and implicitly confirm that the basic thermodynamics of the system is well described by the method.', '1311.6982-4-2-12': 'Moreover, in recent work AdResS has been merged with the MARTINI force field [CITATION].', '1311.6982-4-2-13': 'In this context, the possibility of checking the consistency of a quantity like the chemical potential can be used as a further argument for the validity of the method in applications to large systems of biological interest.', '1311.6982-4-2-14': 'Below we provide the basic technical ingredients of GC-AdResS which are relevant for the calculation of the chemical potential, more specific details can be found in [CITATION].', '1311.6982-4-3-0': '# From AdResS to GC-AdResS', '1311.6982-4-4-0': 'The original idea of AdResS is based on a simple intuitive physical principle:', '1311.6982-4-5-0': 'In [CITATION] we have defined necessary conditions in [MATH] such that the spatial probability distribution of the full-atomistic reference system was reproduced up to a certain (desired) order in the atomistic region of the adaptive system.', '1311.6982-4-5-1': 'We have defined the [MATH]th order of a spatial (configurational) probability distribution of [MATH] molecules, [MATH], as: [EQUATION]', '1311.6982-4-5-2': 'The first order, often mentioned in this work corresponds to the molecular number density [MATH].', '1311.6982-4-5-3': 'Moreover we have shown that, because of the necessary conditions, the accuracy in the atomistic region is independent of the accuracy of the coarse-grained model, thus, in the coarse-grained region, one can use a generic liquid of spheres whose only requirement is that it has the same molecular density of the reference system.', '1311.6982-4-5-4': 'In the simulation set up, [MATH] is calculated via an iterative procedure using the molecular number density in [MATH].', '1311.6982-4-5-5': 'The iterative scheme consists of calculating [MATH] the isothermal compressibility), and the thermodynamic force is considered converged when the target density [MATH] is reached in [MATH].', '1311.6982-4-5-6': 'As a result, [MATH], acting in [MATH], assures that there are no artificial density variations across the system, thus it allows to accurately reproduce the first order of the probability distribution in the atomistic region.', '1311.6982-4-5-7': 'Higher orders can be systematically achieved by imposing in [MATH] a corrective force.', '1311.6982-4-5-8': 'For example, the COM-COM radial distribution function correction for the second order [CITATION].', '1311.6982-4-5-9': 'Next it was proved that indeed the target Grand Canonical distribution, that is the probability distribution of a subsystem (of the size of the atomistic region in GC-AdResS) in a large full atomistic simulation is accurately reproduced.', '1311.6982-4-5-10': 'A large number of tests were performed and the reproduction by GC-AdResS of the probability distribution was numerically proved up to (at least) the third order, more than sufficient in MD simulations.', '1311.6982-4-5-11': 'Within this framework it was finally shown that the sum of work of [MATH] and that of the thermostat corresponds to the difference in chemical potential between the atomistic and coarse-grained resolution; this subject is treated in the next section.', '1311.6982-4-6-0': '# Calculation of Excess Chemical Potential', '1311.6982-4-7-0': 'In Ref. [CITATION] it has been shown that the chemical potential of the atomistic and coarse-grained resolution are related by the following formula: [EQUATION] with [MATH] the chemical potential of the coarse-grained system (in GC-AdResS this corresponds to a liquid of generic spheres), [MATH] the chemical potential of the atomistic system, [MATH] the work of the thermodynamic force in the transition region, [MATH] the work/heat provided by the thermostat in order to slowly equilibrate the inserted/removed degrees of freedom in the transition region.', '1311.6982-4-7-1': '[MATH] is composed by two parts, one, called [MATH], which compensates the dissipation of energy due to the non-conservative interactions in [MATH], and another, [MATH], which is related to the equilibration of the reinserted/removed degrees of freedom (rotational and vibrational).', '1311.6982-4-7-2': 'While the determination of [MATH] is not required for our final aim (that is the calculation of the excess chemical potential, as explained later on), the calculation of [MATH] is very relevant.', '1311.6982-4-7-3': 'However this calculation is not straightforward and we have proposed to introduce an auxiliary Hamiltonian approach where the coarse-grained and atomistic potential are interpolated, and not the forces as in the original AdResS.', '1311.6982-4-7-4': 'Next, we impose that the Hamiltonian system must have the same thermodynamic equilibrium of the original force-based GC-AdResS system; this is done by introducing a thermodynamic force in the auxiliary Hamiltonian approach, which, at the target temperature, keeps the density of particles across the system as in GC-AdResS.', '1311.6982-4-7-5': 'In the auxiliary Hamiltonian approach we have the same equilibrium as the original adaptive (and full atomistic) system and the difference between the work of the original thermodynamic force and the work of the thermodynamic force calculated in the Hamiltonian approach gives [MATH] (further details about this point are given in the Appendix [REF]).', '1311.6982-4-7-6': 'Moreover we have proven numerically, for the case of liquid water, that [MATH], where [MATH] and [MATH] are the atomistic and coarse-grained potential.', '1311.6982-4-7-7': 'It must be noticed that the auxiliary Hamiltonian approach shall not be considered a Hamiltonian approach to adaptive resolution simulation.', '1311.6982-4-7-8': 'In fact, as discussed in Ref.[CITATION] the equilibrium is imposed artificially and per se does not have any physical meaning (for more details, see discussion in the Appendix [REF]).', '1311.6982-4-7-9': 'In the next section of this work we show analytically that the formula above is exact (at least) at the first order w.r.t. the probability distribution of the system as defined in Eq.[REF].', '1311.6982-4-7-10': 'The result above implies that [MATH] can be calculated in a straightforward way during the initial equilibration within in the standard GC-AdResS code.', '1311.6982-4-7-11': 'It must be noticed that, within the AdResS scheme, an approach similar to the auxiliary Hamiltonian has been recently proposed and applied to liquids and mixtures (of toy models so far) by Potestio et al. [CITATION] (see also [CITATION] where such an approach is commented).', '1311.6982-4-7-12': 'At this point according to [REF], if one knows [MATH], then GC-AdResS can automatically provide [MATH].', '1311.6982-4-7-13': 'However we need to do one step more, in fact the quantity of interest is not the total chemical potential, but the excess chemical potential [MATH] which corresponds to the expression of [REF] where the kinetic (ideal gas) part is subtracted.', '1311.6982-4-7-14': 'Regarding the kinetic part, one can notice that the contribution coming from the COM is the same for the coarse-grained and for the atomistic molecules, thus it is automatically removed in the calculation of [REF].', '1311.6982-4-7-15': 'The kinetic part of [MATH] due to the rotational and vibrational degrees of freedom corresponds in our case to [MATH] and in principle can be calculated by hand (chemical potential of an isolated molecules).', '1311.6982-4-7-16': 'However such a calculation may become rather tedious for large and/or complex molecules but in our case it is actually not required.', '1311.6982-4-7-17': 'In fact the Gromacs implementation of AdResS considers the removed degrees of freedom as phantom variables but thermally equilibrate them anyway [CITATION].', '1311.6982-4-7-18': 'Thus the heat provided by the thermostat for the rotational and vibrational part is the same in the atomistic and coarse-graining molecules and is automatically removed in the difference.', '1311.6982-4-7-19': 'Finally, the calculation of [MATH] can be done with standard methods, TI or IPM, which for simple spherical molecules, like those of the coarse-grained system, requires a negligible computational cost.', '1311.6982-4-7-20': 'In conclusion, we have the final expression: [EQUATION]', '1311.6982-4-8-0': '# Analytic Derivation of [MATH]', '1311.6982-4-9-0': 'In this section we derive analytically the equivalence: [MATH] and define its conceptual limitations.', '1311.6982-4-9-1': 'We consider a potential coupling between the atomistic and coarse-grained resolution as the spatial interpolation of the atomistic and coarse-grained potential, as done instead for the forces in the standard AdResS: [EQUATION] where [MATH] and [MATH] are the atomistic and coarse-grained interaction potential between molecule [MATH] and [MATH], respectively, defined by [EQUATION] where [MATH] and [MATH] denotes the atom indexes of the corresponding molecule.', '1311.6982-4-9-2': 'The COM of the molecule is defined as: [EQUATION] where [MATH] is the mass of atom [MATH] of molecule [MATH].', '1311.6982-4-9-3': 'The potential interpolation [REF] provides an auxiliary Hamiltonian to the AdResS system, and the corresponding intermolecular force is given by: [EQUATION]', '1311.6982-4-9-4': 'We refer to the AdResS simulation using force scheme [REF] as auxiliary Hamiltonian AdResS, and all properties of this approach will be added a superscript ""H"".', '1311.6982-4-9-5': 'We define the force of changing representation by [EQUATION]', '1311.6982-4-9-6': 'We use the same notation as in our previous work [CITATION].', '1311.6982-4-9-7': 'The thermodynamic variables for the atomistic and coarse-grained regions are denoted by [MATH] and [MATH], respectively.', '1311.6982-4-9-8': 'We assume that the transition region is an infinitely thin filter (that is a much smaller region than the atomistic and coarse-grained region) that allows molecules to change resolution as they cross it.', '1311.6982-4-9-9': 'Therefore, it is reasonable to assume that: [EQUATION] where [MATH] and [MATH] are the total volume and total number of molecules of the system.', '1311.6982-4-9-10': 'In this work, we adopt the same assumptions as those listed in Sec.III.C of Ref. [CITATION], i.e. we assume the system to be in the thermodynamic limit, and molecules are short-range correlated (short-ranged must be intended as a range comparable to the size of the transition region).', '1311.6982-4-9-11': 'The thermodynamic force for GC-AdResS ([MATH]) and for the auxiliary Hamiltonian AdResS ([MATH]), enforce the system to have a flat density: [EQUATION]', '1311.6982-4-9-12': 'Where [MATH] is the equilibrium number density of the system defined by [MATH].', '1311.6982-4-9-13': 'As shown in Refs. [CITATION], [MATH] provides the balance of the grand potential or equivalently [EQUATION] where [MATH] denotes the integral of the thermodynamic force [MATH].', '1311.6982-4-10-0': ""Instead when we consider the auxiliary Hamiltonian approach, the third term on the R.H.S. of Eq. [REF] is not symmetric w.r.t molecule [MATH] and [MATH], therefore, the Newton's action-reaction law (momentum conservation) does not hold anymore."", '1311.6982-4-10-1': 'As a consequence, the pressure relation between the AT and CG resolution [REF] does not hold and should be derived again.', '1311.6982-4-10-2': 'Now assume, for simplicity and without lost of generality, that the system changes resolution only along the [MATH] direction.', '1311.6982-4-10-3': 'We impose an infinitesimal increment of the volume [MATH] to the AT region, and apply the same decrement of the volume [MATH] to the CG region.', '1311.6982-4-10-4': 'The volume of the transition region is kept constant as if it is an ideal ""piston"" that moves toward the CG region by an amount [MATH].', '1311.6982-4-10-5': 'We assume [MATH], where [MATH] is the cutting surface area.', '1311.6982-4-10-6': 'The displacement [MATH] should be infinitesimal, i.e. much smaller than the size of the transition region.', '1311.6982-4-10-7': 'This is achievable by taking the limit of [MATH], while keeping the system size fixed.', '1311.6982-4-10-8': 'It must be noticed that also the displacements of the molecules are infinitesimal, so it can be reasonably assumed that the resolution of the molecules remains the same under a displacement of [MATH].', '1311.6982-4-10-9': 'Therefore, the change of the free energy of the system is approximately: [EQUATION] where [MATH] and [MATH] are the free energies of the AT and CG region, respectively.', '1311.6982-4-11-0': '[MATH] is the linear dimension of the transition region along [MATH].', '1311.6982-4-11-1': 'Since the resolution changes only along [MATH], the two one-particle forces depend only on [MATH], and only have the component along [MATH].', '1311.6982-4-11-2': 'This can be easily generalized to changing resolution in any direction, i.e., replacing [MATH] by [MATH].', '1311.6982-4-11-3': 'The expression of Eq. [REF] as a sum of different terms is justified by the hypothesis of treating the system in the thermodynamic limit, and by the hypothesis that the interactions are short-ranged compared to the size of the transition region.', '1311.6982-4-11-4': '[MATH] and [MATH] is the numbers of molecules in the AT and CG region, and [MATH] and [MATH] is the volume in the AT and CG region, respectively; [MATH] is the temperature of the system.', '1311.6982-4-11-5': 'The last term is originated by the work done of the ideal piston.', '1311.6982-4-11-6': ""This latter is composed by two parts, the first corresponding to the work done by the thermodynamic force, and the second corresponding to the work done by the force of changing representation (which does not vanish due to the violation of the Newton's action-reaction law)."", '1311.6982-4-11-7': 'The first and second term of Eq. [REF] being forces based on pairwise interactions only, do not contribute to the difference of energy; in fact their total work is zero (as long as the transition region move infinitesimally along [MATH]).', '1311.6982-4-11-8': 'The notation [MATH] in Eq. [REF] denotes the ensemble average, which will be specified soon.', '1311.6982-4-11-9': 'It is straightforward to show that [EQUATION] where [MATH] is the integral of the thermodynamic force [MATH], and [MATH] is the work of changing representation, which can be explicitly written down in a general form as: [EQUATION]', '1311.6982-4-11-10': 'The average is performed over all possible positions of the second molecule (i.e. [MATH]), at fixed position of the first molecule (i.e. [MATH]) in the pairwise interaction.', '1311.6982-4-11-11': 'In case of molecules containing more than one atom, the average is also made over all possible conformations in the atomistic resolution.', '1311.6982-4-11-12': 'In the thermodynamic limit, the equilibrium volume of the AT region maximize the free energy, i.e. [MATH], which yields [EQUATION]', '1311.6982-4-11-13': 'Comparing the expression above with that obtained for GC-AdResS (Eq. [REF]), we have: [EQUATION] which relates the thermodynamic force of the auxiliary Hamiltonian AdResS and the GC-AdResS.', '1311.6982-4-12-0': 'In Ref. [CITATION] we proved that under proper assumptions, when the flat density profile is enforced by the thermodynamic force, the chemical potential difference between the different resolutions is given by [EQUATION]', '1311.6982-4-12-1': 'The same argument can be applied to the auxiliary Hamiltonian approach, and yields the chemical potential difference between the AT and CG resolutions [EQUATION]', '1311.6982-4-12-2': 'In the auxiliary Hamiltonian, we do not have the term [MATH] in the above formula (being the term [MATH] in GC-AdResS, generated by the non-conservative effect of the force interpolation).', '1311.6982-4-12-3': 'By comparing [REF] with [REF], we have the relation [EQUATION] which also relates the thermodynamic force of the auxiliary Hamiltonian AdResS and GC-AdResS.', '1311.6982-4-13-0': 'From Eq. [REF] and [REF], we find the extra work of the thermostat in GC-AdResS being identical to the work of changing representation of the auxiliary Hamiltonian approach: [EQUATION] which basically proves the statement at the beginning of this section.', '1311.6982-4-13-1': 'The ensemble average on the R.H.S. of Eq. [REF] is performed in the ensemble of the system treated with the potential interpolation approach, and the question is if the ensemble average is equivalent if it is performed in the simulation where the force interpolation approach is used.', '1311.6982-4-13-2': 'It is obvious that the spatial probability distribution corresponding to the system treated with the potential interpolation is consistent with the force interpolation at least up to the first order.', '1311.6982-4-13-3': 'It is also possible to systematically obtain equivalence in the ensemble average operation at higher orders of accuracy of the probability distribution, as, for example, it is done for the radial distribution function in Ref. [CITATION].', '1311.6982-4-13-4': 'However, here we do not consider higher order corrections, because it has been numerically shown that actually the ensemble average of [MATH] dose not depend on in which ensemble it is calculated [CITATION].', '1311.6982-4-13-5': 'Therefore, we use Eq. [REF] to calculate [MATH], and measure the ensemble average by the standard AdResS.', '1311.6982-4-13-6': 'As previously discussed, in the Gromacs implementation, the CG molecules also keep the atomistic degrees of freedom even though they are in the CG region, therefore, the kinetic part of [MATH] and [MATH] are identical, and [MATH] vanishes.', '1311.6982-4-13-7': 'Therefore, by inserting Eq. [REF] into [REF], we have [EQUATION]', '1311.6982-4-13-8': 'The extension of Eq. [REF] to multicomponent systems is reported in the Appendix [REF], while in the next section we apply the method to the calculation of [MATH] to liquids and mixtures.', '1311.6982-4-14-0': '# Results and Discussion', '1311.6982-4-15-0': 'We have calculated [MATH] for different liquids and mixtures, choosing cases which are representative of a large class of systems.', '1311.6982-4-15-1': 'Hydrophobic solvation in methane/water and in ethane/water mixtures, hydrophilic solvation in urea/water, a balance of both in water/tert-Butyl alcohol (TBA) mixture, other liquids, e.g. pure methanol and DMSO (and their mixtures with water), non aqueous mixtures in TBA/DMSO and alkane liquids such as methane, ethane and propane.', '1311.6982-4-15-2': 'Moreover, systems as water/urea are commonly used as cosolvent of biological molecules [CITATION] while systems as tert-Butyl alcohol/water play a key role in modern technology [CITATION], thus they are of high interest per se.', '1311.6982-4-15-3': 'All technical details of each simulation are presented in the Appendix [REF].', '1311.6982-4-16-0': 'Results are reported in Table [REF], where the comparison with values obtained using full atomistic TI and available experiments, at the same concentrations, of our calculation is made; in our previous work we have already shown that value of the chemical potential of liquid water obtained with IPM is well reproduced by GC-AdResS, however the computational cost of IMP was very large, thus we do not consider calculations done with IPM in this paper.', '1311.6982-4-16-1': 'The agreement with full atomistic TI simulations is satisfactory in all cases, and thus it proves the solidity of GC-AdResS in describing the essential thermodynamics of a large class of systems.', '1311.6982-4-16-2': 'We also compare the obtained values with those available in literature [CITATION].', '1311.6982-4-16-3': 'Although the concentration of the minor component in the mixtures that we consider, is higher than the concentrations considered in Refs. [CITATION], we are anyway in the very dilute regime and thus the chemical potential should not change in a significant way; we have verified such a supposed consistency for one relevant system (see discussion about Fig. [REF]).', '1311.6982-4-16-4': ""The chemical potential of [MATH]-th liquid's component in a mixture is calculated as (see Appendix [REF]): [EQUATION] where [MATH] is the thermodynamic force applied to the molecules of the [MATH]-th component; this assures that, at the given concentration, the density of molecules of species [MATH], in the transition region, is equivalent to the density of the same liquid's component in a reference full atomistic simulation."", '1311.6982-4-16-5': 'The ensemble average is taken over the position of the second molecule, provided that the first molecule is of species [MATH], and located at position [MATH].', '1311.6982-4-17-0': 'A complementary information to Table [REF] are Fig. [REF] and [REF].', '1311.6982-4-17-1': 'In Fig. [REF] we have studied the behavior of [MATH] as a function of the interaction cut-off.', '1311.6982-4-17-2': 'In fact the current version of GC-AdResS, employs the reaction field method for treating electrostatic interactions in the atomistic region, and the cut off is likely to play a role in some of the systems investigated.', '1311.6982-4-17-3': 'Fig. [REF], for the case of DMSO/water mixture, confirms our intuition and suggests that we could systematically improve the accuracy by increasing the cut-off, and at a value of about [MATH] nm, [MATH] converges.', '1311.6982-4-17-4': 'In any case, at a values of [MATH] nm, which is the one routinely used in full atomistic simulations and used by us, the value obtained with GC-AdResS is already satisfactory.', '1311.6982-4-17-5': 'The cut-off radii used for other systems are reported in Appendix [REF].', '1311.6982-4-17-6': 'A further question that may arise is the capability of our method to predict the behavior of [MATH] as a function of the concentration, above all in the very dilute regime.', '1311.6982-4-17-7': 'In Fig. [REF] we have performed such a study for the case of TBA/water mixture, we show a good agreement between GC-AdResS and TI and show that at a very dilute concentration our calculated value is close to that of experiments, moreover the trend, regarding the TI calculations, is consistent with that reported in Ref. [CITATION].', '1311.6982-4-18-0': 'In essence, according to the results obtained, GC-AdResS allows an on-the-fly determination of [MATH] of each component of a liquid, whenever a simulation is performed, without extra computational costs.', '1311.6982-4-18-1': 'Moreover, Fig.[REF] shows the action of the thermodynamic force and of the thermostat in the transition region [MATH] for TBA-water; the molecular density is sufficiently close to that of reference (the largest difference is below [MATH] and the average difference is below [MATH]), and thus it assures that in the atomistic region there are no (significant) artificial effects on the molecular density due to the perturbation represented by the interpolation of forces in [MATH].', '1311.6982-4-18-2': 'In Fig.[REF] we report various radial distribution functions for TBA-water in the atomistic region of the adaptive set up.', '1311.6982-4-18-3': 'The agreement with data from a full atomistic simulation is highly satisfactory.', '1311.6982-4-18-4': 'Moreover, it must be underlined that, on purpose, we have chosen extreme technical conditions, that is, a very small atomistic and coarse-grained region ([MATH] nm) and a relatively large transition region ([MATH] nm).', '1311.6982-4-18-5': 'Even in these conditions we prove that local properties as those of Fig. [REF] and Fig. [REF], together with a relevant thermodynamics quantity as [MATH] are well reproduced.', '1311.6982-4-18-6': 'This example shows the key features of GC-AdResS, that is, a multiscale simulation where the chemical potential of each component is obtained without extra computational costs and with high accuracy in a simulation where other properties are also calculated with high accuracy.', '1311.6982-4-18-7': 'It must be also noticed that the system corresponding to the figures is, among all the system considered, the case where the action of the thermodynamic force and of the thermostat produces the less accurate agreement with the reference data.', '1311.6982-4-19-0': '# Efficiency', '1311.6982-4-20-0': 'In order to show the numerical efficiency of our approach, we compare the time taken to do a full GC-AdResS simulation and the time for a thermodynamic integration calculation for different systems with varied concentration of TBA in water.', '1311.6982-4-20-1': 'The total time required for an GC-AdResS simulation consists of the time taken to obtain a converged thermodynamic force and the time taken to obtain the coarse-grained chemical potential.', '1311.6982-4-20-2': 'The time taken to complete TI procedure at each value of [MATH] is summed up to obtain the total time.', '1311.6982-4-20-3': 'In this work, the TI is done in two stages, first the van der Waals interactions are coupled followed by the electrostatic coupling.', '1311.6982-4-20-4': 'At each stage, 21 equally distributed values of [MATH] are used, therefore, in total 42 simulations were performed to calculate each TI chemical potential value.', '1311.6982-4-20-5': 'In an AdResS simulation, the initial guess of the thermodynamic force largely determines the time for convergence.', '1311.6982-4-20-6': 'We started with a randomly chosen initial guess ([MATH] kJ/mol, we picked a small value because the TBA molecule is hydrophilic) for system with the highest mole fraction of TBA.', '1311.6982-4-20-7': 'For all the other systems, we used the converged thermodynamic force obtained from the first system as an initial guess.', '1311.6982-4-20-8': 'The convergence was much faster in all the other cases using this approach.', '1311.6982-4-20-9': 'Table [REF] shows the number of iterations required for the thermodynamic force convergence in GC-AdResS and total time required for GC-AdResS and TI calculation.', '1311.6982-4-20-10': 'The advantage of GC-AdResS over TI is that we get two values of excess chemical potential for both solute and solvent in a single calculation, while in TI, the whole process has to be repeated to get the excess chemical potential of other component.', '1311.6982-4-20-11': 'For very dilute systems ([MATH]), however, one has to take a very large systems in GC-AdResS (see the Appendix [REF] for system size).', '1311.6982-4-20-12': 'It takes a large amount of time for the thermodynamic force to reach convergence, and hence TI is always a better option at such low concentrations with a much smaller system size at the same mole fraction.', '1311.6982-4-21-0': '# Current computational convenience: a critical appraisal', '1311.6982-4-22-0': 'The natural question arising from the discussion above is whether or not GC-AdResS is a more convenient technical tool for calculating [MATH] compared to TI.', '1311.6982-4-22-1': 'Currently the answer is neither negative nor positive, although the current work is the first step towards a potentially positive answer for the future.', '1311.6982-4-22-2': 'In fact, the fastest version of AdResS is implemented in the GROMACS code [CITATION]; using the Gromacs version 4.5.1 a speedup of a factor four with respected to full atomistic simulations has been reported for aqueous mixtures [CITATION].', '1311.6982-4-22-3': 'In this case GC-AdResS was more convenient than TI because in one simulation one could obtain the chemical potential of each liquid component and at the same time calculate structural properties (e.g. radial distribution functions).', '1311.6982-4-22-4': 'However in the successive version of GROMACS 4.6.1 the performance of atomistic simulations (above all of SPC/E water) has been highly improved while the corresponding implementation of AdResS is not optimized yet.', '1311.6982-4-22-5': 'At the current state, AdResS can only assure a speed up factor between 2 and 3 for large systems (30000 molecules) compared to full atomistic simulations (except for pure SPC/E water systems).', '1311.6982-4-22-6': 'As a consequence for the calculation of [MATH], TI is in general computationally less demanding than AdResS .', '1311.6982-4-22-7': 'Another point that must be considered (in perspective) for a fair comparison between TI and GC-AdResS, is the following: even if AdResS is optimized, in any code, TI has the advantage that one can use one single molecule in the simulation box to mimic the minor component of a mixture.', '1311.6982-4-22-8': 'In our case, instead, we must treat, technically speaking, a true mixture with a certain number of molecules of the minor component immersed in the liquid of the major component.', '1311.6982-4-22-9': 'Thus, at low concentrations, GC-AdResS simulations require larger systems than those required by TI, moreover, because of the low density of the minor component, the convergence of the corresponding thermodynamic force requires long simulations.', '1311.6982-4-22-10': 'Thus, for very dilute systems, if one is interested only in the chemical potential, TI shall be preferred to GC-AdResS, however if the interest goes beyond the calculation of the chemical potential, (e.g. radial distribution functions) then (optimized) GC-AdResS would still be more convenient.', '1311.6982-4-22-11': 'When the concentration becomes higher, GC-AdResS may become preferable for both tasks: general properties of the mixture and chemical potential, not only because in this case one requires larger systems, but also because the convergence of the thermodynamic force of the minor component is much faster.', '1311.6982-4-22-12': 'Moreover, we would have the flexibility of calculating the chemical potential of both components in one simulation run, whereas in TI, one needs to run two separate simulations in order to get the chemical potential of both components.', '1311.6982-4-22-13': 'The results reported in the previous section about the current efficiency of GC-AdResS are rather encouraging, however currently there is not a clear convenience in using GC-AdResS instead of TI for calculating [MATH]; in any case the technical aspects of code optimization must be reported and we must make clear that the aim of this work is to show that the automatic calculation of [MATH], independently from the simulation code in which is implemented and its computational cost, is a ""conceptual"" feature of GC-AdResS.', '1311.6982-4-23-0': '# Conclusion', '1311.6982-4-24-0': 'We have shown the accuracy of GC-AdResS in calculating the excess chemical potential for a representative class of complex liquids and mixtures.', '1311.6982-4-24-1': 'For any system, the initial equilibration process, that is the determination of the thermodynamic force, automatically delivers the chemical potential.', '1311.6982-4-24-2': 'The only additional calculation required is that of [MATH] which implies the use of IPM or TI, but for a liquid of simple spheres, thus computationally negligible.', '1311.6982-4-24-3': 'The essential message is that GC-AdResS would be, per se, a reliable multiscale technique to calculate the chemical potential and, in perspective, upon computational/technical optimization it may become an efficient tool for calculating [MATH] compared to current techniques in MD such as TI.'}","{'1311.6982-5-0-0': 'We employ the adaptive resolution approach AdResS, in its recently developed Grand Canonical-like version (GC-AdResS) [Wang et al. Phys.Rev.X 3, 011018 (2013)], to calculate the excess chemical potential, [MATH], of various liquids and mixtures.', '1311.6982-5-0-1': 'We compare our results with those obtained from full atomistic simulations using the technique of thermodynamic integration and show a satisfactory agreement.', '1311.6982-5-0-2': 'In GC-AdResS the procedure to calculate [MATH] corresponds to the process of standard initial equilibration of the system; this implies that, independently of the specific aim of the study, [MATH], for each molecular species, is automatically calculated every time a GC-AdResS simulation is performed.', '1311.6982-5-1-0': '# Introduction', '1311.6982-5-2-0': 'The chemical potential represents an important thermodynamic information for any system, in particular for liquids, where the possibility of combining different substances for forming optimal mixtures is strictly related to knowledge of the chemical potential of each component in the mixture environment.', '1311.6982-5-2-1': 'In this perspective, molecular simulation represents a powerful tool for predicting the chemical potential of complex molecular systems.', '1311.6982-5-2-2': 'Popular, well established methodologies in Molecular Dynamics (MD) are Widom particle insertion (IPM) [CITATION] and thermodynamic integration (TI) [CITATION].', '1311.6982-5-2-3': 'IPM is computationally very demanding often beyond a reasonable limit even in presence of large computational resources, but upon convergence, is rather accurate.', '1311.6982-5-2-4': 'TI is computationally convenient but specifically designed to calculate the chemical potential and thus it may not be optimal for employing MD for studying other properties.', '1311.6982-5-2-5': 'In fact TI requires artificial modification of the atomistic interactions (see Appendix).', '1311.6982-5-2-6': 'Recently we have suggested that the chemical potential could be calculated by employing the Adaptive Resolution Simulation method in its Grand Canonical-like formulation (GC-AdResS) [CITATION].', '1311.6982-5-2-7': 'AdResS was originally designed to interface regions of space at different levels of molecular resolution within one simulation set up.', '1311.6982-5-2-8': 'This allows for large and efficient multiscale simulations where the high resolution region is restricted to a small portion of space and the rest of the system is at coarser level.', '1311.6982-5-2-9': 'The recent version of the method, GC-AdResS, given its theoretical framework, should automatically calculate the chemical potential during the process of initial equilibration: in this work we prove that this is indeed the case and report results for the chemical potential for various liquids and mixtures of particular relevance in (bio)-chemistry and material science.', '1311.6982-5-2-10': 'We compare our results with those from full atomistic TI and find a satisfactory agreement.', '1311.6982-5-2-11': 'This agreement allows us to conclude that every time a multiscale GC-AdResS is performed, [MATH] is automatically calculated for each liquid component and implicitly confirm that the basic thermodynamics of the system is well described by the method.', '1311.6982-5-2-12': 'Moreover, in recent work AdResS has been merged with the MARTINI force field [CITATION].', '1311.6982-5-2-13': 'In this context, the possibility of checking the consistency of a quantity like the chemical potential can be used as a further argument for the validity of the method in applications to large systems of biological interest.', '1311.6982-5-2-14': 'Below we provide the basic technical ingredients of GC-AdResS which are relevant for the calculation of the chemical potential, more specific details can be found in [CITATION].', '1311.6982-5-3-0': '# From AdResS to GC-AdResS', '1311.6982-5-4-0': 'The original idea of AdResS is based on a simple intuitive physical principle:', '1311.6982-5-5-0': 'In [CITATION] we have defined necessary conditions in [MATH] such that the spatial probability distribution of the full-atomistic reference system was reproduced up to a certain (desired) order in the atomistic region of the adaptive system.', '1311.6982-5-5-1': 'We have defined the [MATH]th order of a spatial (configurational) probability distribution of [MATH] molecules, [MATH], as: [EQUATION]', '1311.6982-5-5-2': 'The first order, often mentioned in this work, corresponds to the molecular number density [MATH].', '1311.6982-5-5-3': 'Moreover we have shown that, because of the necessary conditions, the accuracy in the atomistic region is independent of the accuracy of the coarse-grained model, thus, in the coarse-grained region, one can use a generic liquid of spheres whose only requirement is that it has the same molecular density of the reference system.', '1311.6982-5-5-4': 'In the simulation set up, [MATH] is calculated via an iterative procedure using the molecular number density in [MATH].', '1311.6982-5-5-5': 'The iterative scheme consists of calculating [MATH] the isothermal compressibility), and the thermodynamic force is considered converged when the target density [MATH] is reached in [MATH].', '1311.6982-5-5-6': 'As a result, [MATH], acting in [MATH], assures that there are no artificial density variations across the system, thus it allows to accurately reproduce the first order of the probability distribution in the atomistic region.', '1311.6982-5-5-7': 'Higher orders can be systematically achieved by imposing in [MATH] a corrective force.', '1311.6982-5-5-8': 'For example, the COM-COM radial distribution function correction for the second order [CITATION].', '1311.6982-5-5-9': 'Next it was proved that indeed the target Grand Canonical distribution, that is the probability distribution of a subsystem (of the size of the atomistic region in GC-AdResS) in a large full atomistic simulation is accurately reproduced.', '1311.6982-5-5-10': 'A large number of tests were performed and the reproduction by GC-AdResS of the probability distribution was numerically proved up to (at least) the third order, more than sufficient in MD simulations.', '1311.6982-5-5-11': 'Within this framework it was finally shown that the sum of work of [MATH] and that of the thermostat corresponds to the difference in chemical potential between the atomistic and coarse-grained resolution; this subject is treated in the next section.', '1311.6982-5-6-0': '# Calculation of Excess Chemical Potential', '1311.6982-5-7-0': 'In Ref. [CITATION] it has been shown that the chemical potential of the atomistic and coarse-grained resolution are related by the following formula: [EQUATION] with [MATH] the chemical potential of the coarse-grained system (in GC-AdResS this corresponds to a liquid of generic spheres), [MATH] the chemical potential of the atomistic system, [MATH] the work of the thermodynamic force in the transition region, [MATH] the work/heat provided by the thermostat in order to slowly equilibrate the inserted/removed degrees of freedom in the transition region.', '1311.6982-5-7-1': '[MATH] is composed by two parts, one, called [MATH], which compensates the dissipation of energy due to the non-conservative interactions in [MATH], and another, [MATH], which is related to the equilibration of the reinserted/removed degrees of freedom (rotational and vibrational).', '1311.6982-5-7-2': 'While the determination of [MATH] is not required for our final aim (that is the calculation of the excess chemical potential, as explained later on), the calculation of [MATH] is very relevant.', '1311.6982-5-7-3': 'However this calculation is not straightforward and we have proposed to introduce an auxiliary Hamiltonian approach where the coarse-grained and atomistic potential are interpolated, and not the forces as in the original AdResS.', '1311.6982-5-7-4': 'Next, we impose that the Hamiltonian system must have the same thermodynamic equilibrium of the original force-based GC-AdResS system; this is done by introducing a thermodynamic force in the auxiliary Hamiltonian approach, which, at the target temperature, keeps the density of particles across the system as in GC-AdResS.', '1311.6982-5-7-5': 'In the auxiliary Hamiltonian approach we have the same equilibrium as the original adaptive (and full atomistic) system and the difference between the work of the original thermodynamic force and the work of the thermodynamic force calculated in the Hamiltonian approach gives [MATH] (further details about this point are given in the Appendix [REF]).', '1311.6982-5-7-6': 'Moreover we have proven numerically, for the case of liquid water, that [MATH], where [MATH] and [MATH] are the atomistic and coarse-grained potential.', '1311.6982-5-7-7': 'It must be noticed that the auxiliary Hamiltonian approach shall not be considered a Hamiltonian approach to adaptive resolution simulation.', '1311.6982-5-7-8': 'In fact, as discussed in Ref.[CITATION] the equilibrium is imposed artificially and per se does not have any physical meaning (for more details, see discussion in the Appendix [REF]).', '1311.6982-5-7-9': 'In the next section of this work we show analytically that the formula above is exact (at least) at the first order w.r.t. the probability distribution of the system as defined in Eq.[REF].', '1311.6982-5-7-10': 'The result above implies that [MATH] can be calculated in a straightforward way during the initial equilibration within in the standard GC-AdResS code.', '1311.6982-5-7-11': 'It must be noticed that, within the AdResS scheme, an approach similar to the auxiliary Hamiltonian has been recently proposed and applied to liquids and mixtures (of toy models so far) by Potestio et al. [CITATION] (see also [CITATION] where such an approach is commented).', '1311.6982-5-7-12': 'At this point according to [REF], if one knows [MATH], then GC-AdResS can automatically provide [MATH].', '1311.6982-5-7-13': 'However we need to do one step more, in fact the quantity of interest is not the total chemical potential, but the excess chemical potential [MATH] which corresponds to the expression of [REF] where the kinetic (ideal gas) part is subtracted.', '1311.6982-5-7-14': 'Regarding the kinetic part, one can notice that the contribution coming from the COM is the same for the coarse-grained and for the atomistic molecules, thus it is automatically removed in the calculation of [REF].', '1311.6982-5-7-15': 'The kinetic part of [MATH] due to the rotational and vibrational degrees of freedom corresponds in our case to [MATH] and in principle can be calculated by hand (chemical potential of an isolated molecules).', '1311.6982-5-7-16': 'However such a calculation may become rather tedious for large and/or complex molecules but in our case it is actually not required.', '1311.6982-5-7-17': 'In fact the Gromacs implementation of AdResS considers the removed degrees of freedom as phantom variables but thermally equilibrate them anyway [CITATION].', '1311.6982-5-7-18': 'Thus the heat provided by the thermostat for the rotational and vibrational part is the same in the atomistic and coarse-graining molecules and is automatically removed in the difference.', '1311.6982-5-7-19': 'Finally, the calculation of [MATH] can be done with standard methods, TI or IPM, which for simple spherical molecules, like those of the coarse-grained system, requires a negligible computational cost.', '1311.6982-5-7-20': 'In conclusion, we have the final expression: [EQUATION]', '1311.6982-5-8-0': '# Analytic Derivation of [MATH]', '1311.6982-5-9-0': 'In this section we derive analytically the equivalence: [MATH] and define its conceptual limitations.', '1311.6982-5-9-1': 'We consider a potential coupling between the atomistic and coarse-grained resolution, that is the spatial interpolation of the atomistic and coarse-grained potential, as done instead for the forces in the standard AdResS: [EQUATION] where [MATH] and [MATH] are the atomistic and coarse-grained interaction potential between molecule [MATH] and [MATH], respectively, defined by [EQUATION] where [MATH] and [MATH] denotes the atom indices of the corresponding molecule.', '1311.6982-5-9-2': 'The COM of the molecule is defined as: [EQUATION] where [MATH] is the mass of atom [MATH] of molecule [MATH].', '1311.6982-5-9-3': 'The potential interpolation [REF] provides an auxiliary Hamiltonian to the AdResS system, and the corresponding intermolecular force is given by: [EQUATION]', '1311.6982-5-9-4': 'We refer to the AdResS simulation using potential scheme [REF] as auxiliary Hamiltonian AdResS, and all properties of this approach will be added a superscript ""H"".', '1311.6982-5-9-5': 'We define the force of changing representation by [EQUATION]', '1311.6982-5-9-6': 'We use the same notation as in our previous work [CITATION].', '1311.6982-5-9-7': 'The thermodynamic variables for the atomistic and coarse-grained regions are denoted by [MATH] and [MATH], respectively.', '1311.6982-5-9-8': 'We assume that the transition region is an infinitely thin filter (that is a much smaller region than the atomistic and coarse-grained region) that allows molecules to change resolution as they cross it.', '1311.6982-5-9-9': 'Therefore, it is reasonable to assume that: [EQUATION] where [MATH] and [MATH] are the total volume and total number of molecules of the system.', '1311.6982-5-9-10': 'In this work, we adopt the same assumptions as those listed in Sec.III.C of Ref. [CITATION], i.e. we assume the system to be in the thermodynamic limit, and molecules are short-range correlated (short-ranged must be intended as a range comparable to the size of the transition region).', '1311.6982-5-9-11': 'The thermodynamic force for GC-AdResS ([MATH]) and for the auxiliary Hamiltonian AdResS ([MATH]), enforce the system to have a flat density: [EQUATION]', '1311.6982-5-9-12': 'Where [MATH] is the equilibrium number density of the system defined by [MATH].', '1311.6982-5-9-13': 'As shown in Refs. [CITATION], [MATH] provides the balance of the grand potential or equivalently [EQUATION] where [MATH] denotes the work of the thermodynamic force [MATH].', '1311.6982-5-10-0': ""Instead when we consider the auxiliary Hamiltonian approach, the third term on the R.H.S. of Eq. [REF] is not symmetric w.r.t molecules [MATH] and [MATH], therefore, the Newton's action-reaction law (momentum conservation) does not hold anymore."", '1311.6982-5-10-1': 'As a consequence, the pressure relation between the AT and CG resolution [REF] does not hold and should be derived again.', '1311.6982-5-10-2': 'Now assume, for simplicity and without lost of generality, that the system changes resolution only along the [MATH] direction.', '1311.6982-5-10-3': 'We impose an infinitesimal increment of the volume [MATH] to the AT region, and apply the same decrement of the volume [MATH] to the CG region.', '1311.6982-5-10-4': 'The volume of the transition region is kept constant as if it is an ideal ""piston"" that moves toward the CG region by an amount [MATH].', '1311.6982-5-10-5': 'We assume [MATH], where [MATH] is the cutting surface area.', '1311.6982-5-10-6': 'The displacement [MATH] should be infinitesimal, i.e. much smaller than the size of the transition region.', '1311.6982-5-10-7': 'This is achievable by taking the limit of [MATH], while keeping the system size fixed.', '1311.6982-5-10-8': 'It must be noticed that also the displacements of the molecules are infinitesimal, so it can be reasonably assumed that the resolution of the molecules remains the same under a displacement of [MATH].', '1311.6982-5-10-9': 'Therefore, the change of the free energy of the system is approximately: [EQUATION] where [MATH] and [MATH] are the free energies of the AT and CG region, respectively.', '1311.6982-5-11-0': '[MATH] is the linear dimension of the transition region along [MATH].', '1311.6982-5-11-1': 'Since the resolution changes only along [MATH], the two one-particle forces depend only on [MATH], and only have the component along [MATH].', '1311.6982-5-11-2': 'This can be easily generalized to changing resolution in any direction, i.e., replacing [MATH] by [MATH].', '1311.6982-5-11-3': 'The expression of Eq. [REF] as a sum of different terms is justified by the hypothesis of treating the system in the thermodynamic limit, and by the hypothesis that the interactions are short-ranged compared to the size of the transition region.', '1311.6982-5-11-4': '[MATH] and [MATH] is the numbers of molecules in the AT and CG region, and [MATH] and [MATH] is the volume in the AT and CG region, respectively; [MATH] is the temperature of the system.', '1311.6982-5-11-5': 'The last term is originated by the work done by the ideal piston.', '1311.6982-5-11-6': ""This term is composed by two parts, the first corresponding to the work done by the thermodynamic force, and the second corresponding to the work done by the force of changing representation (which does not vanish due to the violation of the Newton's action-reaction law)."", '1311.6982-5-11-7': 'The first and second term of Eq. [REF] being forces based on pairwise interactions only, do not contribute to the difference of energy; in fact their total work is zero (as long as the transition region moves infinitesimally along [MATH]).', '1311.6982-5-11-8': 'The notation [MATH] in Eq. [REF] denotes the ensemble average, which will be specified soon.', '1311.6982-5-11-9': 'It is straightforward to show that [EQUATION] where [MATH] is the work of the thermodynamic force [MATH], and [MATH] is the work of changing representation, which can be explicitly written down in a general form as: [EQUATION]', '1311.6982-5-11-10': 'The average is performed over all possible positions of the second molecule (i.e. [MATH]), at fixed position of the first molecule (i.e. [MATH]) in the pairwise interaction.', '1311.6982-5-11-11': 'In case of molecules containing more than one atom, the average is also made over all possible conformations in the atomistic resolution.', '1311.6982-5-11-12': 'In the thermodynamic limit, the equilibrium volume of the AT region maximizes the free energy, i.e. [MATH], which yields [EQUATION]', '1311.6982-5-11-13': 'Comparing the expression above with that obtained for GC-AdResS (Eq. [REF]), we have: [EQUATION] which relates the thermodynamic force of the auxiliary Hamiltonian AdResS and the GC-AdResS.', '1311.6982-5-12-0': 'In Ref. [CITATION] we proved that under proper assumptions, when the flat density profile is enforced by the thermodynamic force, the chemical potential difference between the different resolutions is given by [EQUATION]', '1311.6982-5-12-1': 'The same argument can be applied to the auxiliary Hamiltonian approach, and yields the chemical potential difference between the AT and CG resolutions [EQUATION]', '1311.6982-5-12-2': 'In the auxiliary Hamiltonian, we do not have the term [MATH] in the above formula (being the term [MATH] in GC-AdResS, generated by the non-conservative effect of the force interpolation).', '1311.6982-5-12-3': 'By comparing [REF] with [REF], we have the relation [EQUATION] which also relates the thermodynamic force of the auxiliary Hamiltonian AdResS and GC-AdResS.', '1311.6982-5-13-0': 'From Eq. [REF] and [REF], we find the extra work of the thermostat in GC-AdResS being identical to the work of changing representation of the auxiliary Hamiltonian approach: [EQUATION] which basically proves the statement at the beginning of this section.', '1311.6982-5-13-1': 'The ensemble average on the R.H.S. of Eq. [REF] is performed in the ensemble of the system treated with the potential interpolation approach, and the question is if the ensemble average is equivalent if it is performed in the simulation where the force interpolation approach is used.', '1311.6982-5-13-2': 'It is obvious that the spatial probability distribution corresponding to the system treated with the potential interpolation is consistent with the force interpolation at least up to the first order.', '1311.6982-5-13-3': 'It is also possible to systematically obtain equivalence in the ensemble average operation at higher orders of accuracy of the probability distribution, as, for example, it is done for the radial distribution function in Ref. [CITATION].', '1311.6982-5-13-4': 'However, here we do not consider higher order corrections, because it has been numerically shown that actually the ensemble average of [MATH] dose not depend on in which ensemble it is calculated [CITATION].', '1311.6982-5-13-5': 'Therefore, we use Eq. [REF] to calculate [MATH], and measure the ensemble average by the standard AdResS.', '1311.6982-5-13-6': 'As previously discussed, in the Gromacs implementation, the CG molecules also keep the atomistic degrees of freedom even though they are in the CG region, therefore, the kinetic part of [MATH] and [MATH] are identical, and [MATH] vanishes.', '1311.6982-5-13-7': 'Therefore, by inserting Eq. [REF] into [REF], we have [EQUATION]', '1311.6982-5-13-8': 'The extension of Eq. [REF] to multicomponent systems is reported in the Appendix [REF], while in the next section we apply the method to the calculation of [MATH] to liquids and mixtures.', '1311.6982-5-14-0': '# Results and Discussion', '1311.6982-5-15-0': 'We have calculated [MATH] for different liquids and mixtures, choosing cases which are representative of a large class of systems.', '1311.6982-5-15-1': 'Hydrophobic solvation in methane/water and in ethane/water mixtures, hydrophilic solvation in urea/water, a balance of both in water/tert-Butyl alcohol (TBA) mixture, other liquids, e.g. pure methanol and DMSO (and their mixtures with water), non aqueous mixtures in TBA/DMSO and alkane liquids such as methane, ethane and propane.', '1311.6982-5-15-2': 'Moreover, systems as water/urea are commonly used as cosolvent of biological molecules [CITATION] while systems as tert-Butyl alcohol/water play a key role in modern technology [CITATION], thus they are of high interest per se.', '1311.6982-5-15-3': 'All technical details of each simulation are presented in the Appendix [REF].', '1311.6982-5-16-0': 'Results are reported in Table [REF], where the comparison with values obtained using full atomistic TI and available experiments, at the same concentrations, of our calculation is made; in our previous work we have already shown that value of the chemical potential of liquid water obtained with IPM is well reproduced by GC-AdResS, however the computational cost of IMP was very large, thus we do not consider calculations done with IPM in this paper.', '1311.6982-5-16-1': 'The agreement with full atomistic TI simulations is satisfactory in all cases, and thus it proves the solidity of GC-AdResS in describing the essential thermodynamics of a large class of systems.', '1311.6982-5-16-2': 'We also compare the obtained values with those available in literature [CITATION].', '1311.6982-5-16-3': 'Although the concentration of the minor component in the mixtures that we consider, is higher than the concentrations considered in Refs. [CITATION], we are anyway in the very dilute regime and thus the chemical potential should not change in a significant way; we have verified such a supposed consistency for one relevant system (see discussion about Fig. [REF]).', '1311.6982-5-16-4': ""The chemical potential of [MATH]-th liquid's component in a mixture is calculated as (see Appendix [REF]): [EQUATION] where [MATH] is the thermodynamic force applied to the molecules of the [MATH]-th component; this assures that, at the given concentration, the density of molecules of species [MATH], in the transition region, is equivalent to the density of the same liquid's component in a reference full atomistic simulation."", '1311.6982-5-16-5': 'The ensemble average is taken over the position of the second molecule, provided that the first molecule is of species [MATH], and located at position [MATH].', '1311.6982-5-17-0': 'A complementary information to Table [REF] are Fig. [REF] and [REF].', '1311.6982-5-17-1': 'In Fig. [REF] we have studied the behavior of [MATH] as a function of the interaction cut-off.', '1311.6982-5-17-2': 'In fact the current version of GC-AdResS, employs the reaction field method for treating electrostatic interactions in the atomistic region, and the cut off is likely to play a role in some of the systems investigated.', '1311.6982-5-17-3': 'Fig. [REF], for the case of DMSO/water mixture, confirms our intuition and suggests that we could systematically improve the accuracy by increasing the cut-off, and at a value of about [MATH] nm, [MATH] converges.', '1311.6982-5-17-4': 'In any case, at a values of [MATH] nm, which is the one routinely used in full atomistic simulations and used by us, the value obtained with GC-AdResS is already satisfactory.', '1311.6982-5-17-5': 'The cut-off radii used for other systems are reported in Appendix [REF].', '1311.6982-5-17-6': 'A further question that may arise is the capability of our method to predict the behavior of [MATH] as a function of the concentration, above all in the very dilute regime.', '1311.6982-5-17-7': 'In Fig. [REF] we have performed such a study for the case of TBA/water mixture, we show a good agreement between GC-AdResS and TI and show that at a very dilute concentration our calculated value is close to that of experiments, moreover the trend, regarding the TI calculations, is consistent with that reported in Ref. [CITATION].', '1311.6982-5-18-0': 'In essence, according to the results obtained, GC-AdResS allows an on-the-fly determination of [MATH] of each component of a liquid, whenever a simulation is performed, without extra computational costs.', '1311.6982-5-18-1': 'Moreover, Fig.[REF] shows the action of the thermodynamic force and of the thermostat in the transition region [MATH] for TBA-water; the molecular density is sufficiently close to that of reference (the largest difference is below [MATH] and the average difference is below [MATH]), and thus it assures that in the atomistic region there are no (significant) artificial effects on the molecular density due to the perturbation represented by the interpolation of forces in [MATH].', '1311.6982-5-18-2': 'In Fig.[REF] we report various radial distribution functions for TBA-water in the atomistic region of the adaptive set up.', '1311.6982-5-18-3': 'The agreement with data from a full atomistic simulation is highly satisfactory.', '1311.6982-5-18-4': 'Moreover, it must be underlined that, on purpose, we have chosen extreme technical conditions, that is, a very small atomistic and coarse-grained region ([MATH] nm) and a relatively large transition region ([MATH] nm).', '1311.6982-5-18-5': 'Even in these conditions we prove that local properties as those of Fig. [REF] and Fig. [REF], together with a relevant thermodynamics quantity as [MATH] are well reproduced.', '1311.6982-5-18-6': 'This example shows the key features of GC-AdResS, that is, a multiscale simulation where the chemical potential of each component is obtained without extra computational costs and with high accuracy in a simulation where other properties are also calculated with high accuracy.', '1311.6982-5-18-7': 'It must be also noticed that the system corresponding to the figures is, among all the system considered, the case where the action of the thermodynamic force and of the thermostat produces the less accurate agreement with the reference data.', '1311.6982-5-19-0': '# Efficiency', '1311.6982-5-20-0': 'In order to show the numerical efficiency of our approach, we compare the time taken to do a full GC-AdResS simulation and the time for a thermodynamic integration calculation for different systems with varied concentration of TBA in water.', '1311.6982-5-20-1': 'The total time required for an GC-AdResS simulation consists of the time taken to obtain a converged thermodynamic force and the time taken to obtain the coarse-grained chemical potential.', '1311.6982-5-20-2': 'The time taken to complete TI procedure at each value of [MATH] is summed up to obtain the total time.', '1311.6982-5-20-3': 'In this work, the TI is done in two stages, first the van der Waals interactions are coupled followed by the electrostatic coupling.', '1311.6982-5-20-4': 'At each stage, 21 equally distributed values of [MATH] are used, therefore, in total 42 simulations were performed to calculate each TI chemical potential value.', '1311.6982-5-20-5': 'In an AdResS simulation, the initial guess of the thermodynamic force largely determines the time for convergence.', '1311.6982-5-20-6': 'We started with a randomly chosen initial guess ([MATH] kJ/mol, we picked a small value because the TBA molecule is hydrophilic) for system with the highest mole fraction of TBA.', '1311.6982-5-20-7': 'For all the other systems, we used the converged thermodynamic force obtained from the first system as an initial guess.', '1311.6982-5-20-8': 'The convergence was much faster in all the other cases using this approach.', '1311.6982-5-20-9': 'Table [REF] shows the number of iterations required for the thermodynamic force convergence in GC-AdResS and total time required for GC-AdResS and TI calculation.', '1311.6982-5-20-10': 'The advantage of GC-AdResS over TI is that we get two values of excess chemical potential for both solute and solvent in a single calculation, while in TI, the whole process has to be repeated to get the excess chemical potential of the other component.', '1311.6982-5-20-11': 'For very dilute systems ([MATH]), however, one has to take a very large systems in GC-AdResS (see the Appendix [REF] for system size).', '1311.6982-5-20-12': 'It takes a large amount of time for the thermodynamic force to reach convergence, and hence TI is always a better option at such low concentrations with a much smaller system size at the same mole fraction.', '1311.6982-5-21-0': '# Current computational convenience: a critical appraisal', '1311.6982-5-22-0': 'The natural question arising from the discussion above is whether or not GC-AdResS is a more convenient technical tool for calculating [MATH] compared to TI.', '1311.6982-5-22-1': 'Currently the answer is neither negative nor positive, although the current work is the first step towards a potentially positive answer for the future.', '1311.6982-5-22-2': 'In fact, the fastest version of AdResS is implemented in the GROMACS code [CITATION]; using the Gromacs version 4.5.1 a speedup of a factor four with respected to full atomistic simulations has been reported for aqueous mixtures [CITATION].', '1311.6982-5-22-3': 'In this case GC-AdResS was more convenient than TI because in one simulation one could obtain the chemical potential of each liquid component and at the same time calculate structural properties (e.g. radial distribution functions).', '1311.6982-5-22-4': 'However in the successive version of GROMACS 4.6.1 the performance of atomistic simulations (above all of SPC/E water) has been highly improved while the corresponding implementation of AdResS is not optimized yet.', '1311.6982-5-22-5': 'At the current state, AdResS can only assure a speed up factor between 2 and 3 for large systems (30000 molecules) compared to full atomistic simulations (except for pure SPC/E water systems).', '1311.6982-5-22-6': 'As a consequence for the calculation of [MATH], TI is in general computationally less demanding than AdResS .', '1311.6982-5-22-7': 'Another point that must be considered (in perspective) for a fair comparison between TI and GC-AdResS, is the following: even if AdResS is optimized, in any code, TI has the advantage that one can use one single molecule in the simulation box to mimic the minor component of a mixture.', '1311.6982-5-22-8': 'In our case, instead, we must treat, technically speaking, a true mixture with a certain number of molecules of the minor component immersed in the liquid of the major component.', '1311.6982-5-22-9': 'Thus, at low concentrations, GC-AdResS simulations require larger systems than those required by TI, moreover, because of the low density of the minor component, the convergence of the corresponding thermodynamic force requires long simulations.', '1311.6982-5-22-10': 'Thus, for very dilute systems, if one is interested only in the chemical potential, TI shall be preferred to GC-AdResS, however if the interest goes beyond the calculation of the chemical potential, (e.g. radial distribution functions) then (optimized) GC-AdResS would still be more convenient.', '1311.6982-5-22-11': 'When the concentration becomes higher, GC-AdResS may become preferable for both tasks: general properties of the mixture and chemical potential, not only because in this case one requires larger systems, but also because the convergence of the thermodynamic force of the minor component is much faster.', '1311.6982-5-22-12': 'Moreover, we would have the flexibility of calculating the chemical potential of both components in one simulation run, whereas in TI, one needs to run two separate simulations in order to get the chemical potential of both components.', '1311.6982-5-22-13': 'The results reported in the previous section about the current efficiency of GC-AdResS are rather encouraging, however currently there is not a clear convenience in using GC-AdResS instead of TI for calculating [MATH]; in any case the technical aspects of code optimization must be reported and we must make clear that the aim of this work is to show that the automatic calculation of [MATH], independently from the simulation code in which is implemented and its computational cost, is a ""conceptual"" feature of GC-AdResS.', '1311.6982-5-23-0': '# Conclusion', '1311.6982-5-24-0': 'We have shown the accuracy of GC-AdResS in calculating the excess chemical potential for a representative class of complex liquids and mixtures.', '1311.6982-5-24-1': 'For any system, the initial equilibration process, that is the determination of the thermodynamic force, automatically delivers the chemical potential.', '1311.6982-5-24-2': 'The only additional calculation required is that of [MATH] which implies the use of IPM or TI, but for a liquid of simple spheres, thus computationally negligible.', '1311.6982-5-24-3': 'The essential message is that GC-AdResS would be, per se, a reliable multiscale technique to calculate the chemical potential and, in perspective, upon computational/technical optimization it may become an efficient tool for calculating [MATH] compared to current techniques in MD such as TI.'}"
0902.1111,"{'0902.1111-1-0-0': 'We formulate a general perturbative framework based on a superconducting atomic limit for the description of Andreev bound states (ABS) in interacting quantum dots connected to superconducting leads.', '0902.1111-1-0-1': 'A local effective Hamiltonian for dressed ABS, including both the atomic or molecular levels and the induced proximity effect on the dot is argued to be a natural starting point.', '0902.1111-1-0-2': 'Self-consistent expansions in single-particle tunneling events are shown to provide accurate results in regimes where the superconducting gap is larger than the Kondo temperature, as demonstrated by the comparison to recent Numerical Renomalization Group calculations.', '0902.1111-1-0-3': 'These analytical results may have bearings for interpreting Andreev spectroscopic measurements with STM on carbon nanotubes coupled to superconducting electrodes.', '0902.1111-1-1-0': '# Introduction', '0902.1111-1-2-0': ""When a quantum dot is connected to superconducting electrodes, the proximity effect drastically modifies the dot's electronic structure due to the local formation of Cooper pairs."", '0902.1111-1-2-1': 'The density of states on the dot thus exhibits a gap, so that the formation of discrete sub-gap states arises [CITATION].', '0902.1111-1-2-2': 'These Andreev bound states (ABS) play certainly an important role as they may contribute a large part of the spectral weight [CITATION] and carry most of the Josephson current.', '0902.1111-1-2-3': '[CITATION] A physical understanding of the ABS requires to characterize how these states are connected to the atomic or molecular levels of the uncoupled quantum dot, and to describe quantitatively their evolution as a function of several parameters, such as gate voltage, Coulomb interaction, tunnel couplings, and superconducting gap.', '0902.1111-1-2-4': 'Whereas the ABS have been observed in metal-superconductor hybrid structures [CITATION], no direct spectroscopy has so far been achieved in quantum dot systems.', '0902.1111-1-2-5': 'Andreev bound states come in pairs, one state above and one below the Fermi level, forming a two level system.', '0902.1111-1-2-6': 'Consequently, recent interest in the spectroscopy of the bound states [CITATION] has also been stimulated by proposals to use the latter as a qubit.', '0902.1111-1-2-7': '[CITATION]', '0902.1111-1-3-0': 'Experimentally, superconducting quantum dots can be realized with carbon nanotubes junctions.', '0902.1111-1-3-1': 'It has been shown that nanotubes and InAs quantum dots connected to superconducting electrodes can be tuned from a Coulomb blockade regime, to a Kondo regime,[CITATION] to a weakly interacting Fabry-Perot regime by changing local gate voltages.', '0902.1111-1-3-2': '[CITATION] The Josephson current at zero bias and multiple Andreev reflections at finite bias voltage have been measured in such devices.', '0902.1111-1-3-3': '[CITATION] The transition from a [MATH]-junction to a [MATH]-junction, namely a reversal in the sign of the Josephson current, [CITATION] has also been been observed when a magnetic moment forms on the dot.', '0902.1111-1-3-4': '[CITATION] As a possible application of superconducting junctions, nano-SQUID devices have also been fabricated.', '0902.1111-1-3-5': '[CITATION]', '0902.1111-1-4-0': 'An exact theoretical description of a quantum dot coupled to superconducting leads is only possible when the Coulomb interaction is fully neglected.', '0902.1111-1-4-1': 'The interacting single dot system, as described by the Anderson model with superconducting electrodes, has been so far analyzed by treating the Coulomb interaction either within the mean field approach, [CITATION] the perturbation expansion in the Coulomb interaction [CITATION] or in the tunnel coupling, [CITATION] the Non-Crossing Approximation, [CITATION] Quantum Monte Carlo simulations, [CITATION] or functional renormalization group (FRG) and numerical renormalization group (NRG) calculations.', '0902.1111-1-4-2': '[CITATION]', '0902.1111-1-5-0': 'None of the analytical approaches mentioned above is able to describe entirely the physics of a quantum dot coupled to superconducting leads.', '0902.1111-1-5-1': 'Whereas lowest order perturbation expansions in the tunnel coupling will hardly capture the proximity effect induced by the electrodes, mean field and weak-interaction approaches will miss the Kondo effect.', '0902.1111-1-5-2': 'NRG calculations on the other hand can capture the physics of such a system over a wide range of parameters, but are numerically demanding and not easily portable to more complex molecular systems.', '0902.1111-1-5-3': 'More importantly, in the view of describing the ABS alone, none of these techniques does provide a simple physical picture.', '0902.1111-1-5-4': 'Henceforth we will develop in this paper a new perturbative approach based on an effective local Hamiltonian for dressed ABS, that extends the limit of large superconducting gap proposed previously.', '0902.1111-1-5-5': '[CITATION] This approach will illuminate the nature of the ABS in interacting quantum dots, as well as provide a simple and accurate analytical framework in cases where the gap is not smaller than the Kondo temperature, that may be useful for interpreting future spectroscopic experiments.', '0902.1111-1-5-6': 'In addition, our formalism, which incorporates the atomic or molecular levels from the outset, should easily be extended to describe more complex systems, as for instance superconducting double quantum dots or molecules with more complicated orbital structure (see e.g. Refs. [CITATION]).', '0902.1111-1-6-0': 'We organize our paper as follows.', '0902.1111-1-6-1': 'In Sec. [REF], the system is mapped onto an effective local Hamiltonian, similarly to the widely used atomic limit, but including the proximity effects due to the superconducting leads.', '0902.1111-1-6-2': 'In Sec. [REF], the perturbation theory around this limit is set up and self-consistent equations for the ABS energy renormalizations are derived in order to extend the validity of the bare perturbative approach.', '0902.1111-1-6-3': 'Sec. [REF] illustrates how this expansion can describe ABS in superconducting quantum dots over a wide range of parameters, by a comparison to available NRG data.', '0902.1111-1-6-4': '[CITATION]', '0902.1111-1-7-0': '# Theoretical formulation', '0902.1111-1-8-0': '## Model', '0902.1111-1-9-0': 'We focus in this paper on a single-level quantum dot coupled to superconducting leads, which is relevant experimentally for molecular junctions with large single-electron level spacing.', '0902.1111-1-9-1': 'A simple Hamiltonian able to describe such a system is given by the superconducting Anderson model [EQUATION] where [EQUATION]', '0902.1111-1-9-2': 'In the above equations, [MATH] is the annihilation operator of an electron with spin [MATH] on the dot, [MATH] that of an electron with spin [MATH] and wave vector [MATH] in the lead [MATH], and [MATH].', '0902.1111-1-9-3': 'The leads are assumed to be described by standard s-wave BCS Hamiltonians [MATH] with superconducting gaps [MATH].', '0902.1111-1-9-4': 'The phase difference of the latter is noted [MATH].', '0902.1111-1-9-5': 'Furthermore, the leads are assumed to have flat and symmetric conduction bands, i.e. the kinetic energy [MATH] measured from the Fermi level ranges in [MATH] and the density of states is [MATH].', '0902.1111-1-9-6': 'We assume [MATH]-independent and symmetric tunneling amplitudes [MATH] between the dot and both superconducting leads.', '0902.1111-1-9-7': 'The dot has a level energy [MATH] and Coulomb interaction [MATH].', '0902.1111-1-9-8': 'Experimentally, the crucial characteristic energy scales, namely Coulomb interaction [MATH], total hybridization [MATH] and gap [MATH], are typically all of the same order of magnitude, [CITATION] providing a challenge for analytical methods.', '0902.1111-1-10-0': ""The physics of the quantum dot can be described via its Green's function [EQUATION] where the Nambu spinor [EQUATION] has been introduced."", '0902.1111-1-10-1': 'Because we will only be interested in stationary equilibrium physics, [MATH] shall be computed in the Matsubara frequency formalism.', '0902.1111-1-11-0': '## Effective local Hamiltonian', '0902.1111-1-12-0': 'As the above Hamiltonian has no exact solution, some approximations must be made.', '0902.1111-1-12-1': 'Among the physical ingredients we want to include in a non-perturbative way is the local pairing on the dot that is crucial for the evolution of the Andreev bound states.', '0902.1111-1-12-2': 'Furthermore, the Coulomb interaction shall be taken into account in an exact manner in order to describe the atomic states faithfully, and to highlight how these are adiabatically connected to the ABS.', '0902.1111-1-12-3': 'However, the usual development in weak tunnel coupling [MATH] around the atomic limit [CITATION] is not sufficient to describe the proximity effect at lowest order.', '0902.1111-1-12-4': 'Therefore, we shall consider in what follows an expansion around a superconducting atomic limit.', '0902.1111-1-13-0': 'Such simple solvable limiting case of the model ([REF]) is often referred to as the limit of large gap [MATH], and has been discussed previously [CITATION].', '0902.1111-1-13-1': 'Expansions for finite [MATH] have not however been discussed to our knowledge, and are the topic of this paper.', '0902.1111-1-13-2': 'We emphasize from the outset (see equation ([REF]) below), that the superconducting atomic limit as used normally in the literature corresponds to the limit [MATH] (i.e. infinite electronic bandwidth), taken before [MATH].', '0902.1111-1-13-3': 'The order of the two limits is crucial: if the limit [MATH] was to be taken first, the dot would be completely decoupled from the leads and the proximity effect would be lost, so that the limit of infinite gap would reduce to the usual atomic limit.', '0902.1111-1-13-4': 'As will be shown now, the superconducting atomic limit should rather be interpreted as a low frequency expansion, i.e. a limit where the gap is much larger than the characteristic frequencies of the dot.', '0902.1111-1-14-0': ""We start off by deriving the Green's function defined in Eq. [REF] using the equations of motion."", '0902.1111-1-14-1': 'Thereby, the Coulomb interaction [MATH] will at first be omitted for the sake of clarity.', '0902.1111-1-14-2': 'Note that in the end, [MATH] will simply give an extra contribution which adds to the effective Hamiltonian.', '0902.1111-1-14-3': 'Fourier transformation straightforwardly yields [EQUATION]', '0902.1111-1-14-4': ""In Eq. [REF], [MATH] is a fermionic Matsubara frequency and [MATH] the bare Green's function of electrons with a wave vector [MATH] in the lead [MATH]."", '0902.1111-1-14-5': 'Transforming the sum over wave vectors [MATH] into an integral over energies yields [EQUATION]', '0902.1111-1-14-6': ""In the limit [MATH], the Green's function [REF] becomes purely static and reduces to [EQUATION]"", '0902.1111-1-14-7': ""Note that the low frequency limit we consider here yields a Green's function that indeed depends on the finite bandwidth [MATH], and this shows that the limit [MATH] shall not be taken for the proximity effect to survive."", '0902.1111-1-14-8': 'In what follows, we will therefore keep both [MATH] and [MATH] finite.', '0902.1111-1-14-9': ""Plugging Eq. [REF] into the Green's function [MATH] leads to the same result as would have been obtained with the effective local Hamiltonian [EQUATION] where the local pairing amplitude induced by the leads on the dot reads [EQUATION] which explicitly depends on the ratio [MATH]."", '0902.1111-1-14-10': 'By an appropriate gauge transformation for the operators [MATH], it is always possible to choose [MATH], as shall be done from now on.', '0902.1111-1-14-11': 'The complete local effective Hamiltonian is obtained when the Coulomb interaction is taken into account again.', '0902.1111-1-14-12': 'Defining [MATH], the energy level of the dot is shifted such that the Hamiltonian clearly exhibits particle-hole symmetry for [MATH]: [EQUATION]', '0902.1111-1-14-13': 'The physical interpretation of this effective local Hamiltonian is simple.', '0902.1111-1-14-14': 'For finite gap, the quantum dot is coupled to both the Cooper pairs and the quasiparticles in the leads.', '0902.1111-1-14-15': 'The Cooper pairs, which lie at the Fermi level, are responsible for the proximity effect.', '0902.1111-1-14-16': 'The quasiparticles give rise to conduction electrons excitations with energies higher than the gap [MATH].', '0902.1111-1-14-17': 'In the limit [MATH], the quasiparticles are far in energy and the coupling between them and the dot vanishes, which greatly simplifies the physics and makes an exact solution possible.', '0902.1111-1-14-18': 'Yet, as the dot is still coupled to the Cooper pairs at the Fermi level, the proximity effect survives with a local pairing term proportional to the hybridization [MATH] between dot and leads.', '0902.1111-1-15-0': '## Spectrum of the effective local Hamiltonian', '0902.1111-1-16-0': 'As the Coulomb interaction simply yields an extra energy shift of [MATH] for both empty and doubly occupied dot, the eigenvectors and eigenvalues of the local effective Hamiltonian ([REF]) are readily obtained by a Bogoliubov transformation [CITATION], in perfect analogy with solution of the BCS Hamiltonian.', '0902.1111-1-16-1': '[MATH] has thus four eigenstates, the singly occupied spin [MATH] states [MATH] and [MATH] with energy [MATH], and two BCS-like states given by [EQUATION] where [MATH] denotes the empty dot and [MATH] the doubly occupied dot.', '0902.1111-1-16-2': 'The amplitudes [MATH] and [MATH] can always be chosen to be real with [MATH] and [MATH].', '0902.1111-1-16-3': 'The energies corresponding to these BCS-like states are [MATH].', '0902.1111-1-17-0': 'As [MATH] is always larger than [MATH], the effective local Hamiltonian has two possible ground states: the low energy BCS-like state [MATH] or the degenerate spin [MATH] doublet [MATH].', '0902.1111-1-17-1': 'In the [MATH] state, the energy is minimized for [MATH].', '0902.1111-1-17-2': 'Thus, the spin singlet phase corresponds to a [MATH]-junction (a result well known from the weak coupling limit [CITATION]).', '0902.1111-1-17-3': 'The transition between the singlet phase and the spin [MATH] doublet takes place at [MATH], and Fig. [REF] shows the corresponding phase diagram for variable [MATH], [MATH] and [MATH].', '0902.1111-1-17-4': 'The state adopted by the quantum dot in the large gap limit therefore results from a competition between the local pairing (induced by the proximity effect and characterized by the hybridization [MATH]) and the Coulomb interaction.', '0902.1111-1-18-0': '## Andreev bound states', '0902.1111-1-19-0': 'As outlined in the introduction, the coupling to superconducting leads induces a gap in the spectral function of the dot, inside which discrete Andreev bound states can form.', '0902.1111-1-19-1': 'The spectral function of the dot shows therefore sharp peaks, which could be measured by STM or microwave experiments.', '0902.1111-1-19-2': '[CITATION] These peaks indicate addition energies at which an electron may enter (or leave) the dot, and correspond therefore to transitions between states with [MATH] and [MATH] electrons.', '0902.1111-1-19-3': 'Hence, the ABS peaks in the spectral function may be interpreted as transitions between the superconducting atomic levels of the dot [MATH], possibly renormalized by single-particle tunneling events neglected in [MATH] (to be included in the next section).', '0902.1111-1-19-4': 'Furthermore, transitions from a spin [MATH] doublet to a spin singlet necessarily involve an electron exchange between the dot and the superconducting leads.', '0902.1111-1-19-5': 'As the states [MATH] and [MATH] correspond to the superposition of an empty and doubly occupied dot, this electron exchange and the final singlet states can be understood within the Andreev reflection picture.', '0902.1111-1-20-0': 'Putting everything together, our effective local Hamiltonian in Eq. [REF] describes the energies of the Andreev bound states as transition energies from the spin 1/2 doublet to the spin singlet states.', '0902.1111-1-20-1': '[CITATION] There are thus four Andreev bound states in the large gap limit for the model ([REF]), with energy [MATH] and [MATH] which read: [EQUATION]', '0902.1111-1-20-2': 'The [MATH]/[MATH] transition corresponds to the crossing of the [MATH] and [MATH] states, which occurs for [MATH].', '0902.1111-1-21-0': '# Perturbation expansion around the effective local Hamiltonian', '0902.1111-1-22-0': '## Perturbation theory', '0902.1111-1-23-0': 'The effective Hamiltonian is not sufficient to obtain satisfying results for all regimes of parameters.', '0902.1111-1-23-1': 'First, [MATH] only describes the [MATH]/[MATH]-junction transition due to the competition between a local moment state (stabilized by the Coulomb blockade) and a spin singlet (induced by the proximity effect).', '0902.1111-1-23-2': 'However, if the Coulomb interaction is strong (i.e. [MATH] below the Kondo temperature), the local moment can be screened by the Kondo effect, which will compete with the superconducting gap for the [MATH] transition, so that a typical scaling in the ratio of the Kondo temperature to the gap [MATH] will appear.', '0902.1111-1-23-3': 'Also, the Josephson current in the [MATH]-phase identically vanishes from [MATH], as the spin doublet does not disperse with the superconducting phase difference, a limitation of the large gap limit.', '0902.1111-1-23-4': 'On a more quantitative basis, the experimental gap [MATH] is usually of the order of a few kelvins, which is also the typical scale for both [MATH] and [MATH] in carbon nanotube quantum dot devices.', '0902.1111-1-24-0': ""In order to extend the description of the quantum dot's physics, energy corrections shall be calculated with a perturbation theory around the effective Hamiltonian [REF]."", '0902.1111-1-24-1': 'Once these corrections have been obtained, physical observables like the Josephson current may be computed via the free energy [MATH], with [MATH] the inverse temperature.', '0902.1111-1-24-2': 'Therefore, it is most convenient to work in an action based description, which directly yields the partition function [MATH].', '0902.1111-1-24-3': 'Following Ref. [CITATION], we first integrate over the fermions in the leads.', '0902.1111-1-24-4': 'Omitting the resulting irrelevant constant, the partition function reads [EQUATION] where we have introduced the Grassmann Nambu spinors at Matsubara frequency [MATH], [EQUATION] denoting the Grassmann fields associated with electrons in the dot by [MATH] and [MATH].', '0902.1111-1-25-0': 'The perturbation consists of the terms in Eq. [REF] that are not contained in the action [MATH] corresponding to the effective local Hamiltonian.', '0902.1111-1-25-1': 'A simple identification yields [EQUATION]', '0902.1111-1-25-2': 'Note that [MATH] contains the local pairing term derived in section [REF].', '0902.1111-1-25-3': 'The proximity effect is thus treated non-perturbatively (just like the Coulomb interaction), which is the crucial ingredient of our analytic approach.', '0902.1111-1-25-4': 'The perturbation [MATH] simply corresponds to the tunnel coupling between the dot and the electrodes other than the lowest order proximity effect.', '0902.1111-1-26-0': 'The actual corrections are calculated by expanding the partition function to the first order in [MATH] according to [EQUATION] which we then identify with [EQUATION] where the renormalized superconducting atomic levels [MATH] and [MATH] are obtained from: [EQUATION]', '0902.1111-1-26-1': ""Because the Coulomb interaction is taken into account, Wick's theorem cannot be used to calculate [MATH]."", '0902.1111-1-26-2': 'Instead, expectation values are calculated using Lehmann representation.', '0902.1111-1-26-3': 'Explicit calculations may be found in the appendix.', '0902.1111-1-26-4': 'In the zero temperature limit [MATH], the energy corrections are [EQUATION] with the quasiparticle energy [MATH].', '0902.1111-1-27-0': '## Self-consistent renormalization of the energy', '0902.1111-1-28-0': 'Eqs. [REF]-[REF] yield the first corrections to the energy levels, so that the bound states energies are simply shifted by [MATH] and [MATH].', '0902.1111-1-28-1': 'Obviously, these expressions are logarithmically divergent when the bound states energies [MATH] and [MATH] approach the gap edge, and are therefore only valid as long as e.g. [MATH].', '0902.1111-1-28-2': 'Fortunately, the regime of validity of the above equations can be greatly extended using a self-consistency condition inspired by Brillouin-Wigner perturbation theory, [CITATION], which resums these leading divergences to infinite order.', '0902.1111-1-28-3': 'The resulting self-consistent equations are: [EQUATION] and [EQUATION] where [MATH] have been defined in Eqs [REF] and [REF], and [MATH], [MATH].', '0902.1111-1-28-4': 'Note that terms like [MATH] have no self-consistency because there are no associated divergences.', '0902.1111-1-28-5': 'Eq. [REF] and [REF] clearly now hold as long as the renormalized energies [MATH] and [MATH] are not too close to the gap edge, [MATH] respectively.', '0902.1111-1-29-0': '# Results', '0902.1111-1-30-0': '## Phase diagram', '0902.1111-1-31-0': 'We start by discussing the [MATH] transition line, by comparison to the numerical renormalization group (NRG) data by Bauer et al.[CITATION].', '0902.1111-1-31-1': 'Fig. [REF] shows the extension to smaller gaps [MATH] values of the phase diagram obtained with unrenormalized local superconducting states for infinite gap (Fig. [REF]).', '0902.1111-1-31-2': 'Even though our perturbative approach is fairly simple, the results reproduce nicely the NRG data of Refs. [CITATION] and [CITATION].', '0902.1111-1-31-3': 'The analytically obtained phase diagram is indeed identical to the NRG data for [MATH].', '0902.1111-1-31-4': 'For smaller [MATH], the Kondo effect sets in, but the transition lines remain quantitatively correct for [MATH] near [MATH], with increasing deviations from the NRG calculations close to the particle-hole symmetric point [MATH] at large Coulomb interaction U.', '0902.1111-1-31-5': 'In this regime, the [MATH]-phase possesses a Kondo singlet ground state.', '0902.1111-1-31-6': 'As the leads are superconductors, the formation of a Kondo resonance involves the breaking of Cooper pairs.', '0902.1111-1-31-7': 'Therefore, the transition is now due to the competition between [MATH] and the superconducting gap [MATH], and should occur at [MATH].', '0902.1111-1-32-0': 'Fig. [REF] shows a plot of the transition line for [MATH] as obtained with Eq. [REF] (solid curve).', '0902.1111-1-32-1': 'The vertical, dotted line depicts the asymptote in the effective local limit.', '0902.1111-1-32-2': 'The symbols again correspond to NRG data.[', '0902.1111-1-32-3': '[CITATION] The Kondo temperature is given by [MATH] (see for example Ref. NRG_spectral_Bauer).', '0902.1111-1-32-4': 'The inset shows on a log-log scale that our approach captures an exponential decay of the transition line with the Coulomb interaction.', '0902.1111-1-32-5': 'Nonetheless, the suppression of the BCS-like phase appears quantitatively stronger than expected: a factor 4 instead of 8 is found in the exponential factor of [MATH].', '0902.1111-1-32-6': 'The reason for this is that the vertex renormalizations have not been taken into account, as discussed in the context of U-NCA [CITATION].', '0902.1111-1-32-7': 'Far away from the particle-hole symmetric limit, our results for the Kondo temperature reproduce the lowest-order scaling theory for the infinite-[MATH] Anderson model [CITATION], and are in relatively good agreement with NRG data for all [MATH] values.', '0902.1111-1-33-0': '## Energy renormalizations at particle hole symmetry ([MATH])', '0902.1111-1-34-0': 'While Fig. [REF] only indicates the transition line between the spin [MATH] doublet and the lowest BCS spin singlet, it is also instructive to look at the actual renormalization of the energy levels while varying the gap [MATH] from large values to smaller ones beyond the critical point.', '0902.1111-1-34-1': 'Fig. [REF] indicates the renormalized energies of the two Andreev bound states (i.e. the difference between the spin [MATH] doublet and the two spin singlets energies) for different hybridizations [MATH].', '0902.1111-1-34-2': 'We note that our results are in quantitative agreement with the NRG calculations of Yoshioka and Ohashi.', '0902.1111-1-34-3': '[CITATION] Several regions need to be distinguished.', '0902.1111-1-34-4': 'If the gap [MATH] is much larger than the bandwidth [MATH], all curves collapse at the value [MATH] (left hand side of Fig. [REF]), since there is no hybridization with both quasiparticles and Cooper pairs anymore, and one recovers the bare atomic levels.', '0902.1111-1-34-5': 'When the gap starts to decrease, the proximity effect simply splits the two Andreev bound states according to equations ([REF])-([REF]).', '0902.1111-1-34-6': 'When the gap becomes of the same order as the typical energy scales of the dot [MATH] and [MATH], the superconducting atomic levels start to mix with the electrodes, so that the energies renormalize in a non trivial way.', '0902.1111-1-34-7': 'One can see that the transition involving the highest BCS states ends up touching the gap edge for [MATH], so that half of the ABS are absorbed into the continuum above [MATH], as can be seen in Fig. [REF].', '0902.1111-1-34-8': 'The lowest BCS state follows however a downward renormalization, until the Fermi level is crossed and the ground state becomes the [MATH]-state.', '0902.1111-1-34-9': 'The difference in behavior between the lowest and highest bound states (the former being never allowed to leave the superconducting gap) can be tracked into equations ([REF])-([REF]), where level repulsion effects from the gap edge occur for the low energy level [MATH] but are canceled for the high energy level [MATH], which is hence allowed to penetrate into the continuum.', '0902.1111-1-34-10': 'These considerations unveil how the ABS may be adiabatically connected to the atomic or molecular levels in a complicated fashion.', '0902.1111-1-35-0': 'Again, our simple analytic approach reproduces the NRG results [CITATION] over a vast regime of parameters.', '0902.1111-1-35-1': 'Yet, some deviations are observed in the Kondo regime: we find (for the highest hybridization [MATH]) that the high energy BCS-like state is not absorbed anymore into the continuum of states - an artifact of the limits of our perturbative approach.', '0902.1111-1-35-2': 'Notice also that the energy corrections are too important if the gap becomes very small, an effect actually due to our underestimation of the Kondo temperature at particle-symmetry, as discussed previously.', '0902.1111-1-35-3': 'Finally Fig. [REF] shows that, in the limit of vanishing gap, our approach is only valid as long as [MATH] (as has been mentioned in section [REF]), because the lowest bound state artificially escapes from the gap.', '0902.1111-1-35-4': 'The expected saturation of [MATH] near [MATH] can be restored by adding a further self-consistency for terms such as [MATH] in ([REF]) (not shown here).', '0902.1111-1-36-0': '## Energy renormalizations outside particle hole symmetry ([MATH])', '0902.1111-1-37-0': 'From an experimental point of view, the position of the energy level of the quantum dot is the most controllable parameter of the system (by a simple gate voltage).', '0902.1111-1-37-1': 'Therefore, it is important to analyze the evolution of the Andreev bound states for different values of [MATH].', '0902.1111-1-38-0': 'Fig. [REF] illustrates how the energies of the bound states scale with [MATH] for [MATH] and can be favorably compared to the NRG data by Yoshioka and Ohashi.', '0902.1111-1-38-1': '[CITATION] The more particle-hole symmetry is broken, the more the low energy bound state moves away from the gap edge, ensuring even better convergence of our expansion for a given value of [MATH].', '0902.1111-1-38-2': 'This can be understood given that this bound state corresponds to the transition between [MATH] and the spin [MATH] doublet: outside particle-hole symmetry, the dot either seeks to be as empty as possible (for [MATH]) or as occupied as possible (for [MATH]).', '0902.1111-1-38-3': 'Thus, a BCS-like wave function will be favored.', '0902.1111-1-38-4': 'As a consequence, the Kondo effect (that necessitates a singly occupied dot) is less favored.', '0902.1111-1-38-5': 'This corresponds to a regime where our approximation scheme works at best.', '0902.1111-1-39-0': 'Further understanding can be gained by looking at the energies of the Andreev bound states as a function of [MATH] on Fig. [REF].', '0902.1111-1-39-1': 'We recover the fact that the high energy bound states increases in energy by breaking particle-hole symmetry, whereas the low energy bound state has a decreasing energy.', '0902.1111-1-39-2': 'In addition, Fig. [REF] shows that the dispersion of both ABS weakens for increasing hybridization.', '0902.1111-1-39-3': 'Indeed, the more the dot is hybridized with the leads, the less the Andreev bound state energy is sensitive to the bare values of the dot parameters.', '0902.1111-1-40-0': '## Superconducting correlations on the dot', '0902.1111-1-41-0': 'In order to further analyze the evolution of the states in the dot as a function of the parameters in the model ([REF]), we investigate now the superconducting correlations [MATH] on the dot.', '0902.1111-1-41-1': 'For the effective local Hamiltonian, these correlations are zero in the spin doublet phase.', '0902.1111-1-41-2': 'In the BCS-like phase, the correlations are maximal if the two states [MATH] and [MATH] are equivalent, i.e. at particle hole symmetry.', '0902.1111-1-41-3': 'If the dot level is far from [MATH], the wave function will be predominantly [MATH] (if [MATH] is positive) or [MATH] (if [MATH] is negative).', '0902.1111-1-41-4': 'This obviously kills the superconducting correlations.', '0902.1111-1-42-0': 'As the gap decreases from infinity, the (formerly) singly occupied state will start having a BCS-like admixture and therefore a non zero superconducting correlation.', '0902.1111-1-42-1': 'In contrast, the mixing will result in a decreased correlation in the BCS-like phase.', '0902.1111-1-42-2': 'Nevertheless, if the gap tends to zero, one would expect the correlations to vanish as well.', '0902.1111-1-42-3': 'This is indeed what Fig. [REF] shows.', '0902.1111-1-42-4': 'For large gaps, the dot is in the spin [MATH] phase; the correlations are small, but increase as the states mix.', '0902.1111-1-42-5': 'The transition to the BCS-like phase results in a discontinuous jump in the correlations, before they finally vanish for very small gaps.', '0902.1111-1-42-6': 'It can thus be concluded that the correlations should be normalized by the gap if one is interested in measuring only the mixing effect.', '0902.1111-1-42-7': 'Finally, the two different curves show how hybridization stabilizes the BCS-like state with respect to the spin doublet via the [MATH] transition.', '0902.1111-1-43-0': 'As the Coulomb interaction tries to prevent the formation of a Cooper pair wave function, the transition between the BCS-like phase and the spin doublet can also be achieved if the Coulomb interaction is tuned, as shown in Fig. [REF].', '0902.1111-1-43-1': 'The effect of the mixing is clearly visible by an increase of the correlation [MATH] (now normalized by the gap) while [MATH] is lowered.', '0902.1111-1-43-2': 'We also find that the correlations relative to the gap decrease for higher gaps, which is a simple saturation effect (the highest possible correlations are [MATH]).', '0902.1111-1-43-3': 'Furthermore, our results match reasonably the NRG data from Ref. [CITATION], the mismatch originating probably from the low value of the gap used in this calculation.', '0902.1111-1-44-0': 'Finally, we analyze how the correlations evolve outside particle hole symmetry.', '0902.1111-1-44-1': 'As mentioned above, one expects the correlations to decrease because the dot evolve from a superconducting atomic limit toward a usual atomic limit (i.e. from the states [MATH] toward the states [MATH] and [MATH]).', '0902.1111-1-44-2': 'On the other hand, there will be a transition from the spin doublet to the singlet phase and therefore a mixing effect.', '0902.1111-1-44-3': 'Fig. [REF] shows the competition between the mixing effect (that increases the correlations outside particle hole symmetry) and the evolution toward the normal atomic limit (that lowers the correlations) if [MATH] is increased.', '0902.1111-1-44-4': 'The effect of the Coulomb interaction is once more found to favor the single occupancy.', '0902.1111-1-45-0': '## Josephson current', '0902.1111-1-46-0': 'We now turn to the Josephson current through the quantum dot.', '0902.1111-1-46-1': 'The latter is given by [MATH] (where [MATH] is the free energy).', '0902.1111-1-46-2': 'At zero temperature, the free energy is the same than the level energies, so that the Josephson current can readily be obtained once the renormalized energy levels have been calculated.', '0902.1111-1-47-0': 'Nevertheless, our analytical approach only describes the effective local limit atomic states, and we can therefore only determine the current through the Andreev bound states.', '0902.1111-1-47-1': 'Yet, it is known that the Josephson current also contains a contribution of the continuum of states.', '0902.1111-1-47-2': '[CITATION] The latter can be of the same order and opposite sign as the bound state contribution.', '0902.1111-1-47-3': 'Furthermore, Bauer at al.[CITATION] have shown that the spectral weight of the bound states may vary importantly as a function of the different parameters (like the Coulomb interaction [MATH]), especially in the spin doublet phase.', '0902.1111-1-47-4': 'As we exclusively investigate the effective local limit states, we do not keep track of this effect as well.', '0902.1111-1-47-5': 'Therefore, the Josephson currents obtained in our approach will only provide a rather rough and qualitative idea of the actual total Josephson current.', '0902.1111-1-48-0': 'Fig. [REF] shows the Josephson current calculated as the phase derivative of the ground state energy [MATH], [MATH], for different values of [MATH].', '0902.1111-1-48-1': 'One notices two regimes: If the phase is close to [MATH], the system will be in the BCS-like state.', '0902.1111-1-48-2': 'As there is no magnetic moment in this phase, the ground state corresponds to a [MATH]-junction (i.e. phase difference [MATH]).', '0902.1111-1-48-3': 'If [MATH] increases, the energy of the BCS-like state increases (as can be understood in the effective local limit, where [MATH]).', '0902.1111-1-48-4': 'When the BCS-like state crosses with the spin doublet, the ground state changes and the dot becomes singly occupied.', '0902.1111-1-48-5': 'This magnetic moment leads to a discontinuous jump in the Josephson current and the formation of a [MATH]-junction.', '0902.1111-1-48-6': 'Again, we notice that the spin doublet is stabilized in the particle hole symmetric case.', '0902.1111-1-49-0': '# Conclusion', '0902.1111-1-50-0': 'In this section we summarize our main results.', '0902.1111-1-50-1': 'First, it has been shown how the Hamiltonian of a quantum dot coupled to superconducting leads can be mapped onto an effective local model if the superconducting gap [MATH] is much bigger than the characteristic energy scales of the dot.', '0902.1111-1-50-2': ""This limit can be quite generally regarded as a low frequency expansion of the Green's function of the dot rather than the limit [MATH] used in the literature."", '0902.1111-1-50-3': 'This enabled us to extend the effective local Hamiltonian to leads with a finite electronic bandwidth.', '0902.1111-1-51-0': 'We have then set up a perturbation theory around this local effective Hamiltonian and established self-consistent equations for the energy renormalizations of the Andreev bound states.', '0902.1111-1-51-1': 'We have derived those equations based on the fact that the latter correspond to transitions between different states of the local effective Hamiltonian.', '0902.1111-1-52-0': 'In a last section, we used our formalism to calculate physical quantities such as the Andreev bound state energies or superconducting correlations, and understood how these evolve as a function of gate voltage, hybridization, Coulomb interaction and superconducting gap amplitude.', '0902.1111-1-52-1': 'It has been shown that our simple approach agrees well with NRG data in a vast range of parameters, with the main limitation that the Kondo temperature is not quantitatively described near particle-hole symmetry.', '0902.1111-1-52-2': 'However, most experimentally interesting regimes should be described correctly by the simple equations we have derived.', '0902.1111-1-53-0': 'The simplicity and portability constitute the main advantages of our approach, if one is interested in the Andreev bound states only, compared to extended numerical simulations.', '0902.1111-1-53-1': 'As the perturbative description is analytical and based on atomic-like levels, it should in principle be able to describe more complex systems like multiple quantum dots or molecules with several orbitals coupled to superconducting environments, and be readily applicable to describe future spectrosopic measurements.', '0902.1111-1-54-0': 'We wish to acknowledge stimulating discussions with D. Feinberg and C. Winkelmann, and thank J. Bauer, A. Oguri and A. Hewson for providing us their NRG data.', '0902.1111-1-55-0': '*', '0902.1111-1-56-0': '# Derivation of the energy corrections', '0902.1111-1-57-0': ""The partition function is derived starting from the action's perturbation expansion in section [REF]."", '0902.1111-1-57-1': 'The actual calculations are performed in the operator formalism.', '0902.1111-1-57-2': 'It is very useful to note that the product of two fermionic (or bosonic) Greens functions [MATH] and [MATH] obeys [MATH] (as can be shown using Fourier transformation).', '0902.1111-1-57-3': ""The partition function's perturbation expansion is [EQUATION]"", '0902.1111-1-57-4': 'In the above equation, [MATH] is the Fourier transformed Nambu matrix element [MATH] and the subscript [MATH] indicates that the expectation values are statistical averages calculated in the effective local limit.', '0902.1111-1-57-5': ""The leads' Green's functions are: [EQUATION] with [MATH]."", '0902.1111-1-57-6': 'Furthermore, [MATH] and [MATH].', '0902.1111-1-58-0': ""As one cannot apply Wick's theorem because of the Coulomb interaction, the dot's Green's functions are calculated using Lehmann representation, which yields (for [MATH]) [EQUATION]"", '0902.1111-1-58-1': 'Using [MATH], the partition function becomes: [EQUATION]', '0902.1111-1-58-2': 'As [MATH], terms with an [MATH] are exponentially suppressed for [MATH] and can be omitted.'}","{'0902.1111-2-0-0': 'We formulate a general perturbative framework based on a superconducting atomic limit for the description of Andreev bound states (ABS) in interacting quantum dots connected to superconducting leads.', '0902.1111-2-0-1': 'A local effective Hamiltonian for dressed ABS, including both the atomic or molecular levels and the induced proximity effect on the dot is argued to be a natural starting point.', '0902.1111-2-0-2': 'Self-consistent expansions in single-particle tunneling events are shown to provide accurate results in regimes where the superconducting gap is larger than the Kondo temperature, as demonstrated by the comparison to recent Numerical Renomalization Group calculations.', '0902.1111-2-0-3': 'These analytical results may have bearings for interpreting Andreev spectroscopic measurements with STM on carbon nanotubes coupled to superconducting electrodes.', '0902.1111-2-1-0': '# Introduction', '0902.1111-2-2-0': ""When a quantum dot is connected to superconducting electrodes, the proximity effect drastically modifies the dot's electronic structure due to the local formation of Cooper pairs."", '0902.1111-2-2-1': 'The density of states on the dot thus exhibits a gap, so that the formation of discrete sub-gap states arises [CITATION].', '0902.1111-2-2-2': 'These Andreev bound states (ABS) play certainly an important role as they may contribute a large part of the spectral weight [CITATION] and carry most of the Josephson current.', '0902.1111-2-2-3': '[CITATION] A physical understanding of the ABS requires to characterize how these states are connected to the atomic or molecular levels of the uncoupled quantum dot, and to describe quantitatively their evolution as a function of several parameters, such as gate voltage, Coulomb interaction, tunnel couplings, and superconducting gap.', '0902.1111-2-2-4': 'Whereas the ABS have been observed in metal-superconductor hybrid structures [CITATION], no direct spectroscopy has so far been achieved in quantum dot systems.', '0902.1111-2-2-5': 'Andreev bound states come in pairs, one state above and one below the Fermi level, forming a two level system.', '0902.1111-2-2-6': 'Consequently, recent interest in the spectroscopy of the bound states [CITATION] has also been stimulated by proposals to use the latter as a qubit.', '0902.1111-2-2-7': '[CITATION]', '0902.1111-2-3-0': 'Experimentally, superconducting quantum dots can be realized with carbon nanotubes junctions.', '0902.1111-2-3-1': 'It has been shown that nanotubes and InAs quantum dots connected to superconducting electrodes can be tuned from a Coulomb blockade regime, to a Kondo regime,[CITATION] to a weakly interacting Fabry-Perot regime by changing local gate voltages.', '0902.1111-2-3-2': '[CITATION] The Josephson current at zero bias and multiple Andreev reflections at finite bias voltage have been measured in such devices.', '0902.1111-2-3-3': '[CITATION] The transition from a [MATH]-junction to a [MATH]-junction, namely a reversal in the sign of the Josephson current, [CITATION] has also been been observed when a magnetic moment forms on the dot.', '0902.1111-2-3-4': '[CITATION] As a possible application of superconducting junctions, nano-SQUID devices have also been fabricated.', '0902.1111-2-3-5': '[CITATION]', '0902.1111-2-4-0': 'An exact theoretical description of a quantum dot coupled to superconducting leads is only possible when the Coulomb interaction is fully neglected.', '0902.1111-2-4-1': 'Hence the interacting single dot system, as described by the Anderson model with superconducting electrodes, has been so far analyzed by treating the Coulomb interaction with various analytical schemes, such as the mean field theory, [CITATION] the perturbation expansion in the Coulomb interaction [CITATION] or in the tunnel coupling, [CITATION] the Non-Crossing Approximation (NCA), [CITATION], while numerically simulations based on the numerical renormalization group (NRG), [CITATION] or Quantum Monte Carlo [CITATION] have also been developped.', '0902.1111-2-5-0': 'None of the analytical approaches mentioned above is able to describe entirely the physics of a quantum dot coupled to superconducting leads.', '0902.1111-2-5-1': 'Whereas lowest order perturbation expansions in the tunnel coupling will hardly capture the proximity effect induced by the electrodes [CITATION], mean field and weak-interaction approaches will miss the Kondo effect.', '0902.1111-2-5-2': 'NRG calculations on the other hand can capture the physics of such a system over a wide range of parameters, but are numerically demanding and not easily portable to more complex molecular systems.', '0902.1111-2-5-3': 'More importantly, in the view of describing the ABS alone, none of these techniques does provide a simple physical picture.', '0902.1111-2-5-4': 'Henceforth we will develop in this paper a new perturbative approach based on an effective local Hamiltonian for dressed ABS, that extends the limit of large superconducting gap proposed previously [CITATION] that was used by many authors.', '0902.1111-2-5-5': '[CITATION] This approach will illuminate the nature of the ABS in interacting quantum dots, as well as provide a simple and accurate analytical framework in cases where the gap is not smaller than the Kondo temperature, that may be useful for interpreting future spectroscopic experiments.', '0902.1111-2-5-6': 'In addition, our formalism, which incorporates the atomic or molecular levels from the outset, should easily be extended to describe more complex systems, as for instance superconducting double quantum dots or molecules with more complicated orbital structure (see e.g. Refs. [CITATION]).', '0902.1111-2-6-0': 'We organize our paper as follows.', '0902.1111-2-6-1': 'In Sec. [REF], the system is mapped onto an effective local Hamiltonian, similarly to the widely used atomic limit, but including the proximity effects due to the superconducting leads.', '0902.1111-2-6-2': 'In Sec. [REF], the perturbation theory around this limit is set up and self-consistent equations for the ABS energy renormalizations are derived in order to extend the validity of the bare perturbative approach.', '0902.1111-2-6-3': 'Sec. [REF] illustrates how this expansion can describe ABS in superconducting quantum dots over a wide range of parameters, by a comparison to available NRG data.', '0902.1111-2-6-4': '[CITATION]', '0902.1111-2-7-0': '# Theoretical formulation', '0902.1111-2-8-0': '## Model', '0902.1111-2-9-0': 'We focus in this paper on a single-level quantum dot coupled to superconducting leads, which is relevant experimentally for molecular junctions with large single-electron level spacing.', '0902.1111-2-9-1': 'A simple Hamiltonian able to describe such a system is given by the superconducting Anderson model [EQUATION] where [EQUATION]', '0902.1111-2-9-2': 'In the above equations, [MATH] is the annihilation operator of an electron with spin [MATH] on the dot, [MATH] that of an electron with spin [MATH] and wave vector [MATH] in the lead [MATH], and [MATH].', '0902.1111-2-9-3': 'The leads are assumed to be described by standard s-wave BCS Hamiltonians [MATH] with superconducting gaps [MATH].', '0902.1111-2-9-4': 'The phase difference of the latter is noted [MATH].', '0902.1111-2-9-5': 'Furthermore, the leads are assumed to have flat and symmetric conduction bands, i.e. the kinetic energy [MATH] measured from the Fermi level ranges in [MATH] and the density of states is [MATH].', '0902.1111-2-9-6': 'We assume [MATH]-independent and symmetric tunneling amplitudes [MATH] between the dot and both superconducting leads.', '0902.1111-2-9-7': 'The dot has a level energy [MATH] and Coulomb interaction [MATH].', '0902.1111-2-9-8': 'Experimentally, the crucial characteristic energy scales, namely Coulomb interaction [MATH], total hybridization [MATH] and gap [MATH], are typically all of the same order of magnitude, [CITATION] providing a challenge for analytical methods.', '0902.1111-2-10-0': ""The physics of the quantum dot can be described via its Green's function [EQUATION] where the Nambu spinor [EQUATION] has been introduced."", '0902.1111-2-10-1': 'Because we will only be interested in stationary equilibrium physics, [MATH] shall be computed in the Matsubara frequency formalism.', '0902.1111-2-11-0': '## Effective local Hamiltonian', '0902.1111-2-12-0': 'As the above Hamiltonian has no exact solution, some approximations must be made.', '0902.1111-2-12-1': 'Among the physical ingredients we want to include in a non-perturbative way is the local pairing on the dot that is crucial for the evolution of the Andreev bound states.', '0902.1111-2-12-2': 'Furthermore, the Coulomb interaction shall be taken into account in an exact manner in order to describe the atomic states faithfully, and to highlight how these are adiabatically connected to the ABS.', '0902.1111-2-12-3': 'However, the usual development in weak tunnel coupling [MATH] around the atomic limit [CITATION] is not sufficient to describe the proximity effect at lowest order.', '0902.1111-2-12-4': 'Therefore, we shall consider in what follows an expansion around a superconducting atomic limit.', '0902.1111-2-13-0': 'Such simple solvable limiting case of the model ([REF]) is often referred to as the limit of large gap [MATH], and has been discussed previously [CITATION].', '0902.1111-2-13-1': 'Expansions for finite [MATH] have not however been discussed to our knowledge, and are the topic of this paper.', '0902.1111-2-13-2': 'We emphasize from the outset (see equation ([REF]) below), that the superconducting atomic limit as used normally in the literature corresponds to the limit [MATH] (i.e. infinite electronic bandwidth), taken before [MATH].', '0902.1111-2-13-3': 'The order of the two limits is crucial: if the limit [MATH] was to be taken first, the dot would be completely decoupled from the leads and the proximity effect would be lost, so that the limit of infinite gap would reduce to the usual atomic limit.', '0902.1111-2-13-4': 'As will be shown now, the superconducting atomic limit should rather be interpreted as a low frequency expansion, i.e. a limit where the gap is much larger than the characteristic frequencies of the dot.', '0902.1111-2-14-0': ""We start off by deriving the Green's function defined in Eq. [REF] using the equations of motion."", '0902.1111-2-14-1': 'Thereby, the Coulomb interaction [MATH] will at first be omitted for the sake of clarity.', '0902.1111-2-14-2': 'Note that in the end, [MATH] will simply give an extra contribution which adds to the effective Hamiltonian.', '0902.1111-2-14-3': 'Fourier transformation straightforwardly yields [EQUATION]', '0902.1111-2-14-4': ""In Eq. [REF], [MATH] is a fermionic Matsubara frequency and [MATH] the bare Green's function of electrons with a wave vector [MATH] in the lead [MATH]."", '0902.1111-2-14-5': 'Transforming the sum over wave vectors [MATH] into an integral over energies yields [EQUATION]', '0902.1111-2-14-6': ""In the limit [MATH], the Green's function [REF] becomes purely static and reduces to [EQUATION]"", '0902.1111-2-14-7': ""Note that the low frequency limit we consider here yields a Green's function that indeed depends on the finite bandwidth [MATH], and this shows that the limit [MATH] shall not be taken for the proximity effect to survive."", '0902.1111-2-14-8': 'In what follows, we will therefore keep both [MATH] and [MATH] finite.', '0902.1111-2-14-9': ""Plugging Eq. [REF] into the Green's function [MATH] leads to the same result as would have been obtained with the effective local Hamiltonian [EQUATION] where the local pairing amplitude induced by the leads on the dot reads [EQUATION] which explicitly depends on the ratio [MATH]."", '0902.1111-2-14-10': 'By an appropriate gauge transformation for the operators [MATH], it is always possible to choose [MATH], as shall be done from now on.', '0902.1111-2-14-11': 'The complete local effective Hamiltonian is obtained when the Coulomb interaction is taken into account again.', '0902.1111-2-14-12': 'Defining [MATH], the energy level of the dot is shifted such that the Hamiltonian clearly exhibits particle-hole symmetry for [MATH]: [EQUATION]', '0902.1111-2-14-13': 'The physical interpretation of this effective local Hamiltonian is simple.', '0902.1111-2-14-14': 'For finite gap, the quantum dot is coupled to both the Cooper pairs and the quasiparticles in the leads.', '0902.1111-2-14-15': 'The Cooper pairs, which lie at the Fermi level, are responsible for the proximity effect.', '0902.1111-2-14-16': 'The quasiparticles give rise to conduction electrons excitations with energies higher than the gap [MATH].', '0902.1111-2-14-17': 'In the limit [MATH], the quasiparticles are far in energy and the coupling between them and the dot vanishes, which greatly simplifies the physics and makes an exact solution possible.', '0902.1111-2-14-18': 'Yet, as the dot is still coupled to the Cooper pairs at the Fermi level, the proximity effect survives with a local pairing term proportional to the hybridization [MATH] between dot and leads.', '0902.1111-2-15-0': '## Spectrum of the effective local Hamiltonian', '0902.1111-2-16-0': 'As the Coulomb interaction simply yields an extra energy shift of [MATH] for both empty and doubly occupied dot, the eigenvectors and eigenvalues of the local effective Hamiltonian ([REF]) are readily obtained by a Bogoliubov transformation [CITATION], in perfect analogy with solution of the BCS Hamiltonian.', '0902.1111-2-16-1': '[MATH] has thus four eigenstates, the singly occupied spin [MATH] states [MATH] and [MATH] with energy [MATH], and two BCS-like states given by [EQUATION] where [MATH] denotes the empty dot and [MATH] the doubly occupied dot.', '0902.1111-2-16-2': 'The amplitudes [MATH] and [MATH] can always be chosen to be real with [MATH] and [MATH].', '0902.1111-2-16-3': 'The energies corresponding to these BCS-like states are [MATH].', '0902.1111-2-17-0': 'As [MATH] is always larger than [MATH], the effective local Hamiltonian has two possible ground states: the low energy BCS-like state [MATH] or the degenerate spin [MATH] doublet [MATH].', '0902.1111-2-17-1': 'In the [MATH] state, the energy is minimized for [MATH].', '0902.1111-2-17-2': 'Thus, the spin singlet phase corresponds to a [MATH]-junction (a result well known from the weak coupling limit [CITATION]).', '0902.1111-2-17-3': 'The transition between the singlet phase and the spin [MATH] doublet takes place at [MATH], and Fig. [REF] shows the corresponding phase diagram for variable [MATH], [MATH] and [MATH].', '0902.1111-2-17-4': 'The state adopted by the quantum dot in the large gap limit therefore results from a competition between the local pairing (induced by the proximity effect and characterized by the hybridization [MATH]) and the Coulomb interaction.', '0902.1111-2-18-0': '## Andreev bound states', '0902.1111-2-19-0': 'As outlined in the introduction, the coupling to superconducting leads induces a gap in the spectral function of the dot, inside which discrete Andreev bound states can form.', '0902.1111-2-19-1': 'The spectral function of the dot shows therefore sharp peaks, which could be measured by STM or microwave experiments as proposed recently [CITATION] (noise measurements have also been suggested in [CITATION]).', '0902.1111-2-19-2': 'These peaks indicate addition energies at which an electron may enter (or leave) the dot, and correspond therefore to transitions between states with [MATH] and [MATH] electrons.', '0902.1111-2-19-3': 'Hence, the ABS peaks in the spectral function may be interpreted as transitions between the superconducting atomic levels of the dot [MATH], possibly renormalized by single-particle tunneling events neglected in [MATH] (to be included in the next section).', '0902.1111-2-19-4': 'Furthermore, transitions from a spin [MATH] doublet to a spin singlet necessarily involve an electron exchange between the dot and the superconducting leads.', '0902.1111-2-19-5': 'As the states [MATH] and [MATH] correspond to the superposition of an empty and doubly occupied dot, this electron exchange and the final singlet states can be understood within the Andreev reflection picture.', '0902.1111-2-20-0': 'Putting everything together, our effective local Hamiltonian in Eq. [REF] describes the energies of the Andreev bound states as transition energies from the spin 1/2 doublet to the spin singlet states.', '0902.1111-2-20-1': '[CITATION] There are thus four Andreev bound states in the large gap limit for the model ([REF]), with energy [MATH] and [MATH] which read: [EQUATION]', '0902.1111-2-20-2': 'The [MATH]/[MATH] transition corresponds to the crossing of the [MATH] and [MATH] states, which occurs for [MATH].', '0902.1111-2-21-0': '# Perturbation expansion around the effective local Hamiltonian', '0902.1111-2-22-0': '## Perturbation theory', '0902.1111-2-23-0': 'The effective Hamiltonian is not sufficient to obtain satisfying results for all regimes of parameters.', '0902.1111-2-23-1': 'First, [MATH] only describes the [MATH]/[MATH]-junction transition due to the competition between a local moment state (stabilized by the Coulomb blockade) and a spin singlet (induced by the proximity effect).', '0902.1111-2-23-2': 'However, if the Coulomb interaction is strong (i.e. [MATH] below the Kondo temperature), the local moment can be screened by the Kondo effect, which will compete with the superconducting gap for the [MATH] transition, so that a typical scaling in the ratio of the Kondo temperature to the gap [MATH] will appear.', '0902.1111-2-23-3': 'Also, the Josephson current in the [MATH]-phase identically vanishes from [MATH], as the spin doublet does not disperse with the superconducting phase difference, a limitation of the large gap limit.', '0902.1111-2-23-4': 'On a more quantitative basis, the experimental gap [MATH] is usually of the order of a few kelvins, which is also the typical scale for both [MATH] and [MATH] in carbon nanotube quantum dot devices.', '0902.1111-2-24-0': ""In order to extend the description of the quantum dot's physics, energy corrections shall be calculated with a perturbation theory around the effective Hamiltonian [REF]."", '0902.1111-2-24-1': 'Once these corrections have been obtained, physical observables like the Josephson current may be computed via the free energy [MATH], with [MATH] the inverse temperature.', '0902.1111-2-24-2': 'Therefore, it is most convenient to work in an action based description, which directly yields the partition function [MATH].', '0902.1111-2-24-3': 'Following Ref. [CITATION], we first integrate over the fermions in the leads.', '0902.1111-2-24-4': 'Omitting the resulting irrelevant constant, the partition function reads [EQUATION] where we have introduced the Grassmann Nambu spinors at Matsubara frequency [MATH], [EQUATION] denoting the Grassmann fields associated with electrons in the dot by [MATH] and [MATH].', '0902.1111-2-25-0': 'The perturbation consists of the terms in Eq. [REF] that are not contained in the action [MATH] corresponding to the effective local Hamiltonian.', '0902.1111-2-25-1': 'A simple identification yields [EQUATION]', '0902.1111-2-25-2': 'Note that [MATH] contains the local pairing term derived in section [REF].', '0902.1111-2-25-3': 'The proximity effect is thus treated non-perturbatively (just like the Coulomb interaction), which is the crucial ingredient of our analytic approach.', '0902.1111-2-25-4': 'The perturbation [MATH] simply corresponds to the tunnel coupling between the dot and the electrodes other than the lowest order proximity effect.', '0902.1111-2-26-0': 'The actual corrections are calculated by expanding the partition function to the first order in [MATH] according to [EQUATION] which we then identify with [EQUATION] where the renormalized superconducting atomic levels [MATH] and [MATH] are obtained from: [EQUATION]', '0902.1111-2-26-1': ""Because the Coulomb interaction is taken into account, Wick's theorem cannot be used to calculate [MATH]."", '0902.1111-2-26-2': 'Instead, expectation values are calculated using Lehmann representation.', '0902.1111-2-26-3': 'Explicit calculations may be found in the appendix.', '0902.1111-2-26-4': 'In the zero temperature limit [MATH], the energy corrections are [EQUATION] with the quasiparticle energy [MATH].', '0902.1111-2-27-0': '## Self-consistent renormalization of the energy', '0902.1111-2-28-0': 'Eqs. [REF]-[REF] yield the first corrections to the energy levels, so that the bound states energies [MATH] and [MATH] are simply shifted by [MATH] and [MATH].', '0902.1111-2-28-1': 'Obviously, these expressions are logarithmically divergent when the bound states energies [MATH] and [MATH] approach the gap edge, and are therefore only valid as long as e.g. [MATH].', '0902.1111-2-28-2': 'In the limit of large gap [MATH], these corrections to [MATH] are thus of the order [MATH], so that the small dimensionless parameter is indeed [MATH].', '0902.1111-2-28-3': 'However, this peculiar logarithmic dependence of the ABS energy renormalization shows that doing a straightforward [MATH] expansion around the effective local Hamiltonian will be rapidly uncontrolled, and will have a hard time reproducing the logarithmic singularities at [MATH] close to [MATH].', '0902.1111-2-28-4': 'For this reason, and also because the large gap limit becomes trivial for a finite electronic bandwidth, as discussed in section [REF], it was indeed more appropriate to single out in the total action all terms left over with respect to the local superconducting effective Hamiltonian, see equation ([REF]), and do perturbation theory around these.', '0902.1111-2-29-0': 'Because our lowest-order expansion obviously still breaks down when the gap becomes comparable to the bound state energy, one would naturally seek to resum the leadind logarithmic divergences in equations ([REF])-([REF]).', '0902.1111-2-29-1': 'This can be achieved using a self-consistency condition inspired by Brillouin-Wigner perturbation theory, [CITATION] which allows to extend greatly the regime of validity of the perturbative scheme.', '0902.1111-2-29-2': 'The resulting self-consistent equations that we obtain are: [EQUATION] and [EQUATION] with [MATH], and [MATH] have been defined in Eqs. ([REF])-([REF]), with [MATH], [MATH].', '0902.1111-2-29-3': 'Note that terms like [MATH] have no self-consistency because there are no associated divergences.', '0902.1111-2-29-4': 'Eq. [REF] and [REF] clearly now hold as long as the renormalized energies [MATH] and [MATH] are not too close to the gap edge, [MATH] respectively.', '0902.1111-2-30-0': '# Results', '0902.1111-2-31-0': '## Phase diagram', '0902.1111-2-32-0': 'We start by discussing the [MATH] transition line, by comparison to the numerical renormalization group (NRG) data by Bauer et al.[CITATION].', '0902.1111-2-32-1': 'Fig. [REF] shows the extension to smaller gaps [MATH] values of the phase diagram obtained with unrenormalized local superconducting states for infinite gap (Fig. [REF]).', '0902.1111-2-32-2': 'Even though our perturbative approach is fairly simple, the results reproduce nicely the NRG data of Refs. [CITATION] and [CITATION].', '0902.1111-2-32-3': 'The analytically obtained phase diagram is indeed identical to the NRG data for [MATH].', '0902.1111-2-32-4': 'For smaller [MATH], the Kondo effect sets in, but the transition lines remain quantitatively correct for [MATH] near [MATH], with increasing deviations from the NRG calculations close to the particle-hole symmetric point [MATH] at large Coulomb interaction U.', '0902.1111-2-32-5': 'In this regime, the [MATH]-phase possesses a Kondo singlet ground state.', '0902.1111-2-32-6': 'As the leads are superconductors, the formation of a Kondo resonance involves the breaking of Cooper pairs.', '0902.1111-2-32-7': 'Therefore, the transition is now due to the competition between [MATH] and the superconducting gap [MATH], and should occur at [MATH].', '0902.1111-2-33-0': 'Fig. [REF] shows a plot of the transition line for [MATH] as obtained with Eq. [REF] (solid curve).', '0902.1111-2-33-1': 'The vertical, dotted line depicts the asymptote in the effective local limit.', '0902.1111-2-33-2': 'The symbols again correspond to NRG data.[', '0902.1111-2-33-3': '[CITATION] The Kondo temperature is given by [MATH] (see for example Ref. NRG_spectral_Bauer).', '0902.1111-2-33-4': 'The inset shows on a log-log scale that our approach captures an exponential decay of the transition line with the Coulomb interaction.', '0902.1111-2-33-5': 'Nonetheless, the suppression of the BCS-like phase appears quantitatively stronger than expected: a factor 4 instead of 8 is found in the exponential factor of [MATH].', '0902.1111-2-33-6': 'The reason for this is that the vertex renormalizations have not been taken into account, as discussed in the context of U-NCA [CITATION].', '0902.1111-2-33-7': 'Far away from the particle-hole symmetric limit, our results for the Kondo temperature reproduce the lowest-order scaling theory for the infinite-[MATH] Anderson model [CITATION], and are in relatively good agreement with NRG data for all [MATH] values.', '0902.1111-2-34-0': '## Energy renormalizations at particle hole symmetry ([MATH])', '0902.1111-2-35-0': 'While Fig. [REF] only indicates the transition line between the spin [MATH] doublet and the lowest BCS spin singlet, it is also instructive to look at the actual renormalization of the energy levels while varying the gap [MATH] from large values to smaller ones beyond the critical point.', '0902.1111-2-35-1': 'Fig. [REF] indicates the renormalized energies of the two Andreev bound states (i.e. the difference between the spin [MATH] doublet and the two spin singlets energies) for different hybridizations [MATH].', '0902.1111-2-35-2': 'We note that our results are in quantitative agreement with the NRG calculations of Yoshioka and Ohashi.', '0902.1111-2-35-3': '[CITATION] Several regions need to be distinguished.', '0902.1111-2-35-4': 'If the gap [MATH] is much larger than the bandwidth [MATH], all curves collapse at the value [MATH] (left hand side of Fig. [REF]), since there is no hybridization with both quasiparticles and Cooper pairs anymore, and one recovers the bare atomic levels.', '0902.1111-2-35-5': 'When the gap starts to decrease, the proximity effect simply splits the two Andreev bound states according to equations ([REF])-([REF]).', '0902.1111-2-35-6': 'When the gap becomes of the same order as the typical energy scales of the dot [MATH] and [MATH], the superconducting atomic levels start to mix with the electrodes, so that the energies renormalize in a non trivial way.', '0902.1111-2-35-7': 'One can see that the transition involving the highest BCS states ends up touching the gap edge for [MATH], so that half of the ABS are absorbed into the continuum above [MATH], as can be seen in Fig. [REF].', '0902.1111-2-35-8': 'The lowest BCS state follows however a downward renormalization, until the Fermi level is crossed and the ground state becomes the [MATH]-state.', '0902.1111-2-35-9': 'The difference in behavior between the lowest and highest bound states (the former being never allowed to leave the superconducting gap) can be tracked into equations ([REF])-([REF]), where level repulsion effects from the gap edge occur for the low energy level [MATH] but are canceled for the high energy level [MATH], which is hence allowed to penetrate into the continuum.', '0902.1111-2-35-10': 'These considerations unveil how the ABS may be adiabatically connected to the atomic or molecular levels in a complicated fashion.', '0902.1111-2-36-0': 'Again, our simple analytic approach reproduces the NRG results [CITATION] over a vast regime of parameters.', '0902.1111-2-36-1': 'Yet, some deviations are observed in the Kondo regime: we find (for the highest hybridization [MATH]) that the high energy BCS-like state is not absorbed anymore into the continuum of states - an artifact of the limits of our perturbative approach.', '0902.1111-2-36-2': 'Notice also that the energy corrections are too important if the gap becomes very small, an effect actually due to our underestimation of the Kondo temperature at particle-symmetry, as discussed previously.', '0902.1111-2-36-3': 'Finally Fig. [REF] shows that, in the limit of vanishing gap, our approach is only valid as long as [MATH] (as has been mentioned in section [REF]), because the lowest bound state artificially escapes from the gap.', '0902.1111-2-36-4': 'The expected saturation of [MATH] near [MATH] can be restored by adding a further self-consistency for terms such as [MATH] in ([REF]) (not shown here).', '0902.1111-2-37-0': '## Energy renormalizations outside particle hole symmetry ([MATH])', '0902.1111-2-38-0': 'From an experimental point of view, the position of the energy level of the quantum dot is the most controllable parameter of the system (by a simple gate voltage).', '0902.1111-2-38-1': 'Therefore, it is important to analyze the evolution of the Andreev bound states for different values of [MATH].', '0902.1111-2-39-0': 'Fig. [REF] illustrates how the energies of the bound states scale with [MATH] for [MATH] and can be favorably compared to the NRG data by Yoshioka and Ohashi.', '0902.1111-2-39-1': '[CITATION] The more particle-hole symmetry is broken, the more the low energy bound state moves away from the gap edge, ensuring even better convergence of our expansion for a given value of [MATH].', '0902.1111-2-39-2': 'This can be understood given that this bound state corresponds to the transition between [MATH] and the spin [MATH] doublet: outside particle-hole symmetry, the dot either seeks to be as empty as possible (for [MATH]) or as occupied as possible (for [MATH]).', '0902.1111-2-39-3': 'Thus, a BCS-like wave function will be favored.', '0902.1111-2-39-4': 'As a consequence, the Kondo effect (that necessitates a singly occupied dot) is less favored.', '0902.1111-2-39-5': 'This corresponds to a regime where our approximation scheme works at best.', '0902.1111-2-40-0': 'Further understanding can be gained by looking at the energies of the Andreev bound states as a function of [MATH] on Fig. [REF].', '0902.1111-2-40-1': 'We recover the fact that the high energy bound states increases in energy by breaking particle-hole symmetry, whereas the low energy bound state has a decreasing energy.', '0902.1111-2-40-2': 'In addition, Fig. [REF] shows that the dispersion of both ABS weakens for increasing hybridization.', '0902.1111-2-40-3': 'Indeed, the more the dot is hybridized with the leads, the less the Andreev bound state energy is sensitive to the bare values of the dot parameters.', '0902.1111-2-41-0': '## Superconducting correlations on the dot', '0902.1111-2-42-0': 'In order to further analyze the evolution of the states in the dot as a function of the parameters in the model ([REF]), we investigate now the superconducting correlations [MATH] on the dot.', '0902.1111-2-42-1': 'For the effective local Hamiltonian, these correlations are zero in the spin doublet phase.', '0902.1111-2-42-2': 'In the BCS-like phase, the correlations are maximal if the two states [MATH] and [MATH] are equivalent, i.e. at particle hole symmetry.', '0902.1111-2-42-3': 'If the dot level is far from [MATH], the wave function will be predominantly [MATH] (if [MATH] is positive) or [MATH] (if [MATH] is negative).', '0902.1111-2-42-4': 'This obviously kills the superconducting correlations.', '0902.1111-2-43-0': 'As the gap decreases from infinity, the (formerly) singly occupied state will start having a BCS-like admixture and therefore a non zero superconducting correlation.', '0902.1111-2-43-1': 'In contrast, the mixing will result in a decreased correlation in the BCS-like phase.', '0902.1111-2-43-2': 'Nevertheless, if the gap tends to zero, one would expect the correlations to vanish as well.', '0902.1111-2-43-3': 'This is indeed what Fig. [REF] shows.', '0902.1111-2-43-4': 'For large gaps, the dot is in the spin [MATH] phase; the correlations are small, but increase as the states mix.', '0902.1111-2-43-5': 'The transition to the BCS-like phase results in a discontinuous jump in the correlations, before they finally vanish for very small gaps.', '0902.1111-2-43-6': 'It can thus be concluded that the correlations should be normalized by the gap if one is interested in measuring only the mixing effect.', '0902.1111-2-43-7': 'Finally, the two different curves show how hybridization stabilizes the BCS-like state with respect to the spin doublet via the [MATH] transition.', '0902.1111-2-44-0': 'As the Coulomb interaction tries to prevent the formation of a Cooper pair wave function, the transition between the BCS-like phase and the spin doublet can also be achieved if the Coulomb interaction is tuned, as shown in Fig. [REF].', '0902.1111-2-44-1': 'The effect of the mixing is clearly visible by an increase of the correlation [MATH] (now normalized by the gap) while [MATH] is lowered.', '0902.1111-2-44-2': 'We also find that the correlations relative to the gap decrease for higher gaps, which is a simple saturation effect (the highest possible correlations are [MATH]).', '0902.1111-2-44-3': 'Furthermore, our results match reasonably the NRG data from Ref. [CITATION], the mismatch originating probably from the low value of the gap used in this calculation.', '0902.1111-2-45-0': 'Finally, we analyze how the correlations evolve outside particle hole symmetry.', '0902.1111-2-45-1': 'As mentioned above, one expects the correlations to decrease because the dot evolve from a superconducting atomic limit toward a usual atomic limit (i.e. from the states [MATH] toward the states [MATH] and [MATH]).', '0902.1111-2-45-2': 'On the other hand, there will be a transition from the spin doublet to the singlet phase and therefore a mixing effect.', '0902.1111-2-45-3': 'Fig. [REF] shows the competition between the mixing effect (that increases the correlations outside particle hole symmetry) and the evolution toward the normal atomic limit (that lowers the correlations) if [MATH] is increased.', '0902.1111-2-45-4': 'The effect of the Coulomb interaction is once more found to favor the single occupancy.', '0902.1111-2-46-0': '## Josephson current', '0902.1111-2-47-0': 'We now turn to the Josephson current through the quantum dot.', '0902.1111-2-47-1': 'The latter is given by [MATH] (where [MATH] is the free energy).', '0902.1111-2-47-2': 'At zero temperature, the free energy is the same than the level energies, so that the Josephson current can readily be obtained once the renormalized energy levels have been calculated.', '0902.1111-2-48-0': 'Nevertheless, our analytical approach only describes the effective local limit atomic states, and we can therefore only determine the current through the Andreev bound states.', '0902.1111-2-48-1': 'Yet, it is known that the Josephson current also contains a contribution of the continuum of states.', '0902.1111-2-48-2': '[CITATION] The latter can be of the same order and opposite sign as the bound state contribution.', '0902.1111-2-48-3': 'Furthermore, Bauer at al.[CITATION] have shown that the spectral weight of the bound states may vary importantly as a function of the different parameters (like the Coulomb interaction [MATH]), especially in the spin doublet phase.', '0902.1111-2-48-4': 'As we exclusively investigate the effective local limit states, we do not keep track of this effect as well.', '0902.1111-2-48-5': 'Therefore, the Josephson currents obtained in our approach will only provide a rather rough and qualitative idea of the actual total Josephson current.', '0902.1111-2-49-0': 'Fig. [REF] shows the Josephson current calculated as the phase derivative of the ground state energy [MATH], [MATH], for different values of [MATH].', '0902.1111-2-49-1': 'One notices two regimes: If the phase is close to [MATH], the system will be in the BCS-like state.', '0902.1111-2-49-2': 'As there is no magnetic moment in this phase, the ground state corresponds to a [MATH]-junction (i.e. phase difference [MATH]).', '0902.1111-2-49-3': 'If [MATH] increases, the energy of the BCS-like state increases (as can be understood in the effective local limit, where [MATH]).', '0902.1111-2-49-4': 'When the BCS-like state crosses with the spin doublet, the ground state changes and the dot becomes singly occupied.', '0902.1111-2-49-5': 'This magnetic moment leads to a discontinuous jump in the Josephson current and the formation of a [MATH]-junction.', '0902.1111-2-49-6': 'Again, we notice that the spin doublet is stabilized in the particle hole symmetric case.', '0902.1111-2-50-0': '# Conclusion', '0902.1111-2-51-0': 'In this section we summarize our main results.', '0902.1111-2-51-1': 'First, it has been shown how the Hamiltonian of a quantum dot coupled to superconducting leads can be mapped onto an effective local model if the superconducting gap [MATH] is much bigger than the characteristic energy scales of the dot.', '0902.1111-2-51-2': ""This limit can be quite generally regarded as a low frequency expansion of the Green's function of the dot rather than the limit [MATH] used in the literature."", '0902.1111-2-51-3': 'This enabled us to extend the effective local Hamiltonian to leads with a finite electronic bandwidth.', '0902.1111-2-52-0': 'We have then set up a perturbation theory around this local effective Hamiltonian and established self-consistent equations for the energy renormalizations of the Andreev bound states.', '0902.1111-2-52-1': 'We have derived those equations based on the fact that the latter correspond to transitions between different states of the local effective Hamiltonian.', '0902.1111-2-53-0': 'In a last section, we used our formalism to calculate physical quantities such as the Andreev bound state energies or superconducting correlations, and understood how these evolve as a function of gate voltage, hybridization, Coulomb interaction and superconducting gap amplitude.', '0902.1111-2-53-1': 'It has been shown that our simple approach agrees well with NRG data in a vast range of parameters, with the main limitation that the Kondo temperature is not quantitatively described near particle-hole symmetry.', '0902.1111-2-53-2': 'However, most experimentally interesting regimes should be described correctly by the simple equations we have derived.', '0902.1111-2-54-0': 'The simplicity and portability constitute the main advantages of our approach, if one is interested in the Andreev bound states only, compared to extended numerical simulations.', '0902.1111-2-54-1': 'As the perturbative description is analytical and based on atomic-like levels, it should in principle be able to describe more complex systems like multiple quantum dots or molecules with several orbitals coupled to superconducting environments, and be readily applicable to describe future spectrosopic measurements.', '0902.1111-2-55-0': 'We wish to acknowledge stimulating discussions with D. Feinberg and C. Winkelmann, and thank J. Bauer, A. Oguri and A. Hewson for providing us their NRG data.', '0902.1111-2-56-0': '*', '0902.1111-2-57-0': '# Derivation of the energy corrections', '0902.1111-2-58-0': ""The partition function is derived starting from the action's perturbation expansion in section [REF]."", '0902.1111-2-58-1': 'The actual calculations are performed in the operator formalism.', '0902.1111-2-58-2': 'It is very useful to note that the product of two fermionic (or bosonic) Greens functions [MATH] and [MATH] obeys [MATH] (as can be shown using Fourier transformation).', '0902.1111-2-58-3': ""The partition function's perturbation expansion is [EQUATION]"", '0902.1111-2-58-4': 'In the above equation, [MATH] is the Fourier transformed Nambu matrix element [MATH] and the subscript [MATH] indicates that the expectation values are statistical averages calculated in the effective local limit.', '0902.1111-2-58-5': ""The leads' Green's functions are: [EQUATION] with [MATH]."", '0902.1111-2-58-6': 'Furthermore, [MATH] and [MATH].', '0902.1111-2-59-0': ""As one cannot apply Wick's theorem because of the Coulomb interaction, the dot's Green's functions are calculated using Lehmann representation, which yields (for [MATH]) [EQUATION]"", '0902.1111-2-59-1': 'Using [MATH], the partition function becomes: [EQUATION]', '0902.1111-2-59-2': 'As [MATH], terms with an [MATH] are exponentially suppressed for [MATH] and can be omitted.'}","[['0902.1111-1-20-0', '0902.1111-2-20-0'], ['0902.1111-1-20-1', '0902.1111-2-20-1'], ['0902.1111-1-20-2', '0902.1111-2-20-2'], ['0902.1111-1-51-0', '0902.1111-2-52-0'], ['0902.1111-1-51-1', '0902.1111-2-52-1'], ['0902.1111-1-4-0', '0902.1111-2-4-0'], ['0902.1111-1-16-0', '0902.1111-2-16-0'], ['0902.1111-1-16-1', '0902.1111-2-16-1'], ['0902.1111-1-16-2', '0902.1111-2-16-2'], ['0902.1111-1-16-3', '0902.1111-2-16-3'], ['0902.1111-1-41-0', '0902.1111-2-42-0'], ['0902.1111-1-41-1', '0902.1111-2-42-1'], ['0902.1111-1-41-2', '0902.1111-2-42-2'], ['0902.1111-1-41-3', '0902.1111-2-42-3'], ['0902.1111-1-41-4', '0902.1111-2-42-4'], ['0902.1111-1-23-0', '0902.1111-2-23-0'], ['0902.1111-1-23-1', '0902.1111-2-23-1'], ['0902.1111-1-23-2', '0902.1111-2-23-2'], ['0902.1111-1-23-3', '0902.1111-2-23-3'], ['0902.1111-1-23-4', '0902.1111-2-23-4'], ['0902.1111-1-5-0', '0902.1111-2-5-0'], ['0902.1111-1-5-2', '0902.1111-2-5-2'], ['0902.1111-1-5-3', '0902.1111-2-5-3'], ['0902.1111-1-5-5', '0902.1111-2-5-5'], ['0902.1111-1-5-6', '0902.1111-2-5-6'], ['0902.1111-1-3-0', '0902.1111-2-3-0'], ['0902.1111-1-3-1', '0902.1111-2-3-1'], ['0902.1111-1-3-2', '0902.1111-2-3-2'], ['0902.1111-1-3-3', '0902.1111-2-3-3'], ['0902.1111-1-3-4', '0902.1111-2-3-4'], ['0902.1111-1-54-0', '0902.1111-2-55-0'], ['0902.1111-1-6-0', '0902.1111-2-6-0'], ['0902.1111-1-6-1', '0902.1111-2-6-1'], ['0902.1111-1-6-2', '0902.1111-2-6-2'], ['0902.1111-1-6-3', '0902.1111-2-6-3'], ['0902.1111-1-47-0', '0902.1111-2-48-0'], ['0902.1111-1-47-1', '0902.1111-2-48-1'], ['0902.1111-1-47-2', '0902.1111-2-48-2'], ['0902.1111-1-47-3', '0902.1111-2-48-3'], ['0902.1111-1-47-4', '0902.1111-2-48-4'], ['0902.1111-1-47-5', '0902.1111-2-48-5'], ['0902.1111-1-10-0', '0902.1111-2-10-0'], ['0902.1111-1-10-1', '0902.1111-2-10-1'], ['0902.1111-1-19-0', '0902.1111-2-19-0'], ['0902.1111-1-19-3', '0902.1111-2-19-3'], ['0902.1111-1-19-4', '0902.1111-2-19-4'], ['0902.1111-1-19-5', '0902.1111-2-19-5'], ['0902.1111-1-26-0', '0902.1111-2-26-0'], ['0902.1111-1-26-1', '0902.1111-2-26-1'], ['0902.1111-1-26-2', '0902.1111-2-26-2'], ['0902.1111-1-26-3', '0902.1111-2-26-3'], ['0902.1111-1-26-4', '0902.1111-2-26-4'], ['0902.1111-1-32-0', '0902.1111-2-33-0'], ['0902.1111-1-32-1', '0902.1111-2-33-1'], ['0902.1111-1-32-2', '0902.1111-2-33-2'], ['0902.1111-1-32-3', '0902.1111-2-33-3'], ['0902.1111-1-32-4', '0902.1111-2-33-4'], ['0902.1111-1-32-5', '0902.1111-2-33-5'], ['0902.1111-1-32-6', '0902.1111-2-33-6'], ['0902.1111-1-32-7', '0902.1111-2-33-7'], ['0902.1111-1-57-0', '0902.1111-2-58-0'], ['0902.1111-1-57-1', '0902.1111-2-58-1'], ['0902.1111-1-57-2', '0902.1111-2-58-2'], ['0902.1111-1-57-3', '0902.1111-2-58-3'], ['0902.1111-1-57-4', '0902.1111-2-58-4'], ['0902.1111-1-57-5', '0902.1111-2-58-5'], ['0902.1111-1-57-6', '0902.1111-2-58-6'], ['0902.1111-1-37-0', '0902.1111-2-38-0'], ['0902.1111-1-37-1', '0902.1111-2-38-1'], ['0902.1111-1-34-0', '0902.1111-2-35-0'], ['0902.1111-1-34-1', '0902.1111-2-35-1'], ['0902.1111-1-34-2', '0902.1111-2-35-2'], ['0902.1111-1-34-3', '0902.1111-2-35-3'], ['0902.1111-1-34-4', '0902.1111-2-35-4'], ['0902.1111-1-34-5', '0902.1111-2-35-5'], ['0902.1111-1-34-6', '0902.1111-2-35-6'], ['0902.1111-1-34-7', '0902.1111-2-35-7'], ['0902.1111-1-34-8', '0902.1111-2-35-8'], ['0902.1111-1-34-9', '0902.1111-2-35-9'], ['0902.1111-1-34-10', '0902.1111-2-35-10'], ['0902.1111-1-46-0', '0902.1111-2-47-0'], ['0902.1111-1-46-1', '0902.1111-2-47-1'], ['0902.1111-1-46-2', '0902.1111-2-47-2'], ['0902.1111-1-48-0', '0902.1111-2-49-0'], ['0902.1111-1-48-1', '0902.1111-2-49-1'], ['0902.1111-1-48-2', '0902.1111-2-49-2'], ['0902.1111-1-48-3', '0902.1111-2-49-3'], ['0902.1111-1-48-4', '0902.1111-2-49-4'], ['0902.1111-1-48-5', '0902.1111-2-49-5'], ['0902.1111-1-48-6', '0902.1111-2-49-6'], ['0902.1111-1-0-0', '0902.1111-2-0-0'], ['0902.1111-1-0-1', '0902.1111-2-0-1'], ['0902.1111-1-0-2', '0902.1111-2-0-2'], ['0902.1111-1-0-3', '0902.1111-2-0-3'], ['0902.1111-1-17-0', '0902.1111-2-17-0'], ['0902.1111-1-17-1', '0902.1111-2-17-1'], ['0902.1111-1-17-2', '0902.1111-2-17-2'], ['0902.1111-1-17-3', '0902.1111-2-17-3'], ['0902.1111-1-17-4', '0902.1111-2-17-4'], ['0902.1111-1-14-0', '0902.1111-2-14-0'], ['0902.1111-1-14-1', '0902.1111-2-14-1'], ['0902.1111-1-14-2', '0902.1111-2-14-2'], ['0902.1111-1-14-3', '0902.1111-2-14-3'], ['0902.1111-1-14-4', '0902.1111-2-14-4'], ['0902.1111-1-14-5', '0902.1111-2-14-5'], ['0902.1111-1-14-6', '0902.1111-2-14-6'], ['0902.1111-1-14-7', '0902.1111-2-14-7'], ['0902.1111-1-14-8', '0902.1111-2-14-8'], ['0902.1111-1-14-9', '0902.1111-2-14-9'], ['0902.1111-1-14-10', '0902.1111-2-14-10'], ['0902.1111-1-14-11', '0902.1111-2-14-11'], ['0902.1111-1-14-12', '0902.1111-2-14-12'], ['0902.1111-1-14-13', '0902.1111-2-14-13'], ['0902.1111-1-14-14', '0902.1111-2-14-14'], ['0902.1111-1-14-15', '0902.1111-2-14-15'], ['0902.1111-1-14-16', '0902.1111-2-14-16'], ['0902.1111-1-14-17', '0902.1111-2-14-17'], ['0902.1111-1-14-18', '0902.1111-2-14-18'], ['0902.1111-1-58-0', '0902.1111-2-59-0'], ['0902.1111-1-58-1', '0902.1111-2-59-1'], ['0902.1111-1-58-2', '0902.1111-2-59-2'], ['0902.1111-1-38-0', '0902.1111-2-39-0'], ['0902.1111-1-38-1', '0902.1111-2-39-1'], ['0902.1111-1-38-2', '0902.1111-2-39-2'], ['0902.1111-1-38-3', '0902.1111-2-39-3'], ['0902.1111-1-38-4', '0902.1111-2-39-4'], ['0902.1111-1-38-5', '0902.1111-2-39-5'], ['0902.1111-1-42-0', '0902.1111-2-43-0'], ['0902.1111-1-42-1', '0902.1111-2-43-1'], ['0902.1111-1-42-2', '0902.1111-2-43-2'], ['0902.1111-1-42-3', '0902.1111-2-43-3'], ['0902.1111-1-42-4', '0902.1111-2-43-4'], ['0902.1111-1-42-5', '0902.1111-2-43-5'], ['0902.1111-1-42-6', '0902.1111-2-43-6'], ['0902.1111-1-42-7', '0902.1111-2-43-7'], ['0902.1111-1-25-0', '0902.1111-2-25-0'], ['0902.1111-1-25-1', '0902.1111-2-25-1'], ['0902.1111-1-25-2', '0902.1111-2-25-2'], ['0902.1111-1-25-3', '0902.1111-2-25-3'], ['0902.1111-1-25-4', '0902.1111-2-25-4'], ['0902.1111-1-52-0', '0902.1111-2-53-0'], ['0902.1111-1-52-1', '0902.1111-2-53-1'], ['0902.1111-1-52-2', '0902.1111-2-53-2'], ['0902.1111-1-43-0', '0902.1111-2-44-0'], ['0902.1111-1-43-1', '0902.1111-2-44-1'], ['0902.1111-1-43-2', '0902.1111-2-44-2'], ['0902.1111-1-43-3', '0902.1111-2-44-3'], ['0902.1111-1-50-0', '0902.1111-2-51-0'], ['0902.1111-1-50-1', '0902.1111-2-51-1'], ['0902.1111-1-50-2', '0902.1111-2-51-2'], ['0902.1111-1-50-3', '0902.1111-2-51-3'], ['0902.1111-1-2-0', '0902.1111-2-2-0'], ['0902.1111-1-2-1', '0902.1111-2-2-1'], ['0902.1111-1-2-2', '0902.1111-2-2-2'], ['0902.1111-1-2-3', '0902.1111-2-2-3'], ['0902.1111-1-2-4', '0902.1111-2-2-4'], ['0902.1111-1-2-5', '0902.1111-2-2-5'], ['0902.1111-1-2-6', '0902.1111-2-2-6'], ['0902.1111-1-24-0', '0902.1111-2-24-0'], ['0902.1111-1-24-1', '0902.1111-2-24-1'], ['0902.1111-1-24-2', '0902.1111-2-24-2'], ['0902.1111-1-24-3', '0902.1111-2-24-3'], ['0902.1111-1-24-4', '0902.1111-2-24-4'], ['0902.1111-1-31-0', '0902.1111-2-32-0'], ['0902.1111-1-31-1', '0902.1111-2-32-1'], ['0902.1111-1-31-2', '0902.1111-2-32-2'], ['0902.1111-1-31-3', '0902.1111-2-32-3'], ['0902.1111-1-31-4', '0902.1111-2-32-4'], ['0902.1111-1-31-5', '0902.1111-2-32-5'], ['0902.1111-1-31-6', '0902.1111-2-32-6'], ['0902.1111-1-31-7', '0902.1111-2-32-7'], ['0902.1111-1-53-0', '0902.1111-2-54-0'], ['0902.1111-1-53-1', '0902.1111-2-54-1'], ['0902.1111-1-35-0', '0902.1111-2-36-0'], ['0902.1111-1-35-1', '0902.1111-2-36-1'], ['0902.1111-1-35-2', '0902.1111-2-36-2'], ['0902.1111-1-35-3', '0902.1111-2-36-3'], ['0902.1111-1-35-4', '0902.1111-2-36-4'], ['0902.1111-1-9-0', '0902.1111-2-9-0'], ['0902.1111-1-9-1', '0902.1111-2-9-1'], ['0902.1111-1-9-2', 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'0902.1111-2-29-1'], ['0902.1111-1-28-3', '0902.1111-2-29-2']]",[],"['0902.1111-1-2-7', '0902.1111-1-3-5', '0902.1111-1-4-2', '0902.1111-1-6-4', '0902.1111-1-55-0', '0902.1111-2-2-7', '0902.1111-2-3-5', '0902.1111-2-6-4', '0902.1111-2-56-0', '0902.1111-3-2-7', '0902.1111-3-3-5', '0902.1111-3-6-4', '0902.1111-3-56-0', '0902.1111-4-2-8', '0902.1111-4-3-5', '0902.1111-4-6-4', '0902.1111-4-56-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/0902.1111,"{'0902.1111-3-0-0': 'We formulate a general perturbative framework based on a superconducting atomic limit for the description of Andreev bound states (ABS) in interacting quantum dots connected to superconducting leads.', '0902.1111-3-0-1': 'A local effective Hamiltonian for dressed ABS, including both the atomic or molecular levels and the induced proximity effect on the dot is argued to be a natural starting point.', '0902.1111-3-0-2': 'Self-consistent expansions in single-particle tunneling events are shown to provide accurate results in regimes where the superconducting gap is larger than the Kondo temperature, as demonstrated by the comparison to recent Numerical Renomalization Group calculations.', '0902.1111-3-0-3': 'These analytical results may have bearings for interpreting Andreev spectroscopic measurements with STM on carbon nanotubes coupled to superconducting electrodes.', '0902.1111-3-1-0': '# Introduction', '0902.1111-3-2-0': ""When a quantum dot is connected to superconducting electrodes, the proximity effect drastically modifies the dot's electronic structure due to the local formation of Cooper pairs."", '0902.1111-3-2-1': 'The density of states on the dot thus exhibits a gap, so that the formation of discrete sub-gap states arises [CITATION].', '0902.1111-3-2-2': 'These Andreev bound states (ABS) play certainly an important role as they may contribute a large part of the spectral weight [CITATION] and carry most of the Josephson current.', '0902.1111-3-2-3': '[CITATION] A physical understanding of the ABS requires to characterize how these states are connected to the atomic or molecular levels of the uncoupled quantum dot, and to describe quantitatively their evolution as a function of several parameters, such as gate voltage, Coulomb interaction, tunnel couplings, and superconducting gap.', '0902.1111-3-2-4': 'Whereas the ABS have been observed in metal-superconductor hybrid structures [CITATION], no direct spectroscopy has so far been achieved in quantum dot systems.', '0902.1111-3-2-5': 'Andreev bound states come in pairs, one state above and one below the Fermi level, forming a two level system.', '0902.1111-3-2-6': 'Consequently, recent interest in the spectroscopy of the bound states [CITATION] has also been stimulated by proposals to use the latter as a qubit.', '0902.1111-3-2-7': '[CITATION]', '0902.1111-3-3-0': 'Experimentally, superconducting quantum dots can be realized with carbon nanotubes junctions.', '0902.1111-3-3-1': 'It has been shown that nanotubes and InAs quantum dots connected to superconducting electrodes can be tuned from a Coulomb blockade regime, to a Kondo regime,[CITATION] to a weakly interacting Fabry-Perot regime by changing local gate voltages.', '0902.1111-3-3-2': '[CITATION] The Josephson current at zero bias and multiple Andreev reflections at finite bias voltage have been measured in such devices.', '0902.1111-3-3-3': '[CITATION] The transition from a [MATH]-junction to a [MATH]-junction, namely a reversal in the sign of the Josephson current, [CITATION] has also been been observed when a magnetic moment forms on the dot.', '0902.1111-3-3-4': '[CITATION] As a possible application of superconducting junctions, nano-SQUID devices have also been fabricated.', '0902.1111-3-3-5': '[CITATION]', '0902.1111-3-4-0': 'An exact theoretical description of a quantum dot coupled to superconducting leads is only possible when the Coulomb interaction is fully neglected.', '0902.1111-3-4-1': 'Hence the interacting single dot system, as described by the Anderson model with superconducting electrodes, has been so far analyzed by treating the Coulomb interaction with various analytical schemes, such as the mean field theory, [CITATION] the perturbation expansion in the Coulomb interaction [CITATION] or in the tunnel coupling.', '0902.1111-3-4-2': '[CITATION] Non-perturbative calculations, using the Non-Crossing Approximation (NCA), [CITATION] or the functional Renomalization Group (fRG), [CITATION] as well as numerical simulations based on the numerical renormalization group (NRG), [CITATION] or Quantum Monte Carlo [CITATION] have also been developped.', '0902.1111-3-5-0': 'None of the analytical approaches mentioned above is able to describe entirely the physics of a quantum dot coupled to superconducting leads.', '0902.1111-3-5-1': 'Whereas lowest order perturbation expansions in the tunnel coupling will hardly capture the proximity effect induced by the electrodes [CITATION], mean field and weak-interaction approaches will miss the Kondo effect.', '0902.1111-3-5-2': 'NRG calculations on the other hand can capture the physics of such a system over a wide range of parameters, but are numerically demanding and not easily portable to more complex molecular systems.', '0902.1111-3-5-3': 'More importantly, in the view of describing the ABS alone, none of these techniques does provide a simple physical picture.', '0902.1111-3-5-4': 'Henceforth we will develop in this paper a new perturbative approach based on an effective local Hamiltonian for dressed ABS, that extends the limit of large superconducting gap proposed previously [CITATION] that was used by many authors.', '0902.1111-3-5-5': '[CITATION] This approach will illuminate the nature of the ABS in interacting quantum dots, as well as provide a simple and accurate analytical framework in most relevant cases, that may be useful for interpreting future spectroscopic experiments.', '0902.1111-3-5-6': 'In addition, our formalism, which incorporates the atomic or molecular levels from the outset, should easily be extended to describe more complex systems, as for instance superconducting double quantum dots or molecules with more complicated orbital structure (see e.g. Refs. [CITATION]).', '0902.1111-3-6-0': 'We organize our paper as follows.', '0902.1111-3-6-1': 'In Sec. [REF], the system is mapped onto an effective local Hamiltonian, similarly to the widely used atomic limit, but including the proximity effects due to the superconducting leads.', '0902.1111-3-6-2': 'In Sec. [REF], the perturbation theory around this limit is set up and self-consistent equations for the ABS energy renormalizations are derived in order to extend the validity of the bare perturbative approach.', '0902.1111-3-6-3': 'Sec. [REF] illustrates how this expansion can describe ABS in superconducting quantum dots over a wide range of parameters, by a comparison to available NRG data.', '0902.1111-3-6-4': '[CITATION]', '0902.1111-3-7-0': '# Theoretical formulation', '0902.1111-3-8-0': '## Model', '0902.1111-3-9-0': 'We focus in this paper on a single-level quantum dot coupled to superconducting leads, which is relevant experimentally for molecular junctions with large single-electron level spacing.', '0902.1111-3-9-1': 'A simple Hamiltonian able to describe such a system is given by the superconducting Anderson model [EQUATION] where [EQUATION]', '0902.1111-3-9-2': 'In the above equations, [MATH] is the annihilation operator of an electron with spin [MATH] on the dot, [MATH] that of an electron with spin [MATH] and wave vector [MATH] in the lead [MATH], and [MATH].', '0902.1111-3-9-3': 'The leads are assumed to be described by standard s-wave BCS Hamiltonians [MATH] with superconducting gaps [MATH].', '0902.1111-3-9-4': 'The phase difference of the latter is noted [MATH].', '0902.1111-3-9-5': 'Furthermore, the leads are assumed to have flat and symmetric conduction bands, i.e. the kinetic energy [MATH] measured from the Fermi level ranges in [MATH] and the density of states is [MATH].', '0902.1111-3-9-6': 'We assume [MATH]-independent and symmetric tunneling amplitudes [MATH] between the dot and both superconducting leads.', '0902.1111-3-9-7': 'The dot has a level energy [MATH] and Coulomb interaction [MATH].', '0902.1111-3-9-8': 'Experimentally, the crucial characteristic energy scales, namely Coulomb interaction [MATH], total hybridization [MATH] and gap [MATH], are typically all of the same order of magnitude, [CITATION] providing a challenge for analytical methods.', '0902.1111-3-10-0': ""The physics of the quantum dot can be described via its Green's function [EQUATION] where the Nambu spinor [EQUATION] has been introduced."", '0902.1111-3-10-1': 'Because we will only be interested in stationary equilibrium physics, [MATH] shall be computed in the Matsubara frequency formalism.', '0902.1111-3-11-0': '## Effective local Hamiltonian', '0902.1111-3-12-0': 'As the above Hamiltonian has no exact solution, some approximations must be made.', '0902.1111-3-12-1': 'Among the physical ingredients we want to include in a non-perturbative way is the local pairing on the dot that is crucial for the evolution of the Andreev bound states.', '0902.1111-3-12-2': 'Furthermore, the Coulomb interaction shall be taken into account in an exact manner in order to describe the atomic states faithfully, and to highlight how these are adiabatically connected to the ABS.', '0902.1111-3-12-3': 'However, the usual development in weak tunnel coupling [MATH] around the atomic limit [CITATION] is not sufficient to describe the proximity effect at lowest order.', '0902.1111-3-12-4': 'Therefore, we shall consider in what follows an expansion around a superconducting atomic limit.', '0902.1111-3-13-0': 'Such simple solvable limiting case of the model ([REF]) is often referred to as the limit of large gap [MATH], and has been discussed previously [CITATION].', '0902.1111-3-13-1': 'Expansions for finite [MATH] have not however been discussed to our knowledge, and are the topic of this paper.', '0902.1111-3-13-2': 'We emphasize from the outset (see equation ([REF]) below), that the superconducting atomic limit as used normally in the literature corresponds to the limit [MATH] (i.e. infinite electronic bandwidth), taken before [MATH].', '0902.1111-3-13-3': 'The order of the two limits is crucial: if the limit [MATH] was to be taken first, the dot would be completely decoupled from the leads and the proximity effect would be lost, so that the limit of infinite gap would reduce to the usual atomic limit.', '0902.1111-3-13-4': 'As will be shown now, the superconducting atomic limit should rather be interpreted as a low frequency expansion, i.e. a limit where the gap is much larger than the characteristic frequencies of the dot.', '0902.1111-3-14-0': ""We start off by deriving the Green's function defined in Eq. [REF] using the equations of motion."", '0902.1111-3-14-1': 'Thereby, the Coulomb interaction [MATH] will at first be omitted for the sake of clarity.', '0902.1111-3-14-2': 'Note that in the end, [MATH] will simply give an extra contribution which adds to the effective Hamiltonian.', '0902.1111-3-14-3': 'Fourier transformation straightforwardly yields [EQUATION]', '0902.1111-3-14-4': ""In Eq. [REF], [MATH] is a fermionic Matsubara frequency and [MATH] the bare Green's function of electrons with a wave vector [MATH] in the lead [MATH]."", '0902.1111-3-14-5': 'Transforming the sum over wave vectors [MATH] into an integral over energies yields [EQUATION]', '0902.1111-3-14-6': ""In the limit [MATH], the Green's function [REF] becomes purely static and reduces to [EQUATION]"", '0902.1111-3-14-7': ""Note that the low frequency limit we consider here yields a Green's function that indeed depends on the finite bandwidth [MATH], and this shows that the limit [MATH] shall not be taken for the proximity effect to survive."", '0902.1111-3-14-8': 'In what follows, we will therefore keep both [MATH] and [MATH] finite.', '0902.1111-3-14-9': ""Plugging Eq. [REF] into the Green's function [MATH] leads to the same result as would have been obtained with the effective local Hamiltonian [EQUATION] where the local pairing amplitude induced by the leads on the dot reads [EQUATION] which explicitly depends on the ratio [MATH]."", '0902.1111-3-14-10': 'By an appropriate gauge transformation for the operators [MATH], it is always possible to choose [MATH], as shall be done from now on.', '0902.1111-3-14-11': 'The complete local effective Hamiltonian is obtained when the Coulomb interaction is taken into account again.', '0902.1111-3-14-12': 'Defining [MATH], the energy level of the dot is shifted such that the Hamiltonian clearly exhibits particle-hole symmetry for [MATH]: [EQUATION]', '0902.1111-3-14-13': 'The physical interpretation of this effective local Hamiltonian is simple.', '0902.1111-3-14-14': 'For finite gap, the quantum dot is coupled to both the Cooper pairs and the quasiparticles in the leads.', '0902.1111-3-14-15': 'The Cooper pairs, which lie at the Fermi level, are responsible for the proximity effect.', '0902.1111-3-14-16': 'The quasiparticles give rise to conduction electrons excitations with energies higher than the gap [MATH].', '0902.1111-3-14-17': 'In the limit [MATH], the quasiparticles are far in energy and the coupling between them and the dot vanishes, which greatly simplifies the physics and makes an exact solution possible.', '0902.1111-3-14-18': 'Yet, as the dot is still coupled to the Cooper pairs at the Fermi level, the proximity effect survives with a local pairing term proportional to the hybridization [MATH] between dot and leads.', '0902.1111-3-15-0': '## Spectrum of the effective local Hamiltonian', '0902.1111-3-16-0': 'As the Coulomb interaction simply yields an extra energy shift of [MATH] for both empty and doubly occupied dot, the eigenvectors and eigenvalues of the local effective Hamiltonian ([REF]) are readily obtained by a Bogoliubov transformation [CITATION], in perfect analogy with solution of the BCS Hamiltonian.', '0902.1111-3-16-1': '[MATH] has thus four eigenstates, the singly occupied spin [MATH] states [MATH] and [MATH] with energy [MATH], and two BCS-like states given by [EQUATION] where [MATH] denotes the empty dot and [MATH] the doubly occupied dot.', '0902.1111-3-16-2': 'The amplitudes [MATH] and [MATH] can always be chosen to be real with [MATH] and [MATH].', '0902.1111-3-16-3': 'The energies corresponding to these BCS-like states are [MATH].', '0902.1111-3-17-0': 'As [MATH] is always larger than [MATH], the effective local Hamiltonian has two possible ground states: the low energy BCS-like state [MATH] or the degenerate spin [MATH] doublet [MATH].', '0902.1111-3-17-1': 'In the [MATH] state, the energy is minimized for [MATH].', '0902.1111-3-17-2': 'Thus, the spin singlet phase corresponds to a [MATH]-junction (a result well known from the weak coupling limit [CITATION]).', '0902.1111-3-17-3': 'The transition between the singlet phase and the spin [MATH] doublet takes place at [MATH], and Fig. [REF] shows the corresponding phase diagram for variable [MATH], [MATH] and [MATH].', '0902.1111-3-17-4': 'The state adopted by the quantum dot in the large gap limit therefore results from a competition between the local pairing (induced by the proximity effect and characterized by the hybridization [MATH]) and the Coulomb interaction.', '0902.1111-3-18-0': '## Andreev bound states', '0902.1111-3-19-0': 'As outlined in the introduction, the coupling to superconducting leads induces a gap in the spectral function of the dot, inside which discrete Andreev bound states can form.', '0902.1111-3-19-1': 'The spectral function of the dot shows therefore sharp peaks, which could be measured by STM or microwave experiments as proposed recently [CITATION] (noise measurements have also been suggested in [CITATION]).', '0902.1111-3-19-2': 'These peaks indicate addition energies at which an electron may enter (or leave) the dot, and correspond therefore to transitions between states with [MATH] and [MATH] electrons.', '0902.1111-3-19-3': 'Hence, the ABS peaks in the spectral function may be interpreted as transitions between the superconducting atomic levels of the dot [MATH], possibly renormalized by single-particle tunneling events neglected in [MATH] (to be included in the next section).', '0902.1111-3-19-4': 'Furthermore, transitions from a spin [MATH] doublet to a spin singlet necessarily involve an electron exchange between the dot and the superconducting leads.', '0902.1111-3-19-5': 'As the states [MATH] and [MATH] correspond to the superposition of an empty and doubly occupied dot, this electron exchange and the final singlet states can be understood within the Andreev reflection picture.', '0902.1111-3-20-0': 'Putting everything together, our effective local Hamiltonian in Eq. [REF] describes the energies of the Andreev bound states as transition energies from the spin 1/2 doublet to the spin singlet states.', '0902.1111-3-20-1': '[CITATION] There are thus four Andreev bound states in the large gap limit for the model ([REF]), with energy [MATH] and [MATH] which read: [EQUATION]', '0902.1111-3-20-2': 'The [MATH]/[MATH] transition corresponds to the crossing of the [MATH] and [MATH] states, which occurs for [MATH].', '0902.1111-3-21-0': '# Perturbation expansion around the effective local Hamiltonian', '0902.1111-3-22-0': '## Perturbation theory', '0902.1111-3-23-0': 'The effective Hamiltonian is not sufficient to obtain satisfying results for all regimes of parameters.', '0902.1111-3-23-1': 'First, [MATH] only describes the [MATH]/[MATH]-junction transition due to the competition between a local moment state (stabilized by the Coulomb blockade) and a spin singlet (induced by the proximity effect).', '0902.1111-3-23-2': 'However, if the Coulomb interaction is strong (i.e. [MATH] below the Kondo temperature), the local moment can be screened by the Kondo effect, which will compete with the superconducting gap for the [MATH] transition, so that a typical scaling in the ratio of the Kondo temperature to the gap [MATH] will appear.', '0902.1111-3-23-3': 'Also, the Josephson current in the [MATH]-phase identically vanishes from [MATH], as the spin doublet does not disperse with the superconducting phase difference, a limitation of the large gap limit.', '0902.1111-3-23-4': 'On a more quantitative basis, the experimental gap [MATH] is usually of the order of a few kelvins, which is also the typical scale for both [MATH] and [MATH] in carbon nanotube quantum dot devices.', '0902.1111-3-24-0': ""In order to extend the description of the quantum dot's physics, energy corrections shall be calculated with a perturbation theory around the effective Hamiltonian [REF]."", '0902.1111-3-24-1': 'Once these corrections have been obtained, physical observables like the Josephson current may be computed via the free energy [MATH], with [MATH] the inverse temperature.', '0902.1111-3-24-2': 'Therefore, it is most convenient to work in an action based description, which directly yields the partition function [MATH].', '0902.1111-3-24-3': 'Following Ref. [CITATION], we first integrate over the fermions in the leads.', '0902.1111-3-24-4': 'Omitting the resulting irrelevant constant, the partition function reads [EQUATION] where we have introduced the Grassmann Nambu spinors at Matsubara frequency [MATH], [EQUATION] denoting the Grassmann fields associated with electrons in the dot by [MATH] and [MATH].', '0902.1111-3-25-0': 'The perturbation consists of the terms in Eq. [REF] that are not contained in the action [MATH] corresponding to the effective local Hamiltonian.', '0902.1111-3-25-1': 'A simple identification yields [EQUATION]', '0902.1111-3-25-2': 'Note that [MATH] contains the local pairing term derived in section [REF].', '0902.1111-3-25-3': 'The proximity effect is thus treated non-perturbatively (just like the Coulomb interaction), which is the crucial ingredient of our analytic approach.', '0902.1111-3-25-4': 'The perturbation [MATH] simply corresponds to the tunnel coupling between the dot and the electrodes other than the lowest order proximity effect.', '0902.1111-3-26-0': 'The actual corrections are calculated by expanding the partition function to the first order in [MATH] according to [EQUATION] which we then identify with [EQUATION] where the renormalized superconducting atomic levels [MATH] and [MATH] are obtained from: [EQUATION]', '0902.1111-3-26-1': ""Because the Coulomb interaction is taken into account, Wick's theorem cannot be used to calculate [MATH]."", '0902.1111-3-26-2': 'Instead, expectation values are calculated using Lehmann representation.', '0902.1111-3-26-3': 'Explicit calculations may be found in the appendix.', '0902.1111-3-26-4': 'In the zero temperature limit [MATH], the energy corrections are [EQUATION] with the quasiparticle energy [MATH].', '0902.1111-3-27-0': '## Self-consistent renormalization of the energy', '0902.1111-3-28-0': 'Eqs. [REF]-[REF] yield the first corrections to the energy levels, so that the bound states energies [MATH] and [MATH] are simply shifted by [MATH] and [MATH].', '0902.1111-3-28-1': 'Obviously, these expressions are logarithmically divergent when the bound states energies [MATH] and [MATH] approach the gap edge, and are therefore only valid as long as e.g. [MATH].', '0902.1111-3-28-2': 'In the limit of large gap [MATH], these corrections to [MATH] are thus of the order [MATH], so that the small dimensionless parameter is indeed [MATH].', '0902.1111-3-28-3': 'However, this peculiar logarithmic dependence of the ABS energy renormalization shows that doing a straightforward [MATH] expansion around the effective local Hamiltonian will be rapidly uncontrolled, and will have a hard time reproducing the logarithmic singularities at [MATH] close to [MATH].', '0902.1111-3-28-4': 'For this reason, and also because the large gap limit becomes trivial for a finite electronic bandwidth, as discussed in section [REF], it was indeed more appropriate to single out in the total action all terms left over with respect to the local superconducting effective Hamiltonian, see equation ([REF]), and do perturbation theory around these.', '0902.1111-3-29-0': 'Because our lowest-order expansion obviously still breaks down when the gap becomes comparable to the bound state energy, one would naturally seek to resum the leadind logarithmic divergences in equations ([REF])-([REF]).', '0902.1111-3-29-1': 'This can be achieved using a self-consistency condition inspired by Brillouin-Wigner perturbation theory, [CITATION] which allows to extend greatly the regime of validity of the perturbative scheme.', '0902.1111-3-29-2': 'The resulting self-consistent equations that we obtain are: [EQUATION] and [EQUATION] with [MATH], and [MATH] have been defined in Eqs. ([REF])-([REF]), with [MATH], [MATH].', '0902.1111-3-29-3': 'Note that terms like [MATH] have no self-consistency because there are no associated divergences.', '0902.1111-3-29-4': 'Eq. [REF] and [REF] clearly now hold as long as the renormalized energies [MATH] and [MATH] are not too close to the gap edge, [MATH] respectively.', '0902.1111-3-30-0': '# Results', '0902.1111-3-31-0': '## Phase diagram', '0902.1111-3-32-0': 'We start by discussing the [MATH] transition line, by comparison to the numerical renormalization group (NRG) data by Bauer et al.[CITATION].', '0902.1111-3-32-1': 'Fig. [REF] shows the extension to smaller gaps [MATH] values of the phase diagram obtained with unrenormalized local superconducting states for infinite gap (Fig. [REF]).', '0902.1111-3-32-2': 'Even though our perturbative approach is fairly simple, the results reproduce nicely the NRG data of Refs. [CITATION] and [CITATION].', '0902.1111-3-32-3': 'The analytically obtained phase diagram is indeed identical to the NRG data for [MATH].', '0902.1111-3-32-4': 'For smaller [MATH], the Kondo effect sets in, but the transition lines remain quantitatively correct for [MATH] near [MATH], with increasing deviations from the NRG calculations close to the particle-hole symmetric point [MATH] at large Coulomb interaction U.', '0902.1111-3-32-5': 'In this regime, the [MATH]-phase possesses a Kondo singlet ground state.', '0902.1111-3-32-6': 'As the leads are superconductors, the formation of a Kondo resonance involves the breaking of Cooper pairs.', '0902.1111-3-32-7': 'Therefore, the transition is now due to the competition between [MATH] and the superconducting gap [MATH], and should occur at [MATH].', '0902.1111-3-33-0': 'Fig. [REF] shows a plot of the transition line for [MATH] as obtained with Eq. [REF] (solid curve).', '0902.1111-3-33-1': 'The vertical, dotted line depicts the asymptote in the effective local limit.', '0902.1111-3-33-2': 'The symbols again correspond to NRG data.[', '0902.1111-3-33-3': '[CITATION] The Kondo temperature is given by [MATH] (see for example Ref. NRG_spectral_Bauer).', '0902.1111-3-33-4': 'The inset shows on a log-log scale that our approach captures an exponential decay of the transition line with the Coulomb interaction.', '0902.1111-3-33-5': 'Nonetheless, the suppression of the BCS-like phase appears quantitatively stronger than expected: a factor 4 instead of 8 is found in the exponential factor of [MATH].', '0902.1111-3-33-6': 'The reason for this is that the vertex renormalizations have not been taken into account, as discussed in the context of U-NCA [CITATION].', '0902.1111-3-33-7': 'Far away from the particle-hole symmetric limit, our results for the Kondo temperature reproduce the lowest-order scaling theory for the infinite-[MATH] Anderson model [CITATION], and are in relatively good agreement with NRG data for all [MATH] values.', '0902.1111-3-34-0': '## Energy renormalizations at particle hole symmetry ([MATH])', '0902.1111-3-35-0': 'While Fig. [REF] only indicates the transition line between the spin [MATH] doublet and the lowest BCS spin singlet, it is also instructive to look at the actual renormalization of the energy levels while varying the gap [MATH] from large values to smaller ones beyond the critical point.', '0902.1111-3-35-1': 'Fig. [REF] indicates the renormalized energies of the two Andreev bound states (i.e. the difference between the spin [MATH] doublet and the two spin singlets energies) for different hybridizations [MATH].', '0902.1111-3-35-2': 'We note that our results are in quantitative agreement with the NRG calculations of Yoshioka and Ohashi.', '0902.1111-3-35-3': '[CITATION] Several regions need to be distinguished.', '0902.1111-3-35-4': 'If the gap [MATH] is much larger than the bandwidth [MATH], all curves collapse at the value [MATH] (left hand side of Fig. [REF]), since there is no hybridization with both quasiparticles and Cooper pairs anymore, and one recovers the bare atomic levels.', '0902.1111-3-35-5': 'When the gap starts to decrease, the proximity effect simply splits the two Andreev bound states according to equations ([REF])-([REF]).', '0902.1111-3-35-6': 'When the gap becomes of the same order as the typical energy scales of the dot [MATH] and [MATH], the superconducting atomic levels start to mix with the electrodes, so that the energies renormalize in a non trivial way.', '0902.1111-3-35-7': 'One can see that the transition involving the highest BCS states ends up touching the gap edge for [MATH], so that half of the ABS are absorbed into the continuum above [MATH], as can be seen in Fig. [REF].', '0902.1111-3-35-8': 'The lowest BCS state follows however a downward renormalization, until the Fermi level is crossed and the ground state becomes the [MATH]-state.', '0902.1111-3-35-9': 'The difference in behavior between the lowest and highest bound states (the former being never allowed to leave the superconducting gap) can be tracked into equations ([REF])-([REF]), where level repulsion effects from the gap edge occur for the low energy level [MATH] but are canceled for the high energy level [MATH], which is hence allowed to penetrate into the continuum.', '0902.1111-3-35-10': 'These considerations unveil how the ABS may be adiabatically connected to the atomic or molecular levels in a complicated fashion.', '0902.1111-3-36-0': 'Again, our simple analytic approach reproduces the NRG results [CITATION] over a vast regime of parameters.', '0902.1111-3-36-1': 'Yet, some deviations are observed in the Kondo regime: we find (for the highest hybridization [MATH]) that the high energy BCS-like state is not absorbed anymore into the continuum of states - an artifact of the limits of our perturbative approach.', '0902.1111-3-36-2': 'Notice also that the energy corrections are too important if the gap becomes very small, an effect actually due to our underestimation of the Kondo temperature at particle-symmetry, as discussed previously.', '0902.1111-3-36-3': 'Finally Fig. [REF] shows that, in the limit of vanishing gap, our approach is only valid as long as [MATH] (as has been mentioned in section [REF]), because the lowest bound state artificially escapes from the gap.', '0902.1111-3-36-4': 'The expected saturation of [MATH] near [MATH] can be restored by adding a further self-consistency for terms such as [MATH] in ([REF]) (not shown here).', '0902.1111-3-37-0': '## Energy renormalizations outside particle hole symmetry ([MATH])', '0902.1111-3-38-0': 'From an experimental point of view, the position of the energy level of the quantum dot is the most controllable parameter of the system (by a simple gate voltage).', '0902.1111-3-38-1': 'Therefore, it is important to analyze the evolution of the Andreev bound states for different values of [MATH].', '0902.1111-3-39-0': 'Fig. [REF] illustrates how the energies of the bound states scale with [MATH] for [MATH] and can be favorably compared to the NRG data by Yoshioka and Ohashi.', '0902.1111-3-39-1': '[CITATION] The more particle-hole symmetry is broken, the more the low energy bound state moves away from the gap edge, ensuring even better convergence of our expansion for a given value of [MATH].', '0902.1111-3-39-2': 'This can be understood given that this bound state corresponds to the transition between [MATH] and the spin [MATH] doublet: outside particle-hole symmetry, the dot either seeks to be as empty as possible (for [MATH]) or as occupied as possible (for [MATH]).', '0902.1111-3-39-3': 'Thus, a BCS-like wave function will be favored.', '0902.1111-3-39-4': 'As a consequence, the Kondo effect (that necessitates a singly occupied dot) is less favored.', '0902.1111-3-39-5': 'This corresponds to a regime where our approximation scheme works at best.', '0902.1111-3-40-0': 'Further understanding can be gained by looking at the energies of the Andreev bound states as a function of [MATH] on Fig. [REF].', '0902.1111-3-40-1': 'We recover the fact that the high energy bound states increases in energy by breaking particle-hole symmetry, whereas the low energy bound state has a decreasing energy.', '0902.1111-3-40-2': 'In addition, Fig. [REF] shows that the dispersion of both ABS weakens for increasing hybridization.', '0902.1111-3-40-3': 'Indeed, the more the dot is hybridized with the leads, the less the Andreev bound state energy is sensitive to the bare values of the dot parameters.', '0902.1111-3-41-0': '## Superconducting correlations on the dot', '0902.1111-3-42-0': 'In order to further analyze the evolution of the states in the dot as a function of the parameters in the model ([REF]), we investigate now the superconducting correlations [MATH] on the dot.', '0902.1111-3-42-1': 'For the effective local Hamiltonian, these correlations are zero in the spin doublet phase.', '0902.1111-3-42-2': 'In the BCS-like phase, the correlations are maximal if the two states [MATH] and [MATH] are equivalent, i.e. at particle hole symmetry.', '0902.1111-3-42-3': 'If the dot level is far from [MATH], the wave function will be predominantly [MATH] (if [MATH] is positive) or [MATH] (if [MATH] is negative).', '0902.1111-3-42-4': 'This obviously kills the superconducting correlations.', '0902.1111-3-43-0': 'As the gap decreases from infinity, the (formerly) singly occupied state will start having a BCS-like admixture and therefore a non zero superconducting correlation.', '0902.1111-3-43-1': 'In contrast, the mixing will result in a decreased correlation in the BCS-like phase.', '0902.1111-3-43-2': 'Nevertheless, if the gap tends to zero, one would expect the correlations to vanish as well.', '0902.1111-3-43-3': 'This is indeed what Fig. [REF] shows.', '0902.1111-3-43-4': 'For large gaps, the dot is in the spin [MATH] phase; the correlations are small, but increase as the states mix.', '0902.1111-3-43-5': 'The transition to the BCS-like phase results in a discontinuous jump in the correlations, before they finally vanish for very small gaps.', '0902.1111-3-43-6': 'It can thus be concluded that the correlations should be normalized by the gap if one is interested in measuring only the mixing effect.', '0902.1111-3-43-7': 'Finally, the two different curves show how hybridization stabilizes the BCS-like state with respect to the spin doublet via the [MATH] transition.', '0902.1111-3-44-0': 'As the Coulomb interaction tries to prevent the formation of a Cooper pair wave function, the transition between the BCS-like phase and the spin doublet can also be achieved if the Coulomb interaction is tuned, as shown in Fig. [REF].', '0902.1111-3-44-1': 'The effect of the mixing is clearly visible by an increase of the correlation [MATH] (now normalized by the gap) while [MATH] is lowered.', '0902.1111-3-44-2': 'We also find that the correlations relative to the gap decrease for higher gaps, which is a simple saturation effect (the highest possible correlations are [MATH]).', '0902.1111-3-44-3': 'Furthermore, our results match reasonably the NRG data from Ref. [CITATION], the mismatch originating probably from the low value of the gap used in this calculation.', '0902.1111-3-45-0': 'Finally, we analyze how the correlations evolve outside particle hole symmetry.', '0902.1111-3-45-1': 'As mentioned above, one expects the correlations to decrease because the dot evolve from a superconducting atomic limit toward a usual atomic limit (i.e. from the states [MATH] toward the states [MATH] and [MATH]).', '0902.1111-3-45-2': 'On the other hand, there will be a transition from the spin doublet to the singlet phase and therefore a mixing effect.', '0902.1111-3-45-3': 'Fig. [REF] shows the competition between the mixing effect (that increases the correlations outside particle hole symmetry) and the evolution toward the normal atomic limit (that lowers the correlations) if [MATH] is increased.', '0902.1111-3-45-4': 'The effect of the Coulomb interaction is once more found to favor the single occupancy.', '0902.1111-3-46-0': '## Josephson current', '0902.1111-3-47-0': 'We now turn to the Josephson current through the quantum dot.', '0902.1111-3-47-1': 'The latter is given by [MATH] (where [MATH] is the free energy).', '0902.1111-3-47-2': 'At zero temperature, the free energy is the same than the level energies, so that the Josephson current can readily be obtained once the renormalized energy levels have been calculated.', '0902.1111-3-48-0': 'Nevertheless, our analytical approach only describes the effective local limit atomic states, and we can therefore only determine the current through the Andreev bound states.', '0902.1111-3-48-1': 'Yet, it is known that the Josephson current also contains a contribution of the continuum of states.', '0902.1111-3-48-2': '[CITATION] The latter can be of the same order and opposite sign as the bound state contribution.', '0902.1111-3-48-3': 'Furthermore, Bauer at al.[CITATION] have shown that the spectral weight of the bound states may vary importantly as a function of the different parameters (like the Coulomb interaction [MATH]), especially in the spin doublet phase.', '0902.1111-3-48-4': 'As we exclusively investigate the effective local limit states, we do not keep track of this effect as well.', '0902.1111-3-48-5': 'Therefore, the Josephson currents obtained in our approach will only provide a rather rough and qualitative idea of the actual total Josephson current.', '0902.1111-3-49-0': 'Fig. [REF] shows the Josephson current calculated as the phase derivative of the ground state energy [MATH], [MATH], for different values of [MATH].', '0902.1111-3-49-1': 'One notices two regimes: If the phase is close to [MATH], the system will be in the BCS-like state.', '0902.1111-3-49-2': 'As there is no magnetic moment in this phase, the ground state corresponds to a [MATH]-junction (i.e. phase difference [MATH]).', '0902.1111-3-49-3': 'If [MATH] increases, the energy of the BCS-like state increases (as can be understood in the effective local limit, where [MATH]).', '0902.1111-3-49-4': 'When the BCS-like state crosses with the spin doublet, the ground state changes and the dot becomes singly occupied.', '0902.1111-3-49-5': 'This magnetic moment leads to a discontinuous jump in the Josephson current and the formation of a [MATH]-junction.', '0902.1111-3-49-6': 'Again, we notice that the spin doublet is stabilized in the particle hole symmetric case.', '0902.1111-3-50-0': '# Conclusion', '0902.1111-3-51-0': 'In this section we summarize our main results.', '0902.1111-3-51-1': 'First, it has been shown how the Hamiltonian of a quantum dot coupled to superconducting leads can be mapped onto an effective local model if the superconducting gap [MATH] is much bigger than the characteristic energy scales of the dot.', '0902.1111-3-51-2': ""This limit can be quite generally regarded as a low frequency expansion of the Green's function of the dot rather than the limit [MATH] used in the literature."", '0902.1111-3-51-3': 'This enabled us to extend the effective local Hamiltonian to leads with a finite electronic bandwidth.', '0902.1111-3-52-0': 'We have then set up a perturbation theory around this local effective Hamiltonian and established self-consistent equations for the energy renormalizations of the Andreev bound states.', '0902.1111-3-52-1': 'We have derived those equations based on the fact that the latter correspond to transitions between different states of the local effective Hamiltonian.', '0902.1111-3-53-0': 'In a last section, we used our formalism to calculate physical quantities such as the Andreev bound state energies or superconducting correlations, and understood how these evolve as a function of gate voltage, hybridization, Coulomb interaction and superconducting gap amplitude.', '0902.1111-3-53-1': 'It has been shown that our simple approach agrees well with NRG data in a vast range of parameters, with the main limitation that the Kondo temperature is not quantitatively described near particle-hole symmetry.', '0902.1111-3-53-2': 'However, most experimentally interesting regimes should be described correctly by the simple equations we have derived.', '0902.1111-3-54-0': 'The simplicity and portability constitute the main advantages of our approach, if one is interested in the Andreev bound states only, compared to extended numerical simulations.', '0902.1111-3-54-1': 'As the perturbative description is analytical and based on atomic-like levels, it should in principle be able to describe more complex systems like multiple quantum dots or molecules with several orbitals coupled to superconducting environments, and be readily applicable to describe future spectrosopic measurements.', '0902.1111-3-54-2': 'Extensions of our formalism to the computation of the tunneling current at realistic gap values in three-terminal geometries [CITATION] relevant for STM experiments should certainly deserve further scrutiny.', '0902.1111-3-55-0': 'We wish to acknowledge stimulating discussions with D. Feinberg and C. Winkelmann, and thank J. Bauer, A. Oguri and A. Hewson for providing us their NRG data.', '0902.1111-3-56-0': '*', '0902.1111-3-57-0': '# Derivation of the energy corrections', '0902.1111-3-58-0': ""The partition function is derived starting from the action's perturbation expansion in section [REF]."", '0902.1111-3-58-1': 'The actual calculations are performed in the operator formalism.', '0902.1111-3-58-2': 'It is very useful to note that the product of two fermionic (or bosonic) Greens functions [MATH] and [MATH] obeys [MATH] (as can be shown using Fourier transformation).', '0902.1111-3-58-3': ""The partition function's perturbation expansion is [EQUATION]"", '0902.1111-3-58-4': 'In the above equation, [MATH] is the Fourier transformed Nambu matrix element [MATH] and the subscript [MATH] indicates that the expectation values are statistical averages calculated in the effective local limit.', '0902.1111-3-58-5': ""The leads' Green's functions are: [EQUATION] with [MATH]."", '0902.1111-3-58-6': 'Furthermore, [MATH] and [MATH].', '0902.1111-3-59-0': ""As one cannot apply Wick's theorem because of the Coulomb interaction, the dot's Green's functions are calculated using Lehmann representation, which yields (for [MATH]) [EQUATION]"", '0902.1111-3-59-1': 'Using [MATH], the partition function becomes: [EQUATION]', '0902.1111-3-59-2': 'As [MATH], terms with an [MATH] are exponentially suppressed for [MATH] and can be omitted.'}","{'0902.1111-4-0-0': 'We develop a general perturbative framework based on a superconducting atomic limit for the description of Andreev bound states (ABS) in interacting quantum dots connected to superconducting leads.', '0902.1111-4-0-1': 'A local effective Hamiltonian for dressed ABS, including both the atomic (or molecular) levels and the induced proximity effect on the dot is argued to be a natural starting point.', '0902.1111-4-0-2': 'A self-consistent expansion in single-particle tunneling events is shown to provide accurate results even in regimes where the superconducting gap is smaller than the atomic energies, as demonstrated by a comparison to recent Numerical Renormalization Group calculations.', '0902.1111-4-0-3': 'This simple formulation may have bearings for interpreting Andreev spectroscopic experiments in superconducting devices, such as STM measurements on carbon nanotubes, or radiative emission in optical quantum dots.', '0902.1111-4-1-0': '# Introduction', '0902.1111-4-2-0': ""When a quantum dot is connected to superconducting electrodes, the proximity effect drastically modifies the dot's electronic structure due to the local formation of Cooper pairs."", '0902.1111-4-2-1': 'The density of states on the dot thus exhibits a gap, so that the formation of discrete sub-gap states arises [CITATION].', '0902.1111-4-2-2': 'These Andreev bound states (ABS) play certainly an important role as they may contribute a large part of the spectral weight [CITATION] and carry most of the Josephson current.', '0902.1111-4-2-3': '[CITATION] A physical understanding of the ABS requires to characterize how these states are connected to the atomic (or molecular) levels of the uncoupled quantum dot, and to describe quantitatively their evolution as a function of several parameters, such as gate voltage, Coulomb interaction, tunnel couplings, and superconducting gap.', '0902.1111-4-2-4': 'Whereas the ABS have been observed in metal-superconductor hybrid structures [CITATION], no direct spectroscopy has so far been achieved in quantum dot systems.', '0902.1111-4-2-5': 'Andreev bound states come in pairs, one state above and one below the Fermi level, forming a two level system.', '0902.1111-4-2-6': 'Consequently, recent interest in the spectroscopy of the bound states has also been stimulated by proposals to use the latter as a qubit.', '0902.1111-4-2-7': '[CITATION] At present, several routes have been suggested, such as STM measurements on carbon nanotubes, [CITATION] microwave cavity coupling, [CITATION] visible light emission using a Josephon diode, [CITATION] or noise experiments.', '0902.1111-4-2-8': '[CITATION]', '0902.1111-4-3-0': 'Experimentally, superconducting quantum dots can be realized with carbon nanotubes junctions or semiconducting InAs islands.', '0902.1111-4-3-1': 'It has been shown that quantum dots connected to superconducting electrodes can be tuned from a Coulomb blockade regime, to a Kondo regime,[CITATION] to a weakly interacting Fabry-Perot regime by changing local gate voltages.', '0902.1111-4-3-2': '[CITATION] The Josephson current at zero bias and multiple Andreev reflections at finite bias voltage have been measured in such devices.', '0902.1111-4-3-3': '[CITATION] The transition from a [MATH]-junction to a [MATH]-junction, namely a reversal in the sign of the Josephson current, [CITATION] has also been been observed when a magnetic moment forms on the dot.', '0902.1111-4-3-4': '[CITATION] As a possible application of superconducting junctions, nano-SQUID devices have also been fabricated.', '0902.1111-4-3-5': '[CITATION]', '0902.1111-4-4-0': 'An exact theoretical description of a quantum dot coupled to superconducting leads is only possible when the Coulomb interaction is fully neglected.', '0902.1111-4-4-1': 'Hence the interacting single dot system, as described by the Anderson model with superconducting electrodes, has been so far analyzed by treating the Coulomb interaction with various analytical schemes, such as the mean field theory, [CITATION] the perturbation expansion in the Coulomb interaction [CITATION] or in the tunnel coupling.', '0902.1111-4-4-2': '[CITATION] Non-perturbative calculations, using the Non-Crossing Approximation (NCA), [CITATION] or the functional Renormalization Group (fRG), [CITATION] as well as numerical simulations based on the numerical renormalization group (NRG), [CITATION] or Quantum Monte Carlo [CITATION] have also been developped.', '0902.1111-4-5-0': 'None of the analytical approaches mentioned above is able to describe entirely the physics of a quantum dot coupled to superconducting leads.', '0902.1111-4-5-1': 'Whereas lowest order perturbation expansions in the tunnel coupling will hardly capture the proximity effect induced by the electrodes [CITATION], mean field and weak-interaction approaches will miss the Kondo effect.', '0902.1111-4-5-2': 'NRG calculations on the other hand can capture the physics of such a system over a wide range of parameters, but are numerically demanding and not easily portable to more complex molecular systems.', '0902.1111-4-5-3': 'More importantly, in the view of describing the ABS alone, none of these techniques does provide a simple physical picture.', '0902.1111-4-5-4': 'Henceforth we will develop in this paper a new perturbative approach based on an effective local Hamiltonian for dressed ABS, that extends the limit of large superconducting gap proposed previously [CITATION], which was used by many authors.', '0902.1111-4-5-5': '[CITATION] This approach will illuminate the nature of the ABS in interacting quantum dots, which can be generally viewed as renormalized superconducting atomic states.', '0902.1111-4-5-6': 'This will provide as well a simple analytical framework that is accurate in the most relevant cases, and that may thus be useful for interpreting future spectroscopic experiments.', '0902.1111-4-5-7': 'In particular, calculations provided in the proposals of Refs. [CITATION] and [CITATION] only qualitatively apply in realistic situations where the gap is comparable or smaller that the atomic energies, even when the gap is large compared to the hybridization to the electrodes.', '0902.1111-4-5-8': 'This interesting regime is precisely the one that we want to address in the present work.', '0902.1111-4-5-9': 'In addition, we note that our formalism, which incorporates the atomic (or molecular) levels from the outset, can easily be extended to describe more complex systems, as for instance superconducting double quantum dots or molecules with more complicated orbital structure (see e.g. Refs. [CITATION]).', '0902.1111-4-6-0': 'We organize our paper as follows.', '0902.1111-4-6-1': 'In Sec. [REF], the system is mapped onto an effective local Hamiltonian, similarly to the widely used atomic limit, but including the proximity effects due to the superconducting leads.', '0902.1111-4-6-2': 'In Sec. [REF], the perturbation theory around this limit is set up and self-consistent equations for the ABS energy renormalizations are derived in order to extend the validity of the bare perturbative approach.', '0902.1111-4-6-3': 'Sec. [REF] illustrates how this expansion can describe ABS in superconducting quantum dots over a wide range of parameters, by a comparison to available NRG data.', '0902.1111-4-6-4': '[CITATION]', '0902.1111-4-7-0': '# Theoretical formulation', '0902.1111-4-8-0': '## Model', '0902.1111-4-9-0': 'We focus in this paper on a single-level quantum dot coupled to superconducting leads, which is relevant experimentally for molecular junctions with large single-electron level spacing.', '0902.1111-4-9-1': 'A simple Hamiltonian able to describe such a system is given by the superconducting Anderson model [EQUATION] where [EQUATION]', '0902.1111-4-9-2': 'In the above equations, [MATH] is the annihilation operator of an electron with spin [MATH] on the dot, [MATH] that of an electron with spin [MATH] and wave vector [MATH] in the lead [MATH], and [MATH].', '0902.1111-4-9-3': 'The leads are assumed to be described by standard s-wave BCS Hamiltonians [MATH] with superconducting gaps [MATH].', '0902.1111-4-9-4': 'The phase difference of the latter is noted [MATH].', '0902.1111-4-9-5': 'Furthermore, the leads are assumed to have flat and symmetric conduction bands, i.e. the kinetic energy [MATH] measured from the Fermi level ranges in [MATH] and the density of states is [MATH].', '0902.1111-4-9-6': 'We assume [MATH]-independent and symmetric tunneling amplitudes [MATH] between the dot and both superconducting leads.', '0902.1111-4-9-7': 'The dot has a level energy [MATH] and Coulomb interaction [MATH].', '0902.1111-4-9-8': 'Experimentally, the crucial characteristic energy scales, namely Coulomb interaction [MATH], total hybridization [MATH] and gap [MATH], are typically all of the same order of magnitude, [CITATION] providing a challenge for analytical methods.', '0902.1111-4-10-0': ""The physics of the quantum dot can be described via its Green's function [EQUATION] where the Nambu spinor [EQUATION] has been introduced."", '0902.1111-4-10-1': 'Because we will only be interested in stationary equilibrium physics, [MATH] shall be computed in the Matsubara frequency formalism.', '0902.1111-4-11-0': '## Effective local Hamiltonian', '0902.1111-4-12-0': 'As the above Hamiltonian has no exact solution, some approximations must be made.', '0902.1111-4-12-1': 'Among the physical ingredients we want to include in a non-perturbative way is the local pairing on the dot that is crucial for the evolution of the Andreev bound states.', '0902.1111-4-12-2': 'Furthermore, the Coulomb interaction shall be taken into account in an exact manner in order to describe the atomic states faithfully, and to highlight how these are adiabatically connected to the ABS.', '0902.1111-4-12-3': 'However, the usual development in weak tunnel coupling [MATH] around the atomic limit [CITATION] is not sufficient to describe the proximity effect at lowest order.', '0902.1111-4-12-4': 'Therefore, we shall consider in what follows an expansion around a superconducting atomic limit.', '0902.1111-4-13-0': 'Such simple solvable limiting case of the model ([REF]) is often referred to as the limit of large gap [MATH], and has been discussed previously [CITATION].', '0902.1111-4-13-1': 'Expansions for finite [MATH] have not however been discussed to our knowledge, and are the topic of this paper.', '0902.1111-4-13-2': 'We emphasize from the outset (see equation ([REF]) below), that the superconducting atomic limit as used normally in the literature corresponds to the limit [MATH] (i.e. infinite electronic bandwidth), taken before [MATH].', '0902.1111-4-13-3': 'The order of the two limits is crucial: if the limit [MATH] was to be taken first, the dot would be completely decoupled from the leads and the proximity effect would be lost, so that the limit of infinite gap would reduce to the usual atomic limit.', '0902.1111-4-13-4': 'As will be shown now, the superconducting atomic limit should rather be interpreted as a low frequency expansion, i.e. a limit where the gap is much larger than the characteristic frequencies of the dot.', '0902.1111-4-14-0': ""We start off by deriving the Green's function defined in Eq. [REF] using the equations of motion."", '0902.1111-4-14-1': 'Thereby, the Coulomb interaction [MATH] will at first be omitted for the sake of clarity.', '0902.1111-4-14-2': 'Note that in the end, [MATH] will simply give an extra contribution which adds to the effective Hamiltonian.', '0902.1111-4-14-3': 'Fourier transformation straightforwardly yields [EQUATION]', '0902.1111-4-14-4': ""In Eq. [REF], [MATH] is a fermionic Matsubara frequency, [MATH] the bare Green's function in Nambu space of electrons with a wave vector [MATH] in the lead [MATH], and the Pauli matrices [MATH] have been introduced."", '0902.1111-4-14-5': 'Transforming the sum over wave vectors [MATH] into an integral over energies yields [EQUATION]', '0902.1111-4-14-6': ""In the limit [MATH], the Green's function [REF] becomes purely static and reduces to [EQUATION]"", '0902.1111-4-14-7': ""Note that the low frequency limit we consider here yields a Green's function that indeed depends on the finite bandwidth [MATH], and this shows that the limit [MATH] shall not be taken for the proximity effect to survive."", '0902.1111-4-14-8': 'In what follows, we will therefore keep both [MATH] and [MATH] finite.', '0902.1111-4-14-9': ""Plugging Eq. [REF] into the Green's function [MATH] leads to the same result as would have been obtained with the effective local Hamiltonian [EQUATION] where the local pairing amplitude induced by the leads on the dot reads [EQUATION] which explicitly depends on the ratio [MATH]."", '0902.1111-4-14-10': 'By an appropriate gauge transformation for the operators [MATH], it is always possible to choose [MATH], as shall be done from now on.', '0902.1111-4-14-11': 'The complete local effective Hamiltonian is obtained when the Coulomb interaction is taken into account again.', '0902.1111-4-14-12': 'Defining [MATH], the energy level of the dot is shifted such that the Hamiltonian clearly exhibits particle-hole symmetry for [MATH]: [EQUATION]', '0902.1111-4-14-13': 'The physical interpretation of this effective local Hamiltonian is simple.', '0902.1111-4-14-14': 'For finite gap, the quantum dot is coupled to both the Cooper pairs and the quasiparticles in the leads.', '0902.1111-4-14-15': 'The Cooper pairs, which lie at the Fermi level, are responsible for the proximity effect.', '0902.1111-4-14-16': 'The quasiparticles give rise to conduction electrons excitations with energies higher than the gap [MATH].', '0902.1111-4-14-17': 'In the limit [MATH], the quasiparticles are far in energy and the coupling between them and the dot vanishes, which greatly simplifies the physics and makes an exact solution possible.', '0902.1111-4-14-18': 'Yet, as the dot is still coupled to the Cooper pairs at the Fermi level, the proximity effect survives with a local pairing term proportional to the hybridization [MATH] between dot and leads.', '0902.1111-4-15-0': '## Spectrum of the effective local Hamiltonian', '0902.1111-4-16-0': 'As the Coulomb interaction simply yields an extra energy shift of [MATH] for both empty and doubly occupied dot, the eigenvectors and eigenvalues of the local effective Hamiltonian ([REF]) are readily obtained by a Bogoliubov transformation [CITATION], in perfect analogy with solution of the BCS Hamiltonian.', '0902.1111-4-16-1': '[MATH] has thus four eigenstates, the singly occupied spin [MATH] states [MATH] and [MATH] with energy [MATH], and two BCS-like states given by [EQUATION] where [MATH] denotes the empty dot and [MATH] the doubly occupied dot.', '0902.1111-4-16-2': 'The amplitudes [MATH] and [MATH] can always be chosen to be real with [MATH] and [MATH].', '0902.1111-4-16-3': 'The energies corresponding to these BCS-like states are [MATH].', '0902.1111-4-17-0': 'As [MATH] is always larger than [MATH], the effective local Hamiltonian has two possible ground states: the low energy BCS-like state [MATH] or the degenerate spin [MATH] doublet [MATH].', '0902.1111-4-17-1': 'In the [MATH] state, the energy is minimized for [MATH].', '0902.1111-4-17-2': 'Thus, the spin singlet phase corresponds to a [MATH]-junction (a result well known from the weak coupling limit [CITATION]).', '0902.1111-4-17-3': 'The transition between the singlet phase and the spin [MATH] doublet takes place at [MATH], and Fig. [REF] shows the corresponding phase diagram for variable [MATH], [MATH] and [MATH].', '0902.1111-4-17-4': 'The state adopted by the quantum dot in the large gap limit therefore results from a competition between the local pairing (induced by the proximity effect and characterized by the hybridization [MATH]) and the Coulomb interaction.', '0902.1111-4-18-0': '## Andreev bound states', '0902.1111-4-19-0': 'As outlined in the introduction, the coupling to superconducting leads induces a gap in the spectral function of the dot, inside which discrete Andreev bound states can form.', '0902.1111-4-19-1': 'The spectral function of the dot shows therefore sharp peaks, which could be measured by STM [CITATION] or microwave-[CITATION]optical [CITATION] experiments as proposed recently.', '0902.1111-4-19-2': 'These peaks indicate addition energies at which an electron may enter (or leave) the dot, and correspond therefore to transitions between states with [MATH] and [MATH] electrons.', '0902.1111-4-19-3': 'Hence, the ABS peaks in the spectral function may be interpreted as transitions between the superconducting atomic levels of the dot [MATH], possibly renormalized by single-particle tunneling events neglected in [MATH] (to be included in the next section).', '0902.1111-4-19-4': 'Furthermore, transitions from a spin [MATH] doublet to a spin singlet necessarily involve an electron exchange between the dot and the superconducting leads.', '0902.1111-4-19-5': 'As the states [MATH] and [MATH] correspond to the superposition of an empty and doubly occupied dot, this electron exchange and the final singlet states can be understood within the Andreev reflection picture.', '0902.1111-4-20-0': 'Putting everything together, our effective local Hamiltonian in Eq. [REF] describes the energies of the Andreev bound states as transition energies from the spin 1/2 doublet to the spin singlet states.', '0902.1111-4-20-1': '[CITATION] There are thus four Andreev bound states in the large gap limit for the model ([REF]), with energy [MATH] and [MATH] which read: [EQUATION]', '0902.1111-4-20-2': 'The [MATH]/[MATH] transition corresponds to the crossing of the [MATH] and [MATH] states, which occurs for [MATH].', '0902.1111-4-21-0': '# Perturbation expansion around the effective local Hamiltonian', '0902.1111-4-22-0': '## Perturbation theory', '0902.1111-4-23-0': 'The effective Hamiltonian is not sufficient to obtain satisfying results for all regimes of parameters.', '0902.1111-4-23-1': 'First, [MATH] only describes the [MATH]/[MATH]-junction transition due to the competition between a local moment state (stabilized by the Coulomb blockade) and a spin singlet (induced by the proximity effect).', '0902.1111-4-23-2': 'However, if the Coulomb interaction is strong (i.e. [MATH] below the Kondo temperature), the local moment can be screened by the Kondo effect, which will compete with the superconducting gap for the [MATH] transition, so that a typical scaling in the ratio of the Kondo temperature to the gap [MATH] will appear.', '0902.1111-4-23-3': 'Also, the Josephson current in the [MATH]-phase identically vanishes from [MATH], as the spin doublet does not disperse with the superconducting phase difference, a limitation of the large gap limit.', '0902.1111-4-23-4': 'On a more quantitative basis, the experimental gap [MATH] is usually of the order of a few kelvins, which is also the typical scale for both [MATH] and [MATH] in carbon nanotube quantum dot devices.', '0902.1111-4-24-0': ""In order to extend the description of the quantum dot's physics, energy corrections shall be calculated with a perturbation theory around the effective Hamiltonian [REF]."", '0902.1111-4-24-1': 'Once these corrections have been obtained, physical observables like the Josephson current may be computed via the free energy [MATH], with [MATH] the inverse temperature.', '0902.1111-4-24-2': 'Therefore, it is most convenient to work in an action based description, which directly yields the partition function [MATH].', '0902.1111-4-24-3': 'Following Ref. [CITATION], we first integrate over the fermions in the leads.', '0902.1111-4-24-4': 'Omitting the resulting irrelevant constant, the partition function reads [EQUATION] where we have introduced the Grassmann Nambu spinors at Matsubara frequency [MATH], [EQUATION] denoting the Grassmann fields associated with electrons in the dot by [MATH] and [MATH].', '0902.1111-4-25-0': 'The perturbation consists of the terms in Eq. [REF] that are not contained in the action [MATH] corresponding to the effective local Hamiltonian.', '0902.1111-4-25-1': 'A simple identification yields [EQUATION]', '0902.1111-4-25-2': 'Note that [MATH] contains the local pairing term derived in section [REF].', '0902.1111-4-25-3': 'The proximity effect is thus treated non-perturbatively (just like the Coulomb interaction), which is the crucial ingredient of our analytic approach.', '0902.1111-4-25-4': 'The perturbation [MATH] simply corresponds to the tunnel coupling between the dot and the electrodes other than the lowest order proximity effect.', '0902.1111-4-26-0': 'The actual corrections are calculated by expanding the partition function to the first order in [MATH] according to [EQUATION] which we then identify with [EQUATION] where the renormalized superconducting atomic levels [MATH] and [MATH] are obtained from: [EQUATION]', '0902.1111-4-26-1': ""Because the Coulomb interaction is taken into account, Wick's theorem cannot be used to calculate [MATH]."", '0902.1111-4-26-2': 'Instead, expectation values are calculated using Lehmann representation.', '0902.1111-4-26-3': 'Explicit calculations may be found in the appendix.', '0902.1111-4-26-4': 'In the zero temperature limit [MATH], the energy corrections are [EQUATION] with the quasiparticle energy [MATH].', '0902.1111-4-27-0': '## Self-consistent renormalization of the energy', '0902.1111-4-28-0': 'Eqs. [REF]-[REF] yield the first corrections to the energy levels, so that the bound states energies [MATH] and [MATH] are simply shifted by [MATH] and [MATH].', '0902.1111-4-28-1': 'Obviously, these expressions are logarithmically divergent when the bound states energies [MATH] and [MATH] approach the gap edge, and are therefore only valid as long as e.g. [MATH].', '0902.1111-4-28-2': 'In the limit of large gap [MATH], these corrections to [MATH] are thus of the order [MATH], so that the small dimensionless parameter is indeed [MATH].', '0902.1111-4-28-3': 'However, this peculiar logarithmic dependence of the ABS energy renormalization shows that doing a straightforward [MATH] expansion around the effective local Hamiltonian will be rapidly uncontrolled, and will have a hard time reproducing the logarithmic singularities at [MATH] close to [MATH].', '0902.1111-4-28-4': 'For this reason, and also because the large gap limit becomes trivial for a finite electronic bandwidth, as discussed in section [REF], it was indeed more appropriate to single out in the total action all terms left over with respect to the local superconducting effective Hamiltonian, see equation ([REF]), and do perturbation theory around these.', '0902.1111-4-29-0': 'Because our lowest-order expansion obviously still breaks down when the gap becomes comparable to the bound state energy, one would naturally seek to resum the leadind logarithmic divergences in equations ([REF])-([REF]).', '0902.1111-4-29-1': 'This can be achieved using a self-consistency condition inspired by Brillouin-Wigner perturbation theory, [CITATION] which allows to extend greatly the regime of validity of the perturbative scheme.', '0902.1111-4-29-2': 'The resulting self-consistent equations that we obtain are: [EQUATION] and [EQUATION] with [MATH], and [MATH] have been defined in Eqs. ([REF])-([REF]), with [MATH], [MATH].', '0902.1111-4-29-3': 'Note that terms like [MATH] have no self-consistency because there are no associated divergences.', '0902.1111-4-29-4': 'Eq. [REF] and [REF] clearly now hold as long as the renormalized energies [MATH] and [MATH] are not too close to the gap edge, [MATH] respectively.', '0902.1111-4-30-0': '# Results', '0902.1111-4-31-0': '## Phase diagram', '0902.1111-4-32-0': 'We start by discussing the [MATH] transition line, by comparison to the numerical renormalization group (NRG) data by Bauer et al.[CITATION].', '0902.1111-4-32-1': 'Fig. [REF] shows the extension to smaller gaps [MATH] values of the phase diagram obtained with unrenormalized local superconducting states for infinite gap (Fig. [REF]).', '0902.1111-4-32-2': 'Even though our perturbative approach is fairly simple, the results reproduce nicely the NRG data of Refs. [CITATION] and [CITATION].', '0902.1111-4-32-3': 'The analytically obtained phase diagram is indeed identical to the NRG data for [MATH].', '0902.1111-4-32-4': 'For smaller [MATH], the Kondo effect sets in, but the transition lines remain quantitatively correct for [MATH] near [MATH], with increasing deviations from the NRG calculations close to the particle-hole symmetric point [MATH] at large Coulomb interaction U.', '0902.1111-4-32-5': 'In this regime, the [MATH]-phase possesses a Kondo singlet ground state.', '0902.1111-4-32-6': 'As the leads are superconductors, the formation of a Kondo resonance involves the breaking of Cooper pairs.', '0902.1111-4-32-7': 'Therefore, the transition is now due to the competition between [MATH] and the superconducting gap [MATH], and should occur at [MATH].', '0902.1111-4-33-0': 'Fig. [REF] shows a plot of the transition line for [MATH] as obtained with Eq. [REF] (solid curve).', '0902.1111-4-33-1': 'The vertical, dotted line depicts the asymptote in the effective local limit.', '0902.1111-4-33-2': 'The symbols again correspond to NRG data.[', '0902.1111-4-33-3': '[CITATION] The Kondo temperature is given by [MATH] (see for example Ref. NRG_spectral_Bauer).', '0902.1111-4-33-4': 'The inset shows on a log-log scale that our approach captures an exponential decay of the transition line with the Coulomb interaction.', '0902.1111-4-33-5': 'Nonetheless, the suppression of the BCS-like phase appears quantitatively stronger than expected: a factor 4 instead of 8 is found in the exponential factor of [MATH].', '0902.1111-4-33-6': 'The reason for this is that the vertex renormalizations have not been taken into account, as discussed in the context of U-NCA [CITATION].', '0902.1111-4-33-7': 'Far away from the particle-hole symmetric limit, our results for the Kondo temperature reproduce the lowest-order scaling theory for the infinite-[MATH] Anderson model [CITATION], and are in relatively good agreement with NRG data for all [MATH] values.', '0902.1111-4-34-0': '## Energy renormalizations at particle hole symmetry ([MATH])', '0902.1111-4-35-0': 'While Fig. [REF] only indicates the transition line between the spin [MATH] doublet and the lowest BCS spin singlet, it is also instructive to look at the actual renormalization of the energy levels while varying the gap [MATH] from large values to smaller ones beyond the critical point.', '0902.1111-4-35-1': 'Fig. [REF] indicates the renormalized energies of the two Andreev bound states (i.e. the difference between the spin [MATH] doublet and the two spin singlets energies) for different hybridizations [MATH].', '0902.1111-4-35-2': 'We note that our results are in quantitative agreement with the NRG calculations of Yoshioka and Ohashi.', '0902.1111-4-35-3': '[CITATION] Several regions need to be distinguished.', '0902.1111-4-35-4': 'If the gap [MATH] is much larger than the bandwidth [MATH], all curves collapse at the value [MATH] (left hand side of Fig. [REF]), since there is no hybridization with both quasiparticles and Cooper pairs anymore, and one recovers the bare atomic levels.', '0902.1111-4-35-5': 'When the gap starts to decrease, the proximity effect simply splits the two Andreev bound states according to equations ([REF])-([REF]).', '0902.1111-4-35-6': 'When the gap becomes of the same order as the typical energy scales of the dot [MATH] and [MATH], the superconducting atomic levels start to mix with the electrodes, so that the energies renormalize in a non trivial way.', '0902.1111-4-35-7': 'One can see that the transition involving the highest BCS states ends up touching the gap edge for [MATH], so that half of the ABS are absorbed into the continuum above [MATH], as can be seen in Fig. [REF].', '0902.1111-4-35-8': 'The lowest BCS state follows however a downward renormalization, until the Fermi level is crossed and the ground state becomes the [MATH]-state.', '0902.1111-4-35-9': 'The difference in behavior between the lowest and highest bound states (the former being never allowed to leave the superconducting gap) can be tracked into equations ([REF])-([REF]), where level repulsion effects from the gap edge occur for the low energy level [MATH] but are canceled for the high energy level [MATH], which is hence allowed to penetrate into the continuum.', '0902.1111-4-35-10': 'These considerations unveil how the ABS may be adiabatically connected to the atomic (or molecular) levels in a complicated fashion.', '0902.1111-4-36-0': 'Again, our simple analytic approach reproduces the NRG results [CITATION] over a vast regime of parameters.', '0902.1111-4-36-1': 'Yet, some deviations are observed in the Kondo regime: we find (for the highest hybridization [MATH]) that the high energy BCS-like state is not absorbed anymore into the continuum of states - an artifact of the limits of our perturbative approach.', '0902.1111-4-36-2': 'Notice also that the energy corrections are too important if the gap becomes very small, an effect actually due to our underestimation of the Kondo temperature at particle-symmetry, as discussed previously.', '0902.1111-4-36-3': 'Finally Fig. [REF] shows that, in the limit of vanishing gap, our approach is only valid as long as [MATH] (as has been mentioned in section [REF]), because the lowest bound state artificially escapes from the gap.', '0902.1111-4-36-4': 'The expected saturation of [MATH] near [MATH] can be restored by adding a further self-consistency for terms such as [MATH] in ([REF]) (not shown here).', '0902.1111-4-37-0': '## Energy renormalizations outside particle hole symmetry ([MATH])', '0902.1111-4-38-0': 'From an experimental point of view, the position of the energy level of the quantum dot is the most controllable parameter of the system (by a simple gate voltage).', '0902.1111-4-38-1': 'Therefore, it is important to analyze the evolution of the Andreev bound states for different values of [MATH].', '0902.1111-4-39-0': 'Fig. [REF] illustrates how the energies of the bound states scale with [MATH] for [MATH] and can be favorably compared to the NRG data by Yoshioka and Ohashi.', '0902.1111-4-39-1': '[CITATION] The more particle-hole symmetry is broken, the more the low energy bound state moves away from the gap edge, ensuring even better convergence of our expansion for a given value of [MATH].', '0902.1111-4-39-2': 'This can be understood given that this bound state corresponds to the transition between [MATH] and the spin [MATH] doublet: outside particle-hole symmetry, the dot either seeks to be as empty as possible (for [MATH]) or as occupied as possible (for [MATH]).', '0902.1111-4-39-3': 'Thus, a BCS-like wave function will be favored.', '0902.1111-4-39-4': 'As a consequence, the Kondo effect (that necessitates a singly occupied dot) is less favored.', '0902.1111-4-39-5': 'This corresponds to a regime where our approximation scheme works at best.', '0902.1111-4-40-0': 'Further understanding can be gained by looking at the energies of the Andreev bound states as a function of [MATH] on Fig. [REF].', '0902.1111-4-40-1': 'We recover the fact that the high energy bound states increases in energy by breaking particle-hole symmetry, whereas the low energy bound state has a decreasing energy.', '0902.1111-4-40-2': 'In addition, Fig. [REF] shows that the dispersion of both ABS weakens for increasing hybridization.', '0902.1111-4-40-3': 'Indeed, the more the dot is hybridized with the leads, the less the Andreev bound state energy is sensitive to the bare values of the dot parameters.', '0902.1111-4-41-0': '## Superconducting correlations on the dot', '0902.1111-4-42-0': 'In order to further analyze the evolution of the states in the dot as a function of the parameters in the model ([REF]), we investigate now the superconducting correlations [MATH] on the dot.', '0902.1111-4-42-1': 'For the effective local Hamiltonian, these correlations are zero in the spin doublet phase.', '0902.1111-4-42-2': 'In the BCS-like phase, the correlations are maximal if the two states [MATH] and [MATH] are equivalent, i.e. at particle hole symmetry.', '0902.1111-4-42-3': 'If the dot level is far from [MATH], the wave function will be predominantly [MATH] (if [MATH] is positive) or [MATH] (if [MATH] is negative).', '0902.1111-4-42-4': 'This obviously kills the superconducting correlations.', '0902.1111-4-43-0': 'As the gap decreases from infinity, the (formerly) singly occupied state will start having a BCS-like admixture and therefore a non zero superconducting correlation.', '0902.1111-4-43-1': 'In contrast, the mixing will result in a decreased correlation in the BCS-like phase.', '0902.1111-4-43-2': 'Nevertheless, if the gap tends to zero, one would expect the correlations to vanish as well.', '0902.1111-4-43-3': 'This is indeed what Fig. [REF] shows.', '0902.1111-4-43-4': 'For large gaps, the dot is in the spin [MATH] phase; the correlations are small, but increase as the states mix.', '0902.1111-4-43-5': 'The transition to the BCS-like phase results in a discontinuous jump in the correlations, before they finally vanish for very small gaps.', '0902.1111-4-43-6': 'It can thus be concluded that the correlations should be normalized by the gap if one is interested in measuring only the mixing effect.', '0902.1111-4-43-7': 'Finally, the two different curves show how hybridization stabilizes the BCS-like state with respect to the spin doublet via the [MATH] transition.', '0902.1111-4-44-0': 'As the Coulomb interaction tries to prevent the formation of a Cooper pair wave function, the transition between the BCS-like phase and the spin doublet can also be achieved if the Coulomb interaction is tuned, as shown in Fig. [REF].', '0902.1111-4-44-1': 'The effect of the mixing is clearly visible by an increase of the correlation [MATH] (now normalized by the gap) while [MATH] is lowered.', '0902.1111-4-44-2': 'We also find that the correlations relative to the gap decrease for higher gaps, which is a simple saturation effect (the highest possible correlations are [MATH]).', '0902.1111-4-44-3': 'Furthermore, our results are quantitatively precise if the gap if larger than the hybridization [MATH] for all values of [MATH], while relatively small deviations appear for [MATH], as shown by the comparison to the NRG data from Ref. [CITATION], and to second-order perturbation theory in [MATH] (valid in the singlet phase only, providing accurate results for [MATH] roughly) [CITATION].', '0902.1111-4-45-0': 'Finally, we analyze how the correlations evolve outside particle hole symmetry.', '0902.1111-4-45-1': 'As mentioned above, one expects the correlations to decrease because the dot evolve from a superconducting atomic limit toward a usual atomic limit (i.e. from the states [MATH] toward the states [MATH] and [MATH]).', '0902.1111-4-45-2': 'On the other hand, there will be a transition from the spin doublet to the singlet phase and therefore a mixing effect.', '0902.1111-4-45-3': 'Fig. [REF] shows the competition between the mixing effect (that increases the correlations outside particle hole symmetry) and the evolution toward the normal atomic limit (that lowers the correlations) if [MATH] is increased.', '0902.1111-4-45-4': 'The effect of the Coulomb interaction is once more found to favor the single occupancy.', '0902.1111-4-46-0': '## Josephson current', '0902.1111-4-47-0': 'We now turn to the Josephson current through the quantum dot.', '0902.1111-4-47-1': 'The latter is given by [MATH] (where [MATH] is the free energy).', '0902.1111-4-47-2': 'At zero temperature, the free energy is the same than the level energies, so that the Josephson current can readily be obtained once the renormalized energy levels have been calculated.', '0902.1111-4-48-0': 'Nevertheless, our analytical approach only describes the effective local limit atomic states, and we can therefore only determine the current through the Andreev bound states.', '0902.1111-4-48-1': 'Yet, it is known that the Josephson current also contains a contribution of the continuum of states.', '0902.1111-4-48-2': '[CITATION] The latter can be of the same order and opposite sign as the bound state contribution.', '0902.1111-4-48-3': 'Furthermore, Bauer at al.[CITATION] have shown that the spectral weight of the bound states may vary importantly as a function of the different parameters (like the Coulomb interaction [MATH]), especially in the spin doublet phase.', '0902.1111-4-48-4': 'As we exclusively investigate the effective local limit states, we do not keep track of this effect as well.', '0902.1111-4-48-5': 'Therefore, the Josephson currents obtained in our approach will only provide a rather rough and qualitative idea of the actual total Josephson current.', '0902.1111-4-49-0': 'Fig. [REF] shows the Josephson current calculated as the phase derivative of the ground state energy [MATH], [MATH], for different values of [MATH].', '0902.1111-4-49-1': 'One notices two regimes: If the phase is close to [MATH], the system will be in the BCS-like state.', '0902.1111-4-49-2': 'As there is no magnetic moment in this phase, the ground state corresponds to a [MATH]-junction (i.e. phase difference [MATH]).', '0902.1111-4-49-3': 'If [MATH] increases, the energy of the BCS-like state increases (as can be understood in the effective local limit, where [MATH]).', '0902.1111-4-49-4': 'When the BCS-like state crosses with the spin doublet, the ground state changes and the dot becomes singly occupied.', '0902.1111-4-49-5': 'This magnetic moment leads to a discontinuous jump in the Josephson current and the formation of a [MATH]-junction.', '0902.1111-4-49-6': 'Again, we notice that the spin doublet is stabilized in the particle hole symmetric case.', '0902.1111-4-50-0': '# Conclusion', '0902.1111-4-51-0': 'In this section we summarize our main results.', '0902.1111-4-51-1': 'First, it has been shown how the Hamiltonian of a quantum dot coupled to superconducting leads can be mapped onto an effective local model if the superconducting gap [MATH] is much bigger than the characteristic energy scales of the dot.', '0902.1111-4-51-2': ""This limit can be quite generally regarded as a low frequency expansion of the Green's function of the dot rather than the limit [MATH] used in the literature."", '0902.1111-4-51-3': 'This enabled us to extend the effective local Hamiltonian to leads with a finite electronic bandwidth.', '0902.1111-4-52-0': 'We have then set up a perturbation theory around this local effective Hamiltonian and established self-consistent equations for the energy renormalizations of the Andreev bound states.', '0902.1111-4-52-1': 'We have derived those equations based on the fact that the latter correspond to transitions between different states of the local effective Hamiltonian.', '0902.1111-4-53-0': 'In a last section, we used our formalism to calculate physical quantities such as the Andreev bound state energies or superconducting correlations, and understood how these evolve as a function of gate voltage, hybridization, Coulomb interaction and superconducting gap amplitude.', '0902.1111-4-53-1': 'It has been shown that our simple approach agrees well with NRG data in a vast range of parameters, with the main limitation that the Kondo temperature is not quantitatively described near particle-hole symmetry.', '0902.1111-4-53-2': 'However, most experimentally interesting regimes should be described correctly by the simple equations we have derived.', '0902.1111-4-54-0': 'The simplicity and portability constitute the main advantages of our approach, if one is interested in the Andreev bound states only, compared to extended numerical simulations.', '0902.1111-4-54-1': 'As the perturbative description is analytical and based on atomic-like levels, it should in principle be able to describe more complex systems like multiple quantum dots or molecules with several orbitals coupled to superconducting environments, and be readily applicable to describe future spectrosopic measurements.', '0902.1111-4-54-2': 'Extensions of our formalism to the computation of the tunneling current at realistic gap values in three-terminal geometries [CITATION] relevant for STM experiments should certainly deserve further scrutiny.', '0902.1111-4-55-0': 'We wish to acknowledge stimulating discussions with D. Feinberg and C. Winkelmann, and thank J. Bauer, A. Oguri and A. Hewson for providing us their NRG data.', '0902.1111-4-56-0': '*', '0902.1111-4-57-0': '# Derivation of the energy corrections', '0902.1111-4-58-0': ""The partition function is derived starting from the action's perturbation expansion in section [REF]."", '0902.1111-4-58-1': 'The actual calculations are performed in the operator formalism.', '0902.1111-4-58-2': 'It is very useful to note that the product of two fermionic (or bosonic) Greens functions [MATH] and [MATH] obeys [MATH] (as can be shown using Fourier transformation).', '0902.1111-4-58-3': ""The partition function's perturbation expansion is [EQUATION]"", '0902.1111-4-58-4': 'In the above equation, [MATH] is the Fourier transformed Nambu matrix element [MATH] and the subscript [MATH] indicates that the expectation values are statistical averages calculated in the effective local limit.', '0902.1111-4-58-5': ""The leads' Green's functions are: [EQUATION] with [MATH]."", '0902.1111-4-58-6': 'Furthermore, [MATH] and [MATH].', '0902.1111-4-59-0': ""As one cannot apply Wick's theorem because of the Coulomb interaction, the dot's Green's functions are calculated using Lehmann representation, which yields (for [MATH]) [EQUATION]"", '0902.1111-4-59-1': 'Using [MATH], the partition function becomes: [EQUATION]', '0902.1111-4-59-2': 'As [MATH], terms with an [MATH] are exponentially suppressed for [MATH] and can be omitted.'}",
1411.4901,"{'1411.4901-1-0-0': 'We investigate two competing contact processes on a set of Watts-Strogatz networks with the clustering coefficient tuned by rewiring.', '1411.4901-1-0-1': 'The base for network construction is one-dimensional chain of [MATH] sites, where each site [MATH] is directly linked to nodes labelled as [MATH] and [MATH].', '1411.4901-1-0-2': 'So initially, each node has the same degree [MATH].', '1411.4901-1-0-3': 'The periodic boundary conditions are assumed as well.', '1411.4901-1-1-0': 'For each node [MATH] the links to sites [MATH] and [MATH] are rewired to two randomly selected nodes so far not-connected to node [MATH].', '1411.4901-1-1-1': 'An increase of the rewiring probability [MATH] influences the nodes degree distribution and the network clusterization coefficient [MATH].', '1411.4901-1-1-2': 'For given values of rewiring probability [MATH] the set [MATH] of [MATH] networks is generated.', '1411.4901-1-2-0': ""The network's nodes are decorated with spin-like variables [MATH]."", '1411.4901-1-2-1': 'During simulation each [MATH] node having a [MATH]-site in its neighbourhood converts this neighbour from [MATH] to [MATH] state.', '1411.4901-1-2-2': 'Conversely, a node in [MATH] state having at least one neighbour also in state [MATH]-state converts all nearest-neighbours of this pair into [MATH]-state.', '1411.4901-1-2-3': 'The latter is realized with probability [MATH].', '1411.4901-1-3-0': 'We plot the dependence of the nodes [MATH] final density [MATH] on initial nodes [MATH] fraction [MATH].', '1411.4901-1-3-1': 'Then, we construct the surface of the unstable fixed points in [MATH] space.', '1411.4901-1-3-2': 'The system evolves more often toward [MATH] for [MATH] points situated above this surface while starting simulation with [MATH] parameters situated below this surface leads system to [MATH].', '1411.4901-1-3-3': 'The points on this surface correspond to such value of initial fraction [MATH] of [MATH] nodes (for fixed values [MATH] and [MATH]) for which their final density is [MATH].', '1411.4901-1-4-0': '# Introduction', '1411.4901-1-5-0': 'In computational modeling, the contact processes (CPs) are dynamic systems on discrete media, where the time evolution of a local state towards survival or extinction of particles is determined by the state of the direct neighborhood of a lattice cell or of a network node.', '1411.4901-1-5-1': 'A simple realization is the voter model [CITATION], where a particle creates another particle in its direct neighborhood.', '1411.4901-1-5-2': 'CPs have been introduced in 1974 as a toy model of spread of epidemic on a lattice [CITATION].', '1411.4901-1-5-3': 'Since then, they evolved to a frame for models in different areas, from symbiotic interactions [CITATION] to population [CITATION] or opinion dynamics [CITATION].', '1411.4901-1-5-4': 'Yet, their important role is also to inspire theoretical considerations on non-equilibrium processes [CITATION].', '1411.4901-1-5-5': 'In both these roles, the spectrum of particular realizations of CPs has been remarkably enriched.', '1411.4901-1-5-6': 'In particular, the pair contact processes have been proposed in Ref. [CITATION]; there, a pair of particles annihilate or create a neighbor particle.', '1411.4901-1-6-0': 'The aim of this paper is to report our numerical results on competing CPs of two different kinds.', '1411.4901-1-6-1': 'As far as we know, this case has not been analyzed, with two our texts [CITATION] as an exception.', '1411.4901-1-6-2': 'In literature, applications of competing CPs are of recent interest [CITATION]; yet, in all these approaches the competing processes are of the same kind.', '1411.4901-1-6-3': 'In Ref. [CITATION], the competition has been investigated between the voter model and the pair contact process without annihilation.', '1411.4901-1-6-4': 'The role of network topology has been analyzed by a comparison of results for the Watts-Strogatz network [CITATION] and the Erdos-Renyi network [CITATION], where the clustering coefficient [MATH] has been tuned in both networks.', '1411.4901-1-6-5': 'Our motivation in Ref. [CITATION] was to evaluate the efficiency of the pair contact process by balancing it with the voter model of controlled efficiency; the control was kept by tuning the probability [MATH] of the one-node voter dynamics.', '1411.4901-1-6-6': 'The key result of Ref. [CITATION] was a phase diagram on the plane ([MATH]); below some critical line [MATH], the pair process dominates, while above another critical line [MATH], the voter dynamics prevails.', '1411.4901-1-6-7': 'Between these lines, i.e. for [MATH], the time of relaxation was too long to get a definitive conclusion on the stability of this or that phase.', '1411.4901-1-6-8': 'Both critical lines have been found to depend on the network topology.', '1411.4901-1-7-0': 'In paper [CITATION], preliminary results have been reported on the competition between the voter model dynamics and the Sznajd model dynamics [CITATION].', '1411.4901-1-7-1': 'The latter algorithm bears some resemblance to the pair CP [CITATION]; yet, pairs do not annihilate there, and new particles are created at the whole neighborhood of the pair.', '1411.4901-1-7-2': 'As the result, we have got a slight dependence of the transition line in the phase diagram ([MATH]) on the initial conditions, i.e. on the initial percentage of nodes in the state activated by the voter dynamics.', '1411.4901-1-7-3': 'As we demonstrate below, this result appears to be generic.', '1411.4901-1-7-4': 'The work presented here is entirely devoted to the role of the initial conditions.', '1411.4901-1-7-5': 'This makes the problem more complex; the plane [MATH] to construct the phase diagram is to be substituted by the three-dimensional space [MATH].', '1411.4901-1-8-0': 'The next section ([REF]) is devoted to the model and to the details of our simulation procedure.', '1411.4901-1-8-1': 'The section [REF] provides our numerical results.', '1411.4901-1-8-2': 'In the last section ([REF]) we give a summary, supplemented by a note on a possible application of the scheme presented here.', '1411.4901-1-9-0': '# Model', '1411.4901-1-10-0': '## Network construction', '1411.4901-1-11-0': 'The simulations take place on networks similar to Watts-Strogatz networks [CITATION].', '1411.4901-1-11-1': 'The base for network construction is one-dimensional chain of [MATH] sites, where each site [MATH] is directly linked to nodes labeled as [MATH] and [MATH].', '1411.4901-1-11-2': 'So initially, each node has the same degree [MATH].', '1411.4901-1-11-3': 'The periodic boundary conditions are assumed as well.', '1411.4901-1-12-0': 'For each node [MATH] the links to sites [MATH] and [MATH] are rewired to two randomly selected nodes so far not-connected to node [MATH].', '1411.4901-1-12-1': 'The rewiring procedure occurs with probability [MATH].', '1411.4901-1-12-2': 'The examples of original and rewired network are presented in Fig. [REF].', '1411.4901-1-12-3': 'Increasing rewiring probability [MATH] influence the nodes degree distribution and the network clusterization coefficient.', '1411.4901-1-12-4': 'Please note however, that rewiring procedure does not change average nodes degree, i.e. [MATH].', '1411.4901-1-13-0': 'For given values of rewiring probability [MATH] the set [MATH] of [MATH] networks is generated.', '1411.4901-1-13-1': 'The clusterization coefficient [MATH] for [MATH]-th network is defined as the average over nodes [MATH] of the local coefficient [MATH], where [EQUATION] and [MATH] is the degree of [MATH]-th node, i.e. the number of nodes linked to [MATH], and [MATH] is the actual number of links between these [MATH] nodes [CITATION].', '1411.4901-1-13-2': 'The clusterization coefficients [MATH] for each networks in set [MATH] do not differ more than [MATH] from average values [MATH].', '1411.4901-1-14-0': 'The clusterization coefficient for unrewired [MATH] network is exactly equal to [MATH].', '1411.4901-1-15-0': '## Contact process description', '1411.4901-1-16-0': ""The network's nodes are decorated with spin-like variable [MATH]."", '1411.4901-1-16-1': 'Initially (for [MATH]) the [MATH] value is randomly assigned to the fraction of [MATH] nodes.', '1411.4901-1-16-2': 'The remaining [MATH] nodes are assumed to be in [MATH] state.', '1411.4901-1-17-0': ""Every time step ([MATH]) the random sequence of [MATH] nodes' labels is created by sampling with replacement."", '1411.4901-1-17-1': 'Now, network vertices are visited accordingly to this list.', '1411.4901-1-18-0': 'If the visited node (denoted with double ring in Fig. [REF]) is marked', '1411.4901-1-19-0': 'The simulation time [MATH] should be long enough to ensure reaching stationary state, i.e. [MATH].', '1411.4901-1-20-0': '# Results', '1411.4901-1-21-0': 'In Fig. [REF] the time evolution of fraction of [MATH] nodes [MATH] are presented.', '1411.4901-1-21-1': 'For a given set of [MATH] parameters the results of simulation of the contact process described in Sec. [REF] are averaged over [MATH] networks realizations.', '1411.4901-1-21-2': 'These networks differ both in their topology and initial distribution of [MATH] nodes.', '1411.4901-1-21-3': 'Please note however, that for unrewired network ([MATH], [MATH]) only initial distribution of [MATH] nodes allows for distinguishing among networks.', '1411.4901-1-22-0': 'In Fig. [REF] the dependence of the nodes [MATH] final density [MATH] on initial nodes [MATH] fraction [MATH] are presented.', '1411.4901-1-22-1': 'The curves in Fig. [REF] and [REF] correspond to various network sizes [MATH], 1000, 2000.', '1411.4901-1-22-2': ""With enlarging the system size [MATH] we expect that these curves become more steeper and steeper tending to Heaviside's function [EQUATION] in thermodynamical limit, i.e. for [MATH]."", '1411.4901-1-22-3': 'The common cross point for these curves [MATH] indicate the (unstable) fixed point splitting [MATH] parameter space into two regions:', '1411.4901-1-23-0': 'Please note, that ordinates of these points are equal to [MATH].', '1411.4901-1-23-1': 'This yields a convenient way for rough estimation of abscissas of fixed point basing only on [MATH] vs. [MATH] dependence for single network size (here [MATH]).', '1411.4901-1-24-0': 'The surface of the unstable fixed points in [MATH] space is presented in Fig. [REF].', '1411.4901-1-24-1': 'The system evolves more likely towards [MATH] for [MATH] points situated above this surface while points below this surface lead the system more often to [MATH].', '1411.4901-1-24-2': 'The points on this surface correspond to such value of initial fraction [MATH] of [MATH] nodes (for fixed values [MATH] and [MATH]) for which their final density is [MATH].', '1411.4901-1-24-3': 'Of course, reaching the final concentration [MATH] of [MATH] nodes exactly equal to [MATH] is rather rare.', '1411.4901-1-24-4': 'Thus, we estimate [MATH] as [EQUATION] where [MATH] are the values of [MATH] closest to [MATH] and obeying inequality [MATH] while [MATH] are corresponding initial concentration of [MATH] nodes leading to these values [MATH].', '1411.4901-1-25-0': '# Discussion', '1411.4901-1-26-0': 'Our numerical results indicate that the time evolution drives the system to a homogeneous state where all nodes belong either to [MATH]- or [MATH]-state.', '1411.4901-1-26-1': 'The boundary between the basins of attraction is a surface in the three-dimensional space of parameters: the clustering coefficient [MATH], the probability [MATH] of the [MATH]-process, and the initial concentration of the [MATH]-nodes.', '1411.4901-1-26-2': 'The boundary consists of unstable fixed points.', '1411.4901-1-26-3': 'The data shown in Fig. [REF] indicate, that the transition between two homogeneous states is most sharp for [MATH], i.e. for the Watts-Strogatz network without rewiring.', '1411.4901-1-26-4': 'Once the rewiring introduces some local disorder, the movement of the boundary between the [MATH]-phase and the [MATH]-phase can be stuck on local configurations, and the related metastable states blur the transition.', '1411.4901-1-27-0': 'The advantage of our method of evaluation of the activity of a contact process by counterbalancing it by another contact process is that we get a stationary state which is not frozen absorbing state, but a result of a dynamic equilibrium.', '1411.4901-1-27-1': ""The 'another process' plays a role of a scale, which allows to compare different processes; if a new process appears, its comparison with the voter dynamics allows to evaluate its efficiency with respect to all processes which had been previously compared with the voter model."", '1411.4901-1-28-0': 'Some possible applications of pair processes have been listed already in Ref. [CITATION].', '1411.4901-1-28-1': 'Here we want to add one, related to computer viruses [CITATION].', '1411.4901-1-28-2': 'Namely, the direct competition of the processes considered above can find a counterpart in an algorithm of correcting codes in a network of CP units.', '1411.4901-1-28-3': 'The algorithm could be applied to cure the losses made by destructive viruses, which randomly change data.', '1411.4901-1-28-4': 'Once two neighboring computers compare their versions and find them the same, they can safely share this version with all units connected to the pair, because it is unlikely that errors are found at the same place.', '1411.4901-1-28-5': 'Once they find that their versions are different, both units should be treated as unsafe.', '1411.4901-1-28-6': 'Our results indicate, that the effectiveness of the algorithm depends on the network topology, and in particular on the clustering coefficient.', '1411.4901-1-29-0': 'The work was partially supported by the Polish Ministry of Science and Higher Education and its grants for Scientific Research and by the http://www.plgrid.pl/enPL-Grid Infrastructure.'}","{'1411.4901-2-0-0': 'We investigate two competing contact processes on a set of Watts-Strogatz networks with the clustering coefficient tuned by rewiring.', '1411.4901-2-0-1': 'The base for network construction is one-dimensional chain of [MATH] sites, where each site [MATH] is directly linked to nodes labelled as [MATH] and [MATH].', '1411.4901-2-0-2': 'So initially, each node has the same degree [MATH].', '1411.4901-2-0-3': 'The periodic boundary conditions are assumed as well.', '1411.4901-2-1-0': 'For each node [MATH] the links to sites [MATH] and [MATH] are rewired to two randomly selected nodes so far not-connected to node [MATH].', '1411.4901-2-1-1': 'An increase of the rewiring probability [MATH] influences the nodes degree distribution and the network clusterization coefficient [MATH].', '1411.4901-2-1-2': 'For given values of rewiring probability [MATH] the set [MATH] of [MATH] networks is generated.', '1411.4901-2-2-0': ""The network's nodes are decorated with spin-like variables [MATH]."", '1411.4901-2-2-1': 'During simulation each [MATH] node having a [MATH]-site in its neighbourhood converts this neighbour from [MATH] to [MATH] state.', '1411.4901-2-2-2': 'Conversely, a node in [MATH] state having at least one neighbour also in state [MATH]-state converts all nearest-neighbours of this pair into [MATH]-state.', '1411.4901-2-2-3': 'The latter is realized with probability [MATH].', '1411.4901-2-3-0': 'We plot the dependence of the nodes [MATH] final density [MATH] on initial nodes [MATH] fraction [MATH].', '1411.4901-2-3-1': 'Then, we construct the surface of the unstable fixed points in [MATH] space.', '1411.4901-2-3-2': 'The system evolves more often toward [MATH] for [MATH] points situated above this surface while starting simulation with [MATH] parameters situated below this surface leads system to [MATH].', '1411.4901-2-3-3': 'The points on this surface correspond to such value of initial fraction [MATH] of [MATH] nodes (for fixed values [MATH] and [MATH]) for which their final density is [MATH].', '1411.4901-2-4-0': '# Introduction', '1411.4901-2-5-0': 'In computational modeling, the contact processes (CPs) are dynamic systems on discrete media, where the time evolution of a local state towards survival or extinction of particles is determined by the state of the direct neighborhood of a lattice cell or of a network node.', '1411.4901-2-5-1': 'A simple realization is the voter model [CITATION], where a particle creates another particle in its direct neighborhood.', '1411.4901-2-5-2': 'CPs have been introduced in 1974 as a toy model of spread of epidemic on a lattice [CITATION].', '1411.4901-2-5-3': 'Since then, they evolved to a frame for models in different areas, from symbiotic interactions [CITATION] to population [CITATION] or opinion dynamics [CITATION].', '1411.4901-2-5-4': 'Yet, their important role is also to inspire theoretical considerations on non-equilibrium processes [CITATION].', '1411.4901-2-5-5': 'In both these roles, the spectrum of particular realizations of CPs has been remarkably enriched.', '1411.4901-2-5-6': 'In particular, the pair contact processes have been proposed in Ref. [CITATION]; there, a pair of particles annihilate or create a neighbor particle.', '1411.4901-2-6-0': 'The aim of this paper is to report our numerical results on competing CPs of two different kinds.', '1411.4901-2-6-1': 'As far as we know, this case has not been analyzed, with two our texts [CITATION] as an exception.', '1411.4901-2-6-2': 'In literature, applications of competing CPs are of recent interest [CITATION]; yet, in all these approaches the competing processes are of the same kind.', '1411.4901-2-6-3': 'In Ref. [CITATION], the competition has been investigated between the voter model and the pair contact process without annihilation.', '1411.4901-2-6-4': 'The role of network topology has been analyzed by a comparison of results for the Watts-Strogatz network [CITATION] and the Erdos-Renyi network [CITATION], where the clustering coefficient [MATH] has been tuned in both networks.', '1411.4901-2-6-5': 'Our motivation in Ref. [CITATION] was to evaluate the efficiency of the pair contact process by balancing it with the voter model of controlled efficiency; the control was kept by tuning the probability [MATH] of the one-node voter dynamics.', '1411.4901-2-6-6': 'The key result of Ref. [CITATION] was a phase diagram on the plane ([MATH]); below some critical line [MATH], the pair process dominates, while above another critical line [MATH], the voter dynamics prevails.', '1411.4901-2-6-7': 'Between these lines, i.e. for [MATH], the time of relaxation was too long to get a definitive conclusion on the stability of this or that phase.', '1411.4901-2-6-8': 'Both critical lines have been found to depend on the network topology.', '1411.4901-2-7-0': 'In paper [CITATION], preliminary results have been reported on the competition between the voter model dynamics and the Sznajd model dynamics [CITATION].', '1411.4901-2-7-1': 'The latter algorithm bears some resemblance to the pair CP [CITATION]; yet, pairs do not annihilate there, and new particles are created at the whole neighborhood of the pair.', '1411.4901-2-7-2': 'As the result, we have got a slight dependence of the transition line in the phase diagram ([MATH]) on the initial conditions, i.e. on the initial percentage of nodes in the state activated by the voter dynamics.', '1411.4901-2-7-3': 'As we demonstrate below, this result appears to be generic.', '1411.4901-2-7-4': 'The work presented here is entirely devoted to the role of the initial conditions.', '1411.4901-2-7-5': 'This makes the problem more complex; the plane [MATH] to construct the phase diagram is to be substituted by the three-dimensional space [MATH].', '1411.4901-2-8-0': 'The next section ([REF]) is devoted to the model and to the details of our simulation procedure.', '1411.4901-2-8-1': 'The section [REF] provides our numerical results.', '1411.4901-2-8-2': 'In the last section ([REF]) we give a summary, supplemented by a note on a possible application of the scheme presented here.', '1411.4901-2-9-0': '# Model', '1411.4901-2-10-0': '## Network construction', '1411.4901-2-11-0': 'The simulations take place on networks similar to Watts-Strogatz networks [CITATION].', '1411.4901-2-11-1': 'The base for network construction is one-dimensional chain of [MATH] sites, where each site [MATH] is directly linked to nodes labeled as [MATH] and [MATH].', '1411.4901-2-11-2': 'So initially, each node has the same degree [MATH].', '1411.4901-2-11-3': 'The periodic boundary conditions are assumed as well.', '1411.4901-2-12-0': 'For each node [MATH] the links to sites [MATH] and [MATH] are rewired to two randomly selected nodes so far not-connected to node [MATH].', '1411.4901-2-12-1': 'The rewiring procedure occurs with probability [MATH].', '1411.4901-2-12-2': 'The examples of original and rewired network are presented in Fig. [REF].', '1411.4901-2-12-3': 'Increasing rewiring probability [MATH] influence the nodes degree distribution and the network clusterization coefficient.', '1411.4901-2-12-4': 'Please note however, that rewiring procedure does not change average nodes degree, i.e. [MATH].', '1411.4901-2-13-0': 'For given values of rewiring probability [MATH] the set [MATH] of [MATH] networks is generated.', '1411.4901-2-13-1': 'The clusterization coefficient [MATH] for [MATH]-th network is defined as the average over nodes [MATH] of the local coefficient [MATH], where [EQUATION] and [MATH] is the degree of [MATH]-th node, i.e. the number of nodes linked to [MATH], and [MATH] is the actual number of links between these [MATH] nodes [CITATION].', '1411.4901-2-13-2': 'The clusterization coefficients [MATH] for each networks in set [MATH] do not differ more than [MATH] from average values [MATH].', '1411.4901-2-14-0': 'The clusterization coefficient for unrewired [MATH] network is exactly equal to [MATH].', '1411.4901-2-15-0': '## Contact process description', '1411.4901-2-16-0': ""The network's nodes are decorated with spin-like variable [MATH]."", '1411.4901-2-16-1': 'Initially (for [MATH]) the [MATH] value is randomly assigned to the fraction of [MATH] nodes.', '1411.4901-2-16-2': 'The remaining [MATH] nodes are assumed to be in [MATH] state.', '1411.4901-2-17-0': ""Every time step ([MATH]) the random sequence of [MATH] nodes' labels is created by sampling with replacement."", '1411.4901-2-17-1': 'Now, network vertices are visited accordingly to this list.', '1411.4901-2-18-0': 'If the visited node (denoted with double ring in Fig. [REF]) is marked', '1411.4901-2-19-0': 'The simulation time [MATH] should be long enough to ensure reaching stationary state, i.e. [MATH].', '1411.4901-2-20-0': '# Results', '1411.4901-2-21-0': 'In Fig. [REF] the time evolution of fraction of [MATH] nodes [MATH] are presented.', '1411.4901-2-21-1': 'For a given set of [MATH] parameters the results of simulation of the contact process described in Sec. [REF] are averaged over [MATH] networks realizations.', '1411.4901-2-21-2': 'These networks differ both in their topology and initial distribution of [MATH] nodes.', '1411.4901-2-21-3': 'Please note however, that for unrewired network ([MATH], [MATH]) only initial distribution of [MATH] nodes allows for distinguishing among networks.', '1411.4901-2-22-0': 'In Fig. [REF] the dependence of the nodes [MATH] final density [MATH] on initial nodes [MATH] fraction [MATH] are presented.', '1411.4901-2-22-1': 'The curves in Fig. [REF] and [REF] correspond to various network sizes [MATH], 1000, 2000.', '1411.4901-2-22-2': ""With enlarging the system size [MATH] we expect that these curves become more steeper and steeper tending to Heaviside's function [EQUATION] in thermodynamical limit, i.e. for [MATH]."", '1411.4901-2-22-3': 'The common cross point for these curves [MATH] indicate the (unstable) fixed point splitting [MATH] parameter space into two regions:', '1411.4901-2-23-0': 'Please note, that ordinates of these points are equal to [MATH].', '1411.4901-2-23-1': 'This yields a convenient way for rough estimation of abscissas of fixed point basing only on [MATH] vs. [MATH] dependence for single network size (here [MATH]).', '1411.4901-2-24-0': 'The surface of the unstable fixed points in [MATH] space is presented in Fig. [REF].', '1411.4901-2-24-1': 'The system evolves more likely towards [MATH] for [MATH] points situated above this surface while points below this surface lead the system more often to [MATH].', '1411.4901-2-24-2': 'The points on this surface correspond to such value of initial fraction [MATH] of [MATH] nodes (for fixed values [MATH] and [MATH]) for which their final density is [MATH].', '1411.4901-2-24-3': 'Of course, reaching the final concentration [MATH] of [MATH] nodes exactly equal to [MATH] is rather rare.', '1411.4901-2-24-4': 'Thus, we estimate [MATH] as [EQUATION] where [MATH] are the values of [MATH] closest to [MATH] and obeying inequality [MATH] while [MATH] are corresponding initial concentration of [MATH] nodes leading to these values [MATH].', '1411.4901-2-25-0': '# Discussion', '1411.4901-2-26-0': 'Our numerical results indicate that the time evolution drives the system to a homogeneous state where all nodes belong either to [MATH]- or [MATH]-state.', '1411.4901-2-26-1': 'The boundary between the basins of attraction is a surface in the three-dimensional space of parameters: the clustering coefficient [MATH], the probability [MATH] of the [MATH]-process, and the initial concentration of the [MATH]-nodes.', '1411.4901-2-26-2': 'The boundary consists of unstable fixed points.', '1411.4901-2-26-3': 'The data shown in Fig. [REF] indicate, that the transition between two homogeneous states is most sharp for [MATH], i.e. for the Watts-Strogatz network without rewiring.', '1411.4901-2-26-4': 'Once the rewiring introduces some local disorder, the movement of the boundary between the [MATH]-phase and the [MATH]-phase can be stuck on local configurations, and the related metastable states blur the transition.', '1411.4901-2-27-0': 'The advantage of our method of evaluation of the activity of a contact process by counterbalancing it by another contact process is that we get a stationary state which is not a frozen absorbing state, but a result of a dynamic equilibrium.', '1411.4901-2-27-1': ""The 'another process' plays a role of a scale, which allows to compare different processes; if a new process appears, its comparison with the voter dynamics allows to evaluate its efficiency with respect to all processes which had been previously compared with the voter model."", '1411.4901-2-27-2': 'Also, we evade the method of quasistationary state, which limits the statistics to surviving trials [CITATION].', '1411.4901-2-28-0': 'Some possible applications of pair processes have been listed already in Ref. [CITATION].', '1411.4901-2-28-1': 'Here we want to add one, related to computer viruses [CITATION].', '1411.4901-2-28-2': 'Namely, the direct competition of the processes considered above can find a counterpart in an algorithm of correcting codes in a network of CP units.', '1411.4901-2-28-3': 'The algorithm could be applied to cure the losses made by destructive viruses, which randomly change data.', '1411.4901-2-28-4': 'Once two neighboring computers compare their versions and find them the same, they can safely share this version with all units connected to the pair, because it is unlikely that errors are found at the same place.', '1411.4901-2-28-5': 'Once they find that their versions are different, both units should be treated as unsafe.', '1411.4901-2-28-6': 'Our results indicate, that the effectiveness of the algorithm depends on the network topology, and in particular on the clustering coefficient.', '1411.4901-2-29-0': 'The work was partially supported by the Polish Ministry of Science and Higher Education and its grants for Scientific Research and by the http://www.plgrid.pl/enPL-Grid Infrastructure.'}","[['1411.4901-1-14-0', '1411.4901-2-14-0'], ['1411.4901-1-0-0', '1411.4901-2-0-0'], ['1411.4901-1-0-1', '1411.4901-2-0-1'], ['1411.4901-1-0-2', '1411.4901-2-0-2'], ['1411.4901-1-0-3', '1411.4901-2-0-3'], ['1411.4901-1-21-0', '1411.4901-2-21-0'], ['1411.4901-1-21-1', '1411.4901-2-21-1'], ['1411.4901-1-21-2', '1411.4901-2-21-2'], ['1411.4901-1-21-3', '1411.4901-2-21-3'], ['1411.4901-1-7-0', '1411.4901-2-7-0'], ['1411.4901-1-7-1', '1411.4901-2-7-1'], ['1411.4901-1-7-2', '1411.4901-2-7-2'], ['1411.4901-1-7-3', '1411.4901-2-7-3'], ['1411.4901-1-7-4', '1411.4901-2-7-4'], ['1411.4901-1-7-5', '1411.4901-2-7-5'], ['1411.4901-1-24-0', '1411.4901-2-24-0'], ['1411.4901-1-24-1', '1411.4901-2-24-1'], ['1411.4901-1-24-2', '1411.4901-2-24-2'], ['1411.4901-1-24-3', '1411.4901-2-24-3'], ['1411.4901-1-24-4', '1411.4901-2-24-4'], ['1411.4901-1-1-0', '1411.4901-2-1-0'], ['1411.4901-1-1-1', '1411.4901-2-1-1'], ['1411.4901-1-1-2', '1411.4901-2-1-2'], ['1411.4901-1-26-0', '1411.4901-2-26-0'], ['1411.4901-1-26-1', '1411.4901-2-26-1'], ['1411.4901-1-26-2', '1411.4901-2-26-2'], ['1411.4901-1-26-3', '1411.4901-2-26-3'], ['1411.4901-1-26-4', '1411.4901-2-26-4'], ['1411.4901-1-3-0', '1411.4901-2-3-0'], ['1411.4901-1-3-1', '1411.4901-2-3-1'], ['1411.4901-1-3-2', '1411.4901-2-3-2'], ['1411.4901-1-3-3', '1411.4901-2-3-3'], ['1411.4901-1-11-0', '1411.4901-2-11-0'], ['1411.4901-1-11-1', '1411.4901-2-11-1'], ['1411.4901-1-11-2', '1411.4901-2-11-2'], ['1411.4901-1-11-3', '1411.4901-2-11-3'], ['1411.4901-1-13-0', '1411.4901-2-13-0'], ['1411.4901-1-13-1', '1411.4901-2-13-1'], ['1411.4901-1-13-2', '1411.4901-2-13-2'], ['1411.4901-1-29-0', '1411.4901-2-29-0'], ['1411.4901-1-12-0', '1411.4901-2-12-0'], ['1411.4901-1-12-1', '1411.4901-2-12-1'], ['1411.4901-1-12-2', '1411.4901-2-12-2'], ['1411.4901-1-12-3', '1411.4901-2-12-3'], ['1411.4901-1-12-4', '1411.4901-2-12-4'], 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'1411.4901-2-0-3'], ['1411.4901-1-21-0', '1411.4901-2-21-0'], ['1411.4901-1-21-1', '1411.4901-2-21-1'], ['1411.4901-1-21-2', '1411.4901-2-21-2'], ['1411.4901-1-21-3', '1411.4901-2-21-3'], ['1411.4901-1-7-0', '1411.4901-2-7-0'], ['1411.4901-1-7-1', '1411.4901-2-7-1'], ['1411.4901-1-7-2', '1411.4901-2-7-2'], ['1411.4901-1-7-3', '1411.4901-2-7-3'], ['1411.4901-1-7-4', '1411.4901-2-7-4'], ['1411.4901-1-7-5', '1411.4901-2-7-5'], ['1411.4901-1-24-0', '1411.4901-2-24-0'], ['1411.4901-1-24-1', '1411.4901-2-24-1'], ['1411.4901-1-24-2', '1411.4901-2-24-2'], ['1411.4901-1-24-3', '1411.4901-2-24-3'], ['1411.4901-1-24-4', '1411.4901-2-24-4'], ['1411.4901-1-1-0', '1411.4901-2-1-0'], ['1411.4901-1-1-1', '1411.4901-2-1-1'], ['1411.4901-1-1-2', '1411.4901-2-1-2'], ['1411.4901-1-26-0', '1411.4901-2-26-0'], ['1411.4901-1-26-1', '1411.4901-2-26-1'], ['1411.4901-1-26-2', '1411.4901-2-26-2'], ['1411.4901-1-26-3', '1411.4901-2-26-3'], ['1411.4901-1-26-4', '1411.4901-2-26-4'], ['1411.4901-1-3-0', '1411.4901-2-3-0'], ['1411.4901-1-3-1', '1411.4901-2-3-1'], ['1411.4901-1-3-2', '1411.4901-2-3-2'], ['1411.4901-1-3-3', '1411.4901-2-3-3'], ['1411.4901-1-11-0', '1411.4901-2-11-0'], ['1411.4901-1-11-1', '1411.4901-2-11-1'], ['1411.4901-1-11-2', '1411.4901-2-11-2'], ['1411.4901-1-11-3', '1411.4901-2-11-3'], ['1411.4901-1-13-0', '1411.4901-2-13-0'], ['1411.4901-1-13-1', '1411.4901-2-13-1'], ['1411.4901-1-13-2', '1411.4901-2-13-2'], ['1411.4901-1-29-0', '1411.4901-2-29-0'], ['1411.4901-1-12-0', '1411.4901-2-12-0'], ['1411.4901-1-12-1', '1411.4901-2-12-1'], ['1411.4901-1-12-2', '1411.4901-2-12-2'], ['1411.4901-1-12-3', '1411.4901-2-12-3'], ['1411.4901-1-12-4', '1411.4901-2-12-4'], ['1411.4901-1-6-0', '1411.4901-2-6-0'], ['1411.4901-1-6-1', '1411.4901-2-6-1'], ['1411.4901-1-6-2', '1411.4901-2-6-2'], ['1411.4901-1-6-3', '1411.4901-2-6-3'], ['1411.4901-1-6-4', '1411.4901-2-6-4'], ['1411.4901-1-6-5', '1411.4901-2-6-5'], ['1411.4901-1-6-6', '1411.4901-2-6-6'], ['1411.4901-1-6-7', '1411.4901-2-6-7'], ['1411.4901-1-6-8', '1411.4901-2-6-8'], ['1411.4901-1-22-0', '1411.4901-2-22-0'], ['1411.4901-1-22-1', '1411.4901-2-22-1'], ['1411.4901-1-22-2', '1411.4901-2-22-2'], ['1411.4901-1-17-0', '1411.4901-2-17-0'], ['1411.4901-1-17-1', '1411.4901-2-17-1'], ['1411.4901-1-19-0', '1411.4901-2-19-0'], ['1411.4901-1-23-0', '1411.4901-2-23-0'], ['1411.4901-1-23-1', '1411.4901-2-23-1'], ['1411.4901-1-16-0', '1411.4901-2-16-0'], ['1411.4901-1-16-1', '1411.4901-2-16-1'], ['1411.4901-1-16-2', '1411.4901-2-16-2'], ['1411.4901-1-28-0', '1411.4901-2-28-0'], ['1411.4901-1-28-1', '1411.4901-2-28-1'], ['1411.4901-1-28-2', '1411.4901-2-28-2'], ['1411.4901-1-28-3', '1411.4901-2-28-3'], ['1411.4901-1-28-4', '1411.4901-2-28-4'], ['1411.4901-1-28-5', '1411.4901-2-28-5'], ['1411.4901-1-28-6', '1411.4901-2-28-6'], ['1411.4901-1-5-0', '1411.4901-2-5-0'], ['1411.4901-1-5-1', '1411.4901-2-5-1'], ['1411.4901-1-5-2', '1411.4901-2-5-2'], ['1411.4901-1-5-3', '1411.4901-2-5-3'], ['1411.4901-1-5-4', '1411.4901-2-5-4'], ['1411.4901-1-5-5', '1411.4901-2-5-5'], ['1411.4901-1-5-6', '1411.4901-2-5-6'], ['1411.4901-1-2-0', '1411.4901-2-2-0'], ['1411.4901-1-2-1', '1411.4901-2-2-1'], ['1411.4901-1-2-2', '1411.4901-2-2-2'], ['1411.4901-1-2-3', '1411.4901-2-2-3'], ['1411.4901-1-8-0', '1411.4901-2-8-0'], ['1411.4901-1-8-1', '1411.4901-2-8-1'], ['1411.4901-1-8-2', '1411.4901-2-8-2'], ['1411.4901-1-18-0', '1411.4901-2-18-0'], ['1411.4901-1-27-1', '1411.4901-2-27-1']]","[['1411.4901-1-27-0', '1411.4901-2-27-0']]",[],[],[],"['1411.4901-1-22-3', '1411.4901-2-22-3']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1411.4901,,,
0911.4244,"{'0911.4244-1-0-0': 'We review the quiver matrix model (the ITEP model) in the light of the recent progress on 2d-4d connection of conformal field theories, in particular, the AGT conjecture and the recent proposal by Dijkgraaf and Vafa on the relation between Toda field theories and a class of quiver superconformal gauge theories.', '0911.4244-1-0-1': 'On the basis of the CFT representation of the [MATH] deformation of the model, a quantum spectral curve is introduced as [MATH] at finite [MATH] and for [MATH].', '0911.4244-1-0-2': 'The planar loop equation in the large [MATH] limit follows with the aid of [MATH] constraints.', '0911.4244-1-0-3': 'Residue analysis is provided both for the curve of the matrix model with the ""multi-log"" potential and for the Seiberg-Witten curve in the case of [MATH] with six flavors, leading to the matching of the mass parameters.', '0911.4244-1-1-0': '# Introduction', '0911.4244-1-2-0': 'Matrix models have had successes in several stages of the developments in string theory, gauge theory and related studies of integrable systems.', '0911.4244-1-2-1': 'A list of those in the last twenty years include 2d gravity, exact evaluation of gluino condensate prepotential, topological strings, etc.', '0911.4244-1-3-0': 'Recent progress has been triggered by the construction of a large class of [MATH] superconformal [MATH] ""generalized quiver"" gauge theories in four dimensions by Gaiotto [CITATION].', '0911.4244-1-3-1': '(See also [CITATION]).', '0911.4244-1-3-2': 'Subsequently an interesting conjecture has been made by Alday, Gaiotto and Tachikawa (AGT) [CITATION] (and its [MATH] generalization by [CITATION]) on the equivalence of the Nekrasov partition function [CITATION] and the [MATH] conformal block of the Toda field theory.', '0911.4244-1-3-3': 'These are followed by a number of extensive checks and pieces of supporting evidence [CITATION].', '0911.4244-1-3-4': 'Very recently, Dijkgraaf and Vafa [CITATION] have suggested an explanation of this phenomenon by the so-called quiver matrix model [CITATION].', '0911.4244-1-3-5': '(In this paper, it is occasionally referred to as the ITEP model [CITATION]).', '0911.4244-1-3-6': 'Their reasoning is based on the matrix model realization of type IIB topological strings on a local Calabi-Yau with a local [MATH] singularity which geometrically engineers the gauge theory.', '0911.4244-1-3-7': 'By choosing ""multi-Penner"" potentials [CITATION] for the [MATH] quiver matrix model, they argued that the spectral curve of the matrix model at large [MATH] (the size of the matrix) can be understood as the Seiberg-Witten curve of the attendant [MATH] generalized quiver superconformal gauge theory (SCFT).', '0911.4244-1-4-0': 'The AGT conjecture is regarded as a more concrete realization of the folklore connection between [MATH] gauge theories and the attendant [MATH] sigma models (see [CITATION] for instance), that led to the study of two dimensional quantum integrable field theories in late seventies ([CITATION] for instance).', '0911.4244-1-4-1': 'In the light of potential importance of this subject, we find it useful to devote the substantial part of the present paper in reviewing and reformulating the basic structure of the quiver matrix model [CITATION].', '0911.4244-1-5-0': 'The quiver matrix model associated with Lie algebra [MATH] of ADE type with rank [MATH] is obtained as a solution to extended Virasoro constraints, i.e., [MATH] constraints [CITATION] at finite [MATH] [CITATION].', '0911.4244-1-5-1': 'This fact that the model automatically implements the [MATH] constraints at finite [MATH] is an advantage over more traditional two- and multi-matrix models where finite [MATH] Schwinger-Dyson equations are typically [MATH] type and are more involved [CITATION].', '0911.4244-1-5-2': 'The model is defined by using [MATH] independent free massless chiral bosons in two dimensional CFT with the central charge [MATH] and the final form of the partition function is formulated as the integrations over eigenvalues of the matrices.', '0911.4244-1-5-3': 'A key ingredient of its construction is a set of screening charges of the [MATH] CFT.', '0911.4244-1-5-4': 'By construction, the partition function respects the [MATH], the extended Casimir algebra generated by higher spin currents which commute with the screening charges.', '0911.4244-1-5-5': 'The partition function can be reformulated as integrations over the matrices in the adjoint and bi-fundamental representations.', '0911.4244-1-6-0': 'In the context of the AGT conjecture for more general Toda CFTs, it is natural to consider ""[MATH]-ensembles"" of ADE quiver matrix models.', '0911.4244-1-6-1': 'In this paper, using the [MATH]-ensemble of the [MATH] quiver matrix model at finite [MATH], we define a non-commutative Calabi-Yau threefold with the quantum deformed [MATH] singularity as [EQUATION]', '0911.4244-1-6-2': 'For [MATH], it takes the following form [EQUATION] with the non-commutativity [MATH].', '0911.4244-1-7-0': 'In order to bring these analyses in the more recent context, we take a simple example, namely, [MATH] SQCD with [MATH] flavors to provide a residue analysis of the quiver matrix model curve and that of the Seiberg-Witten curve from type IIA and M-theory consideration.', '0911.4244-1-7-1': 'This leads to the matching of the mass parameters.', '0911.4244-1-8-0': 'This paper is organized as follows.', '0911.4244-1-8-1': 'In section [MATH], after giving several punchlines, we review known facts on the one-matrix model, using the CFT notation and the Virasoro constraints at finite [MATH] [CITATION].', '0911.4244-1-8-2': 'In section [MATH], ordinary ADE quiver matrix models are reviewed in the same spirit and their ""[MATH]-ensemble"" is introduced.', '0911.4244-1-8-3': 'For the case of [MATH] quiver matrix model, we show that the quantum spectral curve [EQUATION] is well-defined within the matrix model integral.', '0911.4244-1-8-4': 'The partition function obeys the [MATH] constraints at finite [MATH] which are the properties of the original matrix integrals.', '0911.4244-1-8-5': 'The planar loop equation follows from these structures together with the large [MATH] factorization.', '0911.4244-1-8-6': ""In section [MATH], we adopt Dijkgraaf-Vafa's recipe to treat [MATH] superconformal gauge theory by using the quiver matrix model with Penner like action."", '0911.4244-1-8-7': 'We concentrate on the [MATH] quiver matrix model and consider the spectral curve.', '0911.4244-1-8-8': 'Section [MATH] treats the Seiberg-Witten curve for [MATH] superconformal gauge theory.', '0911.4244-1-8-9': 'We see that this curve enjoys the same properties as those of the spectral curve.', '0911.4244-1-8-10': 'In particular, we give a matching of the mass parameters.', '0911.4244-1-9-0': '# One-matrix model and the [MATH]-ensemble', '0911.4244-1-10-0': 'In the case of [MATH], the quiver matrix model corresponds to the Hermitian one-matrix model [CITATION] (see also [CITATION]).', '0911.4244-1-10-1': 'The associated CFT is a single free boson with [MATH].', '0911.4244-1-10-2': 'On the other hand, the Liouville CFT which appears in AGT conjecture for [MATH] has the central charge [MATH] with [MATH].', '0911.4244-1-10-3': 'It is known that there is a one-matrix model which has a connection with the CFT with [MATH].', '0911.4244-1-10-4': 'It is the [MATH]-ensemble of one-matrix model with [MATH].', '0911.4244-1-10-5': 'It is easy to deal with the Liouville CFT by introducing the Feign-Fuchs background charge in the CFT notation.', '0911.4244-1-10-6': 'The CFT notation works well for the [MATH]-ensemble of the one-matrix model and the appearance of the CFT with [MATH] in the matrix model is a built-in result.', '0911.4244-1-10-7': 'The case of [MATH] corresponds to the ordinary Hermitian one-matrix model.', '0911.4244-1-10-8': ""The AGT conjecture implies that these deformation parameters are related to Nekrasov's deformation parameters [MATH] and [MATH] by [CITATION] [EQUATION]"", '0911.4244-1-10-9': 'Note that [MATH] and [MATH].', '0911.4244-1-11-0': 'It is known that the [MATH]-ensemble of one-matrix model at finite [MATH] is related to a quantum (non-commutative) spectral curve of the form [EQUATION] whose non-commutativity is given by [EQUATION]', '0911.4244-1-11-1': 'Using the [MATH]-ensemble matrix model, it is possible to define a non-commutative local Calabi-Yau threefold with a quantum deformed [MATH] singularity [EQUATION]', '0911.4244-1-11-2': 'It can be written as the form [EQUATION] where [MATH] is one-matrix model average over the [MATH]-ensemble and [MATH] is the energy momentum tensor of [MATH] CFT expressed as the collective field of the matrix eigenvalues.', '0911.4244-1-11-3': 'Later, we will give exact definitions of these quantities.', '0911.4244-1-12-0': '## Hermitian one-matrix model: undeformed case', '0911.4244-1-13-0': 'In this case the relevant matrix model is the Hermitian one-matrix model.', '0911.4244-1-13-1': 'The partition function takes the form [EQUATION]', '0911.4244-1-13-2': 'Here [MATH] is an [MATH] Hermitian matrix.', '0911.4244-1-13-3': 'Correlation function of this matrix model is defined by [EQUATION]', '0911.4244-1-13-4': 'Here [MATH] is a function of the Hermitian matrix.', '0911.4244-1-14-0': 'The partition function [REF] can be written in terms of the eigenvalues [MATH] of the matrix [MATH]: [EQUATION] where [MATH] is the Vandermonde determinant [EQUATION]', '0911.4244-1-14-1': 'It is well-known that the Hermitian matrix model has a close connection with the [MATH] free chiral boson.', '0911.4244-1-14-2': 'The partition function can be rewritten in terms of CFT operators.', '0911.4244-1-14-3': 'The mode expansion of the chiral boson [MATH] is chosen as follows: [EQUATION] and the non-trivial commutation relations are given by [EQUATION]', '0911.4244-1-14-4': 'Hence, our normalization of the correlator is given by [EQUATION]', '0911.4244-1-14-5': 'The energy momentum tensor with the central charge [MATH] is given by [EQUATION]', '0911.4244-1-14-6': 'The screening charges which commute with the Virasoro generators [MATH] are given by [EQUATION] with a certain integration contour.', '0911.4244-1-15-0': 'The Fock vacuum is given by [EQUATION]', '0911.4244-1-15-1': 'Let [EQUATION]', '0911.4244-1-15-2': 'Then, the partition function [REF] of the Hermitian matrix model can be rewritten in terms of the free chiral boson as follows [EQUATION]', '0911.4244-1-15-3': 'Associated with this expression, for an operator [MATH] constructed from the boson oscillators, we use the following notation [EQUATION]', '0911.4244-1-15-4': 'Within the normal ordering, correlators consisting of the chiral boson [MATH] in CFT have their counterparts in the matrix model correlators [REF]: [EQUATION]', '0911.4244-1-15-5': 'In the matrix model correlator, the chiral boson is realized as a collective field of the eigenvalues: [EQUATION]', '0911.4244-1-15-6': 'It is known that the partition function of the Hermitian matrix model at finite [MATH] obeys the Virasoro constraints [CITATION].', '0911.4244-1-15-7': 'In the CFT language, it follows from the commutativity of the Virasoro generators [MATH] with the screening charge [MATH] and [EQUATION]', '0911.4244-1-15-8': 'In the ITEP construction, the Virasoro constraints manifestly hold at finite [MATH] as [EQUATION]', '0911.4244-1-15-9': 'They are equivalent to the regularity of the correlator of the energy momentum tensor [EQUATION] at [MATH].', '0911.4244-1-16-0': 'Now, using the help of the Hermitian matrix model correlator, a local Calabi-Yau threefold with [MATH] singularity over a Riemann surface [MATH] can be defined by [EQUATION]', '0911.4244-1-16-1': 'Using the collective field expression [EQUATION] we have (for the derivation, see the next subsection, below [REF]) [EQUATION] where [EQUATION]', '0911.4244-1-16-2': 'Hence, the local Calabi-Yau threefold is a surface in [MATH] defined by [EQUATION]', '0911.4244-1-16-3': 'At [MATH], it describes some algebraic curve [MATH] in [MATH]: [EQUATION]', '0911.4244-1-16-4': 'Note that this algebraic curve is well-defined for finite [MATH] due to the Virasoro constraints of the matrix model.', '0911.4244-1-17-0': '## [MATH]-ensemble', '0911.4244-1-18-0': ""Nekrasov's deformation corresponds to the modification of the energy-momentum tensor [REF] by the introduction of the background charge a la Feign-Fuchs: [EQUATION]"", '0911.4244-1-18-1': 'This energy momentum tensor has the central charge [MATH].', '0911.4244-1-18-2': 'Undeformed case is recovered at [MATH], [MATH] and [MATH].', '0911.4244-1-19-0': 'Screening charges for this energy momentum tensor are given by [EQUATION]', '0911.4244-1-19-1': 'Let [EQUATION]', '0911.4244-1-19-2': 'It is natural to consider the following deformation of the partition function [REF]: [EQUATION]', '0911.4244-1-19-3': 'This matrix model is known as the [MATH]-ensemble [CITATION] with [MATH].', '0911.4244-1-19-4': 'For [MATH], it corresponds to the integrations over an orthogonal, hermitian and symplectic matrix respectively.', '0911.4244-1-20-0': 'Instead of the screening charge [MATH], we can use [MATH] to express the partition of the [MATH]-ensemble model.', '0911.4244-1-20-1': 'The corresponding expressions are obtained by replacing [MATH] with [MATH].', '0911.4244-1-21-0': 'It is known that this partition function is related to a non-commutative (or quantum) spectral curve [CITATION].', '0911.4244-1-21-1': 'In this case, the non-commutativity is given by [EQUATION]', '0911.4244-1-21-2': 'For [MATH], [MATH] can be realized as a differential operator [MATH].', '0911.4244-1-22-0': 'Note that the energy-momentum tensor [REF] can be defined by the Miura transformation: [EQUATION]', '0911.4244-1-22-1': 'The collective field expression of the chiral boson [MATH] now becomes [EQUATION]', '0911.4244-1-22-2': 'Note that [EQUATION]', '0911.4244-1-22-3': 'Using the collective field expression, we see that [EQUATION]', '0911.4244-1-22-4': 'The terms in the second line of [REF] correspond to the ""singular"" part of the energy momentum tensor and we can see that [EQUATION] where [EQUATION]', '0911.4244-1-22-5': 'Therefore the Virasoro constraints for the deformed one-matrix model imply that [EQUATION] where [EQUATION]', '0911.4244-1-22-6': 'Here the matrix model average is defined as in the undeformed case.', '0911.4244-1-22-7': 'Explicitly for some function [MATH] of the eigenvalues, we have [EQUATION]', '0911.4244-1-22-8': 'Hence the quantum spectral curve related to the [MATH]-ensemble is defined by [EQUATION]', '0911.4244-1-22-9': 'Explicitly, it is given by [EQUATION]', '0911.4244-1-22-10': 'Note that this quantum spectral curve can be rewritten as [EQUATION]', '0911.4244-1-22-11': 'Therefore, the associated local Calabi-Yau threefold also becomes a non-commutative surface: [EQUATION]', '0911.4244-1-22-12': 'Strictly speaking, for [MATH], [MATH] is a differential operator and the quantum spectral curve is a differential equation for some ""wave function.""', '0911.4244-1-22-13': 'This equation has a close connection with [MATH]-gravity and Hitchin systems.', '0911.4244-1-22-14': 'For recent discussion on this point in the light of the AGT conjecture, see [CITATION] and references therein.', '0911.4244-1-22-15': 'From the point of view of the [MATH]-module, see [CITATION].', '0911.4244-1-22-16': 'In string theory, the non-commutativity corresponds to turn on a constant NS two-form.', '0911.4244-1-23-0': 'Our main concern is the large [MATH] string/gauge duality.', '0911.4244-1-23-1': ""In the planar limit (a large [MATH] limit with the 't Hooft coupling [MATH] kept finite), [MATH] and thus the non-commutativity vanishes : [MATH]."", '0911.4244-1-24-0': '## Large [MATH] limit', '0911.4244-1-25-0': 'The partition function [REF] has a topological expansion [EQUATION]', '0911.4244-1-25-1': ""In the large [MATH] limit with the 't Hooft coupling [MATH] kept finite, leading contribution comes from the planar part [MATH] in [REF] and can be evaluated by the saddle point method."", '0911.4244-1-25-2': 'For simplicity, we assume that the parameter [MATH] is pure imaginary and [MATH].', '0911.4244-1-25-3': ""In this case Nekrasov's deformation parameters [MATH] and [MATH] are real."", '0911.4244-1-26-0': 'The stationary conditions [MATH] yield [EQUATION]', '0911.4244-1-26-1': 'Since we assume that [MATH] is pure imaginary, these stationary equations have real solutions [MATH].', '0911.4244-1-26-2': 'We evaluate the partition function around a classical solution with certain filling fractions [MATH] around the local extrema [MATH].', '0911.4244-1-27-0': 'In the planar limit, the large [MATH] factorization yields [EQUATION] where [EQUATION]', '0911.4244-1-27-1': 'Here [MATH] is the density function of the solution to the stationary conditions [REF].', '0911.4244-1-27-2': '[EQUATION]', '0911.4244-1-27-3': 'The stationary conditions in the large [MATH] limit go to [EQUATION]', '0911.4244-1-27-4': 'Here [MATH] denotes the principal value.', '0911.4244-1-27-5': 'Note that in the stationary conditions and in the definition of [MATH], the parameter [MATH] always appears in the combination [MATH].', '0911.4244-1-27-6': ""Therefore, if we replace the 't Hooft coupling [MATH] by [MATH], we can use the large [MATH] formulas of the undeformed Hermitian one-matrix model."", '0911.4244-1-27-7': ""We will call [MATH] deformed 't Hooft coupling."", '0911.4244-1-28-0': 'Hence, in the large [MATH] limit, the local Calabi-Yau is deformation of [MATH] [EQUATION] and the algebraic curve [MATH] becomes [EQUATION] and the points [MATH] on [MATH] can be covered by two sheets [MATH].', '0911.4244-1-29-0': 'In the large [MATH] limit, the Virasoro constraints become an algebraic equation [EQUATION]', '0911.4244-1-30-0': '# ADE quiver matrix models and their ""[MATH]-ensemble""', '0911.4244-1-31-0': 'In this section, we first briefly review the ADE quiver matrix models [CITATION].', '0911.4244-1-31-1': 'An excellent review for the undeformed case can be found in [CITATION].', '0911.4244-1-31-2': 'Then, we introduce the ""[MATH]-ensemble"" or deformed ADE quiver matrix models.', '0911.4244-1-31-3': 'For [MATH] cases, they can be found in [CITATION].', '0911.4244-1-32-0': 'Using the [MATH]-deformed [MATH] quiver matrix model, we introduce a non-commutative local Calabi-Yau threefold related to deformations of [MATH] singularities.', '0911.4244-1-33-0': '## ADE quiver matrix models and CFT with [MATH]', '0911.4244-1-34-0': 'Let [MATH] be a finite dimensional Lie algebra of ADE type with rank [MATH], [MATH] the Cartan subalgebra of [MATH], and [MATH] its dual.', '0911.4244-1-34-1': 'We sometimes denote the natural pairings between [MATH] and [MATH] by [MATH]: [EQUATION]', '0911.4244-1-34-2': 'Let [MATH] be simple roots of [MATH] and [MATH] is the inner product on [MATH].', '0911.4244-1-34-3': 'Our normalization is chosen as [MATH].', '0911.4244-1-34-4': 'The fundamental weights are denoted by [MATH] [EQUATION]', '0911.4244-1-34-5': 'In the Dynkin diagram of [MATH] we associate [MATH] Hermitian matrices [MATH] with vertices [MATH] for simple roots [MATH], and complex [MATH] matrices [MATH] and their Hermitian conjugate [MATH] with links connecting vertices [MATH] and [MATH].', '0911.4244-1-34-6': 'We label links of the Dynkin diagram by pairs of node label [MATH] with an ordering [MATH].', '0911.4244-1-34-7': 'Let [MATH] and [MATH] be the set of ""edges"" [MATH] (with [MATH]) and the set of ""arrows"" [MATH] respectively: [EQUATION]', '0911.4244-1-35-0': 'The partition function of the quiver matrix model [CITATION] associated with [MATH] is given by [EQUATION] where [EQUATION] with real constants [MATH] obeying the conditions [MATH].', '0911.4244-1-35-1': 'Note that [EQUATION]', '0911.4244-1-35-2': 'The integration measures [MATH] and [MATH] are defined by using the metrics [MATH] and [MATH] respectively.', '0911.4244-1-36-0': 'Integrations over [MATH] are Gauss-Fresnel type and are easily performed: [EQUATION] where [MATH] is the [MATH] identity matrix and [MATH] denotes transposition.', '0911.4244-1-36-1': 'For simplicity we have chosen the normalization of the measure [MATH] to set the proportional constant in the right-handed side of [REF] to be unity.', '0911.4244-1-37-0': 'Now the integrand depends only on the eigenvalues of [MATH] Hermitian matrices [MATH].', '0911.4244-1-37-1': 'Let us denote them by [MATH] and [MATH]).', '0911.4244-1-37-2': 'The partition function of the quiver matrix model (ITEP model) now reduces to the form of integrations over the eigenvalues of [MATH] [CITATION]: [EQUATION] where [MATH] is a potential and [EQUATION]', '0911.4244-1-37-3': 'The partition function [REF] can be rewritten in terms of CFT operators.', '0911.4244-1-37-4': 'Let [MATH] be [MATH]-valued massless chiral field and [MATH].', '0911.4244-1-37-5': 'Their correlators are given by [EQUATION]', '0911.4244-1-37-6': 'The modes [EQUATION] obey the commutation relations [EQUATION]', '0911.4244-1-37-7': 'The Fock vacuum is given by [EQUATION]', '0911.4244-1-37-8': 'Let [EQUATION]', '0911.4244-1-37-9': 'It is convenient to introduce the [MATH]-valued potential [MATH] by [EQUATION]', '0911.4244-1-37-10': 'Note that [MATH].', '0911.4244-1-38-0': 'The energy-momentum tensor is given by [EQUATION] where [MATH] is the Killing form.', '0911.4244-1-38-1': 'Let [MATH]) be an orthonormal basis of the Cartan subalgebra [MATH] with respect to the Killing form: [MATH].', '0911.4244-1-38-2': 'In this basis, the components of the [MATH]-valued chiral boson are just [MATH] independent free chiral bosons: [EQUATION] and the energy-momentum tensor in this basis is given by [EQUATION]', '0911.4244-1-38-3': 'The central charge is [MATH].', '0911.4244-1-39-0': 'Note that for a root [MATH], [MATH] with [MATH].', '0911.4244-1-39-1': 'Then, the bosons [MATH] associated with the simple roots [MATH] are expressed in this basis as follows: [EQUATION]', '0911.4244-1-39-2': 'For roots [MATH] and [MATH], the inner product on the root space is expressed in their components as [MATH].', '0911.4244-1-39-3': 'Here [MATH] and [MATH].', '0911.4244-1-40-0': 'The screening charges associated with the simple roots are defined by [EQUATION] with an appropriate contour integration.', '0911.4244-1-41-0': 'Using these definitions, the partition function [REF] can be written as follows [EQUATION]', '0911.4244-1-41-1': 'The chiral scalar field appears in the matrix model as the collective field of the eigenvalues [EQUATION]', '0911.4244-1-41-2': 'In particular, for [MATH], [EQUATION]', '0911.4244-1-42-0': '## [MATH]-ensemble of ADE quiver matrix model', '0911.4244-1-43-0': 'Inspired by the recent AGT conjecture, we are interested in the conformal Toda field theory based on a finite-dimensional Lie algebra [MATH] of ADE type.', '0911.4244-1-43-1': 'For the conformal Toda field theories, the energy momentum tensor is given by [EQUATION] where [MATH] and [MATH] is the Weyl vector of [MATH], half the sum of the positive roots.', '0911.4244-1-43-2': 'In the orthonormal basis, it takes the form [EQUATION]', '0911.4244-1-43-3': 'The central charge is given by [EQUATION]', '0911.4244-1-43-4': 'Here [MATH] is the Coxeter number of the simply-laced Lie algebra [MATH] whose rank is [MATH].', '0911.4244-1-43-5': 'Explicitly, [MATH] (with [MATH]), [MATH], [MATH], [MATH] and [MATH].', '0911.4244-1-44-0': 'The partition function of the corresponding [MATH]-ensemble quiver matrix model (with [MATH]) is the following deformation of the quiver matrix model [REF] [EQUATION]', '0911.4244-1-44-1': 'At [MATH], it reduces to the original quiver matrix model [REF].', '0911.4244-1-45-0': 'The corresponding collective field realization of chiral scalars are given by [EQUATION]', '0911.4244-1-45-1': 'Now we require the non-commutativity [EQUATION]', '0911.4244-1-45-2': 'If [MATH], [MATH] can be realized as [MATH].', '0911.4244-1-45-3': 'Here [MATH].', '0911.4244-1-46-0': 'Note that in the simple root basis [MATH], the collective fields are given by [EQUATION]', '0911.4244-1-46-1': 'The energy momentum tensor in this basis has the form [EQUATION]', '0911.4244-1-46-2': 'Here we have used [MATH].', '0911.4244-1-46-3': 'Using [REF] and [REF], we can check that the partition function [REF] obey the Virasoro constraints.', '0911.4244-1-47-0': '## [MATH] quiver matrix model and non-commutative spectral curve', '0911.4244-1-48-0': 'In this subsection, we consider the case of [MATH] with its rank [MATH].', '0911.4244-1-48-1': 'The generators of [MATH] algebra in the defining representation are [MATH] traceless Hermitian matrices.', '0911.4244-1-48-2': 'The generators of the Cartan subalgebra [MATH] can be chosen as diagonal ones.', '0911.4244-1-48-3': 'Using the defining representation, the Killing form can be chosen as the trace of [MATH] matrices: [MATH] for [MATH].', '0911.4244-1-49-0': 'In the case of [MATH] quiver matrix model, it is convenient to denote the chiral boson as follows: [EQUATION]', '0911.4244-1-49-1': 'Let [MATH] be a linear map from [MATH] diagonal matrices to [MATH] such that [EQUATION]', '0911.4244-1-49-2': 'The simple roots [MATH] and the fundamental weights [MATH] of [MATH] algebra are given by [EQUATION]', '0911.4244-1-49-3': 'With [MATH], we have the (symmetrized) Cartan matrix of [MATH]: [EQUATION]', '0911.4244-1-49-4': 'It follows that [MATH].', '0911.4244-1-49-5': 'The explicit form of the Weyl vector is given by [EQUATION]', '0911.4244-1-49-6': 'Let [MATH]) be the [MATH] weights in the defining representation with the highest weight [MATH]: [EQUATION]', '0911.4244-1-49-7': 'Note that [MATH].', '0911.4244-1-49-8': 'Using these weights, the components of the chiral boson [REF] can be obtained by [MATH].', '0911.4244-1-49-9': 'The relation between the [MATH] bosons [MATH] associated with the simple roots [MATH] and these [MATH] bosons [MATH] are given by [MATH].', '0911.4244-1-49-10': 'Their correlators are given by [EQUATION]', '0911.4244-1-49-11': 'Let us introduce [MATH] spin-[MATH] currents with one constraint as follows [EQUATION]', '0911.4244-1-49-12': 'Applying [REF] to the weights [MATH] [REF], we can see that the spin-[MATH] currents [MATH] [REF] are, as collective fields of eigenvalues, given by [EQUATION] where [EQUATION]', '0911.4244-1-49-13': 'The partition function of the [MATH] quiver matrix models obeys the [MATH] constraints.', '0911.4244-1-49-14': 'The generators of [MATH] algebra are currents [MATH] with spin [MATH]).', '0911.4244-1-49-15': 'It can be constructed from the spin [MATH]-currents [MATH] by the Miura transformation [CITATION] [EQUATION]', '0911.4244-1-49-16': 'Here [MATH] and we use the following ordering of product for non-commuting objects [MATH]: [EQUATION]', '0911.4244-1-49-17': 'In particular, the current [MATH] is proportional to the energy momentum tensor [MATH] [EQUATION]', '0911.4244-1-49-18': 'In [REF], to go to the third line and the last, we used the traceless condition [MATH] and the explicit form of the Weyl vector [MATH] [REF], respectively.', '0911.4244-1-49-19': 'The central charge of this energy momentum tensor is [MATH].', '0911.4244-1-50-0': 'The [MATH] constraints are equivalent to the regularity of the correlation function [EQUATION]', '0911.4244-1-50-1': 'Let us see this correlator is well-defined at finite [MATH].', '0911.4244-1-50-2': 'The collective field current [MATH] [REF] has simple poles at [MATH] and at [MATH].', '0911.4244-1-50-3': 'As a function of [MATH], [EQUATION] has poles at [MATH] which may cause singularity of its correlator [MATH] at real [MATH].', '0911.4244-1-50-4': 'First, let us examine singularity at [MATH].', '0911.4244-1-50-5': 'Note that from the form of the collective field current [REF], the singularities only come from the factor [MATH].', '0911.4244-1-50-6': 'The other factor [MATH] are independent of [MATH].', '0911.4244-1-50-7': 'It is not difficult to check that near [MATH], [EQUATION] where the effective action [MATH] is defined by [EQUATION]', '0911.4244-1-50-8': 'Similar relations hold near other points [MATH].', '0911.4244-1-50-9': 'Therefore, the singular part of [MATH] is given by [EQUATION]', '0911.4244-1-50-10': 'Hence we have the [MATH] constraints: [EQUATION]', '0911.4244-1-50-11': 'Therefore, we can use this correlator to define a non-commutative Calabi-Yau threefold as follows: [EQUATION] and the associated non-commutative spectral curve [MATH] can be defined by [EQUATION] with the non-commutativity [MATH].', '0911.4244-1-50-12': 'These are well-defined at finite [MATH].', '0911.4244-1-51-0': 'Note that the correlator takes the form [EQUATION]', '0911.4244-1-51-1': 'In principle, the explicit form of [MATH] can be determined by examining regular part of [MATH].', '0911.4244-1-51-2': 'But actual calculation is tedious for general [MATH].', '0911.4244-1-51-3': 'In general, the non-commutative geometry takes the form [EQUATION]', '0911.4244-1-51-4': 'In the next subsection, we give the explicit form of the spectral curve for [MATH].', '0911.4244-1-52-0': 'Before going to the case of [MATH], let us consider the large [MATH] limit.', '0911.4244-1-52-1': ""As in the one-matrix model case, we take [MATH] limit with the deformed 't Hooft couplings [MATH] fixed."", '0911.4244-1-52-2': 'In the large [MATH], the saddle points are determined by the stationary condition [MATH].', '0911.4244-1-52-3': 'In the [MATH] limit, the non-commutativity is lost and the large [MATH] factorization gives [EQUATION] where [EQUATION]', '0911.4244-1-52-4': 'Here the density functions [MATH] are defined by [EQUATION]', '0911.4244-1-52-5': 'In the deformed [MATH] quiver matrix model, the deformed geometry takes the form [EQUATION]', '0911.4244-1-52-6': 'Therefore, the deformed spectral curve [MATH] is covered by [MATH] Riemann sheets and the points on the [MATH]-th sheet can be parametrized by [MATH].', '0911.4244-1-52-7': 'Hence they are fibered over the complex [MATH]-plane.', '0911.4244-1-53-0': 'Generalization to [MATH] and [MATH] cases is straightforward at least in the large [MATH] limit.', '0911.4244-1-53-1': ""Replace the 't Hooft couplings [MATH] by the deformed ones [MATH]."", '0911.4244-1-53-2': 'The finite non-commutative form of the spectral curve would be fixed by requiring the consistency with the symmetry of the partition function [REF].', '0911.4244-1-54-0': '## Deformed [MATH] quiver model and non-commutative curve', '0911.4244-1-55-0': 'For [MATH]-deformed [MATH] quiver matrix model, the explicit forms of the currents are given by [EQUATION] where [EQUATION] and [MATH].', '0911.4244-1-55-1': 'With some work, we can see that at finite [MATH] (with [MATH]), the regular part of the ""spectral determinant"" is given by [EQUATION]', '0911.4244-1-55-2': 'The last two terms in the right-handed side of [REF] do not contribute to the correlator.', '0911.4244-1-55-3': 'Therefore, we have the explicit form of the finite [MATH] spectral curve mentioned in the introduction: [EQUATION] where [EQUATION] with [MATH].', '0911.4244-1-55-4': 'At the commutative point [MATH], the algebraic curve of this form (in the large [MATH] limit) is already known [CITATION].', '0911.4244-1-56-0': 'In the large [MATH] limit with general [MATH], spectral curve factorizes into [EQUATION] where [EQUATION]', '0911.4244-1-56-1': 'Here [MATH] and [EQUATION]', '0911.4244-1-57-0': ""# Dijkgraaf-Vafa's proposal: multi-Penner potential"", '0911.4244-1-58-0': ""In [CITATION], it was pointed out that the correlation function of [MATH] Toda field theory (Liouville theory for [MATH]) in two dimensions can be associated with Nekrasov's partition function [CITATION] of [MATH] quiver gauge theory."", '0911.4244-1-58-1': 'For [MATH] gauge theory with [MATH] hypermultiplets its Nekrasov partition function is identified with the (chiral) four point function of the Toda theory on a sphere: [EQUATION] under the identification ([REF]) between the deformation parameters and the parameter [MATH] appearing in the central charge [REF] of the Toda theory.', '0911.4244-1-59-0': 'The positions of the vertex operators correspond to punctures of the sphere.', '0911.4244-1-59-1': 'Three of them can be chosen at [MATH].', '0911.4244-1-59-2': 'As we will see later, this sphere can be identified with the one on which [MATH] M5-branes wrap.', '0911.4244-1-59-3': 'The vectors [MATH] in the dual space [MATH] of the Cartan subalgebra are expanded as [MATH].', '0911.4244-1-59-4': 'In general, it is possible to consider many types of punctures in the sphere on which M5-branes wrap.', '0911.4244-1-59-5': 'This variety corresponds to the flavor symmetries of the gauge theory [CITATION].', '0911.4244-1-59-6': 'In the Toda theory, this corresponds to choice of the vectors [MATH].', '0911.4244-1-59-7': 'The vertex operator [MATH] corresponding to the ""simple"" puncture associated with the [MATH] flavor symmetry is [CITATION] [EQUATION] where [MATH]) is the highest weight of the (anti-)fundamental representation.', '0911.4244-1-59-8': 'The other types of punctures correspond to more generic choices of [MATH].', '0911.4244-1-59-9': 'The [MATH] gauge theory with [MATH] hypermultiplets has [MATH] flavor symmetry as a subgroup of [MATH].', '0911.4244-1-59-10': 'Therefore, for this gauge theory, the corresponding correlation function is such that two vertex operators are ""simple"" type like ([REF]) while the other two are generic ""[MATH]"" type.', '0911.4244-1-60-0': 'As discussed in [CITATION], by using the relation between [MATH] Toda field theory and the quiver matrix model [CITATION], we reach the matrix model which describes [MATH] quiver gauge theory.', '0911.4244-1-60-1': 'The prescription introduced in [CITATION] is that the original matrix model action is set to zero.', '0911.4244-1-60-2': 'Under zero action, we consider the correlation functions of the vertex operators considered above.', '0911.4244-1-60-3': 'In the correlation function, the vertex operator can be written in terms of the matrices by using ([REF]) as [EQUATION]', '0911.4244-1-60-4': 'With zero action ([MATH]), we define the chiral four-point correlation function which corresponds to [MATH] gauge theory with [MATH] hypermultiplets by [EQUATION] where [MATH].', '0911.4244-1-60-5': 'For later convenience, we set [MATH].', '0911.4244-1-60-6': 'The momentum conservation condition is required [EQUATION]', '0911.4244-1-60-7': 'Using this four-point function, we define the partition function of the deformed [MATH] quiver matrix model by [EQUATION] where the determinantal form of the vertex operators [REF] at punctures [MATH] is converted into a set of new matrix model potentials [MATH] with multi-Penner type interaction: [EQUATION]', '0911.4244-1-60-8': 'Later we will set [MATH], [MATH] and [MATH].', '0911.4244-1-61-0': 'We denote the components of [MATH] in the fundamental weight basis by [MATH]: [MATH].', '0911.4244-1-61-1': 'As explained above, we let the operators at [MATH] be the simple type.', '0911.4244-1-61-2': 'That is, we set [EQUATION]', '0911.4244-1-61-3': 'The operators at [MATH] are the generic type.', '0911.4244-1-62-0': '## [MATH] quiver matrix model corresponding to [MATH] gauge theory', '0911.4244-1-63-0': 'In this subsection, we consider [MATH] quiver matrix model with the action ([REF]).', '0911.4244-1-63-1': 'As seen above, the Nekrasov partition function for [MATH] supersymmetric [MATH] gauge theory with six massive flavors is expected to be described by [MATH] quiver matrix model with the action ([REF]) with [MATH].', '0911.4244-1-64-0': 'We have explicitly seen the spectral curve of [MATH] quiver matrix model in the large [MATH] limit in the subsection [REF].', '0911.4244-1-64-1': 'For convenience, we rewrite [REF] in the following form: [EQUATION] where [EQUATION]', '0911.4244-1-64-2': 'By substituting the action [REF] and ([REF]), the explicit form of [MATH] [REF] are given by [EQUATION] and [MATH].', '0911.4244-1-64-3': 'Here [MATH] are components of [MATH] in the fundamental weight basis: [MATH].', '0911.4244-1-65-0': 'The form of [MATH] [REF] and that of [MATH] [REF] are [EQUATION] where [MATH] and [MATH] are constants.', '0911.4244-1-65-1': 'As seen above, the coordinate [MATH] parametrizes the sphere.', '0911.4244-1-65-2': 'The spectral curve is a triple cover of this sphere and each value of [MATH] at each sheet is given by [MATH].', '0911.4244-1-66-0': 'We consider a one-form [MATH] with [MATH] given by [REF].', '0911.4244-1-66-1': 'Later on this will be identified with the Seiberg-Witten one-form on the [MATH]-th sheet.', '0911.4244-1-66-2': 'From the form of the spectral curve, we can see that the one-form has simple poles at [MATH] and at [MATH].', '0911.4244-1-67-0': 'The residues of [MATH] at the simple punctures [MATH] and [MATH] are given by [EQUATION]', '0911.4244-1-67-1': 'These will be compared with the mass parameter associated with the [MATH] flavor symmetry in the gauge theory.', '0911.4244-1-67-2': 'The residues at [MATH] are of generic type: [EQUATION]', '0911.4244-1-67-3': 'This takes the same form as the mass parameters associated with the [MATH] flavor symmetry, which will be analyzed in the next section.', '0911.4244-1-68-0': 'Using the momentum conservation [REF], the residues at [MATH] are found to be [EQUATION]', '0911.4244-1-68-1': 'This also takes the same form as the [MATH] mass parameters.', '0911.4244-1-69-0': '# [MATH] gauge theory and Seiberg-Witten curve', '0911.4244-1-70-0': 'We consider [MATH] superconformal gauge theory.', '0911.4244-1-70-1': 'The low energy effective theory (and the quantum Coulomb moduli space) of [MATH] supersymmetric gauge theory is described by the Seiberg-Witten curve and the meromorphic one-form called Seiberg-Witten one-form [CITATION].', '0911.4244-1-70-2': 'We will see that the Seiberg-Witten curve of this theory enjoys the same properties as the spectral curve considered above for the [MATH]) case.', '0911.4244-1-71-0': '## [MATH] gauge theory with [MATH]', '0911.4244-1-72-0': 'Consider the [MATH] gauge theory with six massive hypermultiplets.', '0911.4244-1-72-1': 'This theory has an exactly marginal coupling which is the microscopic gauge coupling constant: [EQUATION]', '0911.4244-1-72-2': 'The type IIA brane construction and its M-theory lift lead to the Seiberg-Witten curve which is a hypersurface in 2-complex dimensional space [MATH] [CITATION] [EQUATION] where [MATH]) are the mass parameters of the hypermultiplets and [MATH].', '0911.4244-1-72-3': 'We can see that the coordinate [MATH] has mass dimension one and therefore the constants [MATH], [MATH] and [MATH] have mass dimension one, two and three respectively.', '0911.4244-1-72-4': 'In principle, the constants depend on the mass parameters and the Coulomb moduli parameters [MATH] and [MATH] which correspond to [MATH] and [MATH] at weak coupling.', '0911.4244-1-72-5': 'The dimensional analysis and the regularity constraint in the massless limit show that [MATH] and [MATH] are, respectively, linear in [MATH] and [MATH] and also include the terms [MATH] and [MATH].', '0911.4244-1-72-6': 'Also, [MATH] depends only on the mass parameters.', '0911.4244-1-72-7': 'The curve is translated into the following form [EQUATION] where [MATH], [MATH] and [MATH] are degree two polynomials in [MATH]: [EQUATION]', '0911.4244-1-72-8': 'As in [CITATION], by shifting the coordinate [MATH] appropriately, we can eliminate the quadratic term in [MATH].', '0911.4244-1-72-9': 'Then, by changing the coordinate [MATH], we obtain [CITATION] [EQUATION] where [MATH] and [MATH] are the degree 4 and 6 polynomials respectively and can be written in terms of [MATH], [MATH] and [MATH] [EQUATION]', '0911.4244-1-72-10': 'In this coordinate, the Seiberg-Witten one-form is [MATH].', '0911.4244-1-72-11': 'We denote by [MATH] a coordinate in the sphere on which three M5-branes wrap.', '0911.4244-1-72-12': 'Therefore, [MATH] are local coordinates in the cotangent bundle of this sphere.', '0911.4244-1-72-13': 'We identify the sphere with the one that appeared in the quiver matrix model and Toda theory.', '0911.4244-1-72-14': 'This implies [MATH].', '0911.4244-1-73-0': 'From the expression ([REF]), we can see that the Seiberg-Witten one-form has poles at [MATH] and [MATH].', '0911.4244-1-73-1': 'As found in [CITATION] and we will see below, the structure of those poles corresponds to the flavor symmetry of the quiver gauge theory, because the residues of the Seiberg-Witten one-form are the mass parameters associated with the flavor symmetries.', '0911.4244-1-74-0': 'The residues of the Seiberg-Witten one-form [MATH] at [MATH] and [MATH] are given by [EQUATION] where each value denotes the residue at the [MATH]-th sheet of the Riemann surface.', '0911.4244-1-75-0': 'The residues at [MATH] are found to be [EQUATION]', '0911.4244-1-75-1': 'The flavor symmetry [MATH] associated with the punctures at [MATH] are the symmetry of three hypermultiplets whose masses are [MATH]).', '0911.4244-1-75-2': 'Since the mass associated with [MATH] is [MATH], the residue at [MATH] should be of the form [EQUATION]', '0911.4244-1-75-3': 'Also, the residue at [MATH] should be of the form [EQUATION]', '0911.4244-1-75-4': 'These determine [MATH] completely and lead to [EQUATION]', '0911.4244-1-75-5': 'From this, we can compute the residues at [MATH] as [EQUATION] which corresponds to the Cartan part of [MATH].', '0911.4244-1-75-6': 'At [MATH], we also find the similar form as above.', '0911.4244-1-75-7': 'From the residue analysis alone, we cannot determine the constants [MATH] and [MATH] in ([REF]).', '0911.4244-1-76-0': '## Matching of the matrix model parameters', '0911.4244-1-77-0': 'Obviously, the coordinate [MATH] is identified with [MATH] and [MATH] on the [MATH]-th sheet with [MATH].', '0911.4244-1-77-1': 'Comparing the residues [REF], [REF] with [REF], we find that the parameters in [MATH] quiver matrix model must be chosen as [EQUATION] in order to yield the Seiberg-Witten curve of the gauge theory.', '0911.4244-1-77-2': 'Also, by comparing [REF] with [REF], and [REF] with [REF], we have [EQUATION]', '0911.4244-1-77-3': 'To summarize, we have determined the ""weights"" of the vertex operators [MATH] as follows: [EQUATION]', '0911.4244-1-77-4': 'The matrix model potential [MATH] with the multi-log interaction are finally fixed as (up to constants) [EQUATION]', '0911.4244-1-77-5': ""From the momentum conservation [REF], the deformed 't Hooft couplings [MATH] are related to the mass parameters of the gauge theory as follows: [EQUATION]""}","{'0911.4244-2-0-0': 'We review the quiver matrix model (the ITEP model) in the light of the recent progress on 2d-4d connection of conformal field theories, in particular, on the relation between Toda field theories and a class of quiver superconformal gauge theories.', '0911.4244-2-0-1': 'On the basis of the CFT representation of the [MATH] deformation of the model, a quantum spectral curve is introduced as [MATH] at finite [MATH] and for [MATH].', '0911.4244-2-0-2': 'The planar loop equation in the large [MATH] limit follows with the aid of [MATH] constraints.', '0911.4244-2-0-3': 'Residue analysis is provided both for the curve of the matrix model with the ""multi-log"" potential and for the Seiberg-Witten curve in the case of [MATH] with [MATH] flavors, leading to the matching of the mass parameters.', '0911.4244-2-0-4': 'The isomorphism of the two curves is made manifest.', '0911.4244-2-1-0': '# Introduction', '0911.4244-2-2-0': 'Matrix models have had successes in several stages of the developments in string theory, gauge theory and related studies of integrable systems.', '0911.4244-2-2-1': 'A list of those in the last twenty years include 2d gravity, exact evaluation of gluino condensate prepotential, topological strings, etc.', '0911.4244-2-3-0': 'Recent progress has been triggered by the construction of a large class of [MATH] superconformal [MATH] ""generalized quiver"" gauge theories in four dimensions by Gaiotto [CITATION].', '0911.4244-2-3-1': '(See also [CITATION]).', '0911.4244-2-3-2': 'Subsequently an interesting conjecture has been made by Alday, Gaiotto and Tachikawa (AGT) [CITATION] (and its [MATH] generalization by [CITATION]) on the equivalence of the Nekrasov partition function [CITATION] and the [MATH] conformal block of the Toda field theory.', '0911.4244-2-3-3': 'These are followed by a number of extensive checks and pieces of supporting evidence [CITATION].', '0911.4244-2-3-4': 'Very recently, Dijkgraaf and Vafa [CITATION] have suggested an explanation of this phenomenon by the so-called quiver matrix model [CITATION].', '0911.4244-2-3-5': '(In this paper, it is occasionally referred to as the ITEP model [CITATION]).', '0911.4244-2-3-6': 'Their reasoning is based on the matrix model realization of type IIB topological strings on a local Calabi-Yau with a local [MATH] singularity which geometrically engineers the gauge theory.', '0911.4244-2-3-7': 'By choosing ""multi-Penner"" potentials [CITATION] for the [MATH] quiver matrix model, they argued that the spectral curve of the matrix model at large [MATH] (the size of the matrix) can be understood as the Seiberg-Witten curve of the attendant [MATH] generalized quiver superconformal gauge theory (SCFT).', '0911.4244-2-4-0': 'The AGT conjecture is regarded as a more concrete realization of the folklore connection between [MATH] gauge theories and the attendant [MATH] sigma models (see [CITATION] for instance), that led to the study of two dimensional quantum integrable field theories in late seventies ([CITATION] for instance).', '0911.4244-2-4-1': 'In the light of potential importance of this subject, we find it useful to devote the substantial part of the present paper in reviewing and reformulating the basic structure of the quiver matrix model [CITATION].', '0911.4244-2-5-0': 'The quiver matrix model associated with Lie algebra [MATH] of ADE type with rank [MATH] is obtained as a solution to extended Virasoro constraints, i.e., [MATH] constraints [CITATION] at finite [MATH] [CITATION].', '0911.4244-2-5-1': 'This fact that the model automatically implements the [MATH] constraints at finite [MATH] is an advantage over more traditional two- and multi-matrix models where finite [MATH] Schwinger-Dyson equations are typically [MATH] type and are more involved [CITATION].', '0911.4244-2-5-2': 'The model is defined by using [MATH] independent free massless chiral bosons in two dimensional CFT with the central charge [MATH] and the final form of the partition function is formulated as the integrations over eigenvalues of the matrices.', '0911.4244-2-5-3': 'A key ingredient of its construction is a set of screening charges of the [MATH] CFT.', '0911.4244-2-5-4': 'By construction, the partition function respects the [MATH], the extended Casimir algebra generated by higher spin currents which commute with the screening charges.', '0911.4244-2-5-5': 'The partition function can be reformulated as integrations over the matrices in the adjoint and bi-fundamental representations.', '0911.4244-2-6-0': 'In the context of the AGT conjecture for more general Toda CFTs, it is natural to consider ""[MATH]-ensembles"" of ADE quiver matrix models.', '0911.4244-2-6-1': 'In this paper, using the [MATH]-ensemble of the [MATH] quiver matrix model at finite [MATH], we define a non-commutative Calabi-Yau threefold with the quantum deformed [MATH] singularity as [EQUATION]', '0911.4244-2-6-2': 'For [MATH], it takes the following form [EQUATION] with the non-commutativity [MATH].', '0911.4244-2-7-0': 'In order to bring these analyses in the more recent context, we take a simple example, namely, [MATH] SQCD with [MATH] flavors to provide a residue analysis of the quiver matrix model curve and that of the Seiberg-Witten curve from type IIA and M-theory consideration.', '0911.4244-2-7-1': 'This leads to the matching of the mass parameters.', '0911.4244-2-8-0': 'A main result of this paper is to establish the isomorphism of the spectral curve of the quiver matrix model ([MATH] type and beta deformed in general) in the planar limit and the corresponding Seiberg-Witten curve in the Witten-Gaiotto form.', '0911.4244-2-8-1': 'This has become possible as we have reformulated the matrix model curve, starting from the [MATH] constraints at finite [MATH], proposing the curve as a form of the characteristic equation, and finally using the singlet factorization in the planar limit to write it in a form where the isomorphism is rather manifest.', '0911.4244-2-8-2': 'To the best of our knowledge, such a systematic investigation has not been attempted and only clumsy expressions in the planar limit have been available.', '0911.4244-2-8-3': '(See references in subsequent sections.)', '0911.4244-2-9-0': 'This paper is organized as follows.', '0911.4244-2-9-1': 'In section [MATH], after giving several punchlines, we review known facts on the one-matrix model, using the CFT notation and the Virasoro constraints at finite [MATH] [CITATION].', '0911.4244-2-9-2': 'In section [MATH], ordinary ADE quiver matrix models are reviewed in the same spirit and their ""[MATH]-ensemble"" is introduced.', '0911.4244-2-9-3': 'For the case of [MATH] quiver matrix model, we show that the quantum spectral curve [EQUATION] is well-defined within the matrix model integral.', '0911.4244-2-9-4': 'The partition function obeys the [MATH] constraints at finite [MATH] which are the properties of the original matrix integrals.', '0911.4244-2-9-5': 'The planar loop equation follows from these structures together with the large [MATH] factorization.', '0911.4244-2-9-6': ""In section [MATH], we adopt Dijkgraaf-Vafa's recipe to treat [MATH] superconformal gauge theory by using the quiver matrix model with Penner like action."", '0911.4244-2-9-7': 'We concentrate on the [MATH] quiver matrix model and consider the spectral curve.', '0911.4244-2-9-8': 'Section [MATH] treats the Seiberg-Witten curve for [MATH] gauge theory with [MATH] flavors.', '0911.4244-2-9-9': 'We see that this curve enjoys the same properties as those of the spectral curve.', '0911.4244-2-9-10': 'In particular, we give a matching of the mass parameters first for the case of [MATH].', '0911.4244-2-9-11': 'In order to render the isomorphism of the two curves clearer, a subsection is devoted to establishing this point.', '0911.4244-2-9-12': 'A matching of the mass parameters for general [MATH] is readily given.', '0911.4244-2-10-0': 'The major reason that we emphasized starting with the finite [MATH] spectral curve is already stated above: through this procedure and the large [MATH] factorization of the singlet operators, the isomorphism has been established in this paper.', '0911.4244-2-10-1': 'The continuous flow, beginning with the construction at finite [MATH], and ending with the singlet factorization in the planar limit has been indispensable in order for this paper to be legible and self-contained.', '0911.4244-2-11-0': '# One-matrix model and the [MATH]-ensemble', '0911.4244-2-12-0': 'In the case of [MATH], the quiver matrix model corresponds to the Hermitian one-matrix model [CITATION] (see also [CITATION]).', '0911.4244-2-12-1': 'The associated CFT is a single free boson with [MATH].', '0911.4244-2-12-2': 'On the other hand, the Liouville CFT which appears in AGT conjecture for [MATH] has the central charge [MATH] with [MATH].', '0911.4244-2-12-3': 'It is known that there is a one-matrix model which has a connection with the CFT with [MATH].', '0911.4244-2-12-4': 'It is the [MATH]-ensemble of one-matrix model with [MATH].', '0911.4244-2-12-5': 'It is easy to deal with the Liouville CFT by introducing the Feign-Fuchs background charge in the CFT notation.', '0911.4244-2-12-6': 'The CFT notation works well for the [MATH]-ensemble of the one-matrix model and the appearance of the CFT with [MATH] in the matrix model is a built-in result.', '0911.4244-2-12-7': 'The case of [MATH] corresponds to the ordinary Hermitian one-matrix model.', '0911.4244-2-12-8': ""The AGT conjecture implies that these deformation parameters are related to Nekrasov's deformation parameters [MATH] and [MATH] by [CITATION] [EQUATION]"", '0911.4244-2-12-9': 'Note that [MATH] and [MATH].', '0911.4244-2-13-0': 'It is known that the [MATH]-ensemble of one-matrix model at finite [MATH] is related to a quantum (non-commutative) spectral curve of the form [EQUATION] whose non-commutativity is given by [EQUATION]', '0911.4244-2-13-1': 'Using the [MATH]-ensemble matrix model, it is possible to define a non-commutative local Calabi-Yau threefold with a quantum deformed [MATH] singularity [EQUATION]', '0911.4244-2-13-2': 'It can be written as the form [EQUATION] where [MATH] is one-matrix model average over the [MATH]-ensemble and [MATH] is the energy momentum tensor of [MATH] CFT expressed as the collective field of the matrix eigenvalues.', '0911.4244-2-13-3': 'Later, we will give exact definitions of these quantities.', '0911.4244-2-14-0': '## Hermitian one-matrix model: undeformed case', '0911.4244-2-15-0': 'In this case the relevant matrix model is the Hermitian one-matrix model.', '0911.4244-2-15-1': 'The partition function takes the form [EQUATION]', '0911.4244-2-15-2': 'Here [MATH] is an [MATH] Hermitian matrix.', '0911.4244-2-15-3': 'Correlation function of this matrix model is defined by [EQUATION]', '0911.4244-2-15-4': 'Here [MATH] is a function of the Hermitian matrix.', '0911.4244-2-16-0': 'The partition function [REF] can be written in terms of the eigenvalues [MATH] of the matrix [MATH]: [EQUATION] where [MATH] is the Vandermonde determinant [EQUATION]', '0911.4244-2-16-1': 'It is well-known that the Hermitian matrix model has a close connection with the [MATH] free chiral boson.', '0911.4244-2-16-2': 'The partition function can be rewritten in terms of CFT operators.', '0911.4244-2-16-3': 'The mode expansion of the chiral boson [MATH] is chosen as follows: [EQUATION] and the non-trivial commutation relations are given by [EQUATION]', '0911.4244-2-16-4': 'Hence, our normalization of the correlator is given by [EQUATION]', '0911.4244-2-16-5': 'The energy momentum tensor with the central charge [MATH] is given by [EQUATION]', '0911.4244-2-16-6': 'The screening charges which commute with the Virasoro generators [MATH] are given by [EQUATION] with a certain integration contour.', '0911.4244-2-17-0': 'The Fock vacuum is given by [EQUATION]', '0911.4244-2-17-1': 'Let [EQUATION]', '0911.4244-2-17-2': 'Then, the partition function [REF] of the Hermitian matrix model can be rewritten in terms of the free chiral boson as follows [EQUATION]', '0911.4244-2-17-3': 'Associated with this expression, for an operator [MATH] constructed from the boson oscillators, we use the following notation [EQUATION]', '0911.4244-2-17-4': 'Within the normal ordering, correlators consisting of the chiral boson [MATH] in CFT have their counterparts in the matrix model correlators [REF]: [EQUATION]', '0911.4244-2-17-5': 'In the matrix model correlator, the chiral boson is realized as a collective field of the eigenvalues: [EQUATION]', '0911.4244-2-17-6': 'It is known that the partition function of the Hermitian matrix model at finite [MATH] obeys the Virasoro constraints [CITATION].', '0911.4244-2-17-7': 'In the CFT language, it follows from the commutativity of the Virasoro generators [MATH] with the screening charge [MATH] and [EQUATION]', '0911.4244-2-17-8': 'In the ITEP construction, the Virasoro constraints manifestly hold at finite [MATH] as [EQUATION]', '0911.4244-2-17-9': 'They are equivalent to the regularity of the correlator of the energy momentum tensor [EQUATION] at [MATH].', '0911.4244-2-18-0': 'Now, using the help of the Hermitian matrix model correlator, a local Calabi-Yau threefold with [MATH] singularity over a Riemann surface [MATH] can be defined by [EQUATION]', '0911.4244-2-18-1': 'Using the collective field expression [EQUATION] we have (for the derivation, see the next subsection, below [REF]) [EQUATION] where [EQUATION]', '0911.4244-2-18-2': 'Hence, the local Calabi-Yau threefold is a surface in [MATH] defined by [EQUATION]', '0911.4244-2-18-3': 'At [MATH], it describes some algebraic curve [MATH] in [MATH]: [EQUATION]', '0911.4244-2-18-4': 'Note that this algebraic curve is well-defined for finite [MATH] due to the Virasoro constraints of the matrix model.', '0911.4244-2-19-0': '## [MATH]-ensemble', '0911.4244-2-20-0': ""Nekrasov's deformation corresponds to the modification of the energy-momentum tensor [REF] by the introduction of the background charge a la Feign-Fuchs: [EQUATION]"", '0911.4244-2-20-1': 'This energy momentum tensor has the central charge [MATH].', '0911.4244-2-20-2': 'Undeformed case is recovered at [MATH], [MATH] and [MATH].', '0911.4244-2-21-0': 'Screening charges for this energy momentum tensor are given by [EQUATION]', '0911.4244-2-21-1': 'Let [EQUATION]', '0911.4244-2-21-2': 'It is natural to consider the following deformation of the partition function [REF]: [EQUATION]', '0911.4244-2-21-3': 'This matrix model is known as the [MATH]-ensemble [CITATION] with [MATH].', '0911.4244-2-21-4': 'For [MATH], it corresponds to the integrations over an orthogonal, hermitian and symplectic matrix respectively.', '0911.4244-2-22-0': 'Instead of the screening charge [MATH], we can use [MATH] to express the partition of the [MATH]-ensemble model.', '0911.4244-2-22-1': 'The corresponding expressions are obtained by replacing [MATH] with [MATH].', '0911.4244-2-23-0': 'It is known that this partition function is related to a non-commutative (or quantum) spectral curve [CITATION].', '0911.4244-2-23-1': 'In this case, the non-commutativity is given by [EQUATION]', '0911.4244-2-23-2': 'For [MATH], [MATH] can be realized as a differential operator [MATH].', '0911.4244-2-24-0': 'Note that the energy-momentum tensor [REF] can be defined by the Miura transformation: [EQUATION]', '0911.4244-2-24-1': 'The collective field expression of the chiral boson [MATH] now becomes [EQUATION]', '0911.4244-2-24-2': 'Note that [EQUATION]', '0911.4244-2-24-3': 'Using the collective field expression, we see that [EQUATION]', '0911.4244-2-24-4': 'The terms in the second line of [REF] correspond to the ""singular"" part of the energy momentum tensor and we can see that [EQUATION] where [EQUATION]', '0911.4244-2-24-5': 'Therefore the Virasoro constraints for the deformed one-matrix model imply that [EQUATION] where [EQUATION]', '0911.4244-2-24-6': 'Here the matrix model average is defined as in the undeformed case.', '0911.4244-2-24-7': 'Explicitly for some function [MATH] of the eigenvalues, we have [EQUATION]', '0911.4244-2-24-8': 'Hence the quantum spectral curve related to the [MATH]-ensemble is defined by [EQUATION]', '0911.4244-2-24-9': 'Explicitly, it is given by [EQUATION]', '0911.4244-2-24-10': 'Note that this quantum spectral curve can be rewritten as [EQUATION]', '0911.4244-2-24-11': 'Therefore, the associated local Calabi-Yau threefold also becomes a non-commutative surface: [EQUATION]', '0911.4244-2-24-12': 'Strictly speaking, for [MATH], [MATH] is a differential operator and the quantum spectral curve is a differential equation for some ""wave function.""', '0911.4244-2-24-13': 'This equation has a close connection with [MATH]-gravity and Hitchin systems.', '0911.4244-2-24-14': 'For recent discussion on this point in the light of the AGT conjecture, see [CITATION] and references therein.', '0911.4244-2-24-15': 'From the point of view of the [MATH]-module, see [CITATION].', '0911.4244-2-24-16': 'In string theory, the non-commutativity corresponds to turn on a constant NS two-form.', '0911.4244-2-25-0': 'Our main concern is the large [MATH] string/gauge duality.', '0911.4244-2-25-1': ""In the planar limit (a large [MATH] limit with the 't Hooft coupling [MATH] kept finite), [MATH] and thus the non-commutativity vanishes : [MATH]."", '0911.4244-2-26-0': '## Large [MATH] limit', '0911.4244-2-27-0': 'The partition function [REF] has a topological expansion [EQUATION]', '0911.4244-2-27-1': ""In the large [MATH] limit with the 't Hooft coupling [MATH] kept finite, leading contribution comes from the planar part [MATH] in [REF] and can be evaluated by the saddle point method."", '0911.4244-2-27-2': 'For simplicity, we assume that the parameter [MATH] is pure imaginary and [MATH].', '0911.4244-2-27-3': ""In this case Nekrasov's deformation parameters [MATH] and [MATH] are real."", '0911.4244-2-28-0': 'The stationary conditions [MATH] yield [EQUATION]', '0911.4244-2-28-1': 'Since we assume that [MATH] is pure imaginary, these stationary equations have real solutions [MATH].', '0911.4244-2-28-2': 'We evaluate the partition function around a classical solution with certain filling fractions [MATH] around the local extrema [MATH].', '0911.4244-2-29-0': 'In the planar limit, the large [MATH] factorization yields [EQUATION] where [EQUATION]', '0911.4244-2-29-1': 'Here [MATH] is the density function of the solution to the stationary conditions [REF].', '0911.4244-2-29-2': '[EQUATION]', '0911.4244-2-29-3': 'The stationary conditions in the large [MATH] limit go to [EQUATION]', '0911.4244-2-29-4': 'Here [MATH] denotes the principal value.', '0911.4244-2-29-5': 'Note that in the stationary conditions and in the definition of [MATH], the parameter [MATH] always appears in the combination [MATH].', '0911.4244-2-29-6': ""Therefore, if we replace the 't Hooft coupling [MATH] by [MATH], we can use the large [MATH] formulas of the undeformed Hermitian one-matrix model."", '0911.4244-2-29-7': ""We will call [MATH] deformed 't Hooft coupling."", '0911.4244-2-30-0': 'Hence, in the large [MATH] limit, the local Calabi-Yau is deformation of [MATH] [EQUATION] and the algebraic curve [MATH] becomes [EQUATION] and the points [MATH] on [MATH] can be covered by two sheets [MATH].', '0911.4244-2-31-0': 'In the large [MATH] limit, the Virasoro constraints become an algebraic equation [EQUATION]', '0911.4244-2-32-0': '# ADE quiver matrix models and their ""[MATH]-ensemble""', '0911.4244-2-33-0': 'In this section, we first briefly review the ADE quiver matrix models [CITATION].', '0911.4244-2-33-1': 'An excellent review for the undeformed case can be found in [CITATION].', '0911.4244-2-33-2': 'Then, we introduce the ""[MATH]-ensemble"" or deformed ADE quiver matrix models.', '0911.4244-2-33-3': 'For [MATH] cases, they can be found in [CITATION].', '0911.4244-2-34-0': 'Using the [MATH]-deformed [MATH] quiver matrix model, we introduce a non-commutative local Calabi-Yau threefold related to deformations of [MATH] singularities.', '0911.4244-2-35-0': '## ADE quiver matrix models and CFT with [MATH]', '0911.4244-2-36-0': 'Let [MATH] be a finite dimensional Lie algebra of ADE type with rank [MATH], [MATH] the Cartan subalgebra of [MATH], and [MATH] its dual.', '0911.4244-2-36-1': 'We sometimes denote the natural pairings between [MATH] and [MATH] by [MATH]: [EQUATION]', '0911.4244-2-36-2': 'Let [MATH] be simple roots of [MATH] and [MATH] is the inner product on [MATH].', '0911.4244-2-36-3': 'Our normalization is chosen as [MATH].', '0911.4244-2-36-4': 'The fundamental weights are denoted by [MATH] [EQUATION]', '0911.4244-2-36-5': 'In the Dynkin diagram of [MATH] we associate [MATH] Hermitian matrices [MATH] with vertices [MATH] for simple roots [MATH], and complex [MATH] matrices [MATH] and their Hermitian conjugate [MATH] with links connecting vertices [MATH] and [MATH].', '0911.4244-2-36-6': 'We label links of the Dynkin diagram by pairs of node label [MATH] with an ordering [MATH].', '0911.4244-2-36-7': 'Let [MATH] and [MATH] be the set of ""edges"" [MATH] (with [MATH]) and the set of ""arrows"" [MATH] respectively: [EQUATION]', '0911.4244-2-37-0': 'The partition function of the quiver matrix model [CITATION] associated with [MATH] is given by [EQUATION] where [EQUATION] with real constants [MATH] obeying the conditions [MATH].', '0911.4244-2-37-1': 'Note that [EQUATION]', '0911.4244-2-37-2': 'The integration measures [MATH] and [MATH] are defined by using the metrics [MATH] and [MATH] respectively.', '0911.4244-2-38-0': 'Integrations over [MATH] are Gauss-Fresnel type and are easily performed: [EQUATION] where [MATH] is the [MATH] identity matrix and [MATH] denotes transposition.', '0911.4244-2-38-1': 'For simplicity we have chosen the normalization of the measure [MATH] to set the proportional constant in the right-handed side of [REF] to be unity.', '0911.4244-2-39-0': 'Now the integrand depends only on the eigenvalues of [MATH] Hermitian matrices [MATH].', '0911.4244-2-39-1': 'Let us denote them by [MATH] and [MATH]).', '0911.4244-2-39-2': 'The partition function of the quiver matrix model (ITEP model) now reduces to the form of integrations over the eigenvalues of [MATH] [CITATION]: [EQUATION] where [MATH] is a potential and [EQUATION]', '0911.4244-2-39-3': 'The partition function [REF] can be rewritten in terms of CFT operators.', '0911.4244-2-39-4': 'Let [MATH] be [MATH]-valued massless chiral field and [MATH].', '0911.4244-2-39-5': 'Their correlators are given by [EQUATION]', '0911.4244-2-39-6': 'The modes [EQUATION] obey the commutation relations [EQUATION]', '0911.4244-2-39-7': 'The Fock vacuum is given by [EQUATION]', '0911.4244-2-39-8': 'Let [EQUATION]', '0911.4244-2-39-9': 'It is convenient to introduce the [MATH]-valued potential [MATH] by [EQUATION]', '0911.4244-2-39-10': 'Note that [MATH].', '0911.4244-2-40-0': 'The energy-momentum tensor is given by [EQUATION] where [MATH] is the Killing form.', '0911.4244-2-40-1': 'Let [MATH]) be an orthonormal basis of the Cartan subalgebra [MATH] with respect to the Killing form: [MATH].', '0911.4244-2-40-2': 'In this basis, the components of the [MATH]-valued chiral boson are just [MATH] independent free chiral bosons: [EQUATION] and the energy-momentum tensor in this basis is given by [EQUATION]', '0911.4244-2-40-3': 'The central charge is [MATH].', '0911.4244-2-41-0': 'Note that for a root [MATH], [MATH] with [MATH].', '0911.4244-2-41-1': 'Then, the bosons [MATH] associated with the simple roots [MATH] are expressed in this basis as follows: [EQUATION]', '0911.4244-2-41-2': 'For roots [MATH] and [MATH], the inner product on the root space is expressed in their components as [MATH].', '0911.4244-2-41-3': 'Here [MATH] and [MATH].', '0911.4244-2-42-0': 'The screening charges associated with the simple roots are defined by [EQUATION] with an appropriate contour integration.', '0911.4244-2-43-0': 'Using these definitions, the partition function [REF] can be written as follows [EQUATION]', '0911.4244-2-43-1': 'The chiral scalar field appears in the matrix model as the collective field of the eigenvalues [EQUATION]', '0911.4244-2-43-2': 'In particular, for [MATH], [EQUATION]', '0911.4244-2-44-0': '## [MATH]-ensemble of ADE quiver matrix model', '0911.4244-2-45-0': 'Inspired by the recent AGT conjecture, we are interested in the conformal Toda field theory based on a finite-dimensional Lie algebra [MATH] of ADE type.', '0911.4244-2-45-1': 'For the conformal Toda field theories, the energy momentum tensor is given by [EQUATION] where [MATH] and [MATH] is the Weyl vector of [MATH], half the sum of the positive roots.', '0911.4244-2-45-2': 'In the orthonormal basis, it takes the form [EQUATION]', '0911.4244-2-45-3': 'The central charge is given by [EQUATION]', '0911.4244-2-45-4': 'Here [MATH] is the Coxeter number of the simply-laced Lie algebra [MATH] whose rank is [MATH].', '0911.4244-2-45-5': 'Explicitly, [MATH] (with [MATH]), [MATH], [MATH], [MATH] and [MATH].', '0911.4244-2-46-0': 'The partition function of the corresponding [MATH]-ensemble quiver matrix model (with [MATH]) is the following deformation of the quiver matrix model [REF] [EQUATION]', '0911.4244-2-46-1': 'At [MATH], it reduces to the original quiver matrix model [REF].', '0911.4244-2-47-0': 'The corresponding collective field realization of chiral scalars is given by [EQUATION]', '0911.4244-2-47-1': 'Now we require the non-commutativity [EQUATION]', '0911.4244-2-47-2': 'If [MATH], [MATH] can be realized as [MATH].', '0911.4244-2-47-3': 'Here [MATH].', '0911.4244-2-48-0': 'Note that in the simple root basis [MATH], the collective fields are given by [EQUATION]', '0911.4244-2-48-1': 'The energy momentum tensor in this basis has the form [EQUATION]', '0911.4244-2-48-2': 'Here we have used [MATH].', '0911.4244-2-48-3': 'Using [REF] and [REF], we can check that the partition function [REF] obey the Virasoro constraints.', '0911.4244-2-49-0': '## [MATH] quiver matrix model and non-commutative spectral curve', '0911.4244-2-50-0': 'In this subsection, we consider the case of [MATH] with its rank [MATH].', '0911.4244-2-50-1': 'The generators of [MATH] algebra in the defining representation are [MATH] traceless Hermitian matrices.', '0911.4244-2-50-2': 'The generators of the Cartan subalgebra [MATH] can be chosen as diagonal ones.', '0911.4244-2-50-3': 'Using the defining representation, the Killing form can be chosen as the trace of [MATH] matrices: [MATH] for [MATH].', '0911.4244-2-51-0': 'In the case of [MATH] quiver matrix model, it is convenient to denote the chiral boson as follows: [EQUATION]', '0911.4244-2-51-1': 'Let [MATH] be a linear map from [MATH] diagonal matrices to [MATH] such that [EQUATION]', '0911.4244-2-51-2': 'The simple roots [MATH] and the fundamental weights [MATH] of [MATH] algebra are given by [EQUATION]', '0911.4244-2-51-3': 'With [MATH], we have the (symmetrized) Cartan matrix of [MATH]: [EQUATION]', '0911.4244-2-51-4': 'It follows that [MATH].', '0911.4244-2-51-5': 'The explicit form of the Weyl vector is given by [EQUATION]', '0911.4244-2-51-6': 'Let [MATH]) be the [MATH] weights in the defining representation with the highest weight [MATH]: [EQUATION]', '0911.4244-2-51-7': 'Note that [MATH].', '0911.4244-2-51-8': 'Using these weights, the components of the chiral boson [REF] can be obtained by [MATH].', '0911.4244-2-51-9': 'The relation between the [MATH] bosons [MATH] associated with the simple roots [MATH] and these [MATH] bosons [MATH] are given by [MATH].', '0911.4244-2-51-10': 'Their correlators are given by [EQUATION]', '0911.4244-2-51-11': 'Let us introduce [MATH] spin-[MATH] currents with one constraint as follows [EQUATION]', '0911.4244-2-51-12': 'Applying [REF] to the weights [MATH] [REF], we can see that the spin-[MATH] currents [MATH] [REF] are, as collective fields of eigenvalues, given by [EQUATION] where [EQUATION]', '0911.4244-2-51-13': 'The partition function of the [MATH] quiver matrix models obeys the [MATH] constraints.', '0911.4244-2-51-14': 'The generators of [MATH] algebra are currents [MATH] with spin [MATH]).', '0911.4244-2-51-15': 'It can be constructed from the spin [MATH]-currents [MATH] by the Miura transformation [CITATION] [EQUATION]', '0911.4244-2-51-16': 'Here [MATH] and we use the following ordering of product for non-commuting objects [MATH]: [EQUATION]', '0911.4244-2-51-17': 'In particular, the current [MATH] is proportional to the energy momentum tensor [MATH] [EQUATION]', '0911.4244-2-51-18': 'In [REF], to go to the third line and the last, we used the traceless condition [MATH] and the explicit form of the Weyl vector [MATH] [REF], respectively.', '0911.4244-2-51-19': 'The central charge of this energy momentum tensor is [MATH].', '0911.4244-2-52-0': 'The [MATH] constraints are equivalent to the regularity of the correlation function [EQUATION]', '0911.4244-2-52-1': 'Let us see this correlator is well-defined at finite [MATH].', '0911.4244-2-52-2': 'The collective field current [MATH] [REF] has simple poles at [MATH] and at [MATH].', '0911.4244-2-52-3': 'As a function of [MATH], [EQUATION] has poles at [MATH] which may cause singularity of its correlator [MATH] at real [MATH].', '0911.4244-2-52-4': 'First, let us examine singularity at [MATH].', '0911.4244-2-52-5': 'Note that from the form of the collective field current [REF], the singularities only come from the factor [MATH].', '0911.4244-2-52-6': 'The other factor [MATH] is independent of [MATH].', '0911.4244-2-52-7': 'It is not difficult to check that near [MATH], [EQUATION] where the effective action [MATH] is defined by [EQUATION]', '0911.4244-2-52-8': 'Similar relations hold near other points [MATH].', '0911.4244-2-52-9': 'Therefore, the singular part of [MATH] is given by [EQUATION]', '0911.4244-2-52-10': 'Hence we have the [MATH] constraints: [EQUATION]', '0911.4244-2-52-11': 'Therefore, we can use this correlator to define a non-commutative Calabi-Yau threefold as follows: [EQUATION] and the associated non-commutative spectral curve [MATH] can be defined by [EQUATION] with the non-commutativity [MATH].', '0911.4244-2-52-12': 'These are well-defined at finite [MATH].', '0911.4244-2-53-0': 'Note that the correlator takes the form [EQUATION]', '0911.4244-2-53-1': 'In principle, the explicit form of [MATH] can be determined by examining regular part of [MATH].', '0911.4244-2-53-2': 'But actual calculation is tedious for general [MATH].', '0911.4244-2-53-3': 'In general, the non-commutative geometry takes the form [EQUATION]', '0911.4244-2-53-4': 'In the next subsection, we give the explicit form of the spectral curve for [MATH].', '0911.4244-2-54-0': 'Before going to the case of [MATH], let us consider the large [MATH] limit.', '0911.4244-2-54-1': ""As in the one-matrix model case, we take [MATH] limit with the deformed 't Hooft couplings [MATH] fixed."", '0911.4244-2-54-2': 'In the large [MATH], the saddle points are determined by the stationary condition [MATH].', '0911.4244-2-54-3': 'In the [MATH] limit, the non-commutativity is lost and the large [MATH] factorization gives [EQUATION] where [EQUATION]', '0911.4244-2-54-4': 'Here the density functions [MATH] are defined by [EQUATION]', '0911.4244-2-54-5': 'In the deformed [MATH] quiver matrix model, the deformed geometry takes the form [EQUATION]', '0911.4244-2-54-6': 'Therefore, the deformed spectral curve [MATH] is covered by [MATH] Riemann sheets and the points on the [MATH]-th sheet can be parametrized by [MATH].', '0911.4244-2-54-7': 'Hence they are fibered over the complex [MATH]-plane.', '0911.4244-2-55-0': 'Generalization to [MATH] and [MATH] cases is straightforward at least in the large [MATH] limit.', '0911.4244-2-55-1': ""Replace the 't Hooft couplings [MATH] by the deformed ones [MATH]."", '0911.4244-2-55-2': 'The finite non-commutative form of the spectral curve would be fixed by requiring the consistency with the symmetry of the partition function [REF].', '0911.4244-2-56-0': '## Deformed [MATH] quiver model and non-commutative curve', '0911.4244-2-57-0': 'For [MATH]-deformed [MATH] quiver matrix model, the explicit forms of the currents are given by [EQUATION] where [EQUATION] and [MATH].', '0911.4244-2-57-1': 'With some work, we can see that at finite [MATH] (with [MATH]), the regular part of the ""spectral determinant"" is given by [EQUATION]', '0911.4244-2-57-2': 'The last two terms in the right-handed side of [REF] do not contribute to the correlator.', '0911.4244-2-57-3': 'Therefore, we have the explicit form of the finite [MATH] spectral curve mentioned in the introduction: [EQUATION] where [EQUATION] with [MATH].', '0911.4244-2-57-4': 'At the commutative point [MATH], the algebraic curve of this form (in the large [MATH] limit) is already known [CITATION].', '0911.4244-2-58-0': 'In the large [MATH] limit with general [MATH], spectral curve factorizes into [EQUATION] where [EQUATION]', '0911.4244-2-58-1': 'Here [MATH] and [EQUATION]', '0911.4244-2-59-0': ""# Dijkgraaf-Vafa's proposal: multi-Penner potential"", '0911.4244-2-60-0': ""In [CITATION], it was pointed out that the correlation function of [MATH] Toda field theory (Liouville theory for [MATH]) in two dimensions can be associated with Nekrasov's partition function [CITATION] of [MATH] quiver gauge theory."", '0911.4244-2-60-1': 'For [MATH] gauge theory with [MATH] hypermultiplets its Nekrasov partition function is identified with the (chiral) four point function of the Toda theory on a sphere: [EQUATION] under the identification ([REF]) between the deformation parameters and the parameter [MATH] appearing in the central charge [REF] of the Toda theory.', '0911.4244-2-61-0': 'The positions of the vertex operators correspond to punctures of the sphere.', '0911.4244-2-61-1': 'Three of them can be chosen at [MATH].', '0911.4244-2-61-2': 'As we will see later, this sphere can be identified with the one on which [MATH] M5-branes wrap.', '0911.4244-2-61-3': 'The vectors [MATH] in the dual space [MATH] of the Cartan subalgebra are expanded as [MATH].', '0911.4244-2-61-4': 'In general, it is possible to consider many types of punctures in the sphere on which M5-branes wrap.', '0911.4244-2-61-5': 'This variety corresponds to the flavor symmetries of the gauge theory [CITATION].', '0911.4244-2-61-6': 'In the Toda theory, this corresponds to choice of the vectors [MATH].', '0911.4244-2-61-7': 'The vertex operator [MATH] corresponding to the ""simple"" puncture associated with the [MATH] flavor symmetry is [CITATION] [EQUATION] where [MATH]) is the highest weight of the (anti-)fundamental representation.', '0911.4244-2-61-8': 'The other types of punctures correspond to more generic choices of [MATH].', '0911.4244-2-61-9': 'The [MATH] gauge theory with [MATH] hypermultiplets has [MATH] flavor symmetry as a subgroup of [MATH].', '0911.4244-2-61-10': 'Therefore, for this gauge theory, the corresponding correlation function is such that two vertex operators are of ""simple"" type like ([REF]) while the other two are of generic ""[MATH]"" type.', '0911.4244-2-62-0': 'As discussed in [CITATION], by using the relation between [MATH] Toda field theory and the quiver matrix model [CITATION], we reach the matrix model which describes [MATH] quiver gauge theory.', '0911.4244-2-62-1': 'The prescription introduced in [CITATION] is that the original matrix model action is set to zero.', '0911.4244-2-62-2': 'Under zero action, we consider the correlation functions of the vertex operators considered above.', '0911.4244-2-62-3': 'In the correlation function, the vertex operator can be written in terms of the matrices by using ([REF]) as [EQUATION]', '0911.4244-2-62-4': 'With zero action ([MATH]), we define the chiral four-point correlation function which corresponds to [MATH] gauge theory with [MATH] hypermultiplets by [EQUATION] where [MATH].', '0911.4244-2-62-5': 'For later convenience, we set [MATH].', '0911.4244-2-62-6': 'The momentum conservation condition is required [EQUATION]', '0911.4244-2-62-7': 'Using this four-point function, we define the partition function of the deformed [MATH] quiver matrix model by [EQUATION] where the determinantal form of the vertex operators [REF] at punctures [MATH] is converted into a set of new matrix model potentials [MATH] with multi-Penner type interaction: [EQUATION]', '0911.4244-2-62-8': 'Later we will set [MATH], [MATH] and [MATH].', '0911.4244-2-63-0': 'We denote the components of [MATH] in the fundamental weight basis by [MATH]: [MATH].', '0911.4244-2-63-1': 'As explained above, we let the operators at [MATH] be of the simple type.', '0911.4244-2-63-2': 'That is, we set [EQUATION]', '0911.4244-2-63-3': 'The operators at [MATH] are of the generic type.', '0911.4244-2-64-0': '## [MATH] quiver matrix model corresponding to [MATH] gauge theory', '0911.4244-2-65-0': 'In this subsection, we consider [MATH] quiver matrix model with the action ([REF]).', '0911.4244-2-65-1': 'As seen above, the Nekrasov partition function for [MATH] supersymmetric [MATH] gauge theory with six massive flavors is expected to be described by [MATH] quiver matrix model with the action ([REF]) with [MATH].', '0911.4244-2-66-0': 'We have explicitly seen the spectral curve of [MATH] quiver matrix model in the large [MATH] limit in the subsection [REF].', '0911.4244-2-66-1': 'For convenience, we rewrite [REF] in the following form: [EQUATION] where [EQUATION]', '0911.4244-2-66-2': 'By substituting the action [REF] and ([REF]), the explicit forms of [MATH] [REF] are given by [EQUATION] and [MATH].', '0911.4244-2-66-3': 'Here [MATH] are components of [MATH] in the fundamental weight basis: [MATH].', '0911.4244-2-67-0': 'The form of [MATH] [REF] and that of [MATH] [REF] are [EQUATION] where [MATH] and [MATH] are constants.', '0911.4244-2-67-1': 'As seen above, the coordinate [MATH] parametrizes the sphere.', '0911.4244-2-67-2': 'The spectral curve is a triple cover of this sphere and each value of [MATH] at each sheet is given by [MATH].', '0911.4244-2-68-0': 'We consider a one-form [MATH] with [MATH] given by [REF].', '0911.4244-2-68-1': 'Later on this will be identified with the Seiberg-Witten one-form on the [MATH]-th sheet.', '0911.4244-2-68-2': 'From the form of the spectral curve, we can see that the one-form has simple poles at [MATH] and at [MATH].', '0911.4244-2-69-0': 'The residues of [MATH] at the simple punctures [MATH] and [MATH] are given by [EQUATION]', '0911.4244-2-69-1': 'These will be compared with the mass parameter associated with the [MATH] flavor symmetry in the gauge theory.', '0911.4244-2-69-2': 'The residues at [MATH] are of generic type: [EQUATION]', '0911.4244-2-69-3': 'This takes the same form as the mass parameters associated with the [MATH] flavor symmetry, which will be analyzed in the next section.', '0911.4244-2-70-0': 'Using the momentum conservation [REF], the residues at [MATH] are found to be [EQUATION]', '0911.4244-2-70-1': 'This also takes the same form as the [MATH] mass parameters.', '0911.4244-2-71-0': '# [MATH] gauge theory and Seiberg-Witten curve', '0911.4244-2-72-0': 'We consider [MATH] gauge theory with [MATH].', '0911.4244-2-72-1': 'The low energy effective theory (and the quantum Coulomb moduli space) of [MATH] supersymmetric gauge theory is described by the Seiberg-Witten curve and the meromorphic one-form called Seiberg-Witten one-form [CITATION].', '0911.4244-2-72-2': 'We will see that the Seiberg-Witten curve of this theory enjoys the same properties as the spectral curve considered above for the [MATH]) case.', '0911.4244-2-73-0': '## [MATH] gauge theory with [MATH]', '0911.4244-2-74-0': 'We first consider the [MATH] gauge theory with six massive hypermultiplets.', '0911.4244-2-74-1': 'This theory has an exactly marginal coupling which is the microscopic gauge coupling constant: [EQUATION]', '0911.4244-2-74-2': 'The type IIA brane construction and its M-theory lift lead to the Seiberg-Witten curve which is a hypersurface in 2-complex dimensional space [MATH] [CITATION] [EQUATION] where [MATH]) are the mass parameters of the hypermultiplets and [MATH].', '0911.4244-2-74-3': 'We can see that the coordinate [MATH] has mass dimension one and therefore the constants [MATH], [MATH] and [MATH] have mass dimension one, two and three respectively.', '0911.4244-2-74-4': 'In principle, the constants depend on the mass parameters and the Coulomb moduli parameters [MATH] and [MATH] which correspond to [MATH] and [MATH] at weak coupling.', '0911.4244-2-74-5': 'The dimensional analysis and the regularity constraint in the massless limit show that [MATH] and [MATH] are, respectively, linear in [MATH] and [MATH] and also include the terms [MATH] and [MATH].', '0911.4244-2-74-6': 'Also, [MATH] depends only on the mass parameters.', '0911.4244-2-74-7': 'The curve is translated into the following form [EQUATION] where [MATH], [MATH] and [MATH] are degree two polynomials in [MATH]: [EQUATION]', '0911.4244-2-74-8': 'As in [CITATION], by shifting the coordinate [MATH] appropriately, we can eliminate the quadratic term in [MATH].', '0911.4244-2-74-9': 'Then, by changing the coordinate [MATH], we obtain [CITATION] [EQUATION] where [MATH] and [MATH] are the degree 4 and 6 polynomials respectively and can be written in terms of [MATH], [MATH] and [MATH] [EQUATION]', '0911.4244-2-74-10': 'In this coordinate, the Seiberg-Witten one-form is [MATH].', '0911.4244-2-74-11': 'We denote by [MATH] a coordinate in the sphere on which three M5-branes wrap.', '0911.4244-2-74-12': 'Therefore, [MATH] are local coordinates in the cotangent bundle of this sphere.', '0911.4244-2-74-13': 'We identify the sphere with the one that appeared in the quiver matrix model and Toda theory.', '0911.4244-2-74-14': 'This implies [MATH].', '0911.4244-2-75-0': 'From the expression ([REF]), we can see that the Seiberg-Witten one-form has poles at [MATH] and [MATH].', '0911.4244-2-75-1': 'As found in [CITATION] and we will see below, the structure of those poles corresponds to the flavor symmetry of the quiver gauge theory, because the residues of the Seiberg-Witten one-form are the mass parameters associated with the flavor symmetries.', '0911.4244-2-76-0': 'The residues of the Seiberg-Witten one-form [MATH] at [MATH] and [MATH] are given by [EQUATION] where each value denotes the residue at the [MATH]-th sheet of the Riemann surface.', '0911.4244-2-77-0': 'The residues at [MATH] are found to be [EQUATION]', '0911.4244-2-77-1': 'The flavor symmetry [MATH] associated with the punctures at [MATH] are the symmetry of three hypermultiplets whose masses are [MATH]).', '0911.4244-2-77-2': 'Since the mass associated with [MATH] is [MATH], the residue at [MATH] should be of the form [EQUATION]', '0911.4244-2-77-3': 'Also, the residue at [MATH] should be of the form [EQUATION]', '0911.4244-2-77-4': 'These determine [MATH] completely and lead to [EQUATION]', '0911.4244-2-77-5': 'From this, we can compute the residues at [MATH] as [EQUATION] which corresponds to the Cartan part of [MATH].', '0911.4244-2-77-6': 'At [MATH], we also find the similar form as above.', '0911.4244-2-77-7': 'From the residue analysis alone, we cannot determine the constants [MATH] and [MATH] in ([REF]).', '0911.4244-2-78-0': '### Matching of the matrix model parameters', '0911.4244-2-79-0': 'Obviously, the coordinate [MATH] is identified with [MATH] and [MATH] on the [MATH]-th sheet with [MATH].', '0911.4244-2-79-1': 'Comparing the residues [REF], [REF] with [REF], we find that the parameters in [MATH] quiver matrix model must be chosen as [EQUATION] in order to yield the Seiberg-Witten curve of the gauge theory.', '0911.4244-2-79-2': 'Also, by comparing [REF] with [REF], and [REF] with [REF], we have [EQUATION]', '0911.4244-2-79-3': 'To summarize, we have determined the ""weights"" of the vertex operators [MATH] as follows: [EQUATION]', '0911.4244-2-79-4': 'The matrix model potential [MATH] with the multi-log interaction are finally fixed as (up to constants) [EQUATION]', '0911.4244-2-79-5': ""From the momentum conservation [REF], the deformed 't Hooft couplings [MATH] are related to the mass parameters of the gauge theory as follows: [EQUATION]"", '0911.4244-2-80-0': '## Isomorphism between the matrix model curve and the Seiberg-Witten curve for general [MATH]', '0911.4244-2-81-0': 'It is straightforward to generalize the analysis for [MATH] to general [MATH].', '0911.4244-2-81-1': 'The spectral curve [REF] of the [MATH] quiver matrix model has the form [EQUATION] where [EQUATION]', '0911.4244-2-81-2': 'With the choice of the multi-Penner potential [REF], [MATH] have the following form of the singularities: [EQUATION] for some polynomials [MATH] in [MATH].', '0911.4244-2-82-0': 'On the other hand, The Seiberg-Witten curve [CITATION] for the [MATH] gauge theory with [MATH] massive hypermultiplets is given by [EQUATION]', '0911.4244-2-82-1': 'The mass parameters of the hypermultiplets are denoted by [MATH], and here for simplicity, we have set [MATH].', '0911.4244-2-82-2': 'Also [MATH].', '0911.4244-2-82-3': 'This curve can be rewritten as [EQUATION] where [EQUATION]', '0911.4244-2-82-4': 'Here [MATH] (resp. [MATH]) is the [MATH]-th elementary symmetric polynomial of [MATH] (resp. of [MATH]).', '0911.4244-2-83-0': 'By a change of variable [EQUATION] the Seiberg-Witten curve [REF] turns into the Gaiotto form: [EQUATION] where [MATH] are the following degree [MATH] polynomials in [MATH]: [EQUATION]', '0911.4244-2-83-1': 'By evaluating the residue at [MATH] and at [MATH] as in the case of [MATH], the constant [MATH], which enters in [MATH] [REF], is fixed as [EQUATION]', '0911.4244-2-83-2': 'Now, we can clearly see the similarity between the matrix model curve [REF] in the planar limit and the Seiberg-Witten curve [REF].', '0911.4244-2-84-0': 'Let us compare the residues of one-forms [MATH] and [MATH].', '0911.4244-2-84-1': 'In the [MATH] quiver matrix model, the curve [REF] in the planar limit factorizes into [EQUATION] where [MATH] is defined by [REF].', '0911.4244-2-84-2': 'The residues of [MATH]) at the simple punctures [MATH] and [MATH] are given by [EQUATION]', '0911.4244-2-84-3': 'The residues at generic punctures [MATH] and [MATH] are found to be [EQUATION] for [MATH].', '0911.4244-2-84-4': 'While the residues of the Seiberg-Witten one-form [MATH] at [MATH] and [MATH] are given by [EQUATION] where each value denotes the residue at the [MATH]-th sheet of the Riemann surface.', '0911.4244-2-84-5': 'The residues at [MATH] and at [MATH] on the [MATH]-th sheet ([MATH]) are given by [EQUATION]', '0911.4244-2-84-6': 'For general [MATH], these residues match if the weights of the vertex operators are identified with the mass parameters of the gauge theory as follows [EQUATION]', '0911.4244-2-84-7': 'The matrix model potentials [MATH]) are fixed as [EQUATION]', '0911.4244-2-84-8': 'With this choice of the multi-log potentials, the [MATH] quiver matrix model curve [REF] in the planar limit coincides with the [MATH] Seiberg-Witten curve [REF] with [MATH] massive hypermultiplets.', '0911.4244-2-85-0': 'It is rather simple to provide the counting of the 0d-parameters and 4d-parameters and the matching of the numbers of parameters.', '0911.4244-2-85-1': 'With the rule concerning with the types (simple versus generic) of the punctures [CITATION] explained before, the four of the vertex operators contain [MATH] parameters.', '0911.4244-2-85-2': 'This matches the number of mass parameters in the gauge theory side.', '0911.4244-2-85-3': 'As for the number of the Coulomb moduli, let us first note that, in the planar limit, the [MATH] quiver matrix model develops an array(or ladder) of [MATH] spieces of two-cut eigenvalue distributions.', '0911.4244-2-85-4': 'The Riemann surface developed is genus [MATH] and is depicted as a linear array of [MATH]s with adjacent two (say, [MATH] and [MATH]) connected by two pipes consisting of the [MATH]-th eigenvalue distribution.', '0911.4244-2-85-5': 'Each of the [MATH] Coulomb moduli is obtained simply by integrating the matrix model differential over one of the two pipes and of course represent the filling fraction of the [MATH]-th eigenvalue distribution.', '0911.4244-2-85-6': 'This is an obvious generalization of [MATH] case noted in [CITATION].'}","[['0911.4244-1-26-0', '0911.4244-2-28-0'], ['0911.4244-1-26-1', '0911.4244-2-28-1'], ['0911.4244-1-26-2', '0911.4244-2-28-2'], ['0911.4244-1-73-0', '0911.4244-2-75-0'], ['0911.4244-1-73-1', '0911.4244-2-75-1'], ['0911.4244-1-13-0', '0911.4244-2-15-0'], ['0911.4244-1-13-1', '0911.4244-2-15-1'], ['0911.4244-1-13-2', '0911.4244-2-15-2'], ['0911.4244-1-13-3', '0911.4244-2-15-3'], ['0911.4244-1-13-4', '0911.4244-2-15-4'], ['0911.4244-1-66-0', '0911.4244-2-68-0'], ['0911.4244-1-66-1', '0911.4244-2-68-1'], ['0911.4244-1-66-2', '0911.4244-2-68-2'], ['0911.4244-1-75-0', '0911.4244-2-77-0'], ['0911.4244-1-75-1', '0911.4244-2-77-1'], ['0911.4244-1-75-2', '0911.4244-2-77-2'], ['0911.4244-1-75-3', '0911.4244-2-77-3'], ['0911.4244-1-75-4', '0911.4244-2-77-4'], ['0911.4244-1-75-5', '0911.4244-2-77-5'], ['0911.4244-1-75-6', '0911.4244-2-77-6'], ['0911.4244-1-75-7', '0911.4244-2-77-7'], ['0911.4244-1-40-0', '0911.4244-2-42-0'], ['0911.4244-1-22-0', '0911.4244-2-24-0'], ['0911.4244-1-22-1', '0911.4244-2-24-1'], ['0911.4244-1-22-3', '0911.4244-2-24-3'], ['0911.4244-1-22-4', '0911.4244-2-24-4'], ['0911.4244-1-22-5', '0911.4244-2-24-5'], ['0911.4244-1-22-6', '0911.4244-2-24-6'], ['0911.4244-1-22-7', '0911.4244-2-24-7'], ['0911.4244-1-22-8', '0911.4244-2-24-8'], ['0911.4244-1-22-9', '0911.4244-2-24-9'], ['0911.4244-1-22-10', '0911.4244-2-24-10'], ['0911.4244-1-22-11', '0911.4244-2-24-11'], ['0911.4244-1-22-12', '0911.4244-2-24-12'], ['0911.4244-1-22-13', '0911.4244-2-24-13'], ['0911.4244-1-22-14', '0911.4244-2-24-14'], ['0911.4244-1-22-15', '0911.4244-2-24-15'], ['0911.4244-1-22-16', '0911.4244-2-24-16'], ['0911.4244-1-60-0', '0911.4244-2-62-0'], ['0911.4244-1-60-1', '0911.4244-2-62-1'], ['0911.4244-1-60-2', '0911.4244-2-62-2'], ['0911.4244-1-60-3', '0911.4244-2-62-3'], ['0911.4244-1-60-4', '0911.4244-2-62-4'], ['0911.4244-1-60-5', '0911.4244-2-62-5'], ['0911.4244-1-60-6', '0911.4244-2-62-6'], ['0911.4244-1-60-7', '0911.4244-2-62-7'], ['0911.4244-1-60-8', '0911.4244-2-62-8'], ['0911.4244-1-34-0', '0911.4244-2-36-0'], ['0911.4244-1-34-1', '0911.4244-2-36-1'], ['0911.4244-1-34-2', '0911.4244-2-36-2'], ['0911.4244-1-34-3', '0911.4244-2-36-3'], ['0911.4244-1-34-4', '0911.4244-2-36-4'], ['0911.4244-1-34-5', '0911.4244-2-36-5'], ['0911.4244-1-34-6', '0911.4244-2-36-6'], ['0911.4244-1-34-7', '0911.4244-2-36-7'], ['0911.4244-1-3-0', '0911.4244-2-3-0'], ['0911.4244-1-3-2', '0911.4244-2-3-2'], ['0911.4244-1-3-3', '0911.4244-2-3-3'], ['0911.4244-1-3-4', '0911.4244-2-3-4'], ['0911.4244-1-3-5', '0911.4244-2-3-5'], ['0911.4244-1-3-6', '0911.4244-2-3-6'], ['0911.4244-1-3-7', '0911.4244-2-3-7'], ['0911.4244-1-19-0', '0911.4244-2-21-0'], ['0911.4244-1-19-2', '0911.4244-2-21-2'], ['0911.4244-1-19-3', '0911.4244-2-21-3'], ['0911.4244-1-19-4', '0911.4244-2-21-4'], ['0911.4244-1-25-0', '0911.4244-2-27-0'], ['0911.4244-1-25-1', '0911.4244-2-27-1'], ['0911.4244-1-25-2', '0911.4244-2-27-2'], ['0911.4244-1-25-3', '0911.4244-2-27-3'], ['0911.4244-1-8-0', '0911.4244-2-9-0'], ['0911.4244-1-8-1', '0911.4244-2-9-1'], ['0911.4244-1-8-2', '0911.4244-2-9-2'], ['0911.4244-1-8-3', '0911.4244-2-9-3'], ['0911.4244-1-8-4', '0911.4244-2-9-4'], ['0911.4244-1-8-5', '0911.4244-2-9-5'], ['0911.4244-1-8-6', '0911.4244-2-9-6'], ['0911.4244-1-8-7', '0911.4244-2-9-7'], ['0911.4244-1-8-9', '0911.4244-2-9-9'], ['0911.4244-1-58-0', '0911.4244-2-60-0'], ['0911.4244-1-58-1', '0911.4244-2-60-1'], ['0911.4244-1-29-0', '0911.4244-2-31-0'], ['0911.4244-1-65-0', '0911.4244-2-67-0'], ['0911.4244-1-65-1', '0911.4244-2-67-1'], ['0911.4244-1-65-2', '0911.4244-2-67-2'], ['0911.4244-1-43-0', '0911.4244-2-45-0'], ['0911.4244-1-43-1', '0911.4244-2-45-1'], ['0911.4244-1-43-2', '0911.4244-2-45-2'], ['0911.4244-1-43-3', '0911.4244-2-45-3'], ['0911.4244-1-43-4', '0911.4244-2-45-4'], ['0911.4244-1-0-1', '0911.4244-2-0-1'], ['0911.4244-1-0-2', '0911.4244-2-0-2'], ['0911.4244-1-10-0', '0911.4244-2-12-0'], ['0911.4244-1-10-1', '0911.4244-2-12-1'], ['0911.4244-1-10-2', '0911.4244-2-12-2'], ['0911.4244-1-10-3', '0911.4244-2-12-3'], ['0911.4244-1-10-4', '0911.4244-2-12-4'], ['0911.4244-1-10-5', '0911.4244-2-12-5'], ['0911.4244-1-10-6', '0911.4244-2-12-6'], ['0911.4244-1-10-7', '0911.4244-2-12-7'], ['0911.4244-1-10-8', '0911.4244-2-12-8'], ['0911.4244-1-10-9', '0911.4244-2-12-9'], ['0911.4244-1-41-0', '0911.4244-2-43-0'], ['0911.4244-1-41-1', '0911.4244-2-43-1'], ['0911.4244-1-41-2', '0911.4244-2-43-2'], ['0911.4244-1-64-0', '0911.4244-2-66-0'], ['0911.4244-1-64-1', '0911.4244-2-66-1'], ['0911.4244-1-64-3', '0911.4244-2-66-3'], ['0911.4244-1-14-0', '0911.4244-2-16-0'], ['0911.4244-1-14-1', '0911.4244-2-16-1'], ['0911.4244-1-14-2', '0911.4244-2-16-2'], ['0911.4244-1-14-3', '0911.4244-2-16-3'], ['0911.4244-1-14-4', '0911.4244-2-16-4'], ['0911.4244-1-14-5', '0911.4244-2-16-5'], ['0911.4244-1-14-6', '0911.4244-2-16-6'], ['0911.4244-1-44-0', '0911.4244-2-46-0'], ['0911.4244-1-44-1', '0911.4244-2-46-1'], ['0911.4244-1-49-0', '0911.4244-2-51-0'], ['0911.4244-1-49-1', '0911.4244-2-51-1'], ['0911.4244-1-49-2', '0911.4244-2-51-2'], ['0911.4244-1-49-3', '0911.4244-2-51-3'], ['0911.4244-1-49-4', '0911.4244-2-51-4'], ['0911.4244-1-49-5', '0911.4244-2-51-5'], ['0911.4244-1-49-6', '0911.4244-2-51-6'], ['0911.4244-1-49-8', '0911.4244-2-51-8'], ['0911.4244-1-49-9', '0911.4244-2-51-9'], ['0911.4244-1-49-10', '0911.4244-2-51-10'], ['0911.4244-1-49-11', '0911.4244-2-51-11'], ['0911.4244-1-49-12', '0911.4244-2-51-12'], ['0911.4244-1-49-13', '0911.4244-2-51-13'], ['0911.4244-1-49-14', '0911.4244-2-51-14'], ['0911.4244-1-49-15', '0911.4244-2-51-15'], ['0911.4244-1-49-16', '0911.4244-2-51-16'], ['0911.4244-1-49-17', '0911.4244-2-51-17'], ['0911.4244-1-49-18', '0911.4244-2-51-18'], ['0911.4244-1-49-19', '0911.4244-2-51-19'], ['0911.4244-1-15-0', '0911.4244-2-17-0'], ['0911.4244-1-15-2', '0911.4244-2-17-2'], ['0911.4244-1-15-3', '0911.4244-2-17-3'], ['0911.4244-1-15-4', '0911.4244-2-17-4'], ['0911.4244-1-15-5', '0911.4244-2-17-5'], ['0911.4244-1-15-6', '0911.4244-2-17-6'], ['0911.4244-1-15-7', '0911.4244-2-17-7'], ['0911.4244-1-15-8', '0911.4244-2-17-8'], ['0911.4244-1-15-9', '0911.4244-2-17-9'], ['0911.4244-1-27-0', '0911.4244-2-29-0'], ['0911.4244-1-27-1', '0911.4244-2-29-1'], ['0911.4244-1-27-3', '0911.4244-2-29-3'], ['0911.4244-1-27-4', '0911.4244-2-29-4'], ['0911.4244-1-27-5', '0911.4244-2-29-5'], ['0911.4244-1-27-6', '0911.4244-2-29-6'], ['0911.4244-1-27-7', '0911.4244-2-29-7'], ['0911.4244-1-39-0', '0911.4244-2-41-0'], ['0911.4244-1-39-1', '0911.4244-2-41-1'], ['0911.4244-1-39-2', '0911.4244-2-41-2'], ['0911.4244-1-6-0', '0911.4244-2-6-0'], ['0911.4244-1-6-1', '0911.4244-2-6-1'], ['0911.4244-1-6-2', '0911.4244-2-6-2'], ['0911.4244-1-5-0', '0911.4244-2-5-0'], ['0911.4244-1-5-1', '0911.4244-2-5-1'], ['0911.4244-1-5-2', '0911.4244-2-5-2'], ['0911.4244-1-5-3', '0911.4244-2-5-3'], ['0911.4244-1-5-4', '0911.4244-2-5-4'], ['0911.4244-1-5-5', '0911.4244-2-5-5'], ['0911.4244-1-31-0', '0911.4244-2-33-0'], ['0911.4244-1-31-1', '0911.4244-2-33-1'], ['0911.4244-1-31-2', '0911.4244-2-33-2'], ['0911.4244-1-31-3', '0911.4244-2-33-3'], ['0911.4244-1-46-0', '0911.4244-2-48-0'], ['0911.4244-1-46-1', '0911.4244-2-48-1'], ['0911.4244-1-46-2', '0911.4244-2-48-2'], ['0911.4244-1-46-3', '0911.4244-2-48-3'], ['0911.4244-1-7-0', '0911.4244-2-7-0'], ['0911.4244-1-7-1', '0911.4244-2-7-1'], ['0911.4244-1-48-0', '0911.4244-2-50-0'], ['0911.4244-1-48-1', '0911.4244-2-50-1'], ['0911.4244-1-48-2', '0911.4244-2-50-2'], ['0911.4244-1-48-3', '0911.4244-2-50-3'], ['0911.4244-1-50-0', '0911.4244-2-52-0'], ['0911.4244-1-50-1', '0911.4244-2-52-1'], ['0911.4244-1-50-2', '0911.4244-2-52-2'], ['0911.4244-1-50-3', '0911.4244-2-52-3'], ['0911.4244-1-50-4', '0911.4244-2-52-4'], ['0911.4244-1-50-5', '0911.4244-2-52-5'], ['0911.4244-1-50-7', '0911.4244-2-52-7'], ['0911.4244-1-50-8', '0911.4244-2-52-8'], ['0911.4244-1-50-9', '0911.4244-2-52-9'], ['0911.4244-1-50-10', '0911.4244-2-52-10'], ['0911.4244-1-50-11', '0911.4244-2-52-11'], ['0911.4244-1-50-12', '0911.4244-2-52-12'], ['0911.4244-1-16-0', '0911.4244-2-18-0'], ['0911.4244-1-16-1', '0911.4244-2-18-1'], ['0911.4244-1-16-2', '0911.4244-2-18-2'], ['0911.4244-1-16-3', '0911.4244-2-18-3'], ['0911.4244-1-16-4', '0911.4244-2-18-4'], ['0911.4244-1-51-0', '0911.4244-2-53-0'], ['0911.4244-1-51-1', '0911.4244-2-53-1'], ['0911.4244-1-51-2', '0911.4244-2-53-2'], ['0911.4244-1-51-3', '0911.4244-2-53-3'], ['0911.4244-1-51-4', '0911.4244-2-53-4'], ['0911.4244-1-4-0', '0911.4244-2-4-0'], ['0911.4244-1-4-1', '0911.4244-2-4-1'], ['0911.4244-1-45-1', '0911.4244-2-47-1'], ['0911.4244-1-45-2', '0911.4244-2-47-2'], ['0911.4244-1-63-0', '0911.4244-2-65-0'], ['0911.4244-1-63-1', '0911.4244-2-65-1'], ['0911.4244-1-55-0', '0911.4244-2-57-0'], ['0911.4244-1-55-1', '0911.4244-2-57-1'], ['0911.4244-1-55-2', '0911.4244-2-57-2'], ['0911.4244-1-55-3', '0911.4244-2-57-3'], ['0911.4244-1-55-4', '0911.4244-2-57-4'], ['0911.4244-1-61-0', '0911.4244-2-63-0'], ['0911.4244-1-61-2', '0911.4244-2-63-2'], ['0911.4244-1-21-0', '0911.4244-2-23-0'], ['0911.4244-1-21-1', '0911.4244-2-23-1'], ['0911.4244-1-21-2', '0911.4244-2-23-2'], ['0911.4244-1-32-0', '0911.4244-2-34-0'], ['0911.4244-1-18-0', '0911.4244-2-20-0'], ['0911.4244-1-18-1', '0911.4244-2-20-1'], ['0911.4244-1-18-2', '0911.4244-2-20-2'], ['0911.4244-1-70-1', '0911.4244-2-72-1'], ['0911.4244-1-70-2', '0911.4244-2-72-2'], ['0911.4244-1-36-0', '0911.4244-2-38-0'], ['0911.4244-1-36-1', '0911.4244-2-38-1'], ['0911.4244-1-11-0', '0911.4244-2-13-0'], ['0911.4244-1-11-1', '0911.4244-2-13-1'], ['0911.4244-1-11-2', '0911.4244-2-13-2'], ['0911.4244-1-11-3', '0911.4244-2-13-3'], ['0911.4244-1-52-0', '0911.4244-2-54-0'], ['0911.4244-1-52-1', '0911.4244-2-54-1'], ['0911.4244-1-52-2', '0911.4244-2-54-2'], ['0911.4244-1-52-3', '0911.4244-2-54-3'], ['0911.4244-1-52-4', '0911.4244-2-54-4'], ['0911.4244-1-52-5', 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'0911.4244-2-82-2']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/0911.4244,,,
1801.02073,"{'1801.02073-1-0-0': 'This paper gives comprehensive analyses of corpora based on Wikipedia for several tasks in question answering.', '1801.02073-1-0-1': 'Four recent corpora are collected, WikiQA, SelQA, SQuAD, and InfoboxQA, and first analyzed intrinsically by contextual similarities, question types, and answer categories.', '1801.02073-1-0-2': 'These corpora are then analyzed extrinsically by three question answering tasks, answer retrieval, selection, and triggering.', '1801.02073-1-0-3': 'An indexing-based method for the creation of a silver-standard dataset for answer retrieval using the entire Wikipedia is also presented.', '1801.02073-1-0-4': 'Our analysis shows the uniqueness of these corpora and suggests a better use of them for statistical question answering learning.', '1801.02073-1-1-0': '# Introduction', '1801.02073-1-2-0': 'Question answering (QA) has been a blooming research field for the last decade.', '1801.02073-1-2-1': 'Selection-based QA implies a family of tasks that find answer contexts from large data given questions in natural language.', '1801.02073-1-2-2': 'Three tasks have been proposed for selection-based QA.', '1801.02073-1-2-3': 'Given a document, answer extraction [CITATION] finds answer phrases whereas answer selection [CITATION] and answer triggering [CITATION] find answer sentences instead, although the presence of the answer context is not assumed within the provided document for answer triggering but it is for the other two tasks.', '1801.02073-1-2-4': 'Recently, various QA tasks that are not selection-based have been proposed [CITATION]; however, selection-based QA remains still important because of its practical value to real applications (e.g., IBM Watson, MIT Start).', '1801.02073-1-3-0': 'Several datasets have been released for selection-based QA.', '1801.02073-1-3-1': 'wang:07a created the QASent dataset consisting of 277 questions, which has been widely used for benchmarking the answer selection task.', '1801.02073-1-3-2': 'feng:15a presented InsuranceQA comprising 16K+ questions on insurance contexts.', '1801.02073-1-3-3': 'yang:15a introduced WikiQA for answer selection and triggering.', '1801.02073-1-3-4': 'jurczyk:16 created SelQA for large real-scale answer triggering.', '1801.02073-1-3-5': 'rajpurkar2016squad presented SQuAD for answer extraction and selection as well as for reading comprehension.', '1801.02073-1-3-6': 'Finally, morales-EtAl:2016:EMNLP2016 provided InfoboxQA for answer selection.', '1801.02073-1-4-0': 'These corpora make it possible to evaluate the robustness of statistical question answering learning.', '1801.02073-1-4-1': 'Although all of these corpora target on selection-based QA, they are designed for different purposes such that it is important to understand the nature of these corpora so a better use of them can be made.', '1801.02073-1-4-2': 'In this paper, we make both intrinsic and extrinsic analyses of four latest corpora based on Wikipedia, WikiQA, SelQA, SQuAD, and InfoboxQA.', '1801.02073-1-4-3': 'We first give a thorough intrinsic analysis regarding contextual similarities, question types, and answer categories (Section [REF]).', '1801.02073-1-4-4': 'We then map questions in all corpora to the current version of English Wikipedia and benchmark another selection-based QA task, answer retrieval (Section [REF]).', '1801.02073-1-4-5': 'Finally, we present an extrinsic analysis through a set of experiments cross-testing these corpora using a convolutional neural network architecture (Section [REF]).', '1801.02073-1-5-0': '# Intrinsic Analysis', '1801.02073-1-6-0': 'Four publicly available corpora are selected for our analysis.', '1801.02073-1-6-1': 'These corpora are based on Wikipedia, so more comparable than the others, and have already been used for the evaluation of several QA systems.', '1801.02073-1-7-0': 'WikiQA [CITATION] comprises questions selected from the Bing search queries, where user click data give the questions and their corresponding Wikipedia articles.', '1801.02073-1-7-1': 'The abstracts of these articles are then extracted to create answer candidates.', '1801.02073-1-7-2': 'The assumption is made that if many queries lead to the same article, it must contain the answer context; however, this assumption fails for some occasions, which makes this dataset more challenging.', '1801.02073-1-7-3': 'Since the existence of answer contexts is not guaranteed in this task, it is called answer triggering instead of answer selection.', '1801.02073-1-8-0': 'SelQA [CITATION] is a product of five annotation tasks through crowdsourcing.', '1801.02073-1-8-1': 'It consists of about 8K questions where a half of the questions are paraphrased from the other half, aiming to reduce contextual similarities between questions and answers.', '1801.02073-1-8-2': 'Each question is associated with a section in Wikipedia where the answer context is guaranteed, and also with five sections selected from the entire Wikipedia where the selection is made by the Lucene search engine.', '1801.02073-1-8-3': 'This second dataset does not assume the existence of the answer context, so can be used for the evaluation of answer triggering.', '1801.02073-1-9-0': 'SQuAD [CITATION] presents 107K+ crowdsourced questions on 536 Wikipedia articles, where the answer contexts are guaranteed to exist within the provided paragraph.', '1801.02073-1-9-1': 'It contains annotation of answer phrases as well as the pointers to the sentences including the answer phrases; thus, it can be used for both answer extraction and selection.', '1801.02073-1-9-2': 'This corpus also provides human accuracy on those questions, setting up a reasonable upper bound for machines.', '1801.02073-1-9-3': 'To avoid overfitting, the evaluation set is not publicly available although system outputs can be evaluated by their provided script.', '1801.02073-1-10-0': 'InfoboxQA [CITATION] gives 15K+ questions based on the infoboxes from 150 articles in Wikipedia.', '1801.02073-1-10-1': 'Each question is crowdsourced and associated with an infobox, where each line of the infobox is considered an answer candidate.', '1801.02073-1-10-2': 'This corpus emphasizes the gravity of infoboxes, which summary arguably the most commonly asked information about those articles.', '1801.02073-1-10-3': 'Although the nature of this corpus is different from the others, it can also be used to evaluate answer selection.', '1801.02073-1-11-0': '## Analysis', '1801.02073-1-12-0': 'All corpora provide datasets/splits for answer selection, whereas only (WikiQA, SQuAD) and (WikiQA, SelQA) provide datasets for answer extraction and answer triggering, respectively.', '1801.02073-1-12-1': 'SQuAD is much larger in size although questions in this corpus are often paraphrased multiple times.', '1801.02073-1-12-2': ""On the contrary, SQuAD's average candidates per question ([MATH]) is the smallest because SQuAD extracts answer candidates from paragraphs whereas the others extract them from sections or infoboxes that consist of bigger contexts."", '1801.02073-1-12-3': 'Although InfoboxQA is larger than WikiQA or SelQA, the number of token types ([MATH]) in InfoboxQA is smaller than those two, due to the repetitive nature of infoboxes.', '1801.02073-1-13-0': 'All corpora show similar average answer candidate lengths ([MATH]), except for InfoboxQA where each line in the infobox is considered a candidate.', '1801.02073-1-13-1': 'SelQA and SQuAD show similar average question lengths ([MATH]) because of the similarity between their annotation schemes.', '1801.02073-1-13-2': ""It is not surprising that WikiQA's average question length is the smallest, considering their questions are taken from search queries."", '1801.02073-1-13-3': ""InfoboxQA's average question length is relatively small, due to the restricted information that can be asked from the infoboxes."", '1801.02073-1-13-4': 'InfoboxQA and WikiQA show the least question-answer word overlaps over questions and answers ([MATH] and [MATH] in Table [REF]), respectively.', '1801.02073-1-13-5': 'In terms of the F1-score for overlapping words ([MATH]), SQuAD gives the least portion of overlaps between question-answer pairs although WikiQA comes very close.', '1801.02073-1-14-0': 'Fig. [REF] shows the distributions of seven question types grouped deterministically from the lexicons.', '1801.02073-1-14-1': 'Although these corpora have been independently developed, a general trend is found, where the what question type dominates, followed by how and who, followed by when and where, and so on.', '1801.02073-1-15-0': 'Fig. [REF] shows the distributions of answer categories automatically classified by our Convolutional Neural Network model trained on the data distributed by [CITATION].', '1801.02073-1-15-1': 'Interestingly, each corpus focuses on different categories, Numeric for WikiQA and SelQA, Entity for SQuAD, and Person for InfoboxQA, which gives enough diversities for statistical learning to build robust models.', '1801.02073-1-16-0': '# Answer Retrieval', '1801.02073-1-17-0': 'This section describes another selection-based QA task, called answer retrieval, that finds the answer context from a larger dataset, the entire Wikipedia.', '1801.02073-1-17-1': 'SQuAD provides no mapping of the answer contexts to Wikipedia, whereas WikiQA and SelQA provide mappings; however, their data do not come from the same version of Wikipedia.', '1801.02073-1-17-2': 'We propose an automatic way of mapping the answer contexts from all corpora to the same version of Wikipeda so they can be coherently used for answer retrieval.', '1801.02073-1-18-0': 'Each paragraph in Wikipedia is first indexed by Lucene using 1,2,3-grams, where the paragraphs are separated by WikiExtractor and segmented by NLP4J (28.7M+ paragraphs are indexed).', '1801.02073-1-18-1': 'Each answer sentence from the corpora in Table [REF] is then queried to Lucene, and the top-5 ranked paragraphs are retrieved.', '1801.02073-1-18-2': 'The cosine similarity between each sentence in these paragraphs and the answer sentence is measured for [MATH]-grams, say [MATH].', '1801.02073-1-18-3': 'A weight is assigned to each [MATH]-gram score, say [MATH], and the weighted sum is measured: [MATH].', '1801.02073-1-18-4': 'The fixed weights of [MATH] are used for our experiments, which can be improved.', '1801.02073-1-19-0': 'If there exists a sentence whose [MATH], the paragraph consisting of that sentence is considered the silver-standard answer passage.', '1801.02073-1-19-1': 'Table [REF] shows how robust these silver-standard passages are based on human judgement ([MATH]) and how many passages are collected ([MATH]) for [MATH], where the human judgement is performed on 50 random samples for each case.', '1801.02073-1-19-2': 'For answer retrieval, a dataset is created by [MATH], which gives [MATH] accuracy and [MATH] coverage, respectively.', '1801.02073-1-19-3': 'Finally, each question is queried to Lucene and the top-[MATH] paragraphs are retrieved from the entire Wikipedia.', '1801.02073-1-19-4': 'If the answer sentence exists within those retrieved paragraphs according to the silver-standard, it is considered correct.', '1801.02073-1-20-0': '# Extrinsic Analysis', '1801.02073-1-21-0': '## Answer Selection', '1801.02073-1-22-0': 'Answer selection is evaluated by two metrics, mean average precision (MAP) and mean reciprocal rank (MRR).', '1801.02073-1-22-1': 'The bigram CNN introduced by [CITATION] is used to generate all the results in Table [REF], where models are trained on either single or combined datasets.', '1801.02073-1-22-2': 'Clearly, the questions in WikiQA are the most challenging, and adding more training data from the other corpora hurts accuracy due to the uniqueness of query-based questions in this corpus.', '1801.02073-1-22-3': 'The best model is achieved by training on W+S+Q for SelQA; adding InfoboxQA hurts accuracy for SelQA although it gives a marginal gain for SQuAD.', '1801.02073-1-22-4': 'Just like WikiQA, InfoboxQA performs the best when it is trained on only itself.', '1801.02073-1-22-5': 'From our analysis, we suggest that to use models trained on WikiQA and InfoboxQA for short query-like questions, whereas to use ones trained on SelQA and SQuAD for long natural questions.', '1801.02073-1-23-0': '## Answer Retrieval', '1801.02073-1-24-0': 'Finding a paragraph that includes the answer context out of the entire Wikipedia is an extremely difficult task (128.7M).', '1801.02073-1-24-1': 'The last row of Table [REF] shows results from answer retrieval.', '1801.02073-1-24-2': 'Given [MATH], SelQA and SQuAD show about 34 and 35 accuracy, which are reasonable.', '1801.02073-1-24-3': 'However, WikiQA shows a significantly lower accuracy of 12.47; this is because the questions in WikiQA is about twice shorter than the questions in the other corpora such that not enough lexicons can be extracted from these questions for the Lucene search.', '1801.02073-1-25-0': '## Answer Triggering', '1801.02073-1-26-0': 'The results of [MATH] from the answer retrieval task in Section [REF] are used to create the datasets for answer triggering, where about 65% of the questions are not expected to find their answer contexts from the provided paragraphs for SelQA and SQuAD and 87.5% are not expected for WikiQA.', '1801.02073-1-26-1': 'Answer triggering is evaluated by the F1 scores as presented in Table [REF], where three corpora are cross validated.', '1801.02073-1-26-2': 'The results on WikiQA are pretty low as expected from the poor accuracy on the answer retrieval task.', '1801.02073-1-26-3': 'Training on SelQA gives the best models for both WikiQA and SelQA.', '1801.02073-1-26-4': 'Training on SQuAD gives the best model for SQuAD although the model trained on SelQA is comparable.', '1801.02073-1-26-5': 'Since the answer triggering datasets are about 5 times larger than the answer selection datasets, it is computationally too expensive to combine all data for training.', '1801.02073-1-26-6': 'We plan to find a strong machine to perform this experiment in near future.', '1801.02073-1-27-0': '# Related work', '1801.02073-1-28-0': 'Lately, several deep learning approaches have been proposed for question answering.', '1801.02073-1-28-1': 'yu:14a presented a CNN model that recognizes the semantic similarity between two sentences.', '1801.02073-1-28-2': 'wang-nyberg:2015:ACL-IJCNLP presented a stacked bidirectional LSTM approach to read words in sequence, then outputs their similarity scores.', '1801.02073-1-28-3': 'feng:15a applied a general deep learning framework to non-factoid question answering.', '1801.02073-1-28-4': 'santos:16a introduced an attentive pooling mechanism that led to further improvements in selection-based QA.', '1801.02073-1-29-0': '# Conclusion', '1801.02073-1-30-0': 'We present a comprehensive comparison study of the existing corpora for selection-based question answering.', '1801.02073-1-30-1': 'Our intrinsic analysis provides a better understanding of the uniqueness or similarity between these corpora.', '1801.02073-1-30-2': 'Our extrinsic analysis shows the strength or weakness of combining these corpora together for statistical learning.', '1801.02073-1-30-3': 'Additionally, we create a silver-standard dataset for answer retrieval and triggering, which will be publicly available.', '1801.02073-1-30-4': 'In the future, we will explore different ways of improving the quality of our silver-standard datasets by fine-tuning the hyper-parameters.'}","{'1801.02073-2-0-0': 'This paper gives comprehensive analyses of corpora based on Wikipedia for several tasks in question answering.', '1801.02073-2-0-1': 'Four recent corpora are collected, WikiQA, SelQA, SQuAD, and InfoboxQA, and first analyzed intrinsically by contextual similarities, question types, and answer categories.', '1801.02073-2-0-2': 'These corpora are then analyzed extrinsically by three question answering tasks, answer retrieval, selection, and triggering.', '1801.02073-2-0-3': 'An indexing-based method for the creation of a silver-standard dataset for answer retrieval using the entire Wikipedia is also presented.', '1801.02073-2-0-4': 'Our analysis shows the uniqueness of these corpora and suggests a better use of them for statistical question answering learning.', '1801.02073-2-1-0': '# Introduction', '1801.02073-2-2-0': 'Question answering (QA) has been a blooming research field for the last decade.', '1801.02073-2-2-1': 'Selection-based QA implies a family of tasks that find answer contexts from large data given questions in natural language.', '1801.02073-2-2-2': 'Three tasks have been proposed for selection-based QA.', '1801.02073-2-2-3': 'Given a document, answer extraction [CITATION] finds answer phrases whereas answer selection [CITATION] and answer triggering [CITATION] find answer sentences instead, although the presence of the answer context is not assumed within the provided document for answer triggering but it is for the other two tasks.', '1801.02073-2-2-4': 'Recently, various QA tasks that are not selection-based have been proposed [CITATION]; however, selection-based QA remains still important because of its practical value to real applications (e.g., IBM Watson, MIT Start).', '1801.02073-2-3-0': 'Several datasets have been released for selection-based QA.', '1801.02073-2-3-1': 'wang:07a created the QASent dataset consisting of 277 questions, which has been widely used for benchmarking the answer selection task.', '1801.02073-2-3-2': 'feng:15a presented InsuranceQA comprising 16K+ questions on insurance contexts.', '1801.02073-2-3-3': 'yang:15a introduced WikiQA for answer selection and triggering.', '1801.02073-2-3-4': 'jurczyk:16 created SelQA for large real-scale answer triggering.', '1801.02073-2-3-5': 'rajpurkar2016squad presented SQuAD for answer extraction and selection as well as for reading comprehension.', '1801.02073-2-3-6': 'Finally, morales-EtAl:2016:EMNLP2016 provided InfoboxQA for answer selection.', '1801.02073-2-4-0': 'These corpora make it possible to evaluate the robustness of statistical question answering learning.', '1801.02073-2-4-1': 'Although all of these corpora target on selection-based QA, they are designed for different purposes such that it is important to understand the nature of these corpora so a better use of them can be made.', '1801.02073-2-4-2': 'In this paper, we make both intrinsic and extrinsic analyses of four latest corpora based on Wikipedia, WikiQA, SelQA, SQuAD, and InfoboxQA.', '1801.02073-2-4-3': 'We first give a thorough intrinsic analysis regarding contextual similarities, question types, and answer categories (Section [REF]).', '1801.02073-2-4-4': 'We then map questions in all corpora to the current version of English Wikipedia and benchmark another selection-based QA task, answer retrieval (Section [REF]).', '1801.02073-2-4-5': 'Finally, we present an extrinsic analysis through a set of experiments cross-testing these corpora using a convolutional neural network architecture (Section [REF]).', '1801.02073-2-5-0': '# Intrinsic Analysis', '1801.02073-2-6-0': 'Four publicly available corpora are selected for our analysis.', '1801.02073-2-6-1': 'These corpora are based on Wikipedia, so more comparable than the others, and have already been used for the evaluation of several QA systems.', '1801.02073-2-7-0': 'WikiQA [CITATION] comprises questions selected from the Bing search queries, where user click data give the questions and their corresponding Wikipedia articles.', '1801.02073-2-7-1': 'The abstracts of these articles are then extracted to create answer candidates.', '1801.02073-2-7-2': 'The assumption is made that if many queries lead to the same article, it must contain the answer context; however, this assumption fails for some occasions, which makes this dataset more challenging.', '1801.02073-2-7-3': 'Since the existence of answer contexts is not guaranteed in this task, it is called answer triggering instead of answer selection.', '1801.02073-2-8-0': 'SelQA [CITATION] is a product of five annotation tasks through crowdsourcing.', '1801.02073-2-8-1': 'It consists of about 8K questions where a half of the questions are paraphrased from the other half, aiming to reduce contextual similarities between questions and answers.', '1801.02073-2-8-2': 'Each question is associated with a section in Wikipedia where the answer context is guaranteed, and also with five sections selected from the entire Wikipedia where the selection is made by the Lucene search engine.', '1801.02073-2-8-3': 'This second dataset does not assume the existence of the answer context, so can be used for the evaluation of answer triggering.', '1801.02073-2-9-0': 'SQuAD [CITATION] presents 107K+ crowdsourced questions on 536 Wikipedia articles, where the answer contexts are guaranteed to exist within the provided paragraph.', '1801.02073-2-9-1': 'It contains annotation of answer phrases as well as the pointers to the sentences including the answer phrases; thus, it can be used for both answer extraction and selection.', '1801.02073-2-9-2': 'This corpus also provides human accuracy on those questions, setting up a reasonable upper bound for machines.', '1801.02073-2-9-3': 'To avoid overfitting, the evaluation set is not publicly available although system outputs can be evaluated by their provided script.', '1801.02073-2-10-0': 'InfoboxQA [CITATION] gives 15K+ questions based on the infoboxes from 150 articles in Wikipedia.', '1801.02073-2-10-1': 'Each question is crowdsourced and associated with an infobox, where each line of the infobox is considered an answer candidate.', '1801.02073-2-10-2': 'This corpus emphasizes the gravity of infoboxes, which summary arguably the most commonly asked information about those articles.', '1801.02073-2-10-3': 'Although the nature of this corpus is different from the others, it can also be used to evaluate answer selection.', '1801.02073-2-11-0': '## Analysis', '1801.02073-2-12-0': 'All corpora provide datasets/splits for answer selection, whereas only (WikiQA, SQuAD) and (WikiQA, SelQA) provide datasets for answer extraction and answer triggering, respectively.', '1801.02073-2-12-1': 'SQuAD is much larger in size although questions in this corpus are often paraphrased multiple times.', '1801.02073-2-12-2': ""On the contrary, SQuAD's average candidates per question ([MATH]) is the smallest because SQuAD extracts answer candidates from paragraphs whereas the others extract them from sections or infoboxes that consist of bigger contexts."", '1801.02073-2-12-3': 'Although InfoboxQA is larger than WikiQA or SelQA, the number of token types ([MATH]) in InfoboxQA is smaller than those two, due to the repetitive nature of infoboxes.', '1801.02073-2-13-0': 'All corpora show similar average answer candidate lengths ([MATH]), except for InfoboxQA where each line in the infobox is considered a candidate.', '1801.02073-2-13-1': 'SelQA and SQuAD show similar average question lengths ([MATH]) because of the similarity between their annotation schemes.', '1801.02073-2-13-2': ""It is not surprising that WikiQA's average question length is the smallest, considering their questions are taken from search queries."", '1801.02073-2-13-3': ""InfoboxQA's average question length is relatively small, due to the restricted information that can be asked from the infoboxes."", '1801.02073-2-13-4': 'InfoboxQA and WikiQA show the least question-answer word overlaps over questions and answers ([MATH] and [MATH] in Table [REF]), respectively.', '1801.02073-2-13-5': 'In terms of the F1-score for overlapping words ([MATH]), SQuAD gives the least portion of overlaps between question-answer pairs although WikiQA comes very close.', '1801.02073-2-14-0': 'Fig. [REF] shows the distributions of seven question types grouped deterministically from the lexicons.', '1801.02073-2-14-1': 'Although these corpora have been independently developed, a general trend is found, where the what question type dominates, followed by how and who, followed by when and where, and so on.', '1801.02073-2-15-0': 'Fig. [REF] shows the distributions of answer categories automatically classified by our Convolutional Neural Network model trained on the data distributed by [CITATION].', '1801.02073-2-15-1': 'Interestingly, each corpus focuses on different categories, Numeric for WikiQA and SelQA, Entity for SQuAD, and Person for InfoboxQA, which gives enough diversities for statistical learning to build robust models.', '1801.02073-2-16-0': '# Answer Retrieval', '1801.02073-2-17-0': 'This section describes another selection-based QA task, called answer retrieval, that finds the answer context from a larger dataset, the entire Wikipedia.', '1801.02073-2-17-1': 'SQuAD provides no mapping of the answer contexts to Wikipedia, whereas WikiQA and SelQA provide mappings; however, their data do not come from the same version of Wikipedia.', '1801.02073-2-17-2': 'We propose an automatic way of mapping the answer contexts from all corpora to the same version of Wikipeda so they can be coherently used for answer retrieval.', '1801.02073-2-18-0': 'Each paragraph in Wikipedia is first indexed by Lucene using 1,2,3-grams, where the paragraphs are separated by WikiExtractor and segmented by NLP4J (28.7M+ paragraphs are indexed).', '1801.02073-2-18-1': 'Each answer sentence from the corpora in Table [REF] is then queried to Lucene, and the top-5 ranked paragraphs are retrieved.', '1801.02073-2-18-2': 'The cosine similarity between each sentence in these paragraphs and the answer sentence is measured for [MATH]-grams, say [MATH].', '1801.02073-2-18-3': 'A weight is assigned to each [MATH]-gram score, say [MATH], and the weighted sum is measured: [MATH].', '1801.02073-2-18-4': 'The fixed weights of [MATH] are used for our experiments, which can be improved.', '1801.02073-2-19-0': 'If there exists a sentence whose [MATH], the paragraph consisting of that sentence is considered the silver-standard answer passage.', '1801.02073-2-19-1': 'Table [REF] shows how robust these silver-standard passages are based on human judgement ([MATH]) and how many passages are collected ([MATH]) for [MATH], where the human judgement is performed on 50 random samples for each case.', '1801.02073-2-19-2': 'For answer retrieval, a dataset is created by [MATH], which gives [MATH] accuracy and [MATH] coverage, respectively.', '1801.02073-2-19-3': 'Finally, each question is queried to Lucene and the top-[MATH] paragraphs are retrieved from the entire Wikipedia.', '1801.02073-2-19-4': 'If the answer sentence exists within those retrieved paragraphs according to the silver-standard, it is considered correct.', '1801.02073-2-20-0': '# Extrinsic Analysis', '1801.02073-2-21-0': '## Answer Selection', '1801.02073-2-22-0': 'Answer selection is evaluated by two metrics, mean average precision (MAP) and mean reciprocal rank (MRR).', '1801.02073-2-22-1': 'The bigram CNN introduced by [CITATION] is used to generate all the results in Table [REF], where models are trained on either single or combined datasets.', '1801.02073-2-22-2': 'Clearly, the questions in WikiQA are the most challenging, and adding more training data from the other corpora hurts accuracy due to the uniqueness of query-based questions in this corpus.', '1801.02073-2-22-3': 'The best model is achieved by training on W+S+Q for SelQA; adding InfoboxQA hurts accuracy for SelQA although it gives a marginal gain for SQuAD.', '1801.02073-2-22-4': 'Just like WikiQA, InfoboxQA performs the best when it is trained on only itself.', '1801.02073-2-22-5': 'From our analysis, we suggest that to use models trained on WikiQA and InfoboxQA for short query-like questions, whereas to use ones trained on SelQA and SQuAD for long natural questions.', '1801.02073-2-23-0': '## Answer Retrieval', '1801.02073-2-24-0': 'Finding a paragraph that includes the answer context out of the entire Wikipedia is an extremely difficult task (128.7M).', '1801.02073-2-24-1': 'The last row of Table [REF] shows results from answer retrieval.', '1801.02073-2-24-2': 'Given [MATH], SelQA and SQuAD show about 34 and 35 accuracy, which are reasonable.', '1801.02073-2-24-3': 'However, WikiQA shows a significantly lower accuracy of 12.47; this is because the questions in WikiQA is about twice shorter than the questions in the other corpora such that not enough lexicons can be extracted from these questions for the Lucene search.', '1801.02073-2-25-0': '## Answer Triggering', '1801.02073-2-26-0': 'The results of [MATH] from the answer retrieval task in Section [REF] are used to create the datasets for answer triggering, where about 65% of the questions are not expected to find their answer contexts from the provided paragraphs for SelQA and SQuAD and 87.5% are not expected for WikiQA.', '1801.02073-2-26-1': 'Answer triggering is evaluated by the F1 scores as presented in Table [REF], where three corpora are cross validated.', '1801.02073-2-26-2': 'The results on WikiQA are pretty low as expected from the poor accuracy on the answer retrieval task.', '1801.02073-2-26-3': 'Training on SelQA gives the best models for both WikiQA and SelQA.', '1801.02073-2-26-4': 'Training on SQuAD gives the best model for SQuAD although the model trained on SelQA is comparable.', '1801.02073-2-26-5': 'Since the answer triggering datasets are about 5 times larger than the answer selection datasets, it is computationally too expensive to combine all data for training.', '1801.02073-2-26-6': 'We plan to find a strong machine to perform this experiment in near future.', '1801.02073-2-27-0': '# Related work', '1801.02073-2-28-0': 'Lately, several deep learning approaches have been proposed for question answering.', '1801.02073-2-28-1': 'yu:14a presented a CNN model that recognizes the semantic similarity between two sentences.', '1801.02073-2-28-2': 'wang-nyberg:2015:ACL-IJCNLP presented a stacked bidirectional LSTM approach to read words in sequence, then outputs their similarity scores.', '1801.02073-2-28-3': 'feng:15a applied a general deep learning framework to non-factoid question answering.', '1801.02073-2-28-4': 'santos:16a introduced an attentive pooling mechanism that led to further improvements in selection-based QA.', '1801.02073-2-29-0': '# Conclusion', '1801.02073-2-30-0': 'We present a comprehensive comparison study of the existing corpora for selection-based question answering.', '1801.02073-2-30-1': 'Our intrinsic analysis provides a better understanding of the uniqueness or similarity between these corpora.', '1801.02073-2-30-2': 'Our extrinsic analysis shows the strength or weakness of combining these corpora together for statistical learning.', '1801.02073-2-30-3': 'Additionally, we create a silver-standard dataset for answer retrieval and triggering, which will be publicly available.', '1801.02073-2-30-4': 'In the future, we will explore different ways of improving the quality of our silver-standard datasets by fine-tuning the hyper-parameters.'}","[['1801.02073-1-12-0', '1801.02073-2-12-0'], ['1801.02073-1-12-1', '1801.02073-2-12-1'], ['1801.02073-1-12-2', '1801.02073-2-12-2'], ['1801.02073-1-12-3', '1801.02073-2-12-3'], ['1801.02073-1-14-0', '1801.02073-2-14-0'], ['1801.02073-1-14-1', '1801.02073-2-14-1'], ['1801.02073-1-7-0', '1801.02073-2-7-0'], ['1801.02073-1-7-1', '1801.02073-2-7-1'], ['1801.02073-1-7-2', '1801.02073-2-7-2'], ['1801.02073-1-7-3', '1801.02073-2-7-3'], ['1801.02073-1-10-0', '1801.02073-2-10-0'], ['1801.02073-1-10-1', '1801.02073-2-10-1'], ['1801.02073-1-10-2', '1801.02073-2-10-2'], ['1801.02073-1-10-3', '1801.02073-2-10-3'], ['1801.02073-1-13-0', '1801.02073-2-13-0'], ['1801.02073-1-13-1', '1801.02073-2-13-1'], ['1801.02073-1-13-2', '1801.02073-2-13-2'], ['1801.02073-1-13-3', '1801.02073-2-13-3'], ['1801.02073-1-13-4', '1801.02073-2-13-4'], 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hep-ph-9808250,"{'hep-ph-9808250-1-0-0': '# INTRODUCTION', 'hep-ph-9808250-1-1-0': 'The first approach to the relativistic superfluid mechanics proposed by Israel [CITATION] and Dixon [CITATION] concerns with perfect fluids.', 'hep-ph-9808250-1-1-1': 'The further development of Khalatnikov and Lebedev [CITATION] includes the interaction between the superfluid and normal constituent.', 'hep-ph-9808250-1-1-2': 'The absence of coupling on microscopic level implies thermo-isolation of the constituents [CITATION]; if there is no coupling, a two-constituent system would be an ideal fluid.', 'hep-ph-9808250-1-1-3': 'This principle developed independently by Carter [CITATION] was gained in recent works [CITATION].', 'hep-ph-9808250-1-1-4': 'Although the main attention was directed to the general formalism, including the equations of motion, Carter and Langlois [CITATION] have recently derived the first and second sound speed in a superfluid with a phonon equation of state of the normal constituent.', 'hep-ph-9808250-1-1-5': 'This inspires us to discuss the shock wave propagation in a two-constituent relativistic superfluid.', 'hep-ph-9808250-1-1-6': 'While the relativistic shock waves first considered by Taub [CITATION] have been under detailed discussion [CITATION] and the shock waves in superfluid helium [CITATION] were also investigated, the present problem bears a qualitatively new feature: we work not in the frames of perfect fluid hydrodynamics, for it is impossible to split the conserving particle number current into a conserving ""superfluid"" and ""normal"" parts [CITATION] as it is in the Newtonian limit [CITATION].', 'hep-ph-9808250-1-2-0': 'We use the natural system of units ([MATH]) and the metric corresponding to the Minkowsky space with a metric tensor [MATH].', 'hep-ph-9808250-1-3-0': '# THE CONSERVATION LAWS OF TWO-CONSTITUENT SUPERFLUID DYNAMICS', 'hep-ph-9808250-1-4-0': 'The equations of relativistic superfluid mechanics are determined by the Lagrangian [MATH], whose infinitesimal variation is given by formula [CITATION] [EQUATION] where the particle number vector [MATH] conjugated to the momentum co-vector [MATH] obeys the conservation law [EQUATION] and the entropy vector [MATH] conjugated to the thermal momentum 1-form [MATH] is also conserved [EQUATION] till a shock wave appears.', 'hep-ph-9808250-1-4-1': 'The energy-momentum tensor corresponding to the Lagrangian [MATH] has the form [EQUATION] with the pressure function [EQUATION]', 'hep-ph-9808250-1-4-2': 'The conservation law of the energy-momentum tensor ([REF]) [EQUATION] leads to the equation of motion of the normal constituent [EQUATION] and the irrotationality condition [EQUATION]', 'hep-ph-9808250-1-4-3': 'Vectors [MATH] and [MATH](also [MATH] and [MATH]) are not colinear.', 'hep-ph-9808250-1-4-4': 'For the particle number vector [MATH] determines the Eckart rest frame, while [MATH] determines similarly the superfluid rest frame and that does not coincide with the former.', 'hep-ph-9808250-1-4-5': 'The Lagrangian [MATH] depends on three invariants, them being [EQUATION] for the normal rest frame entropy density; for the cross product given by [EQUATION] and for the effective mass variable (i.e. the chemical potential in the superfluid rest frame) [EQUATION]', 'hep-ph-9808250-1-4-6': 'The secondary variables [MATH] and [MATH] can be expressed through the primary variables [MATH] and [MATH] according to the formula [EQUATION] with the coefficients obtained immediately by differentiation of the Lagrangian [EQUATION]', 'hep-ph-9808250-1-4-7': 'On the other hand we can write [EQUATION] and calculate new coefficients through the pressure function [CITATION] [EQUATION]', 'hep-ph-9808250-1-4-8': 'However, a more convenient way is to obtain all parameters ([REF]) in terms of ([REF]), ([REF]) and ([REF]) and express them in terms of the relative translation speed between the superfluid and normal reference frames [EQUATION] and the effective temperature [EQUATION] instead of invariants [MATH] and [MATH].', 'hep-ph-9808250-1-4-9': 'At low temperature the phonon-like excitations with energetic spectrum [MATH], where [EQUATION] and the latter is the sound speed contribute to thermodynamical functions of the normal constituent.', 'hep-ph-9808250-1-4-10': 'The ""phonon"" Lagrangian has the form [CITATION] [EQUATION] where the pressure of excitations is [EQUATION] and [EQUATION] is the entropy [MATH] in the superfluid reference frame, while [MATH] is a definite constant.', 'hep-ph-9808250-1-4-11': 'The coefficients ([REF]) then are [EQUATION]', 'hep-ph-9808250-1-4-12': 'For an ultra-relativistic spectrum of excitations ([MATH]) the term [MATH] disappears.', 'hep-ph-9808250-1-5-0': '# THE DISCONTINUITIES', 'hep-ph-9808250-1-6-0': 'The discontinuities in superfluid helium have been first discovered by Khalatnikov [CITATION].', 'hep-ph-9808250-1-6-1': 'Discussing the discontinuities in a relativistic superfluid we shall follow the standard formalism of the relativistic shock waves [CITATION].', 'hep-ph-9808250-1-6-2': 'However, since we deal with a strongly self-interacting medium and two-constituent at that, a standard perfect fluid theory is impossible.', 'hep-ph-9808250-1-6-3': 'So we have to think of a new extended formalism, combining the theory of relativistic shock waves [CITATION] and the relativistic superfluid mechanics [CITATION].', 'hep-ph-9808250-1-6-4': 'In the Newtonian limit this method must be reduced to the non-relativistic theory of discontinuities in helium [CITATION], and in the acoustic limit this method must give the first and the second sound in a relativistic superfluid [CITATION].', 'hep-ph-9808250-1-6-5': 'The amplitude of discontinuities is assumed to be not very large, since superfluidity is expected to take place on both sides of the front with no phase transition.', 'hep-ph-9808250-1-7-0': 'Let the hypersurface [MATH] be the front of the discontinuity and vector [MATH] be a unit (space-like) normal to it.', 'hep-ph-9808250-1-7-1': 'The conservation laws ([REF]), ([REF]) entail the conditions [EQUATION] i.e. [EQUATION] where indexes + and - relate to the quantities ahead of and behind the discontinuity, respectively, and the square brackets imply the change across the front of a discontinuity.', 'hep-ph-9808250-1-7-2': 'Eq. ([REF]) conveys the continuity of the orthogonal part (marked always by index [MATH]) of the particle number current.', 'hep-ph-9808250-1-8-0': 'We cannot make use of the entropy conservation law ([REF]) since it does not take place in the shock waves.', 'hep-ph-9808250-1-8-1': 'For that reason the results of Ref. [CITATION] pertain for a most general two-constituent system (but not a superfluid itself), while the discussion of small perturbations suffers no restrictions concerning eq. ([REF]).', 'hep-ph-9808250-1-8-2': 'Indeed, the results for shock waves in a two-constituent system and in a superfluid will coincide when the magnitude of the relevant discontinuities tends to zero.', 'hep-ph-9808250-1-9-0': 'For a two-constituent superfluid the additional equation is the irrotationality condition ([REF]) which yields [EQUATION]', 'hep-ph-9808250-1-9-1': 'Multiplying it by [MATH] and, then, by a unit vector [MATH] orthogonal to [MATH], we get an important relation [EQUATION] (where index [MATH] denotes the tangential part of any quantity, particularly [MATH]).', 'hep-ph-9808250-1-9-2': 'This is the very condition for conservation of irrotational motion passing a plane shock wave [CITATION].', 'hep-ph-9808250-1-9-3': 'It is clear that for a multi-constituent system the irrotational motion conserves at both sides of the shock wave if condition ( [REF]) takes place for each constituent.', 'hep-ph-9808250-1-10-0': 'Substituting the expression ([REF]) and ([REF]) in ([REF]), we obtain [EQUATION]', 'hep-ph-9808250-1-10-1': 'Multiplying this equation by [MATH], we get [EQUATION]', 'hep-ph-9808250-1-10-2': 'Then, multiplying eq. ([REF]) by the unit vector [MATH], in light of ([REF]), we get [EQUATION]', 'hep-ph-9808250-1-10-3': 'This important relation determines the types of discontinuities.', 'hep-ph-9808250-1-10-4': 'Then, the irrotationality condition ([REF]) than coincides with the condition of strong discontinuity in the superfluid constituent which occurs when [MATH].', 'hep-ph-9808250-1-10-5': 'As for the discontinuity in the normal component, in light of ([REF]), it is determined merely by the single relation [EQUATION]', 'hep-ph-9808250-1-10-6': 'Thus the constraint ([REF]) corresponds to ordinary shock waves, while ([REF]) beseems to a ""temperature"" discontinuity of the second sound type.', 'hep-ph-9808250-1-11-0': 'As a particular instance of another type of discontinuity we consider a vortex sheet in superfluid [CITATION].', 'hep-ph-9808250-1-11-1': 'Since the vortex sheet separates the whole space into domains where the superfluidity takes place, the irrotationality condition ([REF]) does not hold in the global sense and we cannot establish the constraint ([REF]) at both sides of the sheet.', 'hep-ph-9808250-1-11-2': 'Therefore, the tangential discontinuities are possible.', 'hep-ph-9808250-1-11-3': 'The conservation law ([REF]) then yields [MATH] that determining weak or slip-stream discontinuity: the particle number flow across the front of discontinuity equals zero, indeed, no matter crosses the hypersurface of the discontinuity, i.e. this hypersurface is made up of streamlines of the fluid.', 'hep-ph-9808250-1-12-0': '# A PLANE SHOCK IN FLAT SPACE', 'hep-ph-9808250-1-13-0': 'Let the discontinuity propagates along the axis [MATH].', 'hep-ph-9808250-1-13-1': 'We choose the unit normal [MATH] - and the medium at rest before the front.', 'hep-ph-9808250-1-13-2': 'As a rule one used to practice with the rest frame co-moving the front, so that the fluid flows in the front with the velocity which is equal to that of a shock wave.', 'hep-ph-9808250-1-13-3': 'Hereby, the relevant vectors and co-vectors may be presented as [EQUATION]', 'hep-ph-9808250-1-13-4': 'Since the medium ahead of front is at rest, the relative velocity [MATH] equals zero, [MATH], while behind the shock [EQUATION]', 'hep-ph-9808250-1-13-5': 'Also [EQUATION]', 'hep-ph-9808250-1-13-6': 'Hence, the velocity of the shock is determined as [EQUATION]', 'hep-ph-9808250-1-13-7': 'Substituting our definitions ([REF]), ([REF]) and ([REF]) in eqs. ([REF]), ([REF]), ([REF]) and ([REF]) we, firstly obtain [EQUATION]', 'hep-ph-9808250-1-13-8': 'The rest equations, in view of ([REF]), will be [EQUATION]', 'hep-ph-9808250-1-13-9': 'The parameter [MATH] incorporates only in eq. ([REF]) and it can be calculated as soon as the rest unknowns are found.', 'hep-ph-9808250-1-13-10': 'Thus, in eqs. ([REF]), ([REF]) and ([REF]) the unknowns are: the four parameters [MATH], [MATH], [MATH], [MATH] and three invariants ([REF]), ([REF]), ([REF]) behind the shock on which the pressure [MATH] depends.', 'hep-ph-9808250-1-13-11': 'The pressure behind the shock can be expressed through [MATH], [MATH], and through the relative velocity for which we use the notation [MATH].', 'hep-ph-9808250-1-13-12': 'The formula [EQUATION] relates the later quantity with [MATH] and [MATH].', 'hep-ph-9808250-1-13-13': 'Our goal is to find the velocity of the shock wave [MATH] for a single parameter given behind the shock.', 'hep-ph-9808250-1-13-14': 'Without the loss of generality [MATH] can be chosen for this parameter.', 'hep-ph-9808250-1-13-15': 'Thus, there are six unknowns in four equations ([REF]), ( [REF]), ([REF]), ([REF]).', 'hep-ph-9808250-1-13-16': 'The rest two relations follow from ([REF]) or ([REF]) [EQUATION] with the coefficients ([REF]) calculated for the state behind the front.', 'hep-ph-9808250-1-14-0': 'The knowledge of the equation of state in explicit form is necessary for calculation of the right-hand side of eq. ([REF]) and the coefficients in eqs. ([REF]) and ([REF]).', 'hep-ph-9808250-1-15-0': '# A LOW-TEMPERATURE CASE', 'hep-ph-9808250-1-16-0': 'The low-temperature equation of state was derived by Carter and Langlois [CITATION].', 'hep-ph-9808250-1-16-1': 'In view of ([REF]), ([REF]), ([REF]) the expressions ([REF]), ([REF]), ([REF]), ([REF]) take the form [EQUATION] where [EQUATION]', 'hep-ph-9808250-1-16-2': 'At low temperature we have the estimations [CITATION] [EQUATION] implying that [MATH] and, hence, equations ([REF]) and ([REF]) approximately (up to the terms [MATH]) coincide with their zero-temperature version.', 'hep-ph-9808250-1-17-0': '# THE SOUND, STRONG AND SMALL-AMPLITUDE SHOCK WAVES', 'hep-ph-9808250-1-18-0': 'If all parameters behind the front tend to their values ahead, the shock becomes a sound wave.', 'hep-ph-9808250-1-18-1': 'Since the entropy in the sound wave is conserved, we can apply formalism [CITATION] achieved for a two-constituent relativistic medium with the conserved particle currents of both constituents.', 'hep-ph-9808250-1-18-2': 'In the linear approximation both methods lead to the same result, namely from the system ([REF]), ([REF]), ([REF]), ([REF]), ( [REF]), ([REF]) we obtain an equation for two branches of sound at arbitrary temperature which is analogous to that derived by Carter [CITATION] and, under assumption [MATH], splits into [EQUATION] and reduces, in the low temperature limit, to the first and the second sound speed, respectively [CITATION]: [MATH], [MATH].', 'hep-ph-9808250-1-18-3': 'Here for any variable [MATH] we used the notation [EQUATION]', 'hep-ph-9808250-1-18-4': 'In order to find the velocity of a small-amplitude shock we rewrite eqs. ([REF]), ([REF]), ([REF]), ([REF]), ([REF]), ( [REF]) in the second-order approximation.', 'hep-ph-9808250-1-18-5': 'After tedious calculations we find the velocity increment [EQUATION] of the shock corresponding to the second sound when the temperature increment [MATH] tends to zero.', 'hep-ph-9808250-1-18-6': 'In turn, the latter gives rise to a finite relative speed [MATH] behind the shock, since [EQUATION]', 'hep-ph-9808250-1-18-7': 'In the non-relativistic limit the equations ([REF]) and ([REF]) yield well known expressions [CITATION].', 'hep-ph-9808250-1-18-8': 'The shock occurs ahead of the second sound, as is in superfluid helium at low temperature.', 'hep-ph-9808250-1-19-0': 'So, in the acoustic limit the solution splits into two independent branches [MATH] and [MATH] corresponding to the first and the second sound.', 'hep-ph-9808250-1-19-1': 'The first branch describes wave propagation through the medium which behaves as a perfect fluid composed of two constituents whose pressure and enthalpy are [MATH] and [MATH] respectively.', 'hep-ph-9808250-1-19-2': 'While the constituents in the waves of the second branch move independently, a counterflow appears: [MATH].', 'hep-ph-9808250-1-19-3': 'In general, a ""mixed"" solution occurs, and the temperature increases together with the chemical potential.', 'hep-ph-9808250-1-20-0': 'The estimations ([REF]) imply that the first sound and the relevant shock wave coincide roughly with usual discontinuity in the cold constituent.A great pressure jump [MATH] is produced inevitably by the change in pressure of the superfluid constituent [MATH], since the contribution of the normal constituent [MATH] is small.', 'hep-ph-9808250-1-20-1': 'Hence, in view of eqs. ([REF] ) and ([REF]) we conclude that strong shock waves at low temperature propagate with the speed [MATH] which approximately equals to the speed of a usual shock wave in cold constituent [MATH], but always [MATH].', 'hep-ph-9808250-1-20-2': 'A more precise result is [EQUATION]', 'hep-ph-9808250-1-20-3': 'The second-sound discontinuities should be regarded as ""moderate"" for intermediate values of [MATH].', 'hep-ph-9808250-1-20-4': 'This takes place if relative changes in the superfluid and normal variables are of the same order and they can be of the same order if they do not access [MATH] greatly.', 'hep-ph-9808250-1-20-5': 'For a superfluid matter of neutron stars and phonon equation of state it is easy to estimate [MATH].', 'hep-ph-9808250-1-20-6': 'For this particular example we performed calculations with an ultra-relativistic superfluid matter.', 'hep-ph-9808250-1-20-7': 'The sound speed in this medium equals exactly to [MATH], and approximately it is the first sound, while the speed of the second sound [MATH].', 'hep-ph-9808250-1-20-8': 'For a not very small pressure change the velocity of the shock wave will be merely [MATH].', 'hep-ph-9808250-1-20-9': 'The dependence of the relative translation speed [MATH] on [MATH] is given in table 1.', 'hep-ph-9808250-1-21-0': 'Although the second sound velocity attains to the saturation value [MATH], the relative translation speed [MATH] grows with the growth of the shock wave intensity.', 'hep-ph-9808250-1-22-0': '# THE FOURTH SOUND', 'hep-ph-9808250-1-23-0': 'The fourth sound takes place when the normal constituent is restrained by some external agent, while the sound propagates through the superfluid constituent.', 'hep-ph-9808250-1-23-1': 'We cannot use the equation of motion ([REF]), but the conservation laws ([REF]), ([REF]) and the irrotationality condition ([REF]) will determine the fourth sound speed.', 'hep-ph-9808250-1-23-2': 'If the sound wave propagates in the direction determined by vector [MATH] the change of gradient of arbitrary quantity [MATH] is proportional to its infinitesimal change [MATH] [CITATION]: [MATH].', 'hep-ph-9808250-1-23-3': 'Thereby, we write - [EQUATION] instead of ([REF]), ([REF]), ([REF]) and ([REF]).', 'hep-ph-9808250-1-23-4': 'In the reference frame co-moving with the normal constituent we put [EQUATION] where [MATH] is the chemical potential in the ""normal"" reference frame.', 'hep-ph-9808250-1-23-5': 'Since the discontinuities propagate through the superfluid constituent, there must be [EQUATION]', 'hep-ph-9808250-1-23-6': 'Equations ([REF]), ([REF]), ([REF]) analogous to the relevant non-relativistic set [CITATION] determine the speed of the fourth sound.', 'hep-ph-9808250-1-23-7': 'Requiring the vanishing determinant of the system ([REF]), ([REF]) we get a quadratic equation for [MATH].', 'hep-ph-9808250-1-23-8': 'While the speed of the first and the second sound is determined by a 4-order system [CITATION], the fourth sound speed follows from two equations.', 'hep-ph-9808250-1-23-9': 'At zero temperature the speed of the first and the fourth sound are obtained by the same equations ([REF]) and ([REF]) and coincide exactly with the sound speed [MATH] in the superfluid constituent.', 'hep-ph-9808250-1-23-10': 'The difference appears at finite temperature on account of the relationship [CITATION] [EQUATION] between the infinitesimal discontinuities in ([REF]), ([REF]) and the temperature dependence of matrices in ([REF]).', 'hep-ph-9808250-1-23-11': 'For the phonon equation of state we get the explicit formula [EQUATION] which generalizes the relevant non-relativistic relation [CITATION] , where [MATH] and [MATH] is the normal and the superfluid energy-density respectively [CITATION].', 'hep-ph-9808250-1-24-0': '# Appendix.', 'hep-ph-9808250-1-24-1': 'The sound in a quark-gluon plasma', 'hep-ph-9808250-1-25-0': 'In principle, the equations for the sound speed are valid without any restriction imposed on the equation of state.', 'hep-ph-9808250-1-25-1': 'If sound propagation is conceivable in the quark-gluon plasma, let us consider briefly this problem.', 'hep-ph-9808250-1-26-0': 'The pressure of partons and gluons is, respectively [CITATION], [EQUATION] where [MATH] is an electric component of the constant background gauge field (multiplied by the coupling constant).', 'hep-ph-9808250-1-26-1': 'Since the massless gluons have the energetic spectrum [MATH], we conclude that a dependence on the cross term [MATH] or [MATH] is absent as it holds for a system of matter and radiation (compare ([REF]) and eq. for spinless bosons ([REF]) at [MATH] ).', 'hep-ph-9808250-1-26-2': 'For calculating the sound speed it is convenient to use instead of ([REF]), ([REF]) its equivalent version written in terms of [MATH][CITATION].', 'hep-ph-9808250-1-26-3': 'It will be [EQUATION] where [MATH] and [MATH] are obtained by formula ([REF]) with [MATH].', 'hep-ph-9808250-1-26-4': 'Hence, the first and the second sound speed are, respectively [EQUATION]', 'hep-ph-9808250-1-26-5': 'It should be noted that when temperature tends to zero, the second sound speed becomes [EQUATION]', 'hep-ph-9808250-1-26-6': 'As a matter of fact, the results of the present section are obtained in the frames of Israel relativistic two-fluid theory [CITATION].', 'hep-ph-9808250-1-26-7': 'However, if we omit the cross terms [MATH] and [MATH], the second sound speed calculated by formula ( [REF]) with the phonon Lagrangian ([REF]) of Carter and Langlois [CITATION] will be [MATH] instead of obvious [MATH].', 'hep-ph-9808250-1-27-0': '# CONCLUSION', 'hep-ph-9808250-1-28-0': 'Summarizing the results obtained in the present study, we emphasize the formulae ([REF]), ([REF]), ([REF]), ([REF]) which determine the propagation of discontinuities through a two-constituent relativistic superfluid in the general case.', 'hep-ph-9808250-1-28-1': 'For a plane shock wave seven equations ([REF]), ([REF]), ([REF]), ([REF]), ([REF]), ([REF]) and ([REF]) with seven unknowns must be solved.', 'hep-ph-9808250-1-28-2': 'In the acoustic limit these equations reduce to formulae ([REF]), ([REF]) for the first and the second sound, demonstrated in calculations of the sound speed ([REF]), ([REF]), ([REF]) in the quark-gluon plasma.', 'hep-ph-9808250-1-28-3': 'At low temperature the system ([REF]), ([REF]), ([REF]), ([REF]), ([REF]), ([REF]) and ([REF]) reduces to ([REF]), ([REF]), ([REF]), ([REF]), ([REF]), ([REF]) and ([REF]).', 'hep-ph-9808250-1-28-4': 'The velocity of strong shock waves is given by ([REF]) and ( [REF]), while ([REF]) and ([REF]) describ the change of parameters in a weak shock wave.', 'hep-ph-9808250-1-28-5': 'As for perspectives and applications, the shock waves and spin-isospin sound in the nuclear matter are worth to be discussed in future.'}","{'hep-ph-9808250-2-0-0': 'We consider the discontinuities in a two-constituent relativistic superfluid.', 'hep-ph-9808250-2-0-1': 'In the acoustic limit they degenerate into the first and second sound which are independent up to the second-order linear approximation.', 'hep-ph-9808250-2-0-2': 'Inclusion of the quadratic deviations relates to the small-amplitude shock.', 'hep-ph-9808250-2-0-3': 'Particularly we consider a plane shock at low temperature when the phonon excitations contribute to the normal constituent.', 'hep-ph-9808250-2-0-4': 'So we found the generalization of the temperature increment and acoustic wave velocity in relativistic superfluid.', 'hep-ph-9808250-2-0-5': 'The fourth sound speed is also calculated.', 'hep-ph-9808250-2-1-0': '# INTRODUCTION', 'hep-ph-9808250-2-2-0': 'The first approach to the relativistic superfluid mechanics proposed by Israel [CITATION] and Dixon [CITATION] concerns with perfect fluids.', 'hep-ph-9808250-2-2-1': 'The method is useful for particular calculations, and as a general model can be applied to relativistic superfluidity.', 'hep-ph-9808250-2-2-2': 'For, strictly speaking, the coupled constituents are not perfect fluids: any coupling results to deviation from ideality.', 'hep-ph-9808250-2-2-3': 'Or: the absence of coupling on microscopic level implies thermo-isolation of the constituents [CITATION].', 'hep-ph-9808250-2-2-4': 'Nevertheless, the attempt of taking into account the deviation from perfect fluid is not senseless.', 'hep-ph-9808250-2-2-5': 'The further development of Khalatnikov and Lebedev [CITATION] includes the interaction between the superfluid and normal constituent.', 'hep-ph-9808250-2-2-6': 'This principle developed independently by Carter [CITATION] was gained in recent works [CITATION].', 'hep-ph-9808250-2-2-7': 'Although the main attention was directed to the general formalism, including the equations of motion, rather than applied problems, Carter and Langlois [CITATION] have recently derived the first and second sound speed in a superfluid with a phonon equation of state of the normal constituent.', 'hep-ph-9808250-2-2-8': 'This inspires us to discuss the shock wave propagation in a two-constituent relativistic superfluid.', 'hep-ph-9808250-2-2-9': 'While the relativistic shock waves first considered by Taub [CITATION] have been under detailed discussion [CITATION] and the shock waves in superfluid helium [CITATION] were also investigated, the present task bears a qualitatively new feature: we work not in the frames of perfect fluid hydrodynamics, for it is impossible to split the conserving particle number current into a conserving ""superfluid"" and ""normal"" parts [CITATION] as it is in the Newtonian limit [CITATION].', 'hep-ph-9808250-2-3-0': 'We use the natural system of units ([MATH]) and the metric corresponding to the Minkowsky space with a metric tensor [MATH].', 'hep-ph-9808250-2-4-0': '# THE CONSERVATION LAWS', 'hep-ph-9808250-2-5-0': 'The equations of relativistic superfluid mechanics are determined by the Lagrangian [MATH], whose infinitesimal variation is given by formula [CITATION] [EQUATION] where the particle number vector [MATH] conjugated to the momentum co-vector [MATH] obeys the conservation law [EQUATION] and the entropy vector [MATH] conjugated to the thermal momentum 1-form [MATH] is also conserved [EQUATION] till a shock wave appears.', 'hep-ph-9808250-2-5-1': 'The energy-momentum tensor corresponding to the Lagrangian [MATH] has the form [EQUATION] with the pressure function [EQUATION]', 'hep-ph-9808250-2-5-2': 'The conservation law of the energy-momentum tensor ([REF]) [EQUATION] leads to the equation of motion of the normal constituent [EQUATION] and the irrotationality condition [EQUATION]', 'hep-ph-9808250-2-5-3': 'Vectors [MATH] and [MATH](also [MATH] and [MATH]) are not colinear.', 'hep-ph-9808250-2-5-4': 'For the particle number vector [MATH] determines the Eckart rest frame, while [MATH] determines similarly the superfluid rest frame and that does not coincide with the former.', 'hep-ph-9808250-2-5-5': 'The Lagrangian [MATH] depends on three invariants, them being [EQUATION] for the normal rest frame entropy density; for the cross product given by [EQUATION] and for the effective mass variable (i.e. the chemical potential in the superfluid rest frame) [EQUATION]', 'hep-ph-9808250-2-5-6': 'The secondary variables [MATH] and [MATH] can be expressed through the primary variables [MATH] and [MATH] according to the formula [EQUATION] with the coefficients obtained immediately by differentiation of the Lagrangian [EQUATION]', 'hep-ph-9808250-2-5-7': 'On the other hand we can write [EQUATION] and calculate new coefficients through the pressure function [CITATION] [EQUATION]', 'hep-ph-9808250-2-5-8': 'However, a more convenient way is to obtain all parameters ([REF]) in terms of ([REF]), ([REF]) and ([REF]) and express them in terms of the relative translation speed between the superfluid and normal reference frames [EQUATION] and the effective temperature [EQUATION] instead of invariants [MATH] and [MATH].', 'hep-ph-9808250-2-5-9': 'At low temperature the phonon-like excitations with energetic spectrum [MATH], where [EQUATION] and the latter is the sound speed contribute to thermodynamical functions of the normal constituent.', 'hep-ph-9808250-2-5-10': 'The ""phonon"" Lagrangian has the form [CITATION] [EQUATION] where the pressure of excitations is [EQUATION] and [EQUATION] is the entropy [MATH] in the superfluid reference frame, while [MATH] is a definite constant.', 'hep-ph-9808250-2-5-11': 'The coefficients ([REF]) then are [EQUATION]', 'hep-ph-9808250-2-5-12': 'For an ultra-relativistic spectrum of excitations ([MATH]) the term [MATH] disappears.', 'hep-ph-9808250-2-6-0': '# THE DISCONTINUITIES', 'hep-ph-9808250-2-7-0': 'The discontinuities in superfluid helium have been first discovered by Khalatnikov [CITATION].', 'hep-ph-9808250-2-7-1': 'Discussing the discontinuities in a relativistic superfluid we shall follow the standard formalism of the relativistic shock waves [CITATION].', 'hep-ph-9808250-2-7-2': 'However, since we deal with a strongly self-interacting medium and two-constituent at that, a standard perfect fluid theory is impossible.', 'hep-ph-9808250-2-7-3': 'So we have to think of a new extended formalism, combining the theory of relativistic shock waves [CITATION] and the relativistic superfluid mechanics [CITATION].', 'hep-ph-9808250-2-7-4': 'In the Newtonian limit this method must be reduced to the non-relativistic theory of discontinuities in helium [CITATION], and in the acoustic limit this method must give the first and the second sound in a relativistic superfluid [CITATION].', 'hep-ph-9808250-2-7-5': 'The amplitude of discontinuities is assumed to be not very large, since superfluidity is expected to take place on both sides of the front with no phase transition.', 'hep-ph-9808250-2-8-0': 'Let the hypersurface [MATH] be the front of the discontinuity and vector [MATH] be a unit (space-like) normal to it.', 'hep-ph-9808250-2-8-1': 'The conservation laws ([REF]), ([REF]) entail the conditions [EQUATION] i.e. [EQUATION] where indexes + and - relate to the quantities ahead of and behind the discontinuity, respectively, and the square brackets imply the change across the front of a discontinuity.', 'hep-ph-9808250-2-8-2': 'Eq. ([REF]) conveys the continuity of the orthogonal part (marked always by index [MATH]) of the particle number current.', 'hep-ph-9808250-2-9-0': 'We cannot make use of the entropy conservation law ([REF]) since it does not take place in the shock waves.', 'hep-ph-9808250-2-9-1': 'For that reason the results of Ref. [CITATION] pertain for a most general two-constituent system (but not a superfluid itself), while the discussion of small perturbations suffers no restrictions concerning eq. ([REF]).', 'hep-ph-9808250-2-9-2': 'Indeed, the results for shock waves in a two-constituent system and in a superfluid will coincide when the magnitude of the relevant discontinuities tends to zero.', 'hep-ph-9808250-2-10-0': 'For a two-constituent superfluid the additional equation is the irrotationality condition ([REF]) which yields [EQUATION]', 'hep-ph-9808250-2-10-1': 'Multiplying it by [MATH] and, then, by a unit vector [MATH] orthogonal to [MATH], we get an important relation [EQUATION] (where index [MATH] denotes the tangential part of any quantity, particularly [MATH]).', 'hep-ph-9808250-2-10-2': 'This is the very condition for conservation of irrotational motion passing a plane shock wave [CITATION].', 'hep-ph-9808250-2-10-3': 'It is clear that for a multi-constituent system the irrotational motion conserves at both sides of the shock wave if condition ( [REF]) takes place for each constituent.', 'hep-ph-9808250-2-11-0': 'Substituting the expression ([REF]) and ([REF]) in ([REF]), we obtain [EQUATION]', 'hep-ph-9808250-2-11-1': 'Multiplying this equation by [MATH], we get [EQUATION]', 'hep-ph-9808250-2-11-2': 'Then, multiplying eq. ([REF]) by the unit vector [MATH], in light of ([REF]), we get [EQUATION]', 'hep-ph-9808250-2-11-3': 'This important relation determines the types of discontinuities.', 'hep-ph-9808250-2-11-4': 'Then, the irrotationality condition ([REF]) than coincides with the condition of strong discontinuity in the superfluid constituent which occurs when [MATH].', 'hep-ph-9808250-2-11-5': 'As for the discontinuity in the normal component, in light of ([REF]), it is determined merely by the single relation [EQUATION]', 'hep-ph-9808250-2-11-6': 'Thus the constraint ([REF]) corresponds to ordinary shock waves, while ([REF]) beseems to a ""temperature"" discontinuity of the second sound type.', 'hep-ph-9808250-2-12-0': 'As a particular instance of another type of discontinuity we consider a vortex sheet in superfluid [CITATION].', 'hep-ph-9808250-2-12-1': 'Since the vortex sheet separates the whole space into domains where the superfluidity takes place, the irrotationality condition ([REF]) does not hold in the global sense and we cannot establish the constraint ([REF]) at both sides of the sheet.', 'hep-ph-9808250-2-12-2': 'Therefore, the tangential discontinuities are possible.', 'hep-ph-9808250-2-12-3': 'The conservation law ([REF]) then yields [MATH] that determining weak or slip-stream discontinuity: the particle number flow across the front of discontinuity equals zero, indeed, no matter crosses the hypersurface of the discontinuity, i.e. this hypersurface is made up of streamlines of the fluid.', 'hep-ph-9808250-2-13-0': '# A PLANE SHOCK IN FLAT SPACE', 'hep-ph-9808250-2-14-0': 'Let the discontinuity propagates along the axis [MATH].', 'hep-ph-9808250-2-14-1': 'We choose the unit normal [MATH] - and the medium at rest before the front.', 'hep-ph-9808250-2-14-2': 'As a rule one used to practice with the rest frame co-moving the front, so that the fluid flows in the front with the velocity which is equal to that of a shock wave.', 'hep-ph-9808250-2-14-3': 'Hereby, the relevant vectors and co-vectors may be presented as [EQUATION]', 'hep-ph-9808250-2-14-4': 'Since the medium ahead of front is at rest, the relative velocity [MATH] equals zero, [MATH], while behind the shock [EQUATION]', 'hep-ph-9808250-2-14-5': 'Also [EQUATION]', 'hep-ph-9808250-2-14-6': 'Hence, the velocity of the shock is determined as [EQUATION]', 'hep-ph-9808250-2-14-7': 'Substituting our definitions ([REF]), ([REF]) and ([REF]) in eqs. ([REF]), ([REF]), ([REF]) and ([REF]) we, firstly obtain [EQUATION]', 'hep-ph-9808250-2-14-8': 'The rest equations, in view of ([REF]), will be [EQUATION]', 'hep-ph-9808250-2-14-9': 'The parameter [MATH] incorporates only in eq. ([REF]) and it can be calculated as soon as the rest unknowns are found.', 'hep-ph-9808250-2-14-10': 'Thus, in eqs. ([REF]), ([REF]) and ([REF]) the unknowns are: the four parameters [MATH], [MATH], [MATH], [MATH] and three invariants ([REF]), ([REF]), ([REF]) behind the shock on which the pressure [MATH] depends.', 'hep-ph-9808250-2-14-11': 'The pressure behind the shock can be expressed through [MATH], [MATH], and through the relative velocity for which we use the notation [MATH].', 'hep-ph-9808250-2-14-12': 'The formula [EQUATION] relates the later quantity with [MATH] and [MATH].', 'hep-ph-9808250-2-14-13': 'Our goal is to find the velocity of the shock wave [MATH] for a single parameter given behind the shock.', 'hep-ph-9808250-2-14-14': 'Without the loss of generality [MATH] can be chosen for this parameter.', 'hep-ph-9808250-2-14-15': 'Thus, there are six unknowns in four equations ([REF]), ( [REF]), ([REF]), ([REF]).', 'hep-ph-9808250-2-14-16': 'The rest two relations follow from ([REF]) or ([REF]) [EQUATION] with the coefficients ([REF]) calculated for the state behind the front.', 'hep-ph-9808250-2-15-0': 'The knowledge of the equation of state in explicit form is necessary for calculation of the right-hand side of eq. ([REF]) and the coefficients in eqs. ([REF]) and ([REF]).', 'hep-ph-9808250-2-16-0': '# A LOW TEMPERATURE CASE', 'hep-ph-9808250-2-17-0': 'The low-temperature equation of state was derived by Carter and Langlois [CITATION].', 'hep-ph-9808250-2-17-1': 'In view of ([REF]), ([REF]), ([REF]) the expressions ([REF]), ([REF]), ([REF]), ([REF]) take the form [EQUATION] where [EQUATION]', 'hep-ph-9808250-2-17-2': 'At low temperature we have the estimations [CITATION] [EQUATION] implying that [MATH] and, hence, equations ([REF]) and ([REF]) approximately (up to the terms [MATH]) coincide with their zero-temperature version.', 'hep-ph-9808250-2-18-0': '# THE SOUND, STRONG AND SMALL-AMPLITUDE SHOCK WAVES', 'hep-ph-9808250-2-19-0': 'If all parameters behind the front tend to their values ahead, the shock becomes a sound wave.', 'hep-ph-9808250-2-19-1': 'Since the entropy in the sound wave is conserved, we can apply formalism [CITATION] achieved for a two-constituent relativistic medium with the conserved particle currents of both constituents.', 'hep-ph-9808250-2-19-2': 'In the linear approximation both methods lead to the same result, namely from the system ([REF]), ([REF]), ([REF]), ([REF]), ( [REF]), ([REF]) we obtain an equation for two branches of sound at arbitrary temperature which is analogous to that derived by Carter [CITATION] and, under assumption [MATH], splits into [EQUATION] and reduces, in the low temperature limit, to the first and the second sound speed, respectively [CITATION]: [MATH], [MATH].', 'hep-ph-9808250-2-19-3': 'Here for any variable [MATH] we used the notation [EQUATION]', 'hep-ph-9808250-2-19-4': 'However, if we omit [MATH], the second sound speed calculated by formula ([REF]) with the phonon Lagrangian ([REF]) of Carter and Langlois [CITATION] will be [MATH] instead of obvious [MATH].', 'hep-ph-9808250-2-19-5': 'Because the Lagrangian ([REF]) is derived for the two-fluid theory with non-zero cross term; while the Lagrangian of thermal excitations of the Israel theory [CITATION] differs from ([REF]); although the relative translation speed [MATH] between the constituents is presented in both approaches.', 'hep-ph-9808250-2-19-6': 'So each Lagrangian is useful in the theory to which it does belong.', 'hep-ph-9808250-2-20-0': 'In order to find the velocity of a small-amplitude shock we rewrite eqs. ([REF]), ([REF]), ([REF]), ([REF]), ([REF]), ( [REF]) in the second-order approximation.', 'hep-ph-9808250-2-20-1': 'After tedious calculations we find the velocity increment [EQUATION] of the shock corresponding to the second sound when the temperature increment [MATH] tends to zero.', 'hep-ph-9808250-2-20-2': 'In turn, the latter gives rise to a finite relative speed [MATH] behind the shock, since [EQUATION]', 'hep-ph-9808250-2-20-3': 'In the non-relativistic limit the equations ([REF]) and ([REF]) yield well known expressions [CITATION].', 'hep-ph-9808250-2-20-4': 'The shock occurs ahead of the second sound, as is in superfluid helium at low temperature.', 'hep-ph-9808250-2-21-0': 'So, in the acoustic limit the solution splits into two independent branches [MATH] and [MATH] corresponding to the first and the second sound.', 'hep-ph-9808250-2-21-1': 'The first branch describes wave propagation through the medium which behaves as a perfect fluid composed of two constituents whose pressure and enthalpy are [MATH] and [MATH] respectively.', 'hep-ph-9808250-2-21-2': 'While the constituents in the waves of the second branch move independently, a counterflow appears: [MATH].', 'hep-ph-9808250-2-21-3': 'In general, a ""mixed"" solution occurs, and the temperature increases together with the chemical potential.', 'hep-ph-9808250-2-22-0': 'The estimations ([REF]) imply that the first sound and the relevant shock wave coincide roughly with usual discontinuity in the cold constituent.A great pressure jump [MATH] is produced inevitably by the change in pressure of the superfluid constituent [MATH], since the contribution of the normal constituent [MATH] is small.', 'hep-ph-9808250-2-22-1': 'Hence, in view of eqs. ([REF]) and ([REF]) we conclude that strong shock waves at low temperature propagate with the speed [MATH] which approximately equals to the speed of a usual shock wave in cold constituent [MATH], but always [MATH].', 'hep-ph-9808250-2-22-2': 'A more precise result is [EQUATION]', 'hep-ph-9808250-2-22-3': 'The second-sound discontinuities should be regarded as ""moderate"" for intermediate values of [MATH].', 'hep-ph-9808250-2-22-4': 'This takes place if relative changes in the superfluid and normal variables are of the same order and they can be of the same order if they do not access [MATH] greatly.', 'hep-ph-9808250-2-22-5': 'For a superfluid matter of neutron stars and phonon equation of state it is easy to estimate [MATH].', 'hep-ph-9808250-2-22-6': 'For this particular example we performed calculations with an ultra-relativistic superfluid matter.', 'hep-ph-9808250-2-22-7': 'The sound speed in this medium equals exactly to [MATH], and approximately it is the first sound, while the speed of the second sound [MATH].', 'hep-ph-9808250-2-22-8': 'For a not very small pressure change the velocity of the shock wave will be merely [MATH].', 'hep-ph-9808250-2-22-9': 'The dependence of the relative translation speed [MATH] on [MATH] is given in table 1.', 'hep-ph-9808250-2-23-0': 'Although the second sound velocity attains to the saturation value [MATH], the relative translation speed [MATH] grows with the growth of the shock wave intensity.', 'hep-ph-9808250-2-24-0': '# THE FOURTH SOUND', 'hep-ph-9808250-2-25-0': 'The fourth sound takes place when the normal constituent is restrained by some external agent, while the sound propagates through the superfluid constituent.', 'hep-ph-9808250-2-25-1': 'We cannot use the equation of motion ([REF]), but the conservation laws ([REF]), ([REF]) and the irrotationality condition ([REF]) will determine the fourth sound speed.', 'hep-ph-9808250-2-25-2': 'If the sound wave propagates in the direction determined by vector [MATH] the change of gradient of arbitrary quantity [MATH] is proportional to its infinitesimal change [MATH] [CITATION]: [MATH].', 'hep-ph-9808250-2-25-3': 'Thereby, we write - [EQUATION] instead of ([REF]), ([REF]), ([REF]) and ([REF]).', 'hep-ph-9808250-2-25-4': 'In the reference frame co-moving with the normal constituent we put [EQUATION] where [MATH] is the chemical potential in the ""normal"" reference frame.', 'hep-ph-9808250-2-25-5': 'Since the discontinuities propagate through the superfluid constituent, there must be [EQUATION]', 'hep-ph-9808250-2-25-6': 'Equations ([REF]), ([REF]), ([REF]) analogous to the relevant non-relativistic set [CITATION] determine the speed of the fourth sound.', 'hep-ph-9808250-2-25-7': 'Requiring the vanishing determinant of the system ([REF]), ([REF]) we get a quadratic equation for [MATH].', 'hep-ph-9808250-2-25-8': 'While the speed of the first and the second sound is determined by a 4-order system [CITATION], the fourth sound speed follows from two equations.', 'hep-ph-9808250-2-25-9': 'At zero temperature the speed of the first and the fourth sound are obtained by the same equations ([REF]) and ([REF]) and coincide exactly with the sound speed [MATH] in the superfluid constituent.', 'hep-ph-9808250-2-25-10': 'The difference appears at finite temperature on account of the relationship [CITATION] [EQUATION] between the infinitesimal discontinuities in ([REF]), ([REF]) and the temperature dependence of matrices in ([REF]).', 'hep-ph-9808250-2-25-11': 'For the phonon equation of state we get the explicit formula [EQUATION] which generalizes the relevant non-relativistic relation [CITATION] , where [MATH] and [MATH] is the normal and the superfluid energy-density respectively [CITATION].', 'hep-ph-9808250-2-26-0': '# CONCLUSION', 'hep-ph-9808250-2-27-0': 'Summarizing the results obtained in the present study, we emphasize the formulae ([REF]), ([REF]), ([REF]), ([REF]) which determine the propagation of discontinuities through a two-constituent relativistic superfluid in the general case.', 'hep-ph-9808250-2-27-1': 'For a plane shock wave seven equations ([REF]), ([REF]), ([REF]), ([REF]), ([REF]), ([REF]) and ([REF]) with seven unknowns must be solved.', 'hep-ph-9808250-2-27-2': 'In the acoustic limit these equations reduce to formulae ([REF]), ([REF]) for the first and the second sound.', 'hep-ph-9808250-2-27-3': 'At low temperature the system ([REF]), ([REF]), ( [REF]), ([REF]), ([REF]), ([REF]) and ([REF]) reduces to ([REF]), ([REF]), ([REF]), ([REF]), ([REF]), ([REF]) and ([REF]).', 'hep-ph-9808250-2-27-4': 'The velocity of strong shock waves is given by ([REF]) and ([REF]), while ([REF]) and ([REF]) describ the change of parameters in a weak shock wave.', 'hep-ph-9808250-2-27-5': 'As for perspectives and applications, the shock waves and spin-isospin sound in the nuclear matter are worth to be discussed in future.'}","[['hep-ph-9808250-1-16-0', 'hep-ph-9808250-2-17-0'], ['hep-ph-9808250-1-16-2', 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'hep-ph-9808250-1-24-0', 'hep-ph-9808250-1-24-1', 'hep-ph-9808250-1-28-3', 'hep-ph-9808250-2-14-5', 'hep-ph-9808250-2-14-7', 'hep-ph-9808250-2-14-15', 'hep-ph-9808250-2-17-1', 'hep-ph-9808250-2-25-3', 'hep-ph-9808250-2-27-3']","{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}",https://arxiv.org/abs/hep-ph/9808250,,,
1309.7626,"{'1309.7626-1-0-0': 'This paper presents an analysis of possible uses of climate policy instruments for the decarbonisation of the global electricity sector in a non-equilibrium economic and technology innovation-diffusion perspective.', '1309.7626-1-0-1': 'Emissions reductions occur through changes in technology and energy consumption; in this context, investment decision-making opportunities occur periodically, which energy policy can incentivise in order to transform energy systems and meet reductions targets.', '1309.7626-1-0-2': ""Energy markets are driven by innovation, dynamic costs and technology diffusion; yet, the incumbent systems optimisation methodology in energy modelling does not address these aspects nor the effectiveness of policy onto decision-making since the dynamics modelled take their source from the top-down 'social-planner' assumption."", '1309.7626-1-0-3': 'This leads to an underestimation of strong technology lock-ins in cost-optimal scenarios of technology.', '1309.7626-1-0-4': 'Our approach explores the global diffusion of low carbon technology in connection to a highly disaggregated sectoral macroeconometric model of the global economy, FTT:Power-E3MG.', '1309.7626-1-0-5': 'A set of ten independent projections to 2050 of the future global power sector in 21 regions based on different combinations of electricity policy instruments are modelled using this framework, with an analysis of their climate impacts.', '1309.7626-1-0-6': 'We show that in an environment emphasising diffusion and learning-by-doing, the impact of combinations of policies does not correspond to the sum of the impacts of individual instruments, but that strong synergies exist between policy schemes.', '1309.7626-1-0-7': 'We show that worldwide carbon pricing on its own is incapable of breaking the current fossil technology lock-in, but that under an elaborate set of policies, the global electricity sector can be decarbonised affordably by 89% by 2050 without early scrapping of capital.', '1309.7626-1-1-0': '# Introduction', '1309.7626-1-2-0': 'The electricity sector emits 38% of global energy-related greenhouse gases (GHGs).', '1309.7626-1-2-1': 'Investment planning in the electricity sector is therefore of critical importance to climate change policy.', '1309.7626-1-2-2': 'Electricity production is the energy sector with the longest time scales for technological change, requiring particularly careful planning in order to avoid locking into heavy emissions systems for many decades, crucial in order to avoid global warming above dangerous levels.', '1309.7626-1-2-3': 'It has been argued that meeting emissions targets to prevent warming beyond 2[MATH]C is possible but restrictive on possible pathways of energy sector development .', '1309.7626-1-2-4': 'However it is also widely accepted that warming beyond 4[MATH]C is likely to lead to catastrophic consequences to global ecosystems and food chains , with important repercussions to global human welfare.', '1309.7626-1-2-5': ""Strong reductions in greenhouse gas emissions involve significant amounts of technology substitution, most probably large scale 'socio-technical transitions' ."", '1309.7626-1-3-0': 'Energy systems being inherently complex, the use of computational tools to inform planning is crucial, and such models must include details of the various energy policy instruments available, and generate as results quantitative changes in emissions for changes in policy.', '1309.7626-1-3-1': 'It is moreover critical to avoid analysing national energy systems in isolation, and thus assessments are ideally carried out using large scale modelling of the global energy and economic system, incorporating endogenously quantities such as energy carrier demand and prices, imports and exports, technology costs, natural resource consumption and economic growth.', '1309.7626-1-3-2': 'Projections of global energy-related GHG emissions can only be performed using multi-sectoral (e.g. power, transport, industry, buildings) and multi-regional models of energy systems in order to cover all emissions sources.', '1309.7626-1-3-3': 'Finally, total emissions depend not only on technology compositions but also on the intensity of fuel use, which depends on energy demand and global economic activity.', '1309.7626-1-4-0': ""Most studies of energy systems are made using cost-optimisation computational models (using the 'social planner' assumption)."", '1309.7626-1-4-1': ""Meanwhile, economics of climate change is generally represented using equilibrium economic theories (an assumption of fully rational behaviour carried out by a 'representative agent'), generally assuming negative macroeconomic impacts of climate change mitigation by construction."", '1309.7626-1-4-2': 'These conceptually related approaches, in which these assumptions simplify greatly the analysis, represent the current methodological standard.', '1309.7626-1-4-3': 'These simplifications however mis-represent aspects of the global energy-economic system that are crucial for climate change mitigation, detailed below, leaving urgent need to improve theory and method beyond these paradigms.', '1309.7626-1-5-0': 'Cost-optimisation technology models, in normative mode, are powerful for finding with outstanding detail lowest cost future technology pathways.', '1309.7626-1-5-1': ""If used in descriptive mode, these assume by construction identical agents who benefit from a degree of information transfer and foresight sufficient to generate cost-optimal pathways, which corresponds to a controlled degree of coordination between all actors involved in the evolution of the system, termed the 'social planner' assumption."", '1309.7626-1-5-2': 'As stated in the Global Energy Assessment , ""A fundamental assumption underlying the pathways is that the coordination required to reach the multiple objectives simultaneously can be achieved"".', '1309.7626-1-5-3': 'While this approach generates a useful simplification to a complex system, it may be argued that such a spontaneous emergence of coordination is somewhat unlikely.', '1309.7626-1-5-4': 'Liberalised energy markets involve actors free to take their investment and consumption decisions based on their particular circumstances, and are only incentivised by policy.', '1309.7626-1-5-5': 'Thus while optimisation frameworks are valuable for identifying cost-effective pathways, they do not suggest how exactly to achieve them, because they do not model decision making.', '1309.7626-1-5-6': 'Strong coordination is difficult to generate from economic policy instruments, leading to technology lock-ins.', '1309.7626-1-6-0': 'Meanwhile, equilibrium economic models impose by construction that climate change mitigation costs are borne strictly at the expense of consumption, an assumption disputed in the economic literature providing little insight on macroeconomic dynamics of mitigation action (e.g. investment, employment), and to which we put forward a striking counterexample .', '1309.7626-1-6-1': 'The assumptions associated with equilibrium economics mean that carbon pricing is emphasised as a single policy tool to fix the climate market failure, even though it is well known to industrial policy-makers that new technologies often cannot successfully diffuse into the marketplace without further government support.', '1309.7626-1-6-2': 'Projecting future greenhouse gas emissions using normative models of energy systems with equilibrium economic models is conceptually inconsistent with the descriptive approach of climate science, misleading if this is done to project environmental impacts as outcomes of energy policy.', '1309.7626-1-7-0': 'A descriptive modelling approach requires using known technology dynamics .', '1309.7626-1-7-1': 'Empirical dynamics are known to exist in scaling up technology systems ([MATH]-shaped diffusion), and their costs (learning curves), which have been extensively studied for decades .', '1309.7626-1-7-2': 'Mathematical generalisations have been suggested involving dynamic differential equations similar to those in population growth mathematical ecology and demography .', '1309.7626-1-7-3': 'In combination with evolutionary dynamics and evolutionary game theory , this offers a artillery of powerful concepts at core of evolutionary economics [CITATION] which emphasises innovation, diffusion and speciation as a source of economic development and growth .', '1309.7626-1-7-4': 'In particular, complexity and path dependence emerge as key concepts to visualise technology dynamics .', '1309.7626-1-7-5': 'Even though [MATH]-shaped curves have been famously shown to arise in energy consumption , such principles have yet to be introduced in energy systems modelling, which would generate a much improved methodological paradigm .', '1309.7626-1-7-6': 'In such a framework, the use of optimisation as a source of dynamic force and actor behaviour would be replaced by such empirically known innovation-selection-diffusion dynamics.', '1309.7626-1-8-0': 'Moving beyond cost-optimisation and equilibrium theory, this work proposes such a paradigm with a non-equilibrium model of energy-economy-environment interactions.', '1309.7626-1-8-1': 'We present an analysis of the global electricity sector using coupled descriptive simulations of technology diffusion dynamics, using FTT:Power, non-equilibrium economics using E3MG, and environmental impacts by combining emulators of the PLASIM-ENTS climate model (PLASIM-ENTSem) and of the GENIE-1 carbon cycle model (CCem) .', '1309.7626-1-8-2': 'It enables to explore the outcomes of particular energy policy tools on technology diffusion, electricity generation, global emissions, climate change and macroeconomic change.', '1309.7626-1-8-3': 'Uncoordinated stochastic dynamics are modelled at the level of profit-seeking investor decisions under bounded rationality , of which the sum does not produce an overall cost optimum, as they disregard in general the effect of these decisions on the wider system or the common good.', '1309.7626-1-8-4': 'The connection a diffusion framework to a non-equilibrium model of the global economy opens a very rich world of macroeconomic dynamics of technological change where the impacts of energy policy reveal complex interactions between the energy sector and the economy.', '1309.7626-1-9-0': 'Ten scenarios of the future global power sector up to 2050 are presented, creating a storyline to provide insight for the construction of effective comprehensive policy portfolios.', '1309.7626-1-9-1': 'Going beyond simple carbon pricing and considering policies that could help trigger the diffusion of new technologies, elaborate combinations of energy policy instruments are found to provide suitable environments to enable fast electricity sector decarbonisation.', '1309.7626-1-9-2': 'Macroeconomic dynamics associated with these scenarios, explored separately by [CITATION], are summarised.', '1309.7626-1-9-3': 'High resolution data in these scenarios are accessible on our website.', '1309.7626-1-9-4': 'Decarbonisation involves the positive externality associated to the global accumulation of knowledge and experience in scaling up, deploying and using new power technologies.', '1309.7626-1-9-5': 'A classic collective action problem emerges: learning cost reductions for new technologies may only become significant and enable cost-effective diffusion when most nations of the World demonstrate strong coordinated dedication to their deployment.', '1309.7626-1-9-6': 'We show that carbon pricing covering all world regions is a necessary but insufficient component for the success of mitigation action in order to break the current fossil fuel technology lock-in.', '1309.7626-1-9-7': 'We show however that in this perspective, reductions in power sector emissions by nearly 90% do not generate significant macroeconomic costs (direct and indirect) but instead, generates additional industrial activity and employment.', '1309.7626-1-10-0': '# The model: FTT:Power-E3MG', '1309.7626-1-11-0': '## Technology diffusion in FTT:Power', '1309.7626-1-12-0': 'Emissions reductions in the energy sector can occur through technology substitution, between technologies that produce the same substitutable service (e.g. electricity, heat, etc), behaviour and practice changes and through reductions in consumption of that service altogether.', '1309.7626-1-12-1': 'Technology, however, comes to life through investments made at specific points in time with an expectation by investors to receive an income or service of which the present value justifies or exceeds the initial investments and the present value of present and future costs [CITATION].', '1309.7626-1-12-2': 'In this context early scrapping of usable capital produces stranded assets that do not produce the expected income, generates losses and requiring new investments earlier than expected.', '1309.7626-1-13-0': 'Where early scrapping is avoided, technological change occurs primarily at the average rate of replacement of existing technology as it ages, which is inversely related to its life span.', '1309.7626-1-13-1': 'However, notwithstanding lifetime considerations, the number of new units of technology of a particular type that can be constructed at any one time can be larger if that industry is in a well established position in the marketplace, with a large production capacity, than if it is emerging.', '1309.7626-1-13-2': 'Thus, even when an emerging technology is very affordable (for instance with strong policy support such as subsidies), it may not necessarily be accessible in every situation where investors are required to make a choice between available options.', '1309.7626-1-13-3': 'Thus in the analysis of the diffusion capacity of technologies and the pace of technological change, not only cost considerations come into play, but also a limited access to technology and information, and these two principles, described in earlier work , form the core of FTT:Power (eq. [REF]).', '1309.7626-1-14-0': 'This decision-making process can be expressed with pairwise comparisons of options.', '1309.7626-1-14-1': 'However, investors do not all face similar situations and do not weigh different aspects in the same way, which cannot possibly be enumerated specifically in a model.', '1309.7626-1-14-2': 'However, when dealing with large numbers of instances of decision making, there will be majority trends in investor choices, who may be assumed, if all relevant considerations are quantified into costs, to be seeking cost minimisation with the goal of profit maximisation for their own respective firms.', '1309.7626-1-14-3': 'Probability distributions may thus be used in order to avoid enumerating the details of all situations faced by investors, and where a particular technology is on average more profitable to use than a second one, there usually exist specific situations where the reverse turns out to be true.', '1309.7626-1-15-0': 'Thus the statistical trend of investor preferences may be expressed as a matrix [MATH] expressing the probability of investor choice between technologies [MATH] and [MATH].', '1309.7626-1-15-1': 'However, constraints of the power system prevent some types of technology to dominate.', '1309.7626-1-15-2': 'These can be expressed with a second matrix [MATH], stochastically stopping investments that lead to stranded or unused assets due to technical problems .', '1309.7626-1-16-0': 'Using the variable [MATH] for the generation capacity market share of a technology, the rate at which shares of one technology ([MATH]) can be replaced by shares of another ([MATH]) is proportional to:', '1309.7626-1-17-0': 'The rate at which units of technology [MATH] come to the end of their working life, with lifetime [MATH], How many old units of [MATH] require replacement, a fraction [MATH] of the total number of replacements.', '1309.7626-1-17-1': 'The rate at which the construction capacity for technology [MATH] can be expanded, with time constant [MATH], The market position of technology [MATH], its share of the market [MATH].', '1309.7626-1-18-0': 'The resulting market dynamics are flows of market shares between existing options, according to choices and constraints [MATH] and [MATH], and is expressed as [EQUATION] using the time scaling matrix [MATH] to express changeover time constants for every pair of technologies.', '1309.7626-1-18-1': 'Adding up substitution flows in both directions between all possible pairs of technologies leads to a family of finite differences equations that tracks the evolution of technological change in a set of substitutable technologies: [EQUATION] with [MATH] expressing investor preferences and [MATH] providing constraints.', '1309.7626-1-18-2': '[MATH] is the average rate of change weighed by the market shares, the scaling of time.', '1309.7626-1-18-3': 'This equation, logistic in the special case of two technologies, is non-linear and generates slow uptakes at small share values, then fast diffusion at intermediate penetration, before a saturation near full penetration.', '1309.7626-1-18-4': 'This system cannot be solved analytically but is straightforward to evaluate numerically using time steps.', '1309.7626-1-18-5': ""It corresponds to the replicator dynamics equation in evolutionary game theory , used to explain Darwin's Theory of Evolution, as well as in evolutionary economics ."", '1309.7626-1-19-0': '## Learning-by-doing and path dependence', '1309.7626-1-20-0': 'Profit-seeking investor choices summarised in [MATH] are driven by cost differences, and these change over time as technologies diffuse in the marketplace and follow their learning curves .', '1309.7626-1-20-1': 'Learning-by-doing cost reductions stem from the accumulation of technical knowledge on production and economies of scale in expansion of productive capacity.', '1309.7626-1-20-2': 'Technology costs are taken here to apply everywhere globally, so that the benefits of learning are global, in a perspective where firms selling technologies operate in a global market.', '1309.7626-1-20-3': 'Emerging technologies have fast cost reductions (e.g. solar panels) while established systems see very little change (e.g. coal systems).', '1309.7626-1-20-4': 'The crucial aspect is that cost reductions are decreasing functions of cumulative investment, not time, and that they do not occur if no investment is made.', '1309.7626-1-20-5': 'Learning thus interacts with diffusion where it incentivises further uptake, which generates further learning and so on, a highly non-linear effect which can lead to sudden technology avalanches.', '1309.7626-1-20-6': ""Policy can generate this, termed 'bridging the techchnology valley of death' ."", '1309.7626-1-20-7': 'Learning curves are connected here only with investor choice probabilities [MATH].', '1309.7626-1-21-0': 'Learning-by-doing introduces the crucial property of path dependence.', '1309.7626-1-21-1': 'As technologies gradually diffuse in the marketplace following investor choices, the full landscape of technology costs continuously changes, and investor preferences thus change.', '1309.7626-1-21-2': 'These changes are permanent and determined completely by past investments, and therefore by the full history of the market.', '1309.7626-1-21-3': 'Thus, from any starting point, future cost landscapes are determined by investment trends at every point in time, and different futures exist, depending strongly on policy choices along the way.', '1309.7626-1-22-0': '## Natural resource use', '1309.7626-1-23-0': 'The diffusion of power systems can only occur in areas where resources are available, for instance windy areas for wind power, or natural water basins and rivers for hydroelectric systems.', '1309.7626-1-23-1': 'Higher productivity sites offer lower costs of electricity production, and tend to be chosen first by profit-seeking investors.', '1309.7626-1-23-2': 'Assuming this, the progression of the development of renewable energy systems produces a tendency towards increasing marginal costs of production for potential new systems as only resources of lower and lower productivity are left to use.', '1309.7626-1-23-3': 'This is well described by functions of increasing marginal costs of production with increasing development, cost-supply curves .', '1309.7626-1-23-4': 'For this purpose, a complete set of such curves was previously estimated from literature data for 190 countries of the world for 13 types of natural resources .', '1309.7626-1-23-5': 'These were aggregated for the 21 regions of E3MG.', '1309.7626-1-23-6': 'Excluding non-renewable resources treated globally, this produced [MATH] cost-supply curves that are used to constrain the expansion of renewable systems in FTT:Power.', '1309.7626-1-23-7': 'The consumption of non-renewable resources is however better represented using dynamic cost-quantity curves, described next.', '1309.7626-1-24-0': '## Fossil fuel price dynamics', '1309.7626-1-25-0': 'Non renewable resources lying in geological formations have an arbitrary value that depends on their cost of extraction, but also on the dynamics of the market.', '1309.7626-1-25-1': 'To their cost of extraction is associated a minimum value that the price of the commodity must take in order for the extraction to be profitable.', '1309.7626-1-25-2': 'These costs are however distributed over a wide range depending on the nature of the geology (e.g. tar sands, ultra-deep offshore, shale oil and gas, etc).', '1309.7626-1-25-3': 'Thus, given a certain demand for the commodity, the price is a function of the extraction cost of the most expensive resource extracted in order to supply the demand, which separates reserves from resources.', '1309.7626-1-25-4': 'As reserves are gradually consumed, this marginal cost increases generating a commodity price increase which unlocks the exploitation of resources situated in locations with higher extraction costs.', '1309.7626-1-25-5': 'For example, tar sands became economical and saw massive expansion above a threshold price of around 85-95/boe.', '1309.7626-1-25-6': 'Thus, to any commodity demand path in time will correspond a path dependent commodity price.', '1309.7626-1-25-7': 'This model is described with an analysis in [CITATION], relying on data from [CITATION].', '1309.7626-1-25-8': 'In FTT:Power, this model is used to determine fuel costs for fossil fuel and nuclear based power technologies.', '1309.7626-1-26-0': '## Timescales of technological change', '1309.7626-1-27-0': 'Innovation generates new technologies that live in niches that protect them from the wider market.', '1309.7626-1-27-1': 'From those niches, in appropriate changes of market conditions, can emerge and diffuse new socio-technical regimes .', '1309.7626-1-27-2': ""After the innovation phase, at the level of diffusion, technologies enter what we termed the 'demographic phase' ."", '1309.7626-1-27-3': 'In scenarios excluding early scrapping, possible rates of change are dictated by the ageing and decommissioning rate of of capital and infrastructure, in other words, the inverse of the life expectancy of the existing capital.', '1309.7626-1-27-4': 'Meanwhile, relative rates of expansion of the productive capacity for new technology determine which technologies can fill in gaps more effectively than others, when other technology is decommissioned.', '1309.7626-1-27-5': 'In many cases of interest, for instance with new technologies, these timescales cannot be reliably measured using small amounts of diffusion data of non-linear nature.', '1309.7626-1-27-6': 'Therefore a theory of diffusion is necessary in order to infer these parameters from technology and industry properties.', '1309.7626-1-27-7': 'This theory is derived and explained in [CITATION].', '1309.7626-1-28-0': '## Modelling the global economy: E3MG', '1309.7626-1-29-0': 'E3MG is an macroeconometric model of the global economy that has been used widely for climate change mitigation economics [CITATION].', '1309.7626-1-29-1': 'It evaluates the parameters of 33 econometric equations onto data between 1971 and 2010, and extrapolates these equations between 2010 and 2050.', '1309.7626-1-29-2': 'The model has a very high resolution: its equations are evaluated for 21 regions of the world, 43 industrial sectors, 28 sectors if consumption, 22 fuel users, and 12 fuel types.', '1309.7626-1-29-3': 'Sectors are interrelated with dynamic input-output tables.', '1309.7626-1-29-4': 'The model incorporates endogenous technological change using accumulated investment and RD spending, particularly in the energy sector as detailed next.', '1309.7626-1-30-0': '## Electricity price-demand interactions', '1309.7626-1-31-0': 'The demand for electricity depends on its price, and it is well known that in situations of high electricity prices, people may strive to find more effective ways to use their income, preferring to invest in more efficient technology, perceived as a worthwhile tradeoff, or to simply reduce their consumption.', '1309.7626-1-31-1': 'When the electricity supply technology mix changes, the minimum price at which electricity can be profitably sold also changes, and with such price changes, the demand for electricity changes.', '1309.7626-1-31-2': 'In particular, when carbon pricing or feed-in tariffs are used to ensure access of expensive renewables into the grid, the price of electricity increases, bearing an externality to all users , affecting consumer demand and behaviour.', '1309.7626-1-31-3': 'Thus, reductions in emissions originate from both a change in the carbon intensity of the power sector and changes in the demand for electricity.', '1309.7626-1-31-4': 'These aspects of energy economics are prominent in this work, responsible for a significant fraction of our projected emissions reductions in scenarios of climate policy.', '1309.7626-1-32-0': 'Electricity demand is modelled in E3MG, using an econometric equation that incorporates endogenous technological change by including accumulated investment and RD expenditures of the form [EQUATION] where for fuel [MATH] and region [MATH], [MATH] is the fuel demand, [MATH] represents output, [MATH] relative prices and [MATH] is the technological progress indicator .', '1309.7626-1-32-1': 'Including accumulated investment and RD makes this equation non-linear, path-dependent and hysteretic.', '1309.7626-1-32-2': 'In particular, prices hikes lead to enhanced energy-saving RD that permanently increases efficiency of fuel use, which would not have occurred without price changes.', '1309.7626-1-32-3': 'Economic feedbacks between FTT:Power and E3MG occurs with four quantities: fuel prices, fuel use, power technology investments and tax revenue recycling .', '1309.7626-1-33-0': '## Climate modelling using PLASIM-ENTSem', '1309.7626-1-34-0': 'The carbon cycle is represented in this work by an emulator of the Grid Enabled Integrated Earth systems model (GENIE-1) Earth System Model .', '1309.7626-1-34-1': 'The carbon cycle emulator (CCem) is designed to be a more sophisticated, but computationally very fast, alternative to simplified climate-carbon cycles in integrated assessment models.', '1309.7626-1-34-2': 'GENIE-1 simulates climate, ocean circulation and sea-ice, together with the terrestrial, oceanic, weathering and sedimentary components of the carbon cycle.', '1309.7626-1-34-3': 'The emulator takes as inputs a time series of anthropogenic carbon emissions and non-CO[MATH] radiative forcing, and outputs a time series of atmospheric CO[MATH] concentration.', '1309.7626-1-34-4': 'Uncertainty in the carbon cycle is captured by varying GENIE-1 parameter inputs, resulting in an emulator ensemble of 86 possible futures.', '1309.7626-1-35-0': 'PLASIM-ENTSem is an emulator of the PLAnet SIMulator - Efficient Numerical Terrestrial Scheme (PLASIM-ENTS) global climate model (GCM).', '1309.7626-1-35-1': 'In the GCM, PLASIM, which includes a Q-flux ocean and a mixed-layer of a given depth , is coupled to a simple surface and vegetation model, ENTS, which represents vegetation and soil carbon through a single plant functional type .', '1309.7626-1-35-2': 'The design of PLASIM-ENTSem is described in detail by [CITATION].', '1309.7626-1-35-3': 'As for CCem, uncertainty is captured by varying PLASIM-ENTS parameter inputs, resulting in a 188-member ensemble of decadally and spatially seasonal climate variables.', '1309.7626-1-36-0': 'The combination of both emulators in the context of this work, with combined uncertainty analysis, is given by [CITATION].', '1309.7626-1-36-1': 'The emissions profiles associated with the scenarios produced by FTT:Power-E3MG are used to provide inputs to CCem, which in turn provides inputs to PLASIM-ENTS.', '1309.7626-1-36-2': 'The median, 5th-95th percentiles of the CCem ensemble output are each used to drive climate simulations using PLASIM-ENTSem, so that uncertainty associated with modelling the carbon cycle is represented in the climate projections.', '1309.7626-1-36-3': 'Based on the median CCem ensemble output, the median, 5th-95th percentiles of warming from the PLASIM-ENTSem ensemble are also calculated; these bounds, therefore, reflect warming uncertainty due to parametric uncertainty in the climate model alone.', '1309.7626-1-36-4': 'The 5th percentile of warming from the PLASIM-ENTSem ensemble based on the 5th percentile of CO2 concentration from the CCem ensemble, and the 95th percentile of warming from the PLASIM-ENTSem ensemble based on the 95th percentile of CO2 concentration from the CCem ensemble are also calculated; this second set of bounds, therefore, reflects warming uncertainty due to parametric uncertainty in the climate model and the carbon cycle model.', '1309.7626-1-37-0': '# Projections to 2050 of the global power sector', '1309.7626-1-38-0': '## Scenario creation and policy instruments', '1309.7626-1-39-0': 'Ten scenarios of electricity policy assumptions of different types and resulting technology mix and emissions up to year 2050 were created with FTT:Power-E3MG, lettered a. to j. (fig. [REF]).', '1309.7626-1-39-1': 'These all lead to different futures for the global power sector and different CO[MATH] emission profiles.', '1309.7626-1-39-2': 'It is impractical to reproduce all the information of these simulations in this paper, and therefore a summary of the results is given here, the details having been made available on the 4CMR website , where they can be displayed in terms of the full resolution of 21 world regions and 24 power technologies, for policy assumptions, electricity generation, emissions and levelised costs.', '1309.7626-1-39-3': 'Four energy policy tools were explored: carbon pricing (CO[MATH]P), technology subsidies (TSs), feed-in tariffs (FiTs) and direct regulations (REGs).', '1309.7626-1-39-4': 'Individual tools and various combinations were explored, a summary given in figure [REF].', '1309.7626-1-39-5': 'By gradually elaborating various policy frameworks, a scenario is found where power sector emissions are reduced by 89% below the 1990 level, involving all four policy instruments used simultaneously.', '1309.7626-1-39-6': 'These are fed to the carbon cycle and climate model emulators CCem and PLASIM-ENTSem in order to determine the resulting atmospheric CO[MATH] concentration and average global warming, for these scenarios where other sectors are not targeted by climate policy.', '1309.7626-1-40-0': 'The nature of FiTs is that access to the grid at a competitive price is ensured (a price higher than the consumer price), the difference being paid either by the grid and passed on to consumers through the price of electricity.', '1309.7626-1-40-1': 'The consumer price of electricity is raised by just the amount that makes this economically viable.', '1309.7626-1-40-2': 'The consumer price in FTT:Power-E3MG is derived from the generation share-weighted average Levelised Cost of Electricity (LCOE) ([MATH]).', '1309.7626-1-40-3': ""The LCOE as perceived by investors when a FIT exists includes an 'effective subsidy' given by the grid that covers a fraction or the whole difference between the levelised cost of these technologies and the consumer price of electricity (investors here may be corporate or, for instance, homeowners)."", '1309.7626-1-40-4': 'In the case of CO[MATH]P, the LCOE calculation that investors are assumed to perform includes possible carbon cost components, and the price of carbon is also passed on to consumers through the price of electricity.', '1309.7626-1-40-5': 'Thus the price of electricity increases with the carbon price unless emitting technologies are phased out.', '1309.7626-1-40-6': 'TSs are fractions of the capital costs of low carbon technologies that are paid by the government, reducing the LCOE that investors see.', '1309.7626-1-40-7': 'These are defined exogenously for every year up to 2050 and were designed to be phased out before then, after which it is hoped that the technology cost landscape becomes permanently altered and that technologies are not indefinitely subsidised.', '1309.7626-1-40-8': 'REGs indicate regulations controlling the construction of additional units of particular technologies, used to phase out particular types of systems.', '1309.7626-1-40-9': 'When a REG is applied to a technology category, no new units are built but existing ones are left to operate until the end of their lifetime.', '1309.7626-1-41-0': 'Fig. [REF] summarises the result of the policy tools exploration.', '1309.7626-1-41-1': 'Electricity generation by technology type is given in the series of panels to the left of each pair, while emissions are given on the right.', '1309.7626-1-41-2': 'The vertical dashed lines indicate the start of the simulations in 2008, and data to the left of this line is historical data from the [CITATION].', '1309.7626-1-41-3': 'The horizontal dashed lines indicate the 1990 levels of electricity demand and emissions.', '1309.7626-1-41-4': 'Dashed curves correspond to the baseline values for comparison.', '1309.7626-1-41-5': 'In all scenarios excluding the baseline, policy schemes generate both a reduction of electricity consumption and emissions.', '1309.7626-1-41-6': 'Consumption reduces due to increases in the price of electricity, through the energy demand econometric equation of E3MG, which contributes significantly to emissions reductions.', '1309.7626-1-41-7': 'All additional emissions reductions are due to changes in fuels used, or, in other words, to changes of technologies.', '1309.7626-1-41-8': 'Emissions reductions in 2050 with respect to the 1990 level are given in percent values.', '1309.7626-1-42-0': '## Climate policy for achieving 90 reductions in power sector emissions', '1309.7626-1-43-0': 'The baseline scenario (fig. [REF] panel a.), which involves maintaining current policies until 2050 (carbon pricing in Europe only), generates global power sector emissions in 2050 of 31 GtCO[MATH]/y, 343 above the 1990 level, total emissions of 68 GtCO[MATH]/y. Cumulative emissions for the time span 2000-2050 amount to 2350 GtCO[MATH].', '1309.7626-1-43-1': 'This pathway is likely to commit the planet to a warming that exceeds 4[MATH]C above pre-industrial levels in around 2100 (fig. [REF] below), consistent with [CITATION] .', '1309.7626-1-43-2': 'In view of finding ways to reduce these emissions and to limit global warming, we searched areas of policy space that reduce these significantly in the short time span, given in fig. [REF].', '1309.7626-1-44-0': 'The first option explored (panel b.) was to use REGs to prevent the construction of new coal power plants worldwide, the systems with highest emissions ([MATH]1 ktCO[MATH]/MWh), unless they are equipped with carbon capture and storage (CCS).', '1309.7626-1-44-1': 'This results mostly in a transfer from a coal lock-in to a gas lock-in, reducing global emissions approximately to the 1990 level, largely insufficient meeting the 2[MATH]C target.', '1309.7626-1-45-0': 'The second option was to use CO[MATH]P as a unique tool, with different price values for different regions covering all world regions shown in fig. [REF], between 100 and 200 2008/tCO[MATH] in 2050 (panel c.) and between 200 and 400 2008/tCO[MATH] (panel d.).', '1309.7626-1-45-1': 'This reduces emissions to around 71% above 1990 and to the 1990 level, respectively.', '1309.7626-1-45-2': 'This modest impact suggests that CO[MATH]P on its own requires very high carbon prices in order to generate significant reductions, or that it is simply insufficient.', '1309.7626-1-45-3': 'However as we show now, combinations of policies achieve this much more effectively.', '1309.7626-1-46-0': 'As a first combination of policies, FiTs (wind and solar) and TSs (all other low carbon technologies except wind and solar) were used without CO[MATH]P (panel e.), of order 30-50 of capital costs for TSs and feed-in prices 5-15 above the electricity price for FiTs (this depends on regions and technologies, for an example see the inset of fig. [REF] or the data on our website).', '1309.7626-1-46-1': 'This generates very modest uptakes of low carbon technologies and thus small emissions reductions, 314 above 1990.', '1309.7626-1-46-2': 'This is due to the very low cost of producing electricity using fossil fuels in comparison to all other technologies, in particular coal, and therefore CO[MATH]P is necessary in order to bridge this cost difference.', '1309.7626-1-47-0': 'Scenario [MATH] shows the use of CO[MATH]P up to 200/tCO[MATH] and feed-in tariffs, the latter generating very little change over scenario c. Using CO[MATH]Ps of up to 200/tCO[MATH] in combination with TSs and FiTs in all world regions (panel g.) yields emissions reductions to 14 above the 1990 level, still insufficient.', '1309.7626-1-47-1': 'With CO[MATH]P of up to 400/tCO[MATH] in combination with the same set of TSs and FiTs (panel h.), reductions are much larger, 57 below the 1990 level.', '1309.7626-1-47-2': 'This indicates how the impact of policy combinations is larger than the sum of the impacts of its components taken separately, offering significant potential synergies.', '1309.7626-1-48-0': 'A scenario was explored in which only the developed world applies the strong energy policy of scenario h (panel i.), in which it is hoped that this generates enough investment to bring the costs of low carbon technologies into the mainstream, thus becoming accessible to developing or under-developed countries.', '1309.7626-1-48-1': 'We see no noticeable uptake of new technology in these countries, costs remaining unaccessible especially in comparison with coal based technologies, and as a consequence global emissions are only reduced to 185% above the 1990 level.', '1309.7626-1-49-0': 'A significant amount of the remaining emissions in scenario h reside in China, where the lock-in of coal technology is very difficult to break given the near absence of alternatives with the exception of hydroelectricity, which is driven to its natural resource limits.', '1309.7626-1-49-1': 'The choice of investors thus needs to be constrained at the expense of having to sell electricity at higher prices.', '1309.7626-1-49-2': 'Therefore REGs were introduced in scenario j in China that prevent the construction of new coal power stations unless they are equipped with CCS.', '1309.7626-1-49-3': 'This additional policy forces additional diversity in the Chinese technology mix, bringing down global emissions to 89% below the 1990 level without early scrapping.', '1309.7626-1-49-4': 'Total cumulative emissions for the time period 2000-2050 in scenario h (in the baseline) are of 1670 Gt (2350 Gt), given that other sectors do not change their technologies significantly, of which 390 Gt (930 Gt) originates from the electricity sector alone.', '1309.7626-1-50-0': '## Climate change projections', '1309.7626-1-51-0': 'Global emissions from all sectors in scenarios [MATH] to [MATH] were fed into the CCem in order to calculate the resulting CO[MATH] concentrations, which themselves were supplied to PLASIM-ENTSem in order to explore their climate change impacts.', '1309.7626-1-51-1': 'Figure [REF], top panel, displays global CO[MATH] emissions for all scenarios from E3MG including all fuel combustion emissions from all users as well as exogenous trends of emissions from all other sectors, obtained from the EDGAR database.', '1309.7626-1-51-2': 'While the changes observed include those in power sector emissions of fig. [REF], they also include modest changes in other sectors (e.g. industry) occurring due to CO[MATH]P for all fuel users subject to the emissions trading scheme (taken to function similarly to the current EU-ETS).', '1309.7626-1-52-0': 'In order to run the climate model emulator, emissions were required up to 2100.', '1309.7626-1-52-1': 'E3MG projections beyond 2050 are fairly uncertain and were not used, and the trends to 2050 were extrapolated to 2100.', '1309.7626-1-52-2': 'Polynomial extrapolations were assumed for scenarios a,e,i while emissions were considered stable beyond 2050 for all other scenarios.', '1309.7626-1-52-3': 'While this is not a particularly accurate method, accuracy is not the primary goal here since the decarbonisation of other sectors of anthropogenic emissions, responsible for most emissions left in scenario j. of order 30 Gt, was not specifically modelled, but represent significant potentials for further reductions (i.e. low carbon technology diffusion in transport and industry).', '1309.7626-1-53-0': 'The middle panel of fig. [REF] shows the resulting atmospheric CO[MATH] concentrations, with uncertainty shown as a blue area.', '1309.7626-1-53-1': 'It was observed that scenario a reaches median values of [MATH] ppm while scenario j reaches [MATH] ppm, uncertainty due to the carbon cycle model.', '1309.7626-1-53-2': 'This is above the generally assumed threshold of 450 ppm for maintaining warming below 2[MATH]C.', '1309.7626-1-53-3': 'These concentrations were fed to the climate model emulator, which yielded global warming median temperature changes of (fig. [REF]) [MATH]C over pre-industrial levels when using the median concentration and only the climate model uncertainty, and [MATH]C when including both carbon cycle and climate model uncertainties.', '1309.7626-1-53-4': 'This therefore could exceed [MATH]C of warming with a 5% chance.', '1309.7626-1-53-5': 'Meanwhile, the electricity decarbonisation scenario yields warming values of [MATH]C with carbon cycle uncertainty only and [MATH]C with both carbon cycle and climate model uncertainties, with a very small probability of not exceeding [MATH]C of warming.', '1309.7626-1-53-6': 'This indicates that the decarbonisation of the power sector by 89% is insufficient if other sectors such as transport and industry are not specifically targeted in climate policy.', '1309.7626-1-54-0': '## Learning cost reductions and energy price dynamics', '1309.7626-1-55-0': 'The uptake of low carbon technologies generate learning cost reductions that alter permanently the technology cost landscape.', '1309.7626-1-55-1': 'Figure [REF] shows world averages of bare technology costs (upper panels) for the baseline and mitigation scenario j, weighed by electricity generation, excluding TSs, CO[MATH]P and FiTs.', '1309.7626-1-55-2': 'These values, when including policy, drive investor choices in both the baseline (left) and the mitigation (right) scenarios.', '1309.7626-1-55-3': 'Roughly speaking, decreases stem from learning-by-doing cost reductions while increases originate from increasing natural resource scarcity with development.', '1309.7626-1-55-4': 'While the cost of PV panels decreases in the baseline scenario mainly due to deployment in Europe, it decreases by more than half its 2008 value in the mitigation scenario where they benefit from FiTs everywhere.', '1309.7626-1-55-5': 'Meanwhile, onshore wind power does come into the mainstream in many regions of the world in the mitigation scenario and does not necessitate support all the way to 2050, where the value of the wind FiTs become near zero or even negative, in which case the policy it is dropped altogether.', '1309.7626-1-55-6': 'In other regions, wind power is limited by resource constrained decreasing capacity factors and corresponding increasing costs.', '1309.7626-1-55-7': 'Other technologies, such as geothermal or wave power (not shown), see very little uptake in this particular mitigation scenario and therefore little cost reductions.', '1309.7626-1-56-0': 'The marginal costs of producing electricity, defined as share-weighted LCOEs, are given for 6 aggregate regions in the lower panels of fig. [REF].', '1309.7626-1-56-1': 'This marginal cost is used in E3MG to construct electricity prices in 21 regions, of which the changes alter electricity consumption .', '1309.7626-1-56-2': 'These are different between regions, stemming from different technology and resource landscapes, where lower marginal costs correspond to higher shares of coal based electricity.', '1309.7626-1-57-0': 'The marginal costs of fossil fuels are calculated using estimates of reserves and resources, described in section [REF], and are not highly affected by changes in policy in these scenarios.', '1309.7626-1-57-1': 'In both scenarios oil and gas costs increase significantly up to 2050 in a similar way, but these increases are dampened by the massive accession to unconventional fossil fuel resources (oil sands, heavy oil and shale gas).', '1309.7626-1-57-2': 'This analysis will be given elsewhere in more detail.', '1309.7626-1-57-3': 'Coal costs are moderately affected by changes in demand due large coal resources.', '1309.7626-1-57-4': 'The price of natural uranium ore is stable until 2035 where an increase is observed, generated by increasing scarcity, and at this level of consumption, U resources are projected to run out before 2100 unless technology changes .', '1309.7626-1-58-0': '## Global economic impacts of an 89% reduction scenario', '1309.7626-1-59-0': 'The macroeconomic impacts of scenarios a and j are explored separately in [CITATION].', '1309.7626-1-59-1': 'It is found there that decarbonising the electricity sector by 89% has economic benefits, generating additional employment, real household income and increases GDP by between 1 and 3% (depending on the regions) in comparison to scenario a.', '1309.7626-1-59-2': 'This is due to two opposing forces acting against one another: the introduction of low carbon technologies force increases in electricity prices (as seen in fig. [REF]), lowering real household disposable income, while low carbon technology production generates further employment in various industrial sectors, increasing household income.', '1309.7626-1-59-3': 'These are shown to approximately cancel, which is possible as long as labour and capital (investment) resources can be made available .', '1309.7626-1-59-4': 'Our assumptions about capital and labour markets are therefore consistent with our assumptions of energy markets, in that they do not automatically produce optimal outcomes.', '1309.7626-1-60-0': 'However, CO[MATH]P generates government income larger than government spending on TSs, the rest being redistributed to households in the form of income tax reductions, increasing further their disposable income.', '1309.7626-1-60-1': 'The resulting impacts are therefore of increased household income and consumption in comparison to the baseline and thus higher GDP.', '1309.7626-1-60-2': 'It is to be noted however that there are winners and losers in this picture both in terms of sectors and world regions, depending how much they depend on activities of the oil, gas and coal sectors.', '1309.7626-1-61-0': '## Local projections of power generation and emissions', '1309.7626-1-62-0': 'It also proves instructive to analyse electricity technology landscapes in individual regions of the world in FTT:Power-E3MG, for policy analysis and for better understanding the nature of technology lock-ins and the restricted local ability to change in a diffusion perspective.', '1309.7626-1-62-1': 'This is presented in fig. [REF] for six key regions or countries: North America, Europe, China, India, Brazil and the Rest of the World, which have different electricity landscapes stemming from differing energy policy strategies and engineering traditions historically, as well as natural resource landscapes.', '1309.7626-1-62-2': ""National strategies, reflecting local engineering specialisation related to technology lock-ins, is a natural outcome of this model's structure (eq. [REF]), which reproduces the better ability of dominating industries to capture the market despite costs."", '1309.7626-1-63-0': 'Renewable energy systems are more exploited in Europe than anywhere else in the world, except in Brazil, where hydroelectricity dominates.', '1309.7626-1-63-1': 'Europe also sees the most diverse electricity sector, with large amounts of wind power already in the baseline scenario, predominantly in northern Europe and the British Isles, large amounts of nuclear power in France, and some solar power in Germany.', '1309.7626-1-63-2': 'Coal fired electricity is mostly phased out before 2050 in the 90% scenario, generating significant emissions reductions.', '1309.7626-1-64-0': 'North America features higher use of fossil fuels for power production than Europe.', '1309.7626-1-64-1': 'However, while E3MG projects a larger potential for consumption reductions, large opportunities for diversification also emerge with significant potentials of renewable energy.', '1309.7626-1-64-2': 'Bioenergy with CCS generates a large contribution to emissions reductions.', '1309.7626-1-65-0': 'China and India have very low technology diversity and important fossil fuel lock-ins.', '1309.7626-1-65-1': 'The amount of coal used in China in the baseline is responsible for emissions of 10 out of 30 Gt of global emissions in 2050.', '1309.7626-1-65-2': 'Diversification proves difficult given the scale of the rate of increase in consumption; breaking the coal lock-in requires REGs in China to phase out existing coal generators.', '1309.7626-1-65-3': 'Large scale diffusion of renewables is slow and retrofitting CCS to coal generators offers a useful alternative.', '1309.7626-1-65-4': 'Electricity demand reductions are very large, which requires further investigations for fuel poverty and further social implications.', '1309.7626-1-66-0': 'In Brazil, even though hydroelectricity is not the least expensive resource, it nevertheless dominates, another form of technology lock-in.', '1309.7626-1-66-1': 'This is typical of a national engineering tradition dominated by this technology for decades.', '1309.7626-1-66-2': 'Brazil is projected to persist developing its hydropower capacity despite higher costs and a decreasing potential, until the cost becomes prohibitively expensive and only less productive sites remain.', '1309.7626-1-67-0': 'The rest of the world includes predominantly countries where the diversity of existing technologies is low, and persists in this direction.', '1309.7626-1-67-1': 'It features large amounts of oil use for electricity despite high oil prices, due to low technology availability or fossil fuel subsidies, which are not successfully phased out despite being the least cost-effective way of producing electricity.', '1309.7626-1-67-2': 'Coal based electricity makes the dominant contribution to emissions in the baseline, the rest divided between oil and gas fired power stations, for a total of 12 out of 30 Gt of global emissions in 2050 in the baseline.', '1309.7626-1-67-3': 'In the mitigation scenario, a significant additional hydroelectricity potential is developed, and coal is replaced by gas turbines, which are eventually retrofitted with CCS.', '1309.7626-1-68-0': '# Policy and strategic consequences', '1309.7626-1-69-0': '## Synergy between policy instruments', '1309.7626-1-70-0': 'This paper shows that in a coupled energy-economy-environment model that does not assume economic equilibrium or technology cost-optimisation, the impact of policy instruments can be different if used individually or in combinations: the impact of combined policy packages does not correspond to the sum of the impacts of individual instruments.', '1309.7626-1-70-1': 'Thus significant synergies between policy instruments.', '1309.7626-1-70-2': 'In this regard we showed that CO[MATH] alone is not likely capable to delivering sufficient emissions reductions, in contrast to the position supported by many policy-makers and economists ; it requires to be combined with TSs, FiTs and REGs.', '1309.7626-1-70-3': 'Relying on CO[MATH]P alone is likely to lead to a status quo in the technology mix while delivering expensive electricity to consumers.', '1309.7626-1-70-4': 'Similarly, TSs and FiTs on their own have little impact unless they are combined with sufficiently high CO[MATH]P.', '1309.7626-1-71-0': 'We furthermore show that elaborate combinations of policy instruments can produce such strong synergy that reductions of electricity sector emissions by 89% by 2050 (58% of cumulative power sector emissions) become possible without early scrapping of electricity generation captital.', '1309.7626-1-71-1': 'Such strong reductions could be complemented by additional reductions in other emissions intensive sectors with additional cross-sectoral synergies: transport, industry and buildings, warranting further work in this area.', '1309.7626-1-72-0': '## The effect of global knowledge spillovers on technology costs: individual vs global coordinated action', '1309.7626-1-73-0': 'Technology systems typically face a vicious cycle: established technologies thrive because they are established, and emerging technologies see barriers to their diffusion due to the lock-in of established technologies.', '1309.7626-1-73-1': 'This is the case unless an emerging technology is a radical improvement over the incumbent, or it benefits from sufficient external support.', '1309.7626-1-73-2': 'Emerging technologies require investment and sales in order to benefit from improvements and economies of scale: repetition, trial and error enables entrepreneurs to improve their products.', '1309.7626-1-73-3': 'They thus require a continuous flow of funds from sales or external investment in order to survive until their products take off on their own in the market.', '1309.7626-1-73-4': 'In the long run, these investments may or may not generate a return, and are thus risky.', '1309.7626-1-73-5': ""Without any investment to bridge the 'technology valley of death', however, they may become failed innovations."", '1309.7626-1-73-6': 'Competitive thresholds are set by incumbent technologies.', '1309.7626-1-74-0': 'Given known learning curves of power systems, a certain additional capacity of emerging technologies such as wind turbines and solar PV panels must be bought and sold in order to bring down their costs to a competitive level set by established technologies such as coal, gas or nuclear power stations.', '1309.7626-1-74-1': 'As we find using FTT:Power-E3MG, this additional capacity necessary to make their cost affordable is very large, and cannot be deployed by a single nation such as Germany or the UK, or even the whole of Europe, for the rest of the world to benefit, particularly true for solar PV.', '1309.7626-1-74-2': 'In contrast, we find that only a concerted global climate policy effort can bring down costs to manageable levels and bring new power technologies into the mainstream, opening very large renewable energy potentials such as that of solar energy.', '1309.7626-1-74-3': 'Such a concerted effort can significantly and permanently alter the global landscape of power technology costs and availability.', '1309.7626-1-74-4': 'We stress that, intellectual property law and border taxes allowing, all countries of the world can benefit from learning cost reductions that may originate from investments and sales occurring elsewhere.', '1309.7626-1-74-5': 'This problem therefore possesses the features of a classic free-rider and collective action problem, where international coordination is the only way by which these cost reductions can take place.', '1309.7626-1-74-6': ""Emerging or developed nations cannot simply 'wait' for climate policy in other nations to generate diffusion and enough learning cost reductions for new technologies to become competitive: without their involvement they might potentially never become competitive."", '1309.7626-1-74-7': 'If the power sector is to decarbonise by 2050, all countries are most likely required to make a contribution to the development of the renewables industry.', '1309.7626-1-75-0': '## Distributional impacts of decarbonisation', '1309.7626-1-76-0': '[CITATION] describes the economic impacts of the decarbonising the power sector; only a summary is provided here.', '1309.7626-1-76-1': 'However, it is important to note that there are some winners and losers in the decarbonisation scenario.', '1309.7626-1-76-2': 'The main winners are the construction and engineering sectors, their workforce and their supply chains.', '1309.7626-1-76-3': 'These all benefit from the higher rates of development and deployment of capital-intensive equipment.', '1309.7626-1-76-4': 'The sectors that lose out are those that supply fossil fuels.', '1309.7626-1-76-5': 'We would also expect reduced rates of economic activity in services sectors, where consumers must spend a larger proportion of income on electricity; this is however compensated by tax reductions originating from government income from carbon pricing.', '1309.7626-1-77-0': 'The outcomes for electricity-intensive sectors are less clear.', '1309.7626-1-77-1': 'Companies in these sectors will face higher costs and may see a loss of demand for their products.', '1309.7626-1-77-2': 'However, as the scenarios are global, there are only limited competitiveness effects.', '1309.7626-1-77-3': 'Many, but by no means all, energy-intensive firms also feature in the supply chains for renewables.', '1309.7626-1-77-4': 'Furthermore, imports and exports of fossil fuels change significantly in any decarbonisation scenario, generating further winners (fuel importers) and losers (fuel exporters).', '1309.7626-1-77-5': 'This aspect requires further investigations into the economic impacts of specific nations.', '1309.7626-1-78-0': 'The most important unknown in the decarbonisation scenarios presented concerns the distributional impact of increasing electricity prices to the less wealthy in all regions of the World.', '1309.7626-1-78-1': 'Depending on government policy, this could trap groups of people in energy poverty, in particular people who cannot afford to change their electricity consuming appliances such as heating or cooling devices.', '1309.7626-1-78-2': 'Therefore, climate policy could be complemented with energy end-use technology subsidies in order to help people change technology for more efficient systems through the transition without significant loss in their standard of living.', '1309.7626-1-78-3': 'Complementary research on the impact of climate policy onto energy access in under-developed regions and the likely strong synergies existing between energy access, climate and economic development policies is likely to shed significant light on these issues.', '1309.7626-1-79-0': '# Conclusion', '1309.7626-1-80-0': 'This work introduces an unprecedented method and analysis of technology diffusion and decarbonisation in the global electricity sector using a coupled model of power technology diffusion, non-equilibrium macroeconomics and climate change.', '1309.7626-1-80-1': 'The use of non-equilibrium models is shown to generate projections significantly different to those created using currently standard cost-optimisation and equilibrium economics approaches.', '1309.7626-1-80-2': 'The relaxation of the cost-optimisation constraint enables scenarios to feature non-optimal technology lock-ins well known to take place in the electricity sector.', '1309.7626-1-80-3': 'Meanwhile, non-equilibrium macroeconometrics enable to explore detailed macroeconomic impacts of energy policy instruments, explored in a sister paper , where investment in low carbon technology generates additional employment and enhanced economic activity as a counterbalance to increasing energy prices during decarbonisation.', '1309.7626-1-81-0': 'Ten scenarios were presented where a rationale is built for the elaboration of composite energy policy instrument combinations with strong synergy.', '1309.7626-1-81-1': 'This involves the analysis of individual policy tools as well as combinations, generating a storyline on how to reduce global emissions to near 90% below their 1990 level.', '1309.7626-1-81-2': 'The demonstration of the existence of strong synergies between policy tools for the diffusion of low carbon technology argues strongly against the use of policy instruments on their own such as carbon pricing, likely to yield very little change in the technology composition and unlikely to break the current fossil technology lock-in.', '1309.7626-1-82-0': 'We presented a decarbonisation scenario where a significant contribution to emissions reductions originates from electricity consumption reductions, which happens through changes the price of electricity as a result of energy policy and changes in production costs.', '1309.7626-1-82-1': 'The remaining emissions reductions stem from changes in the global electricity technology mix itself.', '1309.7626-1-82-2': 'Global warming impacts are provided.', '1309.7626-1-82-3': 'Insight for climate policy-making is given in terms of existing synergies between climate policy instruments and their impact on technology costs through global knowledge spillovers.', '1309.7626-1-82-4': 'This provides a strong argument for global coordination of climate policy in order for all nations to benefit simultaneously from cost reductions for low carbon technologies through research and development and learning-by-doing.'}","{'1309.7626-2-0-0': 'This paper presents an analysis of possible uses of climate policy instruments for the decarbonisation of the global electricity sector in a non-equilibrium economic and technology innovation-diffusion perspective.', '1309.7626-2-0-1': 'Emissions reductions occur through changes in technology and energy consumption; in this context, investment decision-making opportunities occur periodically, which energy policy can incentivise in order to transform energy systems and meet reductions targets.', '1309.7626-2-0-2': ""Energy markets are driven by innovation, dynamic costs and technology diffusion; yet, the incumbent systems optimisation methodology in energy modelling does not address these aspects nor the effectiveness of policy onto decision-making since the dynamics modelled take their source from the top-down 'social-planner' assumption."", '1309.7626-2-0-3': 'This leads to an underestimation of strong technology lock-ins in cost-optimal scenarios of technology.', '1309.7626-2-0-4': 'Breaking this tradition, our approach explores bottom-up investor dynamics led global diffusion of low carbon technology in connection to a highly disaggregated sectoral macroeconometric model of the global economy, FTT:Power-E3MG.', '1309.7626-2-0-5': 'A set of ten different projections to 2050 of the future global power sector in 21 regions based on different combinations of electricity policy instruments are modelled using this framework, with an analysis of their climate impacts.', '1309.7626-2-0-6': 'We show that in an environment emphasising diffusion and learning-by-doing, the impact of combinations of policies does not correspond to the sum of the impacts of individual instruments, but that strong synergies exist between policy schemes.', '1309.7626-2-0-7': 'We show that worldwide carbon pricing on its own is incapable of breaking the current fossil technology lock-in, but that under an elaborate set of policies, the global electricity sector can be decarbonised affordably by 89% by 2050 without early scrapping of capital.', '1309.7626-2-1-0': '# Introduction', '1309.7626-2-2-0': 'The electricity sector emits 38% of global energy-related greenhouse gases .', '1309.7626-2-2-1': 'Investment planning in the electricity sector is therefore of critical importance to climate change policy.', '1309.7626-2-2-2': 'Electricity production is the energy sector with the longest time scales for technological change, requiring particularly careful planning in order to avoid locking into heavy emissions systems for many decades, crucial to avoid global warming above dangerous levels .', '1309.7626-2-2-3': 'It has been argued that meeting emissions targets to prevent warming beyond 2[MATH]C is possible but restrictive on possible pathways of energy sector development .', '1309.7626-2-2-4': 'However it is also widely accepted that warming beyond 4[MATH]C is likely to lead to catastrophic consequences to global ecosystems and food chains , with important repercussions to global human welfare.', '1309.7626-2-2-5': ""Strong reductions in greenhouse gas emissions involve significant amounts of technology substitution, most probably large scale 'socio-technical transitions' ."", '1309.7626-2-3-0': 'Energy systems being inherently complex, the use of computational tools to inform planning is crucial, and such models must include details of the various energy policy instruments available, and generate as results quantitative changes in emissions for changes in policy.', '1309.7626-2-3-1': 'It is moreover critical to avoid analysing national energy systems in isolation, and thus assessments are ideally carried out using large scale modelling of the global energy and economic system, incorporating endogenously quantities such as energy carrier demand and prices, imports and exports, technology costs, natural resource consumption and economic growth.', '1309.7626-2-3-2': 'Projections of global energy-related GHG emissions can only be performed using multi-sectoral (e.g. power, transport, industry, buildings) and multi-regional models of energy systems in order to cover all emissions sources.', '1309.7626-2-3-3': 'Finally, total emissions depend not only on technology compositions but also on the intensity of fuel use, which depends on energy demand and global economic activity.', '1309.7626-2-4-0': ""Most studies of energy systems are made using cost-optimisation computational models (using the 'social planner' assumption)."", '1309.7626-2-4-1': ""Meanwhile, the economics of climate change are generally represented using equilibrium economic theories (an assumption of fully rational behaviour carried out by a 'representative agent'), generally assuming negative macroeconomic impacts of climate change mitigation by construction."", '1309.7626-2-4-2': 'These conceptually related approaches, in which these assumptions simplify greatly the analysis, represent the current methodological standard.', '1309.7626-2-4-3': 'These simplifications however mis-represent aspects of the global energy-economic system that are crucial for climate change mitigation, detailed below, leaving urgent need to improve theory and method beyond these paradigms.', '1309.7626-2-5-0': 'Cost-optimisation technology models, in normative mode, are powerful for finding with outstanding detail lowest cost future technology pathways.', '1309.7626-2-5-1': ""If used in descriptive mode, these assume by construction identical agents who benefit from a degree of information transfer and foresight sufficient to generate cost-optimal pathways, which corresponds to a controlled degree of coordination between all actors involved in the evolution of the system, termed the 'social planner' assumption."", '1309.7626-2-5-2': 'As stated in the Global Energy Assessment , ""A fundamental assumption underlying the pathways is that the coordination required to reach the multiple objectives simultaneously can be achieved"".', '1309.7626-2-5-3': 'While this approach generates a useful simplification to a complex system, it may be argued that such a spontaneous emergence of coordination is somewhat unlikely.', '1309.7626-2-5-4': 'Liberalised energy markets involve actors free to take their investment and consumption decisions based on their particular circumstances, and are only incentivised by policy.', '1309.7626-2-5-5': 'Thus while optimisation frameworks are valuable for identifying cost-effective pathways, they do not suggest how exactly to achieve them, because they do not model decision making.', '1309.7626-2-5-6': 'Strong coordination is difficult to generate from economic policy instruments, leading to technology lock-ins.', '1309.7626-2-6-0': 'Meanwhile, equilibrium economic models impose by construction that climate change mitigation costs are borne strictly at the expense of consumption or investment elsewhere, an assumption disputed in the economic literature , providing little insight on macroeconomic dynamics of mitigation action (e.g. investment, employment), and to which we put forward a striking counterexample .', '1309.7626-2-6-1': 'The assumptions associated with equilibrium economics lead to emphasising carbon pricing as a single policy tool able to fix the climate market failure, even though it is well known to industrial policy-makers that new technologies often cannot successfully diffuse into the marketplace without further government support.', '1309.7626-2-6-2': 'Projecting future greenhouse gas emissions using normative models of energy systems with equilibrium economic models is conceptually inconsistent with the descriptive approach of climate science, misleading if this is done to project environmental impacts as outcomes of energy policy.', '1309.7626-2-7-0': 'A descriptive modelling approach requires using known technology dynamics .', '1309.7626-2-7-1': 'Empirical dynamics are known to exist in scaling up technology systems (e.g. [MATH]-shaped diffusion), and their costs (learning curves), which have been extensively studied for decades .', '1309.7626-2-7-2': 'Mathematical generalisations have been suggested involving dynamic differential equations similar to those in population growth mathematical ecology and demography .', '1309.7626-2-7-3': 'In combination with evolutionary dynamics and evolutionary game theory , this offers a artillery of powerful concepts at core of evolutionary economics [CITATION] which emphasises innovation, diffusion and speciation as a source of economic development and growth .', '1309.7626-2-7-4': ""These are broadly consistent with the 'multi-level perspective' on technology transitions described by [CITATION]."", '1309.7626-2-7-5': 'Furthermore, complexity and path dependence emerge as key concepts to visualise technology dynamics .', '1309.7626-2-7-6': 'Even though [MATH]-shaped curves have been famously shown to arise in energy consumption , such principles have yet to be introduced in energy systems modelling, which would generate a much improved methodological paradigm .', '1309.7626-2-7-7': 'In such a framework, the use of optimisation as a source of dynamic force and actor behaviour would be replaced by empirically known innovation-selection-diffusion dynamics.', '1309.7626-2-8-0': 'Moving beyond cost-optimisation and equilibrium theory, this work proposes such a paradigm along with a non-equilibrium approach to energy-economy-environment interactions.', '1309.7626-2-8-1': 'We present an analysis of the global electricity sector using coupled descriptive simulations of technology diffusion dynamics, using FTT:Power, non-equilibrium economics using E3MG, and environmental impacts by combining emulators of the PLASIM-ENTS climate model (PLASIM-ENTSem) and of the GENIE-1 carbon cycle model (GENIEem) .', '1309.7626-2-8-2': 'It enables to explore the outcomes of particular energy policy tools on technology diffusion, electricity generation, global emissions, climate change and macroeconomic change.', '1309.7626-2-8-3': 'Uncoordinated implicitly stochastic dynamics are modelled at the level of profit-seeking investor decisions under bounded rationality , of which the sum does not produce an overall cost optimum, as they disregard in general the effect of these decisions on the wider system or the common good.', '1309.7626-2-8-4': 'The connection to a diffusion framework to a non-equilibrium model of the global economy opens a very rich world of macroeconomic dynamics of technological change where the impacts of energy policy reveal complex interactions between the energy sector and the economy.', '1309.7626-2-9-0': 'Ten scenarios of the future global power sector up to 2050 are presented, creating a storyline to provide insight for the construction of effective comprehensive policy portfolios.', '1309.7626-2-9-1': 'Going beyond simple carbon pricing and considering policies that could help trigger the diffusion of new technologies, particular combinations of energy policy instruments are found to provide suitable environments to enable fast electricity sector decarbonisation.', '1309.7626-2-9-2': 'Macroeconomic dynamics associated with these scenarios, explored separately by [CITATION], are summarised.', '1309.7626-2-9-3': 'High resolution data in these scenarios are accessible on our website.', '1309.7626-2-9-4': 'Decarbonisation involves the positive externality associated to the global accumulation of knowledge and experience in scaling up, deploying and using new power technologies.', '1309.7626-2-9-5': 'A classic collective action problem emerges: learning cost reductions for new technologies may only become significant and enable cost-effective diffusion when most nations of the World demonstrate strong coordinated dedication to their deployment.', '1309.7626-2-9-6': 'We show that carbon pricing covering all world regions is a necessary but insufficient component for the success of mitigation action in order to break the current fossil fuel technology lock-in.', '1309.7626-2-9-7': 'As we show in a separate paper however , in this perspective, reductions in power sector emissions by nearly 90% do not generate significant macroeconomic costs (direct and indirect) but instead, generates additional industrial activity and employment.', '1309.7626-2-10-0': '# The model: FTT:Power-E3MG', '1309.7626-2-11-0': '## Technology diffusion in FTT:Power', '1309.7626-2-12-0': 'Emissions reductions in the energy sector can occur through technology substitution, between technologies that produce the same substitutable service (e.g. electricity, heat, etc), through behaviour and practice changes and through reductions in the consumption of that service altogether.', '1309.7626-2-12-1': 'Technology, however, comes to life through investments made at specific points in time with an expectation by investors to receive an income or service of which the present value justifies or exceeds the initial investments and the present value of present and future costs [CITATION].', '1309.7626-2-12-2': 'In this context, early scrapping of usable capital produces stranded assets that do not produce the expected income, generating losses and requiring new investments earlier than expected.', '1309.7626-2-13-0': 'Where early scrapping is avoided, technological change occurs primarily at the average rate of replacement of existing technology as it ages, which is inversely related to its life span.', '1309.7626-2-13-1': 'However, notwithstanding lifetime considerations, the number of new units of technology of a particular type that can be constructed at any one time can be larger if that industry is in a well established position in the marketplace, with a large production capacity, than if it is emerging.', '1309.7626-2-13-2': 'Thus, even when an emerging technology is very affordable (for instance with strong policy support such as subsidies), it may not necessarily be accessible in every situation where investors are required to make a choice between available options.', '1309.7626-2-13-3': 'Thus in the analysis of the diffusion capacity of technologies and the pace of technological change, not only cost considerations come into play, but also a limited access to technology and information, and these two principles, described in earlier work , form the core of FTT:Power (eq. [REF]).', '1309.7626-2-14-0': 'This decision-making process can be expressed with pairwise comparisons of options.', '1309.7626-2-14-1': 'However, investors do not all face similar situations and do not weigh different aspects in the same way, which cannot possibly be enumerated specifically in a model.', '1309.7626-2-14-2': 'However, when dealing with large numbers of instances of decision making, there will be majority trends in investor choices, who may be assumed, if all relevant considerations are quantified into costs, to be seeking cost minimisation with the goal of profit maximisation for their own respective firms.', '1309.7626-2-14-3': 'Probability distributions may thus be used in order to avoid enumerating the details of all situations faced by investors, and where a particular technology is on average more profitable to use than a second one, there usually exist specific situations where the reverse turns out to be true.', '1309.7626-2-15-0': 'Thus the statistical trend of investor preferences may be expressed as a matrix [MATH] expressing the probability of investor choice between technologies [MATH] and [MATH].', '1309.7626-2-15-1': 'However, constraints of the power system prevent some types of technology to dominate.', '1309.7626-2-15-2': 'These can be expressed with a second matrix [MATH], stochastically stopping investments that lead to stranded or unused assets due to technical problems .', '1309.7626-2-16-0': 'Using the variable [MATH] for the generation capacity market share of a technology, the rate at which shares of one technology ([MATH]) can be replaced by shares of another ([MATH]) is proportional to:', '1309.7626-2-17-0': 'The rate at which units of technology [MATH] come to the end of their working life, with lifetime [MATH], How many old units of [MATH] require replacement, a fraction [MATH] of the total number of replacements.', '1309.7626-2-17-1': 'The rate at which the construction capacity for technology [MATH] can be expanded, with time constant [MATH], The market position of technology [MATH], its share of the market [MATH].', '1309.7626-2-18-0': 'The resulting market dynamics are flows of market shares between existing options, according to choices and constraints [MATH] and [MATH], and is expressed as [EQUATION] using the time scaling matrix [MATH] to express changeover time constants for every pair of technologies.', '1309.7626-2-18-1': 'Adding up substitution flows in both directions between all possible pairs of technologies leads to a family of finite difference equations that tracks the evolution of technological change in a set of substitutable technologies: [EQUATION] with [MATH] expressing investor preferences and [MATH] providing constraints (see also fig. [REF]).', '1309.7626-2-18-2': '[MATH] is the average rate of change weighed by the market shares, the scaling of time.', '1309.7626-2-18-3': 'This equation, logistic in the special case of two technologies, is non-linear and generates slow uptakes at small share values, then fast diffusion at intermediate penetration, before a saturation near full penetration.', '1309.7626-2-18-4': 'This system cannot be solved analytically but is straightforward to evaluate numerically using time steps.', '1309.7626-2-18-5': ""It corresponds to the replicator dynamics equation in evolutionary game theory , used to explain Darwin's Theory of Evolution, as well as in evolutionary economics ."", '1309.7626-2-19-0': '## Learning-by-doing and path dependence', '1309.7626-2-20-0': 'Profit-seeking investor choices summarised in [MATH] are driven by cost differences, and these change over time as technologies diffuse in the marketplace and follow their learning curves .', '1309.7626-2-20-1': 'Learning-by-doing cost reductions stem from the accumulation of technical knowledge on production and economies of scale in expansion of productive capacity.', '1309.7626-2-20-2': 'Technology costs are taken here to apply everywhere globally, so that the benefits of learning are global, in a perspective where firms selling technologies operate in a global market.', '1309.7626-2-20-3': 'Emerging technologies have fast cost reductions (e.g. solar panels) while established systems see very little change (e.g. coal systems).', '1309.7626-2-20-4': 'The crucial aspect is that cost reductions are decreasing functions of cumulative investment, not time, and that they do not occur if no investment is made.', '1309.7626-2-20-5': 'Learning thus interacts with diffusion where it incentivises further uptake, which generates further learning and so on, a highly non-linear effect which can lead to sudden technology avalanches.', '1309.7626-2-20-6': ""Policy can generate this, termed 'bridging the techchnology valley of death' ."", '1309.7626-2-20-7': 'Learning curves are connected here only with investor choice probabilities [MATH].', '1309.7626-2-21-0': 'Learning-by-doing introduces the crucial property of path dependence.', '1309.7626-2-21-1': 'As technologies gradually diffuse in the marketplace following investor choices, the full landscape of technology costs continuously changes, and investor preferences thus change.', '1309.7626-2-21-2': 'These changes are permanent and determined completely by past investments, and therefore by the full history of the market.', '1309.7626-2-21-3': 'Thus, from any starting point, future cost landscapes are determined by investment trends at every point in time, and different futures exist, depending on investment and policy choices along the way.', '1309.7626-2-22-0': '## Natural resource use', '1309.7626-2-23-0': 'The diffusion of power systems can only occur in areas where resources are available, for instance windy areas for wind power, or natural water basins and rivers for hydroelectric systems.', '1309.7626-2-23-1': 'Higher productivity sites offer lower costs of electricity production, and tend to be chosen first by profit-seeking investors.', '1309.7626-2-23-2': 'Assuming this, the progression of the development of renewable energy systems produces a tendency towards increasing marginal costs of production for potential new systems as only resources of lower and lower productivity are left to use.', '1309.7626-2-23-3': 'This is well described by functions of increasing marginal costs of production with increasing development, cost-supply curves .', '1309.7626-2-23-4': 'For this purpose, a complete set of such curves was previously estimated from literature data for 190 countries of the world for 13 types of natural resources .', '1309.7626-2-23-5': 'These were aggregated for the 21 regions of E3MG.', '1309.7626-2-23-6': 'Excluding non-renewable resources treated globally, this produced [MATH] cost-supply curves that are used to constrain the expansion of renewable systems in FTT:Power.', '1309.7626-2-23-7': 'The consumption of non-renewable resources is however better represented using dynamic cost-quantity curves, described next.', '1309.7626-2-24-0': '## Fossil fuel price dynamics', '1309.7626-2-25-0': 'Non renewable resources lying in geological formations have an arbitrary value that depends on their cost of extraction, but also on the dynamics of the market.', '1309.7626-2-25-1': 'To their cost of extraction is associated a minimum value that the price of the commodity must take in order for the extraction to be profitable.', '1309.7626-2-25-2': 'These costs are however distributed over a wide range depending on the nature of the geology (e.g. tar sands, ultra-deep offshore, shale oil and gas, etc).', '1309.7626-2-25-3': 'Thus, given a certain demand for the commodity, the price is a function of the extraction cost of the most expensive resource extracted in order to supply the demand, which separates reserves from resources.', '1309.7626-2-25-4': 'As reserves are gradually consumed, this marginal cost increases generating a commodity price increase which unlocks the exploitation of resources situated in locations with higher extraction costs.', '1309.7626-2-25-5': 'For example, tar sands became economical and saw massive expansion above a threshold price of around 85-95/boe.', '1309.7626-2-25-6': 'Thus, to any commodity demand path in time will correspond a path dependent commodity price.', '1309.7626-2-25-7': 'This model is described with an analysis in [CITATION], relying on data from [CITATION].', '1309.7626-2-25-8': 'In FTT:Power, this model is used to determine fuel costs for fossil fuel and nuclear based power technologies.', '1309.7626-2-26-0': '## Timescales of technological change', '1309.7626-2-27-0': 'Innovation generates new technologies that live in niches that protect them from the wider market.', '1309.7626-2-27-1': 'From those niches, in appropriate changes of market conditions, can emerge and diffuse new socio-technical regimes .', '1309.7626-2-27-2': ""After the innovation phase, at the level of diffusion, technologies enter what we termed the 'demographic phase' ."", '1309.7626-2-27-3': 'In scenarios excluding early scrapping, possible rates of change are dictated by the ageing and decommissioning rate of of capital and infrastructure, in other words, the inverse of the life expectancy of the existing capital.', '1309.7626-2-27-4': 'Meanwhile, relative rates of expansion of the productive capacity for new technology determine which technologies can fill in gaps more effectively than others, when other technology is decommissioned.', '1309.7626-2-27-5': 'In many cases of interest, for instance with new technologies, these timescales cannot be reliably measured using small amounts of diffusion data of non-linear nature.', '1309.7626-2-27-6': 'Therefore a theory of diffusion is necessary in order to infer these parameters from technology and industry properties.', '1309.7626-2-27-7': 'This theory is derived and explained in [CITATION].', '1309.7626-2-28-0': '## Modelling the global economy: E3MG', '1309.7626-2-29-0': 'E3MG is an macroeconometric model of the global economy that has been used widely for climate change mitigation economics [CITATION].', '1309.7626-2-29-1': 'It evaluates the parameters of 33 econometric equations onto data between 1971 and 2010, and extrapolates these equations between 2010 and 2050.', '1309.7626-2-29-2': 'The model has a very high resolution: its equations are evaluated for 21 regions of the world, 43 industrial sectors, 28 sectors of consumption, 22 fuel users, and 12 fuel types.', '1309.7626-2-29-3': 'Sectors are interrelated with dynamic input-output tables.', '1309.7626-2-29-4': 'The model incorporates endogenous technological change using accumulated investment and RD spending, particularly in the energy sector as detailed next.', '1309.7626-2-30-0': '## Electricity price-demand interactions', '1309.7626-2-31-0': 'The demand for electricity depends on its price, and it is well known that in situations of high electricity prices, people may strive to find more effective ways to use their income, preferring to invest in more efficient technology, perceived as a worthwhile tradeoff, or to simply reduce their consumption.', '1309.7626-2-31-1': 'When the electricity supply technology mix changes, the minimum price at which electricity can be profitably sold also changes, and with such price changes, the demand for electricity changes.', '1309.7626-2-31-2': 'In particular, when carbon pricing or feed-in tariffs are used to ensure access of expensive renewables into the grid, the price of electricity increases, bearing an externality to all users , affecting consumer demand and behaviour.', '1309.7626-2-31-3': 'Thus, reductions in emissions originate from both a change in the carbon intensity of the power sector and changes in the demand for electricity.', '1309.7626-2-31-4': 'These aspects of energy economics are prominent in this work, responsible for a significant fraction of our projected emissions reductions in scenarios of climate policy.', '1309.7626-2-32-0': 'Electricity demand is modelled in E3MG, using an econometric equation that incorporates endogenous technological change by including accumulated investment and RD expenditures of the form [EQUATION] where for fuel [MATH] and region [MATH], [MATH] is the fuel demand, [MATH] represents output, [MATH] relative prices and [MATH] is the technological progress indicator .', '1309.7626-2-32-1': 'Including accumulated investment and RD makes this equation non-linear, path-dependent and hysteretic.', '1309.7626-2-32-2': 'In particular, prices hikes lead to enhanced energy-saving RD that permanently increases efficiency of fuel use, which would not have occurred without price changes.', '1309.7626-2-32-3': 'Economic feedbacks between FTT:Power and E3MG occurs with four quantities: fuel prices, fuel use, power technology investments and tax revenue recycling .', '1309.7626-2-33-0': '## Climate modelling using PLASIM-ENTSem', '1309.7626-2-34-0': 'The carbon cycle is represented in this work by an emulator of the Grid Enabled Integrated Earth systems model (GENIE-1) Earth System Model .', '1309.7626-2-34-1': 'The carbon cycle emulator (GENIEem) is designed to be a more sophisticated, but computationally very fast, alternative to simplified climate-carbon cycles in integrated assessment models.', '1309.7626-2-34-2': 'GENIE-1 simulates climate, ocean circulation and sea-ice, together with the terrestrial, oceanic, weathering and sedimentary components of the carbon cycle.', '1309.7626-2-34-3': 'The emulator takes as inputs a time series of anthropogenic carbon emissions and non-CO[MATH] radiative forcing, and outputs a time series of atmospheric CO[MATH] concentration.', '1309.7626-2-34-4': 'Uncertainty in the carbon cycle is captured by varying GENIE-1 parameter inputs, resulting in an emulator ensemble of 86 possible futures.', '1309.7626-2-35-0': 'PLASIM-ENTSem is an emulator of the PLAnet SIMulator - Efficient Numerical Terrestrial Scheme (PLASIM-ENTS) global climate model (GCM).', '1309.7626-2-35-1': 'In the GCM, PLASIM, which includes a Q-flux ocean and a mixed-layer of a given depth , is coupled to a simple surface and vegetation model, ENTS, which represents vegetation and soil carbon through a single plant functional type .', '1309.7626-2-35-2': 'The design of PLASIM-ENTSem is described in detail by [CITATION].', '1309.7626-2-35-3': 'As for GENIEem, uncertainty is captured by varying PLASIM-ENTS parameter inputs, resulting in a 188-member ensemble of decadally and spatially seasonal climate variables.', '1309.7626-2-36-0': 'The combination of both emulators in the context of this work, with combined uncertainty analysis, is given by [CITATION].', '1309.7626-2-36-1': 'The emissions profiles associated with the scenarios produced by FTT:Power-E3MG are used to provide inputs to GENIEem, which in turn provides inputs to PLASIM-ENTS.', '1309.7626-2-36-2': 'The median, 5th-95th percentiles of the GENIEem ensemble output are each used to drive climate simulations using PLASIM-ENTSem, so that uncertainty associated with modelling the carbon cycle is represented in the climate projections.', '1309.7626-2-36-3': 'Based on the median GENIEem ensemble output, the median, 5th-95th percentiles of warming from the PLASIM-ENTSem ensemble are also calculated; these bounds, therefore, reflect warming uncertainty due to parametric uncertainty in the climate model alone.', '1309.7626-2-36-4': 'The 5th percentile of warming from the PLASIM-ENTSem ensemble based on the 5th percentile of CO2 concentration from the GENIEem ensemble, and the 95th percentile of warming from the PLASIM-ENTSem ensemble based on the 95th percentile of CO2 concentration from the GENIEem ensemble are also calculated; this second set of bounds, therefore, reflects warming uncertainty due to parametric uncertainty in the climate model and the carbon cycle model.', '1309.7626-2-37-0': '# Projections to 2050 of the global power sector', '1309.7626-2-38-0': '## Scenario creation and policy instruments', '1309.7626-2-39-0': 'Ten scenarios of electricity policy assumptions of different types and resulting technology mix and emissions up to year 2050 were created with FTT:Power-E3MG, lettered a. to j. (fig. [REF]).', '1309.7626-2-39-1': 'These all lead to different futures for the global power sector and different CO[MATH] emission profiles.', '1309.7626-2-39-2': 'It is impractical to reproduce all the information of these simulations in this paper, and therefore a summary of the results is given here, the details having been made available on the 4CMR website, where they can be displayed in terms of the full resolution of 21 world regions and 24 power technologies, for policy assumptions, electricity generation, emissions and levelised costs.', '1309.7626-2-39-3': 'Four energy policy tools were explored: carbon pricing (CO[MATH]P), technology subsidies (TSs), feed-in tariffs (FiTs) and direct regulations (REGs).', '1309.7626-2-39-4': 'Individual tools and various combinations were explored, a summary given in figure [REF].', '1309.7626-2-39-5': 'By gradually elaborating various policy frameworks, a scenario was found where power sector emissions are reduced by 89% below the 1990 level, involving all four policy instruments used simultaneously.', '1309.7626-2-39-6': 'These are fed to the carbon cycle and climate model emulators GENIEem and PLASIM-ENTSem in order to determine the resulting atmospheric CO[MATH] concentration and average global warming, for these scenarios where other sectors are not targeted by climate policy.', '1309.7626-2-40-0': 'The nature of FiTs here is that access to the grid at a competitive price is ensured (a price higher than the consumer price), the difference being paid by the grid and passed on to consumers through the price of electricity.', '1309.7626-2-40-1': 'The consumer price of electricity is raised by just the amount that makes this economically viable.', '1309.7626-2-40-2': 'The consumer price in FTT:Power-E3MG is derived from the generation share-weighted average Levelised Cost of Electricity (LCOE) ([MATH]).', '1309.7626-2-40-3': ""The LCOE as perceived by investors when a FIT exists includes an 'effective subsidy' given by the grid that covers a fraction or the whole difference between the levelised cost of these technologies and the consumer price of electricity (investors here may be corporate or, for instance, homeowners)."", '1309.7626-2-40-4': 'In the case of CO[MATH]P, the LCOE calculation that investors are assumed to perform includes possible carbon cost components, and the price of carbon is also passed on to consumers through the price of electricity.', '1309.7626-2-40-5': 'Thus the price of electricity increases with the carbon price unless emitting technologies are phased out.', '1309.7626-2-40-6': 'TSs are fractions of the capital costs of low carbon technologies that are paid by the government, reducing the LCOE that investors see.', '1309.7626-2-40-7': 'These are defined exogenously for every year up to 2050 and were designed to be phased out before then, after which it is hoped that the technology cost landscape becomes permanently altered and that technologies are not indefinitely subsidised.', '1309.7626-2-40-8': 'REGs indicate regulations controlling the construction of additional units of particular technologies, used to phase out particular types of systems.', '1309.7626-2-40-9': 'When a REG is applied to a technology category, no new units are built but existing ones are left to operate until the end of their lifetime.', '1309.7626-2-41-0': 'Fig. [REF] summarises the result of the policy tools exploration.', '1309.7626-2-41-1': 'Electricity generation by technology type is given in the series of panels to the left of each pair, while emissions are given on the right.', '1309.7626-2-41-2': 'The vertical dashed lines indicate the start of the simulations in 2008, and data to the left of this line is historical data from the [CITATION].', '1309.7626-2-41-3': 'The horizontal dashed lines indicate the 1990 levels of electricity demand and emissions.', '1309.7626-2-41-4': 'Dashed curves correspond to the baseline values for comparison.', '1309.7626-2-41-5': 'In all scenarios excluding the baseline, policy schemes generate both a reduction of electricity consumption and emissions.', '1309.7626-2-41-6': 'Consumption reduces due to increases in the price of electricity, through the energy demand econometric equation of E3MG, which contributes significantly to emissions reductions.', '1309.7626-2-41-7': 'All additional emissions reductions are due to changes in fuels used, or, in other words, to changes of technologies.', '1309.7626-2-41-8': 'Emissions reductions in 2050 with respect to the 1990 level are given in percent values.', '1309.7626-2-42-0': '## Climate policy for achieving 90 reductions in power sector emissions', '1309.7626-2-43-0': 'The baseline scenario (fig. [REF] panel a.), which involves maintaining current policies until 2050 (carbon pricing in Europe only), generates global power sector emissions in 2050 of 31 GtCO[MATH]/y, 343 above the 1990 level, and total emissions of 68 GtCO[MATH]/y. Cumulative emissions for the time span 2000-2050 amount to 2350 GtCO[MATH].', '1309.7626-2-43-1': 'This pathway is likely to commit the planet to a warming that exceeds 4[MATH]C above pre-industrial levels in around 2100 (fig. [REF] below), consistent with [CITATION] .', '1309.7626-2-43-2': 'In view of finding ways to reduce these emissions and to limit global warming, we searched areas of policy space that reduce these significantly in the short time span, given in fig. [REF].', '1309.7626-2-44-0': 'The first option explored (panel b.) was to use REGs to prevent the construction of new coal power plants worldwide, the systems with highest emissions ([MATH]1 ktCO[MATH]/MWh), unless they are equipped with carbon capture and storage (CCS).', '1309.7626-2-44-1': 'This results mostly in a transfer from a coal lock-in to a gas lock-in, reducing global emissions approximately to the 1990 level, largely insufficient meeting the 2[MATH]C target.', '1309.7626-2-45-0': 'The second option was to use CO[MATH]P as a unique tool, with different price values for different regions covering all world regions shown in fig. [REF], between 100 and 200 2008/tCO[MATH] in 2050 (panel c.) and between 200 and 400 2008/tCO[MATH] (panel d.).', '1309.7626-2-45-1': 'This reduces emissions to around 71% above 1990 and to the 1990 level, respectively.', '1309.7626-2-45-2': 'This modest impact suggests that CO[MATH]P on its own requires very high carbon prices in order to generate significant reductions, or that it is simply insufficient.', '1309.7626-2-45-3': 'However as we show now, combinations of policies achieve this much more effectively.', '1309.7626-2-46-0': 'As a first combination of policies, FiTs (wind and solar) and TSs (all other low carbon technologies except wind and solar) were used without CO[MATH]P (panel e.), of order 30-50 of capital costs for TSs and feed-in prices 5-15 above the electricity price for FiTs (this depends on regions and technologies, for an example see the inset of fig. [REF] or the data on our website).', '1309.7626-2-46-1': 'This generates very modest uptakes of low carbon technologies and thus small emissions reductions, 314 above 1990.', '1309.7626-2-46-2': 'This is due to the very low cost of producing electricity using fossil fuels in comparison to all other technologies, in particular coal, and therefore CO[MATH]P is necessary in order to bridge this cost difference.', '1309.7626-2-47-0': 'Scenario [MATH] shows the use of CO[MATH]P up to 200/tCO[MATH] and feed-in tariffs, the latter generating very little change over scenario c. Using CO[MATH]Ps of up to 200/tCO[MATH] in combination with TSs and FiTs in all world regions (panel g.) yields emissions reductions to 14 above the 1990 level, still insufficient.', '1309.7626-2-47-1': 'With CO[MATH]P of up to 400/tCO[MATH] in combination with the same set of TSs and FiTs (panel h.), reductions are much larger, 57 below the 1990 level.', '1309.7626-2-47-2': 'This indicates how the impact of policy combinations is larger than the sum of the impacts of its components taken separately, offering significant potential synergies.', '1309.7626-2-48-0': 'A scenario was explored in which only the developed world applies the stringent energy policies of scenario h (panel i.), in which it is hoped that this generates enough investment to bring the costs of low carbon technologies into the mainstream, thus becoming accessible to developing or under-developed countries.', '1309.7626-2-48-1': 'We see no noticeable uptake of new technology in these countries, costs remaining unaccessible especially in comparison with coal based technologies, and as a consequence global emissions are only reduced to 185% above the 1990 level.', '1309.7626-2-49-0': 'A significant amount of the remaining emissions in scenario h reside in China, where the lock-in of coal technology is very difficult to break given the near absence of alternatives with the exception of hydroelectricity, which is driven to its natural resource limits.', '1309.7626-2-49-1': 'The choice of investors thus needs to be constrained at the expense of having to sell electricity at higher prices.', '1309.7626-2-49-2': 'Therefore REGs were introduced in scenario j in China that prevent the construction of new coal power stations unless they are equipped with CCS.', '1309.7626-2-49-3': 'This additional policy forces additional diversity in the Chinese technology mix, bringing down global emissions to 89% below the 1990 level without early scrapping.', '1309.7626-2-49-4': 'Total cumulative emissions for the time period 2000-2050 in scenario j (in the baseline) are of 1670 Gt (2350 Gt), given that other sectors do not change their technologies significantly, of which 390 Gt (930 Gt) originate from the electricity sector alone.', '1309.7626-2-50-0': '## Climate change projections', '1309.7626-2-51-0': 'Global emissions from all sectors in scenarios [MATH] to [MATH] were fed into the GENIEem in order to calculate the resulting CO[MATH] concentrations, which themselves were supplied to PLASIM-ENTSem in order to explore their climate change impacts.', '1309.7626-2-51-1': 'Figure [REF], top panel, displays global CO[MATH] emissions for all scenarios from E3MG including all fuel combustion emissions from all users as well as exogenous trends of emissions from all other sectors, obtained from the EDGAR database.', '1309.7626-2-51-2': 'While the changes observed include those in power sector emissions of fig. [REF], they also include modest changes in other sectors (e.g. industry) occurring due to CO[MATH]P for all fuel users subject to the emissions trading scheme (taken to function similarly to the current EU-ETS).', '1309.7626-2-52-0': 'In order to run the climate model emulator, emissions were required up to 2100.', '1309.7626-2-52-1': 'E3MG projections beyond 2050 are fairly uncertain and were not used, and the trends to 2050 were extrapolated to 2100.', '1309.7626-2-52-2': 'Polynomial extrapolations were assumed for scenarios a,e,i while emissions were considered stable beyond 2050 for all other scenarios.', '1309.7626-2-52-3': 'While this is not a particularly sophisticated extrapolation method, accuracy is not the primary goal here since the decarbonisation of other sectors of anthropogenic emissions, responsible for most emissions left in scenario j. of order 30 Gt, was not specifically modelled, but represent significant potentials for further reductions (i.e. low carbon technology diffusion in transport and industry).', '1309.7626-2-53-0': 'The middle panel of fig. [REF] shows the resulting atmospheric CO[MATH] concentrations, with uncertainty shown as a blue area.', '1309.7626-2-53-1': 'It was observed that scenario a reaches median values of [MATH] ppm while scenario j reaches [MATH] ppm, uncertainty due to the carbon cycle model.', '1309.7626-2-53-2': 'This is above the generally assumed threshold of 450 ppm for maintaining warming below 2[MATH]C.', '1309.7626-2-53-3': 'These concentrations were fed to the climate model emulator, which yielded global warming median temperature changes of (fig. [REF]) [MATH]C over pre-industrial levels when using the median concentration and only the climate model uncertainty, and [MATH]C when including both carbon cycle and climate model uncertainties.', '1309.7626-2-53-4': 'This therefore could exceed [MATH]C of warming with a 5% chance.', '1309.7626-2-53-5': 'Meanwhile, the electricity decarbonisation scenario yields warming values of [MATH]C with carbon cycle uncertainty only and [MATH]C with both carbon cycle and climate model uncertainties, with a very small probability of not exceeding [MATH]C of warming.', '1309.7626-2-53-6': 'This indicates that the decarbonisation of the power sector by 89% is insufficient if other sectors such as transport and industry are not specifically targeted by climate policy.', '1309.7626-2-54-0': '## Learning cost reductions and energy price dynamics', '1309.7626-2-55-0': 'The uptake of low carbon technologies generate learning cost reductions that alter permanently the technology cost landscape.', '1309.7626-2-55-1': 'Figure [REF] shows world averages of bare technology costs (upper panels) for the baseline and mitigation scenario j, weighed by electricity generation, excluding TSs, CO[MATH]P and FiTs.', '1309.7626-2-55-2': 'These values, when including policy, drive investor choices in both the baseline (left) and the mitigation (right) scenarios.', '1309.7626-2-55-3': 'Roughly speaking, decreases stem from learning-by-doing cost reductions while increases originate from increasing natural resource scarcity with development.', '1309.7626-2-55-4': 'While the cost of PV panels decreases in the baseline scenario mainly due to deployment in Europe, it decreases by more than half its 2008 value in the mitigation scenario where they benefit from FiTs everywhere.', '1309.7626-2-55-5': 'Meanwhile, onshore wind power does come into the mainstream in many regions of the world in the mitigation scenario and does not necessitate support all the way to 2050, where the value of the wind FiTs become near zero or even negative, in which case the policy it is dropped altogether.', '1309.7626-2-55-6': 'In other regions, wind power is limited by resource constrained decreasing capacity factors and corresponding increasing costs.', '1309.7626-2-55-7': 'Other technologies, such as geothermal or wave power (not shown), see very little uptake in this particular mitigation scenario and therefore little cost reductions.', '1309.7626-2-56-0': 'The marginal costs of producing electricity, defined as share-weighted LCOEs, are given for 6 aggregate regions in the lower panels of fig. [REF].', '1309.7626-2-56-1': 'This marginal cost is used in E3MG to construct electricity prices in 21 regions, of which the changes alter electricity consumption .', '1309.7626-2-56-2': 'These are different between regions, stemming from different technology and resource landscapes, where lower marginal costs correspond to higher shares of coal based electricity.', '1309.7626-2-57-0': 'The marginal costs of fossil fuels are calculated using estimates of reserves and resources, described in section [REF], and are not highly affected by changes in policy in these scenarios.', '1309.7626-2-57-1': 'In both scenarios oil and gas costs increase significantly up to 2050 in a similar way, but these increases are dampened by the massive accession to unconventional fossil fuel resources (oil sands, heavy oil and shale gas).', '1309.7626-2-57-2': 'This analysis will be given elsewhere in more detail.', '1309.7626-2-57-3': 'Coal costs are moderately affected by changes in demand due large coal resources.', '1309.7626-2-57-4': 'The price of natural uranium ore is stable until 2035 where an increase is observed, generated by increasing scarcity, and at this level of consumption, U resources are projected to run out before 2100 unless technology changes .', '1309.7626-2-58-0': '## Global economic impacts of an 89% reduction scenario', '1309.7626-2-59-0': 'The macroeconomic impacts of scenarios a and j are explored separately in [CITATION].', '1309.7626-2-59-1': 'It is found there that decarbonising the electricity sector by 89% has economic benefits, generating additional employment, real household income and increases GDP by between 1 and 3% (depending on the regions) in comparison to scenario a.', '1309.7626-2-59-2': 'This is due to two opposing forces acting against one another: the introduction of low carbon technologies force increases in electricity prices (as seen in fig. [REF]), lowering real household disposable income, while low carbon technology production generates further employment in various industrial sectors, increasing household income.', '1309.7626-2-59-3': 'These are shown to approximately cancel, which is possible as long as labour and capital (investment) resources can be made available .', '1309.7626-2-59-4': 'Our assumptions about capital and labour markets are therefore consistent with our assumptions of energy markets, in that they do not automatically produce optimal outcomes.', '1309.7626-2-60-0': 'However, CO[MATH]P generates government income larger than government spending on TSs, the rest being redistributed to households in the form of income tax reductions, increasing further their disposable income.', '1309.7626-2-60-1': 'The resulting impacts are therefore of increased household income and consumption in comparison to the baseline and thus higher GDP.', '1309.7626-2-60-2': 'It is to be noted however that there are winners and losers in this picture both in terms of sectors and world regions, depending how much they depend on activities of the oil, gas and coal sectors.', '1309.7626-2-61-0': '## Local projections of power generation and emissions', '1309.7626-2-62-0': 'It also proves instructive to analyse electricity technology landscapes in individual regions of the world in FTT:Power-E3MG, for policy analysis and for better understanding the nature of technology lock-ins and the restricted local ability to change in a diffusion perspective.', '1309.7626-2-62-1': 'This is presented in fig. [REF] for six key regions or countries: North America, Europe, China, India, Brazil and the Rest of the World, which have different electricity landscapes stemming from differing energy policy strategies and engineering traditions historically, as well as natural resource landscapes.', '1309.7626-2-62-2': ""National strategies, reflecting local engineering specialisation related to technology lock-ins, is a natural outcome of this model's structure (eq. [REF]), which reproduces the better ability of dominating industries to capture the market despite costs."", '1309.7626-2-63-0': 'Renewable energy systems are more exploited in Europe than anywhere else in the world, except in Brazil, where hydroelectricity dominates.', '1309.7626-2-63-1': 'Europe also sees the most diverse electricity sector, with large amounts of wind power already in the baseline scenario, predominantly in northern Europe and the British Isles, large amounts of nuclear power in France, and some solar power in Germany.', '1309.7626-2-63-2': 'Coal fired electricity is mostly phased out before 2050 in the 90% scenario, generating significant emissions reductions.', '1309.7626-2-64-0': 'North America features higher use of fossil fuels for power production than Europe.', '1309.7626-2-64-1': 'However, while E3MG projects a larger potential for consumption reductions, large opportunities for diversification also emerge with significant potentials of renewable energy.', '1309.7626-2-64-2': 'Bioenergy with CCS generates a large contribution to emissions reductions.', '1309.7626-2-65-0': 'China and India have very low technology diversity and important fossil fuel lock-ins.', '1309.7626-2-65-1': 'The amount of coal used in China in the baseline is responsible for emissions of 10 out of 30 Gt of global emissions in 2050.', '1309.7626-2-65-2': 'Diversification proves difficult given the scale of the rate of increase in consumption; breaking the coal lock-in requires REGs in China to phase out building new coal generators.', '1309.7626-2-65-3': 'Large scale diffusion of renewables is slow and retrofitting CCS to coal generators offers a useful alternative.', '1309.7626-2-65-4': 'Electricity demand reductions are very large, which requires further investigations for fuel poverty and further social implications.', '1309.7626-2-66-0': 'In Brazil, even though hydroelectricity is not the least expensive resource, it nevertheless dominates, another form of technology lock-in.', '1309.7626-2-66-1': 'This is typical of a national engineering tradition dominated by a technology for decades.', '1309.7626-2-66-2': 'Brazil is projected to persist developing its hydropower capacity despite higher costs and a decreasing potential, until the cost becomes prohibitively expensive and only less productive sites remain.', '1309.7626-2-67-0': 'The rest of the world includes predominantly countries where the diversity of existing technologies is low, and persists in this direction.', '1309.7626-2-67-1': 'It features large amounts of oil use for electricity despite high oil prices, due to low technology availability or fossil fuel subsidies, which are not successfully phased out despite being the least cost-effective way of producing electricity.', '1309.7626-2-67-2': 'Coal based electricity makes the dominant contribution to emissions in the baseline, the rest divided between oil and gas fired power stations, for a total of 12 out of 30 Gt of global emissions in 2050 in the baseline.', '1309.7626-2-67-3': 'In the mitigation scenario, a significant additional hydroelectricity potential is developed, and coal is replaced by gas turbines, which are eventually retrofitted with CCS.', '1309.7626-2-68-0': '# Policy and strategic consequences', '1309.7626-2-69-0': '## Synergy between policy instruments', '1309.7626-2-70-0': 'This paper shows that in a coupled energy-economy-environment model that does not assume economic equilibrium or technology cost-optimisation, the impact of policy instruments can be different if used individually or in combinations: the impact of combined policy packages does not correspond to the sum of the impacts of individual instruments.', '1309.7626-2-70-1': 'Thus significant synergies between policy instruments.', '1309.7626-2-70-2': 'In this regard we showed that CO[MATH] alone is not likely capable of delivering sufficient emissions reductions, in contrast to the position supported by many policy-makers and economists ; it requires to be combined with TSs, FiTs and REGs.', '1309.7626-2-70-3': 'Relying on CO[MATH]P alone even up to 400 2008/tCO[MATH] is likely to lead to a status quo in the technology mix while delivering expensive electricity to consumers.', '1309.7626-2-70-4': 'Similarly, TSs and FiTs on their own have little impact unless they are combined with sufficiently high CO[MATH]P.', '1309.7626-2-71-0': 'We furthermore show that elaborate combinations of policy instruments can produce such strong synergy that reductions of electricity sector emissions by 89% by 2050 (58% of cumulative power sector emissions) become possible without early scrapping of electricity generation captital.', '1309.7626-2-71-1': 'Such strong reductions could be complemented by additional reductions in other emissions intensive sectors with additional cross-sectoral synergies: transport, industry and buildings, warranting further work in this area.', '1309.7626-2-71-2': 'If early scrapping is allowed, these reductions could be achieved even faster, but would most likely involve higher costs, requiring further investigations.', '1309.7626-2-72-0': '## The effect of global knowledge spillovers on technology costs: individual vs global coordinated action', '1309.7626-2-73-0': 'Technology systems typically face a vicious cycle: established technologies thrive because they are established, and emerging technologies see barriers to their diffusion due to the lock-in of established technologies.', '1309.7626-2-73-1': 'This is the case unless an emerging technology is a radical improvement over the incumbent, or it benefits from sufficient external support.', '1309.7626-2-73-2': 'Emerging technologies require investment and sales in order to benefit from improvements and economies of scale: repetition, trial and error enables entrepreneurs to improve their products.', '1309.7626-2-73-3': 'They thus require a continuous flow of funds from sales or external investment in order to survive until their products take off on their own in the market.', '1309.7626-2-73-4': 'In the long run, these investments may or may not generate a return, and are thus risky.', '1309.7626-2-73-5': ""Without any investment to bridge the 'technology valley of death', however, they may become failed innovations."", '1309.7626-2-73-6': 'Competitive thresholds are set by incumbent technologies.', '1309.7626-2-74-0': 'Given known learning curves of power systems, a certain additional capacity of emerging technologies such as wind turbines and solar PV panels must be bought and sold in order to bring down their costs to a competitive level set by established technologies such as coal, gas or nuclear power stations.', '1309.7626-2-74-1': 'As we find using FTT:Power-E3MG, this additional capacity necessary to make their cost affordable is very large, and cannot be deployed by a single nation such as Germany or the UK, or even the whole of Europe, for the rest of the world to benefit, particularly true for solar PV.', '1309.7626-2-74-2': 'In contrast, we find that only a concerted global climate policy effort can bring down costs to manageable levels and bring new power technologies into the mainstream, opening very large renewable energy potentials such as that of solar energy.', '1309.7626-2-74-3': 'Such a concerted effort can significantly and permanently alter the global landscape of power technology costs and availability.', '1309.7626-2-74-4': 'We stress that, intellectual property law and border taxes allowing, all countries of the world can benefit from learning cost reductions that may originate from investments and sales occurring elsewhere.', '1309.7626-2-74-5': 'This problem therefore possesses the features of a classic free-rider and collective action problem, where international coordination is the only way by which these cost reductions can take place.', '1309.7626-2-74-6': ""Emerging or developed nations cannot simply 'wait' for climate policy in other nations to generate diffusion and enough learning cost reductions for new technologies to become competitive: without their involvement they might potentially never become competitive."", '1309.7626-2-74-7': 'If the power sector is to decarbonise by 2050, all countries are most likely required to make a contribution to the development of the renewables industry.', '1309.7626-2-75-0': '## Distributional impacts of decarbonisation', '1309.7626-2-76-0': '[CITATION] describes the economic impacts of the decarbonising the power sector; only a summary is provided here.', '1309.7626-2-76-1': 'However, it is important to note that there are some winners and losers in the decarbonisation scenario.', '1309.7626-2-76-2': 'The main winners are the construction and engineering sectors, their workforce and their supply chains.', '1309.7626-2-76-3': 'These all benefit from the higher rates of development and deployment of capital-intensive equipment.', '1309.7626-2-76-4': 'The sectors that lose out are those that supply fossil fuels.', '1309.7626-2-76-5': 'We would also expect reduced rates of economic activity in services sectors, where consumers must spend a larger proportion of income on electricity; this is however compensated by tax reductions originating from government income from carbon pricing.', '1309.7626-2-77-0': 'The outcomes for electricity-intensive sectors are less clear.', '1309.7626-2-77-1': 'Companies in these sectors will face higher costs and may see a loss of demand for their products.', '1309.7626-2-77-2': 'However, as the scenarios are global, there are only limited competitiveness effects.', '1309.7626-2-77-3': 'Many, but by no means all, energy-intensive firms also feature in the supply chains for renewables.', '1309.7626-2-77-4': 'Furthermore, imports and exports of fossil fuels change significantly in any decarbonisation scenario, generating further winners (fuel importers) and losers (fuel exporters).', '1309.7626-2-77-5': 'This aspect requires further investigations into the economic impacts of specific nations.', '1309.7626-2-78-0': 'The most important unknown in the decarbonisation scenarios presented concerns the distributional impact of increasing electricity prices to the less wealthy in all regions of the World.', '1309.7626-2-78-1': 'Depending on government policy, this could trap groups of people in energy poverty, in particular people who cannot afford to change their electricity consuming appliances such as heating or cooling devices.', '1309.7626-2-78-2': 'Therefore, climate policy could be complemented with energy end-use technology subsidies in order to help people change technology for more efficient systems through the transition without significant loss in their standard of living.', '1309.7626-2-78-3': 'Complementary research on the impact of climate policy onto energy access in under-developed regions and the likely strong synergies existing between energy access, climate and economic development policies is likely to shed significant light on these issues.', '1309.7626-2-79-0': '# Conclusion', '1309.7626-2-80-0': 'This work introduces an unprecedented method and analysis of technology diffusion and decarbonisation in the global electricity sector using a coupled model of power technology diffusion, non-equilibrium macroeconomics and climate change.', '1309.7626-2-80-1': 'The use of non-equilibrium models is shown to generate projections significantly different to those created using currently standard cost-optimisation and equilibrium economics approaches.', '1309.7626-2-80-2': 'The relaxation of the cost-optimisation constraint enables scenarios to feature non-optimal technology lock-ins well known to take place in the electricity sector.', '1309.7626-2-80-3': 'Meanwhile, non-equilibrium macroeconometrics enable to explore detailed macroeconomic impacts of energy policy instruments, explored in a sister paper , where investment in low carbon technology generates additional employment and enhanced economic activity as a counterbalance to increasing energy prices during decarbonisation.', '1309.7626-2-81-0': 'Ten scenarios were presented where a rationale is built for the elaboration of composite energy policy instrument combinations with strong synergy.', '1309.7626-2-81-1': 'This involves the analysis of individual policy tools as well as combinations, generating a storyline on how to reduce global emissions to near 90% below their 1990 level.', '1309.7626-2-81-2': 'The demonstration of the existence of strong synergies between policy tools for the diffusion of low carbon technology argues strongly against the use of policy instruments on their own such as carbon pricing, likely to yield very little change in the technology composition and unlikely to break the current fossil technology lock-in.', '1309.7626-2-82-0': 'We presented a decarbonisation scenario where a significant contribution to emissions reductions originates from electricity consumption reductions, which happens through changes the price of electricity as a result of energy policy and changes in production costs.', '1309.7626-2-82-1': 'The remaining emissions reductions stem from changes in the global electricity technology mix itself.', '1309.7626-2-82-2': 'Global warming impacts are provided.', '1309.7626-2-82-3': 'Insight for climate policy-making is given in terms of existing synergies between climate policy instruments and their impact on technology costs through global knowledge spillovers.', '1309.7626-2-82-4': 'This provides a strong argument for global coordination of climate policy in order for all nations to benefit simultaneously from cost reductions for low carbon technologies through research and development and learning-by-doing.'}","[['1309.7626-1-39-0', '1309.7626-2-39-0'], ['1309.7626-1-39-1', '1309.7626-2-39-1'], ['1309.7626-1-39-3', '1309.7626-2-39-3'], ['1309.7626-1-39-4', '1309.7626-2-39-4'], ['1309.7626-1-43-1', '1309.7626-2-43-1'], ['1309.7626-1-43-2', 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['1309.7626-3-13-2', '1309.7626-4-18-3'], ['1309.7626-3-13-3', '1309.7626-4-18-4']]","[['1309.7626-3-4-0', '1309.7626-4-3-1'], ['1309.7626-3-7-7', '1309.7626-4-16-8'], ['1309.7626-3-59-4', '1309.7626-4-63-0']]","['1309.7626-1-16-0', '1309.7626-2-16-0', '1309.7626-3-16-0', '1309.7626-4-19-1']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1309.7626,"{'1309.7626-3-0-0': 'This paper presents an analysis of climate policy instruments for the decarbonisation of the global electricity sector in a non-equilibrium economic and technology diffusion perspective.', '1309.7626-3-0-1': 'Energy markets are driven by innovation, path-dependent technology costs and diffusion; yet, common optimisation modelling methodologies remain vague on these aspects and have a limited ability to address the effectiveness of policy onto decision-making since the latter is not specifically represented.', '1309.7626-3-0-2': 'This leads to an underestimation of non-cost-optimal technology lock-ins known to occur.', '1309.7626-3-0-3': 'Breaking with tradition, our approach explores bottom-up investor dynamics-led diffusion of low carbon technology with a highly disaggregated sectoral macroeconometric model of the global economy, FTT:Power-E3MG.', '1309.7626-3-0-4': 'A set of ten projections to 2050 of the electricity sector in 21 regions exploring combinations of electricity policy instruments are analysed including climate impacts.', '1309.7626-3-0-5': 'We show that in a diffusion and path-dependent perspective, the impact of combinations of policies does not correspond to the sum of the impacts of individual instruments: synergies exist between policy tools.', '1309.7626-3-0-6': 'We argue that worldwide carbon pricing on its own is incapable of breaking the current fossil technology lock-in, but that under particular sets of policies, the electricity sector can be decarbonised affordably by 90% by 2050 without early scrapping.', '1309.7626-3-1-0': '# Introduction', '1309.7626-3-2-0': 'The electricity sector emits 38% of global energy-related greenhouse gases .', '1309.7626-3-2-1': 'Investment planning in the electricity sector is therefore of critical importance to climate change policy.', '1309.7626-3-2-2': 'Electricity production is the energy sector with the longest time scales for technological change, requiring particularly careful planning in order to avoid locking into heavy emissions systems for many decades, crucial to avoid global warming above dangerous levels .', '1309.7626-3-2-3': 'It has been argued that meeting emissions targets to prevent warming beyond 2[MATH]C is possible but restrictive on possible pathways of energy sector development .', '1309.7626-3-2-4': 'However it is also widely accepted that warming beyond 4[MATH]C is likely to lead to catastrophic consequences to global ecosystems and food chains , with important repercussions to global human welfare.', '1309.7626-3-2-5': 'Large reductions in greenhouse gas emissions involve significant amounts of technology substitution, most likely large scale socio-technical transitions .', '1309.7626-3-3-0': 'Energy systems being inherently complex, the use of computational tools to inform the planning of emissions reductions is crucial, and such models should ideally include details of the various energy policy instruments available, and generate as results quantitative changes in emissions for changes in policy.', '1309.7626-3-3-1': 'It is moreover critical to avoid analysing national energy systems in isolation, and thus assessments are ideally carried out using large scale modelling of the global energy and economic system, incorporating endogenously quantities such as energy carrier demand and prices, imports and exports, technology costs, natural resource consumption and economic growth.', '1309.7626-3-3-2': 'Projections of global energy-related GHG emissions can only be performed using multi-sectoral (e.g. power, transport, industry, buildings) and multi-regional models of energy systems in order to cover all emissions sources.', '1309.7626-3-3-3': 'Finally, total emissions depend not only on technology compositions but also on the intensity of fuel use, which depends on energy demand and global economic activity.', '1309.7626-3-4-0': ""The majority of studies of energy systems are made using cost-optimisation computational models (using a 'social planner' framework)."", '1309.7626-3-4-1': 'Meanwhile, the economics of climate change are often represented using equilibrium economic theories (assumptions of fully rational agent behaviour, perfect foresight and information, carried out by a representative agent, and some variations within these concepts), which tend to yield negative macroeconomic impacts of climate change mitigation, some directly by construction.', '1309.7626-3-4-2': 'In both these approaches the assumptions over the nature of agents enable the optimisation problem involved in the theory to be tractable, and represents the current methodological standard.', '1309.7626-3-4-3': 'These assumptions over the nature of behaviour however may over-simplify aspects of the global energy-economic system that are crucial for climate change mitigation, leaving an urgent need to improve theory and method beyond these paradigms.', '1309.7626-3-5-0': 'Cost-optimisation technology models, in normative mode, are the most powerful for finding with outstanding detail lowest cost future technology pathways that reach particular objectives.', '1309.7626-3-5-1': 'If used for descriptive purposes, they imply a description of agents (investors, consumers) as identical, who possess a degree of information and technology access as well as foresight sufficient to generate pathways that are cost-optimal at the system level, which alternatively corresponds to a controlled degree of coordination between all actors involved in the evolution of the system.', '1309.7626-3-5-2': 'As stated in the Global Energy Assessment , ""A fundamental assumption underlying the pathways is that the coordination required to reach the multiple objectives simultaneously can be achieved"".', '1309.7626-3-5-3': 'While this approach generates a significant simplification to a system highly complex to model, it may be argued that such a spontaneous emergence of coordination is somewhat unlikely.', '1309.7626-3-5-4': 'For instance liberalised energy markets involve actors free to take their investment and consumption decisions based on their particular circumstances, and are only incentivised by policy.', '1309.7626-3-5-5': 'Thus while optimisation frameworks are valuable for identifying feasible and cost-effective pathways that reach particular objectives at the system level, they do not suggest how exactly to achieve them from a policy standpoint, because they do not specifically model decision making by diverse agents.', '1309.7626-3-5-6': 'Strong coordination is difficult to generate from economic policy instruments, leading to sub-optimal outcomes and technology lock-ins.', '1309.7626-3-6-0': 'Meanwhile, equilibrium economic theory implies that climate change mitigation costs are borne at the expense of consumption or investment elsewhere, leading to detrimental economic impacts, which is not generally agreed to occur .', '1309.7626-3-6-1': ""The assumptions associated with equilibrium economics lead to emphasising carbon pricing as a single policy tool able to fix the climate market failure, even though it is well known to industrial policy-makers that new technologies often cannot successfully diffuse into the marketplace without specific government support to 'bridge the technology valley of death' ."", '1309.7626-3-6-2': 'Projecting future greenhouse gas emissions for climate modelling using combinations of models of energy systems designed for normative analyses and equilibrium economic models may be conceptually inconsistent with the simulation approach of climate science, potentially misleading if this is done to project environmental impacts as outcomes of energy policy.', '1309.7626-3-7-0': 'A simulation approach without systems optimisation requires using known technology dynamics and to model decision-making at the firm level by diverse agents .', '1309.7626-3-7-1': 'Empirical dynamics are known to exist in scaling up technology systems (e.g. [MATH]-shaped diffusion), and their costs (learning curves), which have been extensively studied for decades .', '1309.7626-3-7-2': 'Mathematical generalisations have been suggested involving dynamic differential equations similar to those in population growth mathematical ecology and demography .', '1309.7626-3-7-3': ""In combination with evolutionary dynamics and evolutionary game theory , this offers a artillery of powerful concepts at core of evolutionary economics [CITATION] which emphasises innovation, diffusion and speciation as a source of economic development and growth , the clustering of which is possibly responsible for so-called 'Kondratiev cycles' ."", '1309.7626-3-7-4': ""These are broadly consistent with the 'multi-level perspective' on technology transitions described by [CITATION]."", '1309.7626-3-7-5': 'Furthermore, complexity and path dependence emerge as key concepts to visualise technology dynamics .', '1309.7626-3-7-6': 'Even though [MATH]-shaped curves have been famously shown to arise in energy consumption , such principles have yet to be introduced in mainstream energy systems modelling , which would generate an improved methodological paradigm .', '1309.7626-3-7-7': 'In such a framework, the use of optimisation as a source of dynamic force and actor behaviour would be replaced by empirically known innovation-selection-diffusion dynamics.', '1309.7626-3-8-0': 'Breaking with tradition, this work proposes such a paradigm along with a non-equilibrium approach to energy-economy-environment interactions.', '1309.7626-3-8-1': 'We present an analysis of the global electricity sector using coupled simulations of technology diffusion dynamics, using FTT:Power, non-equilibrium economics using E3MG, and environmental impacts by combining emulators of the PLASIM-ENTS climate model (PLASIM-ENTSem) and of the GENIE-1 carbon cycle model (GENIEem) .', '1309.7626-3-8-2': 'It enables to explore the outcomes of particular energy policy tools on technology diffusion, electricity generation, global emissions, climate change and macroeconomic change.', '1309.7626-3-8-3': 'Uncoordinated dynamics are modelled at the level of diverse profit-seeking investor decisions under bounded rationality , of which the sum does not produce an overall cost optimum, as investors disregard in general the collective effect of their decisions on the wider system or the common good.', '1309.7626-3-8-4': 'The connection of a diffusion framework to a non-equilibrium model of the global economy opens a very rich world of macroeconomic dynamics of technological change where the impacts of energy policy reveal complex interactions between the energy sector and the economy.', '1309.7626-3-9-0': 'Ten scenarios of the future global power sector up to 2050 are presented, creating a storyline to provide insight for the construction of effective comprehensive policy portfolios.', '1309.7626-3-9-1': 'Going beyond simple carbon pricing and considering policies that could help trigger the diffusion of new technologies, particular combinations of energy policy instruments are found to provide suitable environments to enable fast electricity sector decarbonisation.', '1309.7626-3-9-2': 'Macroeconomic dynamics associated with these scenarios, which will be explored separately elsewhere, are summarised.', '1309.7626-3-9-3': 'High resolution data in these scenarios are accessible on our website at http://www.4cmr.group.cam.ac.uk/research/FTT/fttviewer, where costs, electricity generation and emissions can be explored in 21 world regions and 24 technologies.', '1309.7626-3-9-4': 'Decarbonisation involves the positive externality associated to the global accumulation of knowledge and experience in scaling up, deploying and using new power technologies.', '1309.7626-3-9-5': 'A classic collective action problem emerges: learning cost reductions for new technologies may only become significant and enable cost-effective diffusion when most nations of the World demonstrate strong coordinated dedication to their deployment.', '1309.7626-3-9-6': 'We show that carbon pricing covering all world regions is a necessary but insufficient component for the success of mitigation action in order to break the current fossil fuel technology lock-in.', '1309.7626-3-9-7': 'As has been shown earlier however , in this perspective, reductions in power sector emissions may not necessarily imply significant macroeconomic costs (direct and indirect) but instead, could actually generate additional industrial activity and employment.', '1309.7626-3-10-0': '# The model: FTT:Power-E3MG', '1309.7626-3-11-0': '## Technology diffusion in FTT:Power', '1309.7626-3-12-0': 'Emissions reductions in the energy sector can occur through technology substitution, between technologies that produce the same substitutable service (e.g. electricity, heat, etc), through behaviour and practice changes and through reductions in the consumption of that service altogether.', '1309.7626-3-12-1': 'Technology, however, comes to life through investments made at specific points in time with an expectation by investors to receive an income or service of which the present value justifies or exceeds the initial investments and the present value of present and future costs [CITATION].', '1309.7626-3-12-2': 'In this context, early scrapping of usable capital produces stranded assets that do not produce the expected income, generating losses and requiring new investments earlier than expected.', '1309.7626-3-13-0': 'Where early scrapping is avoided, technological change occurs primarily at the average rate of replacement of existing technology as it ages, which is inversely related to its life span.', '1309.7626-3-13-1': 'However, notwithstanding lifetime considerations, the number of new units of technology of a particular type that can be constructed at any one time can be larger if that industry is in a well established position in the marketplace, with a large production capacity, than if it is emerging.', '1309.7626-3-13-2': 'Thus, even when an emerging technology is very affordable (for instance with strong policy support such as subsidies), it may not necessarily be accessible in every situation where investors are required to make a choice between available options.', '1309.7626-3-13-3': 'Thus in the analysis of the diffusion capacity of technologies and the pace of technological change, not only cost considerations come into play, but also a limited access to technology and information, and these two principles, described in earlier work , form the core of FTT:Power (eq. [REF]).', '1309.7626-3-14-0': 'This decision-making process by diverse agents can be expressed with probabilistic pairwise comparisons of options.', '1309.7626-3-14-1': 'Investors do not all face similar situations and do not weigh different aspects in the same way, which cannot possibly be enumerated specifically in a model.', '1309.7626-3-14-2': 'However, when dealing with large numbers of instances of decision making, there will be majority trends in investor choices, who may be assumed, if all relevant considerations are quantified into costs, to be seeking cost minimisation with the goal of profit maximisation for their own respective firms.', '1309.7626-3-14-3': 'Data derived probability distributions may thus be used in order to avoid enumerating the details of all situations faced by investors, and where a particular technology is on average more profitable to use than a second one, there usually exist specific situations where the reverse turns out to be true.', '1309.7626-3-14-4': 'This provides a crucial simple representation of diversity in decision-making.', '1309.7626-3-15-0': 'Thus the statistical trend of investor preferences may be expressed as a matrix [MATH] expressing the relative fractions of investor choices between two technologies [MATH] and [MATH] out of a set.', '1309.7626-3-15-1': 'However, constraints of the power system prevent some types of technology to dominate.', '1309.7626-3-15-2': 'These can be expressed with a second matrix [MATH], stopping investments that lead to stranded or unused assets due to technical problems .', '1309.7626-3-16-0': 'Using the variable [MATH] for the generation capacity market share of a technology, the rate at which shares of one technology ([MATH]) can be replaced by shares of another ([MATH]) is proportional to:', '1309.7626-3-17-0': 'The rate at which units of technology [MATH] come to the end of their working life, with life expectancy [MATH], How many old units of [MATH] require replacement, a fraction [MATH] of the total number of replacements.', '1309.7626-3-17-1': 'The rate at which the construction capacity for technology [MATH] can be expanded, with time constant [MATH], The market position of technology [MATH], its share of the market [MATH].', '1309.7626-3-18-0': 'The resulting market dynamics are flows of market shares between existing options, according to choices and constraints [MATH] and [MATH], and is expressed as [EQUATION] using the time scaling matrix [MATH] to express changeover time constants for every pair of technologies.', '1309.7626-3-18-1': 'Adding up substitution flows in both directions between all possible pairs of technologies leads to a family of finite difference equations that tracks the evolution of technological change in a set of substitutable technologies: [EQUATION] with [MATH] expressing investor preferences and [MATH] providing constraints (see also fig. [REF]).', '1309.7626-3-18-2': '[MATH] is the average rate of change weighed by the market shares, the scaling of time.', '1309.7626-3-18-3': 'This equation, logistic in the special case of two technologies, is non-linear and generates slow uptakes at small share values, then fast diffusion at intermediate penetration, before a saturation near full penetration.', '1309.7626-3-18-4': 'This system cannot be solved analytically but is straightforward to evaluate numerically using time steps.', '1309.7626-3-18-5': 'It corresponds to the replicator dynamics equation in evolutionary game theory , also used in mathematical genetics as well as in evolutionary economics .', '1309.7626-3-19-0': '## Learning-by-doing and path dependence', '1309.7626-3-20-0': 'Profit-seeking investor choices summarised in [MATH] are driven by cost differences, and these change over time as technologies diffuse in the marketplace and follow their learning curves .', '1309.7626-3-20-1': 'Learning-by-doing cost reductions stem from the accumulation of technical knowledge on production and economies of scale in expansion of productive capacity.', '1309.7626-3-20-2': 'Technology costs are taken here to apply everywhere globally, so that the benefits of learning are global, in a perspective where firms selling technologies operate in a global market.', '1309.7626-3-20-3': 'Emerging technologies have fast cost reductions (e.g. solar panels) while established systems see very little change (e.g. coal systems).', '1309.7626-3-20-4': 'The crucial aspect is that cost reductions are decreasing functions of cumulative investment, not time, and that they do not occur if no investment is made.', '1309.7626-3-20-5': 'Learning thus interacts with diffusion where it incentivises further uptake, which generates further learning and so on, a highly non-linear effect which can lead to sudden technology avalanches.', '1309.7626-3-20-6': ""Policy can generate this, termed 'bridging the techchnology valley of death' ."", '1309.7626-3-20-7': 'Learning curves are connected here only with investor choice probabilities [MATH].', '1309.7626-3-21-0': 'Learning-by-doing introduces the crucial property of path dependence.', '1309.7626-3-21-1': 'As technologies gradually diffuse in the marketplace following investor choices, the full landscape of technology costs continuously changes, and investor preferences thus change.', '1309.7626-3-21-2': 'These changes are permanent and determined completely by past investments, and therefore by the full history of the market.', '1309.7626-3-21-3': 'Thus, from any starting point, future cost landscapes are determined by investment trends at every point in time, and different futures exist, depending on investment and policy choices along the way.', '1309.7626-3-21-4': 'Technology costs and learning rates are given in [CITATION], with more detail on the 4CMR website.', '1309.7626-3-22-0': '## Natural resource use', '1309.7626-3-23-0': 'The diffusion of power systems can only occur in areas where resources are available, for instance windy areas for wind power, or natural water basins and rivers for hydroelectric systems.', '1309.7626-3-23-1': 'Higher productivity sites offer lower costs of electricity production, and tend to be chosen first by profit-seeking investors.', '1309.7626-3-23-2': 'Assuming this, the progression of the development of renewable energy systems produces a tendency towards increasing marginal costs of production for potential new systems as only resources of lower and lower productivity are left to use.', '1309.7626-3-23-3': 'This is well described by functions of increasing marginal costs of production with increasing development, cost-supply curves .', '1309.7626-3-23-4': 'For this purpose, a complete set of such curves was previously estimated from literature data for 190 countries of the world for 13 types of natural resources .', '1309.7626-3-23-5': 'These were aggregated for the 21 regions of E3MG.', '1309.7626-3-23-6': 'Excluding non-renewable resources treated globally, this produced [MATH] cost-supply curves that are used to constrain the expansion of renewable systems in FTT:Power.', '1309.7626-3-23-7': 'The consumption of non-renewable resources is however better represented using dynamic cost-quantity curves, described next.', '1309.7626-3-24-0': '## Fossil fuel price dynamics', '1309.7626-3-25-0': 'Non renewable resources lying in geological formations have an arbitrary value that depends on their cost of extraction, but also on the dynamics of the market.', '1309.7626-3-25-1': 'To their cost of extraction is associated a minimum value that the price of the commodity must take in order for the extraction to be profitable.', '1309.7626-3-25-2': 'These costs are however distributed over a wide range depending on the nature of the geology (e.g. tar sands, ultra-deep offshore, shale oil and gas, etc).', '1309.7626-3-25-3': 'Thus, given a certain demand for the commodity, the price is a function of the extraction cost of the most expensive resource extracted in order to supply the demand, which separates reserves from resources.', '1309.7626-3-25-4': 'As reserves are gradually consumed, this marginal cost increases generating a commodity price increase which unlocks the exploitation of resources situated in locations with higher extraction costs.', '1309.7626-3-25-5': 'For example, tar sands became economical and saw massive expansion above a threshold price of around 85-95/boe.', '1309.7626-3-25-6': 'Thus, to any commodity demand path in time will correspond a path dependent commodity price.', '1309.7626-3-25-7': 'This model is described with an analysis in [CITATION], relying on data from [CITATION].', '1309.7626-3-25-8': 'In FTT:Power, this model is used to determine fuel costs for fossil fuel and nuclear based power technologies.', '1309.7626-3-26-0': '## Timescales of technological change', '1309.7626-3-27-0': 'Innovation generates new technologies that live in niches that protect them from the wider market.', '1309.7626-3-27-1': 'From those niches, in appropriate changes of market conditions, can emerge and diffuse new socio-technical regimes .', '1309.7626-3-27-2': ""After the innovation phase, at the level of diffusion, technologies enter what we termed the 'demographic phase' ."", '1309.7626-3-27-3': 'In scenarios excluding early scrapping, possible rates of change are dictated by the ageing and decommissioning rate of of capital and infrastructure, in other words, the inverse of the life expectancy of the existing capital.', '1309.7626-3-27-4': 'Meanwhile, relative rates of expansion of the productive capacity for new technology determine which technologies can fill in gaps more effectively than others, when other technology is decommissioned.', '1309.7626-3-27-5': 'In many cases of interest, for instance with new technologies, these timescales cannot be reliably measured using small amounts of diffusion data of non-linear nature.', '1309.7626-3-27-6': 'Therefore a theory of diffusion is necessary in order to infer these parameters from technology and industry properties.', '1309.7626-3-27-7': 'This theory is derived and explained in [CITATION].', '1309.7626-3-28-0': '## Modelling the global economy: E3MG', '1309.7626-3-29-0': 'E3MG (and variant E3ME) is an out-of-equilibrium macroeconometric model of the global economy that has been used widely for studies of climate change mitigation macroeconomics [CITATION].', '1309.7626-3-29-1': 'It evaluates the parameters of 28 econometric equations onto data between 1971 and 2010, and extrapolates these equations between 2010 and 2050.', '1309.7626-3-29-2': 'The model features a high resolution: its equations are evaluated for 21 regions of the world, 43 industrial sectors, 28 sectors of consumption, 22 fuel users, and 12 fuel types.', '1309.7626-3-29-3': 'Sectors are interrelated with dynamic input-output tables.', '1309.7626-3-29-4': 'The model incorporates endogenous growth and endogenous technological change using accumulated investment and RD spending, particularly in the energy sector as detailed next, which makes the model strongly path-dependent and far from equilibrium.', '1309.7626-3-30-0': '## Electricity price-demand interactions', '1309.7626-3-31-0': 'The demand for electricity depends on its price, and it is well known that in situations of high electricity prices, people may strive to find more effective ways to use their income, preferring to invest in more efficient technology, perceived as a worthwhile tradeoff, or to simply reduce their consumption.', '1309.7626-3-31-1': 'When the electricity supply technology mix changes, the minimum price at which electricity can be profitably sold also changes, and with such price changes, the demand for electricity changes.', '1309.7626-3-31-2': 'In particular, when carbon pricing or feed-in tariffs are used to ensure access of expensive renewables into the grid, the price of electricity increases, bearing an externality to all users , affecting consumer demand and behaviour.', '1309.7626-3-31-3': 'Thus, reductions in emissions originate from both a change in the carbon intensity of the power sector and changes in the demand for electricity.', '1309.7626-3-31-4': 'These aspects of energy economics are prominent in this work, responsible for a significant fraction of our projected emissions reductions in scenarios of climate policy.', '1309.7626-3-32-0': 'Electricity demand is modelled in E3MG, using an econometric equation that incorporates endogenous technological change by including accumulated investment and RD expenditures of the form [EQUATION] where for fuel [MATH] and region [MATH], [MATH] is the fuel demand, [MATH] represents output, [MATH] relative prices and [MATH] is the technological progress indicator .', '1309.7626-3-32-1': 'Including accumulated investment and RD makes this equation non-linear, path-dependent and hysteretic.', '1309.7626-3-32-2': 'In particular, prices hikes lead to enhanced energy-saving RD that permanently increases efficiency of fuel use, which would not have occurred without price changes.', '1309.7626-3-32-3': 'Economic feedbacks between FTT:Power and E3MG occurs with four quantities: fuel prices, fuel use, power technology investments and tax revenue recycling.', '1309.7626-3-33-0': '## Climate modelling using PLASIM-ENTSem', '1309.7626-3-34-0': 'The carbon cycle is represented in this work by an emulator of the Grid Enabled Integrated Earth systems model (GENIE-1) Earth System Model .', '1309.7626-3-34-1': 'The carbon cycle emulator (GENIEem) is designed to be a more sophisticated, but computationally very fast, alternative to simplified climate-carbon cycles in integrated assessment models.', '1309.7626-3-34-2': 'GENIE-1 simulates climate, ocean circulation and sea-ice, together with the terrestrial, oceanic, weathering and sedimentary components of the carbon cycle.', '1309.7626-3-34-3': 'The emulator takes as inputs a time series of anthropogenic carbon emissions and non-CO[MATH] radiative forcing, and outputs a time series of atmospheric CO[MATH] concentration.', '1309.7626-3-34-4': 'Uncertainty in the carbon cycle is captured by varying GENIE-1 parameter inputs, resulting in an emulator ensemble of 86 possible futures.', '1309.7626-3-35-0': 'PLASIM-ENTSem is an emulator of the PLAnet SIMulator - Efficient Numerical Terrestrial Scheme (PLASIM-ENTS) global climate model (GCM).', '1309.7626-3-35-1': 'In the GCM, PLASIM, which includes a Q-flux ocean and a mixed-layer of a given depth , is coupled to a simple surface and vegetation model, ENTS, which represents vegetation and soil carbon through a single plant functional type .', '1309.7626-3-35-2': 'The design of PLASIM-ENTSem is described in detail by [CITATION].', '1309.7626-3-35-3': 'As for GENIEem, uncertainty is captured by varying PLASIM-ENTS parameter inputs, resulting in a 188-member ensemble of decadally and spatially seasonal climate variables.', '1309.7626-3-36-0': 'The combination of both emulators in the context of this work, with combined uncertainty analysis, is given by [CITATION].', '1309.7626-3-36-1': 'The emissions profiles associated with the scenarios produced by FTT:Power-E3MG are used to provide inputs to GENIEem, which in turn provides inputs to PLASIM-ENTS.', '1309.7626-3-36-2': 'The median, 5th-95th percentiles of the GENIEem ensemble output are each used to drive climate simulations using PLASIM-ENTSem, so that uncertainty associated with modelling the carbon cycle is represented in the climate projections.', '1309.7626-3-36-3': 'Based on the median GENIEem ensemble output, the median, 5th-95th percentiles of warming from the PLASIM-ENTSem ensemble are also calculated; these bounds, therefore, reflect warming uncertainty due to parametric uncertainty in the climate model alone.', '1309.7626-3-36-4': 'The 5th percentile of warming from the PLASIM-ENTSem ensemble based on the 5th percentile of CO2 concentration from the GENIEem ensemble, and the 95th percentile of warming from the PLASIM-ENTSem ensemble based on the 95th percentile of CO2 concentration from the GENIEem ensemble are also calculated; this second set of bounds, therefore, reflects warming uncertainty due to parametric uncertainty in the climate model and the carbon cycle model.', '1309.7626-3-37-0': '# Projections to 2050 of the global power sector', '1309.7626-3-38-0': '## Scenario creation and policy instruments', '1309.7626-3-39-0': 'Ten scenarios of electricity policy assumptions of different types and resulting technology mix and emissions up to year 2050 were created with FTT:Power-E3MG, lettered a to j (fig. [REF]).', '1309.7626-3-39-1': 'These all lead to different futures for the global power sector and different CO[MATH] emission profiles.', '1309.7626-3-39-2': 'It is impractical to reproduce all the information of these simulations in this paper, and therefore a summary of the results is given here, the details having been made available on the 4CMR website, where they can be displayed in terms of the full resolution of 21 world regions and 24 power technologies, for policy assumptions, electricity generation, emissions and levelised costs.', '1309.7626-3-39-3': 'Four energy policy tools were explored: carbon pricing/taxing (CO[MATH]P), technology subsidies (TSs), feed-in tariffs (FiTs) and direct regulations (REGs).', '1309.7626-3-39-4': 'Individual tools and various combinations were explored, a summary given in figure [REF].', '1309.7626-3-39-5': 'By gradually elaborating various policy frameworks, a scenario was found where power sector emissions are reduced by 90% below the 1990 level, involving all four policy instruments used simultaneously.', '1309.7626-3-39-6': 'Emissions are fed to the carbon cycle and climate model emulators GENIEem and PLASIM-ENTSem in order to determine the resulting atmospheric CO[MATH] concentration and average global warming, for these scenarios where other sectors are not targeted by climate policy.', '1309.7626-3-40-0': 'The nature of FiTs here is that access to the grid at a competitive price is ensured (a price higher than the consumer price), the difference being paid by the grid and passed on to consumers through the price of electricity.', '1309.7626-3-40-1': 'The consumer price of electricity is raised by just the amount that makes this economically viable.', '1309.7626-3-40-2': 'The consumer price (P) in FTT:Power-E3MG is derived from the generation (G) share-weighted average Levelised Cost of Electricity (LCOE), [MATH].', '1309.7626-3-40-3': ""The LCOE as perceived by investors when a FiT exists includes an 'effective subsidy' given by the grid that covers a fraction or the whole difference between the levelised cost of these technologies and the consumer price of electricity (investors here may be corporate or, for instance, homeowners)."", '1309.7626-3-40-4': 'In the case of CO[MATH]P, the LCOE calculation that investors are assumed to perform includes possible carbon cost components, and the price of carbon is also passed on to consumers through the price of electricity.', '1309.7626-3-40-5': 'Thus the price of electricity increases with the carbon price unless emitting technologies are phased out.', '1309.7626-3-40-6': 'TSs are fractions of the capital costs of low carbon technologies that are paid by the government, reducing the LCOE that investors see.', '1309.7626-3-40-7': 'These are defined exogenously for every year up to 2050 and were designed to be phased out before then, after which it is hoped that the technology cost landscape becomes permanently altered and that technologies are not indefinitely subsidised.', '1309.7626-3-40-8': 'REGs indicate regulations controlling the construction of additional units of particular technologies, used to phase out particular types of systems.', '1309.7626-3-40-9': 'When a REG is applied to a technology category, no new units are built but existing ones are left to operate until the end of their lifetime.', '1309.7626-3-41-0': 'Fig. [REF] summarises the result of the policy tools exploration.', '1309.7626-3-41-1': 'Electricity generation by technology type is given in the series of panels to the left of each pair, while emissions are given on the right.', '1309.7626-3-41-2': 'The vertical dashed lines indicate the start of the simulations in 2008, and data to the left of this line is historical data from the [CITATION].', '1309.7626-3-41-3': 'The horizontal dashed lines indicate the 1990 levels of electricity demand and emissions.', '1309.7626-3-41-4': 'Dashed curves correspond to the baseline values for comparison.', '1309.7626-3-41-5': 'In all scenarios excluding the baseline, policy schemes generate both a reduction of electricity consumption and emissions.', '1309.7626-3-41-6': 'Consumption reduces due to increases in the price of electricity, through the energy demand econometric equation of E3MG, which contributes significantly to emissions reductions.', '1309.7626-3-41-7': 'All additional emissions reductions are due to changes in fuels used, or, in other words, to changes of technologies.', '1309.7626-3-41-8': 'Emissions reductions in 2050 with respect to the 1990 level are given in percent values.', '1309.7626-3-42-0': '## Climate policy for achieving 90 reductions in power sector emissions', '1309.7626-3-43-0': 'The baseline scenario (fig. [REF] panel a), which involves maintaining current policies until 2050 (carbon pricing in Europe only), generates global power sector emissions in 2050 of 30 GtCO[MATH]/y, 318 above the 1990 level, and total emissions of 65 GtCO[MATH]/y. Cumulative emissions for the time span 2000-2050 amount to 2321 GtCO[MATH].', '1309.7626-3-43-1': 'This pathway is likely to commit the planet to a warming that exceeds 4[MATH]C above pre-industrial levels in around 2100 (fig. [REF] below), consistent with [CITATION] .', '1309.7626-3-43-2': 'In view of finding ways to reduce these emissions and to limit global warming, we searched areas of policy space that reduce these significantly in the short time span, given in fig. [REF].', '1309.7626-3-44-0': 'The first option explored (panel b) was to use REGs to prevent the construction of new coal power plants worldwide, the systems with highest emissions ([MATH]1 ktCO[MATH]/MWh), unless they are equipped with carbon capture and storage (CCS).', '1309.7626-3-44-1': 'This results mostly in a transfer from a coal lock-in to a gas lock-in, reducing global emissions approximately to the 1990 level, largely insufficient for meeting the 2[MATH]C target.', '1309.7626-3-45-0': 'The second option was to use CO[MATH]P as a unique tool, with different price values for different regions covering all world regions shown in fig. [REF], between 100 and 200 2008/tCO[MATH] in 2050 (panel c) and between 200 and 400 2008/tCO[MATH] (panel d).', '1309.7626-3-45-1': 'This measure, mostly generating reductions in electricity consumption due to higher electricity prices, reduces emissions to around 65% above the 1990 level and to 9% below the 1990 level, respectively.', '1309.7626-3-45-2': 'This modest impact suggests that CO[MATH]P on its own requires very high carbon prices in order to generate significant reductions, or that it is simply insufficient.', '1309.7626-3-45-3': 'However as we show now, combinations of policies achieve this much more effectively.', '1309.7626-3-46-0': 'As a first combination of policies, FiTs (wind and solar) and TSs (all other low carbon technologies except wind and solar) were introduced without CO[MATH]P (panel e), of order 30-50 of capital costs for TSs and feed-in prices 5-15 above the electricity price for FiTs (this depends on regions and technologies, see the inset of fig. [REF] for a world average or the data on our website for details).', '1309.7626-3-46-1': 'This generates very modest uptakes of low carbon technologies and thus small emissions reductions, 276 above 1990.', '1309.7626-3-46-2': 'This is due to the low cost of producing electricity using fossil fuels in comparison to all other technologies, in particular coal, and therefore without very high subsidies, CO[MATH]P is necessary in order to bridge this cost difference.', '1309.7626-3-47-0': 'Scenario [MATH] shows the use of CO[MATH]P up to 200/tCO[MATH] and feed-in tariffs, the latter generating very little change over scenario c. Using CO[MATH]Ps of up to 200/tCO[MATH] in combination with TSs and FiTs in all world regions (panel g) yields emissions reductions to 32 above the 1990 level, still insufficient.', '1309.7626-3-47-1': 'With CO[MATH]P of up to 400/tCO[MATH] in combination with the same set of TSs and FiTs (panel h), reductions are much larger, 46 below the 1990 level.', '1309.7626-3-47-2': 'This indicates how the impact of policy combinations is larger than the sum of the impacts of its components taken separately, offering significant potential synergies.', '1309.7626-3-48-0': 'A scenario was explored where only the developed world applies the stringent climate policies of scenario h (panel i), in which it is hoped that this generates enough investment to bring the costs of low carbon technologies down into the mainstream, thus becoming accessible to developing or under-developed countries.', '1309.7626-3-48-1': 'We see no noticeable uptake of new technology in these countries, costs remaining unaccessible especially in comparison with coal based technologies, and as a consequence global emissions are only reduced to 204% above the 1990 level.', '1309.7626-3-49-0': 'A significant amount of the remaining emissions in scenario h reside in China, where the lock-in of coal technology is very difficult to break given the near absence of alternatives with the exception of hydroelectricity, which is driven to its natural resource limits.', '1309.7626-3-49-1': 'The choice of investors thus needs to be constrained at the expense of having to sell electricity at higher prices.', '1309.7626-3-49-2': 'Therefore REGs were introduced in scenario j in China that prevent the construction of new coal power stations unless they are equipped with CCS.', '1309.7626-3-49-3': 'This additional policy forces additional diversity in the Chinese technology mix, bringing down global emissions to 90% below the 1990 level without early scrapping.', '1309.7626-3-49-4': 'Note that it is possible that under different scenarios of TSs, FiTs and REGs, the CO[MATH]P necessary for these emissions reductions could be lower, requiring further investigations in this complex parameter space.', '1309.7626-3-49-5': 'Total cumulative emissions for the time period 2000-2050 in scenario j (in the baseline) are of 1603 Gt (2321 Gt), given that other sectors do not change their technologies significantly, of which 350 Gt (893 Gt) originate from the electricity sector alone.', '1309.7626-3-50-0': '## Climate change projections', '1309.7626-3-51-0': 'Global emissions from all sectors in scenarios [MATH] to [MATH] were fed into the GENIEem in order to calculate the resulting CO[MATH] concentrations, which themselves were supplied to PLASIM-ENTSem in order to explore their climate change impacts.', '1309.7626-3-51-1': 'Figure [REF], top panel, displays global CO[MATH] emissions for all scenarios from E3MG including all fuel combustion emissions from all users as well as exogenous trends of emissions from all other sectors, obtained from the EDGAR database.', '1309.7626-3-51-2': 'While the changes observed include those in power sector emissions of fig. [REF], they also include modest changes in other sectors (e.g. industry) occurring due to CO[MATH]P for all fuel users subject to the emissions trading scheme (taken to function similarly to the current EU-ETS).', '1309.7626-3-52-0': 'In order to run the climate model emulator, emissions were required up to 2100.', '1309.7626-3-52-1': 'E3MG projections beyond 2050 are fairly uncertain and were not used, and the trends to 2050 were extrapolated to 2100.', '1309.7626-3-52-2': 'Polynomial extrapolations were assumed for scenarios a,e,i while emissions were considered stable beyond 2050 for all other scenarios.', '1309.7626-3-52-3': 'While this is not a particularly sophisticated extrapolation method, accuracy is not the primary goal here since the decarbonisation of other sectors of anthropogenic emissions, responsible for most emissions left in scenario j of order 30 Gt, was not specifically modelled, but represent significant potentials for further reductions (i.e. low carbon technology diffusion in transport and industry).', '1309.7626-3-53-0': 'The middle panel of fig. [REF] shows the resulting atmospheric CO[MATH] concentrations, with uncertainty shown as a blue area.', '1309.7626-3-53-1': 'It was observed that scenario a reaches median values of [MATH] ppm in 2050 while scenario j reaches [MATH] ppm, uncertainty due to the carbon cycle model.', '1309.7626-3-53-2': 'This is above the generally assumed threshold of 450 ppm for maintaining warming below 2[MATH]C.', '1309.7626-3-53-3': 'These concentrations were fed to the climate model emulator, which yielded global warming median temperature changes of (fig. [REF]) between [MATH]C and [MATH]C over pre-industrial levels with a median value of [MATH]C when using the median concentration and only the climate model uncertainty, and between [MATH]C and [MATH]C with the same median when including both carbon cycle and climate model uncertainties.', '1309.7626-3-53-4': 'This therefore could exceed [MATH]C of warming with a 5% chance.', '1309.7626-3-53-5': 'Meanwhile, the electricity decarbonisation scenario yields warming values of between [MATH]C and [MATH]C, median of [MATH]C, with carbon cycle uncertainty only and between [MATH]C and [MATH]C, same median, with both carbon cycle and climate model uncertainties.', '1309.7626-3-53-6': 'The electricity sector decarbonisation scenario thus has a negligible probability of not exceeding [MATH]C of warming.', '1309.7626-3-53-7': ""This indicates that the decarbonisation of the power sector by as much as 90% is insufficient if other sectors such as transport and industry are not specifically targeted by climate policy, in order to avoid 'dangerous' climate change."", '1309.7626-3-54-0': '## Learning cost reductions and energy price dynamics', '1309.7626-3-55-0': 'The uptake of low carbon technologies generate learning cost reductions that alter permanently the technology cost landscape.', '1309.7626-3-55-1': 'Figure [REF] shows world averages of bare technology costs (upper panels) for the baseline and mitigation scenario j, regionally weighed by electricity generation, excluding TSs, CO[MATH]P and FiTs.', '1309.7626-3-55-2': 'These values, when including policy, drive investor choices in both the baseline (left) and the mitigation (right) scenarios.', '1309.7626-3-55-3': 'Roughly speaking, decreases stem from learning-by-doing cost reductions while increases originate from increasing natural resource scarcity with development.', '1309.7626-3-55-4': 'While the cost of PV panels decreases in the baseline scenario mainly due to deployment in Europe, it decreases by more than half its 2008 value in the mitigation scenario where they benefit from FiTs everywhere.', '1309.7626-3-55-5': 'Meanwhile, onshore wind power does come into the mainstream in many regions of the world in the mitigation scenario and does not necessitate support all the way to 2050, where the value of the wind FiTs become near zero or even negative, in which case the policy it is dropped altogether.', '1309.7626-3-55-6': 'In other regions, wind power is limited by resource constrained decreasing capacity factors and corresponding increasing costs.', '1309.7626-3-55-7': 'Other technologies, such as geothermal or wave power (not shown), see very little uptake in this particular mitigation scenario and therefore little cost reductions.', '1309.7626-3-56-0': 'The marginal costs of producing electricity, defined as share-weighted LCOEs, are given for 6 aggregate regions in the lower panels of fig. [REF].', '1309.7626-3-56-1': 'This marginal cost is used in E3MG to construct electricity prices in 21 regions, of which the changes alter electricity consumption.', '1309.7626-3-56-2': 'These are different between regions, stemming from different technology and resource landscapes, where lower marginal costs correspond to higher shares of coal based electricity.', '1309.7626-3-56-3': 'Significant increases are observed in the 90% decarbonisation scenario in all regions, reflecting the cost of the energy transition passed-on to consumers.', '1309.7626-3-57-0': 'The marginal costs of fossil fuels are calculated using estimates of reserves and resources, described in section [REF], and are not highly affected by changes in policy in these scenarios.', '1309.7626-3-57-1': 'In both scenarios oil and gas costs increase significantly up to 2050 in a similar way, but these increases are dampened by the massive accession to unconventional fossil fuel resources (oil sands, heavy oil and shale gas).', '1309.7626-3-57-2': 'This analysis will be given elsewhere in more detail.', '1309.7626-3-57-3': 'Coal costs are moderately affected by changes in demand due large coal resources.', '1309.7626-3-57-4': 'The price of natural uranium ore is stable until 2035 where an increase is observed, generated by increasing scarcity, and at this level of consumption, U resources are projected to run out before 2100 unless technology changes .', '1309.7626-3-58-0': '## Global economic impacts of an 90% reduction scenario', '1309.7626-3-59-0': 'The macroeconomic impacts of scenarios a and j in E3MG is a vast subject beyond the scope of the present paper, and will be explored in detail elsewhere.', '1309.7626-3-59-1': 'In the current analysis, we find that decarbonising the electricity sector by 90% has small economic benefits, generating additional employment, real household income and increases GDP by between 1 and 3% (depending on the region) in comparison to scenario a, broadly consistent with previous similar analyses performed with our model .', '1309.7626-3-59-2': 'This is due to two opposing forces acting against one another: the introduction of low carbon technologies force increases in electricity prices (as seen in fig. [REF]), lowering real household disposable income, while low carbon technology production generates further employment in various industrial sectors, increasing household income.', '1309.7626-3-59-3': 'These are shown to approximately cancel, which is possible as long as labour and capital (investment) resources can be made available .', '1309.7626-3-59-4': 'Our assumptions about capital and labour markets are therefore consistent with our assumptions of energy markets, in that these resources are not assumed currently used optimally and their markets do not automatically produce optimal outcomes.', '1309.7626-3-60-0': 'However, CO[MATH]P generates government income larger than government spending on TSs, the rest being redistributed to households in the form of income tax reductions, increasing further their disposable income.', '1309.7626-3-60-1': 'The resulting impacts are therefore of increased household income and consumption in comparison to the baseline and thus higher GDP.', '1309.7626-3-60-2': 'It is to be noted however that there are winners and losers in this picture both in terms of sectors and world regions, depending how much they depend on activities of the oil, gas and coal sectors.', '1309.7626-3-61-0': '## Local projections of power generation and emissions', '1309.7626-3-62-0': 'It proves instructive to analyse electricity technology landscapes in individual regions of the world in FTT:Power-E3MG, for policy analysis and for better understanding the nature of technology lock-ins and the restricted local ability to change in a diffusion perspective.', '1309.7626-3-62-1': 'This is presented in fig. [REF] for six key regions or countries: North America, Europe, China, India, Brazil and the Rest of the World, which have different electricity landscapes stemming from differing energy policy strategies and engineering traditions historically, as well as natural resource landscapes.', '1309.7626-3-62-2': ""National strategies, reflecting local engineering specialisation related to technology lock-ins, is a natural outcome of this model's structure (eq. [REF]), which reproduces the better ability of dominating industries to capture the market despite costs."", '1309.7626-3-63-0': 'Renewable energy systems are more exploited in Europe than anywhere else in the world, except in Brazil, where hydroelectricity dominates.', '1309.7626-3-63-1': 'Europe also sees the most diverse electricity sector, with large amounts of wind power already in the baseline scenario, predominantly in northern Europe and the British Isles, large amounts of nuclear power in France, and some solar power in Germany.', '1309.7626-3-63-2': 'Coal fired electricity is mostly phased out before 2050 in the 90% scenario, generating significant emissions reductions.', '1309.7626-3-64-0': 'North America features higher use of fossil fuels for power production than Europe.', '1309.7626-3-64-1': 'However, while E3MG projects a larger potential for consumption reductions, large opportunities for diversification also emerge with significant potentials of renewable energy.', '1309.7626-3-64-2': 'Bioenergy with CCS generates a large contribution to emissions reductions.', '1309.7626-3-65-0': 'China and India have very low technology diversity and important fossil fuel lock-ins.', '1309.7626-3-65-1': 'The amount of coal used in China in the baseline is responsible for emissions of 10 out of 30 Gt of global emissions in 2050.', '1309.7626-3-65-2': 'Diversification proves difficult given the scale of the rate of increase in consumption; breaking the coal lock-in requires REGs in China to phase out building new coal generators.', '1309.7626-3-65-3': 'Large scale diffusion of renewables is slow and retrofitting CCS to coal generators offers a useful alternative.', '1309.7626-3-65-4': 'Electricity demand reductions are very large, which requires further investigations for fuel poverty and other social implications.', '1309.7626-3-66-0': 'In Brazil, even though hydroelectricity is not the least expensive resource, it nevertheless dominates, another form of technology lock-in.', '1309.7626-3-66-1': 'This is typical of a national engineering tradition dominated by a technology for decades.', '1309.7626-3-66-2': 'Brazil is projected to persist developing its hydropower capacity despite higher costs and a decreasing potential, until the cost becomes prohibitively expensive and only less productive hydro resources remain.', '1309.7626-3-67-0': 'The rest of the world includes predominantly countries where the diversity of existing technologies is low, and persists in this direction.', '1309.7626-3-67-1': 'It features large amounts of oil use for electricity despite high oil prices, due to restricted access to technology or fossil fuel subsidies, which are not successfully phased out despite being the least cost-effective way of producing electricity.', '1309.7626-3-67-2': 'Coal based electricity makes the dominant contribution to emissions in the baseline, the rest divided between oil and gas fired power stations, for a total of 12 out of 30 Gt of global emissions in 2050 in the baseline.', '1309.7626-3-67-3': 'In the mitigation scenario, a significant additional hydroelectricity potential is developed, and coal is replaced by gas turbines, which are eventually retrofitted with CCS.', '1309.7626-3-68-0': '# Policy and strategic consequences', '1309.7626-3-69-0': '## Synergy between policy instruments', '1309.7626-3-70-0': 'This paper shows that in a coupled energy-economy-environment model that does not assume economic equilibrium or use technology cost-optimisation, the impact of policy instruments can be different if used individually or in combinations: the impact of combined policy packages does not correspond to the sum of the impacts of individual instruments.', '1309.7626-3-70-1': 'Thus significant synergies exist between policy instruments.', '1309.7626-3-70-2': 'In this regard we showed that CO[MATH] alone is not likely capable of delivering sufficient emissions reductions, in contrast to the position supported by policy-makers and economists following the neoclassical environmental economics view of the pricing the externality ; it requires to be combined with TSs, FiTs and REGs.', '1309.7626-3-70-3': 'Relying on CO[MATH]P alone even up to 400 2008/tCO[MATH] is likely to lead to a status quo in the technology mix while delivering very expensive electricity to consumers.', '1309.7626-3-70-4': 'Similarly, TSs and FiTs on their own have little impact unless they are combined with sufficiently high CO[MATH]P.', '1309.7626-3-71-0': 'We furthermore suggest that particular combinations of policy instruments can produce such strong synergy that reductions of electricity sector emissions by 90% by 2050 (61% of 2000-2050 cumulative power sector emissions) become possible without early scrapping of electricity generation capital.', '1309.7626-3-71-1': 'Such strong reductions could be complemented by additional reductions in other emissions intensive sectors with additional cross-sectoral synergies: transport, industry and buildings, warranting further work in this area.', '1309.7626-3-71-2': 'If early scrapping is allowed, these reductions could be achieved even faster, but would most likely involve higher costs, requiring further investigations.', '1309.7626-3-71-3': 'Finally, different combinations of the policies analysed here could also lead to 90% emissions reductions, for example with lower CO[MATH]Ps and higher TSs, FiTs and/or more REGs.', '1309.7626-3-72-0': '## The effect of global knowledge spillovers on technology costs: individual vs global coordinated action', '1309.7626-3-73-0': 'Technology systems typically face a vicious cycle: established technologies thrive because they are established, and emerging technologies see barriers to their diffusion due to the lock-in of established technologies.', '1309.7626-3-73-1': 'This is the case unless an emerging technology is a radical improvement over the incumbent, or it benefits from sufficient external support.', '1309.7626-3-73-2': 'Emerging technologies require investment and sales in order to benefit from improvements and economies of scale: repetition, trial and error enables entrepreneurs to improve their products.', '1309.7626-3-73-3': 'They thus require a continuous flow of funds from sales or external investment in order to survive until their products take off on their own in the market.', '1309.7626-3-73-4': 'In the long run, these investments may or may not generate a return, and are thus risky.', '1309.7626-3-73-5': ""Without any investment to bridge the 'technology valley of death', however, they may become failed innovations."", '1309.7626-3-73-6': 'Competitive thresholds are set by incumbent technologies.', '1309.7626-3-74-0': 'Given known learning curves of power systems, a certain additional capacity of emerging technologies such as wind turbines and solar PV panels must be bought and sold in order to bring down their costs to a competitive level set by established technologies such as coal, gas or nuclear power stations.', '1309.7626-3-74-1': 'As we find using FTT:Power-E3MG, this additional capacity necessary to make their cost affordable is very large, and cannot be deployed by a single nation such as Germany or the UK, or even the whole of Europe, for the rest of the world to benefit, particularly true for solar PV.', '1309.7626-3-74-2': 'In contrast, we find that only a concerted global climate policy effort can bring down costs to manageable levels and bring new power technologies into the mainstream, opening very large renewable energy potentials such as that of solar energy.', '1309.7626-3-74-3': 'Such a concerted effort can significantly and permanently alter the global landscape of power technology costs and availability.', '1309.7626-3-74-4': 'We stress that, intellectual property law and border taxes allowing, all countries of the world can benefit from learning cost reductions that may originate from investments and sales occurring elsewhere.', '1309.7626-3-74-5': 'This problem therefore possesses the features of a classic free-rider and collective action problem, where international coordination is the only way by which these cost reductions can take place.', '1309.7626-3-74-6': ""Emerging or developed nations cannot simply 'wait' for climate policy in other nations to generate diffusion and enough learning cost reductions for new technologies to become competitive: without their involvement they might potentially never become competitive."", '1309.7626-3-74-7': 'If the power sector is to decarbonise by 2050, all countries are most likely required to make a contribution to the development of the renewables industry.', '1309.7626-3-75-0': '## Distributional impacts of decarbonisation', '1309.7626-3-76-0': 'A summary of the economic impacts of the decarbonising the power sector was given above, where climate policy moderately benefits the global economy.', '1309.7626-3-76-1': 'It is important to note that there are some winners and losers in the decarbonisation scenario.', '1309.7626-3-76-2': 'The main winners are the construction and engineering sectors, their workforce and their supply chains.', '1309.7626-3-76-3': 'These all benefit from the higher rates of development and deployment of capital-intensive equipment.', '1309.7626-3-76-4': 'The sectors that lose out are those that supply fossil fuels.', '1309.7626-3-76-5': 'We would also expect reduced rates of economic activity in services sectors, where consumers must spend a larger proportion of income on electricity; this is however compensated by tax reductions originating from government income from carbon pricing.', '1309.7626-3-77-0': 'The outcomes for electricity-intensive sectors are less clear.', '1309.7626-3-77-1': 'Companies in these sectors will face higher costs and may see a loss of demand for their products.', '1309.7626-3-77-2': 'However, as the scenarios are global, there are only limited competitiveness effects.', '1309.7626-3-77-3': 'Many, but by no means all, energy-intensive firms also feature in the supply chains for renewables.', '1309.7626-3-77-4': 'Furthermore, imports and exports of fossil fuels change significantly in any decarbonisation scenario, generating further winners (fuel importers) and losers (fuel exporters).', '1309.7626-3-77-5': 'This aspect requires further investigations into the economic impacts of specific nations.', '1309.7626-3-78-0': 'The most important unknown in the decarbonisation scenarios presented concerns the distributional impact of increasing electricity prices to the less wealthy in all regions of the World.', '1309.7626-3-78-1': 'Depending on government policy, this could trap groups of people in energy poverty, in particular people who cannot afford to change their electricity consuming appliances such as heating or cooling devices.', '1309.7626-3-78-2': 'Therefore, climate policy could be complemented with energy end-use technology access programs in order to help people change technology for more efficient systems through the transition without significant loss in their standard of living.', '1309.7626-3-78-3': 'Complementary research on the impact of climate policy onto energy access and the likely strong synergies existing between energy access, climate and economic development policies is likely to shed significant light on these issues.', '1309.7626-3-79-0': '# Conclusion', '1309.7626-3-80-0': 'This work introduces a novel method and analysis of technology diffusion and decarbonisation in the global electricity sector using a coupled model of power technology diffusion, non-equilibrium macroeconomics and climate change.', '1309.7626-3-80-1': 'The use of non-equilibrium models is shown to generate projections significantly different to those created using currently standard cost-optimisation and equilibrium economics approaches.', '1309.7626-3-80-2': 'The relaxation of the cost-optimisation constraint enables scenarios to feature non-optimal technology lock-ins well known to take place in the electricity sector.', '1309.7626-3-80-3': 'Meanwhile, non-equilibrium macroeconometrics enable to explore macroeconomic impacts of energy policy instruments, where investment in low carbon technology generates additional employment and enhanced economic activity as a counterbalance to increasing energy prices during decarbonisation.', '1309.7626-3-81-0': 'Ten scenarios were presented where a rationale is built for the elaboration of composite energy policy instrument combinations with strong synergy.', '1309.7626-3-81-1': 'This involves the analysis of individual policy tools as well as combinations, generating a storyline on how to reduce global emissions to near 90% below their 1990 level.', '1309.7626-3-81-2': 'The demonstration of the existence of strong synergies between policy tools for the diffusion of low carbon technology argues strongly against the use of policy instruments on their own such as carbon pricing, likely to yield little change in the technology composition and unlikely to break the current fossil technology lock-in.', '1309.7626-3-82-0': 'We presented a decarbonisation scenario where a significant contribution to emissions reductions originates from electricity consumption reductions, which happens through changes in the price of electricity as a result of energy policy and changes in production costs.', '1309.7626-3-82-1': 'The remaining emissions reductions stem from changes in the global electricity technology mix itself.', '1309.7626-3-82-2': 'Global warming impacts are provided.', '1309.7626-3-82-3': 'Insight for climate policy-making is given in terms of existing synergies between climate policy instruments and their impact on technology costs through global knowledge spillovers.', '1309.7626-3-82-4': 'This provides a strong argument for global coordination of climate policy in order for all nations to benefit simultaneously from cost reductions for low carbon technologies through research and development and learning-by-doing.'}","{'1309.7626-4-0-0': 'This paper presents an analysis of climate policy instruments for the decarbonisation of the global electricity sector in a non-equilibrium economic and technology diffusion perspective.', '1309.7626-4-0-1': 'Energy markets are driven by innovation, path-dependent technology choices and diffusion.', '1309.7626-4-0-2': 'However, conventional optimisation models lack detail on these aspects and have limited ability to address the effectiveness of policy interventions because they do not represent decision-making.', '1309.7626-4-0-3': 'As a result, known effects of technology lock-ins are liable to be underestimated.', '1309.7626-4-0-4': 'In contrast, our approach places investor decision-making at the core of the analysis and investigates how it drives the diffusion of low-carbon technology in a highly disaggregated, hybrid, global macroeconometric model, FTT:Power-E3MG.', '1309.7626-4-0-5': 'Ten scenarios to 2050 of the electricity sector in 21 regions exploring combinations of electricity policy instruments are analysed, including their climate impacts.', '1309.7626-4-0-6': 'We show that in a diffusion and path-dependent perspective, the impact of combinations of policies does not correspond to the sum of impacts of individual instruments: synergies exist between policy tools.', '1309.7626-4-0-7': 'We argue that the carbon price required to break the current fossil technology lock-in can be much lower when combined with other policies, and that a 90% decarbonisation of the electricity sector by 2050 is affordable without early scrapping.', '1309.7626-4-1-0': '# Introduction', '1309.7626-4-2-0': 'The electricity sector emits 38% of global energy-related greenhouse gases .', '1309.7626-4-2-1': 'Investment planning in the electricity sector is therefore of critical importance to climate-change policy.', '1309.7626-4-2-2': 'Electricity production is an energy sector with some of the longest time scales for technological change, requiring particularly careful planning in order to avoid locking in, for many decades, to heavily emitting systems that could commit society to dangerous levels of global warming .', '1309.7626-4-2-3': 'Meeting emissions targets to prevent warming beyond 2[MATH]C significantly restricts the number of possible pathways of energy sector development .', '1309.7626-4-2-4': 'However warming beyond 2[MATH]C is likely to lead to catastrophic consequences for global ecosystems and food chains, with important repercussions for global human welfare .', '1309.7626-4-2-5': 'Large reductions in greenhouse gas emissions involve significant amounts of technology substitution, most likely large scale socio-technical transitions .', '1309.7626-4-3-0': 'Results of techno-economic studies of climate change mitigation depend strongly on assumptions made concerning technology dynamics .', '1309.7626-4-3-1': 'The majority of studies of energy systems are made using either bottom-up cost-optimisation, or top-down general equilibrium utility optimisation (equilibrium) computational models, or a combination of both.', '1309.7626-4-3-2': 'In these models however, dynamics result mostly from the assumptions about optimisation that underpin the modelling approaches.', '1309.7626-4-4-0': 'In stark contrast with more traditional optimisation-based approaches, this work proposes a new modelling paradigm based exclusively on non-equilibrium dynamics to simulate the impacts of specific policy frameworks, through the economy, onto the environment.', '1309.7626-4-4-1': ""We present an analysis of the global electricity sector with high resolution simulations of technology diffusion dynamics, using the 'Future Technology Transformations' framework (FTT:Power), coupled with non-equilibrium macroeconomics (E3MG), and environmental impacts derived by combining emulators of the climate system (PLASIM-ENTSem) and the carbon cycle (GENIEem)."", '1309.7626-4-5-0': 'Uncoordinated technology dynamics are modelled at the level of diverse profit-seeking investor decisions incentivised by policy under bounded rationality, as opposed to system level optimisation.', '1309.7626-4-5-1': 'This setup enables us to explore the outcomes of particular energy policy tools for technology diffusion, electricity generation, global emissions, climate change and macroeconomic change.', '1309.7626-4-5-2': 'The connection of a diffusion framework to a non-equilibrium model of the global economy opens a very rich world of macroeconomic dynamics and technological change where the impacts of energy policy reveal complex interactions between the energy sector and the economy.', '1309.7626-4-6-0': 'Ten scenarios of the future global power sector up to 2050 are presented, creating a storyline to provide insight for the construction of effective comprehensive energy policy portfolios in the context of non-equilibrium dynamics.', '1309.7626-4-6-1': 'Going beyond carbon pricing only and considering other policies that could help trigger the diffusion of new technologies, particular combinations are found to feature mutual synergies that provide suitable environments for fast electricity sector decarbonisation: up to 90% by 2050.', '1309.7626-4-6-2': 'Macroeconomic dynamics in these scenarios are summarised.', '1309.7626-4-6-3': 'High resolution scenario data are accessible on our website at http://www.4cmr.group.cam.ac.uk/research/FTT/fttviewerwww.4cmr.group.cam.ac.uk/research/FTT/fttviewer, where costs, electricity generation and emissions can be explored in 21 world regions and 24 technologies.', '1309.7626-4-7-0': 'Decarbonisation involves the positive externality associated to the global accumulation of knowledge and experience in scaling up, deploying and using new power technologies.', '1309.7626-4-7-1': 'A classic collective action problem emerges: learning cost reductions for new technologies may only become significant and enable cost-effective diffusion when most nations of the World demonstrate strong coordinated dedication to their deployment.', '1309.7626-4-7-2': 'We show that carbon pricing covering all world regions is a necessary but insufficient component for the success of mitigation action in order to break the current fossil fuel technology lock-in, unless the price is very high.', '1309.7626-4-8-0': 'As has been shown earlier , in the disequilibrium perspective, reductions in power sector emissions may not necessarily imply significant macroeconomic costs (direct and indirect) but instead, could generate additional industrial activity and employment.', '1309.7626-4-9-0': '# Material and methods', '1309.7626-4-10-0': '## Review of the literature', '1309.7626-4-11-0': 'The great majority of studies of energy systems at the global scale are made using cost-optimisation computational models (a social planner approach).', '1309.7626-4-11-1': 'Meanwhile, the economics of climate change are often represented using general equilibrium economic theory , which tend to yield negative macroeconomic impacts of climate change mitigation, one could argue, by construction.', '1309.7626-4-11-2': 'In both these approaches, which together represent the current methodological standard, the assumptions about the nature of agents make the optimisation problem involved tractable.', '1309.7626-4-11-3': 'These assumptions about the nature of behaviour, however, may over-simplify aspects of an inherently complex global energy-economic system that are crucial for climate change mitigation, leaving open the question as to how much results stem from these simplifications, and whether relaxing these constraints changes perspectives.', '1309.7626-4-12-0': 'Cost-optimisation technology models, in normative mode, are still the most powerful tools for finding detailed, lowest-cost future technology pathways that reach particular objectives.', '1309.7626-4-12-1': 'If used for descriptive purposes, they imply a description of agents (investors, consumers) as identical, who possess a degree of information and technology access as well as foresight sufficient to generate pathways that are cost-optimal at the system level, which alternatively corresponds to a controlled degree of coordination between all actors involved in the evolution of the system.', '1309.7626-4-12-2': 'As stated in the Global Energy Assessment , ""A fundamental assumption underlying the pathways is that the coordination required to reach the multiple objectives simultaneously can be achieved"".', '1309.7626-4-12-3': 'While this approach generates a significant simplification to a highly complex system, it may be argued that such a spontaneous emergence of coordination is somewhat unlikely.', '1309.7626-4-12-4': 'For instance liberalised energy markets involve actors free to take their investment and consumption decisions based on their particular circumstances, and are only incentivised by policy.', '1309.7626-4-12-5': 'Thus while optimisation frameworks are valuable for identifying feasible and cost-effective pathways that reach particular objectives at the system level, they do not suggest how exactly to achieve them from a policy standpoint, because they do not specifically model decision making by diverse agents.', '1309.7626-4-12-6': 'Strong coordination is difficult to generate from economic policy instruments, leading to sub-optimal outcomes and technology lock-ins.', '1309.7626-4-13-0': 'Meanwhile, equilibrium economic theory implies that climate change mitigation costs are borne at the expense of consumption or investment elsewhere, leading to detrimental economic impacts, which is not universally agreed to occur .', '1309.7626-4-13-1': 'In particular, equilibrium theory relies on decreasing or constant returns to scale, becoming unstable in the presence of processes with increasing returns such as induced technological change .', '1309.7626-4-13-2': 'Increasing returns also imply the property of path dependence and involve complexity, where new ordering principles can emerge from the interactions between system parts .', '1309.7626-4-13-3': 'Since path dependent systems may not return to equilibrium after disturbances, scenarios diverge from each other for small differences of starting parameters, in a similar way to physical models of the climate.', '1309.7626-4-14-0': 'Equilibrium economic analyses recommend carbon pricing as the single most efficient policy tool to fix the climate market failure, when equated to the social cost of emitting carbon.', '1309.7626-4-14-1': ""However, it is recognised that some new technologies might not successfully bridge the technology innovation 'valley of death' to the marketplace at politically practicable carbon prices without further government support ."", '1309.7626-4-14-2': 'Deriving future scenarios using normative models and equilibrium economics is conceptually inconsistent with the simulation approach of climate science, and hence potentially misleading for many important stakeholders.', '1309.7626-4-15-0': 'A simulation approach without systems optimisation is possible using known technology dynamics and a model for decision-making at the firm level by diverse agents .', '1309.7626-4-15-1': 'Empirically repeatable dynamics are known to exist in scaling up technology systems (e.g. [MATH]-shaped diffusion), and their costs (learning curves), which have been extensively studied for decades .', '1309.7626-4-15-2': 'Declining costs of technology with cumulative experience in scaling them up can be modelled either at the bottom-up scale (learning curves, e.g. , criticised by ), or at the aggregate scale , aspects reviewed by [CITATION].', '1309.7626-4-15-3': 'Meanwhile, mathematical generalisations of diffusion dynamics have been suggested involving dynamic differential equations similar to those in population growth mathematical ecology and demography .', '1309.7626-4-15-4': 'In combination with evolutionary dynamics and evolutionary game theory , this offers an artillery of powerful concepts at the core of evolutionary economics [CITATION].', '1309.7626-4-15-5': ""This emphasises innovation, diffusion and speciation as a source of economic development and growth , the clustering of which is possibly responsible for 'Kondratiev cycles' ."", '1309.7626-4-15-6': ""These are broadly consistent with the 'multi-level perspective' on technology transitions described by [CITATION]."", '1309.7626-4-15-7': 'Furthermore, complexity and path dependence emerge as key concepts to envisage technology dynamics .', '1309.7626-4-16-0': 'Intermediate scale models do exist that introduce investor/consumer diversity in technology choices driving changes in energy supply, end-use and emissions.', '1309.7626-4-16-1': 'For example, agent-based models can be used to represent the multi-level perspective on technology transitions .', '1309.7626-4-16-2': 'Meanwhile models using multinomial logit structures parameterised by survey data provide a natural representation of diversity based on discrete choice theory .', '1309.7626-4-16-3': 'Both approaches provide an appropriate replacement for the neoclassical representative agent.', '1309.7626-4-16-4': 'However, even models with detailed behavioural treatments do not currently have a complete representation of empirically known technology diffusion patterns as arise for instance in energy systems , which stem from both the diversity of choice and industrial dynamics .', '1309.7626-4-16-5': 'Conversely, evolutionary models of technology innovation-diffusion do not have detailed representations of consumer choice and diversity.', '1309.7626-4-16-6': 'But critically, both principles have not yet diffused widely into mainstream global scale integrated climate-energy systems modelling used for informing climate policy .', '1309.7626-4-16-7': 'Including these would generate an improved methodological paradigm , which we propose here.', '1309.7626-4-16-8': 'In such a framework, optimisation as a source of dynamic force and representative agent behaviour is replaced by empirically known innovation-selection-diffusion dynamics with behaviour diversity, which are not characterised by equilibria, but feature complex dynamics.', '1309.7626-4-17-0': '## Technology diffusion in FTT:Power', '1309.7626-4-18-0': 'Emissions reductions in the energy sector can occur through technology substitution, between technologies that produce the same substitutable service (e.g. electricity, heat, etc), through behaviour and practice changes and through reductions in the consumption of that service altogether.', '1309.7626-4-18-1': 'Technological change occurs primarily at the average rate of replacement of existing technology as it ages, which is inversely related to its life span.', '1309.7626-4-18-2': 'However, notwithstanding lifetime considerations, the number of new units of technology of a particular type that can be constructed at any one time can be larger if that industry is in a well established position in the marketplace, with a large production capacity, than if it is emerging.', '1309.7626-4-18-3': 'Even when an emerging technology is (or is made) very affordable, it may not always be accessible to every investor making a choice between available options.', '1309.7626-4-18-4': 'Thus in the analysis of the diffusion capacity of technologies, not only cost considerations as seen by diverse agents come into play, but also a limited access to technology and information, and these principles form the core of FTT:Power (Fig. [REF], see for a general model description, and for a detailed mathematical derivation).', '1309.7626-4-19-0': '[REF] provides a mathematical derivation of the technology dynamics at the heart of the FTT model, given here.', '1309.7626-4-19-1': 'Using the variable [MATH] for the generation capacity market share of a technology, the rate at which shares of one technology type ([MATH]) can be replaced by shares of another type ([MATH]) is proportional to:', '1309.7626-4-20-0': 'The rate at which units of technology [MATH] come to the end of their working life, with death rate [MATH], How many old units of [MATH] require replacement, a fraction [MATH] of the total share of replacements.', '1309.7626-4-20-1': 'The rate at which the construction capacity for technology [MATH] can be expanded, with growth rate [MATH], The market position of technology [MATH], its share of the market [MATH].', '1309.7626-4-21-0': 'This implies the following dynamical equation, [EQUATION] with matrix elements [MATH] expressing the rate of technology diffusion from industrial dynamics, [MATH] expressing the probability of investor preferences and [MATH] providing technical system constraints.', '1309.7626-4-21-1': '[MATH] is the average sectoral rate of technology turnover.', '1309.7626-4-21-2': 'This equation solves to the classic logistic function of time in the special case of two interacting technologies, with diffusion rate equal to [MATH].', '1309.7626-4-21-3': 'But generally, it is complex and non-linear, and generates slow uptakes at small penetration, then fast diffusion at intermediate penetration, before a saturation near full penetration, the three matrices together determining the pace of change.', '1309.7626-4-21-4': 'This complex system cannot be solved analytically but is straightforward to evaluate numerically using time steps.', '1309.7626-4-21-5': 'It corresponds to the replicator dynamics equation in evolutionary theory , equivalent to a Lotka-Volterra set of equations of population dynamics for competing species, also used in mathematical genetics as well as in evolutionary economics : an ordering principle emerging from technology interactions.', '1309.7626-4-22-0': '## Learning-by-doing and path dependence', '1309.7626-4-23-0': 'Profit-seeking investor choices [MATH] are driven by cost differences, and these decrease over time as technologies diffuse and follow learning curves , generating increasing returns to adoption.', '1309.7626-4-23-1': 'Learning-by-doing cost reductions stem from the accumulation of technical knowledge on production and economies of scale in expansion of productive capacity for specific technologies.', '1309.7626-4-23-2': 'Technology costs are taken here to apply in globalised firms and markets.', '1309.7626-4-23-3': 'Emerging technologies have fast cost reductions (e.g. solar panels) while established systems see very little change (e.g. coal plants).', '1309.7626-4-23-4': 'Cost reductions are decreasing functions of cumulative investment, not time, and they do not occur if no investment is made.', '1309.7626-4-23-5': 'Learning thus interacts with diffusion where it incentivises further uptake, which generates further learning and so on, a highly self-propagating effect which can lead to sudden technology avalanches.', '1309.7626-4-24-0': 'Such increasing returns to adoption give the crucial property of path dependence to FTT:Power .', '1309.7626-4-24-1': 'As technologies diffuse following investor choices, the full landscape of technology costs continuously changes, and investor preferences thus change.', '1309.7626-4-24-2': 'These changes are permanent and determined by past investments, and therefore by the full history of the market, and different futures emerge, depending on investment and policy choices along the way.', '1309.7626-4-24-3': 'Technology costs and learning rates are given in [CITATION], with more detail on the 4CMR website.', '1309.7626-4-25-0': '## Natural resource use', '1309.7626-4-26-0': 'The diffusion of power systems can only occur in areas where energy resources are available, for instance windy areas for wind power, or natural water basins and rivers for hydroelectric dams.', '1309.7626-4-26-1': 'Higher productivity sites offer lower costs of electricity production, and tend to be chosen first by developers.', '1309.7626-4-26-2': 'Assuming this, the progression of renewable energy systems development follows increasing marginal costs of production for potential new systems as only resources of ever lower productivity are left to use (decreasing returns to adoption).', '1309.7626-4-26-3': 'This is well described by cost-supply curves .', '1309.7626-4-26-4': 'For this purpose, a complete set of curves was previously estimated from combined literature and data for 190 countries and 9 types of renewable resources .', '1309.7626-4-26-5': 'These were aggregated for the 21 regions of E3MG ([REF]).', '1309.7626-4-26-6': 'This produced 189 cost-supply curves that are used to constrain the expansion of renewable systems in FTT:Power.', '1309.7626-4-26-7': 'The consumption of non-renewable resources is however better represented using a depletion algorithm, described next.', '1309.7626-4-27-0': '## Fossil fuel cost dynamics', '1309.7626-4-28-0': 'Non-renewable energy resources lying in geological formations have an arbitrary value that depends on their cost of extraction, but also on the dynamics of the market.', '1309.7626-4-28-1': 'To their cost of extraction is associated a minimum value that the price of the commodity must take in order for the extraction to be profitable.', '1309.7626-4-28-2': 'These costs are however distributed over a wide range depending on the nature of the geology (e.g. tar sands, ultra-deep offshore, shale oil and gas, etc).', '1309.7626-4-28-3': 'Thus, given a certain demand for the commodity, the price is a function of the extraction cost of the most expensive resource extracted in order to supply the demand, and it separates what is considered reserves from resources.', '1309.7626-4-28-4': 'As reserves are gradually consumed, the marginal cost increases generating a commodity price increase that unlocks the exploitation of resources situated in ever more difficult locations with higher extraction costs.', '1309.7626-4-28-5': 'For example, tar sands became economical and saw massive expansion above a threshold oil price of around 85-95/boe .', '1309.7626-4-28-6': 'Thus, to any commodity demand path in time will correspond a path dependent commodity price.', '1309.7626-4-28-7': 'The algorithm used here is described with an analysis in [CITATION], relying on data from [CITATION].', '1309.7626-4-28-8': 'In FTT:Power, this model is used to determine fuel costs for fossil fuel and nuclear based power technologies in global markets.', '1309.7626-4-29-0': '## Modelling the global economy: E3MG', '1309.7626-4-30-0': 'E3MG (and variant E3ME) is an out-of-equilibrium macroeconometric model of the global economy that has been used widely for studies of climate change mitigation macroeconomics [CITATION].', '1309.7626-4-30-1': 'It evaluates the parameters of 28 econometric equations using data from 1971 to 2010, and extrapolates these equations between 2010 and 2050.', '1309.7626-4-30-2': 'The model features a high resolution: its equations are evaluated for 21 regions of the world, 43 industrial sectors, 28 sectors of consumption, 22 fuel users, and 12 fuel types.', '1309.7626-4-30-3': 'Sectors are interrelated with dynamic input-output tables.', '1309.7626-4-30-4': 'The model does not optimise economic resources but incorporates endogenous growth and endogenous technical change.', '1309.7626-4-30-5': ""This is done following Kaldor's theory of cumulative causation (, see also ), where Technology Progress Indicators (TPIs) are created by cumulating past investments [MATH] and RD spending using a relative time-weighting, [EQUATION]"", '1309.7626-4-30-6': 'Such TPIs are used in the industrial prices, international trade and employment regressions.', '1309.7626-4-30-7': 'Lower prices incentivises higher consumption, and thus industrial investment and RD expenditures for production capacity expansions, which lead to lower prices and so on, producing a self-reinforcing cycle of cumulative causation of knowledge accumulation.', '1309.7626-4-30-8': 'Including accumulated investment and RD makes E3MG non-linear, path-dependent and hysteretic, and thus far from equilibrium.', '1309.7626-4-30-9': 'E3MG region definitions are given in [REF].', '1309.7626-4-31-0': '## Endogenous technical change and energy price-demand interactions', '1309.7626-4-32-0': 'The demand for electricity depends on its price, and it is well known that in situations of high electricity prices, people may strive to find more effective ways to use their income, preferring to invest in more efficient technology, perceived as a worthwhile tradeoff, or to simply reduce their consumption.', '1309.7626-4-32-1': 'When the electricity supply technology mix changes, the minimum price at which electricity can be profitably sold also changes, and with such price changes, the demand for electricity changes.', '1309.7626-4-32-2': 'For example, when carbon pricing or feed-in tariffs are used to ensure access of expensive renewables into the grid, the price of electricity increases, affecting consumer demand and behaviour.', '1309.7626-4-32-3': 'Thus, reductions in emissions originate from both a change in the carbon intensity of the power sector and changes in the demand for electricity.', '1309.7626-4-32-4': 'These aspects of energy economics are prominent in this work, responsible for a significant fraction of our calculated emissions reductions in scenarios of climate policy.', '1309.7626-4-33-0': 'Electricity demand is modelled in E3MG using an econometric equation that incorporates a contribution from spillovers from investment and RD spending in other sectors (, see also the E3ME manual, ).', '1309.7626-4-33-1': 'Since new investments tend to involve technologies with higher energy efficiencies and because the turnover of capital does not allow return to old technology, here the TPI is formed by cumulating positive increases in investment and RD, which thus cannot decrease.', '1309.7626-4-33-2': 'The energy demand equation takes the form [EQUATION] where for fuel [MATH] and region [MATH], [MATH] is (in log space) the fuel demand, [MATH] represents sectoral output, [MATH] relative prices.', '1309.7626-4-33-3': 'Economic feedbacks between FTT:Power and E3MG occur with four quantities: electricity prices, fuel use, power technology investments and tax revenue recycling.', '1309.7626-4-34-0': '## Emulating large models of the natural world', '1309.7626-4-35-0': 'Detailed models of the Earth system are highly dynamical, complex and computationally demanding.', '1309.7626-4-35-1': ""An efficient way to integrate dynamic responses to inputs from other models is to create reduced-order statistical representations of model outputs ('emulators') which can be used as surrogate models for coupling applications."", '1309.7626-4-35-2': 'Emulators provide a method of analysing the otherwise intractable cascade of uncertainty across multiple complex systems.', '1309.7626-4-35-3': ""This approach was used here in order to obtain representations of the planet's carbon cycle and its climate system, by emulating data produced by the large models PLASIM-ENTS and GENIE-1 ."", '1309.7626-4-35-4': '[REF] provides a detailed account of the procedure.', '1309.7626-4-36-0': 'E3MG-FTT emissions in each scenario were fed to the carbon cycle emulator, in order to obtain a trace of CO[MATH] concentrations within an uncertainty range.', '1309.7626-4-36-1': 'This trace with uncertainty was fed to the emulator of the climate system in order to obtain a trace of future global warming and climate change within an uncertainty range, which thus cascades the uncertainty of both models.', '1309.7626-4-37-0': '# Results and discussion', '1309.7626-4-38-0': '## Scenario creation and policy instruments', '1309.7626-4-39-0': 'Ten scenarios of electricity policy assumptions of different types and resulting technology mix and emissions up to year 2050 were created with FTT:Power-E3MG, lettered a to j (Fig. [REF]).', '1309.7626-4-39-1': 'These all lead to different futures for the global power sector and different CO[MATH] emission profiles.', '1309.7626-4-39-2': 'It is impractical to reproduce all the information of these simulations in this paper, and therefore a summary of the results is given here, the details having been made available on the 4CMR website, where they can be displayed in terms of the full resolution of 21 world regions and 24 power technologies, for policy assumptions, electricity generation, emissions and levelised costs.', '1309.7626-4-39-3': 'Four energy policy tools were explored: carbon pricing/taxing, technology subsidies, feed-in tariffs (FiTs) and direct regulations.', '1309.7626-4-39-4': 'Individual tools and various combinations were explored, a summary given in figure [REF].', '1309.7626-4-39-5': 'By gradually elaborating various policy frameworks, a scenario was found where power sector emissions are reduced by 90% below the 1990 level, involving all four policy instruments used simultaneously.', '1309.7626-4-39-6': 'Emissions are fed to the carbon cycle and climate model emulators GENIEem and PLASIM-ENTSem in order to determine the resulting atmospheric CO[MATH] concentration and average global warming, for these scenarios where other sectors are not targeted by climate policy.', '1309.7626-4-40-0': 'The nature of FiTs here is that access to the grid at a competitive price is ensured (a price higher than the consumer price), the difference being paid by the grid and passed on to consumers through the price of electricity.', '1309.7626-4-40-1': 'The consumer price of electricity is raised by just the amount that makes this economically viable.', '1309.7626-4-40-2': 'The consumer price in the model is derived from an averaged Levelised Cost of Electricity (LCOE), [EQUATION] with [MATH] the electricity generation.', '1309.7626-4-40-3': ""The LCOE as perceived by investors when a FiT exists includes an 'effective subsidy' given by the grid that covers the difference between the levelised cost of a technology and the consumer price of electricity plus a margin (investors here may be corporate or homeowners)."", '1309.7626-4-40-4': 'In the case of carbon pricing, the LCOE calculation that investors are assumed to perform includes a carbon cost component, and the price of carbon is passed on to consumers through the price of electricity.', '1309.7626-4-40-5': 'Thus the price of electricity also increases with the carbon price unless emitting technologies are phased out.', '1309.7626-4-41-0': 'Technology subsidies are fractions of the capital costs of low carbon technologies that are paid by the government, reducing the LCOE that investors face.', '1309.7626-4-41-1': 'These are defined exogenously for every year up to 2050 and are phased out before then, after which it is hoped that the technology cost landscape becomes permanently altered such that technologies do not need to be indefinitely subsidised.', '1309.7626-4-41-2': 'Regulations refer to controlling the construction of new units of particular technologies, and can be used to phase out particular types of systems.', '1309.7626-4-41-3': 'When a regulation is applied to a technology category, no new units are built but existing ones are left to operate until the end of their lifetime.', '1309.7626-4-42-0': 'Fig. [REF] summarises the result of the policy tools exploration.', '1309.7626-4-42-1': 'Electricity generation by technology type is given in the series of panels to the left of each pair, while emissions are given on the right.', '1309.7626-4-42-2': 'The vertical dashed lines indicate the start of the simulations in 2008, and trends to the left of this line are historical data from the [CITATION].', '1309.7626-4-42-3': 'The horizontal dashed lines indicate the 1990 levels of electricity demand and emissions.', '1309.7626-4-42-4': 'Dashed curves correspond to the baseline values for comparison.', '1309.7626-4-42-5': 'In all scenarios excluding the baseline, policy schemes generate both a reduction of electricity consumption and emissions.', '1309.7626-4-42-6': 'Consumption reduces due to increases in the price of electricity, through the energy demand econometric equation of E3MG, which contributes significantly to emissions reductions.', '1309.7626-4-42-7': 'All additional emissions reductions are due to changes in fuels used associated to changes of technologies.', '1309.7626-4-42-8': 'CO[MATH] emission levels in 2050 with respect to the 1990 level are given in percent values.', '1309.7626-4-43-0': '## Climate policy for achieving 90 reductions in power sector emissions', '1309.7626-4-44-0': 'The baseline scenario (Fig. [REF] panel a), which involves maintaining current policies until 2050, leads to global power sector emissions in 2050 of 30 GtCO[MATH]/y, 318 above the 1990 level, and total emissions of 65 GtCO[MATH]/y. Cumulative emissions for the time span 2000-2050 amount to 2321 GtCO[MATH].', '1309.7626-4-44-1': 'According to the model, this pathway is likely to commit the planet to a warming that exceeds 4[MATH]C above pre-industrial levels in around 2100 (Fig. [REF] and section [REF] below, with high probability), consistent with [CITATION] and [CITATION].', '1309.7626-4-44-2': ""In view of finding ways to reduce the power sector's share of these emissions and to limit global warming, we searched areas of policy space for effective abatement in the short time span."", '1309.7626-4-45-0': 'The first option explored (panel b) was to use regulations to prevent the construction of new coal power plants worldwide, the systems with highest emissions ([MATH]1 ktCO[MATH]/GWh), unless they are equipped with carbon capture and storage (CCS).', '1309.7626-4-45-1': 'This results mostly in a transfer from a coal lock-in to a gas lock-in, reducing global emissions approximately to the 1990 level, largely insufficient for meeting the 2[MATH]C target.', '1309.7626-4-46-0': 'The second option was to use carbon pricing as a unique tool, with different price values for different regions covering all world regions shown in Fig. [REF], between 100 and 200 2008/tCO[MATH] in 2050 (panel c) and between 200 and 400 2008/tCO[MATH] (panel d).', '1309.7626-4-46-1': 'This measure, mostly generating reductions in electricity consumption due to higher electricity prices, yields emissions of around 65% above the 1990 level and to 9% below the 1990 level, respectively.', '1309.7626-4-46-2': 'This modest impact suggests that carbon pricing on its own requires very high carbon prices in order to generate significant reductions, or that it is simply insufficient.', '1309.7626-4-46-3': 'However as we show now, combinations of policies achieve this much more effectively.', '1309.7626-4-47-0': 'As a first combination of policies, FiTs (wind and solar) and technology subsidies (all other low carbon technologies except wind and solar) were introduced without carbon pricing (panel e), of order 30-50% of capital costs for technology subsidies and feed-in prices 5-15% above the electricity price for FiTs.', '1309.7626-4-47-1': 'The fine details depend on regions and technologies, see the inset of Fig. [REF] for a world average or the data on our website for details.', '1309.7626-4-47-2': 'This generates very modest uptakes of low carbon technologies and thus small changes in emissions compared to the baseline, 276% above 1990.', '1309.7626-4-47-3': 'This is due to the low cost of producing electricity using fossil fuels in comparison to all other technologies, in particular coal, and therefore without very high subsidies, carbon pricing is necessary in order to bridge this cost difference.', '1309.7626-4-48-0': 'Scenario [MATH] shows the use of carbon pricing up to 200/tCO[MATH] with FiTs, the latter generating very little change over scenario c. Using carbon pricings of up to 200/tCO[MATH] in combination with technology subsidies and FiTs in all world regions (panel g) yields emissions of 32 above the 1990 level, still insufficient.', '1309.7626-4-48-1': 'With carbon pricing of up to 400/tCO[MATH] in combination with the same set of technology subsidies and FiTs (panel h), reductions are much larger, 46 below the 1990 level.', '1309.7626-4-48-2': 'This indicates how the impact of policy combinations may be larger than the sum of the impacts of its components taken separately, offering significant potential synergies.', '1309.7626-4-49-0': 'A scenario was explored where only the developed world applies the stringent climate policies of scenario h (panel i), in which it is hoped that this generates enough investment to bring the costs of low carbon technologies down into the mainstream, thus becoming accessible to developing or under-developed countries.', '1309.7626-4-49-1': 'We see no noticeable uptake of new technology in these countries, costs remaining unaccessible especially in comparison with coal based technologies, and as a consequence global emissions remain at 204 above the 1990 level.', '1309.7626-4-50-0': 'A significant amount of the remaining power sector emissions in scenario h reside in China (79), where the lock-in of coal technology is very difficult to break given the near absence of alternatives with the exception of hydroelectricity, which is driven to its natural resource limits.', '1309.7626-4-50-1': 'The choice of investors thus needs to be constrained at the expense of having to sell electricity at higher prices.', '1309.7626-4-50-2': 'Therefore regulations were introduced in scenario j in China that prevent the construction of new coal power stations unless they are equipped with CCS.', '1309.7626-4-50-3': 'This additional policy forces additional diversity in the Chinese technology mix, bringing down global emissions to 90% below the 1990 level without early scrapping.', '1309.7626-4-50-4': 'Note that it is possible that under different scenarios of technology subsidies, FiTs and regulations, the carbon price necessary for these emissions reductions could be lower, requiring further investigations in this complex parameter space.', '1309.7626-4-50-5': 'Total cumulative emissions for the time period 2000-2050 in scenario j (in the baseline) are of 1603 Gt (2321 Gt), given that other sectors do not change their technologies significantly, of which 350 Gt (893 Gt) originate from the electricity sector alone.', '1309.7626-4-51-0': '## Climate change projections', '1309.7626-4-52-0': 'Global emissions from all sectors in scenarios [MATH] to [MATH] were fed into the carbon cycle emulator (GENIEem) in order to calculate the resulting CO[MATH] concentrations with their uncertainty range, which themselves were supplied to the climate system emulator (PLASIM-ENTSem) in order to find out their climate change impacts with climate uncertainty, cascading the uncertainty of the carbon cycle into that of the climate system.', '1309.7626-4-52-1': 'Figure [REF], top panel, displays global CO[MATH] emissions for all scenarios of Fig. [REF], including however all fuel combustion emissions from endogenous sources in the model as well as exogenous trends of emissions for non-fuel-related sectors (e.g. land use), obtained from the EDGAR database.', '1309.7626-4-52-2': 'While the changes modelled include those in power sector emissions of Fig. [REF], they also include modest changes in other sectors (e.g. industry) occurring due to carbon pricing for all fuel users subject to the emissions trading scheme and due to changes in economic activity.', '1309.7626-4-53-0': 'In order to run the climate model emulator, emissions were required up to 2100.', '1309.7626-4-53-1': 'In complex models such as E3MG or climate models, uncertainty increases with time span from the present.', '1309.7626-4-53-2': 'This currently makes convergence more difficult in E3MG beyond 2050, especially in scenarios of stringent climate policy which may lead the model to venture near the boundary of the behavioural space defined by its econometric relationships prescribed by data prior to 2010.', '1309.7626-4-53-3': 'However the primary interest in this work resides in assessing the impacts of near term policy action on the future state of the world.', '1309.7626-4-53-4': 'Since the baseline scenario emissions trend is very nearly linear, it was extrapolated with a polynomial to 2100, as well as those of scenarios [MATH] and [MATH].', '1309.7626-4-53-5': 'All other scenarios feature stabilised emissions in 2050 ([MATH], [MATH], [MATH], [MATH], [MATH], [MATH] and [MATH]), and thus their 2050 emissions values were assumed to be maintained constant up to 2100.', '1309.7626-4-54-0': 'The middle panel of Fig. [REF] shows the resulting atmospheric CO[MATH] concentrations, with uncertainty given as a blue area.', '1309.7626-4-54-1': 'It was observed that scenario a already reaches a median value of [MATH] ppm in 2050 while scenario j reaches [MATH] ppm.', '1309.7626-4-54-2': 'This is above the generally agreed threshold of 450 ppm for maintaining warming below 2[MATH]C.', '1309.7626-4-54-3': 'These concentrations with uncertainty were fed to the climate model emulator (Fig. [REF], bottom).', '1309.7626-4-54-4': 'For the baseline scenario, this yielded global warming median temperature changes of between [MATH]C and [MATH]C over pre-industrial levels with a median value of [MATH]C when using the median concentration and only the climate model uncertainty, and between [MATH]C and [MATH]C with the same median when cascading the carbon cycle uncertainty into that of the climate model.', '1309.7626-4-54-5': 'This therefore could in principle exceed as high as [MATH]C of warming with a low probability.', '1309.7626-4-55-0': 'Meanwhile, the electricity decarbonisation scenario [MATH] yields warming values of between [MATH]C and [MATH]C, median of [MATH]C, with carbon cycle uncertainty only and between [MATH]C and [MATH]C, same median, with both carbon cycle and climate model uncertainties.', '1309.7626-4-55-1': 'The electricity sector decarbonisation scenario thus has a negligible probability of not exceeding [MATH]C of warming.', '1309.7626-4-55-2': ""This indicates that the decarbonisation of the power sector by as much as 90% is insufficient if other sectors such as transport and industry are not specifically targeted by climate policy, in order to avoid 'dangerous' climate change."", '1309.7626-4-56-0': '## Learning cost reductions and energy price dynamics', '1309.7626-4-57-0': 'The uptake of low carbon technologies generate learning cost reductions that alter permanently the technology cost landscape.', '1309.7626-4-57-1': 'Figure [REF] shows world averages of bare technology costs (upper panels) for the baseline and mitigation scenario j, regionally weighted by electricity generation, excluding technology subsidies, the carbon price and FiTs.', '1309.7626-4-57-2': 'These values, when including policy, drive investor choices in both the baseline (left) and the mitigation (right) scenarios.', '1309.7626-4-57-3': 'Roughly speaking, decreases stem from learning-by-doing cost reductions while increases originate from increasing natural resource scarcity with development.', '1309.7626-4-57-4': 'While the cost of PV panels decreases in the baseline scenario mainly due to deployment in Europe, it decreases by more than half its 2008 value in the mitigation scenario where they benefit from FiTs everywhere.', '1309.7626-4-57-5': 'Meanwhile, onshore wind power does come into the mainstream in many regions of the world in the mitigation scenario and does not necessitate support all the way to 2050, where the value of the wind FiTs become near zero or even negative, in which case the policy is dropped altogether.', '1309.7626-4-57-6': 'In other regions, wind power is limited by resource constrained decreasing capacity factors and corresponding increasing costs.', '1309.7626-4-57-7': 'Other technologies, such as geothermal or wave power (not shown), see very little uptake in this particular mitigation scenario and therefore little cost reductions.', '1309.7626-4-58-0': 'The costs of producing electricity, defined as share-weighted average LCOEs, are given for 6 aggregate regions in the lower panels of Fig. [REF].', '1309.7626-4-58-1': 'Such a marginal cost is used in E3MG to construct electricity prices in 21 regions, of which the changes alter electricity consumption.', '1309.7626-4-58-2': 'These are different between regions, stemming from different technology and resource landscapes, where lower marginal costs correspond to higher shares of coal based electricity.', '1309.7626-4-58-3': 'Significant increases are observed in the 90% decarbonisation scenario in all regions, reflecting the cost of the energy transition passed-on to consumers.', '1309.7626-4-59-0': 'The marginal costs of fossil fuels are calculated using estimates of reserves and resources, described in section [REF], and are not highly affected by changes in policy in these scenarios.', '1309.7626-4-59-1': 'In both scenarios oil and gas costs increase significantly up to 2050 in a similar way, but these increases are dampened by the massive accession to unconventional fossil fuel resources (oil sands, heavy oil and shale gas).', '1309.7626-4-59-2': 'This analysis will be expanded elsewhere.', '1309.7626-4-59-3': 'Coal costs are only moderately affected by changes in demand due to large coal resources.', '1309.7626-4-59-4': 'The cost of natural uranium ore is stable until 2035 where an increase is observed, generated by increasing scarcity, and at this level of consumption, U resources are projected to run out before 2100 unless technology changes .', '1309.7626-4-60-0': '## Global economic impacts of a 90% reduction scenario', '1309.7626-4-61-0': 'The macroeconomic impacts of scenarios a and j in E3MG is a vast subject beyond the scope of the present paper, and will only be summarised here.', '1309.7626-4-61-1': 'We find that decarbonising the electricity sector by 90% has moderate economic benefits, generating additional employment, real household income and increases in GDP of between 1 and 3% (depending on the region) in comparison to scenario a, broadly consistent with previous similar analyses performed with the model .', '1309.7626-4-61-2': 'This is due to two opposing forces acting against one another: the introduction of low carbon technologies force increases in electricity prices (as seen in Fig. [REF]), lowering real household disposable income, while low carbon technology production generates further employment in various industrial sectors, increasing household income.', '1309.7626-4-61-3': 'These were observed to roughly cancel each other out, which is possible as long as labour and capital (investment) resources can be made available .', '1309.7626-4-62-0': 'In our scenarios, carbon pricing generates government income larger than government spending on technology subsidies, the rest being redistributed to households in the form of income tax reductions, increasing further their disposable income.', '1309.7626-4-62-1': 'The resulting impacts are therefore of increased household income and consumption in comparison to the baseline and thus higher GDP.', '1309.7626-4-62-2': 'It is to be noted however that there are winners and losers in this picture both in terms of sectors and world regions, depending how much they depend on activities of the oil, gas and coal sectors.', '1309.7626-4-63-0': 'In this disequilibrium demand-led perspective, our assumptions about capital and labour markets are consistent with our assumptions of energy markets, in that these resources are not assumed currently used optimally and their markets do not automatically produce optimal outcomes.', '1309.7626-4-63-1': 'This means that excess finance as well as unemployment exist in the model.', '1309.7626-4-63-2': 'This contrasts with the results of many other economic models used to assess the economic impacts of climate change mitigation , which tend to give moderate negative impacts.', '1309.7626-4-63-3': 'The main difference stems precisely from assumptions over economic resources: in general equilibrium theory, displacing economic resources that are optimally used can only lead to effects detrimental to the economy (crowding out effects).', '1309.7626-4-63-4': 'However unemployment does exist in the world economy, and it is not generally agreed that investment resources are currently used to their utmost potential .', '1309.7626-4-63-5': 'Further research into this issue is crucial and requires modelling the global financial sector, absent in equilibrium theory, but also not specifically treated in our model.', '1309.7626-4-64-0': '## Local scenarios of power generation and emissions', '1309.7626-4-65-0': 'It proves instructive to analyse electricity technology landscapes in individual regions of the world in FTT:Power-E3MG, for policy analysis and for better understanding the nature of technology lock-ins and the restricted local ability to change in a diffusion perspective.', '1309.7626-4-65-1': 'This is presented in Fig. [REF] for six key regions or countries: North America, Europe, China, India, Brazil and the Rest of the World, which have different electricity landscapes stemming from differing energy policy strategies and engineering traditions historically, as well as natural resource endowments.', '1309.7626-4-65-2': ""National strategies, reflecting local engineering specialisation related to technology lock-ins, is a natural outcome of this model's structure (eq. [REF]), which reproduces the better ability of dominating industries to capture the market despite costs."", '1309.7626-4-66-0': 'Renewable energy systems are more exploited in Europe than anywhere else in the world, except in Brazil, where hydroelectricity dominates.', '1309.7626-4-66-1': 'Europe also sees the most diverse electricity sector, with large amounts of wind power already in the baseline scenario, predominantly in northern Europe and the British Isles, large amounts of nuclear power in France, and some solar power in Germany.', '1309.7626-4-66-2': 'Coal fired electricity is mostly phased out before 2050 in the 90% scenario, resulting in significant emissions reductions.', '1309.7626-4-67-0': 'North America features higher use of fossil fuels for power production than Europe.', '1309.7626-4-67-1': 'However, while E3MG projects a larger potential for consumption reductions, large opportunities for diversification also emerge with significant potentials of renewable energy.', '1309.7626-4-67-2': 'Bioenergy with CCS generates a large contribution to American emissions reductions.', '1309.7626-4-68-0': 'China and India have very low technology diversity and important fossil fuel lock-ins.', '1309.7626-4-68-1': 'The amount of coal used in China in the baseline is responsible for 10 out of 30 Gt of global emissions in 2050.', '1309.7626-4-68-2': 'Diversification proves difficult given the scale of the rate of increase in consumption; breaking the coal lock-in requires regulations in China to phase out building new coal generators.', '1309.7626-4-68-3': 'Large scale diffusion of renewables is slow and retrofitting CCS to coal generators offers a useful alternative.', '1309.7626-4-68-4': 'Electricity demand reductions are very large, which requires further investigations for fuel poverty and other social implications.', '1309.7626-4-69-0': 'In Brazil, even though hydroelectricity is not the least expensive resource, it nevertheless dominates, another form of technology lock-in.', '1309.7626-4-69-1': 'This is typical of a national engineering tradition dominated by a technology for decades.', '1309.7626-4-69-2': 'Brazil is projected to persist in developing its hydropower capacity despite higher costs and a decreasing potential, until the cost becomes prohibitively high and only less productive hydro resources remain.', '1309.7626-4-70-0': 'The rest of the world includes predominantly countries where the diversity of existing technologies is low, and persists in this direction.', '1309.7626-4-70-1': 'It features large amounts of oil use for electricity despite high oil prices, due to restricted access to technology or fossil fuel subsidies, which are not successfully phased out despite being the least cost-effective way of producing electricity.', '1309.7626-4-70-2': 'Coal based electricity makes the dominant contribution to emissions in the baseline, the rest divided between oil and gas fired power stations, for a total of 12 out of 30 Gt of global emissions in 2050 in the baseline.', '1309.7626-4-70-3': 'In the mitigation scenario, a significant additional hydroelectricity potential is developed, and coal is replaced by gas turbines, which are eventually retrofitted with CCS.', '1309.7626-4-71-0': '# Conclusions and policy implications', '1309.7626-4-72-0': '## Synergy between policy instruments', '1309.7626-4-73-0': 'This paper shows that in a coupled energy-economy-environment model that does not assume economic equilibrium or use technology cost-optimisation, the impact of policy instruments can be different if used individually or in combinations: the impact of combined policy packages does not correspond to the sum of the impacts of individual instruments.', '1309.7626-4-73-1': 'Thus significant synergies exist between policy instruments.', '1309.7626-4-73-2': 'In this regard we showed that in a technology diffusion perspective, carbon pricing alone is not likely capable of delivering sufficient emissions reductions unless it is unrealistically high; it requires to be combined with technology subsidies, FiTs and regulations.', '1309.7626-4-73-3': 'This can be ascribed largely to the inertia of diffusion, and contrasts with the neoclassical environmental economics view that pricing the externality generates the desired outcome most efficiently .', '1309.7626-4-73-4': 'Our model results indicate that relying on carbon pricing alone even up to 400 2008/tCO[MATH] is likely to lead to a status quo in the technology mix while delivering very expensive electricity to consumers.', '1309.7626-4-73-5': 'Similarly, technology subsidies and FiTs on their own have little impact unless they are combined with sufficiently high carbon pricing.', '1309.7626-4-74-0': 'We furthermore suggested that particular combinations of policy instruments can produce such strong synergy that reductions of electricity sector emissions of 90% by 2050 (61% of 2000-2050 cumulative baseline power sector emissions) become possible without early scrapping of electricity generation capital.', '1309.7626-4-74-1': 'Such strong reductions could be complemented by additional reductions in other emissions intensive sectors with additional cross-sectoral synergies: transport, industry and buildings, warranting further work in this area.', '1309.7626-4-74-2': 'If early scrapping is allowed, these reductions could be achieved even faster, but would most likely involve higher costs.', '1309.7626-4-74-3': 'Finally, different combinations of the policies analysed here could also lead to 90% emissions reductions, for example with lower carbon prices and higher technology subsidies, FiTs and/or more regulations.', '1309.7626-4-75-0': '## The effect of global knowledge spillovers on technology costs: individual vs global coordinated action', '1309.7626-4-76-0': 'Technology systems typically face a vicious cycle: established technologies thrive because they are established, and emerging technologies see barriers to their diffusion due to the lock-in of established technologies.', '1309.7626-4-76-1': 'This is the case unless an emerging technology is a radical improvement over the incumbent, or it benefits from sufficient external support.', '1309.7626-4-76-2': 'Emerging technologies require investment and sales in order to benefit from improvements and economies of scale: repetition, trial and error enables entrepreneurs to improve their products.', '1309.7626-4-76-3': 'They thus require a continuous flow of funds from sales or external investment in order to survive until their products take off on their own in the market.', '1309.7626-4-76-4': 'In the long run, these investments may or may not generate a return, and are thus risky.', '1309.7626-4-76-5': ""Without any investment to bridge the 'technology valley of death', however, they may become failed innovations."", '1309.7626-4-77-0': 'Given estimated learning curves of power systems, a certain additional capacity of emerging technologies such as wind turbines and solar PV panels must be deployed in order to bring down their costs to a competitive level set by incumbent technologies.', '1309.7626-4-77-1': 'As we find here, this additional capacity is very large, and cannot be deployed by a single nation such as Germany or even the whole of Europe, for the rest of the world to benefit.', '1309.7626-4-77-2': 'In contrast, we find that only a concerted global climate policy effort can bring down costs to manageable levels and bring new power technologies into the mainstream, opening very large renewable energy potentials such as that of solar energy.', '1309.7626-4-77-3': 'Such a concerted effort can significantly and permanently alter the global landscape of power technology costs and availability.', '1309.7626-4-77-4': 'We stress that all countries of the world can benefit from learning cost reductions that originate from investments and sales occurring in various locations.', '1309.7626-4-77-5': 'This problem therefore possesses the features of a classic free-rider and collective action problem, where international coordination is the only way by which these cost reductions can take place.', '1309.7626-4-77-6': ""Emerging or developed nations cannot simply 'wait' for climate policy in other nations to generate diffusion and enough learning cost reductions for new technologies to become competitive: without their involvement they might potentially never become competitive."", '1309.7626-4-77-7': 'If the power sector is to decarbonise by 2050, all countries are most likely required to make a contribution to the development of the renewables industry.'}",
quant-ph-9911100,"{'quant-ph-9911100-1-0-0': 'We consider the case when decoherence is due to the fluctuations of some classical variable or parameter of a system and not to its entanglement with the environment.', 'quant-ph-9911100-1-0-1': 'Under few and quite general assumptions, we derive a model-independent formalism for this non-dissipative decoherence, and we apply it to explain the decoherence observed in some recent experiments in cavity QED and on trapped ions.', 'quant-ph-9911100-1-1-0': '# Introduction', 'quant-ph-9911100-1-2-0': 'Decoherence is the rapid transformation of a pure linear superposition state into the corresponding statistical mixture [EQUATION] this process does not preserve the purity of the state, that is, [MATH], and therefore it has to be described in terms of a non-unitary evolution.', 'quant-ph-9911100-1-2-1': 'The most common approach is the so-called environment-induced decoherence [CITATION] which is based on the consideration that it is extremely difficult to isolate perfectly a system from uncontrollable degrees of freedom (the ""environment"").', 'quant-ph-9911100-1-2-2': 'The non-unitary evolution of the system of interest is obtained by considering the interaction with these uncontrolled degrees of freedom and tracing over them.', 'quant-ph-9911100-1-2-3': 'In this approach, decoherence is caused by the entanglement of the two states of the superposition with two approximately orthogonal states of the environment [MATH] and [MATH] [EQUATION]', 'quant-ph-9911100-1-2-4': 'Tracing over the environment and using [MATH], one gets [EQUATION] where [MATH] is defined in Eq. ([REF]).', 'quant-ph-9911100-1-2-5': 'The environment behaves as a measurement apparatus because the states [MATH] behave as ""pointer states"" associated with [MATH]; in this way the environment acquires ""information"" on the system state and therefore decoherence is described as an irreversible flow of information from the system into the environment [CITATION].', 'quant-ph-9911100-1-2-6': 'In this approach, the system energy is usually not conserved and the interaction with the environment also accounts for the irreversible thermalization of the system of interest.', 'quant-ph-9911100-1-2-7': 'However this approach is inevitably model-dependent, because one has to assume a model Hamiltonian for the environment and the interaction between system and environment.', 'quant-ph-9911100-1-2-8': 'This modelization, and therefore any quantitative prediction, becomes problematic whenever the environmental degrees of freedom responsible for decoherence are not easily recognizable.', 'quant-ph-9911100-1-3-0': 'Decoherence is not always necessarily due to the entanglement with an environment, but it may be due, as well, to the fluctuations of some classical parameter or internal variable of the system.', 'quant-ph-9911100-1-3-1': 'This kind of decoherence is present even in isolated systems, where environment-induced decoherence has to be neglected.', 'quant-ph-9911100-1-3-2': 'In these cases the system energy is conserved, and one has a different form of decoherence, which we shall call ""non-dissipative decoherence"".', 'quant-ph-9911100-1-3-3': 'In such cases, every single experimental run is characterized by the usual unitary evolution generated by the system Hamiltonian.', 'quant-ph-9911100-1-3-4': 'However, definite statistical prediction are obtained only repeating the experiment many times and this is when decoherence takes place, because each run corresponds to a different random value or stochastic realization of the fluctuating classical variable.', 'quant-ph-9911100-1-3-5': 'The experimental results correspond therefore to an average over these fluctuations and they will describe in general an effective non-unitary evolution.', 'quant-ph-9911100-1-4-0': 'In this paper we shall present a quite general theory of non-dissipative decoherence for isolated systems which can be applied for two different kinds of fluctuating variables or parameters: the case of a random evolution time and the case of a fluctuating Rabi frequency yielding a fluctuation of the Hamiltonian.', 'quant-ph-9911100-1-4-1': 'In both cases one has random phases [MATH] in the energy eigenstates basis that, once averaged over many experimental runs, lead to the decay of off-diagonal matrix elements of the density operator, while leaving the diagonal ones unchanged.', 'quant-ph-9911100-1-5-0': 'The outline of the paper is as follows.', 'quant-ph-9911100-1-5-1': 'In Section II we shall derive the theory under general assumptions, following closely the original derivation presented in [CITATION].', 'quant-ph-9911100-1-5-2': 'In Section III we shall apply this theory in order to describe the decoherence effects observed in two cavity QED experiments performed in Paris, one describing Rabi oscillations associated with the resonant interaction between a Rydberg atom and a microwave cavity mode [CITATION], and the second one a Ramsey interferometry experiment using a dispersive interaction between the cavity mode and the atom [CITATION].', 'quant-ph-9911100-1-5-3': 'In Section IV we shall apply our approach to a Rabi oscillation experiment for trapped ions [CITATION], and Section V is for concluding remarks.', 'quant-ph-9911100-1-6-0': '# The general formalism', 'quant-ph-9911100-1-7-0': 'The formalism describing non-dissipative decoherence of isolated systems has been derived in [CITATION] by considering the case of a system with random evolution time.', 'quant-ph-9911100-1-7-1': 'The evolution time may be random because of the finite time needed to prepare the initial state of the system, because of the randomness of the detection time, as well as many other reasons.', 'quant-ph-9911100-1-7-2': 'For example, in cavity QED experiments, the evolution time is the interaction time, which is determined by the time of flight of the atoms within the cavity and this time can be random due to atomic velocity dispersion.', 'quant-ph-9911100-1-8-0': 'In these cases, the experimental observations are not described by the usual density matrix of the whole system [MATH], but by its time averaged counterpart [CITATION] [EQUATION] where [MATH] is the usual unitarily evolved density operator from the initial state and [MATH].', 'quant-ph-9911100-1-8-1': 'Therefore [MATH] denotes the random evolution time, while [MATH] is a parameter describing the usual ""clock"" time.', 'quant-ph-9911100-1-8-2': 'Using Eq. ([REF]), one can write [EQUATION] where [EQUATION] is the evolution operator for the averaged state of the system.', 'quant-ph-9911100-1-8-3': 'Following Ref. [CITATION], we determine the function [MATH] by imposing the following plausible conditions: i) [MATH] must be a density operator, i.e. it must be self-adjoint, positive-definite, and with unit-trace.', 'quant-ph-9911100-1-8-4': 'This leads to the condition that [MATH] must be non-negative and normalized, i.e a probability density in [MATH] so that Eq. ([REF]) is a completely positive mapping.', 'quant-ph-9911100-1-8-5': 'ii) [MATH] satisfies the semigroup property [MATH], with [MATH].', 'quant-ph-9911100-1-9-0': 'The semigroup condition is satisfied by an exponential dependence on [MATH] [EQUATION] where [MATH] naturally appears as a scaling time.', 'quant-ph-9911100-1-9-1': 'A solution satisfying all the conditions we have imposed can be found by separating [MATH] in its hermitian and antihermitian part [MATH] and by considering the Gamma function integral identity [CITATION] [EQUATION]', 'quant-ph-9911100-1-9-2': 'Now the right hand side of Eq. ([REF]) can be identified with the right hand side of Eq. ([REF]) if we impose the following conditions: [MATH], where [MATH] is another scaling time, generally different from [MATH]; [MATH] in order to make the exponential terms identical, and [MATH] in order to get a normalized probability distribution [MATH].', 'quant-ph-9911100-1-9-3': 'This choice yields the following expressions for the evolution operator for the averaged density matrix [MATH] and for the probability density [MATH] [CITATION] [EQUATION]', 'quant-ph-9911100-1-9-4': 'Notice that the ordinary quantum evolution is recovered when [MATH]; in this limit [MATH] so that [MATH] and [MATH] is the usual unitary evolution.', 'quant-ph-9911100-1-9-5': 'Moreover, it can be seen that Eq. ([REF]) implies that [MATH] satisfies a finite difference equation [CITATION].', 'quant-ph-9911100-1-9-6': 'The semigroup condition leads to the form of the probability distribution [MATH] we use to perform the average on the fluctuating evolution times.', 'quant-ph-9911100-1-9-7': 'However, notice that this probability distribution depends on both the two scaling times [MATH] and [MATH] only apparently.', 'quant-ph-9911100-1-9-8': 'In fact, if we change variable in the time integral, [MATH], it is possible to rewrite the integral expression for [MATH] in the following way [EQUATION] where [EQUATION]', 'quant-ph-9911100-1-9-9': 'This probability density depends only on [MATH].', 'quant-ph-9911100-1-9-10': 'However Eq. ([REF]) contains an effective rescaled time evolution generator [MATH].', 'quant-ph-9911100-1-9-11': 'The physical meaning of the probability distribution of Eq. ([REF]), of the rescaled evolution operator, and of the two scaling times can be understood if we consider the following simple example.', 'quant-ph-9911100-1-9-12': 'Let us consider a system with Hamiltonian [MATH], where [EQUATION] ([MATH] is the Heaviside step function), that is, a system with Hamiltonian [MATH] which is periodically applied for a time [MATH], with time period [MATH]) and which is ""turned off"" otherwise.', 'quant-ph-9911100-1-9-13': 'The unitary evolution operator for this system is [MATH], where [MATH] and [EQUATION] which can be however well approximated by the ""rescaled"" evolution operator [MATH].', 'quant-ph-9911100-1-9-14': 'In fact, the maximum relative error in replacing [MATH] with [MATH] is [MATH] and becomes negligible at large times (see Fig. 1).', 'quant-ph-9911100-1-9-15': 'This fact suggests to interpret the time average of Eq. ([REF]) as an average over unitary evolutions generated by [MATH], taking place randomly in time, with mean time width [MATH], and separated by a mean time interval [MATH].', 'quant-ph-9911100-1-9-16': 'This interpretation is confirmed by the fact that when [MATH], for integer [MATH], the probability distribution [MATH] of Eq. ([REF]) is a known statistical distribution giving the probability density that the waiting time for [MATH] independent events is [MATH] when [MATH] is the mean time interval between two events.', 'quant-ph-9911100-1-9-17': 'A particularly clear example of the random process in time implied by the above equations is provided by the micromaser [CITATION] in which a microwave cavity is crossed by a beam of resonant atoms with mean injection rate [MATH], and a mean interaction time within the cavity corresponding to [MATH].', 'quant-ph-9911100-1-9-18': 'In the micromaser theory, the non unitary operator [MATH] describing the effective dynamics of the microwave mode during each atomic crossing replaces the evolution operator [MATH] [CITATION].', 'quant-ph-9911100-1-9-19': 'Another example of interrupted evolution is provided by the experimental scheme proposed in [CITATION] for the quantum non-demolition (QND) measurement [CITATION] of the photon number in a high-Q cavity.', 'quant-ph-9911100-1-9-20': 'In this proposal, the photon number is determined by measuring the phase shift induced on a train of Rydberg atoms sent through the microwave cavity with mean rate [MATH], and interacting dispersively with the cavity mode.', 'quant-ph-9911100-1-9-21': 'These two examples show that the two scaling times [MATH] and [MATH] have not to be considered as new universal constants, but as two characteristic times of the system under study.', 'quant-ph-9911100-1-10-0': 'However, in most cases, one does not have an interrupted evolution as in micromaser-like situations, but a standard, continuous evolution generated by an Hamiltonian [MATH].', 'quant-ph-9911100-1-10-1': 'In this case the ""scaled"" effective evolution operator has to coincide with the usual one, [MATH], and this is possible only if [MATH].', 'quant-ph-9911100-1-10-2': 'In this case [MATH] is simply the parameter characterizing the strength of the fluctuations of the random evolution time.', 'quant-ph-9911100-1-10-3': 'This meaning of the parameter [MATH] in the case of equal scaling times is confirmed by the expressions of the mean and the variance of the probability distribution of Eq. ([REF]) [EQUATION]', 'quant-ph-9911100-1-10-4': 'When [MATH], the mean evolution time coincide with the ""clock \'\' time [MATH], while the variance of the evolution time becomes [MATH].', 'quant-ph-9911100-1-10-5': 'In the rest of the paper we shall always consider the standard situation of an isolated system with Hamiltonian [MATH], continuously evolving in time, and we shall always assume [MATH].', 'quant-ph-9911100-1-11-0': 'When [MATH], [MATH] is the usual unitary evolution.', 'quant-ph-9911100-1-11-1': 'For finite [MATH], on the contrary, the evolution equation ([REF]) describes a decay of the off-diagonal matrix elements in the energy representation, whereas the diagonal matrix elements remain constant, i.e. the energy is still a constant of motion.', 'quant-ph-9911100-1-11-2': 'In fact, in the energy eigenbasis, Eqs. ([REF]) and ([REF]) yield [EQUATION] where [MATH] and [EQUATION]', 'quant-ph-9911100-1-11-3': 'This means that, in general, the effect of the average over the fluctuating evolution time yields an exponential decay and a frequency shift [MATH] of every term oscillating in time with frequency [MATH].', 'quant-ph-9911100-1-12-0': 'The phase diffusion aspects of the present approach can also be seen if the evolution equation of the averaged density matrix [MATH] is considered.', 'quant-ph-9911100-1-12-1': 'In fact, by differentiating with respect to time Eq. ([REF]) and using ([REF]), one gets the following master equation for [MATH] (we consider the case [MATH]) [EQUATION] expanding the logarithm at second order in [MATH], one obtains [EQUATION] which is the well-known phase-destroying master equation [CITATION].', 'quant-ph-9911100-1-12-2': 'Hence Eq. ([REF]) appears as a generalized phase-destroying master equation taking into account higher order terms in [MATH].', 'quant-ph-9911100-1-12-3': 'Notice, however, that the present approach is different from the usual master equation approach in the sense that it is model-independent and no perturbative and specific statistical assumptions are made.', 'quant-ph-9911100-1-12-4': 'The solution of Eq. ([REF]) gives an expression for [MATH] similar to that of Eq. ([REF]), but with [CITATION] [EQUATION] which are nonetheless obtained also as a first order expansion in [MATH] of Eqs. ([REF]) and ([REF]).', 'quant-ph-9911100-1-12-5': 'The opposite limit [MATH] has been discussed in detail in Ref. [CITATION].', 'quant-ph-9911100-1-13-0': 'Finally a comment concerning the form of the evolution operator for the averaged density matrix [MATH] of Eq. ([REF]).', 'quant-ph-9911100-1-13-1': 'At first sight it seems that [MATH] is in general a multivalued function of the Liouvillian [MATH], and that [MATH] is uniquely defined only when [MATH], [MATH] integer.', 'quant-ph-9911100-1-13-2': 'However, this form for [MATH] is a consequence of the time average over [MATH] of Eq. ([REF]), which is a properly defined, non-negative probability distribution only if the algebraic definition of the power law function [MATH] is assumed.', 'quant-ph-9911100-1-13-3': 'This means that in Eq. ([REF]) one has to take the first determination of the power-law function and in this way [MATH] is univocally defined.', 'quant-ph-9911100-1-14-0': '# Application to cavity QED experiments', 'quant-ph-9911100-1-15-0': 'A first experimental situation in which the above formalism can be applied is the Rabi oscillation experiment of Ref. [CITATION], in which the resonant interaction between a quantized mode in a high-Q microwave cavity (with annihilation operator [MATH]) and two circular Rydberg states ([MATH] and [MATH]) of a Rb atom has been studied.', 'quant-ph-9911100-1-15-1': 'This interaction is well described by the usual Jaynes-Cummings [CITATION] model, which in the interaction picture reads [EQUATION] where [MATH] is the Rabi frequency.', 'quant-ph-9911100-1-16-0': 'The Rabi oscillations describing the exchange of excitations between atom and cavity mode are studied by injecting the velocity-selected Rydberg atom, prepared in the excited state [MATH], in the high-Q cavity and measuring the population of the lower atomic level [MATH], [MATH], as a function of the interaction time [MATH], which is varied by changing the Rydberg atom velocity.', 'quant-ph-9911100-1-16-1': 'Different initial states of the cavity mode have been considered in [CITATION].', 'quant-ph-9911100-1-16-2': 'We shall restrict only to the case of vacuum state induced Rabi oscillations, where the decoherence effect is particularly evident.', 'quant-ph-9911100-1-16-3': 'The Hamiltonian evolution according to Eq. ([REF]) predicts in this case Rabi oscillations of the form [EQUATION]', 'quant-ph-9911100-1-16-4': 'Experimentally instead, damped oscillations are observed, which are well fitted by [EQUATION] where the decay time fitting the experimental data is [MATH]sec [CITATION] and the corresponding Rabi frequency is [MATH] Khz (see Fig. 2).', 'quant-ph-9911100-1-16-5': 'This decay of quantum coherence cannot be associated with photon leakage out of the cavity because the cavity relaxation time is larger ([MATH]sec) and also because in this case one would have an asymptotic limit [MATH].', 'quant-ph-9911100-1-16-6': 'Therefore decoherence in this case has certainly a non dissipative origin, and dark counts of the atomic detectors, dephasing collisions with background gas or stray magnetic fields within the cavity have been suggested as possible sources of the damped oscillations.', 'quant-ph-9911100-1-16-7': '[CITATION].', 'quant-ph-9911100-1-17-0': 'The damped behavior of Eq. ([REF]) can be easily obtained if one applies the formalism described above.', 'quant-ph-9911100-1-17-1': 'In fact, from the linearity of Eq. ([REF]), one has that the time averaging procedure is also valid for mean values and matrix elements of each subsystem.', 'quant-ph-9911100-1-17-2': 'Therefore one has [EQUATION]', 'quant-ph-9911100-1-17-3': 'Using Eqs. ([REF]), ([REF]), ([REF]) and ([REF]), Eq. ([REF]) can be rewritten in the same form of Eq. ([REF]) [EQUATION] where, using Eqs. ([REF]) and ([REF]), [EQUATION]', 'quant-ph-9911100-1-17-4': 'If the characteristic time [MATH] is sufficiently small, i.e. [MATH], there is no phase shift, [MATH], and [EQUATION] (see also Eqs. ([REF]) and ([REF])).', 'quant-ph-9911100-1-17-5': 'The fact that in Ref. [CITATION] the Rabi oscillation frequency essentially coincides with the theoretically expected one, suggests that the time [MATH] characterizing the fluctuations of the interaction time is sufficiently small so that it is reasonable to use Eq. ([REF]).', 'quant-ph-9911100-1-17-6': 'Using the above values for [MATH] and [MATH], one can derive an estimate for [MATH], so to get [MATH]sec. This estimate is consistent with the assumption [MATH] we have made, but, more importantly, it turns out to be comparable to the experimental value of the uncertainty in the interaction time.', 'quant-ph-9911100-1-17-7': 'In fact, the fluctuations of the interaction time are mainly due to the experimental uncertainty of the atomic velocity [MATH], that is [MATH] (see Ref. [CITATION]), and taking an average interaction time [MATH]sec, one gets [MATH]sec, which is just the estimate we have derived from the experimental values.', 'quant-ph-9911100-1-17-8': 'This simple argument supports the interpretation that the decoherence observed in [CITATION] is essentially due to the randomness of the interaction time.', 'quant-ph-9911100-1-17-9': 'In fact, in our opinion, the other effects proposed as possible sources of decoherence, such as dark counts of the atomic detectors, dephasing collisions with background gas or stray magnetic fields within the cavity, would give an overall, time-independent, contrast reduction of the Rabi oscillations, different from the observed exponential decay.', 'quant-ph-9911100-1-18-0': 'Results similar to that of Ref. [CITATION] have been very recently obtained by H. Walther group at the Max Planck Institut fur Quantenoptik, in a Rabi oscillation experiment involving again a high-Q microwave cavity mode resonantly interacting with Rydberg atoms [CITATION].', 'quant-ph-9911100-1-18-1': 'In this case, three different initial Fock states [MATH] of the cavity mode, [MATH], have been studied, and preliminary results show a good quantitative agreement of the experimental data with our theoretical approach based on the dispersion of the interaction times.', 'quant-ph-9911100-1-19-0': 'Another cavity QED experiment in which the observed decay of quantum coherence can be, at least partially, explained with our formalism in terms of a random interaction time, is the Ramsey interferometry experiment of M. Brune et al. [CITATION].', 'quant-ph-9911100-1-19-1': 'In this experiment, a QND measurement of the mean photon number of a microwave cavity mode is obtained by measuring, in a Ramsey interferometry scheme, the dispersive light shifts produced on circular Rydberg states by a nonresonant microwave field.', 'quant-ph-9911100-1-19-2': 'The experimental scheme in this case is similar to that of the Rabi oscillation experiment, with two main differences: i) two low-Q microwave cavities [MATH] and [MATH], which can be fed by a classical source [MATH] with frequency [MATH], are added just before and after the cavity of interest [MATH]; ii) the cavity mode is highly detuned from the atomic transition ([MATH]), so to work in the dispersive regime.', 'quant-ph-9911100-1-19-3': 'In the interaction picture with respect to [EQUATION] (we use the classical field as reference for the atomic phases), the Hamiltonian has the following dispersive form [CITATION] [EQUATION] where [MATH], the Rabi frequency within the classical cavities [MATH] is nonzero only when the atom is in [MATH] and [MATH], and [MATH] is nonzero only within [MATH].', 'quant-ph-9911100-1-19-4': 'In the experiment, single circular Rydberg atoms are sent through the apparatus initially prepared in the state [MATH], and let us assume that the microwave cavity mode in [MATH] is in a generic state [MATH].', 'quant-ph-9911100-1-19-5': 'The atom is subject to a [MATH] pulse in [MATH], so that [EQUATION]', 'quant-ph-9911100-1-19-6': 'Then the atom crosses the cavity [MATH] with an interaction time [MATH] and the dispersive interaction yields [EQUATION]', 'quant-ph-9911100-1-19-7': 'Finally the atom is subject to the second [MATH] pulse in the second Ramsey zone [MATH] and the joint state of the Rydberg atom and the cavity mode becomes [EQUATION] where [MATH] is the time of flight from [MATH] to [MATH].', 'quant-ph-9911100-1-19-8': 'The experimentally interesting quantity is the probability to find at the end the atom in the [MATH] state, [MATH], whose theoretical expression according to Eq. ([REF]) is [EQUATION] where the photon number-dependent frequency shift [MATH] is given by [EQUATION]', 'quant-ph-9911100-1-19-9': 'In Eqs. ([REF]) and ([REF]) we have used the fact that [MATH] is equal to the ratio between the waist of the cavity mode [MATH] and the distance between the two Ramsey cavities [MATH].', 'quant-ph-9911100-1-19-10': 'The actual experiment of Ref. [CITATION] has been performed in the bad cavity limity [MATH] in which the cavity [MATH] relaxation time [MATH] is smaller than the atom-cavity interaction time.', 'quant-ph-9911100-1-19-11': 'In this case, the cavity photon number randomly changes during [MATH] and in the corresponding expression ([REF]) for the frequency shift [MATH], the photon number [MATH] has to replaced by the mean value [MATH].', 'quant-ph-9911100-1-19-12': 'The Ramsey fringes are observed by sweeping the frequency of the classical source [MATH] around resonance, that is, studying [MATH] as a function of the detuning [MATH].', 'quant-ph-9911100-1-19-13': 'The experimentally observed Ramsey fringes show a reduced contrast, which moreover decreases for increasing detunings [MATH] (see Fig. 2 of Ref. [CITATION]).', 'quant-ph-9911100-1-19-14': 'Therefore one can try to explain the reduced contrast, i.e., the loss of quantum coherence, in terms of a fluctuating evolution time, which in this case means a random time of flight [MATH] originated again by the dispersion of the atomic velocities.', 'quant-ph-9911100-1-19-15': 'We average again the quantity [MATH] of Eq. ([REF]) over the probability distribution [MATH] derived in Section II, replacing [MATH] with a random time of flight [MATH], and we obtain [EQUATION] where, using Eq. ([REF]), the fringe visibility function [MATH] is given by [EQUATION] and [MATH] is the frequency shift [EQUATION]', 'quant-ph-9911100-1-19-16': 'The parameter [MATH] characterizing the strength of the fluctuations of the time of flight [MATH] can be estimated with arguments similar to those considered for the Rabi oscillation experiment.', 'quant-ph-9911100-1-19-17': 'Since [MATH] and [MATH]sec (see Ref. [CITATION]), one has [MATH]sec. For the interesting range of detunings [MATH], one has [MATH], so that one can neglect again the frequency shift ([REF]) and approximate the fringe visibility function ([REF]) with a gaussian function, that is, [EQUATION]', 'quant-ph-9911100-1-19-18': 'This gaussian modulation of the Ramsey fringes with a width [MATH] is consistent with the typical experimental Ramsey fringe signal (see Fig. 2 of Ref. [CITATION]), but it is not able to completely account for the observed modulation and contrast reduction of the fringes.', 'quant-ph-9911100-1-19-19': 'This means that, contrary to the case of the Rabi oscillation experiment, in this case the role of other experimental imperfections such as random phases due to stray fields, imperfect [MATH] pulses in [MATH] and [MATH] and detection errors, is as relevant as that of the dispersion of atomic velocities and these other effects have to be taken into account to get an exhaustive explanation of the observed decoherence.', 'quant-ph-9911100-1-20-0': '# Rabi oscillation experiments in trapped ions', 'quant-ph-9911100-1-21-0': 'Another interesting Rabi oscillation experiment, performed on a different system, that is, a trapped ion [CITATION], has recently observed a decoherence effect which cannot be attributed to dissipation.', 'quant-ph-9911100-1-21-1': 'In the trapped ion experiment of Ref. [CITATION], the interaction between two internal states ([MATH] and [MATH]) of a Be ion and the center-of-mass vibrations in the [MATH] direction, induced by two driving Raman lasers is studied.', 'quant-ph-9911100-1-21-2': 'In the interaction picture with respect to the free vibrational and internal Hamiltonian, this interaction is described by the following Hamiltonian [CITATION] [EQUATION] where [MATH] denotes the annihiliation operator for the vibrations along the [MATH] direction, [MATH] is the corresponding frequency and [MATH] is the detuning between the internal transition and the frequency difference between the two Raman lasers.', 'quant-ph-9911100-1-21-3': 'The Rabi frequency [MATH] is proportional to the two Raman laser intensities, and [MATH] is the Lamb-Dicke parameter [CITATION].', 'quant-ph-9911100-1-21-4': 'When the two Raman lasers are tuned to the first blue sideband, i.e. [MATH], Hamiltonian ([REF]) predicts Rabi oscillations between [MATH] and [MATH] is a vibrational Fock state) with a frequency [CITATION] [EQUATION] where [MATH] is the generalized Laguerre polynomial.', 'quant-ph-9911100-1-21-5': 'These Rabi oscillations have been experimentally verified by preparing the initial state [MATH], (with [MATH] ranging from [MATH] to [MATH]) and measuring the probability [MATH] as a function of the interaction time [MATH], which is varied by changing the duration of the Raman laser pulses.', 'quant-ph-9911100-1-21-6': 'Again, as in the cavity QED experiment of [CITATION], the experimental Rabi oscillations are damped and well fitted by [CITATION] [EQUATION] where the measured oscillation frequencies [MATH] are in very good agreement with the theoretical prediction ([REF]) corresponding to the measured Lamb-Dicke parameter [MATH] [CITATION].', 'quant-ph-9911100-1-21-7': 'As concerns the decay rates [MATH], the experimental values are fitted in [CITATION] by [EQUATION] where [MATH] Khz.', 'quant-ph-9911100-1-21-8': 'This power-law scaling has attracted the interest of a number of authors and it has been investigated in Refs. [CITATION], even if a clear explanation of this behavior of the decay rates is still lacking.', 'quant-ph-9911100-1-21-9': 'On the contrary, the scaling law ([REF]) can be simply accounted for in the previous formalism if we consider the small [MATH] limit of Eq. ([REF]), which is again suggested by the fact that the experimental and theoretical predictions for the frequencies [MATH] agree.', 'quant-ph-9911100-1-21-10': 'In fact, the [MATH]-dependence of the theoretical prediction of Eq. ([REF]) for [MATH] is well approximated, within 10 %, by the power law dependence (see Fig. 3) [EQUATION] so that, using Eq. ([REF]), one has immediately the power law dependence [MATH] of Eq. ([REF]).', 'quant-ph-9911100-1-21-11': 'The value of the parameter [MATH] can be obtained by matching the values corresponding to [MATH], and using Eq. ([REF]), that is [MATH] sec, where we have used the experimental value [MATH] Khz.', 'quant-ph-9911100-1-22-0': 'However, this value of the parameter [MATH] cannot be explained in terms of some interaction time uncertainty, such as the time jitter of the Raman laser pulses, which is experimentally found to be much smaller [CITATION].', 'quant-ph-9911100-1-22-1': 'In this case, instead, the observed decoherence can be attributed, as already suggested in [CITATION], to the fluctuation of the Raman laser intensities, yielding a fluctuating Rabi frequency parameter [MATH] of the Hamiltonian ([REF]).', 'quant-ph-9911100-1-22-2': 'In this case, the evolution is driven by a fluctuating Hamiltonian [MATH], where [MATH] in Eq. ([REF]), so that [EQUATION] where [MATH], and we have defined the positive dimensionless random variable [MATH], which is proportional to the pulse area.', 'quant-ph-9911100-1-22-3': 'It is now easy to understand that the physical situation is analogous to that characterized by a random interaction time considered in the preceding sections, with [MATH] replaced by [MATH] and [MATH] by [MATH].', 'quant-ph-9911100-1-22-4': 'It is therefore straightforward to adapt the formalism developed in Section II to this case, in which the fluctuating quantity is the pulse area [MATH], yielding again random phases in the energy basis representation.', 'quant-ph-9911100-1-22-5': 'In analogy with Eq. ([REF]), one considers an averaged density matrix [EQUATION]', 'quant-ph-9911100-1-22-6': 'Imposing again that [MATH] must be a density operator and the semigroup property, one finds results analogous to Eqs. ([REF]) and ([REF]) [EQUATION]', 'quant-ph-9911100-1-22-7': 'Here, the parameters [MATH] and [MATH] are introduced as scaling parameters, but they have a clear meaning, as it can be easily seen by considering the mean and the variance of the probability distribution of Eq. ([REF]), [EQUATION] implying that [MATH] has now to be meant as a mean Rabi frequency, and that [MATH] quantifies the strength of [MATH] fluctuations.', 'quant-ph-9911100-1-22-8': 'It is interesting to note that these first two moments of [MATH] determine the properties of the fluctuating Rabi frequency [MATH], which can be written as [EQUATION] that is, the Rabi frequency [MATH] is a white, non-gaussian (due to the non-gaussian form of [MATH]) stochastic process.', 'quant-ph-9911100-1-22-9': 'In fact, the semigroup assumption we have made implies a Markovian treatment in which the spectrum of the laser intensity fluctuations is flat in the relevant frequency range.', 'quant-ph-9911100-1-22-10': 'This in particular implies that we are neglecting the dynamics at small times, of the order of the correlation time of the laser intensity fluctuations.', 'quant-ph-9911100-1-23-0': 'The estimated value of [MATH] gives a reasonable estimate of the pulse area fluctuations, since it corresponds to a fractional error of the pulse area [MATH] of [MATH] for a pulse duration of [MATH]sec, and which is decreasing for increasing pulse durations.', 'quant-ph-9911100-1-24-0': 'The present analysis shows many similarities with that of Ref. [CITATION] which also tries to explain the decay of the Rabi oscillations in the ion trap experiments of [CITATION] in terms of laser intensity fluctuations.', 'quant-ph-9911100-1-24-1': 'The authors of Ref. [CITATION] in fact use a phase destroying master equation coinciding with the second-order expansion ([REF]) of our generalized master equation of Eq. ([REF]) (see Eq. (16) of Ref. [CITATION] with the identifications [MATH] and [MATH]) and moreover derive the same numerical estimate for the pulse area fluctuation strength [MATH].', 'quant-ph-9911100-1-24-2': 'Despite this similarities, they do not recover the scaling ([REF]) of the decay rates [MATH] only because they do not use the general expression of the Rabi frequency ([REF]), (and which is well approximated by the power law ([REF])) but its Lamb-Dicke limit [MATH], which is valid only when [MATH].', 'quant-ph-9911100-1-24-3': 'There is however another, more fundamental, difference between our approach and that of Ref. [CITATION].', 'quant-ph-9911100-1-24-4': 'They assume from the beginning that the laser intensity fluctuations have a white and gaussian character, while we make no a priori assumption on the statistical properties of the pulse area [MATH].', 'quant-ph-9911100-1-24-5': 'We derive these properties, i.e. the probability distribution ([REF]), only from the semigroup condition, and it is interesting to note that this condition yields a gaussian probability distribution for the pulse area only as a limiting case.', 'quant-ph-9911100-1-24-6': 'In fact, from Eq. ([REF]) one can see that [MATH] tends to become a gaussian with the same mean value [MATH] and the same width [MATH] only in the large time limit [MATH] [EQUATION]', 'quant-ph-9911100-1-24-7': 'The non-gaussian character of [MATH] can be traced back to the fact that [MATH] must be definite and normalized in the interval [MATH] and not in [MATH].', 'quant-ph-9911100-1-24-8': 'Notice that at [MATH], Eq. ([REF]) assumes the exponential form [MATH].', 'quant-ph-9911100-1-24-9': 'Only at large times [MATH] the random variable [MATH] becomes the sum of many independent contributions and assumes the gaussian form.', 'quant-ph-9911100-1-25-0': 'Due to the non-gaussian nature of the random variable [MATH], we find that the more generally valid phase-destroying master equation is given by Eq. ([REF]) (with [MATH] replaced by [MATH]).', 'quant-ph-9911100-1-25-1': 'The predictions of Eq. ([REF]) significantly depart from its second order expansion in [MATH], Eq. ([REF]), corresponding to the gaussian limit, as soon as [MATH] becomes comparable with the typical timescale of the system under study, which, in the present case, is the inverse of the Rabi frequency.', 'quant-ph-9911100-1-26-0': 'The present analysis of the Rabi oscillation experiment of Ref. [CITATION] can be repeated for the very recent experiment with trapped ions performed in Innsbruck [CITATION], in which Rabi oscillations involving the vibrational levels and an optical quadrupole transition of a single [MATH]Ca[MATH] ion have been observed.', 'quant-ph-9911100-1-26-1': 'Damped oscillations corresponding to initial vibrational numbers [MATH] and [MATH] are reported.', 'quant-ph-9911100-1-26-2': 'From the data with [MATH], [MATH] Khz and [MATH] Khz, we get [MATH] and this estimate is consistent with attributing again the decoherence to the fluctuations of the Rabi frequency caused by laser intensity fluctuations.', 'quant-ph-9911100-1-26-3': 'Moreover in this case, the experiment is performed in the Lamb-Dicke limit [MATH], and therefore, using again Eq. ([REF]), we expect, in this case, a linear scaling with the vibrational number, [MATH].', 'quant-ph-9911100-1-27-0': '# Concluding remarks', 'quant-ph-9911100-1-28-0': 'Decoherence is not always necessarily due to the entanglement with an environment, but it may be due, as well, to the fluctuations of some classical parameter or internal variable of a system.', 'quant-ph-9911100-1-28-1': 'This is a different form of decoherence, which is present even in isolated systems, and that we have called non-dissipative decoherence.', 'quant-ph-9911100-1-28-2': 'In this paper we have presented a model-independent theory for non-dissipative decoherence, which can be applied in the case of a random evolution time or in the case of a fluctuating Hamiltonian.', 'quant-ph-9911100-1-28-3': 'This approach proves to be a flexible tool, able to give a quantitative understanding of the decoherence caused by the fluctuations of classical quantities.', 'quant-ph-9911100-1-28-4': 'In fact, in this paper we have given a simple and unified description of the decoherence phenomenon observed in recent Rabi oscillation experiments performed in a cavity QED configuration [CITATION] and on a trapped ion [CITATION].', 'quant-ph-9911100-1-28-5': 'In particular, this approach has allowed us to explain for the first time in simple terms, the power-law scaling of the coherence decay rates of Eq. ([REF]), observed in the trapped ion experiment.', 'quant-ph-9911100-1-29-0': 'The relevant aspect of the approach applied here, and introduced in Ref. [CITATION], is its model-independence.', 'quant-ph-9911100-1-29-1': 'The formalism is in fact derived starting from few, very general assumptions: i) the average density matrix [MATH] has all the usual properties of a density matrix; ii) the semigroup property for the time evolution generator [MATH] for [MATH].', 'quant-ph-9911100-1-29-2': 'With this respect, this approach seems to provide a very general description of non-dissipative decoherence, in which the random properties of the fluctuating classical variables are characterized by the two, system-dependent, time parameters [MATH] and [MATH].', 'quant-ph-9911100-1-29-3': 'As we have seen in section II, in the cases where one has a standard, continuous evolution, the two times coincide [MATH].', 'quant-ph-9911100-1-29-4': 'Under ideal conditions of no fluctuating classical variable or parameter, one would have [MATH], and the usual unitary evolution of an isolated system in quantum mechanics would be recovered.', 'quant-ph-9911100-1-29-5': 'However, the generality of the approach suggests in some way the possibility that the parameter [MATH], even though system-dependent, might have a lower nonzero limit, which would be reached just in the case of no fluctuations of experimental origin.', 'quant-ph-9911100-1-29-6': 'This would mean a completely new description of time in quantum mechanics.', 'quant-ph-9911100-1-29-7': 'In fact, the evolution time of a system [MATH] (and not the ""clock"" time [MATH]) would become an intrinsically random variable with a well defined probability distribution, without the difficulty of introducing an evolution time operator.', 'quant-ph-9911100-1-29-8': 'In Ref. [CITATION] it is suggested a relation of the nonzero limit for [MATH] with the ""energy-time"" [MATH] where [MATH] is the uncertainty in energy.', 'quant-ph-9911100-1-29-9': 'This would give a precise meaning to the time-energy uncertainty relation because now [MATH] rules the width of the time distribution function.', 'quant-ph-9911100-1-29-10': 'However, this ""intrinsic assumption"" is not necessarily implied by the formalism developed in [CITATION] and applied, with a more pragmatic attitude, in the present paper.'}","{'quant-ph-9911100-2-0-0': 'We consider the case when decoherence is due to the fluctuations of some classical variable or parameter of a system and not to its entanglement with the environment.', 'quant-ph-9911100-2-0-1': 'Under few and quite general assumptions, we derive a model-independent formalism for this non-dissipative decoherence, and we apply it to explain the decoherence observed in some recent experiments in cavity QED and on trapped ions.', 'quant-ph-9911100-2-1-0': '# Introduction', 'quant-ph-9911100-2-2-0': 'Decoherence is the rapid transformation of a pure linear superposition state into the corresponding statistical mixture [EQUATION] this process does not preserve the purity of the state, that is, [MATH], and therefore it has to be described in terms of a non-unitary evolution.', 'quant-ph-9911100-2-2-1': 'The most common approach is the so-called environment-induced decoherence [CITATION] which is based on the consideration that it is extremely difficult to isolate perfectly a system from uncontrollable degrees of freedom (the ""environment"").', 'quant-ph-9911100-2-2-2': 'The non-unitary evolution of the system of interest is obtained by considering the interaction with these uncontrolled degrees of freedom and tracing over them.', 'quant-ph-9911100-2-2-3': 'In this approach, decoherence is caused by the entanglement of the two states of the superposition with two approximately orthogonal states of the environment [MATH] and [MATH] [EQUATION]', 'quant-ph-9911100-2-2-4': 'Tracing over the environment and using [MATH], one gets [EQUATION] where [MATH] is defined in Eq. ([REF]).', 'quant-ph-9911100-2-2-5': 'The environment behaves as a measurement apparatus because the states [MATH] behave as ""pointer states"" associated with [MATH]; in this way the environment acquires ""information"" on the system state and therefore decoherence is described as an irreversible flow of information from the system into the environment [CITATION].', 'quant-ph-9911100-2-2-6': 'In this approach, the system energy is usually not conserved and the interaction with the environment also accounts for the irreversible thermalization of the system of interest.', 'quant-ph-9911100-2-2-7': 'However this approach is inevitably model-dependent, because one has to assume a model Hamiltonian for the environment and the interaction between system and environment.', 'quant-ph-9911100-2-2-8': 'This modelization, and therefore any quantitative prediction, becomes problematic whenever the environmental degrees of freedom responsible for decoherence are not easily recognizable.', 'quant-ph-9911100-2-3-0': 'Decoherence is not always necessarily due to the entanglement with an environment, but it may be due, as well, to the fluctuations of some classical parameter or internal variable of the system.', 'quant-ph-9911100-2-3-1': 'This kind of decoherence is present even in isolated systems, where environment-induced decoherence has to be neglected.', 'quant-ph-9911100-2-3-2': 'In these cases the system energy is conserved, and one has a different form of decoherence, which we shall call ""non-dissipative decoherence"".', 'quant-ph-9911100-2-3-3': 'In such cases, every single experimental run is characterized by the usual unitary evolution generated by the system Hamiltonian.', 'quant-ph-9911100-2-3-4': 'However, definite statistical prediction are obtained only repeating the experiment many times and this is when decoherence takes place, because each run corresponds to a different random value or stochastic realization of the fluctuating classical variable.', 'quant-ph-9911100-2-3-5': 'The experimental results correspond therefore to an average over these fluctuations and they will describe in general an effective non-unitary evolution.', 'quant-ph-9911100-2-4-0': 'In this paper we shall present a quite general theory of non-dissipative decoherence for isolated systems which can be applied for two different kinds of fluctuating variables or parameters: the case of a random evolution time and the case of a fluctuating Rabi frequency yielding a fluctuation of the Hamiltonian.', 'quant-ph-9911100-2-4-1': 'In both cases one has random phases [MATH] in the energy eigenstates basis that, once averaged over many experimental runs, lead to the decay of off-diagonal matrix elements of the density operator, while leaving the diagonal ones unchanged.', 'quant-ph-9911100-2-5-0': 'The outline of the paper is as follows.', 'quant-ph-9911100-2-5-1': 'In Section II we shall derive the theory under general assumptions, following closely the original derivation presented in [CITATION].', 'quant-ph-9911100-2-5-2': 'In Section III we shall apply this theory in order to describe the decoherence effects observed in two cavity QED experiments performed in Paris, one describing Rabi oscillations associated with the resonant interaction between a Rydberg atom and a microwave cavity mode [CITATION], and the second one a Ramsey interferometry experiment using a dispersive interaction between the cavity mode and the atom [CITATION].', 'quant-ph-9911100-2-5-3': 'In Section IV we shall apply our approach to a Rabi oscillation experiment for trapped ions [CITATION], and Section V is for concluding remarks.', 'quant-ph-9911100-2-6-0': '# The general formalism', 'quant-ph-9911100-2-7-0': 'The formalism describing non-dissipative decoherence of isolated systems has been derived in [CITATION] by considering the case of a system with random evolution time.', 'quant-ph-9911100-2-7-1': 'The evolution time may be random because of the finite time needed to prepare the initial state of the system, because of the randomness of the detection time, as well as many other reasons.', 'quant-ph-9911100-2-7-2': 'For example, in cavity QED experiments, the evolution time is the interaction time, which is determined by the time of flight of the atoms within the cavity and this time can be random due to atomic velocity dispersion.', 'quant-ph-9911100-2-8-0': 'In these cases, the experimental observations are not described by the usual density matrix of the whole system [MATH], but by its time averaged counterpart [CITATION] [EQUATION] where [MATH] is the usual unitarily evolved density operator from the initial state and [MATH].', 'quant-ph-9911100-2-8-1': 'Therefore [MATH] denotes the random evolution time, while [MATH] is a parameter describing the usual ""clock"" time.', 'quant-ph-9911100-2-8-2': 'Using Eq. ([REF]), one can write [EQUATION] where [EQUATION] is the evolution operator for the averaged state of the system.', 'quant-ph-9911100-2-8-3': 'Following Ref. [CITATION], we determine the function [MATH] by imposing the following plausible conditions: i) [MATH] must be a density operator, i.e. it must be self-adjoint, positive-definite, and with unit-trace.', 'quant-ph-9911100-2-8-4': 'This leads to the condition that [MATH] must be non-negative and normalized, i.e a probability density in [MATH] so that Eq. ([REF]) is a completely positive mapping.', 'quant-ph-9911100-2-8-5': 'ii) [MATH] satisfies the semigroup property [MATH], with [MATH].', 'quant-ph-9911100-2-9-0': 'The semigroup condition is satisfied by an exponential dependence on [MATH] [EQUATION] where [MATH] naturally appears as a scaling time.', 'quant-ph-9911100-2-9-1': 'A solution satisfying all the conditions we have imposed can be found by separating [MATH] in its hermitian and antihermitian part [MATH] and by considering the Gamma function integral identity [CITATION] [EQUATION]', 'quant-ph-9911100-2-9-2': 'Now the right hand side of Eq. ([REF]) can be identified with the right hand side of Eq. ([REF]) if we impose the following conditions: [MATH], where [MATH] is another scaling time, generally different from [MATH]; [MATH] in order to make the exponential terms identical, and [MATH] in order to get a normalized probability distribution [MATH].', 'quant-ph-9911100-2-9-3': 'This choice yields the following expressions for the evolution operator for the averaged density matrix [MATH] and for the probability density [MATH] [CITATION] [EQUATION]', 'quant-ph-9911100-2-9-4': 'Notice that the ordinary quantum evolution is recovered when [MATH]; in this limit [MATH] so that [MATH] and [MATH] is the usual unitary evolution.', 'quant-ph-9911100-2-9-5': 'Moreover, it can be seen that Eq. ([REF]) implies that [MATH] satisfies a finite difference equation [CITATION].', 'quant-ph-9911100-2-9-6': 'The semigroup condition leads to the form of the probability distribution [MATH] we use to perform the average on the fluctuating evolution times.', 'quant-ph-9911100-2-9-7': 'However, notice that this probability distribution depends on both the two scaling times [MATH] and [MATH] only apparently.', 'quant-ph-9911100-2-9-8': 'In fact, if we change variable in the time integral, [MATH], it is possible to rewrite the integral expression for [MATH] in the following way [EQUATION] where [EQUATION]', 'quant-ph-9911100-2-9-9': 'This probability density depends only on [MATH].', 'quant-ph-9911100-2-9-10': 'However Eq. ([REF]) contains an effective rescaled time evolution generator [MATH].', 'quant-ph-9911100-2-9-11': 'The physical meaning of the probability distribution of Eq. ([REF]), of the rescaled evolution operator, and of the two scaling times can be understood if we consider the following simple example.', 'quant-ph-9911100-2-9-12': 'Let us consider a system with Hamiltonian [MATH], where [EQUATION] ([MATH] is the Heaviside step function), that is, a system with Hamiltonian [MATH] which is periodically applied for a time [MATH], with time period [MATH]) and which is ""turned off"" otherwise.', 'quant-ph-9911100-2-9-13': 'The unitary evolution operator for this system is [MATH], where [MATH] and [EQUATION] which can be however well approximated by the ""rescaled"" evolution operator [MATH].', 'quant-ph-9911100-2-9-14': 'In fact, the maximum relative error in replacing [MATH] with [MATH] is [MATH] and becomes negligible at large times (see Fig. 1).', 'quant-ph-9911100-2-9-15': 'This fact suggests to interpret the time average of Eq. ([REF]) as an average over unitary evolutions generated by [MATH], taking place randomly in time, with mean time width [MATH], and separated by a mean time interval [MATH].', 'quant-ph-9911100-2-9-16': 'This interpretation is confirmed by the fact that when [MATH], for integer [MATH], the probability distribution [MATH] of Eq. ([REF]) is a known statistical distribution giving the probability density that the waiting time for [MATH] independent events is [MATH] when [MATH] is the mean time interval between two events.', 'quant-ph-9911100-2-9-17': 'A particularly clear example of the random process in time implied by the above equations is provided by the micromaser [CITATION] in which a microwave cavity is crossed by a beam of resonant atoms with mean injection rate [MATH], and a mean interaction time within the cavity corresponding to [MATH].', 'quant-ph-9911100-2-9-18': 'In the micromaser theory, the non unitary operator [MATH] describing the effective dynamics of the microwave mode during each atomic crossing replaces the evolution operator [MATH] [CITATION].', 'quant-ph-9911100-2-9-19': 'Another example of interrupted evolution is provided by the experimental scheme proposed in [CITATION] for the quantum non-demolition (QND) measurement [CITATION] of the photon number in a high-Q cavity.', 'quant-ph-9911100-2-9-20': 'In this proposal, the photon number is determined by measuring the phase shift induced on a train of Rydberg atoms sent through the microwave cavity with mean rate [MATH], and interacting dispersively with the cavity mode.', 'quant-ph-9911100-2-9-21': 'These two examples show that the two scaling times [MATH] and [MATH] have not to be considered as new universal constants, but as two characteristic times of the system under study.', 'quant-ph-9911100-2-10-0': 'However, in most cases, one does not have an interrupted evolution as in micromaser-like situations, but a standard, continuous evolution generated by an Hamiltonian [MATH].', 'quant-ph-9911100-2-10-1': 'In this case the ""scaled"" effective evolution operator has to coincide with the usual one, [MATH], and this is possible only if [MATH].', 'quant-ph-9911100-2-10-2': 'In this case [MATH] is simply the parameter characterizing the strength of the fluctuations of the random evolution time.', 'quant-ph-9911100-2-10-3': 'This meaning of the parameter [MATH] in the case of equal scaling times is confirmed by the expressions of the mean and the variance of the probability distribution of Eq. ([REF]) [EQUATION]', 'quant-ph-9911100-2-10-4': 'When [MATH], the mean evolution time coincide with the ""clock \'\' time [MATH], while the variance of the evolution time becomes [MATH].', 'quant-ph-9911100-2-10-5': 'In the rest of the paper we shall always consider the standard situation of an isolated system with Hamiltonian [MATH], continuously evolving in time, and we shall always assume [MATH].', 'quant-ph-9911100-2-11-0': 'When [MATH], [MATH] is the usual unitary evolution.', 'quant-ph-9911100-2-11-1': 'For finite [MATH], on the contrary, the evolution equation ([REF]) describes a decay of the off-diagonal matrix elements in the energy representation, whereas the diagonal matrix elements remain constant, i.e. the energy is still a constant of motion.', 'quant-ph-9911100-2-11-2': 'In fact, in the energy eigenbasis, Eqs. ([REF]) and ([REF]) yield [EQUATION] where [MATH] and [EQUATION]', 'quant-ph-9911100-2-11-3': 'This means that, in general, the effect of the average over the fluctuating evolution time yields an exponential decay and a frequency shift [MATH] of every term oscillating in time with frequency [MATH].', 'quant-ph-9911100-2-12-0': 'The phase diffusion aspects of the present approach can also be seen if the evolution equation of the averaged density matrix [MATH] is considered.', 'quant-ph-9911100-2-12-1': 'In fact, by differentiating with respect to time Eq. ([REF]) and using ([REF]), one gets the following master equation for [MATH] (we consider the case [MATH]) [EQUATION] expanding the logarithm at second order in [MATH], one obtains [EQUATION] which is the well-known phase-destroying master equation [CITATION].', 'quant-ph-9911100-2-12-2': 'Hence Eq. ([REF]) appears as a generalized phase-destroying master equation taking into account higher order terms in [MATH].', 'quant-ph-9911100-2-12-3': 'Notice, however, that the present approach is different from the usual master equation approach in the sense that it is model-independent and no perturbative and specific statistical assumptions are made.', 'quant-ph-9911100-2-12-4': 'The solution of Eq. ([REF]) gives an expression for [MATH] similar to that of Eq. ([REF]), but with [CITATION] [EQUATION] which are nonetheless obtained also as a first order expansion in [MATH] of Eqs. ([REF]) and ([REF]).', 'quant-ph-9911100-2-12-5': 'The opposite limit [MATH] has been discussed in detail in Ref. [CITATION].', 'quant-ph-9911100-2-13-0': 'Finally a comment concerning the form of the evolution operator for the averaged density matrix [MATH] of Eq. ([REF]).', 'quant-ph-9911100-2-13-1': 'At first sight it seems that [MATH] is in general a multivalued function of the Liouvillian [MATH], and that [MATH] is uniquely defined only when [MATH], [MATH] integer.', 'quant-ph-9911100-2-13-2': 'However, this form for [MATH] is a consequence of the time average over [MATH] of Eq. ([REF]), which is a properly defined, non-negative probability distribution only if the algebraic definition of the power law function [MATH] is assumed.', 'quant-ph-9911100-2-13-3': 'This means that in Eq. ([REF]) one has to take the first determination of the power-law function and in this way [MATH] is univocally defined.', 'quant-ph-9911100-2-14-0': '# Application to cavity QED experiments', 'quant-ph-9911100-2-15-0': 'A first experimental situation in which the above formalism can be applied is the Rabi oscillation experiment of Ref. [CITATION], in which the resonant interaction between a quantized mode in a high-Q microwave cavity (with annihilation operator [MATH]) and two circular Rydberg states ([MATH] and [MATH]) of a Rb atom has been studied.', 'quant-ph-9911100-2-15-1': 'This interaction is well described by the usual Jaynes-Cummings [CITATION] model, which in the interaction picture reads [EQUATION] where [MATH] is the Rabi frequency.', 'quant-ph-9911100-2-16-0': 'The Rabi oscillations describing the exchange of excitations between atom and cavity mode are studied by injecting the velocity-selected Rydberg atom, prepared in the excited state [MATH], in the high-Q cavity and measuring the population of the lower atomic level [MATH], [MATH], as a function of the interaction time [MATH], which is varied by changing the Rydberg atom velocity.', 'quant-ph-9911100-2-16-1': 'Different initial states of the cavity mode have been considered in [CITATION].', 'quant-ph-9911100-2-16-2': 'We shall restrict only to the case of vacuum state induced Rabi oscillations, where the decoherence effect is particularly evident.', 'quant-ph-9911100-2-16-3': 'The Hamiltonian evolution according to Eq. ([REF]) predicts in this case Rabi oscillations of the form [EQUATION]', 'quant-ph-9911100-2-16-4': 'Experimentally instead, damped oscillations are observed, which are well fitted by [EQUATION] where the decay time fitting the experimental data is [MATH]sec [CITATION] and the corresponding Rabi frequency is [MATH] Khz (see Fig. 2).', 'quant-ph-9911100-2-16-5': 'This decay of quantum coherence cannot be associated with photon leakage out of the cavity because the cavity relaxation time is larger ([MATH]sec) and also because in this case one would have an asymptotic limit [MATH].', 'quant-ph-9911100-2-16-6': 'Therefore decoherence in this case has certainly a non dissipative origin, and dark counts of the atomic detectors, dephasing collisions with background gas or stray magnetic fields within the cavity have been suggested as possible sources of the damped oscillations.', 'quant-ph-9911100-2-16-7': '[CITATION].', 'quant-ph-9911100-2-17-0': 'The damped behavior of Eq. ([REF]) can be easily obtained if one applies the formalism described above.', 'quant-ph-9911100-2-17-1': 'In fact, from the linearity of Eq. ([REF]), one has that the time averaging procedure is also valid for mean values and matrix elements of each subsystem.', 'quant-ph-9911100-2-17-2': 'Therefore one has [EQUATION]', 'quant-ph-9911100-2-17-3': 'Using Eqs. ([REF]), ([REF]), ([REF]) and ([REF]), Eq. ([REF]) can be rewritten in the same form of Eq. ([REF]) [EQUATION] where, using Eqs. ([REF]) and ([REF]), [EQUATION]', 'quant-ph-9911100-2-17-4': 'If the characteristic time [MATH] is sufficiently small, i.e. [MATH], there is no phase shift, [MATH], and [EQUATION] (see also Eqs. ([REF]) and ([REF])).', 'quant-ph-9911100-2-17-5': 'The fact that in Ref. [CITATION] the Rabi oscillation frequency essentially coincides with the theoretically expected one, suggests that the time [MATH] characterizing the fluctuations of the interaction time is sufficiently small so that it is reasonable to use Eq. ([REF]).', 'quant-ph-9911100-2-17-6': 'Using the above values for [MATH] and [MATH], one can derive an estimate for [MATH], so to get [MATH]sec. This estimate is consistent with the assumption [MATH] we have made, but, more importantly, it turns out to be comparable to the experimental value of the uncertainty in the interaction time.', 'quant-ph-9911100-2-17-7': 'In fact, the fluctuations of the interaction time are mainly due to the experimental uncertainty of the atomic velocity [MATH], that is [MATH] (see Ref. [CITATION]), and taking an average interaction time [MATH]sec, one gets [MATH]sec, which is just the estimate we have derived from the experimental values.', 'quant-ph-9911100-2-17-8': 'This simple argument supports the interpretation that the decoherence observed in [CITATION] is essentially due to the randomness of the interaction time.', 'quant-ph-9911100-2-17-9': 'In fact, in our opinion, the other effects proposed as possible sources of decoherence, such as dark counts of the atomic detectors, dephasing collisions with background gas or stray magnetic fields within the cavity, would give an overall, time-independent, contrast reduction of the Rabi oscillations, different from the observed exponential decay.', 'quant-ph-9911100-2-18-0': 'Results similar to that of Ref. [CITATION] have been very recently obtained by H. Walther group at the Max Planck Institut fur Quantenoptik, in a Rabi oscillation experiment involving again a high-Q microwave cavity mode resonantly interacting with Rydberg atoms [CITATION].', 'quant-ph-9911100-2-18-1': 'In this case, three different initial Fock states [MATH] of the cavity mode, [MATH], have been studied, and preliminary results show a good quantitative agreement of the experimental data with our theoretical approach based on the dispersion of the interaction times.', 'quant-ph-9911100-2-19-0': 'Another cavity QED experiment in which the observed decay of quantum coherence can be, at least partially, explained with our formalism in terms of a random interaction time, is the Ramsey interferometry experiment of M. Brune et al. [CITATION].', 'quant-ph-9911100-2-19-1': 'In this experiment, a QND measurement of the mean photon number of a microwave cavity mode is obtained by measuring, in a Ramsey interferometry scheme, the dispersive light shifts produced on circular Rydberg states by a nonresonant microwave field.', 'quant-ph-9911100-2-19-2': 'The experimental scheme in this case is similar to that of the Rabi oscillation experiment, with two main differences: i) two low-Q microwave cavities [MATH] and [MATH], which can be fed by a classical source [MATH] with frequency [MATH], are added just before and after the cavity of interest [MATH]; ii) the cavity mode is highly detuned from the atomic transition ([MATH]), so to work in the dispersive regime.', 'quant-ph-9911100-2-19-3': 'In the interaction picture with respect to [EQUATION] (we use the classical field as reference for the atomic phases), the Hamiltonian has the following dispersive form [CITATION] [EQUATION] where [MATH], the Rabi frequency within the classical cavities [MATH] is nonzero only when the atom is in [MATH] and [MATH], and [MATH] is nonzero only within [MATH].', 'quant-ph-9911100-2-19-4': 'In the experiment, single circular Rydberg atoms are sent through the apparatus initially prepared in the state [MATH], and let us assume that the microwave cavity mode in [MATH] is in a generic state [MATH].', 'quant-ph-9911100-2-19-5': 'The atom is subject to a [MATH] pulse in [MATH], so that [EQUATION]', 'quant-ph-9911100-2-19-6': 'Then the atom crosses the cavity [MATH] with an interaction time [MATH] and the dispersive interaction yields [EQUATION]', 'quant-ph-9911100-2-19-7': 'Finally the atom is subject to the second [MATH] pulse in the second Ramsey zone [MATH] and the joint state of the Rydberg atom and the cavity mode becomes [EQUATION] where [MATH] is the time of flight from [MATH] to [MATH].', 'quant-ph-9911100-2-19-8': 'The experimentally interesting quantity is the probability to find at the end the atom in the [MATH] state, [MATH], whose theoretical expression according to Eq. ([REF]) is [EQUATION] where the photon number-dependent frequency shift [MATH] is given by [EQUATION]', 'quant-ph-9911100-2-19-9': 'In Eqs. ([REF]) and ([REF]) we have used the fact that [MATH] is equal to the ratio between the waist of the cavity mode [MATH] and the distance between the two Ramsey cavities [MATH].', 'quant-ph-9911100-2-19-10': 'The actual experiment of Ref. [CITATION] has been performed in the bad cavity limity [MATH] in which the cavity [MATH] relaxation time [MATH] is smaller than the atom-cavity interaction time.', 'quant-ph-9911100-2-19-11': 'In this case, the cavity photon number randomly changes during [MATH] and in the corresponding expression ([REF]) for the frequency shift [MATH], the photon number [MATH] has to replaced by the mean value [MATH].', 'quant-ph-9911100-2-19-12': 'The Ramsey fringes are observed by sweeping the frequency of the classical source [MATH] around resonance, that is, studying [MATH] as a function of the detuning [MATH].', 'quant-ph-9911100-2-19-13': 'The experimentally observed Ramsey fringes show a reduced contrast, which moreover decreases for increasing detunings [MATH] (see Fig. 2 of Ref. [CITATION]).', 'quant-ph-9911100-2-19-14': 'Therefore one can try to explain the reduced contrast, i.e., the loss of quantum coherence, in terms of a fluctuating evolution time, which in this case means a random time of flight [MATH] originated again by the dispersion of the atomic velocities.', 'quant-ph-9911100-2-19-15': 'We average again the quantity [MATH] of Eq. ([REF]) over the probability distribution [MATH] derived in Section II, replacing [MATH] with a random time of flight [MATH], and we obtain [EQUATION] where, using Eq. ([REF]), the fringe visibility function [MATH] is given by [EQUATION] and [MATH] is the frequency shift [EQUATION]', 'quant-ph-9911100-2-19-16': 'The parameter [MATH] characterizing the strength of the fluctuations of the time of flight [MATH] can be estimated with arguments similar to those considered for the Rabi oscillation experiment.', 'quant-ph-9911100-2-19-17': 'Since [MATH] and [MATH]sec (see Ref. [CITATION]), one has [MATH]sec. For the interesting range of detunings [MATH], one has [MATH], so that one can neglect again the frequency shift ([REF]) and approximate the fringe visibility function ([REF]) with a gaussian function, that is, [EQUATION]', 'quant-ph-9911100-2-19-18': 'This gaussian modulation of the Ramsey fringes with a width [MATH] is consistent with the typical experimental Ramsey fringe signal (see Fig. 2 of Ref. [CITATION]), but it is not able to completely account for the observed modulation and contrast reduction of the fringes.', 'quant-ph-9911100-2-19-19': 'This means that, contrary to the case of the Rabi oscillation experiment, in this case the role of other experimental imperfections such as random phases due to stray fields, imperfect [MATH] pulses in [MATH] and [MATH] and detection errors, is as relevant as that of the dispersion of atomic velocities and these other effects have to be taken into account to get an exhaustive explanation of the observed decoherence.', 'quant-ph-9911100-2-20-0': '# Rabi oscillation experiments in trapped ions', 'quant-ph-9911100-2-21-0': 'Another interesting Rabi oscillation experiment, performed on a different system, that is, a trapped ion [CITATION], has recently observed a decoherence effect which cannot be attributed to dissipation.', 'quant-ph-9911100-2-21-1': 'In the trapped ion experiment of Ref. [CITATION], the interaction between two internal states ([MATH] and [MATH]) of a Be ion and the center-of-mass vibrations in the [MATH] direction, induced by two driving Raman lasers is studied.', 'quant-ph-9911100-2-21-2': 'In the interaction picture with respect to the free vibrational and internal Hamiltonian, this interaction is described by the following Hamiltonian [CITATION] [EQUATION] where [MATH] denotes the annihiliation operator for the vibrations along the [MATH] direction, [MATH] is the corresponding frequency and [MATH] is the detuning between the internal transition and the frequency difference between the two Raman lasers.', 'quant-ph-9911100-2-21-3': 'The Rabi frequency [MATH] is proportional to the two Raman laser intensities, and [MATH] is the Lamb-Dicke parameter [CITATION].', 'quant-ph-9911100-2-21-4': 'When the two Raman lasers are tuned to the first blue sideband, i.e. [MATH], Hamiltonian ([REF]) predicts Rabi oscillations between [MATH] and [MATH] is a vibrational Fock state) with a frequency [CITATION] [EQUATION] where [MATH] is the generalized Laguerre polynomial.', 'quant-ph-9911100-2-21-5': 'These Rabi oscillations have been experimentally verified by preparing the initial state [MATH], (with [MATH] ranging from [MATH] to [MATH]) and measuring the probability [MATH] as a function of the interaction time [MATH], which is varied by changing the duration of the Raman laser pulses.', 'quant-ph-9911100-2-21-6': 'Again, as in the cavity QED experiment of [CITATION], the experimental Rabi oscillations are damped and well fitted by [CITATION] [EQUATION] where the measured oscillation frequencies [MATH] are in very good agreement with the theoretical prediction ([REF]) corresponding to the measured Lamb-Dicke parameter [MATH] [CITATION].', 'quant-ph-9911100-2-21-7': 'As concerns the decay rates [MATH], the experimental values are fitted in [CITATION] by [EQUATION] where [MATH] Khz.', 'quant-ph-9911100-2-21-8': 'This power-law scaling has attracted the interest of a number of authors and it has been investigated in Refs. [CITATION], even if a clear explanation of this behavior of the decay rates is still lacking.', 'quant-ph-9911100-2-21-9': 'On the contrary, the scaling law ([REF]) can be simply accounted for in the previous formalism if we consider the small [MATH] limit of Eq. ([REF]), which is again suggested by the fact that the experimental and theoretical predictions for the frequencies [MATH] agree.', 'quant-ph-9911100-2-21-10': 'In fact, the [MATH]-dependence of the theoretical prediction of Eq. ([REF]) for [MATH] is well approximated, within 10 %, by the power law dependence (see Fig. 3) [EQUATION] so that, using Eq. ([REF]), one has immediately the power law dependence [MATH] of Eq. ([REF]).', 'quant-ph-9911100-2-21-11': 'The value of the parameter [MATH] can be obtained by matching the values corresponding to [MATH], and using Eq. ([REF]), that is [MATH] sec, where we have used the experimental value [MATH] Khz.', 'quant-ph-9911100-2-22-0': 'However, this value of the parameter [MATH] cannot be explained in terms of some interaction time uncertainty, such as the time jitter of the Raman laser pulses, which is experimentally found to be much smaller [CITATION].', 'quant-ph-9911100-2-22-1': 'In this case, instead, the observed decoherence can be attributed, as already suggested in [CITATION], to the fluctuation of the Raman laser intensities, yielding a fluctuating Rabi frequency parameter [MATH] of the Hamiltonian ([REF]).', 'quant-ph-9911100-2-22-2': 'In this case, the evolution is driven by a fluctuating Hamiltonian [MATH], where [MATH] in Eq. ([REF]), so that [EQUATION] where [MATH], and we have defined the positive dimensionless random variable [MATH], which is proportional to the pulse area.', 'quant-ph-9911100-2-22-3': 'It is now easy to understand that the physical situation is analogous to that characterized by a random interaction time considered in the preceding sections, with [MATH] replaced by [MATH] and [MATH] by [MATH].', 'quant-ph-9911100-2-22-4': 'It is therefore straightforward to adapt the formalism developed in Section II to this case, in which the fluctuating quantity is the pulse area [MATH], yielding again random phases in the energy basis representation.', 'quant-ph-9911100-2-22-5': 'In analogy with Eq. ([REF]), one considers an averaged density matrix [EQUATION]', 'quant-ph-9911100-2-22-6': 'Imposing again that [MATH] must be a density operator and the semigroup property, one finds results analogous to Eqs. ([REF]) and ([REF]) [EQUATION]', 'quant-ph-9911100-2-22-7': 'Here, the parameters [MATH] and [MATH] are introduced as scaling parameters, but they have a clear meaning, as it can be easily seen by considering the mean and the variance of the probability distribution of Eq. ([REF]), [EQUATION] implying that [MATH] has now to be meant as a mean Rabi frequency, and that [MATH] quantifies the strength of [MATH] fluctuations.', 'quant-ph-9911100-2-22-8': 'It is interesting to note that these first two moments of [MATH] determine the properties of the fluctuating Rabi frequency [MATH], which can be written as [EQUATION] that is, the Rabi frequency [MATH] is a white, non-gaussian (due to the non-gaussian form of [MATH]) stochastic process.', 'quant-ph-9911100-2-22-9': 'In fact, the semigroup assumption we have made implies a Markovian treatment in which the spectrum of the laser intensity fluctuations is flat in the relevant frequency range.', 'quant-ph-9911100-2-22-10': 'This in particular implies that we are neglecting the dynamics at small times, of the order of the correlation time of the laser intensity fluctuations.', 'quant-ph-9911100-2-23-0': 'The estimated value of [MATH] gives a reasonable estimate of the pulse area fluctuations, since it corresponds to a fractional error of the pulse area [MATH] of [MATH] for a pulse duration of [MATH]sec, and which is decreasing for increasing pulse durations.', 'quant-ph-9911100-2-24-0': 'The present analysis shows many similarities with that of Ref. [CITATION] which also tries to explain the decay of the Rabi oscillations in the ion trap experiments of [CITATION] in terms of laser intensity fluctuations.', 'quant-ph-9911100-2-24-1': 'The authors of Ref. [CITATION] in fact use a phase destroying master equation coinciding with the second-order expansion ([REF]) of our generalized master equation of Eq. ([REF]) (see Eq. (16) of Ref. [CITATION] with the identifications [MATH] and [MATH]) and moreover derive the same numerical estimate for the pulse area fluctuation strength [MATH].', 'quant-ph-9911100-2-24-2': 'Despite this similarities, they do not recover the scaling ([REF]) of the decay rates [MATH] only because they do not use the general expression of the Rabi frequency ([REF]), (and which is well approximated by the power law ([REF])) but its Lamb-Dicke limit [MATH], which is valid only when [MATH].', 'quant-ph-9911100-2-24-3': 'There is however another, more fundamental, difference between our approach and that of Ref. [CITATION].', 'quant-ph-9911100-2-24-4': 'They assume from the beginning that the laser intensity fluctuations have a white and gaussian character, while we make no a priori assumption on the statistical properties of the pulse area [MATH].', 'quant-ph-9911100-2-24-5': 'We derive these properties, i.e. the probability distribution ([REF]), only from the semigroup condition, and it is interesting to note that this condition yields a gaussian probability distribution for the pulse area only as a limiting case.', 'quant-ph-9911100-2-24-6': 'In fact, from Eq. ([REF]) one can see that [MATH] tends to become a gaussian with the same mean value [MATH] and the same width [MATH] only in the large time limit [MATH] [EQUATION]', 'quant-ph-9911100-2-24-7': 'The non-gaussian character of [MATH] can be traced back to the fact that [MATH] must be definite and normalized in the interval [MATH] and not in [MATH].', 'quant-ph-9911100-2-24-8': 'Notice that at [MATH], Eq. ([REF]) assumes the exponential form [MATH].', 'quant-ph-9911100-2-24-9': 'Only at large times [MATH] the random variable [MATH] becomes the sum of many independent contributions and assumes the gaussian form.', 'quant-ph-9911100-2-25-0': 'Due to the non-gaussian nature of the random variable [MATH], we find that the more generally valid phase-destroying master equation is given by Eq. ([REF]) (with [MATH] replaced by [MATH]).', 'quant-ph-9911100-2-25-1': 'The predictions of Eq. ([REF]) significantly depart from its second order expansion in [MATH], Eq. ([REF]), corresponding to the gaussian limit, as soon as [MATH] becomes comparable with the typical timescale of the system under study, which, in the present case, is the inverse of the Rabi frequency.', 'quant-ph-9911100-2-26-0': 'The present analysis of the Rabi oscillation experiment of Ref. [CITATION] can be repeated for the very recent experiment with trapped ions performed in Innsbruck [CITATION], in which Rabi oscillations involving the vibrational levels and an optical quadrupole transition of a single [MATH]Ca[MATH] ion have been observed.', 'quant-ph-9911100-2-26-1': 'Damped oscillations corresponding to initial vibrational numbers [MATH] and [MATH] are reported.', 'quant-ph-9911100-2-26-2': 'From the data with [MATH], [MATH] Khz and [MATH] Khz, we get [MATH] and this estimate is consistent with attributing again the decoherence to the fluctuations of the Rabi frequency caused by laser intensity fluctuations.', 'quant-ph-9911100-2-26-3': 'Moreover in this case, the experiment is performed in the Lamb-Dicke limit [MATH], and therefore, using again Eq. ([REF]), we expect, in this case, a linear scaling with the vibrational number, [MATH].', 'quant-ph-9911100-2-27-0': '# Concluding remarks', 'quant-ph-9911100-2-28-0': 'Decoherence is not always necessarily due to the entanglement with an environment, but it may be due, as well, to the fluctuations of some classical parameter or internal variable of a system.', 'quant-ph-9911100-2-28-1': 'This is a different form of decoherence, which is present even in isolated systems, and that we have called non-dissipative decoherence.', 'quant-ph-9911100-2-28-2': 'In this paper we have presented a model-independent theory for non-dissipative decoherence, which can be applied in the case of a random evolution time or in the case of a fluctuating Hamiltonian.', 'quant-ph-9911100-2-28-3': 'This approach proves to be a flexible tool, able to give a quantitative understanding of the decoherence caused by the fluctuations of classical quantities.', 'quant-ph-9911100-2-28-4': 'In fact, in this paper we have given a simple and unified description of the decoherence phenomenon observed in recent Rabi oscillation experiments performed in a cavity QED configuration [CITATION] and on a trapped ion [CITATION].', 'quant-ph-9911100-2-28-5': 'In particular, this approach has allowed us to explain for the first time in simple terms, the power-law scaling of the coherence decay rates of Eq. ([REF]), observed in the trapped ion experiment.', 'quant-ph-9911100-2-29-0': 'The relevant aspect of the approach applied here, and introduced in Ref. [CITATION], is its model-independence.', 'quant-ph-9911100-2-29-1': 'The formalism is in fact derived starting from few, very general assumptions: i) the average density matrix [MATH] has all the usual properties of a density matrix; ii) the semigroup property for the time evolution generator [MATH] for [MATH].', 'quant-ph-9911100-2-29-2': 'With this respect, this approach seems to provide a very general description of non-dissipative decoherence, in which the random properties of the fluctuating classical variables are characterized by the two, system-dependent, time parameters [MATH] and [MATH].', 'quant-ph-9911100-2-29-3': 'As we have seen in section II, in the cases where one has a standard, continuous evolution, the two times coincide [MATH].', 'quant-ph-9911100-2-29-4': 'Under ideal conditions of no fluctuating classical variable or parameter, one would have [MATH], and the usual unitary evolution of an isolated system in quantum mechanics would be recovered.', 'quant-ph-9911100-2-29-5': 'However, the generality of the approach suggests in some way the possibility that the parameter [MATH], even though system-dependent, might have a lower nonzero limit, which would be reached just in the case of no fluctuations of experimental origin.', 'quant-ph-9911100-2-29-6': 'This would mean a completely new description of time in quantum mechanics.', 'quant-ph-9911100-2-29-7': 'In fact, the evolution time of a system [MATH] (and not the ""clock"" time [MATH]) would become an intrinsically random variable with a well defined probability distribution, without the difficulty of introducing an evolution time operator.', 'quant-ph-9911100-2-29-8': 'In Ref. [CITATION] it is suggested a relation of the nonzero limit for [MATH] with the ""energy-time"" [MATH] where [MATH] is the uncertainty in energy.', 'quant-ph-9911100-2-29-9': 'This would give a precise meaning to the time-energy uncertainty relation because now [MATH] rules the width of the time distribution function.', 'quant-ph-9911100-2-29-10': 'However, this ""intrinsic assumption"" is not necessarily implied by the formalism developed in [CITATION] and applied, with a more pragmatic attitude, in the present paper.'}","[['quant-ph-9911100-1-8-0', 'quant-ph-9911100-2-8-0'], ['quant-ph-9911100-1-8-1', 'quant-ph-9911100-2-8-1'], ['quant-ph-9911100-1-8-2', 'quant-ph-9911100-2-8-2'], ['quant-ph-9911100-1-8-3', 'quant-ph-9911100-2-8-3'], ['quant-ph-9911100-1-8-4', 'quant-ph-9911100-2-8-4'], ['quant-ph-9911100-1-8-5', 'quant-ph-9911100-2-8-5'], ['quant-ph-9911100-1-11-0', 'quant-ph-9911100-2-11-0'], ['quant-ph-9911100-1-11-1', 'quant-ph-9911100-2-11-1'], ['quant-ph-9911100-1-11-2', 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'quant-ph-9911100-3-9-16'], ['quant-ph-9911100-2-9-17', 'quant-ph-9911100-3-9-17'], ['quant-ph-9911100-2-9-18', 'quant-ph-9911100-3-9-18'], ['quant-ph-9911100-2-9-19', 'quant-ph-9911100-3-9-19'], ['quant-ph-9911100-2-9-20', 'quant-ph-9911100-3-9-20'], ['quant-ph-9911100-2-9-21', 'quant-ph-9911100-3-9-21'], ['quant-ph-9911100-2-10-0', 'quant-ph-9911100-3-10-0'], ['quant-ph-9911100-2-10-1', 'quant-ph-9911100-3-10-1'], ['quant-ph-9911100-2-10-2', 'quant-ph-9911100-3-10-2'], ['quant-ph-9911100-2-10-3', 'quant-ph-9911100-3-10-3'], ['quant-ph-9911100-2-10-4', 'quant-ph-9911100-3-10-4'], ['quant-ph-9911100-2-10-5', 'quant-ph-9911100-3-10-5'], ['quant-ph-9911100-2-11-0', 'quant-ph-9911100-3-11-0'], ['quant-ph-9911100-2-11-1', 'quant-ph-9911100-3-11-1'], ['quant-ph-9911100-2-11-2', 'quant-ph-9911100-3-11-2'], ['quant-ph-9911100-2-11-3', 'quant-ph-9911100-3-11-3'], ['quant-ph-9911100-2-5-0', 'quant-ph-9911100-3-5-0'], ['quant-ph-9911100-2-5-1', 'quant-ph-9911100-3-5-1'], ['quant-ph-9911100-2-5-2', 'quant-ph-9911100-3-5-2'], ['quant-ph-9911100-2-5-3', 'quant-ph-9911100-3-5-3']]",[],[],[],[],"['quant-ph-9911100-1-16-7', 'quant-ph-9911100-1-17-3', 'quant-ph-9911100-2-16-7', 'quant-ph-9911100-2-17-3', 'quant-ph-9911100-3-16-7', 'quant-ph-9911100-3-17-3']","{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}",https://arxiv.org/abs/quant-ph/9911100,"{'quant-ph-9911100-3-0-0': 'We consider the case when decoherence is due to the fluctuations of some classical variable or parameter of a system and not to its entanglement with the environment.', 'quant-ph-9911100-3-0-1': 'Under few and quite general assumptions, we derive a model-independent formalism for this non-dissipative decoherence, and we apply it to explain the decoherence observed in some recent experiments in cavity QED and on trapped ions.', 'quant-ph-9911100-3-1-0': '# Introduction', 'quant-ph-9911100-3-2-0': 'Decoherence is the rapid transformation of a pure linear superposition state into the corresponding statistical mixture [EQUATION] this process does not preserve the purity of the state, that is, [MATH], and therefore it has to be described in terms of a non-unitary evolution.', 'quant-ph-9911100-3-2-1': 'The most common approach is the so-called environment-induced decoherence [CITATION] which is based on the consideration that it is extremely difficult to isolate perfectly a system from uncontrollable degrees of freedom (the ""environment"").', 'quant-ph-9911100-3-2-2': 'The non-unitary evolution of the system of interest is obtained by considering the interaction with these uncontrolled degrees of freedom and tracing over them.', 'quant-ph-9911100-3-2-3': 'In this approach, decoherence is caused by the entanglement of the two states of the superposition with two approximately orthogonal states of the environment [MATH] and [MATH] [EQUATION]', 'quant-ph-9911100-3-2-4': 'Tracing over the environment and using [MATH], one gets [EQUATION] where [MATH] is defined in Eq. ([REF]).', 'quant-ph-9911100-3-2-5': 'The environment behaves as a measurement apparatus because the states [MATH] behave as ""pointer states"" associated with [MATH]; in this way the environment acquires ""information"" on the system state and therefore decoherence is described as an irreversible flow of information from the system into the environment [CITATION].', 'quant-ph-9911100-3-2-6': 'In this approach, the system energy is usually not conserved and the interaction with the environment also accounts for the irreversible thermalization of the system of interest.', 'quant-ph-9911100-3-2-7': 'However this approach is inevitably model-dependent, because one has to assume a model Hamiltonian for the environment and the interaction between system and environment.', 'quant-ph-9911100-3-2-8': 'This modelization, and therefore any quantitative prediction, becomes problematic whenever the environmental degrees of freedom responsible for decoherence are not easily recognizable.', 'quant-ph-9911100-3-3-0': 'Decoherence is not always necessarily due to the entanglement with an environment, but it may be due, as well, to the fluctuations of some classical parameter or internal variable of the system.', 'quant-ph-9911100-3-3-1': 'This kind of decoherence is present even in isolated systems, where environment-induced decoherence has to be neglected.', 'quant-ph-9911100-3-3-2': 'In these cases the system energy is conserved, and one has a different form of decoherence, which we shall call ""non-dissipative decoherence"".', 'quant-ph-9911100-3-3-3': 'In such cases, every single experimental run is characterized by the usual unitary evolution generated by the system Hamiltonian.', 'quant-ph-9911100-3-3-4': 'However, definite statistical prediction are obtained only repeating the experiment many times and this is when decoherence takes place, because each run corresponds to a different random value or stochastic realization of the fluctuating classical variable.', 'quant-ph-9911100-3-3-5': 'The experimental results correspond therefore to an average over these fluctuations and they will describe in general an effective non-unitary evolution.', 'quant-ph-9911100-3-4-0': 'In this paper we shall present a quite general theory of non-dissipative decoherence for isolated systems which can be applied for two different kinds of fluctuating variables or parameters: the case of a random evolution time and the case of a fluctuating Rabi frequency yielding a fluctuation of the Hamiltonian.', 'quant-ph-9911100-3-4-1': 'In both cases one has random phases [MATH] in the energy eigenstates basis that, once averaged over many experimental runs, lead to the decay of off-diagonal matrix elements of the density operator, while leaving the diagonal ones unchanged.', 'quant-ph-9911100-3-5-0': 'The outline of the paper is as follows.', 'quant-ph-9911100-3-5-1': 'In Section II we shall derive the theory under general assumptions, following closely the original derivation presented in [CITATION].', 'quant-ph-9911100-3-5-2': 'In Section III we shall apply this theory in order to describe the decoherence effects observed in two cavity QED experiments performed in Paris, one describing Rabi oscillations associated with the resonant interaction between a Rydberg atom and a microwave cavity mode [CITATION], and the second one a Ramsey interferometry experiment using a dispersive interaction between the cavity mode and the atom [CITATION].', 'quant-ph-9911100-3-5-3': 'In Section IV we shall apply our approach to a Rabi oscillation experiment for trapped ions [CITATION], and Section V is for concluding remarks.', 'quant-ph-9911100-3-6-0': '# The general formalism', 'quant-ph-9911100-3-7-0': 'The formalism describing non-dissipative decoherence of isolated systems has been derived in [CITATION] by considering the case of a system with random evolution time.', 'quant-ph-9911100-3-7-1': 'The evolution time may be random because of the finite time needed to prepare the initial state of the system, because of the randomness of the detection time, as well as many other reasons.', 'quant-ph-9911100-3-7-2': 'For example, in cavity QED experiments, the evolution time is the interaction time, which is determined by the time of flight of the atoms within the cavity and this time can be random due to atomic velocity dispersion.', 'quant-ph-9911100-3-8-0': 'In these cases, the experimental observations are not described by the usual density matrix of the whole system [MATH], but by its time averaged counterpart [CITATION] [EQUATION] where [MATH] is the usual unitarily evolved density operator from the initial state and [MATH].', 'quant-ph-9911100-3-8-1': 'Therefore [MATH] denotes the random evolution time, while [MATH] is a parameter describing the usual ""clock"" time.', 'quant-ph-9911100-3-8-2': 'Using Eq. ([REF]), one can write [EQUATION] where [EQUATION] is the evolution operator for the averaged state of the system.', 'quant-ph-9911100-3-8-3': 'Following Ref. [CITATION], we determine the function [MATH] by imposing the following plausible conditions: i) [MATH] must be a density operator, i.e. it must be self-adjoint, positive-definite, and with unit-trace.', 'quant-ph-9911100-3-8-4': 'This leads to the condition that [MATH] must be non-negative and normalized, i.e a probability density in [MATH] so that Eq. ([REF]) is a completely positive mapping.', 'quant-ph-9911100-3-8-5': 'ii) [MATH] satisfies the semigroup property [MATH], with [MATH].', 'quant-ph-9911100-3-9-0': 'The semigroup condition is satisfied by an exponential dependence on [MATH] [EQUATION] where [MATH] naturally appears as a scaling time.', 'quant-ph-9911100-3-9-1': 'A solution satisfying all the conditions we have imposed can be found by separating [MATH] in its hermitian and antihermitian part [MATH] and by considering the Gamma function integral identity [CITATION] [EQUATION]', 'quant-ph-9911100-3-9-2': 'Now the right hand side of Eq. ([REF]) can be identified with the right hand side of Eq. ([REF]) if we impose the following conditions: [MATH], where [MATH] is another scaling time, generally different from [MATH]; [MATH] in order to make the exponential terms identical, and [MATH] in order to get a normalized probability distribution [MATH].', 'quant-ph-9911100-3-9-3': 'This choice yields the following expressions for the evolution operator for the averaged density matrix [MATH] and for the probability density [MATH] [CITATION] [EQUATION]', 'quant-ph-9911100-3-9-4': 'Notice that the ordinary quantum evolution is recovered when [MATH]; in this limit [MATH] so that [MATH] and [MATH] is the usual unitary evolution.', 'quant-ph-9911100-3-9-5': 'Moreover, it can be seen that Eq. ([REF]) implies that [MATH] satisfies a finite difference equation [CITATION].', 'quant-ph-9911100-3-9-6': 'The semigroup condition leads to the form of the probability distribution [MATH] we use to perform the average on the fluctuating evolution times.', 'quant-ph-9911100-3-9-7': 'However, notice that this probability distribution depends on both the two scaling times [MATH] and [MATH] only apparently.', 'quant-ph-9911100-3-9-8': 'In fact, if we change variable in the time integral, [MATH], it is possible to rewrite the integral expression for [MATH] in the following way [EQUATION] where [EQUATION]', 'quant-ph-9911100-3-9-9': 'This probability density depends only on [MATH].', 'quant-ph-9911100-3-9-10': 'However Eq. ([REF]) contains an effective rescaled time evolution generator [MATH].', 'quant-ph-9911100-3-9-11': 'The physical meaning of the probability distribution of Eq. ([REF]), of the rescaled evolution operator, and of the two scaling times can be understood if we consider the following simple example.', 'quant-ph-9911100-3-9-12': 'Let us consider a system with Hamiltonian [MATH], where [EQUATION] ([MATH] is the Heaviside step function), that is, a system with Hamiltonian [MATH] which is periodically applied for a time [MATH], with time period [MATH]) and which is ""turned off"" otherwise.', 'quant-ph-9911100-3-9-13': 'The unitary evolution operator for this system is [MATH], where [MATH] and [EQUATION] which can be however well approximated by the ""rescaled"" evolution operator [MATH].', 'quant-ph-9911100-3-9-14': 'In fact, the maximum relative error in replacing [MATH] with [MATH] is [MATH] and becomes negligible at large times (see Fig. 1).', 'quant-ph-9911100-3-9-15': 'This fact suggests to interpret the time average of Eq. ([REF]) as an average over unitary evolutions generated by [MATH], taking place randomly in time, with mean time width [MATH], and separated by a mean time interval [MATH].', 'quant-ph-9911100-3-9-16': 'This interpretation is confirmed by the fact that when [MATH], for integer [MATH], the probability distribution [MATH] of Eq. ([REF]) is a known statistical distribution giving the probability density that the waiting time for [MATH] independent events is [MATH] when [MATH] is the mean time interval between two events.', 'quant-ph-9911100-3-9-17': 'A particularly clear example of the random process in time implied by the above equations is provided by the micromaser [CITATION] in which a microwave cavity is crossed by a beam of resonant atoms with mean injection rate [MATH], and a mean interaction time within the cavity corresponding to [MATH].', 'quant-ph-9911100-3-9-18': 'In the micromaser theory, the non unitary operator [MATH] describing the effective dynamics of the microwave mode during each atomic crossing replaces the evolution operator [MATH] [CITATION].', 'quant-ph-9911100-3-9-19': 'Another example of interrupted evolution is provided by the experimental scheme proposed in [CITATION] for the quantum non-demolition (QND) measurement [CITATION] of the photon number in a high-Q cavity.', 'quant-ph-9911100-3-9-20': 'In this proposal, the photon number is determined by measuring the phase shift induced on a train of Rydberg atoms sent through the microwave cavity with mean rate [MATH], and interacting dispersively with the cavity mode.', 'quant-ph-9911100-3-9-21': 'These two examples show that the two scaling times [MATH] and [MATH] have not to be considered as new universal constants, but as two characteristic times of the system under study.', 'quant-ph-9911100-3-10-0': 'However, in most cases, one does not have an interrupted evolution as in micromaser-like situations, but a standard, continuous evolution generated by an Hamiltonian [MATH].', 'quant-ph-9911100-3-10-1': 'In this case the ""scaled"" effective evolution operator has to coincide with the usual one, [MATH], and this is possible only if [MATH].', 'quant-ph-9911100-3-10-2': 'In this case [MATH] is simply the parameter characterizing the strength of the fluctuations of the random evolution time.', 'quant-ph-9911100-3-10-3': 'This meaning of the parameter [MATH] in the case of equal scaling times is confirmed by the expressions of the mean and the variance of the probability distribution of Eq. ([REF]) [EQUATION]', 'quant-ph-9911100-3-10-4': 'When [MATH], the mean evolution time coincide with the ""clock \'\' time [MATH], while the variance of the evolution time becomes [MATH].', 'quant-ph-9911100-3-10-5': 'In the rest of the paper we shall always consider the standard situation of an isolated system with Hamiltonian [MATH], continuously evolving in time, and we shall always assume [MATH].', 'quant-ph-9911100-3-11-0': 'When [MATH], [MATH] is the usual unitary evolution.', 'quant-ph-9911100-3-11-1': 'For finite [MATH], on the contrary, the evolution equation ([REF]) describes a decay of the off-diagonal matrix elements in the energy representation, whereas the diagonal matrix elements remain constant, i.e. the energy is still a constant of motion.', 'quant-ph-9911100-3-11-2': 'In fact, in the energy eigenbasis, Eqs. ([REF]) and ([REF]) yield [EQUATION] where [MATH] and [EQUATION]', 'quant-ph-9911100-3-11-3': 'This means that, in general, the effect of the average over the fluctuating evolution time yields an exponential decay and a frequency shift [MATH] of every term oscillating in time with frequency [MATH].', 'quant-ph-9911100-3-12-0': 'The phase diffusion aspects of the present approach can also be seen if the evolution equation of the averaged density matrix [MATH] is considered.', 'quant-ph-9911100-3-12-1': 'In fact, by differentiating with respect to time Eq. ([REF]) and using ([REF]), one gets the following master equation for [MATH] (we consider the case [MATH]) [EQUATION] expanding the logarithm at second order in [MATH], one obtains [EQUATION] which is the well-known phase-destroying master equation [CITATION].', 'quant-ph-9911100-3-12-2': 'Hence Eq. ([REF]) appears as a generalized phase-destroying master equation taking into account higher order terms in [MATH].', 'quant-ph-9911100-3-12-3': 'Notice, however, that the present approach is different from the usual master equation approach in the sense that it is model-independent and no perturbative and specific statistical assumptions are made.', 'quant-ph-9911100-3-12-4': 'The solution of Eq. ([REF]) gives an expression for [MATH] similar to that of Eq. ([REF]), but with [CITATION] [EQUATION] which are nonetheless obtained also as a first order expansion in [MATH] of Eqs. ([REF]) and ([REF]).', 'quant-ph-9911100-3-12-5': 'The opposite limit [MATH] has been discussed in detail in Ref. [CITATION].', 'quant-ph-9911100-3-13-0': 'Finally a comment concerning the form of the evolution operator for the averaged density matrix [MATH] of Eq. ([REF]).', 'quant-ph-9911100-3-13-1': 'At first sight it seems that [MATH] is in general a multivalued function of the Liouvillian [MATH], and that [MATH] is uniquely defined only when [MATH], [MATH] integer.', 'quant-ph-9911100-3-13-2': 'However, this form for [MATH] is a consequence of the time average over [MATH] of Eq. ([REF]), which is a properly defined, non-negative probability distribution only if the algebraic definition of the power law function [MATH] is assumed.', 'quant-ph-9911100-3-13-3': 'This means that in Eq. ([REF]) one has to take the first determination of the power-law function and in this way [MATH] is univocally defined.', 'quant-ph-9911100-3-14-0': '# Application to cavity QED experiments', 'quant-ph-9911100-3-15-0': 'A first experimental situation in which the above formalism can be applied is the Rabi oscillation experiment of Ref. [CITATION], in which the resonant interaction between a quantized mode in a high-Q microwave cavity (with annihilation operator [MATH]) and two circular Rydberg states ([MATH] and [MATH]) of a Rb atom has been studied.', 'quant-ph-9911100-3-15-1': 'This interaction is well described by the usual Jaynes-Cummings [CITATION] model, which in the interaction picture reads [EQUATION] where [MATH] is the Rabi frequency.', 'quant-ph-9911100-3-16-0': 'The Rabi oscillations describing the exchange of excitations between atom and cavity mode are studied by injecting the velocity-selected Rydberg atom, prepared in the excited state [MATH], in the high-Q cavity and measuring the population of the lower atomic level [MATH], [MATH], as a function of the interaction time [MATH], which is varied by changing the Rydberg atom velocity.', 'quant-ph-9911100-3-16-1': 'Different initial states of the cavity mode have been considered in [CITATION].', 'quant-ph-9911100-3-16-2': 'We shall restrict only to the case of vacuum state induced Rabi oscillations, where the decoherence effect is particularly evident.', 'quant-ph-9911100-3-16-3': 'The Hamiltonian evolution according to Eq. ([REF]) predicts in this case Rabi oscillations of the form [EQUATION]', 'quant-ph-9911100-3-16-4': 'Experimentally instead, damped oscillations are observed, which are well fitted by [EQUATION] where the decay time fitting the experimental data is [MATH]sec [CITATION] and the corresponding Rabi frequency is [MATH] Khz (see Fig. 2).', 'quant-ph-9911100-3-16-5': 'This decay of quantum coherence cannot be associated with photon leakage out of the cavity because the cavity relaxation time is larger ([MATH]sec) and also because in this case one would have an asymptotic limit [MATH].', 'quant-ph-9911100-3-16-6': 'Therefore decoherence in this case has certainly a non dissipative origin, and dark counts of the atomic detectors, dephasing collisions with background gas or stray magnetic fields within the cavity have been suggested as possible sources of the damped oscillations.', 'quant-ph-9911100-3-16-7': '[CITATION].', 'quant-ph-9911100-3-17-0': 'The damped behavior of Eq. ([REF]) can be easily obtained if one applies the formalism described above.', 'quant-ph-9911100-3-17-1': 'In fact, from the linearity of Eq. ([REF]), one has that the time averaging procedure is also valid for mean values and matrix elements of each subsystem.', 'quant-ph-9911100-3-17-2': 'Therefore one has [EQUATION]', 'quant-ph-9911100-3-17-3': 'Using Eqs. ([REF]), ([REF]), ([REF]) and ([REF]), Eq. ([REF]) can be rewritten in the same form of Eq. ([REF]) [EQUATION] where, using Eqs. ([REF]) and ([REF]), [EQUATION]', 'quant-ph-9911100-3-17-4': 'If the characteristic time [MATH] is sufficiently small, i.e. [MATH], there is no phase shift, [MATH], and [EQUATION] (see also Eqs. ([REF]) and ([REF])).', 'quant-ph-9911100-3-17-5': 'The fact that in Ref. [CITATION] the Rabi oscillation frequency essentially coincides with the theoretically expected one, suggests that the time [MATH] characterizing the fluctuations of the interaction time is sufficiently small so that it is reasonable to use Eq. ([REF]).', 'quant-ph-9911100-3-17-6': 'Using the above values for [MATH] and [MATH], one can derive an estimate for [MATH], so to get [MATH]sec. This estimate is consistent with the assumption [MATH] we have made, but, more importantly, it turns out to be comparable to the experimental value of the uncertainty in the interaction time.', 'quant-ph-9911100-3-17-7': 'In fact, the fluctuations of the interaction time are mainly due to the experimental uncertainty of the atomic velocity [MATH], that is [MATH] (see Ref. [CITATION]), and taking an average interaction time [MATH]sec, one gets [MATH]sec, which is just the estimate we have derived from the experimental values.', 'quant-ph-9911100-3-17-8': 'This simple argument supports the interpretation that the decoherence observed in [CITATION] is essentially due to the randomness of the interaction time.', 'quant-ph-9911100-3-17-9': 'In fact, in our opinion, the other effects proposed as possible sources of decoherence, such as dark counts of the atomic detectors, dephasing collisions with background gas or stray magnetic fields within the cavity, would give an overall, time-independent, contrast reduction of the Rabi oscillations, different from the observed exponential decay.', 'quant-ph-9911100-3-18-0': 'Results similar to that of Ref. [CITATION] have been very recently obtained by H. Walther group at the Max Planck Institut fur Quantenoptik, in a Rabi oscillation experiment involving again a high-Q microwave cavity mode resonantly interacting with Rydberg atoms [CITATION].', 'quant-ph-9911100-3-18-1': 'In this case, three different initial Fock states [MATH] of the cavity mode, [MATH], have been studied, and preliminary results show a good quantitative agreement of the experimental data with our theoretical approach based on the dispersion of the interaction times.', 'quant-ph-9911100-3-19-0': 'Another cavity QED experiment in which the observed decay of quantum coherence can be, at least partially, explained with our formalism in terms of a random interaction time, is the Ramsey interferometry experiment of M. Brune et al. [CITATION].', 'quant-ph-9911100-3-19-1': 'In this experiment, a QND measurement of the mean photon number of a microwave cavity mode is obtained by measuring, in a Ramsey interferometry scheme, the dispersive light shifts produced on circular Rydberg states by a nonresonant microwave field.', 'quant-ph-9911100-3-19-2': 'The experimental scheme in this case is similar to that of the Rabi oscillation experiment, with two main differences: i) two low-Q microwave cavities [MATH] and [MATH], which can be fed by a classical source [MATH] with frequency [MATH], are added just before and after the cavity of interest [MATH]; ii) the cavity mode is highly detuned from the atomic transition ([MATH]), so to work in the dispersive regime.', 'quant-ph-9911100-3-19-3': 'In the interaction picture with respect to [EQUATION] (we use the classical field as reference for the atomic phases), the Hamiltonian has the following dispersive form [CITATION] [EQUATION] where [MATH], the Rabi frequency within the classical cavities [MATH] is nonzero only when the atom is in [MATH] and [MATH], and [MATH] is nonzero only within [MATH].', 'quant-ph-9911100-3-19-4': 'In the experiment, single circular Rydberg atoms are sent through the apparatus initially prepared in the state [MATH], and let us assume that the microwave cavity mode in [MATH] is in a generic state [MATH].', 'quant-ph-9911100-3-19-5': 'The atom is subject to a [MATH] pulse in [MATH], so that [EQUATION]', 'quant-ph-9911100-3-19-6': 'Then the atom crosses the cavity [MATH] with an interaction time [MATH] and the dispersive interaction yields [EQUATION]', 'quant-ph-9911100-3-19-7': 'Finally the atom is subject to the second [MATH] pulse in the second Ramsey zone [MATH] and the joint state of the Rydberg atom and the cavity mode becomes [EQUATION] where [MATH] is the time of flight from [MATH] to [MATH].', 'quant-ph-9911100-3-19-8': 'The experimentally interesting quantity is the probability to find at the end the atom in the [MATH] state, [MATH], whose theoretical expression according to Eq. ([REF]) is [EQUATION] where the photon number-dependent frequency shift [MATH] is given by [EQUATION]', 'quant-ph-9911100-3-19-9': 'In Eqs. ([REF]) and ([REF]) we have used the fact that [MATH] is equal to the ratio between the waist of the cavity mode [MATH] and the distance between the two Ramsey cavities [MATH].', 'quant-ph-9911100-3-19-10': 'The actual experiment of Ref. [CITATION] has been performed in the bad cavity limity [MATH] in which the cavity [MATH] relaxation time [MATH] is smaller than the atom-cavity interaction time.', 'quant-ph-9911100-3-19-11': 'In this case, the cavity photon number randomly changes during [MATH] and in the corresponding expression ([REF]) for the frequency shift [MATH], the photon number [MATH] has to replaced by the mean value [MATH].', 'quant-ph-9911100-3-19-12': 'The Ramsey fringes are observed by sweeping the frequency of the classical source [MATH] around resonance, that is, studying [MATH] as a function of the detuning [MATH].', 'quant-ph-9911100-3-19-13': 'The experimentally observed Ramsey fringes show a reduced contrast, which moreover decreases for increasing detunings [MATH] (see Fig. 2 of Ref. [CITATION]).', 'quant-ph-9911100-3-19-14': 'Therefore one can try to explain the reduced contrast, i.e., the loss of quantum coherence, in terms of a fluctuating evolution time, which in this case means a random time of flight [MATH] originated again by the dispersion of the atomic velocities.', 'quant-ph-9911100-3-19-15': 'We average again the quantity [MATH] of Eq. ([REF]) over the probability distribution [MATH] derived in Section II, replacing [MATH] with a random time of flight [MATH], and we obtain [EQUATION] where, using Eq. ([REF]), the fringe visibility function [MATH] is given by [EQUATION] and [MATH] is the frequency shift [EQUATION]', 'quant-ph-9911100-3-19-16': 'The parameter [MATH] characterizing the strength of the fluctuations of the time of flight [MATH] can be estimated with arguments similar to those considered for the Rabi oscillation experiment.', 'quant-ph-9911100-3-19-17': 'Since [MATH] and [MATH]sec (see Ref. [CITATION]), one has [MATH]sec. For the interesting range of detunings [MATH], one has [MATH], so that one can neglect again the frequency shift ([REF]) and approximate the fringe visibility function ([REF]) with a gaussian function, that is, [EQUATION]', 'quant-ph-9911100-3-19-18': 'This gaussian modulation of the Ramsey fringes with a width [MATH] is consistent with the typical experimental Ramsey fringe signal (see Fig. 2 of Ref. [CITATION]), but it is not able to completely account for the observed modulation and contrast reduction of the fringes.', 'quant-ph-9911100-3-19-19': 'This means that, contrary to the case of the Rabi oscillation experiment, in this case the role of other experimental imperfections such as random phases due to stray fields, imperfect [MATH] pulses in [MATH] and [MATH] and detection errors, is as relevant as that of the dispersion of atomic velocities and these other effects have to be taken into account to get an exhaustive explanation of the observed decoherence.', 'quant-ph-9911100-3-20-0': '# Rabi oscillation experiments in trapped ions', 'quant-ph-9911100-3-21-0': 'Another interesting Rabi oscillation experiment, performed on a different system, that is, a trapped ion [CITATION], has recently observed a decoherence effect which cannot be attributed to dissipation.', 'quant-ph-9911100-3-21-1': 'In the trapped ion experiment of Ref. [CITATION], the interaction between two internal states ([MATH] and [MATH]) of a Be ion and the center-of-mass vibrations in the [MATH] direction, induced by two driving Raman lasers is studied.', 'quant-ph-9911100-3-21-2': 'In the interaction picture with respect to the free vibrational and internal Hamiltonian, this interaction is described by the following Hamiltonian [CITATION] [EQUATION] where [MATH] denotes the annihiliation operator for the vibrations along the [MATH] direction, [MATH] is the corresponding frequency and [MATH] is the detuning between the internal transition and the frequency difference between the two Raman lasers.', 'quant-ph-9911100-3-21-3': 'The Rabi frequency [MATH] is proportional to the two Raman laser intensities, and [MATH] is the Lamb-Dicke parameter [CITATION].', 'quant-ph-9911100-3-21-4': 'When the two Raman lasers are tuned to the first blue sideband, i.e. [MATH], Hamiltonian ([REF]) predicts Rabi oscillations between [MATH] and [MATH] is a vibrational Fock state) with a frequency [CITATION] [EQUATION] where [MATH] is the generalized Laguerre polynomial.', 'quant-ph-9911100-3-21-5': 'These Rabi oscillations have been experimentally verified by preparing the initial state [MATH], (with [MATH] ranging from [MATH] to [MATH]) and measuring the probability [MATH] as a function of the interaction time [MATH], which is varied by changing the duration of the Raman laser pulses.', 'quant-ph-9911100-3-21-6': 'Again, as in the cavity QED experiment of [CITATION], the experimental Rabi oscillations are damped and well fitted by [CITATION] [EQUATION] where the measured oscillation frequencies [MATH] are in very good agreement with the theoretical prediction ([REF]) corresponding to the measured Lamb-Dicke parameter [MATH] [CITATION].', 'quant-ph-9911100-3-21-7': 'As concerns the decay rates [MATH], the experimental values are fitted in [CITATION] by [EQUATION] where [MATH] Khz.', 'quant-ph-9911100-3-21-8': 'This power-law scaling has attracted the interest of a number of authors and it has been investigated in Refs. [CITATION], even if a clear explanation of this behavior of the decay rates is still lacking.', 'quant-ph-9911100-3-21-9': 'On the contrary, the scaling law ([REF]) can be simply accounted for in the previous formalism if we consider the small [MATH] limit of Eq. ([REF]), which is again suggested by the fact that the experimental and theoretical predictions for the frequencies [MATH] agree.', 'quant-ph-9911100-3-21-10': 'In fact, the [MATH]-dependence of the theoretical prediction of Eq. ([REF]) for [MATH] is well approximated, within 10 %, by the power law dependence (see Fig. 3) [EQUATION] so that, using Eq. ([REF]), one has immediately the power law dependence [MATH] of Eq. ([REF]).', 'quant-ph-9911100-3-21-11': 'The value of the parameter [MATH] can be obtained by matching the values corresponding to [MATH], and using Eq. ([REF]), that is [MATH] sec, where we have used the experimental value [MATH] Khz.', 'quant-ph-9911100-3-22-0': 'However, this value of the parameter [MATH] cannot be explained in terms of some interaction time uncertainty, such as the time jitter of the Raman laser pulses, which is experimentally found to be much smaller [CITATION].', 'quant-ph-9911100-3-22-1': 'In this case, instead, the observed decoherence can be attributed, as already suggested in [CITATION], to the fluctuation of the Raman laser intensities, yielding a fluctuating Rabi frequency parameter [MATH] of the Hamiltonian ([REF]).', 'quant-ph-9911100-3-22-2': 'In this case, the evolution is driven by a fluctuating Hamiltonian [MATH], where [MATH] in Eq. ([REF]), so that [EQUATION] where [MATH], and we have defined the positive dimensionless random variable [MATH], which is proportional to the pulse area.', 'quant-ph-9911100-3-22-3': 'It is now easy to understand that the physical situation is analogous to that characterized by a random interaction time considered in the preceding sections, with [MATH] replaced by [MATH] and [MATH] by [MATH].', 'quant-ph-9911100-3-22-4': 'It is therefore straightforward to adapt the formalism developed in Section II to this case, in which the fluctuating quantity is the pulse area [MATH], yielding again random phases in the energy basis representation.', 'quant-ph-9911100-3-22-5': 'In analogy with Eq. ([REF]), one considers an averaged density matrix [EQUATION]', 'quant-ph-9911100-3-22-6': 'Imposing again that [MATH] must be a density operator and the semigroup property, one finds results analogous to Eqs. ([REF]) and ([REF]) [EQUATION]', 'quant-ph-9911100-3-22-7': 'Here, the parameters [MATH] and [MATH] are introduced as scaling parameters, but they have a clear meaning, as it can be easily seen by considering the mean and the variance of the probability distribution of Eq. ([REF]), [EQUATION] implying that [MATH] has now to be meant as a mean Rabi frequency, and that [MATH] quantifies the strength of [MATH] fluctuations.', 'quant-ph-9911100-3-22-8': 'It is interesting to note that these first two moments of [MATH] determine the properties of the fluctuating Rabi frequency [MATH], which can be written as [EQUATION] that is, the Rabi frequency [MATH] is a white, non-gaussian (due to the non-gaussian form of [MATH]) stochastic process.', 'quant-ph-9911100-3-22-9': 'In fact, the semigroup assumption we have made implies a Markovian treatment in which the spectrum of the laser intensity fluctuations is flat in the relevant frequency range.', 'quant-ph-9911100-3-22-10': 'This in particular implies that we are neglecting the dynamics at small times, of the order of the correlation time of the laser intensity fluctuations.', 'quant-ph-9911100-3-23-0': 'The estimated value of [MATH] gives a reasonable estimate of the pulse area fluctuations, since it corresponds to a fractional error of the pulse area [MATH] of [MATH] for a pulse duration of [MATH]sec, and which is decreasing for increasing pulse durations.', 'quant-ph-9911100-3-24-0': 'The present analysis shows many similarities with that of Ref. [CITATION] which also tries to explain the decay of the Rabi oscillations in the ion trap experiments of [CITATION] in terms of laser intensity fluctuations.', 'quant-ph-9911100-3-24-1': 'The authors of Ref. [CITATION] in fact use a phase destroying master equation coinciding with the second-order expansion ([REF]) of our generalized master equation of Eq. ([REF]) (see Eq. (16) of Ref. [CITATION] with the identifications [MATH] and [MATH]) and moreover derive the same numerical estimate for the pulse area fluctuation strength [MATH].', 'quant-ph-9911100-3-24-2': 'Despite this similarities, they do not recover the scaling ([REF]) of the decay rates [MATH] only because they do not use the general expression of the Rabi frequency ([REF]), (and which is well approximated by the power law ([REF])) but its Lamb-Dicke limit [MATH], which is valid only when [MATH].', 'quant-ph-9911100-3-24-3': 'There is however another, more fundamental, difference between our approach and that of Ref. [CITATION].', 'quant-ph-9911100-3-24-4': 'They assume from the beginning that the laser intensity fluctuations have a white and gaussian character, while we make no a priori assumption on the statistical properties of the pulse area [MATH].', 'quant-ph-9911100-3-24-5': 'We derive these properties, i.e. the probability distribution ([REF]), only from the semigroup condition, and it is interesting to note that this condition yields a gaussian probability distribution for the pulse area only as a limiting case.', 'quant-ph-9911100-3-24-6': 'In fact, from Eq. ([REF]) one can see that [MATH] tends to become a gaussian with the same mean value [MATH] and the same width [MATH] only in the large time limit [MATH] [EQUATION]', 'quant-ph-9911100-3-24-7': 'The non-gaussian character of [MATH] can be traced back to the fact that [MATH] must be definite and normalized in the interval [MATH] and not in [MATH].', 'quant-ph-9911100-3-24-8': 'Notice that at [MATH], Eq. ([REF]) assumes the exponential form [MATH].', 'quant-ph-9911100-3-24-9': 'Only at large times [MATH] the random variable [MATH] becomes the sum of many independent contributions and assumes the gaussian form.', 'quant-ph-9911100-3-25-0': 'Due to the non-gaussian nature of the random variable [MATH], we find that the more generally valid phase-destroying master equation is given by Eq. ([REF]) (with [MATH] replaced by [MATH]).', 'quant-ph-9911100-3-25-1': 'The predictions of Eq. ([REF]) significantly depart from its second order expansion in [MATH], Eq. ([REF]), corresponding to the gaussian limit, as soon as [MATH] becomes comparable with the typical timescale of the system under study, which, in the present case, is the inverse of the Rabi frequency.', 'quant-ph-9911100-3-26-0': 'The present analysis of the Rabi oscillation experiment of Ref. [CITATION] can be repeated for the very recent experiment with trapped ions performed in Innsbruck [CITATION], in which Rabi oscillations involving the vibrational levels and an optical quadrupole transition of a single [MATH]Ca[MATH] ion have been observed.', 'quant-ph-9911100-3-26-1': 'Damped oscillations corresponding to initial vibrational numbers [MATH] and [MATH] are reported.', 'quant-ph-9911100-3-26-2': 'From the data with [MATH], [MATH] Khz and [MATH] Khz, we get [MATH] and this estimate is consistent with attributing again the decoherence to the fluctuations of the Rabi frequency caused by laser intensity fluctuations.', 'quant-ph-9911100-3-26-3': 'Moreover in this case, the experiment is performed in the Lamb-Dicke limit [MATH], and therefore, using again Eq. ([REF]), we expect, in this case, a linear scaling with the vibrational number, [MATH].', 'quant-ph-9911100-3-27-0': '# Concluding remarks', 'quant-ph-9911100-3-28-0': 'Decoherence is not always necessarily due to the entanglement with an environment, but it may be due, as well, to the fluctuations of some classical parameter or internal variable of a system.', 'quant-ph-9911100-3-28-1': 'This is a different form of decoherence, which is present even in isolated systems, and that we have called non-dissipative decoherence.', 'quant-ph-9911100-3-28-2': 'In this paper we have presented a model-independent theory for non-dissipative decoherence, which can be applied in the case of a random evolution time or in the case of a fluctuating Hamiltonian.', 'quant-ph-9911100-3-28-3': 'This approach proves to be a flexible tool, able to give a quantitative understanding of the decoherence caused by the fluctuations of classical quantities.', 'quant-ph-9911100-3-28-4': 'In fact, in this paper we have given a simple and unified description of the decoherence phenomenon observed in recent Rabi oscillation experiments performed in a cavity QED configuration [CITATION] and on a trapped ion [CITATION].', 'quant-ph-9911100-3-28-5': 'In particular, this approach has allowed us to explain for the first time in simple terms, the power-law scaling of the coherence decay rates of Eq. ([REF]), observed in the trapped ion experiment.', 'quant-ph-9911100-3-29-0': 'The relevant aspect of the approach applied here, and introduced in Ref. [CITATION], is its model-independence.', 'quant-ph-9911100-3-29-1': 'The formalism is in fact derived starting from few, very general assumptions: i) the average density matrix [MATH] has all the usual properties of a density matrix; ii) the semigroup property for the time evolution generator [MATH] for [MATH].', 'quant-ph-9911100-3-29-2': 'With this respect, this approach seems to provide a very general description of non-dissipative decoherence, in which the random properties of the fluctuating classical variables are characterized by the two, system-dependent, time parameters [MATH] and [MATH].', 'quant-ph-9911100-3-29-3': 'As we have seen in section II, in the cases where one has a standard, continuous evolution, the two times coincide [MATH].', 'quant-ph-9911100-3-29-4': 'Under ideal conditions of no fluctuating classical variable or parameter, one would have [MATH], and the usual unitary evolution of an isolated system in quantum mechanics would be recovered.', 'quant-ph-9911100-3-29-5': 'However, the generality of the approach suggests in some way the possibility that the parameter [MATH], even though system-dependent, might have a lower nonzero limit, which would be reached just in the case of no fluctuations of experimental origin.', 'quant-ph-9911100-3-29-6': 'This would mean a completely new description of time in quantum mechanics.', 'quant-ph-9911100-3-29-7': 'In fact, the evolution time of a system [MATH] (and not the ""clock"" time [MATH]) would become an intrinsically random variable with a well defined probability distribution, without the difficulty of introducing an evolution time operator.', 'quant-ph-9911100-3-29-8': 'In Ref. [CITATION] it is suggested a relation of the nonzero limit for [MATH] with the ""energy-time"" [MATH] where [MATH] is the uncertainty in energy.', 'quant-ph-9911100-3-29-9': 'This would give a precise meaning to the time-energy uncertainty relation because now [MATH] rules the width of the time distribution function.', 'quant-ph-9911100-3-29-10': 'However, this ""intrinsic assumption"" is not necessarily implied by the formalism developed in [CITATION] and applied, with a more pragmatic attitude, in the present paper.'}",,
1804.11061,"{'1804.11061-1-0-0': 'By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored.', '1804.11061-1-0-1': 'They give some new hypergeometric series identities when the parameters are specified.', '1804.11061-1-1-0': '# Introduction', '1804.11061-1-2-0': 'For a complex variable [MATH] and an nonnegative integer [MATH], define the shifted-factorial to be u\'u\'u""o\'o\'o\'and(x)_n=x(x+1)(x+n-1) whenn=1,2,.u', '1804.11061-1-2-1': '(b_1)_k(b_s)_kz^k,u\'u\'u""o\'o\'o\'where [MATH] and [MATH] are complex parameters such that no zero factors appear in the denominators of the summand on the right hand side.', '1804.11061-1-2-2': 'Then Whipple\'s [MATH]-series identity (cf. [CITATION]) can be stated as [EQUATION] where [MATH] and [MATH] is the well-known gamma function u\'u\'u""o\'o\'o\'-5.0mmGamma(x)=_0^t^x-1e^-tdtwithRe(x)>0.u', '1804.11061-1-2-3': 'u\'u\'u""o\'o\'o\'About the research for the generalizations of Watson\'s [MATH]-series identity and [REF] with integer parameters, the reader may refer to the papers [CITATION].', '1804.11061-1-2-4': 'The corresponding [MATH]-analogues can be found in Wei and Wang [CITATION].', '1804.11061-1-3-0': 'Recall Fox-Wright function [MATH] (cf. [CITATION]; see also [CITATION]), which is defined by [EQUATION] and regarded as a generalization of hypergeometric series, where the coefficients [MATH] and [MATH] are positive real numbers such that u\'u\'u""o\'o\'o\'j=1^qB_j-_i=1^pA_i0.u\'u\'u""o\'o\'o\'Naturally, the domain where [MATH] and [MATH] take on values can be extended to the complex field and the upper coefficient relation is ignored when the series is terminating.', '1804.11061-1-3-1': 'For simplifying the expression, we shall frequently use the symbol u\'u\'u""o\'o\'o\'i=1^r(x_i+Ak)u\'u\'u""o\'o\'o\'in the Fox-Wright function.', '1804.11061-1-4-0': 'The importance of Fox-Wright function lies in that it can be applied to many fields.', '1804.11061-1-4-1': 'Miller and Moskowitz [CITATION] offered the applications of Fox-Wright function to the solution of algebraic trinomial equations and to a problem of information theory.', '1804.11061-1-4-2': 'Mainardi and Pagnini [CITATION] told us that Fox-Wright function plays an important role in finding the fundamental solution of the factional diffusion equation of distributed order in time.', '1804.11061-1-4-3': 'More applications of the function can be found in [CITATION].', '1804.11061-1-4-4': 'The following fact should be mentioned.', '1804.11061-1-4-5': 'Aomoto and Iguchi [CITATION] introduced the quasi hypergeometric function, which is exactly multiple Fox-Wright function, and showed that it satisfies a system of difference-differential equations.', '1804.11061-1-5-0': 'One pair of inverse relations implied in the works of Bressould [CITATION] and Gasper [CITATION] (see also [CITATION]) can be expressed as follows.', '1804.11061-1-6-0': 'Let [MATH] all be complex numbers.', '1804.11061-1-6-1': 'Then the system of equations [EQUATION] is equivalent to the system of equations [EQUATION]', '1804.11061-1-6-2': 'Although the importance of Fox-Wright function has been realized for many years, there are only a small number of summation formulas for this function to our knowledge.', '1804.11061-1-6-3': ""The corresponding results, which can be seen in the papers [CITATION], are all from inversion techniques and identities related to Saalschutz's theorem."", '1804.11061-1-6-4': 'The reader may refer to [CITATION] for more details on inversion techniques.', '1804.11061-1-7-0': 'Inspired by the importance of Fox-Wright function and the lack of summation formulas for this function, we shall derive several summation formulas for Fox-Wright function according to Lemma [REF], [REF] and some other hypergeometric series identities in Sections 2-3.', '1804.11061-1-8-0': '# Whipple-type series and summation formulasu\'u\'u""o\'o\'o\'for Fox-Wright function', '1804.11061-1-9-0': 'Let [MATH] and [MATH] both be complex numbers.', '1804.11061-1-9-1': 'The case [MATH] of [REF] reads as [EQUATION]', '1804.11061-1-9-2': 'Perform the replacements [MATH] and [MATH] to obtain [EQUATION] which fits to [REF] with [EQUATION]', '1804.11061-1-9-3': 'Then [REF] produces the dual relation [EQUATION]', '1804.11061-1-9-4': 'Rewrite it in accordance with Fox-Wright function, we get Theorem [REF].', '1804.11061-1-10-0': ""When [MATH], Theorem [REF] reduces to the hypergeometric series identity [EQUATION] which is a special case of Dougall's [MATH]-series identity (cf. [CITATION]): [EQUATION] provided that [MATH]."", '1804.11061-1-11-0': 'Other two hypergeometric series identities from Theorem [REF] can be displayed as follows.', '1804.11061-1-12-0': 'Let [MATH] and [MATH] both be complex numbers.', '1804.11061-1-12-1': 'Then [EQUATION] where the expression on the left hand side is [EQUATION]', '1804.11061-1-12-2': 'Remark: In the symbol [MATH], the parameters on the first two lines are numerators and other ones are denominators.', '1804.11061-1-12-3': 'This remark is also applicative to Theorem [REF].', '1804.11061-1-13-0': 'Lemma 1 of Chu [CITATION] gives [EQUATION]', '1804.11061-1-13-1': 'Calculating the [MATH]-series on the right hand side by [REF] and then setting [MATH] in the resulting identity, we have [EQUATION]', '1804.11061-1-13-2': 'Employ the substitutions [MATH] and [MATH] to gain [EQUATION] which suits to [REF] with [EQUATION]', '1804.11061-1-13-3': 'Then [REF] offers the dual relation [EQUATION]', '1804.11061-1-13-4': 'Interchanging the summation order, we achieve Theorem [REF] after some simplifications.', '1804.11061-1-14-0': 'When [MATH], Theorem [REF] reduces to Theorem [REF].', '1804.11061-1-14-1': 'Performing the the replacements [MATH] and [MATH] in the case [MATH] of Theorem [REF], we attain the following reciprocal formula after some reformulations.', '1804.11061-1-15-0': 'Let [MATH] and [MATH] both be complex numbers.', '1804.11061-1-15-1': 'Then [EQUATION] where the symbol on the left hand side stands for [EQUATION]', '1804.11061-1-15-2': 'Two hypergeometric series identities from Corollary [REF] can be laid out as follows.', '1804.11061-1-16-0': 'Let [MATH] and [MATH] both be complex numbers.', '1804.11061-1-16-1': 'Then [EQUATION] where the expression on the left hand side is [EQUATION]', '1804.11061-1-16-2': 'Utilizing Lemma 2 of Chu [CITATION], it is not difficult to obtain [EQUATION]', '1804.11061-1-16-3': 'Evaluating the [MATH]-series on the right hand side by [REF] and then taking [MATH] in the resulting identity, we have [EQUATION]', '1804.11061-1-16-4': 'Employ the substitutions [MATH] and [MATH] to get [EQUATION] which satisfies [REF] with [EQUATION]', '1804.11061-1-16-5': 'Then [REF] produces the dual relation [EQUATION]', '1804.11061-1-16-6': 'Interchanging the summation order, we gain Theorem [REF] after some simplifications.', '1804.11061-1-17-0': 'When [MATH], Theorem [REF] also reduces to Theorem [REF].', '1804.11061-1-17-1': 'Performing the the replacements [MATH] and [MATH] in the case [MATH] of Theorem [REF], we achieve the following reciprocal formula after some reformulations.', '1804.11061-1-18-0': 'Let [MATH] and [MATH] both be complex numbers.', '1804.11061-1-18-1': 'Then [EQUATION] where the symbol on the left hand side stands for [EQUATION]', '1804.11061-1-18-2': 'Two hypergeometric series identities from Corollary [REF] can be displayed as follows.', '1804.11061-1-19-0': '# Two different summation formulas for Fox-Wright function', '1804.11061-1-20-0': 'Let [MATH], [MATH] and [MATH] all be complex numbers.', '1804.11061-1-20-1': 'A [MATH]-series identity due to Chu and Wang [CITATION] can be expressed as [EQUATION]', '1804.11061-1-20-2': 'Employ the substitution [MATH] to attain [EQUATION] which fits to [REF] with [EQUATION]', '1804.11061-1-20-3': 'Then [REF] gives the dual relation [EQUATION]', '1804.11061-1-20-4': 'Rewriting it in terms of Fox-Wright function, we obtain Theorem [REF].', '1804.11061-1-21-0': 'Let [MATH] and [MATH] both be complex numbers.', '1804.11061-1-21-1': 'Fixing [MATH] in Corollary [REF], we get the following identity.', '1804.11061-1-22-0': 'Let [MATH] be a complex number and [MATH] a nonnegative integer.', '1804.11061-1-22-1': 'Let [MATH] and [MATH] both be complex numbers.', '1804.11061-1-22-2': ""Setting [MATH] and [MATH] in the transformation formula (cf. [CITATION]) [EQUATION] and calculating the series on the right hand side by Dixon's [MATH]-series identity(cf. [CITATION]): [EQUATION] where [MATH], we gain [EQUATION] provided that [MATH]."", '1804.11061-1-22-3': 'The case [MATH] of it can be manipulated as [EQUATION] which suits to [REF] with [EQUATION]', '1804.11061-1-22-4': 'Then [REF] produces the dual relation [EQUATION]', '1804.11061-1-22-5': 'Rewriting it according to Fox-Wright function, we achieve Theorem [REF].', '1804.11061-1-23-0': 'The case [MATH] of [REF] reads as [EQUATION] with [MATH].', '1804.11061-1-23-1': 'When [MATH], Theorem [REF] reduces to the special case of it: [EQUATION]', '1804.11061-1-23-2': 'Taking [MATH] in Theorem [REF], we attain the following result.', '1804.11061-1-24-0': 'Let [MATH] be a complex number and [MATH] a nonnegative integer.'}","{'1804.11061-2-0-0': 'By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored.', '1804.11061-2-0-1': 'They give some new hypergeometric series identities when the parameters are specified.', '1804.11061-2-1-0': '# Introduction', '1804.11061-2-2-0': 'For a complex variable [MATH] and an nonnegative integer [MATH], define the shifted-factorial to be u\'u\'u""o\'o\'o\'and(x)_n=x(x+1)(x+n-1) whenn=1,2,.u', '1804.11061-2-2-1': '(b_1)_k(b_s)_kz^k,u\'u\'u""o\'o\'o\'where [MATH] and [MATH] are complex parameters such that no zero factors appear in the denominators of the summand on the right hand side.', '1804.11061-2-2-2': 'Then Whipple\'s [MATH]-series identity (cf. [CITATION]) can be stated as [EQUATION] where [MATH] and [MATH] is the well-known gamma function u\'u\'u""o\'o\'o\'-5.0mmGamma(x)=_0^t^x-1e^-tdtwithRe(x)>0.u', '1804.11061-2-2-3': 'u\'u\'u""o\'o\'o\'About the research for the generalizations of Watson\'s [MATH]-series identity and [REF] with integer parameters, the reader may refer to the papers [CITATION].', '1804.11061-2-2-4': 'The corresponding [MATH]-analogues can be found in Wei and Wang [CITATION].', '1804.11061-2-3-0': 'Recall Fox-Wright function [MATH] (cf. [CITATION]; see also [CITATION]), which is defined by [EQUATION] and regarded as a generalization of hypergeometric series, where the coefficients [MATH] and [MATH] are positive real numbers such that u\'u\'u""o\'o\'o\'j=1^qB_j-_i=1^pA_i0.u\'u\'u""o\'o\'o\'Naturally, the domain where [MATH] and [MATH] take on values can be extended to the complex field and the upper coefficient relation is ignored when the series is terminating.', '1804.11061-2-3-1': 'For convenience, we shall frequently use the symbol u\'u\'u""o\'o\'o\'i=1^r(x_i+Ak)u\'u\'u""o\'o\'o\'in the Fox-Wright function.', '1804.11061-2-4-0': 'The importance of Fox-Wright function lies in that it can be applied to many fields.', '1804.11061-2-4-1': 'Miller and Moskowitz [CITATION] offered the applications of Fox-Wright function to the solution of algebraic trinomial equations and to a problem of information theory.', '1804.11061-2-4-2': 'Mainardi and Pagnini [CITATION] told us that Fox-Wright function plays an important role in finding the fundamental solution of the factional diffusion equation of distributed order in time.', '1804.11061-2-4-3': 'More applications of the function can be found in [CITATION].', '1804.11061-2-4-4': 'The following fact should be mentioned.', '1804.11061-2-4-5': 'Aomoto and Iguchi [CITATION] introduced the quasi hypergeometric function, which is exactly multiple Fox-Wright function, and showed that it satisfies a system of difference-differential equations.', '1804.11061-2-5-0': 'One pair of inverse relations implied in the works of Bressould [CITATION] and Gasper [CITATION] (see also [CITATION]) can be expressed as follows.', '1804.11061-2-6-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-2-6-1': 'Then the system of equations [EQUATION] is equivalent to the system of equations [EQUATION]', '1804.11061-2-6-2': 'Although the importance of Fox-Wright function has been realized for many years, there are only a small number of summation formulas for this function to our knowledge.', '1804.11061-2-6-3': ""The corresponding results, which can be seen in the papers [CITATION], are all from inversion techniques and identities related to Saalschutz's theorem."", '1804.11061-2-6-4': 'The reader may refer to [CITATION] for more details on inversion techniques.', '1804.11061-2-7-0': 'Inspired by the importance of Fox-Wright function and the lack of summation formulas for this function, we shall derive several summation formulas for Fox-Wright function according to Lemma [REF], [REF] and some other hypergeometric series identities in Sections 2-3.', '1804.11061-2-8-0': '# Whipple-type series and summation formulasu\'u\'u""o\'o\'o\'for Fox-Wright function', '1804.11061-2-9-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-2-9-1': 'The case [MATH] of [REF] reads [EQUATION]', '1804.11061-2-9-2': 'Perform the replacements [MATH] and [MATH] to obtain [EQUATION] which fits to [REF] with [EQUATION]', '1804.11061-2-9-3': 'Then [REF] produces the dual relation [EQUATION]', '1804.11061-2-9-4': 'Rewrite it in accordance with Fox-Wright function, we get Theorem [REF].', '1804.11061-2-10-0': ""When [MATH], Theorem [REF] reduces to the hypergeometric series identity [EQUATION] which is a special case of Dougall's [MATH]-series identity (cf. [CITATION]): [EQUATION] provided that [MATH]."", '1804.11061-2-11-0': 'Other two hypergeometric series identities from Theorem [REF] can be displayed as follows.', '1804.11061-2-12-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-2-12-1': 'Then [EQUATION] where the expression on the left hand side is [EQUATION]', '1804.11061-2-12-2': 'Remark: In the symbol [MATH], the parameters on the first two lines are numerators and other ones are denominators.', '1804.11061-2-12-3': 'This remark is also applicative to Theorem [REF].', '1804.11061-2-13-0': 'Lemma 1 of Chu [CITATION] gives [EQUATION]', '1804.11061-2-13-1': 'Calculating the [MATH]-series on the right hand side by [REF] and then setting [MATH] in the resulting identity, we have [EQUATION]', '1804.11061-2-13-2': 'Employ the substitutions [MATH] and [MATH] to gain [EQUATION] which suits to [REF] with [EQUATION]', '1804.11061-2-13-3': 'Then [REF] offers the dual relation [EQUATION]', '1804.11061-2-13-4': 'Interchanging the summation order, we achieve Theorem [REF] after some simplifications.', '1804.11061-2-14-0': 'When [MATH], Theorem [REF] reduces to Theorem [REF].', '1804.11061-2-14-1': 'Performing the the replacements [MATH] and [MATH] in the case [MATH] of Theorem [REF], we attain the following reciprocal formula after some reformulations.', '1804.11061-2-15-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-2-15-1': 'Then [EQUATION] where the symbol on the left hand side stands for [EQUATION]', '1804.11061-2-15-2': 'Two hypergeometric series identities from Corollary [REF] can be laid out as follows.', '1804.11061-2-16-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-2-16-1': 'Then [EQUATION] where the expression on the left hand side is [EQUATION]', '1804.11061-2-16-2': 'Utilizing Lemma 2 of Chu [CITATION], it is not difficult to obtain [EQUATION]', '1804.11061-2-16-3': 'Evaluating the [MATH]-series on the right hand side by [REF] and then taking [MATH] in the resulting identity, we have [EQUATION]', '1804.11061-2-16-4': 'Employ the substitutions [MATH] and [MATH] to get [EQUATION] which satisfies [REF] with [EQUATION]', '1804.11061-2-16-5': 'Then [REF] produces the dual relation [EQUATION]', '1804.11061-2-16-6': 'Interchanging the summation order, we gain Theorem [REF] after some simplifications.', '1804.11061-2-17-0': 'When [MATH], Theorem [REF] also reduces to Theorem [REF].', '1804.11061-2-17-1': 'Performing the the replacements [MATH] and [MATH] in the case [MATH] of Theorem [REF], we achieve the following reciprocal formula after some reformulations.', '1804.11061-2-18-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-2-18-1': 'Then [EQUATION] where the symbol on the left hand side stands for [EQUATION]', '1804.11061-2-18-2': 'Two hypergeometric series identities from Corollary [REF] can be displayed as follows.', '1804.11061-2-19-0': '# Two different summation formulas for Fox-Wright function', '1804.11061-2-20-0': 'Let [MATH], [MATH] and [MATH] be complex numbers.', '1804.11061-2-20-1': 'A [MATH]-series identity due to Chu and Wang [CITATION] can be expressed as [EQUATION]', '1804.11061-2-20-2': 'Employ the substitution [MATH] to attain [EQUATION] which fits to [REF] with [EQUATION]', '1804.11061-2-20-3': 'Then [REF] gives the dual relation [EQUATION]', '1804.11061-2-20-4': 'Writing it in terms of Fox-Wright function, we obtain Theorem [REF].', '1804.11061-2-21-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-2-21-1': 'Fixing [MATH] in Corollary [REF], we get the following identity.', '1804.11061-2-22-0': 'Let [MATH] be a complex number and [MATH] a nonnegative integer.', '1804.11061-2-22-1': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-2-22-2': ""Setting [MATH] and [MATH] in the transformation formula (cf. [CITATION]) [EQUATION] and calculating the series on the right hand side by Dixon's [MATH]-series identity(cf. [CITATION]): [EQUATION] where [MATH], we gain [EQUATION] provided that [MATH]."", '1804.11061-2-22-3': 'The case [MATH] of it can be manipulated as [EQUATION] which suits to [REF] with [EQUATION]', '1804.11061-2-22-4': 'Then [REF] produces the dual relation [EQUATION]', '1804.11061-2-22-5': 'Writing it according to Fox-Wright function, we achieve Theorem [REF].', '1804.11061-2-23-0': 'The case [MATH] of [REF] reads [EQUATION] with [MATH].', '1804.11061-2-23-1': 'When [MATH], Theorem [REF] reduces to the special case of it: [EQUATION]', '1804.11061-2-23-2': 'Taking [MATH] in Theorem [REF], we attain the following result.', '1804.11061-2-24-0': 'Let [MATH] be a complex number and [MATH] a nonnegative integer.'}","[['1804.11061-1-13-0', '1804.11061-2-13-0'], ['1804.11061-1-13-1', '1804.11061-2-13-1'], ['1804.11061-1-13-2', '1804.11061-2-13-2'], ['1804.11061-1-13-3', '1804.11061-2-13-3'], ['1804.11061-1-13-4', '1804.11061-2-13-4'], ['1804.11061-1-10-0', '1804.11061-2-10-0'], 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['1804.11061-2-9-2', '1804.11061-3-9-2'], ['1804.11061-2-9-3', '1804.11061-3-9-3'], ['1804.11061-2-9-4', '1804.11061-3-9-4'], ['1804.11061-2-20-0', '1804.11061-3-21-0'], ['1804.11061-2-20-2', '1804.11061-3-21-2'], ['1804.11061-2-20-3', '1804.11061-3-21-3'], ['1804.11061-2-20-4', '1804.11061-3-21-4'], ['1804.11061-2-15-0', '1804.11061-3-15-0'], ['1804.11061-2-15-1', '1804.11061-3-15-1'], ['1804.11061-2-15-2', '1804.11061-3-15-2'], ['1804.11061-2-21-0', '1804.11061-3-23-0'], ['1804.11061-2-21-1', '1804.11061-3-23-2'], ['1804.11061-2-18-0', '1804.11061-3-18-0'], ['1804.11061-2-18-1', '1804.11061-3-18-1'], ['1804.11061-2-18-2', '1804.11061-3-18-2'], ['1804.11061-2-24-0', '1804.11061-3-26-0'], ['1804.11061-2-17-0', '1804.11061-3-17-0'], ['1804.11061-2-17-1', '1804.11061-3-17-1'], ['1804.11061-2-6-0', '1804.11061-3-6-0'], ['1804.11061-2-6-1', '1804.11061-3-6-1'], ['1804.11061-2-6-2', '1804.11061-3-6-2'], ['1804.11061-2-6-3', '1804.11061-3-6-3'], ['1804.11061-2-6-4', '1804.11061-3-6-4']]","[['1804.11061-1-15-0', '1804.11061-2-15-0'], ['1804.11061-1-16-0', '1804.11061-2-16-0'], ['1804.11061-1-23-0', '1804.11061-2-23-0'], ['1804.11061-1-12-0', '1804.11061-2-12-0'], ['1804.11061-1-20-0', '1804.11061-2-20-0'], ['1804.11061-1-20-4', '1804.11061-2-20-4'], ['1804.11061-1-21-0', '1804.11061-2-21-0'], ['1804.11061-1-22-1', '1804.11061-2-22-1'], ['1804.11061-1-22-5', '1804.11061-2-22-5'], ['1804.11061-1-3-1', '1804.11061-2-3-1'], ['1804.11061-1-6-0', '1804.11061-2-6-0'], ['1804.11061-1-18-0', '1804.11061-2-18-0'], ['1804.11061-1-9-0', '1804.11061-2-9-0'], ['1804.11061-1-9-1', '1804.11061-2-9-1'], ['1804.11061-2-2-0', '1804.11061-3-2-0'], ['1804.11061-2-2-3', '1804.11061-3-2-3'], ['1804.11061-2-2-4', '1804.11061-3-2-4'], ['1804.11061-2-22-2', '1804.11061-3-24-2'], ['1804.11061-2-10-0', '1804.11061-3-10-0'], ['1804.11061-2-16-2', '1804.11061-3-16-2'], ['1804.11061-2-13-0', '1804.11061-3-13-0'], ['1804.11061-2-4-1', '1804.11061-3-4-1'], ['1804.11061-2-4-2', '1804.11061-3-4-2'], ['1804.11061-2-4-5', '1804.11061-3-4-5'], ['1804.11061-2-5-0', '1804.11061-3-5-0'], ['1804.11061-2-3-1', '1804.11061-3-3-2']]",[],"[['1804.11061-2-2-1', '1804.11061-3-2-1'], ['1804.11061-2-2-2', '1804.11061-3-2-2'], ['1804.11061-2-20-1', '1804.11061-3-21-1'], ['1804.11061-2-3-0', '1804.11061-3-3-0']]",[],"['1804.11061-3-19-0', '1804.11061-3-22-0', '1804.11061-3-23-1', '1804.11061-3-26-1', '1804.11061-3-27-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1804.11061,"{'1804.11061-3-0-0': 'By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored.', '1804.11061-3-0-1': 'They give some new hypergeometric series identities when the parameters are specified.', '1804.11061-3-1-0': '# Introduction', '1804.11061-3-2-0': 'For a complex variable [MATH] and an nonnegative integer [MATH], define the shifted-factorial to be [EQUATION].', '1804.11061-3-2-1': 'Following [CITATION], define the hypergeometric series by [EQUATION] where [MATH] and [MATH] are complex parameters such that no zero factors appear in the denominators of the summand on the right hand side.', '1804.11061-3-2-2': ""Then Whipple's [MATH]-series identity (cf. [CITATION][p. 149]) can be stated as [EQUATION] where [MATH] and [MATH] is the well-known gamma function [EQUATION]."", '1804.11061-3-2-3': ""About the research for the generalizations of Watson's [MATH]-series identity and [REF] with integer parameters, the reader may refer to the papers [CITATION]."", '1804.11061-3-2-4': 'The corresponding [MATH]-analogues can be found in [CITATION].', '1804.11061-3-2-5': 'Some strange evaluations of hypergeometric series can be seen in the papers [CITATION].', '1804.11061-3-3-0': 'Recall Fox-Wright function [MATH] (cf. [CITATION]; see also [CITATION][p. 21], which is defined by [EQUATION] and regarded as a generalization of hypergeometric series, where the coefficients [MATH] and [MATH] are positive real numbers such that [EQUATION].', '1804.11061-3-3-1': 'Naturally, the domain where [MATH] and [MATH] take on values can be extended to the complex field and the upper coefficient relation is ignored when the series is terminating.', '1804.11061-3-3-2': 'For convenience, we shall frequently use the symbol [EQUATION] in the Fox-Wright function.', '1804.11061-3-4-0': 'The importance of Fox-Wright function lies in that it can be applied to many fields.', '1804.11061-3-4-1': '[CITATION] offered the applications of Fox-Wright function to the solution of algebraic trinomial equations and to a problem of information theory.', '1804.11061-3-4-2': '[CITATION] told us that Fox-Wright function plays an important role in finding the fundamental solution of the factional diffusion equation of distributed order in time.', '1804.11061-3-4-3': 'More applications of the function can be found in [CITATION].', '1804.11061-3-4-4': 'The following fact should be mentioned.', '1804.11061-3-4-5': '[CITATION] introduced the quasi hypergeometric function, which is exactly multiple Fox-Wright function, and showed that it satisfies a system of difference-differential equations.', '1804.11061-3-5-0': 'One pair of inverse relations implied in the works of [CITATION] and [CITATION] (see also [CITATION][p. 17]) can be expressed as follows.', '1804.11061-3-6-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-3-6-1': 'Then the system of equations [EQUATION] is equivalent to the system of equations [EQUATION]', '1804.11061-3-6-2': 'Although the importance of Fox-Wright function has been realized for many years, there are only a small number of summation formulas for this function to our knowledge.', '1804.11061-3-6-3': ""The corresponding results, which can be seen in the papers [CITATION], are all from inversion techniques and identities related to Saalschutz's theorem."", '1804.11061-3-6-4': 'The reader may refer to [CITATION] for more details on inversion techniques.', '1804.11061-3-7-0': 'Inspired by the importance of Fox-Wright function and the lack of summation formulas for this function, we shall derive several summation formulas for Fox-Wright function according to Lemma [REF], [REF] and some other hypergeometric series identities in Sections 2-3.', '1804.11061-3-8-0': '# Whipple-type series and summation formulas for Fox-Wright function', '1804.11061-3-9-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-3-9-1': 'The case [MATH] of [REF] reads [EQUATION]', '1804.11061-3-9-2': 'Perform the replacements [MATH] and [MATH] to obtain [EQUATION] which fits to [REF] with [EQUATION]', '1804.11061-3-9-3': 'Then [REF] produces the dual relation [EQUATION]', '1804.11061-3-9-4': 'Rewrite it in accordance with Fox-Wright function, we get Theorem [REF].', '1804.11061-3-10-0': ""When [MATH], Theorem [REF] reduces to the hypergeometric series identity [EQUATION] which is a special case of Dougall's [MATH]-series identity (cf. [CITATION][p. 71]): [EQUATION] provided that [MATH]."", '1804.11061-3-11-0': 'Other two hypergeometric series identities from Theorem [REF] can be displayed as follows.', '1804.11061-3-12-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-3-12-1': 'Then [EQUATION] where the expression on the left hand side is [EQUATION]', '1804.11061-3-12-2': 'Remark: In the symbol [MATH], the parameters on the first two lines are numerators and other ones are denominators.', '1804.11061-3-12-3': 'This remark is also applicative to Theorem [REF].', '1804.11061-3-13-0': 'Lemma 1 of [CITATION] gives [EQUATION]', '1804.11061-3-13-1': 'Calculating the [MATH]-series on the right hand side by [REF] and then setting [MATH] in the resulting identity, we have [EQUATION]', '1804.11061-3-13-2': 'Employ the substitutions [MATH] and [MATH] to gain [EQUATION] which suits to [REF] with [EQUATION]', '1804.11061-3-13-3': 'Then [REF] offers the dual relation [EQUATION]', '1804.11061-3-13-4': 'Interchanging the summation order, we achieve Theorem [REF] after some simplifications.', '1804.11061-3-14-0': 'When [MATH], Theorem [REF] reduces to Theorem [REF].', '1804.11061-3-14-1': 'Performing the the replacements [MATH] and [MATH] in the case [MATH] of Theorem [REF], we attain the following reciprocal formula after some reformulations.', '1804.11061-3-15-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-3-15-1': 'Then [EQUATION] where the symbol on the left hand side stands for [EQUATION]', '1804.11061-3-15-2': 'Two hypergeometric series identities from Corollary [REF] can be laid out as follows.', '1804.11061-3-16-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-3-16-1': 'Then [EQUATION] where the expression on the left hand side is [EQUATION]', '1804.11061-3-16-2': 'Utilizing Lemma 2 of [CITATION], it is not difficult to obtain [EQUATION]', '1804.11061-3-16-3': 'Evaluating the [MATH]-series on the right hand side by [REF] and then taking [MATH] in the resulting identity, we have [EQUATION]', '1804.11061-3-16-4': 'Employ the substitutions [MATH] and [MATH] to get [EQUATION] which satisfies [REF] with [EQUATION]', '1804.11061-3-16-5': 'Then [REF] produces the dual relation [EQUATION]', '1804.11061-3-16-6': 'Interchanging the summation order, we gain Theorem [REF] after some simplifications.', '1804.11061-3-17-0': 'When [MATH], Theorem [REF] also reduces to Theorem [REF].', '1804.11061-3-17-1': 'Performing the the replacements [MATH] and [MATH] in the case [MATH] of Theorem [REF], we achieve the following reciprocal formula after some reformulations.', '1804.11061-3-18-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-3-18-1': 'Then [EQUATION] where the symbol on the left hand side stands for [EQUATION]', '1804.11061-3-18-2': 'Two hypergeometric series identities from Corollary [REF] can be displayed as follows.', '1804.11061-3-19-0': '[[MATH] in Corollary [REF]] [EQUATION] [[MATH] in Corollary [REF]] [EQUATION]', '1804.11061-3-20-0': '# Two different summation formulas for Fox-Wright function', '1804.11061-3-21-0': 'Let [MATH], [MATH] and [MATH] be complex numbers.', '1804.11061-3-21-1': 'A [MATH]-series identity due to [CITATION][Corollary 16] can be expressed as [EQUATION]', '1804.11061-3-21-2': 'Employ the substitution [MATH] to attain [EQUATION] which fits to [REF] with [EQUATION]', '1804.11061-3-21-3': 'Then [REF] gives the dual relation [EQUATION]', '1804.11061-3-21-4': 'Writing it in terms of Fox-Wright function, we obtain Theorem [REF].', '1804.11061-3-22-0': 'When [MATH], Theorem [REF] offers the following formula.', '1804.11061-3-23-0': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-3-23-1': 'Then [EQUATION]', '1804.11061-3-23-2': 'Fixing [MATH] in Corollary [REF], we get the following identity.', '1804.11061-3-24-0': 'Let [MATH] be a complex number and [MATH] a nonnegative integer.', '1804.11061-3-24-1': 'Let [MATH] and [MATH] be complex numbers.', '1804.11061-3-24-2': ""Setting [MATH] and [MATH] in the transformation formula (cf. [CITATION][p. 147]) [EQUATION] and calculating the series on the right hand side by Dixon's [MATH]-series identity(cf. [CITATION][p. 72]): [EQUATION] where [MATH], we gain [EQUATION] provided that [MATH]."", '1804.11061-3-24-3': 'The case [MATH] of it can be manipulated as [EQUATION] which suits to [REF] with [EQUATION]', '1804.11061-3-24-4': 'Then [REF] produces the dual relation [EQUATION]', '1804.11061-3-24-5': 'Writing it according to Fox-Wright function, we achieve Theorem [REF].', '1804.11061-3-25-0': 'The case [MATH] of [REF] reads [EQUATION] with [MATH].', '1804.11061-3-25-1': 'When [MATH], Theorem [REF] reduces to the special case of it: [EQUATION]', '1804.11061-3-25-2': 'Taking [MATH] in Theorem [REF], we attain the following result.', '1804.11061-3-26-0': 'Let [MATH] be a complex number and [MATH] a nonnegative integer.', '1804.11061-3-26-1': 'Then [EQUATION]', '1804.11061-3-27-0': 'The authors are grateful to the reviewer for helpful comments.'}",,
1203.3215,"{'1203.3215-1-0-0': 'Discovery of VHE emission towards the Carina arm region with the H.E.S.S. telescope array: HESSJ1018-589', '1203.3215-1-1-0': 'The Carina arm region, containing the supernova remnant SNRG284.3-1.8, the high-energy (HE; E[MATH]100 MeV) binary 1FGLJ1018.6-5856 and the energetic pulsar PSRJ1016-5857 and its nebula, has been observed with the H.E.S.S. telescope array.', '1203.3215-1-1-1': 'The observational coverage of the region in very-high-energy (VHE; E[MATH]0.1TeV) [MATH]-rays benefits from deep exposure (40h) of the neighboring open cluster Westerlund 2.', '1203.3215-1-1-2': 'The observations have revealed a new extended region of VHE [MATH]-ray emission.', '1203.3215-1-1-3': 'The new VHE source HESSJ1018-589 shows a bright, point-like emission region positionally coincident with SNRG284.3-1.8 and 1FGLJ1018.6-5856 and a diffuse extension towards the direction of PSRJ1016-5857.', '1203.3215-1-1-4': 'A soft ([MATH]=2.7[MATH]0.5[MATH]) photon index, with a differential flux at 1TeV of N[MATH]=(4.2[MATH]1.1)[MATH]TeV[MATH] cm[MATH] s[MATH] is found for the point-like source, whereas the total emission region including the diffuse emission region is well fit by a power-law function with spectral index [MATH]=2.9[MATH]0.4[MATH] and differential flux at 1TeV of N[MATH]=(6.8[MATH]1.6)[MATH]TeV[MATH] cm[MATH] s[MATH].', '1203.3215-1-1-5': 'This H.E.S.S. detection motivated follow-up X-ray observations with the XMM-Newton satellite to investigate the origin of the VHE emission.', '1203.3215-1-1-6': 'The analysis of the XMM-Newton data resulted in the discovery of a bright, non-thermal point-like source (XMMU J101855.4-58564) with a photon index of [MATH]=1.65[MATH]0.08 in the center of SNRG284.3-1.8, and a thermal, extended emission region coincident with its bright northern filament.', '1203.3215-1-1-7': 'The characteristics of this thermal emission are used to estimate the plasma density in the region as n[MATH]0.5 cm[MATH](2.9kpc/d)[MATH].', '1203.3215-1-1-8': 'The position of XMMUJ101855.4-58564 is compatible with the position reported by the Fermi-LAT collaboration for the binary system 1FGLJ1018.6-5856 and the variable Swift XRT source identified with it.', '1203.3215-1-1-9': 'The new X-ray data are used alongside archival multi-wavelength data to investigate the relationship between the VHE [MATH]-ray emission from HESSJ1018-589 and the various potential counterparts in the Carina arm region.', '1203.3215-1-2-0': 'H.E.S.S. detection of HESSJ1018-589', '1203.3215-1-3-0': '# Introduction', '1203.3215-1-4-0': 'The H.E.S.S. (High Energy Stereoscopic System) collaboration has carried out observations of the Carina arm as part of the Galactic Plane Survey .', '1203.3215-1-4-1': 'The observed region includes three potential VHE [MATH]-ray emitters, SNRG284.3-1.8 (MSH 10-5[MATH]), the high spin-down luminosity pulsar PSRJ1016-5857 and the Fermi Large Area Telescope (Fermi-LAT) [MATH]-ray binary 1FGLJ1018.6-5856 .', '1203.3215-1-4-2': 'SNRG284.3-1.8 has an incomplete radio shell (Fig. [REF]) with a non-thermal radio spectrum and a flux density of (5.4[MATH]0.8)Jy at 8.4GHz .', '1203.3215-1-4-3': 'Evidence of interaction with molecular clouds (MC; [MATH]CO J=1-0) has been reported by [CITATION] using observations with the 1.2m Columbia Millimeter-Wave Telescope.', '1203.3215-1-4-4': 'The MC content integrated over velocities in the local standard of rest from -14.95kms[MATH] to -21.45kms[MATH] traces the radio shell shape of the SNR.', '1203.3215-1-5-0': 'The shell-like SNR shows a bright, narrow filament, coincident with an H[MATH] filament to the East.', '1203.3215-1-5-1': 'Optical and CO observations imply that SNRG284.3-1.8 is most likely a 10[MATH]yr old remnant of a type II supernova with a massive stellar progenitor and that it is located at a distance of d[MATH]=2.9kpc (with an error of [MATH]20%) .', '1203.3215-1-5-2': 'The western edge of the shell shows a so-called finger emission region (see Fig. [REF]) extended towards the direction of PSRJ1016-5857.', '1203.3215-1-5-3': 'This Vela-like pulsar and its associated X-ray pulsar wind nebula (PWN), located [MATH] away from the geometrical center of the SNR, were discovered by [CITATION] in a search for counterparts of the unidentified source 3EGJ1013-5915 with the Parkes telescope.', '1203.3215-1-5-4': 'The energetic pulsar, formerly associated with the SNRG284.3-1.8, has a rotation period of 107ms, a characteristic age [MATH]=21kyr and a spin-down luminosity of 2.6[MATH]ergs[MATH].', '1203.3215-1-5-5': ""Its distance can be estimated from the pulsar's dispersion measure to be d[MATH]kpc or d[MATH]0.03[MATH]."", '1203.3215-1-6-0': 'The two-emission-regions hypothesis is strengthened when the projection of a rectangular region on the uncorrelated excess map along the extension of the image (with a width of twice the H.E.S.S. mean PSF) is evaluated.', '1203.3215-1-6-1': 'The fit of a single Gaussian function to the projection yields a [MATH]/[MATH] of 21.04/13 (null hypothesis probability P=0.072) whereas the fit to a double Gaussian function leads to a better [MATH]/[MATH] of 9.6/10 (P=0.476).', '1203.3215-1-6-2': 'Fig. [REF] shows the profile of the rectangular region (in white in the inset).', '1203.3215-1-6-3': 'The single and double Gaussian function fits are shown in dashed and solid lines, respectively.', '1203.3215-1-7-0': 'The energy spectrum (shown in Fig. [REF]) for the VHE excess was computed by means of a forward-folding maximum likelihood fit .', '1203.3215-1-7-1': 'To estimate the background the reflected background method was used, in which symmetric regions, not contaminated by known sources, are used to extract the background .', '1203.3215-1-7-2': 'To derive the source spectrum two regions were selected, one around the point-like source and a second region of 0.30[MATH] radius accounting for the total VHE [MATH]-ray emission centered on [MATH]45.6[MATH] and [MATH]=-59[MATH] (J2000).', '1203.3215-1-7-3': 'The photon spectra in both cases are well represented with a simple power-law function dN/dE=N[MATH](E/1TeV)[MATH].', '1203.3215-1-7-4': 'The point-like source has a photon index of [MATH]=2.7[MATH]0.5[MATH]0.2[MATH] statistically compatible with the total (0.30[MATH]) VHE [MATH]-ray emission region, [MATH]=2.9[MATH]0.4[MATH]0.2[MATH].', '1203.3215-1-7-5': 'The normalization constants at 1TeV are N[MATH]=(4.2[MATH]1.1)[MATH]TeV[MATH]cm[MATH]s[MATH] and N[MATH]=(6.8[MATH]1.6)[MATH]TeV[MATH]cm[MATH]s[MATH] for the point-like source and the total emission region, respectively.', '1203.3215-1-7-6': 'The systematic error on the normalization constant N[MATH] is estimated from simulated data to be 20% .', '1203.3215-1-8-0': 'Finally, Fig. [REF] shows the light curve of the point-like emission region, integrated between 0.6TeV and 10TeV.', '1203.3215-1-8-1': 'The Lomb-Scargle test was applied to the data to search for periodicity or variability without positive results for this data set.', '1203.3215-1-8-2': 'Fitting a constant to the integrated flux yields a [MATH]/[MATH] of 67.2/66, equivalent to a variable integral flux on a run-by-run basis at the level of 1.0[MATH].', '1203.3215-1-8-3': 'The lack of orbital coverage prevents any firm conclusion on variability of the TeV emission at the Fermi-LAT reported period (16.58days).', '1203.3215-1-9-0': '# X-ray observations with XMM-Newton', '1203.3215-1-10-0': 'XMM-Newton observations were acquired (ID: 0604700101, PI.', '1203.3215-1-10-1': 'E. de Ona Wilhelmi) to investigate the origin of the VHE emission region.', '1203.3215-1-10-2': 'The field was observed on the 22[MATH] August 2009, with a total integration time of 20ksec. The observations were centered on [MATH]55.60[MATH] and [MATH]=-58[MATH]56.8[MATH] (J2000) and acquired with the EPIC-PN and EPIC-MOS cameras in full-frame mode with a medium filter in a single pointing.', '1203.3215-1-10-3': 'This position allowed a good coverage of the whole SNR structure in the EPIC cameras.', '1203.3215-1-11-0': 'The data were analyzed using the XMM Science Analysis System (SAS v11.0.0 ) and calibration files valid at September 2011.', '1203.3215-1-11-1': 'To exclude high background flares, which could potentially affect the observations, light curves were extracted above 10keV for the entire FoV of the EPIC cameras, but no contamination was found.', '1203.3215-1-11-2': 'Therefore the full data set was used for the image and spectral analysis.', '1203.3215-1-11-3': 'To create images, spectra, and light curves, events with FLAG=0, and PATTERN=12 (MOS) and 4 (PN) were selected.', '1203.3215-1-11-4': 'Hereafter clean event files in the 0.3 to 8keV energy band are used.', '1203.3215-1-12-0': 'Images combining the different EPIC instruments (see Fig. [REF] a) and b)), vignetting-corrected and subtracted for particle induced and soft proton background, were produced using the ESAS analysis package (integrated in SAS).', '1203.3215-1-12-1': 'In the 0.3keV to 2keV energy range a bright source is detected (see Fig. [REF] a)).', '1203.3215-1-12-2': 'The radial profile of this central source was derived from the three EPIC cameras and fit with the corresponding PSF, confirming its point-like nature within the instrument angular resolution and observation sensitivity.', '1203.3215-1-12-3': 'This source is surrounded by diffuse emission extending up to the radio shell.', '1203.3215-1-12-4': 'A strong enhancement of the diffuse emission is visible just downstream of the radio and H[MATH] filament (see Fig. [REF] a) and d)).', '1203.3215-1-12-5': 'At higher energies (2 to 8keV, see Fig. [REF] b)), the diffuse emission is strongly suppressed suggesting a thermal nature of the emission pervading the SNR, the only significant feature being the bright point-like central source.', '1203.3215-1-13-0': 'This source, dubbed XMMU J101855.4-58564, is located at [MATH]55.40[MATH] and [MATH]=-58[MATH]45.6 (J2000) with statistical error of [MATH]0.25 in each coordinate (derived using the SAS task edetect).', '1203.3215-1-13-1': 'The position is compatible with the one derived by [CITATION] The photon spectrum of the point-like source was derived integrating over a 20 circle around the fit position and the background was estimated from a circle of 40 located in the vicinity.', '1203.3215-1-13-2': 'The spectrum is well fit ([MATH]=0.97, [MATH]=159) by an absorbed power-law function in the 0.5 to 7.5keV energy range, with a photon index of [MATH]=1.66[MATH]0.11[MATH] and an integrated flux of F[MATH]=(6.5[MATH]0.7[MATH]ergcm[MATH]s[MATH].', '1203.3215-1-13-3': 'The absorption column density N[MATH] is (6.6[MATH]0.8[MATH] cm[MATH], supporting a Galactic origin of the source.', '1203.3215-1-13-4': 'Other models such as a black-body model give fits that are statistically inadequate.', '1203.3215-1-13-5': 'Fig. [REF] shows the measured spectrum for MOS1, MOS2 and PN (in black, red and green respectively).', '1203.3215-1-14-0': 'Archival 2MASS (Two Micron All Sky Survey) data of the region show a bright star (with magnitudes J=10.44[MATH]0.02, H=10.14[MATH]0.02 and K=10.02[MATH]0.02) dubbed 2MASS 10185560-5856459, located at [MATH]55.6[MATH] and [MATH]=-58[MATH]46 (J2000), 1.3 away from XMMUJ101855.4-58564, the likely counterpart in the binary system.', '1203.3215-1-14-1': 'This source also appears in the USNO catalog with magnitudes B=12.76 and R=11.16.', '1203.3215-1-14-2': 'A distance of d[MATH]=5.4[MATH]kpc to the 2MASS star has been estimated through photometry by [CITATION].', '1203.3215-1-14-3': 'The position of XMMUJ101855.4-58564 is also in agreement with the variable compact object detected by Swift XRT at [MATH]55.54[MATH] and [MATH]=-58[MATH]45.9 (J2000) and the Fermi-LAT source 1FGLJ1018.6-5856 .', '1203.3215-1-15-0': 'To the North-East of XMMUJ101855.4-58564 a faint extended emission region located just downstream of the radio and H[MATH] filament of the remnant is visible at low energy (E[MATH]2keV) (Fig. [REF] a) and d)).', '1203.3215-1-15-1': 'To extract the X-ray spectrum, the background was modeled using the ESAS software following the approach of [CITATION].', '1203.3215-1-15-2': 'This background model is subsequently used to fit the signal region.', '1203.3215-1-15-3': 'The final shell spectrum is well represented by an absorbed non-equilibrium ionization (PSHOCK) thermal model with a temperature of kT[MATH]0.5keV and a column density of 8[MATH]cm[MATH] (in Fig. [REF]).', '1203.3215-1-15-4': 'The normalization factor A, defined as [EQUATION] is 1.5[MATH]cm[MATH].', '1203.3215-1-15-5': 'From this value, assuming a spherical volume (V) corresponding to the 2 source extraction region (see Fig. [REF]a)) and a fully ionized gas (n[MATH]=1.2[MATH]n), a plasma density can be derived to be n[MATH]0.5cm[MATH](2.9kpc/d)[MATH].', '1203.3215-1-16-0': '# Discussion', '1203.3215-1-17-0': 'A new VHE [MATH]-ray source, HESSJ1018-589, has been discovered in the vicinity of the shell-like remnant SNRG284.3-1.8.', '1203.3215-1-17-1': 'The extended TeV emission region coincides with SNRG284.3-1.8 and PSRJ1016-5857, both viable candidates to explain the observed VHE [MATH]-ray emission.', '1203.3215-1-17-2': 'This emission region is also positionally compatible with the new HE [MATH]-ray binary 1FGLJ1018.6-5856 reported recently by the Fermi-LAT collaboration .', '1203.3215-1-17-3': 'Different possible scenarios in the context of multi-wavelength observations, including new results derived from XMM-Newton observations, are discussed in the following.', '1203.3215-1-18-0': 'Although the spectral characteristics and light curve of the H.E.S.S. source do not yet allow a firm identification of the origin of the VHE [MATH]-ray emission, the morphology of the source is considered to clarify the situation.', '1203.3215-1-18-1': 'Two distinct emission regions are detected in the H.E.S.S data, one point-like emission region (A) located in the center of SNRG284.3-1.8 with a centroid compatible with the 95% confidence contour of 1FGLJ1018.6-5856, and a diffuse emission region (B) extending towards the direction of PSRJ1016-5857, with its centroid compatible with the position of the pulsar.', '1203.3215-1-19-0': 'The new HE [MATH]-ray binary 1FGLJ1018.6-5856 shares many characteristics with the VHE [MATH]-ray binary LS5039 .', '1203.3215-1-19-1': 'The XMM-Newton observations presented here reveal a bright non-thermal point-like source, XMMUJ101855.4-58564, in the center of the SNR and compatible with the position of the binary system and the H.E.S.S. point-like emission.', '1203.3215-1-19-2': 'The X-ray photon spectrum resembles that of pulsars, with a photon spectral index of 1.67 and a column density of (6.6[MATH]0.8)[MATH]cm[MATH], compatible with that from the thermal emission region coincident with the bright radio filament (7.9[MATH]cm[MATH]).', '1203.3215-1-19-3': 'The position of XMMUJ101855.4-58564 centered on the SNR and the similar distance (also compatible with the distance to the associated 2MASS star) suggest a physical association between the two objects (rather than an association between the SNR and PSRJ1016-5857), namely that the compact object within the binary system was the stellar progenitor for the type II SN explosion (see e. g. ).', '1203.3215-1-19-4': 'No extended emission or putative PWN around the point-like source is observed in the present data at the level of the XMM-Newton observation sensitivity.', '1203.3215-1-19-5': 'The light curve of the X-ray emission does not show any indication of variability or periodicity on short time scales, for the time resolution of the MOS and PN cameras in full-frame mode (2.6s and 73.4ms respectively).', '1203.3215-1-19-6': 'XMMUJ101855.4-58564 has been associated with the Swift XRT source, which itself has been identified as a counterpart to the HE binary system.', '1203.3215-1-19-7': 'The best-fit position of HESSJ1018-589 is compatible within less than 1[MATH] with both the position of XMMUJ101855.4-58564 and the variable HE source, whose position has been determined accurately using timing analysis.', '1203.3215-1-19-8': 'The spectral type of the possible companion star, O6V((f)) is similar to the one in the VHE [MATH]-ray binary LS5039.', '1203.3215-1-20-0': 'In a binary scenario composed of a massive star and a pulsar or a black hole, modulated VHE [MATH]-rays can be produced by different mechanisms, namely inverse Compton emission or pions produced by high-energy protons interacting with the stellar wind .', '1203.3215-1-20-1': 'Similar to previously detected VHE binary systems such as LS5039, LSI +61 303 and PSRB1259-63 periodic emission is found at HE.', '1203.3215-1-20-2': 'Despite the similarities with other VHE binaries and in particular LS5039, and the good positional agreement with the Fermi-LAT source, the association of the H.E.S.S. source (A) with 1FGLJ1018.6-5856 is still uncertain.', '1203.3215-1-20-3': 'No flux variability has been observed yet and the Lomb-Scargle test does not recover the Fermi-LAT reported 16.58 days modulation of the HE signal.', '1203.3215-1-20-4': 'Nevertheless it should be noted that the non-detection might be due to possible contamination from the neighboring diffuse emission, statistics of the data set, and and inadequate time sampling of the orbit.', '1203.3215-1-21-0': 'HESSJ1018-589 (A) is also coincident with the SNRG284.3-1.8.', '1203.3215-1-21-1': 'SNRs are believed to be sites of particle acceleration up to at least a few tens of TeV.', '1203.3215-1-21-2': 'Two types of VHE [MATH]-ray emission associated with SNRs have been discovered with IACTs, VHE [MATH]-ray emission from shell-like SNRs such as RXJ1713.7-3947 or SN 1006 , in which in general the VHE morphology is in good agreement with the synchrotron X-ray emission; and VHE [MATH]-ray radiation which seems to originate through proton-proton (p-p) interaction of cosmic rays (CR) accelerated in the SNR interacting with local MCs in the vicinity, such as W28 .', '1203.3215-1-21-3': '[CITATION] reported evidence of interaction of SNRG284.3-1.8 with embedded MC, constraining the distance to the SNR to [MATH]2.9kpc.', '1203.3215-1-21-4': 'The analysis performed in the context of this work of public [MATH]CO (J =1-0) data from the CfA 1.2 m Millimeter-Wave Telescope yields an estimate of the MC mass of [MATH]M[MATH].', '1203.3215-1-21-5': '[CITATION] also reported on optical observations in the direction of SNRG284.3-1.8 and associated a bright optical filament (Fig. [REF] d)) coincident with the brightest shell structure to the North-East, indicating collisional excitation of the ISM, swept by the expanding SNR shock wave.', '1203.3215-1-21-6': 'The observed MCs, if indeed physically associated with the SNR, could provide enough target material to explain the VHE emission in a scenario in which the [MATH]-ray are produced via p-p interaction.', '1203.3215-1-21-7': 'However, contrary to some other SNRs at VHE, the emission detected with H.E.S.S. does not match the shell-type morphology within the present statistics.', '1203.3215-1-22-0': 'The diffuse emission detected with H.E.S.S. (B) extends towards the direction of PSRJ1016-5857.', '1203.3215-1-22-1': 'PSRJ1016-5857 was detected as a bright EGRET source and pulsed emission at HE has been reported by the AGILE and the Fermi-LAT collaborations.', '1203.3215-1-22-2': 'With a spin-down luminosity of 2.6[MATH]ergs[MATH], the radio, HE and X-ray pulsar PSRJ1016-5857 is energetic enough to power the entire H.E.S.S. source, assuming a dispersion-measured estimated distance of 9kpc.', '1203.3215-1-22-3': 'In this scenario, particles are accelerated in the wind termination shock and produce VHE [MATH]-ray emission by inverse Compton (IC) processes as they propagate away from the pulsar.', '1203.3215-1-22-4': 'As a result of the interactions of relativistic leptons with the local magnetic field and low-energy radiation, non-thermal radiation is produced up to [MATH]100TeV (for a recent review see ).', '1203.3215-1-22-5': 'Assuming a distance of 9kpc, the total VHE luminosity in the 1 to 10TeV energy range is 9.7[MATH](d/9kpc)[MATH]ergs[MATH], implying a maximum conversion from rotational energy into non-thermal emission with efficiency 0.4[MATH], with similar features to other well-established VHE PWNe, such as Vela X or HESSJ1026-582 .', '1203.3215-1-22-6': 'The associated X-ray nebula has been detected with Chandra in the 0.8 to 7keV energy range with a size of 3[MATH].', '1203.3215-1-22-7': 'The different size of the VHE and X-ray nebula, [MATH](d/9kpc)pc and [MATH](d/9kpc)pc respectively, can be easily accommodated in a relic nebula scenario, and explained by the different energies (and hence cooling times) of the electron population emitting X-rays and VHE [MATH]-rays as seen, e. g. in HESSJ1825-137 for a low magnetic field of the order of a few [MATH]G.', '1203.3215-1-23-0': '# Conclusions', '1203.3215-1-24-0': 'A new VHE [MATH]-ray source dubbed HESSJ1018-589 has been detected with the H.E.S.S. telescope array with a significance of 8.3[MATH].', '1203.3215-1-24-1': 'The complex VHE morphology and faint VHE emission prevent a unequivocal identification of the source given the presence of several possible counterparts.', '1203.3215-1-24-2': 'The H.E.S.S. source seems to be composed of two emission regions but the statistics are still too low to make firm conclusions about the origin of those.', '1203.3215-1-25-0': 'Several counterparts are discussed using energetics arguments as to the possible origin of either part or all of the emission.', '1203.3215-1-25-1': 'In a SNR/MC scenario, SNRG284.3-1.8 could partially explain the VHE [MATH]-ray emission via p-p interactions with the associated MCs.', '1203.3215-1-25-2': 'However the fact that the VHE emission does not trace either the irregular shell or the cloud morphology disfavors SNRG284.3-1.8 as the only counterpart.', '1203.3215-1-26-0': 'The morphology and good positional agreement between the H.E.S.S. best-fit position A and the new Fermi-LAT binary 1FGLJ1018.6-5856 suggest a common origin.', '1203.3215-1-26-1': 'The analysis of the XMM-Newton observations revealed a non-thermal point-like source, XMMUJ101855.4-58564, with photon spectral index of 1.67, similar to the compact object found in LS 5039.', '1203.3215-1-26-2': 'Likewise, the spectral class of the massive star companion listed in the 2MASS and USNO catalogs is similar to the one in LS5039.', '1203.3215-1-26-3': 'However, no variability has been found in the H.E.S.S. light curve.', '1203.3215-1-26-4': 'A dedicated observation campaign at VHE should help to clarify whether or not the two sources are indeed associated.', '1203.3215-1-27-0': 'The energetic pulsar PSRJ1016-5857, also recently detected in Fermi-LAT and AGILE data, and its X-ray nebula seem the most likely candidate to power the extended VHE [MATH]-ray source, given the high spin-down luminosity (2.6[MATH]ergs[MATH]) and X-ray nebula, which implies a population of high energy electrons able to up-scatter soft photon fields to VHE.', '1203.3215-1-27-1': 'The estimated age of the pulsar (21kyr) would also explain the large size of the VHE nebula, similar to other systems such as Vela X .', '1203.3215-1-28-0': 'Finally, XMM-Newton observations also revealed thermal emission behind the brightest synchrotron part of the radio shell of SNRG284.3-1.8, which might be associated with shock heated interstellar matter.', '1203.3215-1-28-1': 'The column density is statistically compatible with the one derived from the direction of XMMUJ101855.4-58564.', '1203.3215-1-28-2': 'The similar column density and the position of the pulsar candidate with respect to the center of the SNR could indicate a common origin, where XMMUJ101855.4-58564 is interpreted as the pulsar left behind after the supernova explosion.', '1203.3215-1-29-0': 'The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia.', '1203.3215-1-29-1': 'We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.', '1203.3215-1-29-2': 'This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.'}","{'1203.3215-2-0-0': 'Discovery of VHE emission towards the Carina arm region with the H.E.S.S. telescope array: HESSJ1018-589', '1203.3215-2-1-0': 'The Carina arm region, containing the supernova remnant SNRG284.3-1.8, the high-energy (HE; E[MATH]100 MeV) binary 1FGLJ1018.6-5856 and the energetic pulsar PSRJ1016-5857 and its nebula, has been observed with the H.E.S.S. telescope array.', '1203.3215-2-1-1': 'The observational coverage of the region in very-high-energy (VHE; E[MATH]0.1TeV) [MATH]-rays benefits from deep exposure (40h) of the neighboring open cluster Westerlund 2.', '1203.3215-2-1-2': 'The observations have revealed a new extended region of VHE [MATH]-ray emission.', '1203.3215-2-1-3': 'The new VHE source HESSJ1018-589 shows a bright, point-like emission region positionally coincident with SNRG284.3-1.8 and 1FGLJ1018.6-5856 and a diffuse extension towards the direction of PSRJ1016-5857.', '1203.3215-2-1-4': 'A soft ([MATH]=2.7[MATH]0.5[MATH]) photon index, with a differential flux at 1TeV of N[MATH]=(4.2[MATH]1.1)[MATH]TeV[MATH] cm[MATH] s[MATH] is found for the point-like source, whereas the total emission region including the diffuse emission region is well fit by a power-law function with spectral index [MATH]=2.9[MATH]0.4[MATH] and differential flux at 1TeV of N[MATH]=(6.8[MATH]1.6)[MATH]TeV[MATH] cm[MATH] s[MATH].', '1203.3215-2-1-5': 'This H.E.S.S. detection motivated follow-up X-ray observations with the XMM-Newton satellite to investigate the origin of the VHE emission.', '1203.3215-2-1-6': 'The analysis of the XMM-Newton data resulted in the discovery of a bright, non-thermal point-like source (XMMU J101855.4-58564) with a photon index of [MATH]=1.65[MATH]0.08 in the center of SNRG284.3-1.8, and a thermal, extended emission region coincident with its bright northern filament.', '1203.3215-2-1-7': 'The characteristics of this thermal emission are used to estimate the plasma density in the region as n[MATH]0.5 cm[MATH](2.9kpc/d)[MATH].', '1203.3215-2-1-8': 'The position of XMMUJ101855.4-58564 is compatible with the position reported by the Fermi-LAT collaboration for the binary system 1FGLJ1018.6-5856 and the variable Swift XRT source identified with it.', '1203.3215-2-1-9': 'The new X-ray data are used alongside archival multi-wavelength data to investigate the relationship between the VHE [MATH]-ray emission from HESSJ1018-589 and the various potential counterparts in the Carina arm region.', '1203.3215-2-2-0': 'H.E.S.S. detection of HESSJ1018-589', '1203.3215-2-3-0': '# Introduction', '1203.3215-2-4-0': 'The H.E.S.S. (High Energy Stereoscopic System) collaboration has carried out observations of the Carina arm as part of the Galactic Plane Survey .', '1203.3215-2-4-1': 'The observed region includes three potential VHE [MATH]-ray emitters, SNRG284.3-1.8 (MSH 10-5[MATH]), the high spin-down luminosity pulsar PSRJ1016-5857 and the Fermi Large Area Telescope (Fermi-LAT) [MATH]-ray binary 1FGLJ1018.6-5856 .', '1203.3215-2-4-2': 'SNRG284.3-1.8 has an incomplete radio shell (Fig. [REF]) with a non-thermal radio spectrum and a flux density of (5.4[MATH]0.8)Jy at 8.4GHz .', '1203.3215-2-4-3': 'Evidence of interaction with molecular clouds (MC; [MATH]CO J=1-0) has been reported by [CITATION] using observations with the 1.2m Columbia Millimeter-Wave Telescope.', '1203.3215-2-4-4': 'The MC content integrated over velocities in the local standard of rest from -14.95kms[MATH] to -21.45kms[MATH] traces the radio shell shape of the SNR.', '1203.3215-2-5-0': 'The shell-like SNR shows a bright, narrow filament, coincident with an H[MATH] filament to the East.', '1203.3215-2-5-1': 'Optical and CO observations imply that SNRG284.3-1.8 is most likely a 10[MATH]yr old remnant of a type II supernova with a massive stellar progenitor and that it is located at a distance of d[MATH]=2.9kpc (with an error of [MATH]20%) .', '1203.3215-2-5-2': 'The western edge of the shell shows a so-called finger emission region (see Fig. [REF]) extended towards the direction of PSRJ1016-5857.', '1203.3215-2-5-3': 'This Vela-like pulsar and its associated X-ray pulsar wind nebula (PWN), located [MATH] away from the geometrical center of the SNR, were discovered by [CITATION] in a search for counterparts of the unidentified source 3EGJ1013-5915 with the Parkes telescope.', '1203.3215-2-5-4': 'The energetic pulsar, formerly associated with the SNRG284.3-1.8, has a rotation period of 107ms, a characteristic age [MATH]=21kyr and a spin-down luminosity of 2.6[MATH]ergs[MATH].', '1203.3215-2-5-5': ""Its distance can be estimated from the pulsar's dispersion measure to be d[MATH]kpc or d[MATH]0.03[MATH]."", '1203.3215-2-6-0': 'The two-emission-regions hypothesis is strengthened when the projection of a rectangular region on the uncorrelated excess map along the extension of the image (with a width of twice the H.E.S.S. mean PSF) is evaluated.', '1203.3215-2-6-1': 'The fit of a single Gaussian function to the projection yields a [MATH]/[MATH] of 21.04/13 (null hypothesis probability P=0.072) whereas the fit to a double Gaussian function leads to a better [MATH]/[MATH] of 9.6/10 (P=0.476).', '1203.3215-2-6-2': 'Fig. [REF] shows the profile of the rectangular region (in white in the inset).', '1203.3215-2-6-3': 'The single and double Gaussian function fits are shown in dashed and solid lines, respectively.', '1203.3215-2-7-0': 'The energy spectrum (shown in Fig. [REF]) for the VHE excess was computed by means of a forward-folding maximum likelihood fit .', '1203.3215-2-7-1': 'To estimate the background the reflected background method was used, in which symmetric regions, not contaminated by known sources, are used to extract the background .', '1203.3215-2-7-2': 'To derive the source spectrum two regions were selected, one around the point-like source and a second region of 0.30[MATH] radius accounting for the total VHE [MATH]-ray emission centered on [MATH]45.6[MATH] and [MATH]=-59[MATH] (J2000).', '1203.3215-2-7-3': 'The photon spectra in both cases are well represented with a simple power-law function dN/dE=N[MATH](E/1TeV)[MATH].', '1203.3215-2-7-4': 'The point-like source has a photon index of [MATH]=2.7[MATH]0.5[MATH]0.2[MATH] statistically compatible with the total (0.30[MATH]) VHE [MATH]-ray emission region, [MATH]=2.9[MATH]0.4[MATH]0.2[MATH].', '1203.3215-2-7-5': 'The normalization constants at 1TeV are N[MATH]=(4.2[MATH]1.1)[MATH]TeV[MATH]cm[MATH]s[MATH] and N[MATH]=(6.8[MATH]1.6)[MATH]TeV[MATH]cm[MATH]s[MATH] for the point-like source and the total emission region, respectively.', '1203.3215-2-7-6': 'The systematic error on the normalization constant N[MATH] is estimated from simulated data to be 20% .', '1203.3215-2-8-0': 'Finally, Fig. [REF] shows the light curve of the point-like emission region, integrated between 0.6TeV and 10TeV.', '1203.3215-2-8-1': 'The Lomb-Scargle test was applied to the data to search for periodicity or variability without positive results for this data set.', '1203.3215-2-8-2': 'Fitting a constant to the integrated flux yields a [MATH]/[MATH] of 67.2/66, equivalent to a variable integral flux on a run-by-run basis at the level of 1.0[MATH].', '1203.3215-2-8-3': 'The lack of orbital coverage prevents any firm conclusion on variability of the TeV emission at the Fermi-LAT reported period (16.58days).', '1203.3215-2-9-0': '# X-ray observations with XMM-Newton', '1203.3215-2-10-0': 'XMM-Newton observations were acquired (ID: 0604700101, PI.', '1203.3215-2-10-1': 'E. de Ona Wilhelmi) to investigate the origin of the VHE emission region.', '1203.3215-2-10-2': 'The field was observed on the 22[MATH] August 2009, with a total integration time of 20ksec. The observations were centered on [MATH]55.60[MATH] and [MATH]=-58[MATH]56.8[MATH] (J2000) and acquired with the EPIC-PN and EPIC-MOS cameras in full-frame mode with a medium filter in a single pointing.', '1203.3215-2-10-3': 'This position allowed a good coverage of the whole SNR structure in the EPIC cameras.', '1203.3215-2-11-0': 'The data were analyzed using the XMM Science Analysis System (SAS v11.0.0 ) and calibration files valid at September 2011.', '1203.3215-2-11-1': 'To exclude high background flares, which could potentially affect the observations, light curves were extracted above 10keV for the entire FoV of the EPIC cameras, but no contamination was found.', '1203.3215-2-11-2': 'Therefore the full data set was used for the image and spectral analysis.', '1203.3215-2-11-3': 'To create images, spectra, and light curves, events with FLAG=0, and PATTERN=12 (MOS) and 4 (PN) were selected.', '1203.3215-2-11-4': 'Hereafter clean event files in the 0.3 to 8keV energy band are used.', '1203.3215-2-12-0': 'Images combining the different EPIC instruments (see Fig. [REF] a) and b)), vignetting-corrected and subtracted for particle induced and soft proton background, were produced using the ESAS analysis package (integrated in SAS).', '1203.3215-2-12-1': 'In the 0.3keV to 2keV energy range a bright source is detected (see Fig. [REF] a)).', '1203.3215-2-12-2': 'The radial profile of this central source was derived from the three EPIC cameras and fit with the corresponding PSF, confirming its point-like nature within the instrument angular resolution and observation sensitivity.', '1203.3215-2-12-3': 'This source is surrounded by diffuse emission extending up to the radio shell.', '1203.3215-2-12-4': 'A strong enhancement of the diffuse emission is visible just downstream of the radio and H[MATH] filament (see Fig. [REF] a) and d)).', '1203.3215-2-12-5': 'At higher energies (2 to 8keV, see Fig. [REF] b)), the diffuse emission is strongly suppressed suggesting a thermal nature of the emission pervading the SNR, the only significant feature being the bright point-like central source.', '1203.3215-2-13-0': 'This source, dubbed XMMU J101855.4-58564, is located at [MATH]55.40[MATH] and [MATH]=-58[MATH]45.6 (J2000) with statistical error of [MATH]0.25 in each coordinate (derived using the SAS task edetect).', '1203.3215-2-13-1': 'The position is compatible with the one derived by [CITATION] The photon spectrum of the point-like source was derived integrating over a 20 circle around the fit position and the background was estimated from a circle of 40 located in the vicinity.', '1203.3215-2-13-2': 'The spectrum is well fit ([MATH]=0.97, [MATH]=159) by an absorbed power-law function in the 0.5 to 7.5keV energy range, with a photon index of [MATH]=1.66[MATH]0.11[MATH] and an integrated flux of F[MATH]=(6.5[MATH]0.7[MATH]ergcm[MATH]s[MATH].', '1203.3215-2-13-3': 'The absorption column density N[MATH] is (6.6[MATH]0.8[MATH] cm[MATH], supporting a Galactic origin of the source.', '1203.3215-2-13-4': 'Other models such as a black-body model give fits that are statistically inadequate.', '1203.3215-2-13-5': 'Fig. [REF] shows the measured spectrum for MOS1, MOS2 and PN (in black, red and green respectively).', '1203.3215-2-14-0': 'Archival 2MASS (Two Micron All Sky Survey) data of the region show a bright star (with magnitudes J=10.44[MATH]0.02, H=10.14[MATH]0.02 and K=10.02[MATH]0.02) dubbed 2MASS 10185560-5856459, located at [MATH]55.6[MATH] and [MATH]=-58[MATH]46 (J2000), 1.3 away from XMMUJ101855.4-58564, the likely counterpart in the binary system.', '1203.3215-2-14-1': 'This source also appears in the USNO catalog with magnitudes B=12.76 and R=11.16.', '1203.3215-2-14-2': 'A distance of d[MATH]=5.4[MATH]kpc to the 2MASS star has been estimated through photometry by [CITATION].', '1203.3215-2-14-3': 'The position of XMMUJ101855.4-58564 is also in agreement with the variable compact object detected by Swift XRT at [MATH]55.54[MATH] and [MATH]=-58[MATH]45.9 (J2000) and the Fermi-LAT source 1FGLJ1018.6-5856 .', '1203.3215-2-15-0': 'To the North-East of XMMUJ101855.4-58564 a faint extended emission region located just downstream of the radio and H[MATH] filament of the remnant is visible at low energy (E[MATH]2keV) (Fig. [REF] a) and d)).', '1203.3215-2-15-1': 'To extract the X-ray spectrum, the background was modeled using the ESAS software following the approach of [CITATION].', '1203.3215-2-15-2': 'This background model is subsequently used to fit the signal region.', '1203.3215-2-15-3': 'The final shell spectrum is well represented by an absorbed non-equilibrium ionization (PSHOCK) thermal model with a temperature of kT[MATH]0.5keV and a column density of 8[MATH]cm[MATH] (in Fig. [REF]).', '1203.3215-2-15-4': 'The normalization factor A, defined as [EQUATION] is 1.5[MATH]cm[MATH].', '1203.3215-2-15-5': 'From this value, assuming a spherical volume (V) corresponding to the 2 source extraction region (see Fig. [REF]a)) and a fully ionized gas (n[MATH]=1.2[MATH]n), a plasma density can be derived to be n[MATH]0.5cm[MATH](2.9kpc/d)[MATH].', '1203.3215-2-16-0': '# Discussion', '1203.3215-2-17-0': 'A new VHE [MATH]-ray source, HESSJ1018-589, has been discovered in the vicinity of the shell-like remnant SNRG284.3-1.8.', '1203.3215-2-17-1': 'The extended TeV emission region coincides with SNRG284.3-1.8 and PSRJ1016-5857, both viable candidates to explain the observed VHE [MATH]-ray emission.', '1203.3215-2-17-2': 'This emission region is also positionally compatible with the new HE [MATH]-ray binary 1FGLJ1018.6-5856 reported recently by the Fermi-LAT collaboration .', '1203.3215-2-17-3': 'Different possible scenarios in the context of multi-wavelength observations, including new results derived from XMM-Newton observations, are discussed in the following.', '1203.3215-2-18-0': 'Although the spectral characteristics and light curve of the H.E.S.S. source do not yet allow a firm identification of the origin of the VHE [MATH]-ray emission, the morphology of the source is considered to clarify the situation.', '1203.3215-2-18-1': 'Two distinct emission regions are detected in the H.E.S.S data, one point-like emission region (A) located in the center of SNRG284.3-1.8 with a centroid compatible with the 95% confidence contour of 1FGLJ1018.6-5856, and a diffuse emission region (B) extending towards the direction of PSRJ1016-5857, with its centroid compatible with the position of the pulsar.', '1203.3215-2-19-0': 'The new HE [MATH]-ray binary 1FGLJ1018.6-5856 shares many characteristics with the VHE [MATH]-ray binary LS5039 .', '1203.3215-2-19-1': 'The XMM-Newton observations presented here reveal a bright non-thermal point-like source, XMMUJ101855.4-58564, in the center of the SNR and compatible with the position of the binary system and the H.E.S.S. point-like emission.', '1203.3215-2-19-2': 'The X-ray photon spectrum resembles that of pulsars, with a photon spectral index of 1.67 and a column density of (6.6[MATH]0.8)[MATH]cm[MATH], compatible with that from the thermal emission region coincident with the bright radio filament (7.9[MATH]cm[MATH]).', '1203.3215-2-19-3': 'The position of XMMUJ101855.4-58564 centered on the SNR and the similar distance (also compatible with the distance to the associated 2MASS star) suggest a physical association between the two objects (rather than an association between the SNR and PSRJ1016-5857), namely that the compact object within the binary system was the stellar progenitor for the type II SN explosion (see e. g. ).', '1203.3215-2-19-4': 'No extended emission or putative PWN around the point-like source is observed in the present data at the level of the XMM-Newton observation sensitivity.', '1203.3215-2-19-5': 'The light curve of the X-ray emission does not show any indication of variability or periodicity on short time scales, for the time resolution of the MOS and PN cameras in full-frame mode (2.6s and 73.4ms respectively).', '1203.3215-2-19-6': 'XMMUJ101855.4-58564 has been associated with the Swift XRT source, which itself has been identified as a counterpart to the HE binary system.', '1203.3215-2-19-7': 'The best-fit position of HESSJ1018-589 is compatible within less than 1[MATH] with both the position of XMMUJ101855.4-58564 and the variable HE source, whose position has been determined accurately using timing analysis.', '1203.3215-2-19-8': 'The spectral type of the possible companion star, O6V((f)) is similar to the one in the VHE [MATH]-ray binary LS5039.', '1203.3215-2-20-0': 'In a binary scenario composed of a massive star and a pulsar or a black hole, modulated VHE [MATH]-rays can be produced by different mechanisms, namely inverse Compton emission or pions produced by high-energy protons interacting with the stellar wind .', '1203.3215-2-20-1': 'Similar to previously detected VHE binary systems such as LS5039, LSI +61 303 and PSRB1259-63 periodic emission is found at HE.', '1203.3215-2-20-2': 'Despite the similarities with other VHE binaries and in particular LS5039, and the good positional agreement with the Fermi-LAT source, the association of the H.E.S.S. source (A) with 1FGLJ1018.6-5856 is still uncertain.', '1203.3215-2-20-3': 'No flux variability has been observed yet and the Lomb-Scargle test does not recover the Fermi-LAT reported 16.58 days modulation of the HE signal.', '1203.3215-2-20-4': 'Nevertheless it should be noted that the non-detection might be due to possible contamination from the neighboring diffuse emission, statistics of the data set, and and inadequate time sampling of the orbit.', '1203.3215-2-21-0': 'HESSJ1018-589 (A) is also coincident with the SNRG284.3-1.8.', '1203.3215-2-21-1': 'SNRs are believed to be sites of particle acceleration up to at least a few tens of TeV.', '1203.3215-2-21-2': 'Two types of VHE [MATH]-ray emission associated with SNRs have been discovered with IACTs, VHE [MATH]-ray emission from shell-like SNRs such as RXJ1713.7-3947 or SN 1006 , in which in general the VHE morphology is in good agreement with the synchrotron X-ray emission; and VHE [MATH]-ray radiation which seems to originate through proton-proton (p-p) interaction of cosmic rays (CR) accelerated in the SNR interacting with local MCs in the vicinity, such as W28 .', '1203.3215-2-21-3': '[CITATION] reported evidence of interaction of SNRG284.3-1.8 with embedded MC, constraining the distance to the SNR to [MATH]2.9kpc.', '1203.3215-2-21-4': 'The analysis performed in the context of this work of public [MATH]CO (J =1-0) data from the CfA 1.2 m Millimeter-Wave Telescope yields an estimate of the MC mass of [MATH]M[MATH].', '1203.3215-2-21-5': '[CITATION] also reported on optical observations in the direction of SNRG284.3-1.8 and associated a bright optical filament (Fig. [REF] d)) coincident with the brightest shell structure to the North-East, indicating collisional excitation of the ISM, swept by the expanding SNR shock wave.', '1203.3215-2-21-6': 'The observed MCs, if indeed physically associated with the SNR, could provide enough target material to explain the VHE emission in a scenario in which the [MATH]-ray are produced via p-p interaction.', '1203.3215-2-21-7': 'However, contrary to some other SNRs at VHE, the emission detected with H.E.S.S. does not match the shell-type morphology within the present statistics.', '1203.3215-2-22-0': 'The diffuse emission detected with H.E.S.S. (B) extends towards the direction of PSRJ1016-5857.', '1203.3215-2-22-1': 'PSRJ1016-5857 was detected as a bright EGRET source and pulsed emission at HE has been reported by the AGILE and the Fermi-LAT collaborations.', '1203.3215-2-22-2': 'With a spin-down luminosity of 2.6[MATH]ergs[MATH], the radio, HE and X-ray pulsar PSRJ1016-5857 is energetic enough to power the entire H.E.S.S. source, assuming a dispersion-measured estimated distance of 9kpc.', '1203.3215-2-22-3': 'In this scenario, particles are accelerated in the wind termination shock and produce VHE [MATH]-ray emission by inverse Compton (IC) processes as they propagate away from the pulsar.', '1203.3215-2-22-4': 'As a result of the interactions of relativistic leptons with the local magnetic field and low-energy radiation, non-thermal radiation is produced up to [MATH]100TeV (for a recent review see ).', '1203.3215-2-22-5': 'Assuming a distance of 9kpc, the total VHE luminosity in the 1 to 10TeV energy range is 9.7[MATH](d/9kpc)[MATH]ergs[MATH], implying a maximum conversion from rotational energy into non-thermal emission with efficiency 0.4[MATH], with similar features to other well-established VHE PWNe, such as Vela X or HESSJ1026-582 .', '1203.3215-2-22-6': 'The associated X-ray nebula has been detected with Chandra in the 0.8 to 7keV energy range with a size of 3[MATH].', '1203.3215-2-22-7': 'The different size of the VHE and X-ray nebula, [MATH](d/9kpc)pc and [MATH](d/9kpc)pc respectively, can be easily accommodated in a relic nebula scenario, and explained by the different energies (and hence cooling times) of the electron population emitting X-rays and VHE [MATH]-rays as seen, e. g. in HESSJ1825-137 for a low magnetic field of the order of a few [MATH]G.', '1203.3215-2-23-0': '# Conclusions', '1203.3215-2-24-0': 'A new VHE [MATH]-ray source dubbed HESSJ1018-589 has been detected with the H.E.S.S. telescope array with a significance of 8.3[MATH].', '1203.3215-2-24-1': 'The complex VHE morphology and faint VHE emission prevent a unequivocal identification of the source given the presence of several possible counterparts.', '1203.3215-2-24-2': 'The H.E.S.S. source seems to be composed of two emission regions but the statistics are still too low to make firm conclusions about the origin of those.', '1203.3215-2-25-0': 'Several counterparts are discussed using energetics arguments as to the possible origin of either part or all of the emission.', '1203.3215-2-25-1': 'In a SNR/MC scenario, SNRG284.3-1.8 could partially explain the VHE [MATH]-ray emission via p-p interactions with the associated MCs.', '1203.3215-2-25-2': 'However the fact that the VHE emission does not trace either the irregular shell or the cloud morphology disfavors SNRG284.3-1.8 as the only counterpart.', '1203.3215-2-26-0': 'The morphology and good positional agreement between the H.E.S.S. best-fit position A and the new Fermi-LAT binary 1FGLJ1018.6-5856 suggest a common origin.', '1203.3215-2-26-1': 'The analysis of the XMM-Newton observations revealed a non-thermal point-like source, XMMUJ101855.4-58564, with photon spectral index of 1.67, similar to the compact object found in LS 5039.', '1203.3215-2-26-2': 'Likewise, the spectral class of the massive star companion listed in the 2MASS and USNO catalogs is similar to the one in LS5039.', '1203.3215-2-26-3': 'However, no variability has been found in the H.E.S.S. light curve.', '1203.3215-2-26-4': 'A dedicated observation campaign at VHE should help to clarify whether or not the two sources are indeed associated.', '1203.3215-2-27-0': 'The energetic pulsar PSRJ1016-5857, also recently detected in Fermi-LAT and AGILE data, and its X-ray nebula seem the most likely candidate to power the extended VHE [MATH]-ray source, given the high spin-down luminosity (2.6[MATH]ergs[MATH]) and X-ray nebula, which implies a population of high energy electrons able to up-scatter soft photon fields to VHE.', '1203.3215-2-27-1': 'The estimated age of the pulsar (21kyr) would also explain the large size of the VHE nebula, similar to other systems such as Vela X .', '1203.3215-2-28-0': 'Finally, XMM-Newton observations also revealed thermal emission behind the brightest synchrotron part of the radio shell of SNRG284.3-1.8, which might be associated with shock heated interstellar matter.', '1203.3215-2-28-1': 'The column density is statistically compatible with the one derived from the direction of XMMUJ101855.4-58564.', '1203.3215-2-28-2': 'The similar column density and the position of the pulsar candidate with respect to the center of the SNR could indicate a common origin, where XMMUJ101855.4-58564 is interpreted as the pulsar left behind after the supernova explosion.', '1203.3215-2-29-0': 'The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia.', '1203.3215-2-29-1': 'We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.', '1203.3215-2-29-2': 'This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.'}","[['1203.3215-1-21-1', '1203.3215-2-21-1'], ['1203.3215-1-21-2', '1203.3215-2-21-2'], ['1203.3215-1-21-3', 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'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1203.3215,,,
1107.4048,"{'1107.4048-1-0-0': 'We discuss the phase structure of [MATH] D4 branes wrapped on a temporal (Euclidean) and a spatial circle, in terms of the near-horizon AdS geometries.', '1107.4048-1-0-1': 'This system has been studied previously to understand four dimensional pure [MATH] Yang-Mills theory (YM4) through holography.', '1107.4048-1-0-2': 'In the usual treatment of the subject, the phase transition between the AdS soliton and the black D4 brane is interpreted as the strong coupling continuation of the confinement/deconfinement transition in YM4.', '1107.4048-1-0-3': 'We show that this interpretation is not valid, since the black D4 brane and the deconfinement phase of YM4 have different order parameters and cannot be identified.', '1107.4048-1-0-4': 'We propose an alternative gravity dual of the confinement/deconfinement transition in terms of a Gregory-Laflamme transition of the AdS soliton in the IIB frame, where the strong coupling continuation of the deconfinement phase of YM4 is a localized D3 soliton.', '1107.4048-1-0-5': 'Our proposal offers a new explanation of several aspects the thermodynamics of holographic QCD.', '1107.4048-1-0-6': 'As an example, we show a new mechanism of the chiral symmetry restoration in the Sakai-Sugimoto model.', '1107.4048-1-0-7': 'The issues discussed in this paper pertain to gravity duals of non-supersymmetric gauge theories in general.', '1107.4048-1-1-0': '# Introduction', '1107.4048-1-2-0': 'Strongly coupled field theories such as QCD and various condensed matter systems are not amenable to perturbative calculations and require numerical or other non-perturbative tools.', '1107.4048-1-2-1': 'A powerful new tool in this context is holography, which maps gauge theories to gravity in a higher dimension [CITATION].', '1107.4048-1-2-2': 'Holography, however, is most reliable for a certain class of supersymmetric gauge theories; in the generic case, including for pure Yang Mills theories, the results of the gravity analysis are to be interpreted with care to figure things out on the gauge theory side.', '1107.4048-1-3-0': 'Recently, it was pointed out in [CITATION] that there are problems with a naive application of holography to a two dimensional large [MATH] bosonic gauge theory at finite temperature.', '1107.4048-1-3-1': 'The dual of this system is based on gravity backgrounds involving [MATH] D2 branes on [MATH].', '1107.4048-1-3-2': 'The phase diagram of gravity, interpreted naively, does not admit a unique continuation to the regime of gauge theory, owing to its dependence on the boundary condition for fermions on the brane.', '1107.4048-1-3-3': 'However, it was shown in [CITATION] that the phase structures in gravity and gauge theory could be smoothly connected through an appropriate choice of fermion boundary conditions.', '1107.4048-1-4-0': 'In this paper, we will show that similar problems arise when we holographically analyse more general non-supersymmetric gauge theories at finite temperature.', '1107.4048-1-4-1': 'The particular example we will focus on is that of holographic QCD from D4 branes [CITATION].', '1107.4048-1-4-2': 'We will discuss some problems with the usual correspondence between the confinement/deconfinement transition in QCD and the Scherk-Schwarz transition between an AdS soliton and a black D4 brane in the gravity dual.', '1107.4048-1-4-3': 'Some of these problems were first discussed in [CITATION] (see section [REF] for further details).', '1107.4048-1-4-4': 'Especially we will show that the black D4 brane cannot be identified with the (strong coupling continuation of the) deconfinement phase in QCD in four dimensions.', '1107.4048-1-4-5': 'As a resolution of these problems, we will propose an alternative scenario in which the confinement/deconfinement transition corresponds to a Gregory-Laflamme transition [CITATION] between a uniformly distributed AdS soliton and a localized AdS soliton in the IIB frame.', '1107.4048-1-5-0': 'The scenario we propose suggests that we need to reconsider several previous results in holographic QCD including the Sakai-Sugimoto model [CITATION].', '1107.4048-1-5-1': 'One important ingredient in the Sakai-Sugimoto model is the mechanism of chiral symmetry restoration at high temperatures [CITATION].', '1107.4048-1-5-2': 'We will propose a new mechanism for chiral symmetry restoration in our framework.', '1107.4048-1-6-0': 'The plan of the paper is as follows.', '1107.4048-1-7-0': 'Section [REF] contains a short review of finite temperature QCD and the holographic approach to four dimensional Yang Mills theory (YM4) using D4 branes.', '1107.4048-1-7-1': ""In Section [REF] we discuss the gravity theory at a finite temperature and a 'Scherk-Schwarz' transition between a solitonic D4 brane and a black D4 brane."", '1107.4048-1-7-2': 'In Section [REF] we recall the conventional correspondence between this transition and the confinement/deconfinement transition of the Yang Mills theory, and discuss problems with this correspondence, with special emphasis on the mismatch of [MATH] centre symmetry between the black D4 phase and the deconfined 4 dimensional YM theory.', '1107.4048-1-7-3': 'In Section [REF], we discuss the phase structure of the D4 branes using an unconventional (periodic) fermion boundary condition along the Euclidean time circle and show that the high temperature in gravity is described by a localized solitonic D3 brane (or its T-dual) whose centre symmetry precisely matches with the deconfinement phase of four dimensional YM theory.', '1107.4048-1-7-4': 'This leads us, in Section [REF], to propose a new representation of the confinement/deconfinement phase transition in terms of a GL transition between the solitonic D4 and the (T-dual of) the localized solitonic D3 (see Table [REF]).', '1107.4048-1-7-5': 'In Section [REF], we show some new correspondences between phenomena in QCD and their counterparts in gravity following our proposal.', '1107.4048-1-7-6': 'In Section [REF], we suggest a new mechanism of chiral symmetry restoration in the Sakai-Sugimoto model in keeping with our proposal.', '1107.4048-1-7-7': 'Section [REF] contains the concluding remarks and some open problems.', '1107.4048-1-8-0': '# Review of QCD and its gravity dual', '1107.4048-1-9-0': '## Finite temperature QCD', '1107.4048-1-10-0': 'In this section, we briefly review some salient properties of four dimensional [MATH] pure Yang-Mills theory at a finite temperature at large [MATH], a system which we will subsequently investigate through holography.', '1107.4048-1-11-0': 'An important symmetry in this theory is the [MATH] symmetry along the temporal cycle, which is the centre of the [MATH] symmetry.', '1107.4048-1-11-1': 'The order parameter of this symmetry is the temporal Polyakov loop operator [EQUATION]', '1107.4048-1-11-2': 'If this operator is zero, the [MATH] symmetry is preserved and, if not, it is broken.', '1107.4048-1-11-3': ""Especially, if the [MATH] symmetry is preserved, physical quantities do not depend on the temporal radius [MATH] at [MATH] order due to 'large [MATH] volume independence' [CITATION]."", '1107.4048-1-11-4': 'If the gauge theory is on a torus [MATH], the large [MATH] volume independence is generalized and physical quantities do not depend on [MATH], if the [MATH] symmetry along the [MATH]-th direction is preserved [CITATION].', '1107.4048-1-12-0': 'Let us consider the phase structure of the Yang-Mills theory.', '1107.4048-1-12-1': 'At low temperatures, the confinement phase is the dominant thermodynamic phase.', '1107.4048-1-12-2': 'In this phase, the [MATH] symmetry is preserved ([MATH]).', '1107.4048-1-12-3': 'At high temperatures, the deconfinement phase, in which the [MATH] symmetry is broken ([MATH]), dominates.', '1107.4048-1-12-4': 'There is a confinement/deconfinement transition at an intermediate temperature.', '1107.4048-1-12-5': 'From studies in large [MATH] lattice gauge theories, this is expected to be a first order phase transition [CITATION].', '1107.4048-1-13-0': ""As we remarked above, the free energy in the confinement phase does not depend on the temperature through the 'large [MATH] volume independence'."", '1107.4048-1-13-1': 'It implies that the entropy becomes zero at [MATH].', '1107.4048-1-13-2': 'On the other hand, the entropy in the deconfinement phase would be [MATH].', '1107.4048-1-13-3': 'This has a simple physical interpretation: in the confinement phase, the spectrum excludes gluon states, which have [MATH] degree of the freedom, and consists only of gauge singlet states like glueballs, leading to an [MATH] entropy.', '1107.4048-1-14-0': '## Holographic QCD', '1107.4048-1-15-0': 'In this section, we will review the construction of four dimensional [MATH] pure Yang-Mills theory from [MATH] D4 branes [CITATION].', '1107.4048-1-15-1': 'Let us first consider a 10 dimensional Euclidean spacetime with an [MATH] and consider D4 branes wrapping on the [MATH].', '1107.4048-1-15-2': 'We define the coordinate along this [MATH] as [MATH] and its periodicity as [MATH].', '1107.4048-1-15-3': 'The effective theory on this brane is a 5 dimensional supersymmetric Yang-Mills theory on the [MATH].', '1107.4048-1-15-4': 'For the fermions on the brane, the boundary condition along the circle can be AP (antiperiodic) or P (periodic); to specify the theory, we must pick one of these two boundary conditions.', '1107.4048-1-15-5': 'Let us take the AP boundary condition.', '1107.4048-1-15-6': 'This gives rise to fermion masses proportional to the Kaluza-Klein scale [MATH], leading to supersymmetry breaking (this is called the SS - Scherk-Schwarz- mechanism).', '1107.4048-1-15-7': 'This, in turn, induces masses for the adjoint scalars and for [MATH], which are proportional to [MATH] at one-loop.', '1107.4048-1-15-8': 'Therefore, if [MATH] is sufficiently small and the dynamical scale [MATH] is much less than both the above mass scales, then the fermions and adjoint scalars are decoupled and the 5 dimensional supersymmetric Yang-Mills theory is reduced to a four dimensional pure Yang-Mills theory.', '1107.4048-1-16-0': 'By taking the large [MATH] limit of this system a la Maldacena at low temperatures, we obtain the dual gravity description of the compactified 5 dimensional SYM theory [CITATION], which consists, at low temperatures, of a solitonic D[MATH] brane solution wrapping the [MATH].', '1107.4048-1-16-1': 'The explicit metric is given by [EQUATION]', '1107.4048-1-16-2': 'There is also a non-trivial value of the five form potential which we do not write explicitly.', '1107.4048-1-16-3': 'Here [MATH] is the YM coupling on the D[MATH] world-volume and [MATH] is found by putting [MATH] in the general formula [EQUATION]', '1107.4048-1-16-4': 'Since the [MATH]-cycle shrinks to zero at [MATH], in order to avoid possible conical singularities we must choose the asymptotic radius [MATH] as follows [EQUATION]', '1107.4048-1-16-5': ""With this choice, the contractible [MATH]-cycle, together with the radial direction [MATH], forms a so-called 'cigar' geometry which is topologically a disc."", '1107.4048-1-16-6': 'In order that the fermions are well-defined on this geometry, they must obey the AP boundary condition.', '1107.4048-1-16-7': 'The AP boundary condition along [MATH] is, of course, consistent with the choice of fermion b.c. in the boundary theory.', '1107.4048-1-17-0': 'The leading order gravity solution described above is not always valid.', '1107.4048-1-17-1': 'E.g. in order that the stringy modes can be ignored, we should ensure that the curvature in string units must be small.', '1107.4048-1-17-2': 'In other words, the typical length scale of this solution near [MATH], viz, [MATH], must satisfy [MATH] [CITATION].', '1107.4048-1-17-3': 'This condition turns out to be equivalent to [EQUATION]', '1107.4048-1-17-4': 'This is opposite to the condition which we found for the validity of the four-dimensional gauge theory description (see footnote [REF]).', '1107.4048-1-17-5': 'Thus, this gravity solution can describe the 5 dimensional SYM but cannot directly describe the 4 dimensional YM theory, inferences about which can only be obtained through extrapolation.', '1107.4048-1-17-6': 'This is a common problem in the construction of holographic duals of non-supersymmetric gauge theories.', '1107.4048-1-17-7': 'This has been discussed at length; in particular, the gravity description, which necessitates extrapolation to strong coupling, has been likened (cf. [CITATION] and [CITATION], p. 196-197) to strong coupling lattice gauge theory.', '1107.4048-1-17-8': 'Many interesting results, including the qualitative predictions in [CITATION], have been obtained using this prescription.', '1107.4048-1-18-0': 'Let us explore some properties of the soliton solution ([REF]).', '1107.4048-1-18-1': 'This solution is expected, on the basis of several arguments, to correspond to the confinement phase in the four dimensional gauge theory.', '1107.4048-1-18-2': 'For example, it can be shown that the classical action of this solution at a finite temperature [MATH] is [CITATION] [EQUATION] where [MATH] is the volume of the three spatial dimensions ([MATH]) and [MATH] is a constant which is obtainable from the formula below [CITATION] by putting [MATH] [EQUATION]', '1107.4048-1-18-3': 'Thus the free energy of this solution, [MATH], is independent of temperature, and hence the entropy is zero at [MATH] order.', '1107.4048-1-18-4': 'These facts are consistent with the interpretation of this solution as the confinement phase in the large [MATH] gauge theory, in which the temporal [MATH] symmetry is preserved.', '1107.4048-1-19-0': 'Note that the action ([REF]) divided by [MATH] does depend on [MATH], implying that the [MATH] symmetry along the [MATH]-cycle is broken.', '1107.4048-1-19-1': 'Although this observation might appear irrelevant for the 4 dimensional Yang-Mills theory, it is this symmetry breaking which allows a conventional KK reduction to four dimensions (see the next two sections for more details).', '1107.4048-1-20-0': '# Finite temperature holographic QCD', '1107.4048-1-21-0': '## Confinement/deconfinement transition in Holographic QCD in the standard scenario', '1107.4048-1-22-0': 'In existing holographic studies of the thermodynamics in the Yang-Mills theory, it is assumed that the black D4 brane solution corresponds to the deconfinement phase and the Scherk-Schwarz transition in the gravity is related to the confinement/deconfinement transition.', '1107.4048-1-22-1': 'In this section, we review this scenario and we will explain its problems in section [REF].', '1107.4048-1-23-0': 'To discuss holographic QCD at finite temperatures, we begin by compactifying Euclidean time in the boundary theory on a circle with periodicity [MATH].', '1107.4048-1-23-1': 'To permit description in terms of four dimensional gauge theory, the temperature should, of course, be kept far below the KK scale, [MATH] .', '1107.4048-1-23-2': 'In order to determine the gravitational theory, we need to fix the periodicity of fermions in the gauge theory along the time cycle.', '1107.4048-1-23-3': 'In the existing literature, the fermion boundary condition is chosen to be AP according to the usual practice for thermal fermions (we will see in section [REF] that this choice is not imperative here).', '1107.4048-1-23-4': ""The resulting fermionic bc's: (AP, AP) along [MATH] lead to a [MATH] symmetry of the system under [MATH]."", '1107.4048-1-23-5': 'In the gravity dual, we must therefore, include the [MATH]-transform of the solution ([REF]), which is a black D4 solution: [EQUATION]', '1107.4048-1-23-6': 'Here [MATH] in this metric is related to [MATH] as in ([REF]) [EQUATION]', '1107.4048-1-23-7': 'This solution has a contractible [MATH]-cycle and the fermions must obey the anti-periodic boundary condition (see footnote [REF]), consistent with the AP b.c. in the gauge theory.', '1107.4048-1-23-8': 'Analogously to ([REF]), the gravity description of this solution is reliable if [MATH].', '1107.4048-1-24-0': 'The classical action density can be evaluated, as before, yielding [EQUATION]', '1107.4048-1-24-1': 'In contrast with ([REF]), the free energy now is a function of [MATH] and hence the entropy is [MATH] which is appropriate for a description of the deconfinement phase in the gauge theory.', '1107.4048-1-25-0': 'By comparing ([REF]) and ([REF]), we see that at low temperatures (large [MATH]) the solitonic D4 solution dominates, while at high temperatures the black brane dominates.', '1107.4048-1-25-1': 'The transition between the two solutions, which we will call the ""Scherk-Schwarz (SS) transition"", occurs at [CITATION] [EQUATION]', '1107.4048-1-25-2': 'In the standard discussions of holographic QCD [CITATION], this transition is interpreted as the (continuation of the) confinement/deconfinement transition in the gauge theory.', '1107.4048-1-25-3': 'In the following sections, we will explain various problems with this interpretation and present a resolution by proposing an alternative scenario.', '1107.4048-1-26-0': '## Problems with the standard gravity analysis', '1107.4048-1-27-0': 'The previous subsection describes the usual relation between the confinement/deconfinement transition in 4 dimensional Yang-Mills theory and the SS transition in holographic QCD.', '1107.4048-1-27-1': 'However as we will now discuss, there are several problems associated with this correspondence.', '1107.4048-1-27-2': 'Some of these problems were mentioned earlier in [CITATION].', '1107.4048-1-28-0': 'In order to understand these problems, it is convenient to consider first the phase structure of the 5 dimensional SYM on the [MATH].', '1107.4048-1-28-1': 'See Figure [REF].', '1107.4048-1-28-2': 'As we have seen, the gravity description is valid in the strong coupling regime of this theory (the blue region in Figure [REF]) which is characterized by the solitonic D4 solution and the black D4 solution.', '1107.4048-1-28-3': 'On the other hand, the 4 dimensional Yang-Mills theory is realized at weak coupling ([MATH]) and low temperature ([MATH]) (the upper green region in Figure [REF]) which is characterized by the confinement phase and the deconfinement phase.', '1107.4048-1-28-4': 'The usual proposal is that the solitonic D4 and black D4 solutions correspond to the confinement and deconfinement phases respectively.', '1107.4048-1-29-0': 'To examine this proposal in detail, it is useful to consider the realization of the [MATH] centre symmetry in various phases of the five-dimensional SYM theory along [MATH] and [MATH].', '1107.4048-1-29-1': 'The [MATH] symmetry along [MATH] is characterized by the order parameter [MATH] which is the temporal Polyakov loop ([REF]).', '1107.4048-1-29-2': 'Similarly the [MATH] symmetry along [MATH] is characterized by the order parameter [MATH] which is the Polyakov loop around the [MATH]-cycle: [EQUATION]', '1107.4048-1-29-3': 'In Table [REF] we have collected values of [MATH] for the phases of 5D SYM discussed in this paper .', '1107.4048-1-30-0': 'Before testing the above proposal for holographic QCD in terms of these order parameters, let us see how to obtain this table.', '1107.4048-1-30-1': ""The 'confinement' phase denotes a phase of the 5D SYM whose KK compactification, at small enough [MATH], coincides with the usual confinement phase of YM4."", '1107.4048-1-30-2': 'The latter, clearly has [MATH].', '1107.4048-1-30-3': 'Now, the fact that we have obtained the 4D phase through a KK reduction, along [MATH], implies that the corresponding phase of the 5D theory must have [MATH].', '1107.4048-1-30-4': 'This follows from the fact [CITATION] that in a [MATH] phase, KK reduction does not work since the effective KK scale in a large [MATH] gauge theory becomes [MATH] in stead of the usual [MATH], and goes to zero at large [MATH] .', '1107.4048-1-30-5': ""Thus, the 'confinement' phase must have [MATH], as shown in the first row of Table [REF]."", '1107.4048-1-30-6': ""By similar arguments, the 'deconfinement' phase of 5D SYM which coincides with the deconfinement phase of YM4 must have [MATH] (for validity of the KK reduction) and [MATH] (to exhibit 4D deconfinement)."", '1107.4048-1-30-7': 'This is shown in the second row of Table [REF].', '1107.4048-1-31-0': 'The next two rows in Table [REF] describe two additional phases of SYM5, which appear only for the (AP,AP) b.c. via the [MATH] symmetry.', '1107.4048-1-31-1': 'The third row describes the [MATH] mirror of the confinement phase of YM4, which is characterized by [MATH] and [MATH].', '1107.4048-1-31-2': 'Since [MATH], this phase is a fully 5 dimensional object and is not related to the YM4 (indeed, so far as the Polyakov loop order parameters are concerned, this phase can be identified with a deconfinement phase of SYM5).', '1107.4048-1-31-3': 'Another phase is the mirror of the deconfinement phase in YM4, which is characterized by [MATH] and [MATH].', '1107.4048-1-31-4': 'Since this phase has the same order parameters as the original deconfinement phase, it is not obvious whether these two phases are distinct or smoothly connected through a cross over (we represent the possible phase boundary/crossover by the horizontal dotted line in Figure [REF]).', '1107.4048-1-32-0': 'To describe the next three rows in Table [REF], note that the order parameters [MATH] have a manifestation in gravity.', '1107.4048-1-32-1': ""Thus, e.g. if the [MATH] symmetry along [MATH] is preserved ([MATH]), then, due to the 'large [MATH] volume independence' mentioned in Section [REF], the free energy [MATH] should be independent of [MATH] (equivalently, the action should be linear in [MATH])."", '1107.4048-1-32-2': 'This is satisfied by the action in ([REF]).', '1107.4048-1-32-3': 'Hence for the parameter regime in which the solitonic D4 brane is the dominant phase of the theory, the 5 dim SYM dual exists in a phase characterized by [MATH].', '1107.4048-1-32-4': 'Similarly, the non-linear dependence of the action ([REF]) on [MATH] implies that the gauge theory dual has [MATH].', '1107.4048-1-32-5': 'These results are represented by the 5th row of Table [REF].', '1107.4048-1-32-6': 'One can similarly deduce by looking at the [MATH] dependence of ([REF]) that for the phase of the gauge theory represented by the black D4 solution, [MATH], which could also be inferred by the [MATH] symmetry.', '1107.4048-1-32-7': 'This is represented by the 6th row of Table [REF].', '1107.4048-1-32-8': 'In the last row, we have included a phase of SYM5 with (P, AP) fermion b.c. along ([MATH]) which corresponds to a localized D3 soliton, to be discussed in the next section.', '1107.4048-1-33-0': 'We now come back to the conventional proposal of holographic QCD discussed in the beginning of this section; let us define this proposal as a combination of three propositions:', '1107.4048-1-34-0': '(a) the solitonic D4 brane phase of gravity corresponds to the confinement phase of YM4,', '1107.4048-1-35-0': '(b) the black D4 corresponds to the deconfinement phase of YM4, and', '1107.4048-1-36-0': '(c) the SS transition between the two gravity solutions corresponds to the confinement/deconfinement transition of YM4.', '1107.4048-1-37-0': 'In terms of Table [REF], both the solitonic D4 and the confinement phase of YM4 (regarded as a phase of SYM) have [MATH]; hence both of these phases have the [MATH] symmetry and (a) can in principle be valid.', '1107.4048-1-37-1': 'However, (b) cannot be valid, since the black D4 phase ([MATH]) and the deconfinement phase of YM4 ([MATH]) have different [MATH] symmetry.', '1107.4048-1-37-2': 'In particular, the black D4 corresponds to a phase that cannot be KK reduced to YM4 at all because of vanishing [MATH].', '1107.4048-1-37-3': 'In fact, it would correspond to the [MATH] mirror of the confinement phase of YM4 (which can be identified with a deconfinement phase of SYM5 which is intrinsically 5 dimensional).', '1107.4048-1-37-4': 'Thus, there must be at least one phase boundary between the deconfinement phase of YM4 and the black D4 phase (see Figure [REF] for one possibility).', '1107.4048-1-37-5': 'Regarding (c), if the SS transition persists even at weak coupling, then it clearly cannot coincide with the confinement/deconfinement transition.', '1107.4048-1-37-6': 'The reason is that, even at weak coupling the SS transition can only occur at [MATH] (since even stringy effects should satisfy the [MATH] symmetry), whereas it is known that the confinement/deconfinement phase transition temperature of YM4 goes down ([MATH] goes up) at smaller values of [MATH] (see Figure [REF]).', '1107.4048-1-38-0': 'Through the arguments in this section, it is obvious that the black D4 brane solution does not correspond to the deconfinement phase in the 4 dimensional Yang-Mills theory and the SS transition does not correspond to the confinement/deconfinement transition.', '1107.4048-1-38-1': 'In the next section, we propose an alternative correspondence, which resolves these problems.', '1107.4048-1-39-0': '# Our proposal for the confinement/deconfinement transition in holographic QCD', '1107.4048-1-40-0': 'In the previous section, we considered the conventional picture of holographic QCD at finite temperature and discussed some of the problems in describing phases of YM4.', '1107.4048-1-40-1': 'In this section, we propose an alternative picture to resolve these problems.', '1107.4048-1-41-0': 'In the above discussion, we described finite temperature gauge theory by choosing AP b.c. for fermions on the brane along the compactified Euclidean time.', '1107.4048-1-41-1': 'However, let us recall that the gauge theory of interest here is pure Yang Mills theory in four dimensions, which does not have fermions.', '1107.4048-1-41-2': 'Fermions reappear when the validity condition ([REF]) is enforced and [MATH] goes above the KK scales [MATH].', '1107.4048-1-41-3': 'In this sense, fermions are an artifact of the holographic method and in the region of validity of the pure YM theory, the periodicity of the fermion should not affect the gauge theory results.', '1107.4048-1-42-0': 'If we start with the 5 dimensional SYM on [MATH] with a periodic temporal circle, the partition function is the twisted one: Tr [MATH], and is not the standard thermal partition function.', '1107.4048-1-42-1': 'However, under the limit [MATH] and [MATH], in which the 4 dimensional YM theory appears, the fermions are effectively decoupled; hence [MATH].', '1107.4048-1-42-2': ""Thus in this limit the 'twisted' partition becomes the usual thermal partition function, even with the periodic b.c. for the fermions: [EQUATION]"", '1107.4048-1-42-3': 'As a consequence, it is pertinent to study the phase structure of the 5 dimensional SYM on [MATH] with (P,AP) boundary condition through holography.', '1107.4048-1-42-4': 'Indeed, in this sector, we will find a phase transition in gravity, which continues naturally, at weak coupling, to the confinement/deconfinement transition in the 4D Yang-Mills theory.', '1107.4048-1-43-0': '## GL transition in the gravity theory with (P,AP) boundary condition', '1107.4048-1-44-0': 'In this section, we reconsider the gravity theory described in Section [REF], now compactified on [MATH] with (P,AP) b.c. for fermions.', '1107.4048-1-44-1': 'In this case, contrary to the (AP,AP) case, the black D4 brane solution ([REF]) is not allowed (see the comment below ([REF])).', '1107.4048-1-44-2': 'However a non-trivial phase boundary still exists, as we will see shortly.', '1107.4048-1-44-3': 'The detailed phase structure for this boundary condition is summarized in [CITATION] (with [MATH] and [MATH] exchanged); see also [CITATION].', '1107.4048-1-44-4': 'We will now outline the salient features.', '1107.4048-1-45-0': 'Let us start from large [MATH].', '1107.4048-1-45-1': 'In this region the solitonic D4 brane solution ([REF]) is thermodynamically dominant.', '1107.4048-1-45-2': 'As we decrease [MATH], at some point winding modes of the string wrapping on the [MATH]-cycle would be excited; we can estimate such a temperature as follows.', '1107.4048-1-45-3': 'Since the mass of the winding mode in the metric ([REF]) is given as [MATH], this winding mode would be relevant if [MATH], where [MATH] is defined above ([REF]) [CITATION].', '1107.4048-1-45-4': 'Thus, below the inverse temperature [EQUATION] winding modes would be excited and the gravity description given by ([REF]) would be invalid.', '1107.4048-1-45-5': 'In order to avoid this problem, we perform a T-duality along the [MATH]-cycle and go to the IIB frame, where the solitonic D4 solution becomes a solitonic D3 brane uniformly smeared on the dual [MATH]-cycle.', '1107.4048-1-45-6': 'The metric describing this solution is given by [EQUATION]', '1107.4048-1-45-7': 'Here [MATH] is the dual of [MATH], hence [MATH] has a periodicity [MATH].', '1107.4048-1-45-8': 'By considering winding modes on this metric, we can see that the gravity description is now valid if [MATH].', '1107.4048-1-45-9': 'Thus even when [MATH] is below ([REF]), the gravity analysis in the IIB frame is possible.', '1107.4048-1-46-0': 'As [MATH] decreases (hence the dual radius in the IIB frame increases), the uniformly smeared solitonic D3 brane becomes thermodynamically unstable at a certain temperature.', '1107.4048-1-46-1': 'This temperature, called the Gregory-Laflamme (GL) instability point, is numerically given by [CITATION]: [EQUATION]', '1107.4048-1-46-2': 'It is expected that, even before [MATH] is lowered all the way to [MATH], the smeared solitonic D3 brane becomes meta-stable and undergoes a first order Gregory-Laflamme (GL) transition at an inverse temperature [MATH], leading to a more stable, solitonic D3 brane localized on the dual cycle.', '1107.4048-1-46-3': 'This localized solution spontaneously breaks translation symmetry along the temporal direction.', '1107.4048-1-46-4': 'Although it is difficult to derive the critical temperature [MATH] precisely, we can approximately evaluate it as follows [CITATION].', '1107.4048-1-46-5': 'The metric of the localized solitonic D3 brane is approximately given by that of the solitonic D3 on [MATH] for a sufficiently large radius [MATH] of the dual cycle [CITATION], [EQUATION]', '1107.4048-1-46-6': 'Here [MATH] (see Eq. ([REF])).', '1107.4048-1-46-7': 'The value of [MATH] is determined by the smoothness condition (as in ([REF])).', '1107.4048-1-46-8': '[MATH] is regarded as a radial component of a spherical coordinate and is related to the cylindrical coordinates [MATH] of ([REF]) by roughly [MATH].', '1107.4048-1-46-9': 'Here we implicitly assume that the soliton is localized at [MATH].', '1107.4048-1-46-10': 'This approximation is valid if [MATH] and [MATH] are sufficiently small such that we can ignore the finite size effect associated with the dual circle [CITATION].', '1107.4048-1-47-0': 'The classical action of this solution turns out to be [EQUATION]', '1107.4048-1-47-1': 'By comparing this classical action with ([REF]) , we can estimate that the GL transition will happen around [EQUATION]', '1107.4048-1-47-2': 'Note that, near this critical point, [MATH] is similar in magnitude to, but less than, the dual radius [CITATION], so we expect ([REF])-([REF]) to receive some corrections from the finite size effect.', '1107.4048-1-48-0': 'In addition to these two solutions, ([REF]) and ([REF]), there is another solution: solitonic D3 brane non-uniformly smeared on the dual cycle.', '1107.4048-1-48-1': 'The metric of this solution is perturbatively derived in [CITATION].', '1107.4048-1-48-2': 'This non-uniform solution arises at the GL instability point and is always unstable.', '1107.4048-1-48-3': 'Although the complete behaviour of the non-uniform solution has not been explored, it is expected that it merges with the localized solitonic D3 as shown in Figure [REF] [CITATION].', '1107.4048-1-48-4': 'The fact that the GL instability temperature ([REF]) is higher than the approximately obtained GL critical temperature ([REF]) is consistent with this expectation.', '1107.4048-1-49-0': 'Let us make a few comments on the localized solitonic D3 solution ([REF]).', '1107.4048-1-49-1': 'Firstly, through a calculation similar to ([REF]), the gravity description can be shown to be valid (i.e. stringy effects can be ignored) if [MATH].', '1107.4048-1-49-2': 'The excitation of the winding modes can be ignored if [MATH] [CITATION].', '1107.4048-1-49-3': 'Secondly, this solution ceases to exist if [MATH] is too large, as shown in Figure [REF]; intuitively, if [MATH] were too large, the dual cycle would become smaller than the size of the localized soliton, which is not possible.', '1107.4048-1-49-4': 'Thirdly, the free energy [MATH] of this solution, unlike that of ([REF]), is proportional to temperature; hence the entropy of this solution is non-zero at [MATH] and the temporal [MATH] symmetry is broken.', '1107.4048-1-49-5': 'In addition, because of the non-trivial [MATH]-dependence of the classical action ([REF]) we can see that the [MATH] symmetry along the [MATH]-cycle is also broken.', '1107.4048-1-49-6': 'Thus the Polyakov loop [MATH] and [MATH] in this solution are both non-zero as shown in Table [REF] and, contrary to the black D4 solution, the localized solitonic D3 solution is appropriate for a description of the deconfinement phase in the dual gauge theory (which also has [MATH]).', '1107.4048-1-49-7': 'We will build on this observation in the next subsection.', '1107.4048-1-50-0': '## GL as a confinement/deconfinement transition', '1107.4048-1-51-0': 'The phase structure of the 5 dimensional SYM on [MATH] with the (P,AP) boundary condition is shown in Figure [REF].', '1107.4048-1-51-1': 'The strong coupling region (the blue region in Figure [REF]) described by type II supergravity and is characterized by the GL phase transition which occurs at a temperature given by ([REF]).', '1107.4048-1-51-2': 'In the weak coupling region (the green region in Figure [REF]), the 4 dimensional Yang-Mills is realized at low temperatures ([MATH]).', '1107.4048-1-51-3': 'Although this region is common to the (P,AP) phase diagram in Figure [REF]) and the (AP,AP) phase diagram in Figure [REF], the mirror of this region under the [MATH]) does not exist in Figure [REF]) since the (P,AP) b.c is not [MATH]-symmetric.', '1107.4048-1-52-0': 'The main point to emphasize here it that: contrary to the previous phase structure in the (AP,AP) case, now the localized solitonic D3 phase has the same order parameters ([MATH]) as the deconfinement phase, thus making it plausible that these two phases are smoothly connected.', '1107.4048-1-52-1': 'In Figure [REF], we have indicated this by assuming the simplest extrapolation through the region of the intermediate coupling.', '1107.4048-1-53-0': 'Our proposal', '1107.4048-1-54-0': 'In view of the above observations, we propose a strong coupling continuation of weakly coupled 4-dimensional Yang Mills theory as shown in Table [REF].', '1107.4048-1-55-0': 'Of course, our proposal is based on a simple extrapolation between the intermediate coupling regime, and the real story there could be more involved.', '1107.4048-1-55-1': 'However the mere existence of such a simple extrapolation is a significant improvement over the previous proposal in the (AP,AP) case, where we are certain that there has to be at least one phase boundary between the deconfinement phase and black D4 solution (for the simple reason that their order parameters have different values).', '1107.4048-1-55-2': 'It is clearly important, therefore, to further investigate the nature of the deconfinement phase through the solitonic D3 solution based on the correspondence outlined above.', '1107.4048-1-56-0': '# New correspondences in holographic QCD', '1107.4048-1-57-0': 'Using our proposed correspondence, we can explain anew several phenomena in the gauge theory from gravity.', '1107.4048-1-57-1': 'In this section, we list some of these phenomena.', '1107.4048-1-58-0': '## Polyakov loop and D3 brane distribution', '1107.4048-1-59-0': 'In our proposal, we identified the GL transition in the IIB frame to the confinement/deconfinement transition in the gauge theory.', '1107.4048-1-59-1': 'In the GL transition, the distribution of the D3 branes on the dual circle changes from a uniform distribution at low temperatures to a localized one at high temperatures.', '1107.4048-1-59-2': 'In this subsection, we explain what the corresponding phenomenon is in the gauge theory.', '1107.4048-1-60-0': 'Since the original five-dimensional gauge theory (of the D4 branes) appears in the IIA frame, in order to understand the role of the D3 branes, we need to consider a T-duality along the temporal direction.', '1107.4048-1-60-1': 'Under this T-duality, [MATH], which is transverse to the D3 brane, is mapped to the gauge potential [MATH] on the D4 brane.', '1107.4048-1-60-2': ""Thus the D3 brane distribution on the dual circle is related to a distribution of [MATH]'s, which are defined as the eigenvalues of the Polyakov loop operator [MATH]."", '1107.4048-1-61-0': 'If the D3 branes are uniformly distributed, the [MATH] are also uniformly distributed.', '1107.4048-1-61-1': 'In that case, by choosing an appropriate gauge, we can take [MATH].', '1107.4048-1-61-2': 'Then the temporal Polyakov loop operator ([REF]) becomes [EQUATION]', '1107.4048-1-61-3': 'Hence this distribution corresponds to the confinement phase.', '1107.4048-1-61-4': 'On the other hand, if the D3 branes are localized, then [MATH] are also localized and the Polyakov loop becomes non-zero, which characterizes the deconfinement phase.', '1107.4048-1-61-5': 'This observation is consistent with the entropy arguments in section [REF].', '1107.4048-1-62-0': 'The above discussion shows a direct relation between the D3 brane distribution and the eigenvalue distribution of the Polyakov loop operator.', '1107.4048-1-62-1': 'This latter quantity can sometimes be explicitly evaluated [CITATION].', '1107.4048-1-62-2': 'E.g. in [CITATION], for four dimensional Yang Mills theory on a small [MATH], we found that the uniform, non-uniform and localized distribution of [MATH], all appear, with obvious correspondence to similar gravitational solutions.', '1107.4048-1-62-3': 'Especially the free energies of these three solutions show a ""swallow tail"" relation similar to Figure [REF].', '1107.4048-1-62-4': '(See Figure 3 in [CITATION].)', '1107.4048-1-62-5': 'This correspondence strongly supports our proposal.', '1107.4048-1-63-0': '## GL transition as a Hagedorn transition', '1107.4048-1-64-0': 'Our proposal opens up the interesting possibility of a relation between the GL transition and the Hagedorn transition.', '1107.4048-1-65-0': 'From ([REF]), ([REF]) and ([REF]), we see that the GL transition in the IIB supergravity analysis happens near a temperature where the winding modes around the temporal cycle start becoming light in the IIA frame.', '1107.4048-1-65-1': 'This may indicate that the GL transition is associated with the excitation of the winding modes of the IIA string.', '1107.4048-1-65-2': 'Indeed, the KK modes of the graviton along the dual temporal circle, which cause the GL instability in the IIB description, are mapped to winding modes around on the temporal circle through the T-duality [CITATION].', '1107.4048-1-65-3': 'This is similar to the Hagedorn transition in string theory [CITATION], where the temporal winding modes cause the instability.', '1107.4048-1-65-4': 'Thus the GL transition in the IIB description might correspond to the Hagedorn transition in the IIA description.', '1107.4048-1-66-0': 'Note that on the large [MATH] gauge theory side also, the confinement/deconfinement transition has been shown to be related to the Hagedorn transition [CITATION].', '1107.4048-1-66-1': 'This makes it plausible that the Hagedorn transition in the Yang-Mills theory continues to the Hagedorn transition in the IIA string, which, as we argued above, is possibly the dual of the GL transition in the IIB supergravity.', '1107.4048-1-67-0': '# Chiral symmetry restoration in Sakai-Sugimoto model', '1107.4048-1-68-0': 'In the previous sections, we have seen that the conventional holographic representation of the confinement/deconfinement transition as the SS transition is fraught with problems, to circumvent which we proposed in Section [REF] a different interpretation in terms of a GL transition.', '1107.4048-1-68-1': 'However, the SS transition has widely been employed in holographic QCD and purports to explain several phenomena in real QCD.', '1107.4048-1-68-2': 'In particular, the chiral symmetry restoration in the Sakai-Sugimoto model was neatly explained in [CITATION].', '1107.4048-1-68-3': 'In their scenario the black D4 brane plays a crucial role.', '1107.4048-1-68-4': 'However, since in our proposal the black branes do not appear anymore, we need to find an alternative idea for realizing chiral symmetry restoration.', '1107.4048-1-68-5': 'In this section, we discuss how chiral symmetry restoration can happen in the localized solitonic D3 background.', '1107.4048-1-69-0': '## Sakai-Sugimoto model and chiral symmetry breaking', '1107.4048-1-70-0': 'The Sakai-Sugimoto model [CITATION] was proposed to describe low energy hadron physics in holographic QCD and elegantly reproduces many aspects of the real QCD.', '1107.4048-1-70-1': 'We first briefly review this model and show how chiral symmetry breaking at low temperatures is realized in terms of the dual gravity.', '1107.4048-1-71-0': 'The Sakai-Sugimoto model is an extension of the holographic model of QCD discussed in section [REF].', '1107.4048-1-71-1': 'Sakai and Sugimoto added, to the [MATH] D4 brane system, [MATH] D8 and [MATH] branes which are localized on the [MATH] circle and fill all other directions as follows [EQUATION]', '1107.4048-1-71-2': 'Here [MATH] denotes the compactified directions.', '1107.4048-1-71-3': '(See Figure [REF] (a).)', '1107.4048-1-71-4': 'This model has [MATH] gauge symmetry on the [MATH] D8 and [MATH] branes, which can be interpreted as a chiral [MATH] flavor symmetry in QCD.', '1107.4048-1-71-5': 'As we will see soon, in some situations, the D8 and [MATH] branes merge and the chiral symmetry is broken to a single [MATH].', '1107.4048-1-71-6': 'Sakai and Sugimoto proposed that this is the holographic realization of chiral symmetry breaking [CITATION].', '1107.4048-1-72-0': 'Let us take a large [MATH] limit a la Maldacena and, according to the principle of holography, replace the [MATH] D4 branes with a corresponding gravity solution.', '1107.4048-1-72-1': 'Here we will keep [MATH] such that we can ignore the back-reaction of the D8/[MATH] branes onto the background geometry (probe approximation).', '1107.4048-1-72-2': 'In that case, the background (D4 brane) geometry is determined thermodynamically as we have studied in section [REF] and [REF], and the D8/[MATH] brane configuration coupled to this background, is determined dynamically, with the given asymptotic ([MATH]) location of the D8 and [MATH] on the [MATH] as a boundary condition.', '1107.4048-1-73-0': 'Chiral symmetry breaking in this model happens as follows.', '1107.4048-1-73-1': 'At sufficiently low temperatures, the favoured geometric background is that of solitonic D4 solution ([REF]), as we have seen in section [REF] and [REF].', '1107.4048-1-73-2': 'On this background, the D8 and [MATH] cannot extend separately and need to merge as shown in Figure [REF] (b).', '1107.4048-1-73-3': 'As a result, the [MATH] gauge symmetry on the D8 and [MATH] is broken to [MATH], representing chiral symmetry breaking ([MATH]SB) in the dual gauge theory.', '1107.4048-1-74-0': '## Chiral symmetry restoration in the black D4 brane background', '1107.4048-1-75-0': 'In the gauge theory, it is expected that chiral symmetry is restored at a sufficiently high temperature.', '1107.4048-1-75-1': 'Thus if holographic QCD is to work, there should be a corresponding phenomena in the dual gravity description.', '1107.4048-1-76-0': 'In [CITATION], a mechanism for chiral symmetry restoration was suggested by considering D8 and [MATH] in the black D4 brane background ([REF]) in the (AP,AP) case.', '1107.4048-1-76-1': ""Contrary to the solitonic D4 brane geometry ([REF]), the [MATH] plane of the black D4 background has a 'cigar' geometry."", '1107.4048-1-76-2': 'Thus the D8 and [MATH] branes can wrap the cigar separately as shown in Figure [REF] (c).', '1107.4048-1-76-3': 'If such a configuration is energetically favoured, [MATH] gauge symmetry is preserved and chiral symmetry is restored.', '1107.4048-1-77-0': 'In addition to this configuration, another configuration shown in Figure [REF] (d) is possible in the black D4 background.', '1107.4048-1-77-1': 'This configuration is similar to Figure [REF] (b) and chiral symmetry is broken.', '1107.4048-1-77-2': 'In [CITATION], energies of these two configurations are compared and it was found that the configuration in (c) is favoured at a higher temperature.', '1107.4048-1-77-3': 'This is consistent with gauge theory.', '1107.4048-1-77-4': 'The configuration in (d) is favoured at a lower temperature (but higher than the SS critical temperature ([REF]) such that the favoured background geometry is black D4), only if the distance between D8 and [MATH] at [MATH] is smaller than a critical value.', '1107.4048-1-77-5': 'Note that the configuration (d) would correspond to a deconfinement and chiral symmetry breaking phase in the gauge theory.', '1107.4048-1-78-0': '## Chiral symmetry restoration in the localized solitonic D3 brane.', '1107.4048-1-79-0': 'Although the chiral symmetry restoration was explained, as above, in the black D4 brane background, our main thesis in this paper is that the black D4 solution itself is fraught with problems, if we interpret this solution as the deconfinement phase of the four dimensional gauge theory (see Section [REF]).', '1107.4048-1-79-1': 'The alternative we proposed in Section [REF] is that the localized solitonic D3 brane solution ([REF]) in the (P,AP) b.c. should be taken as the correct representation of the deconfinement phase in the dual gauge theory.', '1107.4048-1-79-2': 'In view of this, we need to understand chiral symmetry restoration in this solution instead of in the black D4 solution.', '1107.4048-1-80-0': 'One subtlety in the application of our proposal to the Sakai-Sugimoto model is the existence of the fundamental quarks from the open string between the D4 and D8/[MATH] brane.', '1107.4048-1-80-1': 'These quarks are, of course, not decoupled under the 4 dimensional limit [MATH].', '1107.4048-1-80-2': 'Thus we have to impose the AP b.c. on the temporal cycle to investigate its thermodynamics.', '1107.4048-1-80-3': 'This is a problem since we need to use the P b.c. in our proposal.', '1107.4048-1-80-4': 'However as far as we take [MATH] and ignore the back reaction of D8, we can ignore the contribution of the quarks to the D4 geometries.', '1107.4048-1-80-5': 'Thus we can use the P b.c. to investigate the background metric.', '1107.4048-1-80-6': 'In addition, when we determine the profile of the probe D8/[MATH] branes on the background geometry, only bosonic modes are relevant [CITATION] and fermions does not play any role.', '1107.4048-1-80-7': 'Therefore if our interest is only in the mechanism of the chiral symmetry restoration, the periodicity of the fermions must be irrelevant and we use the P b.c. to investigate it.', '1107.4048-1-81-0': 'In our proposal, the gravity analysis has been done in IIB supergravity by performing a T-duality along the [MATH]-cycle: [MATH].', '1107.4048-1-81-1': 'Thus it is convenient to dualize the brane configuration of the Sakai-Sugimoto model to the IIB frame.', '1107.4048-1-81-2': 'Since all the D branes in the Sakai-Sugimoto model wrap the [MATH], the above T-duality maps the D4 branes and D8/[MATH] branes to D3 and D7/[MATH] branes spreading [EQUATION]', '1107.4048-1-81-3': 'See Figure [REF] (e) also.', '1107.4048-1-81-4': 'The distribution along [MATH] of the D branes in the IIB description (related to the gauge field [MATH] in the IIA description), is determined dynamically.', '1107.4048-1-81-5': 'In the probe approximation, according to the analyses in section [REF], at temperatures below the GL critical temperature ([REF]), the D3 branes are distributed uniformly and, above it, they are localized on the [MATH].', '1107.4048-1-81-6': 'The fate of the chiral gauge group in the gravity representation depends on the stable configurations of the probe D7/[MATH] branes in these backgrounds.', '1107.4048-1-82-0': 'In the uniformly smeared solitonic D3 brane geometry ([REF]) shown in Figure [REF] (f), the situation is similar to (b) in Figure [REF].', '1107.4048-1-82-1': 'Since the [MATH] direction is smoothly pinched off at [MATH], the D7 and [MATH] have to merge and chiral symmetry is broken.', '1107.4048-1-83-0': 'On the other hand, in the localized solitonic D3 brane geometry, the horizontal direction is not fully pinched off.', '1107.4048-1-83-1': 'Recall that the geometry ([REF]) is pinched off at [MATH], where [MATH] and [MATH].', '1107.4048-1-83-2': 'Thus [MATH] can reach zero.', '1107.4048-1-83-3': 'As a result, D7 and [MATH] can extend separately as shown in Figure [REF] (g).', '1107.4048-1-83-4': 'This configuration is similar to (e) in Figure [REF] and would restore chiral symmetry.', '1107.4048-1-83-5': 'In addition to this configuration, a chiral symmetry broken configuration is also possible as shown in Figure [REF] (h).', '1107.4048-1-84-0': 'The last task is the evaluation of the stability of these two configurations at temperatures higher than [MATH] ([REF]).', '1107.4048-1-84-1': 'In the flat space, the force between a single D3 and single D7 (or between a single D3 and a single [MATH]) in our configuration is repulsive, since the number of the Neumann-Dirichlet open strings between them is 6 [CITATION].', '1107.4048-1-84-2': 'Thus we expect that even after taking the large [MATH] and near horizon limits, the force may be repulsive.', '1107.4048-1-84-3': 'As a result, the D7/[MATH] branes, which are separated from the localized SD3 in the [MATH]-direction would try to move away as far as possible in this direction.', '1107.4048-1-84-4': 'However, since the [MATH]-direction is compactified on a circle, the D7/[MATH] branes should then end as being fixed at the point on the [MATH]-circle which is antipodal to the localized SD3.', '1107.4048-1-84-5': 'E.g. if we put the localized SD3 brane at [MATH], the D7 as well as the [MATH] branes will both be at [MATH].', '1107.4048-1-84-6': 'See Figure [REF].', '1107.4048-1-84-7': 'In this case, we can effectively restrict the dynamics of these branes to ([MATH]) plane (given by [MATH]).', '1107.4048-1-84-8': 'Thus the problem reduces to finding classical solutions [MATH] with a boundary condition: [MATH]), where we put the D7 at [MATH] and [MATH] at [MATH].', '1107.4048-1-84-9': '( [MATH] is the distance between the D7 and [MATH] branes on the boundary and should be taken as [MATH]. )', '1107.4048-1-85-0': 'The problem stated above is difficult to solve precisely near the GL transition since the background metric ([REF]) is only an approximate description.', '1107.4048-1-85-1': 'The metric around the D7/[MATH] brane (at [MATH]) becomes more and more accurate, however, when [MATH].', '1107.4048-1-85-2': 'As a result, we solve for the stable configuration in this limit.', '1107.4048-1-85-3': 'Details of the calculation are presented in appendix [REF].', '1107.4048-1-85-4': 'We find two solutions corresponding to Figure [REF] (g) and (h), as shown in Figure [REF].', '1107.4048-1-85-5': 'In the appendix, we compare the classical actions of these two solutions numerically and observe that the [MATH] solution is favoured at higher temperature.', '1107.4048-1-85-6': 'Therefore, even in our proposal, similarly to the case of the black D4 [CITATION], we can explain chiral symmetry restoration at high enough temperatures.', '1107.4048-1-86-0': '# Conclusions', '1107.4048-1-87-0': 'In this paper, we showed that the conventional representation of the confinement/deconfinement transition in holographic QCD has several problems and proposed an alternative representation which resolves these problems.', '1107.4048-1-88-0': 'As mentioned earlier, problems similar to the above had also been encountered in the study of two dimensional bosonic gauge theory in [CITATION].', '1107.4048-1-88-1': 'This indicates that the issues addressed in this paper are rather general in the discussion of holography for non-supersymmetric gauge theories at finite temperatures.', '1107.4048-1-88-2': 'To elaborate, in the standard holographic procedure, a [MATH]-dimensional non-supersymmetric gauge theory is first constructed through the KK reduction of a [MATH]-dimensional super Yang-Mills theory on a Scherk-Schwarz circle.', '1107.4048-1-88-3': 'The [MATH]-dimensional SYM at large [MATH] can be mapped to a scaling limit of D[MATH] brane geometries [CITATION].', '1107.4048-1-88-4': 'At finite temperatures, because of the two compact cycles (temporal and Scherk-Schwarz), several distinct solutions (depending on boundary conditions) appear in gravity as shown in section [REF] and [REF]: solitonic D[MATH] (equivalently, uniformly smeared solitonic D[MATH]), localized solitonic D[MATH], and black D[MATH].', '1107.4048-1-88-5': 'The black D[MATH] brane solution appears at high temperatures in the (AP,AP) case, while the localized solitonic D[MATH] brane solution is the high temperature phase in the (P,AP) case.', '1107.4048-1-88-6': 'A table similar to Table [REF] can again be constructed, where the appropriate order parameters would be [MATH]; these would again appear to favour the localized solitonic D[MATH] phase as the more suitable representation of the deconfinement of [MATH]-dimensional YM theory (rather than the more conventional black D[MATH] phase which appears only in the (AP,AP) b.c.).', '1107.4048-1-88-7': 'Following this logic, a [MATH]-dimensional analogue of our proposal (see Section [REF]) would appear to give a better description of holographic QCD in [MATH] dimensions.', '1107.4048-1-88-8': 'In particular, we believe that the Gregory-Laflamme transition between the solitonic D[MATH] and localized solitonic D[MATH], would, as in this paper, be related to the confinement/deconfinement transition in the [MATH]-dimensional gauge theory.', '1107.4048-1-89-0': '## Further questions', '1107.4048-1-90-0': 'In order to further understand holographic QCD through the above proposal, it would be of interest to address the following questions.', '1107.4048-1-91-0': 'Transition temperature', '1107.4048-1-92-0': 'From ([REF]), the critical temperature of the confinement/deconfinement transition (the GL transition) would be [MATH].', '1107.4048-1-92-1': 'On the other hand, holographic QCD seems to predict that the square root of the QCD string tension is [MATH] whereas the glueball masses are [MATH] [CITATION].', '1107.4048-1-92-2': 'It would be important to understand the reason for the separation of these scales and how they evolve from strong coupling to weak coupling.', '1107.4048-1-93-0': 'Quantitative correspondence', '1107.4048-1-94-0': 'Although our new proposal reproduces the known qualitative features of 4 dimensional Yang-Mills theory, it does not automatically lead to a quantitative agreement.', '1107.4048-1-94-1': 'For example, the free energy of the deconfinement phase of the YM theory at a sufficiently high temperature must be proportional to [MATH], since the coupling becomes weak and the theory becomes approximately conformal.', '1107.4048-1-94-2': 'However the free energy of the localized solitonic D3 is proportional to [MATH] (as can be seen from ([REF])).', '1107.4048-1-94-3': 'This is not entirely surprising since the functional form of [MATH] can change as one evolves from weak coupling to strong coupling.', '1107.4048-1-94-4': 'Furthermore, one has to exercise caution in defining a high temperature limit of YM4 in the holographic context since the temperature must always remain much smaller than the KK scale.', '1107.4048-1-95-0': 'YM4 from SYM5 with (AP,AP) In this article, we have emphasized on the correspondence between YM4 and SYM5 with (P,AP) b.c..', '1107.4048-1-95-1': 'However, as we pointed out in footnote [REF], the 5 dimensional SYM with (AP,AP) b.c. should as well be related to YM4, since the boundary condition becomes irrelevant in the limit [MATH] and [MATH].', '1107.4048-1-95-2': 'A possible way this correspondence may work is as follows.', '1107.4048-1-95-3': 'In the (AP,AP) case, if we treat the black D4 solution (the blue region in the bottom half of Figure [REF]), which is not related to YM4, as irrelevant, and focus on the solitonic D4 brane, its winding modes would appear to be light around the temperature ([REF]).', '1107.4048-1-95-4': 'A Hagedorn transition parallel to the (P,AP) case might occur around here and might continue to the confinement/deconfinement transition in YM4.', '1107.4048-1-95-5': 'However, in order to investigate it further through gravity, we need to understand the gauge/gravity correspondence in the 0B frame as mentioned in footnote [REF].', '1107.4048-1-96-0': 'Real time', '1107.4048-1-97-0': 'It would be important to explore what geometry corresponds to the localized solitonic D3 brane ([REF]) in the real time formalism.', '1107.4048-1-97-1': 'An understanding of this would allow us to address dynamical properties of the deconfinement phase, e.g. transport properties.', '1107.4048-1-97-2': 'Since previous results in this area were based on the black brane solutions, it would be important to see how well-known results such as the viscosity bound [CITATION] can be derived in our proposal.', '1107.4048-1-98-0': 'AdS/CMT and chemical potential dependence', '1107.4048-1-99-0': 'The SS transition for D3 branes has been studied in the context of the AdS/CMT correspondence to investigate the superconductor/insulator transition in 2+1 dimension [CITATION].', '1107.4048-1-99-1': 'In these studies, a chemical potential for a [MATH] charge was introduced and the phase structure involving this chemical potential has been derived.', '1107.4048-1-99-2': 'It would be interesting to ask whether our proposal has any bearing on these studies, e.g., whether the GL transition analogous to the one discussed here can be a candidate for the superconductor/insulator or some other transition.', '1107.4048-1-99-3': 'A possible line of investigation could be to study the chemical potential dependence of [MATH] for the GL transition vis-a-vis that for the SS transition and see whether any qualitative differences appear.'}","{'1107.4048-2-0-0': 'We discuss the phase structure of [MATH] D4 branes wrapped on a temporal (Euclidean) and a spatial circle, in terms of the near-horizon AdS geometries.', '1107.4048-2-0-1': 'This system has been studied previously to understand four dimensional pure [MATH] Yang-Mills theory (YM4) through holography.', '1107.4048-2-0-2': 'In the usual treatment of the subject, the phase transition between the AdS soliton and the black D4 brane is interpreted as the strong coupling continuation of the confinement/deconfinement transition in YM4.', '1107.4048-2-0-3': 'We show that this interpretation is not valid, since the black D4 brane and the deconfinement phase of YM4 have different realizations of the [MATH] centre symmetry and cannot be identified.', '1107.4048-2-0-4': 'We propose an alternative gravity dual of the confinement/deconfinement transition in terms of a Gregory-Laflamme transition of the AdS soliton in the IIB frame, where the strong coupling continuation of the deconfinement phase of YM4 is a localized D3 soliton.', '1107.4048-2-0-5': 'Our proposal offers a new explanation of several aspects of the thermodynamics of holographic QCD.', '1107.4048-2-0-6': 'As an example, we show a new mechanism of chiral symmetry restoration in the Sakai-Sugimoto model.', '1107.4048-2-0-7': 'The issues discussed in this paper pertain to gravity duals of non-supersymmetric gauge theories in general.', '1107.4048-2-1-0': '# Introduction', '1107.4048-2-2-0': 'Strongly coupled field theories such as QCD and various condensed matter systems are not amenable to perturbative calculations and require numerical or other non-perturbative tools.', '1107.4048-2-2-1': 'A powerful new tool in this context is holography, which maps gauge theories to gravity in a higher dimension [CITATION].', '1107.4048-2-2-2': 'Holography, however, is most reliable and quantitative for special supersymmetric gauge theories.', '1107.4048-2-2-3': 'In the generic case, including for pure Yang Mills theories, inferences from gravity tend to be less quantitative due to various degrees of extrapolations, although such exercises have proved to be rather valuable, e.g., in the context of holographic QCD [CITATION].', '1107.4048-2-3-0': 'Recently, it was pointed out in [CITATION] that there are problems with a naive application of holography to a two dimensional large [MATH] bosonic gauge theory at finite temperature.', '1107.4048-2-3-1': 'The dual of this system is based on gravity backgrounds involving [MATH] D2 branes on [MATH].', '1107.4048-2-3-2': 'The phase diagram of gravity, interpreted naively, does not admit a unique continuation to the regime of gauge theory, owing to its dependence on the boundary condition for fermions on the brane.', '1107.4048-2-3-3': 'However, it was shown in [CITATION] that the phase structures in gravity and gauge theory could be smoothly connected through an appropriate choice of fermion boundary conditions.', '1107.4048-2-4-0': 'In this paper, we will show that similar problems arise when we holographically analyse more general non-supersymmetric gauge theories at finite temperature.', '1107.4048-2-4-1': 'The particular example we will focus on is that of holographic QCD from D4 branes [CITATION].', '1107.4048-2-4-2': 'We will discuss some problems with the usual correspondence between the confinement/deconfinement transition in QCD and the Scherk-Schwarz transition between an AdS soliton and a black D4 brane in the gravity dual.', '1107.4048-2-4-3': 'Some of these problems were first discussed in [CITATION] (see section [REF] for further details).', '1107.4048-2-4-4': 'Especially we will show that the black D4 brane cannot be identified with the (strong coupling continuation of the) deconfinement phase in QCD in four dimensions.', '1107.4048-2-4-5': 'As a resolution of these problems, we will propose an alternative scenario in which the confinement/deconfinement transition corresponds to a Gregory-Laflamme transition [CITATION] between a uniformly distributed AdS soliton and a localized AdS soliton in the IIB frame.', '1107.4048-2-5-0': 'The scenario we propose suggests that we need to reconsider several previous results in holographic QCD including the Sakai-Sugimoto model [CITATION].', '1107.4048-2-5-1': 'One important ingredient in the Sakai-Sugimoto model is the mechanism of chiral symmetry restoration at high temperatures [CITATION].', '1107.4048-2-5-2': 'We will propose a new mechanism for chiral symmetry restoration in our framework.', '1107.4048-2-6-0': 'The plan of the paper is as follows.', '1107.4048-2-7-0': 'Section [REF] contains a short review of finite temperature QCD and the holographic approach to four dimensional Yang Mills theory (YM4) using D4 branes.', '1107.4048-2-7-1': ""In Section [REF] we discuss the gravity theory at a finite temperature and a 'Scherk-Schwarz' transition between a solitonic D4 brane and a black D4 brane."", '1107.4048-2-7-2': 'In Section [REF] we recall the conventional correspondence between this transition and the confinement/deconfinement transition of the Yang Mills theory, and discuss problems with this correspondence, with special emphasis on the mismatch of [MATH] centre symmetry between the black D4 phase and the deconfined 4 dimensional YM theory.', '1107.4048-2-7-3': 'In Section [REF], we discuss the phase structure of the D4 branes using an unconventional (periodic) fermion boundary condition along the Euclidean time circle and show that the high temperature in gravity is described by a localized solitonic D3 brane (or its T-dual) whose centre symmetry precisely matches with the deconfinement phase of four dimensional YM theory.', '1107.4048-2-7-4': 'This leads us, in Section [REF], to propose a new representation of the confinement/deconfinement phase transition in terms of a GL transition between the solitonic D4 and the (T-dual of) the localized solitonic D3 (see Table [REF]).', '1107.4048-2-7-5': 'In Section [REF], we show some new correspondences between phenomena in QCD and their counterparts in gravity following our proposal.', '1107.4048-2-7-6': 'In Section [REF], we suggest a new mechanism of chiral symmetry restoration in the Sakai-Sugimoto model in keeping with our proposal.', '1107.4048-2-7-7': 'Section [REF] contains the concluding remarks and some open problems.', '1107.4048-2-8-0': '# Review of QCD and its gravity dual', '1107.4048-2-9-0': '## Finite temperature QCD', '1107.4048-2-10-0': 'In this section, we briefly review some salient properties of four dimensional [MATH] pure Yang-Mills theory at a finite temperature at large [MATH], a system which we will subsequently investigate through holography.', '1107.4048-2-11-0': 'An important symmetry in this theory is the [MATH] symmetry along the temporal cycle, which is the centre of the [MATH] symmetry.', '1107.4048-2-11-1': 'The order parameter of this symmetry is the temporal Polyakov loop operator [EQUATION]', '1107.4048-2-11-2': 'If this operator is zero, the [MATH] symmetry is preserved and, if not, it is broken.', '1107.4048-2-11-3': ""Especially, if the [MATH] symmetry is preserved, physical quantities do not depend on the temporal radius [MATH] at [MATH] order due to 'large [MATH] volume independence' [CITATION]."", '1107.4048-2-11-4': 'If the gauge theory is on a torus [MATH], the large [MATH] volume independence is generalized and physical quantities do not depend on [MATH], if the [MATH] symmetry along the [MATH]-th direction is preserved [CITATION].', '1107.4048-2-12-0': 'Let us consider the phase structure of the Yang-Mills theory.', '1107.4048-2-12-1': 'At low temperatures, the confinement phase is the dominant thermodynamic phase.', '1107.4048-2-12-2': 'In this phase, the [MATH] symmetry is preserved ([MATH]).', '1107.4048-2-12-3': 'At high temperatures, the deconfinement phase, in which the [MATH] symmetry is broken ([MATH]), dominates.', '1107.4048-2-12-4': 'There is a confinement/deconfinement transition at an intermediate temperature.', '1107.4048-2-12-5': 'From studies in large [MATH] lattice gauge theories, this is expected to be a first order phase transition [CITATION].', '1107.4048-2-13-0': ""As we remarked above, the free energy in the confinement phase does not depend on the temperature through the 'large [MATH] volume independence'."", '1107.4048-2-13-1': 'It implies that the entropy becomes zero at [MATH].', '1107.4048-2-13-2': 'On the other hand, the entropy in the deconfinement phase would be [MATH].', '1107.4048-2-13-3': 'This has a simple physical interpretation: in the confinement phase, the spectrum excludes gluon states, which have [MATH] degree of the freedom, and consists only of gauge singlet states like glueballs, leading to an [MATH] entropy.', '1107.4048-2-14-0': '## Holographic QCD', '1107.4048-2-15-0': 'In this section, we will review the construction of four dimensional [MATH] pure Yang-Mills theory from [MATH] D4 branes [CITATION].', '1107.4048-2-15-1': 'Let us first consider a 10 dimensional Euclidean spacetime with an [MATH] and consider D4 branes wrapping on the [MATH].', '1107.4048-2-15-2': 'We define the coordinate along this [MATH] as [MATH] and its periodicity as [MATH].', '1107.4048-2-15-3': 'The effective theory on this brane is a 5 dimensional supersymmetric Yang-Mills theory on the [MATH].', '1107.4048-2-15-4': 'For the fermions on the brane, the boundary condition along the circle can be AP (antiperiodic) or P (periodic); to specify the theory, we must pick one of these two boundary conditions.', '1107.4048-2-15-5': 'Let us take the AP boundary condition.', '1107.4048-2-15-6': 'This gives rise to fermion masses proportional to the Kaluza-Klein scale [MATH], leading to supersymmetry breaking (this is called the SS - Scherk-Schwarz- mechanism).', '1107.4048-2-15-7': 'This, in turn, induces masses for the adjoint scalars and for [MATH], which are proportional to [MATH] at one-loop.', '1107.4048-2-15-8': 'Therefore, if [MATH] is sufficiently small and the dynamical scale [MATH] is much less than both the above mass scales, then the fermions and adjoint scalars are decoupled and the 5 dimensional supersymmetric Yang-Mills theory is reduced to a four dimensional pure Yang-Mills theory.', '1107.4048-2-16-0': 'By taking the large [MATH] limit of this system a la Maldacena at low temperatures, we obtain the dual gravity description of the compactified 5 dimensional SYM theory [CITATION], which consists, at low temperatures, of a solitonic D[MATH] brane solution wrapping the [MATH].', '1107.4048-2-16-1': 'The explicit metric is given by [EQUATION]', '1107.4048-2-16-2': 'There is also a non-trivial value of the five form potential which we do not write explicitly.', '1107.4048-2-16-3': 'Here [MATH] is the YM coupling on the D[MATH] world-volume and [MATH] is found by putting [MATH] in the general formula [EQUATION]', '1107.4048-2-16-4': 'Since the [MATH]-cycle shrinks to zero at [MATH], in order to avoid possible conical singularities we must choose the asymptotic radius [MATH] as follows [EQUATION]', '1107.4048-2-16-5': ""With this choice, the contractible [MATH]-cycle, together with the radial direction [MATH], forms a so-called 'cigar' geometry which is topologically a disc."", '1107.4048-2-16-6': 'In order that the fermions are well-defined on this geometry, they must obey the AP boundary condition.', '1107.4048-2-16-7': 'The AP boundary condition along [MATH] is, of course, consistent with the choice of fermion b.c. in the boundary theory.', '1107.4048-2-17-0': 'The leading order gravity solution described above is not always valid.', '1107.4048-2-17-1': 'E.g. in order that the stringy modes can be ignored, we should ensure that the curvature in string units must be small.', '1107.4048-2-17-2': 'In other words, the typical length scale of this solution near [MATH], viz, [MATH], must satisfy [MATH] [CITATION].', '1107.4048-2-17-3': 'This condition turns out to be equivalent to [EQUATION]', '1107.4048-2-17-4': 'This is opposite to the condition which we found for the validity of the four-dimensional gauge theory description (see footnote [REF]).', '1107.4048-2-17-5': 'Thus, this gravity solution can describe the 5 dimensional SYM but cannot directly describe the 4 dimensional YM theory, inferences about which can only be obtained through extrapolation.', '1107.4048-2-17-6': 'This is a common problem in the construction of holographic duals of non-supersymmetric gauge theories.', '1107.4048-2-17-7': 'This has been discussed at length; in particular, the gravity description, which necessitates extrapolation to strong coupling, has been likened (cf. [CITATION] and [CITATION], p. 196-197) to strong coupling lattice gauge theory.', '1107.4048-2-17-8': 'Many interesting results, including the qualitative predictions in [CITATION], have been obtained using this prescription.', '1107.4048-2-18-0': 'Let us explore some properties of the soliton solution ([REF]).', '1107.4048-2-18-1': 'This solution is expected, on the basis of several arguments, to correspond to the confinement phase in the four dimensional gauge theory.', '1107.4048-2-18-2': 'For example, it can be shown that the classical action of this solution at a finite temperature [MATH] is [CITATION] [EQUATION] where [MATH] is the volume of the three spatial dimensions ([MATH]) and [MATH] is a constant which is obtainable from the formula below [CITATION] by putting [MATH] [EQUATION]', '1107.4048-2-18-3': 'Thus the free energy of this solution, [MATH], is independent of temperature, and hence the entropy is zero at [MATH] order.', '1107.4048-2-18-4': 'These facts are consistent with the interpretation of this solution as the confinement phase in the large [MATH] gauge theory, in which the temporal [MATH] symmetry is preserved.', '1107.4048-2-19-0': 'Note that the action ([REF]) divided by [MATH] does depend on [MATH], implying that the [MATH] symmetry along the [MATH]-cycle is broken.', '1107.4048-2-19-1': 'Although this observation might appear irrelevant for the 4 dimensional Yang-Mills theory, it is this symmetry breaking which allows a conventional KK reduction to four dimensions (see the next two sections for more details).', '1107.4048-2-20-0': '# Finite temperature holographic QCD', '1107.4048-2-21-0': '## Confinement/deconfinement transition in Holographic QCD in the standard scenario', '1107.4048-2-22-0': 'In existing holographic studies of the thermodynamics in the Yang-Mills theory, it is assumed that the black D4 brane solution corresponds to the deconfinement phase and the Scherk-Schwarz transition in the gravity is related to the confinement/deconfinement transition.', '1107.4048-2-22-1': 'In this section, we review this scenario and we will explain its problems in section [REF].', '1107.4048-2-23-0': 'To discuss holographic QCD at finite temperatures, we begin by compactifying Euclidean time in the boundary theory on a circle with periodicity [MATH].', '1107.4048-2-23-1': 'To permit description in terms of four dimensional gauge theory, the temperature should, of course, be kept far below the KK scale, [MATH] .', '1107.4048-2-23-2': 'In order to determine the gravitational theory, we need to fix the periodicity of fermions in the gauge theory along the time cycle.', '1107.4048-2-23-3': 'In the existing literature, the fermion boundary condition is chosen to be AP according to the usual practice for thermal fermions (we will see in section [REF] that this choice is not imperative here).', '1107.4048-2-23-4': ""The resulting fermionic bc's: (AP, AP) along [MATH] lead to a [MATH] symmetry of the system under [MATH]."", '1107.4048-2-23-5': 'In the gravity dual, we must therefore, include the [MATH]-transform of the solution ([REF]), which is a black D4 solution: [EQUATION]', '1107.4048-2-23-6': 'Here [MATH] in this metric is related to [MATH] as in ([REF]) [EQUATION]', '1107.4048-2-23-7': 'This solution has a contractible [MATH]-cycle and the fermions must obey the anti-periodic boundary condition (see footnote [REF]), consistent with the AP b.c. in the gauge theory.', '1107.4048-2-23-8': 'Analogously to ([REF]), the gravity description of this solution is reliable if [MATH].', '1107.4048-2-24-0': 'The classical action density can be evaluated, as before, yielding [EQUATION]', '1107.4048-2-24-1': 'In contrast with ([REF]), the free energy now is a function of [MATH] and hence the entropy is [MATH] which is appropriate for a description of the deconfinement phase in the gauge theory.', '1107.4048-2-25-0': 'By comparing ([REF]) and ([REF]), we see that at low temperatures (large [MATH]) the solitonic D4 solution dominates, while at high temperatures the black brane dominates.', '1107.4048-2-25-1': 'The transition between the two solutions, which we will call the ""Scherk-Schwarz (SS) transition"", occurs at [CITATION] [EQUATION]', '1107.4048-2-25-2': 'In the standard discussions of holographic QCD [CITATION], this transition is interpreted as the (continuation of the) confinement/deconfinement transition in the gauge theory.', '1107.4048-2-25-3': 'In the following sections, we will explain various problems with this interpretation and present a resolution by proposing an alternative scenario.', '1107.4048-2-26-0': '## Problems with the standard gravity analysis', '1107.4048-2-27-0': 'The previous subsection describes the usual relation between the confinement/deconfinement transition in 4 dimensional Yang-Mills theory and the SS transition in holographic QCD.', '1107.4048-2-27-1': 'However as we will now discuss, there are several problems associated with this correspondence.', '1107.4048-2-27-2': 'Some of these problems were mentioned earlier in [CITATION].', '1107.4048-2-28-0': 'In order to understand these problems, it is convenient to consider first the phase structure of the 5 dimensional SYM on the [MATH].', '1107.4048-2-28-1': 'See Figure [REF].', '1107.4048-2-28-2': 'As we have seen, the gravity description is valid in the strong coupling regime of this theory (the blue region in Figure [REF]) which is characterized by the solitonic D4 solution and the black D4 solution.', '1107.4048-2-28-3': 'On the other hand, the 4 dimensional Yang-Mills theory is realized at weak coupling ([MATH]) and low temperature ([MATH]) (the upper green region in Figure [REF]) which is characterized by the confinement phase and the deconfinement phase.', '1107.4048-2-28-4': 'The usual proposal is that the solitonic D4 and black D4 solutions correspond to the confinement and deconfinement phases respectively.', '1107.4048-2-29-0': 'To examine this proposal in detail, it is useful to consider the realization of the [MATH] centre symmetry in various phases of the five-dimensional SYM theory along [MATH] and [MATH].', '1107.4048-2-29-1': 'The [MATH] symmetry along [MATH] is characterized by the order parameter [MATH] which is the temporal Polyakov loop ([REF]).', '1107.4048-2-29-2': 'Similarly the [MATH] symmetry along [MATH] is characterized by the order parameter [MATH] which is the Polyakov loop around the [MATH]-cycle: [EQUATION]', '1107.4048-2-29-3': 'In Table [REF] we have collected values of [MATH] for the phases of 5D SYM discussed in this paper.', '1107.4048-2-30-0': 'Before testing the above proposal for holographic QCD in terms of these order parameters, let us see how to obtain this table.', '1107.4048-2-30-1': ""The 'confinement' phase denotes a phase of the 5D SYM whose KK compactification, at small enough [MATH], coincides with the usual confinement phase of YM4."", '1107.4048-2-30-2': 'The latter, clearly has [MATH].', '1107.4048-2-30-3': 'Now, the fact that we have obtained the 4D phase through a KK reduction, along [MATH], implies that the corresponding phase of the 5D theory must have [MATH].', '1107.4048-2-30-4': 'This follows from the fact [CITATION] that in a [MATH] phase, KK reduction does not work since the effective KK scale in a large [MATH] gauge theory becomes [MATH] in stead of the usual [MATH], and goes to zero at large [MATH] .', '1107.4048-2-30-5': ""Thus, the 'confinement' phase must have [MATH], as shown in the first row of Table [REF]."", '1107.4048-2-30-6': ""By similar arguments, the 'deconfinement' phase of 5D SYM which coincides with the deconfinement phase of YM4 must have [MATH] (for validity of the KK reduction) and [MATH] (to exhibit 4D deconfinement)."", '1107.4048-2-30-7': 'This is shown in the second row of Table [REF].', '1107.4048-2-31-0': 'The next two rows in Table [REF] describe two additional phases of SYM5, which appear only for the (AP,AP) b.c. via the [MATH] symmetry.', '1107.4048-2-31-1': 'The third row describes the [MATH] mirror of the confinement phase of YM4, which is characterized by [MATH] and [MATH].', '1107.4048-2-31-2': 'Since [MATH], this phase is a fully 5 dimensional object and is not related to the YM4 (indeed, so far as the Polyakov loop order parameters are concerned, this phase can be identified with a deconfinement phase of SYM5).', '1107.4048-2-31-3': 'Another phase is the mirror of the deconfinement phase in YM4, which is characterized by [MATH] and [MATH].', '1107.4048-2-31-4': 'Since this phase has the same order parameters as the original deconfinement phase, it is not obvious whether these two phases are distinct or smoothly connected through a cross over (we represent the possible phase boundary/crossover by the horizontal dotted line in Figure [REF]).', '1107.4048-2-32-0': 'To describe the next three rows in Table [REF], note that the order parameters [MATH] have a manifestation in gravity.', '1107.4048-2-32-1': ""Thus, e.g. if the [MATH] symmetry along [MATH] is preserved ([MATH]), then, due to the 'large [MATH] volume independence' mentioned in Section [REF], the free energy [MATH] should be independent of [MATH] (equivalently, the action should be linear in [MATH])."", '1107.4048-2-32-2': 'This is satisfied by the action in ([REF]).', '1107.4048-2-32-3': 'Hence for the parameter regime in which the solitonic D4 brane is the dominant phase of the theory, the 5 dim SYM dual exists in a phase characterized by [MATH].', '1107.4048-2-32-4': 'Similarly, the non-linear dependence of the action ([REF]) on [MATH] implies that the gauge theory dual has [MATH].', '1107.4048-2-32-5': 'These results are represented by the 5th row of Table [REF].', '1107.4048-2-32-6': 'One can similarly deduce by looking at the [MATH] dependence of ([REF]) that for the phase of the gauge theory represented by the black D4 solution, [MATH], which could also be inferred by the [MATH] symmetry.', '1107.4048-2-32-7': 'This is represented by the 6th row of Table [REF].', '1107.4048-2-32-8': 'In the last row, we have included a phase of SYM5 with (P, AP) fermion b.c. along ([MATH]) which corresponds to a localized D3 soliton, to be discussed in the next section.', '1107.4048-2-33-0': 'We now come back to the conventional proposal of holographic QCD discussed in the beginning of this section; let us define this proposal as a combination of three propositions:', '1107.4048-2-34-0': '(a) the solitonic D4 brane phase of gravity corresponds to the confinement phase of YM4,', '1107.4048-2-35-0': '(b) the black D4 corresponds to the deconfinement phase of YM4, and', '1107.4048-2-36-0': '(c) the SS transition between the two gravity solutions corresponds to the confinement/deconfinement transition of YM4.', '1107.4048-2-37-0': 'In terms of Table [REF], both the solitonic D4 and the confinement phase of YM4 (regarded as a phase of SYM) have [MATH]; hence both of these phases have the same [MATH] symmetry and (a) can in principle be valid.', '1107.4048-2-37-1': 'However, (b) cannot be valid, since the black D4 phase ([MATH]) and the deconfinement phase of YM4 ([MATH]) have different [MATH] symmetry.', '1107.4048-2-37-2': 'In particular, the black D4 corresponds to a phase that cannot be KK reduced to YM4 at all because of vanishing [MATH].', '1107.4048-2-37-3': 'In fact, it would correspond to the [MATH] mirror of the confinement phase of YM4 (which can be identified with a deconfinement phase of SYM5 which is intrinsically 5 dimensional).', '1107.4048-2-37-4': 'Thus, there must be at least one phase boundary between the deconfinement phase of YM4 and the black D4 phase (see Figure [REF] for one possibility).', '1107.4048-2-37-5': 'Regarding (c), if the SS transition persists even at weak coupling, then it clearly cannot coincide with the confinement/deconfinement transition.', '1107.4048-2-37-6': 'The reason is that, even at weak coupling the SS transition can only occur at [MATH] (since even stringy effects should satisfy the [MATH] symmetry), whereas it is known that the confinement/deconfinement phase transition temperature of YM4 goes down ([MATH] goes up) at smaller values of [MATH] (see Figure [REF]).', '1107.4048-2-38-0': 'Through the arguments in this section, it is obvious that the black D4 brane solution does not correspond to the deconfinement phase in the 4 dimensional Yang-Mills theory and the SS transition does not correspond to the confinement/deconfinement transition.', '1107.4048-2-38-1': 'In the next section, we propose an alternative correspondence, which resolves these problems.', '1107.4048-2-39-0': '# Our proposal for the confinement/deconfinement transition in holographic QCD', '1107.4048-2-40-0': 'In the previous section, we considered the conventional picture of holographic QCD at finite temperature and discussed some of the problems in describing phases of YM4.', '1107.4048-2-40-1': 'In this section, we propose an alternative picture to resolve these problems.', '1107.4048-2-41-0': 'In the above discussion, we described finite temperature gauge theory by choosing AP b.c. for fermions on the brane along the compactified Euclidean time.', '1107.4048-2-41-1': 'However, let us recall that the gauge theory of interest here is pure Yang Mills theory in four dimensions, which does not have fermions.', '1107.4048-2-41-2': 'Fermions reappear when the validity condition ([REF]) is enforced and [MATH] goes above the KK scales [MATH].', '1107.4048-2-41-3': 'In this sense, fermions are an artifact of the holographic method and in the region of validity of the pure YM theory, the periodicity of the fermion should not affect the gauge theory results.', '1107.4048-2-42-0': 'If we start with the 5 dimensional SYM on [MATH] with a periodic temporal circle, the partition function is the twisted one: Tr [MATH], and is not the standard thermal partition function.', '1107.4048-2-42-1': 'However, under the limit [MATH] and [MATH], in which the 4 dimensional YM theory appears, the fermions are effectively decoupled; hence [MATH].', '1107.4048-2-42-2': ""Thus in this limit the 'twisted' partition becomes the usual thermal partition function, even with the periodic b.c. for the fermions: [EQUATION]"", '1107.4048-2-42-3': 'As a consequence, it is pertinent to study the phase structure of the 5 dimensional SYM on [MATH] with (P,AP) boundary condition through holography.', '1107.4048-2-42-4': 'Indeed, in this sector, we will find a phase transition in gravity, which continues naturally, at weak coupling, to the confinement/deconfinement transition in the 4D Yang-Mills theory.', '1107.4048-2-43-0': '## Gregory-Laflamme transition in the gravity theory with (P,AP) boundary condition', '1107.4048-2-44-0': 'In this section, we reconsider the gravity theory described in Section [REF], now compactified on [MATH] with (P,AP) b.c. for fermions.', '1107.4048-2-44-1': 'In this case, contrary to the (AP,AP) case, the black D4 brane solution ([REF]) is not allowed (see the comment below ([REF])).', '1107.4048-2-44-2': 'However a non-trivial phase boundary still exists, as we will see shortly.', '1107.4048-2-44-3': 'The detailed phase structure for this boundary condition is summarized in [CITATION] (with [MATH] and [MATH] exchanged); see also [CITATION].', '1107.4048-2-44-4': 'We will now outline the salient features.', '1107.4048-2-45-0': 'Let us start from large [MATH].', '1107.4048-2-45-1': 'In this region the solitonic D4 brane solution ([REF]) is thermodynamically dominant.', '1107.4048-2-45-2': 'As we decrease [MATH], at some point winding modes of the string wrapping on the [MATH]-cycle would be excited; we can estimate such a temperature as follows.', '1107.4048-2-45-3': 'Since the mass of the winding mode in the metric ([REF]) is given as [MATH], this winding mode would be relevant if [MATH], where [MATH] is defined above ([REF]) [CITATION].', '1107.4048-2-45-4': 'Thus, below the inverse temperature [EQUATION] winding modes would be excited and the gravity description given by ([REF]) would be invalid.', '1107.4048-2-45-5': 'In order to avoid this problem, we perform a T-duality along the [MATH]-cycle and go to the IIB frame, where the solitonic D4 solution becomes a solitonic D3 brane uniformly smeared on the dual [MATH]-cycle.', '1107.4048-2-45-6': 'The metric describing this solution is given by [EQUATION]', '1107.4048-2-45-7': 'Here [MATH] is the dual of [MATH], hence [MATH] has a periodicity [MATH].', '1107.4048-2-45-8': 'By considering winding modes on this metric, we can see that the gravity description is now valid if [MATH].', '1107.4048-2-45-9': 'Thus even when [MATH] is below ([REF]), the gravity analysis in the IIB frame is possible.', '1107.4048-2-46-0': 'As [MATH] decreases (hence the dual radius in the IIB frame increases), the uniformly smeared solitonic D3 brane becomes thermodynamically unstable at a certain temperature.', '1107.4048-2-46-1': 'This temperature, called the Gregory-Laflamme (GL) instability point, is numerically given by [CITATION]: [EQUATION]', '1107.4048-2-46-2': 'It is expected that, even before [MATH] is lowered all the way to [MATH], the smeared solitonic D3 brane becomes meta-stable and undergoes a first order Gregory-Laflamme (GL) transition at an inverse temperature [MATH], leading to a more stable, solitonic D3 brane localized on the dual cycle.', '1107.4048-2-46-3': 'This localized solution spontaneously breaks translation symmetry along the temporal direction.', '1107.4048-2-46-4': 'Although it is difficult to derive the critical temperature [MATH] precisely, we can approximately evaluate it as follows [CITATION].', '1107.4048-2-46-5': 'The metric of the localized solitonic D3 brane is approximately given by that of the solitonic D3 on [MATH] for a sufficiently large radius [MATH] of the dual cycle [CITATION], [EQUATION]', '1107.4048-2-46-6': 'Here [MATH] (see Eq. ([REF])).', '1107.4048-2-46-7': 'The value of [MATH] is determined by the smoothness condition (as in ([REF])).', '1107.4048-2-46-8': '[MATH] is regarded as a radial component of a spherical coordinate and is related to the cylindrical coordinates [MATH] of ([REF]) by roughly [MATH].', '1107.4048-2-46-9': 'Here we implicitly assume that the soliton is localized at [MATH].', '1107.4048-2-46-10': 'This approximation is valid if [MATH] and [MATH] are sufficiently small such that we can ignore the finite size effect associated with the dual circle [CITATION].', '1107.4048-2-47-0': 'The classical action of this solution turns out to be [EQUATION]', '1107.4048-2-47-1': 'By comparing this classical action with ([REF]) , we can estimate that the GL transition will happen around [EQUATION]', '1107.4048-2-47-2': 'Note that, near this critical point, [MATH] is similar in magnitude to, but less than, the dual radius [CITATION], so we expect ([REF])-([REF]) to receive some corrections from the finite size effect.', '1107.4048-2-48-0': 'In addition to these two solutions, ([REF]) and ([REF]), there is another solution: solitonic D3 brane non-uniformly smeared on the dual cycle.', '1107.4048-2-48-1': 'The metric of this solution is perturbatively derived in [CITATION].', '1107.4048-2-48-2': 'This non-uniform solution arises at the GL instability point and is always unstable.', '1107.4048-2-48-3': 'Although the complete behaviour of the non-uniform solution has not been explored, it is expected that it merges with the localized solitonic D3 as shown in Figure [REF] [CITATION].', '1107.4048-2-48-4': 'The fact that the GL instability temperature ([REF]) is higher than the approximately obtained GL critical temperature ([REF]) is consistent with this expectation.', '1107.4048-2-49-0': 'Let us make a few comments on the localized solitonic D3 solution ([REF]).', '1107.4048-2-49-1': 'Firstly, through a calculation similar to ([REF]), the gravity description can be shown to be valid (i.e. stringy effects can be ignored) if [MATH].', '1107.4048-2-49-2': 'The excitation of the winding modes can be ignored if [MATH] [CITATION].', '1107.4048-2-49-3': 'Secondly, this solution ceases to exist if [MATH] is too large, as shown in Figure [REF]; intuitively, if [MATH] were too large, the dual cycle would become smaller than the size of the localized soliton, which is not possible.', '1107.4048-2-49-4': 'Thirdly, the free energy [MATH] of this solution, unlike that of ([REF]), is proportional to temperature; hence the entropy of this solution is non-zero at [MATH] and the temporal [MATH] symmetry is broken.', '1107.4048-2-49-5': 'In addition, because of the non-trivial [MATH]-dependence of the classical action ([REF]) we can see that the [MATH] symmetry along the [MATH]-cycle is also broken.', '1107.4048-2-49-6': 'Thus the Polyakov loop [MATH] and [MATH] in this solution are both non-zero as shown in Table [REF] and, contrary to the black D4 solution, the localized solitonic D3 solution is appropriate for a description of the deconfinement phase in the dual gauge theory (which also has [MATH]).', '1107.4048-2-49-7': 'We will build on this observation in the next subsection.', '1107.4048-2-50-0': '## Gregory-Laflamme transition as a confinement/deconfinement transition', '1107.4048-2-51-0': 'The phase structure of the 5 dimensional SYM on [MATH] with the (P,AP) boundary condition is shown in Figure [REF].', '1107.4048-2-51-1': 'The strong coupling region (the blue region in Figure [REF]) described by type II supergravity and is characterized by the GL phase transition which occurs at a temperature given by ([REF]).', '1107.4048-2-51-2': 'In the weak coupling region (the green region in Figure [REF]), the 4 dimensional Yang-Mills is realized at low temperatures ([MATH]).', '1107.4048-2-51-3': 'Although this region is common to the (P,AP) phase diagram in Figure [REF] and the (AP,AP) phase diagram in Figure [REF], the mirror of this region under the [MATH]) does not exist in Figure [REF] since the (P,AP) b.c is not [MATH]-symmetric.', '1107.4048-2-52-0': 'The main point to emphasize here it that: contrary to the previous phase structure in the (AP,AP) case, now the localized solitonic D3 phase has the same order parameters ([MATH]) as the deconfinement phase, thus making it plausible that these two phases are smoothly connected.', '1107.4048-2-52-1': 'In Figure [REF], we have indicated this by assuming the simplest extrapolation through the region of the intermediate coupling.', '1107.4048-2-53-0': 'Our proposal', '1107.4048-2-54-0': 'In view of the above observations, we propose a strong coupling continuation of weakly coupled 4-dimensional Yang Mills theory as shown in Table [REF].', '1107.4048-2-55-0': 'Of course, our proposal is based on a simple extrapolation between the intermediate coupling regime, and the real story there could be more involved.', '1107.4048-2-55-1': 'However the mere existence of such a simple extrapolation is a significant improvement over the previous proposal in the (AP,AP) case, where we are certain that there has to be at least one phase boundary between the deconfinement phase and black D4 solution (for the simple reason that their order parameters have different values).', '1107.4048-2-55-2': 'It is clearly important, therefore, to further investigate the nature of the deconfinement phase in terms of the localized solitonic D3 solution based on the correspondence outlined above.', '1107.4048-2-56-0': '# New correspondences in holographic QCD', '1107.4048-2-57-0': 'Using our proposed correspondence, we can explain anew several phenomena in the gauge theory from gravity.', '1107.4048-2-57-1': 'In this section, we list some of these phenomena.', '1107.4048-2-58-0': '## Polyakov loop and D3 brane distribution', '1107.4048-2-59-0': 'In our proposal, we identified the GL transition in the IIB frame to the confinement/deconfinement transition in the gauge theory.', '1107.4048-2-59-1': 'In the GL transition, the distribution of the D3 branes on the dual circle changes from a uniform distribution at low temperatures to a localized one at high temperatures.', '1107.4048-2-59-2': 'In this subsection, we explain what the corresponding phenomenon is in the gauge theory.', '1107.4048-2-60-0': 'Since the original five-dimensional gauge theory (of the D4 branes) appears in the IIA frame, in order to understand the role of the D3 branes, we need to consider a T-duality along the temporal direction.', '1107.4048-2-60-1': 'Under this T-duality, [MATH], which is transverse to the D3 brane, is mapped to the gauge potential [MATH] on the D4 brane.', '1107.4048-2-60-2': ""Thus the D3 brane distribution on the dual circle is related to a distribution of [MATH]'s, which are defined as the eigenvalues of the Polyakov loop operator [MATH]."", '1107.4048-2-61-0': 'If the D3 branes are uniformly distributed, the [MATH] are also uniformly distributed.', '1107.4048-2-61-1': 'In that case, by choosing an appropriate gauge, we can take [MATH].', '1107.4048-2-61-2': 'Then the temporal Polyakov loop operator ([REF]) becomes [EQUATION]', '1107.4048-2-61-3': 'Hence this distribution corresponds to the confinement phase.', '1107.4048-2-61-4': 'On the other hand, if the D3 branes are localized, then [MATH] are also localized and the Polyakov loop becomes non-zero, which characterizes the deconfinement phase.', '1107.4048-2-61-5': 'This observation is consistent with the entropy arguments in section [REF].', '1107.4048-2-62-0': 'The above discussion shows a direct relation between the D3 brane distribution and the eigenvalue distribution of the Polyakov loop operator.', '1107.4048-2-62-1': 'This latter quantity can sometimes be explicitly evaluated [CITATION].', '1107.4048-2-62-2': 'E.g. in [CITATION], for four dimensional Yang Mills theory on a small [MATH], we found that the uniform, non-uniform and localized distribution of [MATH], all appear, with obvious correspondence to similar gravitational solutions.', '1107.4048-2-62-3': 'Especially the free energies of these three solutions show a ""swallow tail"" relation similar to Figure [REF].', '1107.4048-2-62-4': '(See Figure 3 in [CITATION].)', '1107.4048-2-62-5': 'This correspondence strongly supports our proposal.', '1107.4048-2-63-0': '## Gregory-Laflamme transition as a Hagedorn transition', '1107.4048-2-64-0': 'Our proposal opens up the interesting possibility of a relation between the GL transition and the Hagedorn transition.', '1107.4048-2-65-0': 'From ([REF]), ([REF]) and ([REF]), we see that the GL transition in the IIB supergravity analysis happens near a temperature where the winding modes around the temporal cycle start becoming light in the IIA frame.', '1107.4048-2-65-1': 'This may indicate that the GL transition is associated with the excitation of the winding modes of the IIA string.', '1107.4048-2-65-2': 'Indeed, the KK modes of the graviton along the dual temporal circle, which cause the GL instability in the IIB description, are mapped to winding modes around on the temporal circle through the T-duality [CITATION].', '1107.4048-2-65-3': 'This is similar to the Hagedorn transition in string theory [CITATION], where the temporal winding modes cause the instability.', '1107.4048-2-65-4': 'Thus the GL transition in the IIB description might correspond to the Hagedorn transition in the IIA description.', '1107.4048-2-66-0': 'Note that on the large [MATH] gauge theory side also, the confinement/deconfinement transition has been shown to be related to the Hagedorn transition [CITATION].', '1107.4048-2-66-1': 'This makes it plausible that the Hagedorn transition in the Yang-Mills theory continues to the Hagedorn transition in the IIA string, which, as we argued above, is possibly the dual of the GL transition in the IIB supergravity.', '1107.4048-2-67-0': '# Chiral symmetry restoration in Sakai-Sugimoto model', '1107.4048-2-68-0': 'In the previous sections, we have seen that the conventional holographic representation of the confinement/deconfinement transition as the SS transition is fraught with problems, to circumvent which we proposed in Section [REF] a different interpretation in terms of a GL transition.', '1107.4048-2-68-1': 'However, the SS transition has widely been employed in holographic QCD and purports to explain several phenomena in real QCD.', '1107.4048-2-68-2': 'In particular, the chiral symmetry restoration in the Sakai-Sugimoto model was neatly explained in [CITATION].', '1107.4048-2-68-3': 'In their scenario the black D4 brane plays a crucial role.', '1107.4048-2-68-4': 'However, since in our proposal the black branes do not appear anymore, we need to find an alternative idea for realizing chiral symmetry restoration.', '1107.4048-2-68-5': 'In this section, we discuss how chiral symmetry restoration can happen in the localized solitonic D3 background.', '1107.4048-2-69-0': '## Sakai-Sugimoto model and chiral symmetry breaking', '1107.4048-2-70-0': 'The Sakai-Sugimoto model [CITATION] was proposed to describe low energy hadron physics in holographic QCD and elegantly reproduces many aspects of the real QCD.', '1107.4048-2-70-1': 'We first briefly review this model and show how chiral symmetry breaking at low temperatures is realized in terms of the dual gravity.', '1107.4048-2-71-0': 'The Sakai-Sugimoto model is an extension of the holographic model of QCD discussed in section [REF].', '1107.4048-2-71-1': 'Sakai and Sugimoto added, to the [MATH] D4 brane system, [MATH] D8 and [MATH] branes which are localized on the [MATH] circle and fill all other directions as follows [EQUATION]', '1107.4048-2-71-2': 'Here [MATH] denotes the compactified directions.', '1107.4048-2-71-3': '(See Figure [REF] (a).)', '1107.4048-2-71-4': 'This model has a free parameter [MATH], which is the asymptotic ([MATH]) distance between the D8 and [MATH] brane on the [MATH] circle.', '1107.4048-2-71-5': 'In the original Sakai-Sugimoto model [CITATION] [MATH] was taken to be [MATH]; however, we will let it be general and take values in [MATH] (the other half, [MATH] is related by a reflection and need not be considered separately).', '1107.4048-2-72-0': 'This model has [MATH] gauge symmetry on the [MATH] D8 and [MATH] branes, which can be interpreted as a chiral [MATH] flavor symmetry in QCD.', '1107.4048-2-72-1': 'As we will see soon, in some situations, the D8 and [MATH] branes merge and the chiral symmetry is broken to a single [MATH].', '1107.4048-2-72-2': 'Sakai and Sugimoto proposed that this is the holographic realization of chiral symmetry breaking [CITATION].', '1107.4048-2-73-0': 'Let us take a large [MATH] limit a la Maldacena and, according to the principle of holography, replace the [MATH] D4 branes with a corresponding gravity solution.', '1107.4048-2-73-1': ""Here we will keep [MATH] such that we can ignore the back-reaction of the D8/[MATH] branes onto the background geometry (this is the so-called 'probe approximation')."", '1107.4048-2-73-2': 'In that case, the background (D4 brane) geometry is determined thermodynamically (as the dominant classical solution at a given temperature, as we have done in sections [REF] and [REF]), and the D8/[MATH] brane configuration coupled to this background, is determined dynamically, with the given distance [MATH] as a boundary condition.', '1107.4048-2-74-0': 'Chiral symmetry breaking in this model happens as follows.', '1107.4048-2-74-1': 'At sufficiently low temperatures, the favoured geometric background is that of solitonic D4 solution ([REF]), as we have seen in section [REF] and [REF].', '1107.4048-2-74-2': 'On this background, the D8 and [MATH] cannot extend separately and need to merge as shown in Figure [REF] (b).', '1107.4048-2-74-3': 'As a result, the [MATH] gauge symmetry on the D8 and [MATH] is broken to [MATH], representing chiral symmetry breaking ([MATH]SB) in the dual gauge theory.', '1107.4048-2-75-0': '## Chiral symmetry restoration in the black D4 brane background', '1107.4048-2-76-0': 'In the gauge theory, it is expected that chiral symmetry is restored at a sufficiently high temperature.', '1107.4048-2-76-1': 'Thus if holographic QCD is to work, there should be a corresponding phenomena in the dual gravity description.', '1107.4048-2-77-0': 'In [CITATION], a mechanism for chiral symmetry restoration was suggested by considering D8 and [MATH] in the black D4 brane background ([REF]) in the (AP,AP) case.', '1107.4048-2-77-1': ""Contrary to the solitonic D4 brane geometry ([REF]), the [MATH] plane of the black D4 background has a 'cigar' geometry."", '1107.4048-2-77-2': 'Thus the D8 and [MATH] branes can wrap the cigar separately as shown in Figure [REF] (c).', '1107.4048-2-77-3': 'If such a configuration is energetically favoured, [MATH] gauge symmetry is preserved and chiral symmetry is restored.', '1107.4048-2-78-0': 'In addition to this configuration, another configuration shown in Figure [REF] (d) is possible in the black D4 background.', '1107.4048-2-78-1': 'This configuration is similar to Figure [REF] (b) and chiral symmetry is broken.', '1107.4048-2-78-2': 'In [CITATION], energies of these two configurations are compared with the black D4 background.', '1107.4048-2-78-3': 'It was found (see Figure 7 of [CITATION]) that for [MATH], it is configuration (c) which is always favoured, indicating a concurrence of deconfinement and chiral symmetry restoration transitions at [MATH], while for [MATH], a new window [MATH] opens up where configuration (d) is the favoured one, indicating coexistence of deconfinement and broken chiral symmetry.', '1107.4048-2-79-0': '## Chiral symmetry restoration in the localized solitonic D3 brane.', '1107.4048-2-80-0': 'Although the chiral symmetry restoration was explained, as above, in the black D4 brane background, our main thesis in this paper is that the black D4 solution itself is fraught with problems, if we interpret this solution as the deconfinement phase of the four dimensional gauge theory (see Section [REF]).', '1107.4048-2-80-1': 'The alternative we proposed in Section [REF] is that the localized solitonic D3 brane solution ([REF]) in the (P,AP) b.c. should be taken as the correct representation of the deconfinement phase in the dual gauge theory.', '1107.4048-2-80-2': 'In view of this, we need to understand chiral symmetry restoration in this solution instead of in the black D4 solution.', '1107.4048-2-81-0': 'One subtlety in the application of our proposal to the Sakai-Sugimoto model is the existence of the fundamental quarks from the open string between the D4 and D8/[MATH] brane.', '1107.4048-2-81-1': 'These quarks are, of course, not decoupled under the 4 dimensional limit [MATH].', '1107.4048-2-81-2': 'Thus we have to impose the AP b.c. on the temporal cycle to investigate its thermodynamics.', '1107.4048-2-81-3': 'This is a problem since we need to use the P b.c. in our proposal.', '1107.4048-2-81-4': 'However as far as we take [MATH] and ignore the back reaction of D8, we can ignore the contribution of the quarks to the D4 geometries.', '1107.4048-2-81-5': 'Thus we can use the P b.c. to investigate the background metric.', '1107.4048-2-81-6': 'In addition, when we determine the profile of the probe D8/[MATH] branes on the background geometry, only bosonic modes are relevant [CITATION] and fermions does not play any role.', '1107.4048-2-81-7': 'Therefore if our interest is only in the mechanism of the chiral symmetry restoration, the periodicity of the fermions must be irrelevant and we use the P b.c. to investigate it.', '1107.4048-2-82-0': 'In our proposal, the gravity analysis has been done in IIB supergravity by performing a T-duality along the [MATH]-cycle: [MATH].', '1107.4048-2-82-1': 'Thus it is convenient to dualize the brane configuration of the Sakai-Sugimoto model to the IIB frame.', '1107.4048-2-82-2': 'Since all the D branes in the Sakai-Sugimoto model wrap the [MATH], the above T-duality maps the D4 branes and D8/[MATH] branes to D3 and D7/[MATH] branes spreading [EQUATION]', '1107.4048-2-82-3': 'See Figure [REF] (e) also.', '1107.4048-2-82-4': 'The distribution along [MATH] of the D branes in the IIB description (related to the gauge field [MATH] in the IIA description), is determined dynamically.', '1107.4048-2-82-5': 'In the probe approximation, according to the analyses in section [REF], at temperatures below the GL critical temperature ([REF]), the D3 branes are distributed uniformly and, above it, they are localized on the [MATH].', '1107.4048-2-82-6': 'The fate of the chiral gauge group in the gravity representation depends on the stable configurations of the probe D7/[MATH] branes in these backgrounds.', '1107.4048-2-83-0': 'In the uniformly smeared solitonic D3 brane geometry ([REF]) shown in Figure [REF] (f), the situation is similar to (b) in Figure [REF].', '1107.4048-2-83-1': 'Since the [MATH] direction is smoothly pinched off at [MATH], the D7 and [MATH] have to merge and chiral symmetry is broken.', '1107.4048-2-84-0': 'On the other hand, in the localized solitonic D3 brane geometry, the horizontal direction is not fully pinched off.', '1107.4048-2-84-1': 'Recall that the geometry ([REF]) is pinched off at [MATH], where [MATH] and [MATH].', '1107.4048-2-84-2': 'Thus [MATH] can reach zero.', '1107.4048-2-84-3': 'As a result, D7 and [MATH] can extend separately as shown in Figure [REF] (g).', '1107.4048-2-84-4': 'This configuration is similar to (e) in Figure [REF] and would restore chiral symmetry.', '1107.4048-2-84-5': 'In addition to this configuration, a chiral symmetry broken configuration is also possible as shown in Figure [REF] (h).', '1107.4048-2-85-0': 'The last task is the evaluation of the stability of these two configurations at temperatures higher than [MATH] ([REF]).', '1107.4048-2-85-1': 'In the flat space, the force between a single D3 and single D7 (or between a single D3 and a single [MATH]) in our configuration is repulsive, since the number of the Neumann-Dirichlet open strings between them is 6 [CITATION].', '1107.4048-2-85-2': 'Thus we expect that even after taking the large [MATH] and near horizon limits, the force may be repulsive.', '1107.4048-2-85-3': 'As a result, the D7/[MATH] branes, which are separated from the localized SD3 in the [MATH]-direction, would try to move away as far as possible in this direction.', '1107.4048-2-85-4': 'However, since the [MATH]-direction is compactified on a circle, the D7/[MATH] branes should then end as being fixed at the point on the [MATH]-circle which is antipodal to the localized SD3.', '1107.4048-2-85-5': 'E.g. if we put the localized SD3 brane at [MATH], the D7 as well as the [MATH] branes will both be at [MATH].', '1107.4048-2-85-6': 'See the left diagram of Figure [REF].', '1107.4048-2-85-7': 'In this case, we can effectively restrict the dynamics of these branes to ([MATH]) plane (given by [MATH]).', '1107.4048-2-85-8': 'Thus the problem reduces to finding classical solutions [MATH] with a boundary condition: [MATH]), where we put the D7 at [MATH] and [MATH] at [MATH].', '1107.4048-2-86-0': 'The problem stated above is difficult to solve precisely near the GL transition since the background metric ([REF]) is only an approximate description.', '1107.4048-2-86-1': 'The metric around the D7/[MATH] brane (at [MATH]) becomes more and more accurate, however, when [MATH].', '1107.4048-2-86-2': 'As a result, we solve for the stable configuration in this limit.', '1107.4048-2-86-3': 'Details of the calculation are presented in appendix [REF].', '1107.4048-2-86-4': 'We find three solutions, corresponding to the D brane configurations of Figures [REF] (g) and (h), as shown in Figure [REF].', '1107.4048-2-86-5': 'In the appendix, we compare the classical DBI actions of these solutions numerically, as depicted in Figure [REF] (a) (see also the phase diagram in Figure [REF]): as one increases temperature beyond a certain value, chiral symmetry is restored, the transition being of first order.', '1107.4048-2-86-6': 'Therefore, one can see that even in our proposal, similarly to the case of the black D4 [CITATION], we can explain chiral symmetry restoration at high enough temperatures.', '1107.4048-2-87-0': '# Conclusions', '1107.4048-2-88-0': 'In this paper, we showed that the conventional representation of the confinement/deconfinement transition in holographic QCD has several problems and proposed an alternative representation which resolves these problems.', '1107.4048-2-89-0': 'As mentioned earlier, problems similar to the above had also been encountered in the study of two dimensional bosonic gauge theory in [CITATION].', '1107.4048-2-89-1': 'This indicates that the issues addressed in this paper are rather general in the discussion of holography for non-supersymmetric gauge theories at finite temperatures.', '1107.4048-2-89-2': 'To elaborate, in the standard holographic procedure, a [MATH]-dimensional non-supersymmetric gauge theory is first constructed through the KK reduction of a [MATH]-dimensional super Yang-Mills theory on a Scherk-Schwarz circle.', '1107.4048-2-89-3': 'The [MATH]-dimensional SYM at large [MATH] can be mapped to a scaling limit of D[MATH] brane geometries [CITATION].', '1107.4048-2-89-4': 'At finite temperatures, because of the two compact cycles (temporal and Scherk-Schwarz), several distinct solutions (depending on boundary conditions) appear in gravity as shown in section [REF] and [REF]: solitonic D[MATH] (equivalently, uniformly smeared solitonic D[MATH]), localized solitonic D[MATH], and black D[MATH].', '1107.4048-2-89-5': 'The black D[MATH] brane solution appears at high temperatures in the (AP,AP) case, while the localized solitonic D[MATH] brane solution is the high temperature phase in the (P,AP) case.', '1107.4048-2-89-6': 'A table similar to Table [REF] can again be constructed, where the appropriate order parameters would be [MATH]; these would again appear to favour the localized solitonic D[MATH] phase as the more suitable representation of the deconfinement of [MATH]-dimensional YM theory (rather than the more conventional black D[MATH] phase which appears only in the (AP,AP) b.c.).', '1107.4048-2-89-7': 'Following this logic, a [MATH]-dimensional analogue of our proposal (see Section [REF]) would appear to give a better description of holographic QCD in [MATH] dimensions.', '1107.4048-2-89-8': 'In particular, we believe that the Gregory-Laflamme transition between the solitonic D[MATH] and localized solitonic D[MATH], would, as in this paper, be related to the confinement/deconfinement transition in the [MATH]-dimensional gauge theory.', '1107.4048-2-90-0': '## Further questions', '1107.4048-2-91-0': 'In order to further understand holographic QCD through the above proposal, it would be of interest to address the following questions.', '1107.4048-2-92-0': 'Transition temperature', '1107.4048-2-93-0': 'From ([REF]), the critical temperature of the confinement/deconfinement transition (the GL transition) would be [MATH].', '1107.4048-2-93-1': 'On the other hand, holographic QCD seems to predict that the square root of the QCD string tension is [MATH] whereas the glueball masses are [MATH] [CITATION].', '1107.4048-2-93-2': 'It would be important to understand the reason for the separation of these scales and how they evolve from strong coupling to weak coupling.', '1107.4048-2-94-0': 'Quantitative correspondence', '1107.4048-2-95-0': 'Although our new proposal reproduces the known qualitative features of 4 dimensional Yang-Mills theory, it does not automatically lead to a quantitative agreement.', '1107.4048-2-95-1': 'For example, the free energy of the deconfinement phase of the YM theory at a sufficiently high temperature must be proportional to [MATH], since the coupling becomes weak and the theory becomes approximately conformal.', '1107.4048-2-95-2': 'However the free energy of the localized solitonic D3 is proportional to [MATH] (as can be seen from ([REF])).', '1107.4048-2-95-3': 'This is not entirely surprising since the functional form of [MATH] can change as one evolves from weak coupling to strong coupling.', '1107.4048-2-95-4': 'Furthermore, one has to exercise caution in defining a high temperature limit of YM4 in the holographic context since the temperature must always remain much smaller than the KK scale.', '1107.4048-2-96-0': 'YM4 from SYM5 with (AP,AP) In this article, we have emphasized the correspondence between YM4 and SYM5 with (P,AP) b.c..', '1107.4048-2-96-1': 'However, as we pointed out in footnote [REF], the 5 dimensional SYM with (AP,AP) b.c. should also be related to YM4, since the boundary condition becomes irrelevant in the limit [MATH] and [MATH].', '1107.4048-2-96-2': 'A possible way this correspondence may work is as follows.', '1107.4048-2-96-3': 'In the (AP,AP) case, if we treat the black D4 solution (the blue region in the bottom half of Figure [REF]), which is not related to YM4, as irrelevant, and focus on the solitonic D4 brane, its winding modes would appear to be light around the temperature ([REF]).', '1107.4048-2-96-4': 'A Hagedorn transition parallel to the (P,AP) case might occur around here and might continue to the confinement/deconfinement transition in YM4.', '1107.4048-2-96-5': 'However, in order to investigate it further through gravity, we need to understand the gauge/gravity correspondence in the 0B frame as mentioned in footnote [REF].', '1107.4048-2-97-0': 'Real time It would be important to explore what geometry corresponds to the deconfinement phase in the real time formalism.', '1107.4048-2-97-1': 'An understanding of this would allow us to address dynamical properties of the deconfinement phase, e.g. transport properties.', '1107.4048-2-97-2': 'Since previous results in this area were based on the black brane solutions, it would be important to see how well-known results such as the viscosity bound [CITATION] can be derived in our proposal.', '1107.4048-2-98-0': 'AdS/CMT and chemical potential dependence', '1107.4048-2-99-0': 'The SS transition for D3 branes has been studied in the context of the AdS/CMT correspondence to investigate the superconductor/insulator transition in 2+1 dimension [CITATION].', '1107.4048-2-99-1': 'In these studies, a chemical potential for a [MATH] charge was introduced and the phase structure involving this chemical potential has been derived.', '1107.4048-2-99-2': 'It would be interesting to ask whether our proposal has any bearing on these studies, e.g., whether the GL transition analogous to the one discussed here can be a candidate for the superconductor/insulator or some other transition.', '1107.4048-2-99-3': 'A possible line of investigation could be to study the chemical potential dependence of [MATH] for the GL transition vis-a-vis that for the SS transition and see whether any qualitative differences appear.'}","[['1107.4048-1-13-0', '1107.4048-2-13-0'], ['1107.4048-1-13-1', '1107.4048-2-13-1'], ['1107.4048-1-13-2', '1107.4048-2-13-2'], ['1107.4048-1-13-3', '1107.4048-2-13-3'], ['1107.4048-1-70-0', '1107.4048-2-70-0'], ['1107.4048-1-70-1', '1107.4048-2-70-1'], ['1107.4048-1-24-0', '1107.4048-2-24-0'], ['1107.4048-1-24-1', '1107.4048-2-24-1'], ['1107.4048-1-97-1', '1107.4048-2-97-1'], ['1107.4048-1-97-2', '1107.4048-2-97-2'], ['1107.4048-1-49-0', '1107.4048-2-49-0'], ['1107.4048-1-49-1', '1107.4048-2-49-1'], ['1107.4048-1-49-2', '1107.4048-2-49-2'], ['1107.4048-1-49-3', '1107.4048-2-49-3'], ['1107.4048-1-49-4', '1107.4048-2-49-4'], ['1107.4048-1-49-5', '1107.4048-2-49-5'], ['1107.4048-1-49-6', '1107.4048-2-49-6'], ['1107.4048-1-49-7', '1107.4048-2-49-7'], ['1107.4048-1-73-0', '1107.4048-2-74-0'], ['1107.4048-1-73-1', '1107.4048-2-74-1'], ['1107.4048-1-73-2', '1107.4048-2-74-2'], ['1107.4048-1-73-3', '1107.4048-2-74-3'], ['1107.4048-1-31-0', '1107.4048-2-31-0'], ['1107.4048-1-31-1', 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'1107.4048-2-53-0', '1107.4048-2-62-4', '1107.4048-2-71-3', '1107.4048-2-92-0', '1107.4048-2-94-0', '1107.4048-2-98-0', '1107.4048-3-6-0', '1107.4048-3-28-1', '1107.4048-3-33-0', '1107.4048-3-34-0', '1107.4048-3-47-6', '1107.4048-3-54-0', '1107.4048-3-63-4', '1107.4048-3-72-3', '1107.4048-3-93-0', '1107.4048-3-95-0', '1107.4048-3-99-0', '1107.4048-4-6-0', '1107.4048-4-28-1', '1107.4048-4-33-0', '1107.4048-4-34-0', '1107.4048-4-47-6', '1107.4048-4-54-0', '1107.4048-4-63-4', '1107.4048-4-72-3', '1107.4048-4-93-0', '1107.4048-4-95-0', '1107.4048-4-99-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1107.4048,"{'1107.4048-3-0-0': 'We discuss the phase structure of [MATH] D4 branes wrapped on a temporal (Euclidean) and a spatial circle, in terms of the near-horizon AdS geometries.', '1107.4048-3-0-1': 'This system has been studied previously to understand four dimensional pure [MATH] Yang-Mills theory (YM4) through holography.', '1107.4048-3-0-2': 'In the usual treatment of the subject, the phase transition between the AdS soliton and the black D4 brane is interpreted as the strong coupling continuation of the confinement/deconfinement transition in YM4.', '1107.4048-3-0-3': 'We show that this interpretation is not valid, since the black D4 brane and the deconfinement phase of YM4 have different realizations of the [MATH] centre symmetry and cannot be identified.', '1107.4048-3-0-4': 'We propose an alternative gravity dual of the confinement/deconfinement transition in terms of a Gregory-Laflamme transition of the AdS soliton in the IIB frame, where the strong coupling continuation of the deconfinement phase of YM4 is a localized D3 soliton.', '1107.4048-3-0-5': 'Our proposal offers a new explanation of several aspects of the thermodynamics of holographic QCD.', '1107.4048-3-0-6': 'As an example, we show a new mechanism of chiral symmetry restoration in the Sakai-Sugimoto model.', '1107.4048-3-0-7': 'The issues discussed in this paper pertain to gravity duals of non-supersymmetric gauge theories in general.', '1107.4048-3-1-0': '# Introduction', '1107.4048-3-2-0': 'Strongly coupled field theories such as QCD and various condensed matter systems are not amenable to perturbative calculations and require numerical or other non-perturbative tools.', '1107.4048-3-2-1': 'A powerful new tool in this context is holography, which maps gauge theories to gravity in a higher dimension [CITATION].', '1107.4048-3-2-2': 'Holography, however, is most reliable and quantitative for special supersymmetric gauge theories.', '1107.4048-3-2-3': 'In the generic case, including for pure Yang Mills theories, inferences from gravity tend to be less quantitative due to various degrees of extrapolations, although such exercises have proved to be rather valuable, e.g., in the context of holographic QCD [CITATION].', '1107.4048-3-3-0': 'Recently, it was pointed out in [CITATION] that there are problems with a naive application of holography to a two dimensional large [MATH] bosonic gauge theory at finite temperature.', '1107.4048-3-3-1': 'The dual of this system is based on gravity backgrounds involving [MATH] D2 branes on [MATH].', '1107.4048-3-3-2': 'The phase diagram of gravity, interpreted naively, does not admit a unique continuation to the regime of gauge theory, owing to its dependence on the boundary condition for fermions on the brane.', '1107.4048-3-3-3': 'However, it was shown in [CITATION] that the phase structures in gravity and gauge theory could be smoothly connected through an appropriate choice of fermion boundary conditions.', '1107.4048-3-4-0': 'In this paper, we will show that similar problems arise when we holographically analyse more general non-supersymmetric gauge theories at finite temperature.', '1107.4048-3-4-1': 'The particular example we will focus on is that of holographic QCD from D4 branes [CITATION].', '1107.4048-3-4-2': 'We will discuss some problems with the usual correspondence between the confinement/deconfinement transition in QCD and the Scherk-Schwarz transition between an AdS soliton and a black D4 brane in the gravity dual.', '1107.4048-3-4-3': 'Some of these problems were first discussed in [CITATION] (see section [REF] for further details).', '1107.4048-3-4-4': 'Especially we will show that the black D4 brane cannot be identified with the (strong coupling continuation of the) deconfinement phase in QCD in four dimensions.', '1107.4048-3-4-5': 'As a resolution of these problems, we will propose an alternative scenario in which the confinement/deconfinement transition corresponds to a Gregory-Laflamme transition [CITATION] between a uniformly distributed AdS soliton and a localized AdS soliton in the IIB frame.', '1107.4048-3-5-0': 'The scenario we propose suggests that we need to reconsider several previous results in holographic QCD including the Sakai-Sugimoto model [CITATION].', '1107.4048-3-5-1': 'One important ingredient in the Sakai-Sugimoto model is the mechanism of chiral symmetry restoration at high temperatures [CITATION].', '1107.4048-3-5-2': 'We will propose a new mechanism for chiral symmetry restoration in our framework.', '1107.4048-3-6-0': 'The plan of the paper is as follows.', '1107.4048-3-7-0': 'Section [REF] contains a short review of finite temperature QCD and the holographic approach to four dimensional Yang Mills theory (YM4) using D4 branes.', '1107.4048-3-7-1': ""In Section [REF] we discuss the gravity theory at a finite temperature and a 'Scherk-Schwarz' transition between a solitonic D4 brane and a black D4 brane."", '1107.4048-3-7-2': 'In Section [REF] we recall the conventional correspondence between this transition and the confinement/deconfinement transition of the Yang Mills theory, and discuss problems with this correspondence, with special emphasis on the mismatch of [MATH] centre symmetry between the black D4 phase and the deconfined 4 dimensional YM theory.', '1107.4048-3-7-3': 'In Section [REF], we discuss the phase structure of the D4 branes using an unconventional (periodic) fermion boundary condition along the Euclidean time circle and show that the high temperature in gravity is described by a localized solitonic D3 brane (or its T-dual) whose centre symmetry precisely matches with the deconfinement phase of four dimensional YM theory.', '1107.4048-3-7-4': 'This leads us, in Section [REF], to propose a new representation of the confinement/deconfinement phase transition in terms of a GL transition between the solitonic D4 and the (T-dual of) the localized solitonic D3 (see Table [REF]).', '1107.4048-3-7-5': 'In Section [REF], we show some new correspondences between phenomena in QCD and their counterparts in gravity following our proposal.', '1107.4048-3-7-6': 'In Section [REF], we suggest a new mechanism of chiral symmetry restoration in the Sakai-Sugimoto model in keeping with our proposal.', '1107.4048-3-7-7': 'Section [REF] contains the concluding remarks and some open problems.', '1107.4048-3-8-0': '# Review of QCD and its gravity dual', '1107.4048-3-9-0': '## Finite temperature QCD', '1107.4048-3-10-0': 'In this section, we briefly review some salient properties of four dimensional [MATH] pure Yang-Mills theory at a finite temperature at large [MATH], a system which we will subsequently investigate through holography.', '1107.4048-3-11-0': 'An important symmetry in this theory is the [MATH] symmetry along the temporal cycle, which is the centre of the [MATH] symmetry.', '1107.4048-3-11-1': 'The order parameter of this symmetry is the temporal Polyakov loop operator [EQUATION]', '1107.4048-3-11-2': 'If this operator is zero, the [MATH] symmetry is preserved and, if not, it is broken.', '1107.4048-3-11-3': ""Especially, if the [MATH] symmetry is preserved, physical quantities do not depend on the temporal radius [MATH] at [MATH] order due to 'large [MATH] volume independence' [CITATION]."", '1107.4048-3-11-4': 'If the gauge theory is on a torus [MATH], the large [MATH] volume independence is generalized and physical quantities do not depend on [MATH], if the [MATH] symmetry along the [MATH]-th direction is preserved [CITATION].', '1107.4048-3-12-0': 'Let us consider the phase structure of the Yang-Mills theory.', '1107.4048-3-12-1': 'At low temperatures, the confinement phase is the dominant thermodynamic phase.', '1107.4048-3-12-2': 'In this phase, the [MATH] symmetry is preserved ([MATH]).', '1107.4048-3-12-3': 'At high temperatures, the deconfinement phase, in which the [MATH] symmetry is broken ([MATH]), dominates.', '1107.4048-3-12-4': 'There is a confinement/deconfinement transition at an intermediate temperature.', '1107.4048-3-12-5': 'From studies in large [MATH] lattice gauge theories, this is expected to be a first order phase transition [CITATION].', '1107.4048-3-13-0': ""As we remarked above, the free energy in the confinement phase does not depend on the temperature through the 'large [MATH] volume independence'."", '1107.4048-3-13-1': 'It implies that the entropy becomes zero at [MATH].', '1107.4048-3-13-2': 'On the other hand, the entropy in the deconfinement phase would be [MATH].', '1107.4048-3-13-3': 'This has a simple physical interpretation: in the confinement phase, the spectrum excludes gluon states, which have [MATH] degree of the freedom, and consists only of gauge singlet states like glueballs, leading to an [MATH] entropy.', '1107.4048-3-14-0': '## Holographic QCD', '1107.4048-3-15-0': 'In this section, we will review the construction of four dimensional [MATH] pure Yang-Mills theory from [MATH] D4 branes [CITATION].', '1107.4048-3-15-1': 'Let us first consider a 10 dimensional Euclidean spacetime with an [MATH] and consider D4 branes wrapping on the [MATH].', '1107.4048-3-15-2': 'We define the coordinate along this [MATH] as [MATH] and its periodicity as [MATH].', '1107.4048-3-15-3': 'The effective theory on this brane is a 5 dimensional supersymmetric Yang-Mills theory on the [MATH].', '1107.4048-3-15-4': 'For the fermions on the brane, the boundary condition along the circle can be AP (antiperiodic) or P (periodic); to specify the theory, we must pick one of these two boundary conditions.', '1107.4048-3-15-5': 'Let us take the AP boundary condition.', '1107.4048-3-15-6': 'This gives rise to fermion masses proportional to the Kaluza-Klein scale [MATH], leading to supersymmetry breaking (this is called the SS - Scherk-Schwarz- mechanism).', '1107.4048-3-15-7': 'This, in turn, induces masses for the adjoint scalars and for [MATH], which are proportional to [MATH] at one-loop.', '1107.4048-3-15-8': 'Therefore, if [MATH] is sufficiently small and the dynamical scale [MATH] is much less than both the above mass scales, then the fermions and adjoint scalars are decoupled and the 5 dimensional supersymmetric Yang-Mills theory is reduced to a four dimensional pure Yang-Mills theory.', '1107.4048-3-16-0': 'By taking the large [MATH] limit of this system a la Maldacena at low temperatures, we obtain the dual gravity description of the compactified 5 dimensional SYM theory [CITATION], which consists, at low temperatures, of a solitonic D[MATH] brane solution wrapping the [MATH].', '1107.4048-3-16-1': 'The explicit metric is given by [EQUATION]', '1107.4048-3-16-2': 'There is also a non-trivial value of the five form potential which we do not write explicitly.', '1107.4048-3-16-3': 'Here [MATH] is the YM coupling on the D[MATH] world-volume and [MATH] is found by putting [MATH] in the general formula [EQUATION]', '1107.4048-3-16-4': 'Since the [MATH]-cycle shrinks to zero at [MATH], in order to avoid possible conical singularities we must choose the asymptotic radius [MATH] as follows [EQUATION]', '1107.4048-3-16-5': ""With this choice, the contractible [MATH]-cycle, together with the radial direction [MATH], forms a so-called 'cigar' geometry which is topologically a disc."", '1107.4048-3-16-6': 'In order that the fermions are well-defined on this geometry, they must obey the AP boundary condition.', '1107.4048-3-16-7': 'The AP boundary condition along [MATH] is, of course, consistent with the choice of fermion b.c. in the boundary theory.', '1107.4048-3-17-0': 'The leading order gravity solution described above is not always valid.', '1107.4048-3-17-1': 'E.g. in order that the stringy modes can be ignored, we should ensure that the curvature in string units must be small.', '1107.4048-3-17-2': 'In other words, the typical length scale of this solution near [MATH], viz, [MATH], must satisfy [MATH] [CITATION].', '1107.4048-3-17-3': 'This condition turns out to be equivalent to [EQUATION]', '1107.4048-3-17-4': 'This is opposite to the condition which we found for the validity of the four-dimensional gauge theory description (see footnote [REF]).', '1107.4048-3-17-5': 'Thus, this gravity solution can describe the 5 dimensional SYM but cannot directly describe the 4 dimensional YM theory, inferences about which can only be obtained through extrapolation.', '1107.4048-3-17-6': 'This is a common problem in the construction of holographic duals of non-supersymmetric gauge theories.', '1107.4048-3-17-7': 'This has been discussed at length; in particular, the gravity description, which necessitates extrapolation to strong coupling, has been likened (cf. [CITATION] and [CITATION], p. 196-197) to strong coupling lattice gauge theory.', '1107.4048-3-17-8': 'Many interesting results, including the qualitative predictions in [CITATION], have been obtained using this prescription.', '1107.4048-3-18-0': 'Let us explore some properties of the soliton solution ([REF]).', '1107.4048-3-18-1': 'This solution is expected, on the basis of several arguments, to correspond to the confinement phase in the four dimensional gauge theory.', '1107.4048-3-18-2': 'For example, it can be shown that the classical action of this solution at a finite temperature [MATH] is [CITATION] [EQUATION] where [MATH] is the volume of the three spatial dimensions ([MATH]) and [MATH] is a constant which is obtainable from the formula below [CITATION] by putting [MATH] [EQUATION]', '1107.4048-3-18-3': 'Thus the free energy of this solution, [MATH], is independent of temperature, and hence the entropy is zero at [MATH] order.', '1107.4048-3-18-4': 'These facts are consistent with the interpretation of this solution as the confinement phase in the large [MATH] gauge theory, in which the temporal [MATH] symmetry is preserved.', '1107.4048-3-19-0': 'Note that the action ([REF]) divided by [MATH] does depend on [MATH], implying that the [MATH] symmetry along the [MATH]-cycle is broken.', '1107.4048-3-19-1': 'Although this observation might appear irrelevant for the 4 dimensional Yang-Mills theory, it is this symmetry breaking which allows a conventional KK reduction to four dimensions (see the next two sections for more details).', '1107.4048-3-20-0': '# Finite temperature holographic QCD', '1107.4048-3-21-0': '## Confinement/deconfinement transition in Holographic QCD in the standard scenario', '1107.4048-3-22-0': 'In existing holographic studies of the thermodynamics in the Yang-Mills theory, it is assumed that the black D4 brane solution corresponds to the deconfinement phase and the Scherk-Schwarz transition in the gravity is related to the confinement/deconfinement transition.', '1107.4048-3-22-1': 'In this section, we review this scenario and we will explain its problems in section [REF].', '1107.4048-3-23-0': 'To discuss holographic QCD at finite temperatures, we begin by compactifying Euclidean time in the boundary theory on a circle with periodicity [MATH].', '1107.4048-3-23-1': 'To permit description in terms of four dimensional gauge theory, the temperature should, of course, be kept far below the KK scale, [MATH] .', '1107.4048-3-23-2': 'In order to determine the gravitational theory, we need to fix the periodicity of fermions in the gauge theory along the time cycle.', '1107.4048-3-23-3': 'In the existing literature, the fermion boundary condition is chosen to be AP according to the usual practice for thermal fermions (we will see in section [REF] that this choice is not imperative here).', '1107.4048-3-23-4': ""The resulting fermionic bc's: (AP, AP) along [MATH] lead to a [MATH] symmetry of the system under [MATH]."", '1107.4048-3-23-5': 'In the gravity dual, we must therefore, include the [MATH]-transform of the solution ([REF]), which is a black D4 solution: [EQUATION]', '1107.4048-3-23-6': 'Here [MATH] in this metric is related to [MATH] as in ([REF]) [EQUATION]', '1107.4048-3-23-7': 'This solution has a contractible [MATH]-cycle and the fermions must obey the anti-periodic boundary condition (see footnote [REF]), consistent with the AP b.c. in the gauge theory.', '1107.4048-3-23-8': 'Analogously to ([REF]), the gravity description of this solution is reliable if [MATH].', '1107.4048-3-24-0': 'The classical action density can be evaluated, as before, yielding [EQUATION]', '1107.4048-3-24-1': 'In contrast with ([REF]), the free energy now is a function of [MATH] and hence the entropy is [MATH] which is appropriate for a description of the deconfinement phase in the gauge theory.', '1107.4048-3-25-0': 'By comparing ([REF]) and ([REF]), we see that at low temperatures (large [MATH]) the solitonic D4 solution dominates, while at high temperatures the black brane dominates.', '1107.4048-3-25-1': 'The transition between the two solutions, which we will call the ""Scherk-Schwarz (SS) transition"", occurs at [CITATION] [EQUATION]', '1107.4048-3-25-2': 'In the standard discussions of holographic QCD [CITATION], this transition is interpreted as the (continuation of the) confinement/deconfinement transition in the gauge theory.', '1107.4048-3-25-3': 'In the following sections, we will explain various problems with this interpretation and present a resolution by proposing an alternative scenario.', '1107.4048-3-26-0': '## Problems with the standard gravity analysis', '1107.4048-3-27-0': 'The previous subsection describes the usual relation between the confinement/deconfinement transition in 4 dimensional Yang-Mills theory and the SS transition in holographic QCD.', '1107.4048-3-27-1': 'However as we will now discuss, there are several problems associated with this correspondence.', '1107.4048-3-27-2': 'Some of these problems were mentioned earlier in [CITATION].', '1107.4048-3-28-0': 'In order to understand these problems, it is convenient to consider first the phase structure of the 5 dimensional SYM on the [MATH].', '1107.4048-3-28-1': 'See Figure [REF].', '1107.4048-3-28-2': 'As we have seen, the gravity description is valid in the strong coupling regime of this theory (the blue region in Figure [REF]) which is characterized by the solitonic D4 solution and the black D4 solution.', '1107.4048-3-28-3': 'On the other hand, the 4 dimensional Yang-Mills theory is realized at weak coupling ([MATH]) and low temperature ([MATH]) (the upper green region in Figure [REF]) which is characterized by the confinement phase and the deconfinement phase.', '1107.4048-3-28-4': 'The usual proposal is that the solitonic D4 and black D4 solutions correspond to the confinement and deconfinement phases respectively.', '1107.4048-3-29-0': 'To examine this proposal in detail, it is useful to consider the realization of the [MATH] centre symmetry in various phases of the five-dimensional SYM theory along [MATH] and [MATH].', '1107.4048-3-29-1': 'The [MATH] symmetry along [MATH] is characterized by the order parameter [MATH] which is the temporal Polyakov loop ([REF]).', '1107.4048-3-29-2': 'Similarly the [MATH] symmetry along [MATH] is characterized by the order parameter [MATH] which is the Polyakov loop around the [MATH]-cycle: [EQUATION]', '1107.4048-3-29-3': 'In Table [REF] we have collected values of [MATH] for the phases of 5D SYM discussed in this paper.', '1107.4048-3-30-0': 'Before testing the above proposal for holographic QCD in terms of these order parameters, let us see how to obtain this table.', '1107.4048-3-30-1': ""The 'confinement' phase denotes a phase of the 5D SYM whose KK compactification, at small enough [MATH], coincides with the usual confinement phase of YM4."", '1107.4048-3-30-2': 'The latter, clearly has [MATH].', '1107.4048-3-30-3': 'Now, the fact that we have obtained the 4D phase through a KK reduction, along [MATH], implies that the corresponding phase of the 5D theory must have [MATH].', '1107.4048-3-30-4': 'This follows from the fact [CITATION] that in a [MATH] phase, KK reduction does not work since the effective KK scale in a large [MATH] gauge theory becomes [MATH] in stead of the usual [MATH], and goes to zero at large [MATH] .', '1107.4048-3-30-5': ""Thus, the 'confinement' phase must have [MATH], as shown in the first row of Table [REF]."", '1107.4048-3-30-6': ""By similar arguments, the 'deconfinement' phase of 5D SYM which coincides with the deconfinement phase of YM4 must have [MATH] (for validity of the KK reduction) and [MATH] (to exhibit 4D deconfinement)."", '1107.4048-3-30-7': 'This is shown in the second row of Table [REF].', '1107.4048-3-31-0': 'The next two rows in Table [REF] describe two additional phases of SYM5, which appear only for the (AP,AP) b.c. via the [MATH] symmetry.', '1107.4048-3-31-1': 'The third row describes the [MATH] mirror of the confinement phase of YM4, which is characterized by [MATH] and [MATH].', '1107.4048-3-31-2': 'Since [MATH], this phase is a fully 5 dimensional object and is not related to the YM4 (indeed, so far as the Polyakov loop order parameters are concerned, this phase can be identified with a deconfinement phase of SYM5).', '1107.4048-3-31-3': 'Another phase is the mirror of the deconfinement phase in YM4, which is characterized by [MATH] and [MATH].', '1107.4048-3-31-4': 'Since this phase has the same order parameters as the original deconfinement phase, it is not obvious whether these two phases are distinct or smoothly connected through a cross over (we represent the possible phase boundary/crossover by the horizontal dotted line in Figure [REF]).', '1107.4048-3-32-0': 'To describe the next three rows in Table [REF], note that the order parameters [MATH] have a manifestation in gravity.', '1107.4048-3-32-1': ""Thus, e.g. if the [MATH] symmetry along [MATH] is preserved ([MATH]), then, due to the 'large [MATH] volume independence' mentioned in Section [REF], the free energy [MATH] should be independent of [MATH] (equivalently, the action should be linear in [MATH])."", '1107.4048-3-32-2': 'This is satisfied by the action in ([REF]).', '1107.4048-3-32-3': 'Hence for the parameter regime in which the solitonic D4 brane is the dominant phase of the theory, the 5 dim SYM dual exists in a phase characterized by [MATH].', '1107.4048-3-32-4': 'Similarly, the non-linear dependence of the action ([REF]) on [MATH] implies that the gauge theory dual has [MATH].', '1107.4048-3-32-5': 'These results are represented by the 5th row of Table [REF].', '1107.4048-3-32-6': 'One can similarly deduce by looking at the [MATH] dependence of ([REF]) that for the phase of the gauge theory represented by the black D4 solution, [MATH], which could also be inferred by the [MATH] symmetry.', '1107.4048-3-32-7': 'This is represented by the 6th row of Table [REF].', '1107.4048-3-32-8': 'In the last row, we have included a phase of SYM5 with (P, AP) fermion b.c. along ([MATH]) which corresponds to a localized D3 soliton, to be discussed in the next section.', '1107.4048-3-33-0': 'We now come back to the conventional proposal of holographic QCD discussed in the beginning of this section; let us define this proposal as a combination of three propositions:', '1107.4048-3-34-0': '(a) the solitonic D4 brane phase of gravity corresponds to the confinement phase of YM4,', '1107.4048-3-35-0': '(b) the black D4 corresponds to the deconfinement phase of YM4, and', '1107.4048-3-36-0': '(c) the SS transition between the two gravity solutions corresponds to the confinement/deconfinement transition of YM4.', '1107.4048-3-37-0': 'In terms of Table [REF], both the solitonic D4 and the confinement phase of YM4 (regarded as a phase of SYM) have [MATH]; hence both of these phases have the same [MATH] symmetry and (a) can in principle be valid.', '1107.4048-3-37-1': 'However, (b) cannot be valid, since the black D4 phase ([MATH]) and the deconfinement phase of YM4 ([MATH]) have different [MATH] symmetry.', '1107.4048-3-37-2': 'In particular, the black D4 corresponds to a phase that cannot be KK reduced to YM4 at all because of vanishing [MATH].', '1107.4048-3-37-3': 'In fact, it would correspond to the [MATH] mirror of the confinement phase of YM4 (which can be identified with a deconfinement phase of SYM5 which is intrinsically 5 dimensional).', '1107.4048-3-37-4': 'Thus, there must be at least one phase boundary between the deconfinement phase of YM4 and the black D4 phase (see Figure [REF] for one possibility).', '1107.4048-3-37-5': 'Regarding (c), if the SS transition persists even at weak coupling, then it clearly cannot coincide with the confinement/deconfinement transition.', '1107.4048-3-37-6': 'The reason is that, even at weak coupling the SS transition can only occur at [MATH] (since even stringy effects should satisfy the [MATH] symmetry), whereas it is known that the confinement/deconfinement phase transition temperature of YM4 goes down ([MATH] goes up) at smaller values of [MATH] (see Figure [REF]).', '1107.4048-3-38-0': 'Through the arguments in this section, it is obvious that the black D4 brane solution does not correspond to the deconfinement phase in the 4 dimensional Yang-Mills theory and the SS transition does not correspond to the confinement/deconfinement transition.', '1107.4048-3-38-1': 'In the next section, we propose an alternative correspondence, which resolves these problems.', '1107.4048-3-39-0': '# Our proposal for the confinement/deconfinement transition in holographic QCD', '1107.4048-3-40-0': 'In the previous section, we considered the conventional picture of holographic QCD at finite temperature and discussed some of the problems in describing phases of YM4.', '1107.4048-3-40-1': 'In this section, we propose an alternative picture to resolve these problems.', '1107.4048-3-41-0': 'In the above discussion, we described finite temperature gauge theory by choosing AP b.c. for fermions on the brane along the compactified Euclidean time.', '1107.4048-3-41-1': 'However, let us recall that the gauge theory of interest here is pure Yang Mills theory in four dimensions, which does not have fermions.', '1107.4048-3-41-2': 'Fermions reappear when the validity condition ([REF]) is enforced and [MATH] goes above the KK scales [MATH].', '1107.4048-3-41-3': 'In this sense, fermions are an artifact of the holographic method and in the region of validity of the pure YM theory, the periodicity of the fermion should not affect the gauge theory results.', '1107.4048-3-42-0': 'If we start with the 5 dimensional SYM on [MATH] with a periodic temporal circle, the partition function is the twisted one: Tr [MATH], and is not the standard thermal partition function.', '1107.4048-3-42-1': 'However, under the limit [MATH] and [MATH], in which the 4 dimensional YM theory appears, the fermions are effectively decoupled; hence [MATH].', '1107.4048-3-42-2': ""Thus in this limit the 'twisted' partition becomes the usual thermal partition function, even with the periodic b.c. for the fermions."", '1107.4048-3-42-3': 'More precisely, we have [EQUATION]', '1107.4048-3-42-4': 'Thus, independent of the boundary condition along the [MATH], we obtain the thermal partition function of YM4.', '1107.4048-3-43-0': 'As a consequence, it is pertinent to study the phase structure of the 5 dimensional SYM on [MATH] with (P,AP) boundary condition through holography.', '1107.4048-3-43-1': 'Indeed, in this sector, we will find a phase transition in gravity, which continues naturally, at weak coupling, to the confinement/deconfinement transition in the 4D Yang-Mills theory.', '1107.4048-3-44-0': '## Gregory-Laflamme transition in the gravity theory with (P,AP) boundary condition', '1107.4048-3-45-0': 'In this section, we reconsider the gravity theory described in Section [REF], now compactified on [MATH] with (P,AP) b.c. for fermions.', '1107.4048-3-45-1': 'In this case, contrary to the (AP,AP) case, the black D4 brane solution ([REF]) is not allowed (see the comment below ([REF])).', '1107.4048-3-45-2': 'However a non-trivial phase boundary still exists, as we will see shortly.', '1107.4048-3-45-3': 'The detailed phase structure for this boundary condition is summarized in [CITATION] (with [MATH] and [MATH] exchanged); see also [CITATION].', '1107.4048-3-45-4': 'We will now outline the salient features.', '1107.4048-3-46-0': 'Let us start from large [MATH].', '1107.4048-3-46-1': 'In this region the solitonic D4 brane solution ([REF]) is thermodynamically dominant.', '1107.4048-3-46-2': 'As we decrease [MATH], it reaches [MATH], which is the order of the effective string length at [MATH] [CITATION].', '1107.4048-3-46-3': 'Below this value, winding modes of the string wrapping on the [MATH]-cycle would be excited.', '1107.4048-3-46-4': 'Thus the gravity description given by ([REF]) would be valid only if [EQUATION]', '1107.4048-3-46-5': 'In order to avoid this problem, we perform a T-duality along the [MATH]-cycle and go to the IIB frame, where the solitonic D4 solution becomes a solitonic D3 brane uniformly smeared on the dual [MATH]-cycle.', '1107.4048-3-46-6': 'The metric describing this solution is given by [EQUATION]', '1107.4048-3-46-7': 'Here [MATH] is the dual of [MATH], hence [MATH] has a periodicity [MATH].', '1107.4048-3-46-8': 'By considering winding modes on this metric, we can see that the gravity description is now valid if [MATH].', '1107.4048-3-46-9': 'Thus even when [MATH] is below ([REF]), the gravity analysis in the IIB frame is possible.', '1107.4048-3-47-0': 'As [MATH] decreases (hence the dual radius in the IIB frame increases), the uniformly smeared solitonic D3 brane becomes thermodynamically unstable at a certain temperature.', '1107.4048-3-47-1': 'This temperature, called the Gregory-Laflamme (GL) instability point, is numerically given by [CITATION]: [EQUATION]', '1107.4048-3-47-2': 'It is expected that, even before [MATH] is lowered all the way to [MATH], the smeared solitonic D3 brane becomes meta-stable and undergoes a first order Gregory-Laflamme (GL) transition at an inverse temperature [MATH], leading to a more stable, solitonic D3 brane localized on the dual cycle.', '1107.4048-3-47-3': 'This localized solution spontaneously breaks translation symmetry along the temporal direction.', '1107.4048-3-47-4': 'Although it is difficult to derive the critical temperature [MATH] precisely, we can approximately evaluate it as follows [CITATION].', '1107.4048-3-47-5': 'The metric of the localized solitonic D3 brane is approximately given by that of the solitonic D3 on [MATH] for a sufficiently large radius [MATH] of the dual cycle [CITATION], [EQUATION]', '1107.4048-3-47-6': 'Here [MATH] (see Eq. ([REF])).', '1107.4048-3-47-7': 'The value of [MATH] is determined by the smoothness condition (as in ([REF])).', '1107.4048-3-47-8': '[MATH] is regarded as a radial component of a spherical coordinate and is related to the cylindrical coordinates [MATH] of ([REF]) by roughly [MATH].', '1107.4048-3-47-9': 'Here we implicitly assume that the soliton is localized at [MATH].', '1107.4048-3-47-10': 'This approximation is valid if [MATH] and [MATH] are sufficiently small such that we can ignore the finite size effect associated with the dual circle [CITATION].', '1107.4048-3-48-0': 'The classical action of this solution turns out to be [EQUATION]', '1107.4048-3-48-1': 'By comparing this classical action with ([REF]) , we can estimate that the GL transition will happen around [EQUATION]', '1107.4048-3-48-2': 'Note that, near this critical point, [MATH] is similar in magnitude to, but less than, the dual radius [CITATION], so we expect ([REF])-([REF]) to receive some corrections from the finite size effect.', '1107.4048-3-49-0': 'In addition to these two solutions, ([REF]) and ([REF]), there is another solution: solitonic D3 brane non-uniformly smeared on the dual cycle.', '1107.4048-3-49-1': 'The metric of this solution is perturbatively derived in [CITATION].', '1107.4048-3-49-2': 'This non-uniform solution arises at the GL instability point and is always unstable.', '1107.4048-3-49-3': 'Although the complete behaviour of the non-uniform solution has not been explored, it is expected that it merges with the localized solitonic D3 as shown in Figure [REF] [CITATION].', '1107.4048-3-49-4': 'The fact that the GL instability temperature ([REF]) is higher than the approximately obtained GL critical temperature ([REF]) is consistent with this expectation.', '1107.4048-3-50-0': 'Let us make a few comments on the localized solitonic D3 solution ([REF]).', '1107.4048-3-50-1': 'Firstly, through a calculation similar to ([REF]), the gravity description can be shown to be valid (i.e. stringy effects can be ignored) if [MATH].', '1107.4048-3-50-2': 'Secondly, this solution ceases to exist if [MATH] is too large, as shown in Figure [REF]; intuitively, if [MATH] were too large, the dual cycle would become smaller than the size of the localized soliton, which is not possible.', '1107.4048-3-50-3': 'Thirdly, the free energy [MATH] of this solution, unlike that of ([REF]), is proportional to temperature; hence the entropy of this solution is non-zero at [MATH] and the temporal [MATH] symmetry is broken.', '1107.4048-3-50-4': 'In addition, because of the non-trivial [MATH]-dependence of the classical action ([REF]) we can see that the [MATH] symmetry along the [MATH]-cycle is also broken.', '1107.4048-3-50-5': 'Thus the Polyakov loop [MATH] and [MATH] in this solution are both non-zero as shown in Table [REF] and, contrary to the black D4 solution, the localized solitonic D3 solution is appropriate for a description of the deconfinement phase in the dual gauge theory (which also has [MATH]).', '1107.4048-3-50-6': 'We will build on this observation in the next subsection.', '1107.4048-3-51-0': '## Gregory-Laflamme transition as a confinement/deconfinement transition', '1107.4048-3-52-0': 'The phase structure of the 5 dimensional SYM on [MATH] with the (P,AP) boundary condition is shown in Figure [REF].', '1107.4048-3-52-1': 'The strong coupling region (the blue region in Figure [REF]) described by type II supergravity and is characterized by the GL phase transition which occurs at a temperature given by ([REF]).', '1107.4048-3-52-2': 'In the weak coupling region (the green region in Figure [REF]), the 4 dimensional Yang-Mills is realized at low temperatures ([MATH]).', '1107.4048-3-52-3': 'Although this region is common to the (P,AP) phase diagram in Figure [REF] and the (AP,AP) phase diagram in Figure [REF], the mirror of this region under the [MATH]) does not exist in Figure [REF] since the (P,AP) b.c is not [MATH]-symmetric.', '1107.4048-3-53-0': 'The main point to emphasize here it that: contrary to the previous phase structure in the (AP,AP) case, now the localized solitonic D3 phase has the same order parameters ([MATH]) as the deconfinement phase, thus making it plausible that these two phases are smoothly connected.', '1107.4048-3-53-1': 'In Figure [REF], we have indicated this by assuming the simplest extrapolation through the region of the intermediate coupling.', '1107.4048-3-54-0': 'Our proposal', '1107.4048-3-55-0': 'In view of the above observations, we propose a strong coupling continuation of weakly coupled 4-dimensional Yang Mills theory as shown in Table [REF].', '1107.4048-3-56-0': 'Of course, our proposal is based on a simple extrapolation between the intermediate coupling regime, and the real story there could be more involved.', '1107.4048-3-56-1': 'However the mere existence of such a simple extrapolation is a significant improvement over the previous proposal in the (AP,AP) case, where we are certain that there has to be at least one phase boundary between the deconfinement phase and black D4 solution (for the simple reason that their order parameters have different values).', '1107.4048-3-56-2': 'It is clearly important, therefore, to further investigate the nature of the deconfinement phase in terms of the localized solitonic D3 solution based on the correspondence outlined above.', '1107.4048-3-57-0': '# New correspondences in holographic QCD', '1107.4048-3-58-0': 'Using our proposed correspondence, we can explain anew several phenomena in the gauge theory from gravity.', '1107.4048-3-58-1': 'In this section, we list some of these phenomena.', '1107.4048-3-59-0': '## Polyakov loop and D3 brane distribution', '1107.4048-3-60-0': 'In our proposal, we identified the GL transition in the IIB frame to the confinement/deconfinement transition in the gauge theory.', '1107.4048-3-60-1': 'In the GL transition, the distribution of the D3 branes on the dual circle changes from a uniform distribution at low temperatures to a localized one at high temperatures.', '1107.4048-3-60-2': 'In this subsection, we explain what the corresponding phenomenon is in the gauge theory.', '1107.4048-3-61-0': 'Since the original five-dimensional gauge theory (of the D4 branes) appears in the IIA frame, in order to understand the role of the D3 branes, we need to consider a T-duality along the temporal direction.', '1107.4048-3-61-1': 'Under this T-duality, [MATH], which is transverse to the D3 brane, is mapped to the gauge potential [MATH] on the D4 brane.', '1107.4048-3-61-2': ""Thus the D3 brane distribution on the dual circle is related to a distribution of [MATH]'s, which are defined as the eigenvalues of the Polyakov loop operator [MATH]."", '1107.4048-3-62-0': 'If the D3 branes are uniformly distributed, the [MATH] are also uniformly distributed.', '1107.4048-3-62-1': 'In that case, by choosing an appropriate gauge, we can take [MATH].', '1107.4048-3-62-2': 'Then the temporal Polyakov loop operator ([REF]) becomes [EQUATION]', '1107.4048-3-62-3': 'Hence this distribution corresponds to the confinement phase.', '1107.4048-3-62-4': 'On the other hand, if the D3 branes are localized, then [MATH] are also localized and the Polyakov loop becomes non-zero, which characterizes the deconfinement phase.', '1107.4048-3-62-5': 'This observation is consistent with the entropy arguments in section [REF].', '1107.4048-3-63-0': 'The above discussion shows a direct relation between the D3 brane distribution and the eigenvalue distribution of the Polyakov loop operator.', '1107.4048-3-63-1': 'This latter quantity can sometimes be explicitly evaluated [CITATION].', '1107.4048-3-63-2': 'E.g. in [CITATION], for four dimensional Yang Mills theory on a small [MATH], we found that the uniform, non-uniform and localized distribution of [MATH], all appear, with obvious correspondence to similar gravitational solutions.', '1107.4048-3-63-3': 'Especially the free energies of these three solutions show a ""swallow tail"" relation similar to Figure [REF].', '1107.4048-3-63-4': '(See Figure 3 in [CITATION].)', '1107.4048-3-63-5': 'This correspondence strongly supports our proposal.', '1107.4048-3-64-0': '## Gregory-Laflamme transition as a Hagedorn transition', '1107.4048-3-65-0': 'Our proposal opens up the interesting possibility of a relation between the GL transition and the Hagedorn transition.', '1107.4048-3-66-0': 'It is known that the GL instability is an instability of the KK modes of the graviton along the compact circle [CITATION].', '1107.4048-3-66-1': 'In our case, the KK modes along the dual temporal circle, which are associated with the GL instability at ([REF]) in the IIB description, are mapped to winding modes around the temporal circle through the T-duality [CITATION].', '1107.4048-3-66-2': 'This then indicates that the type IIB GL transition is associated with the excitation of the winding modes of the IIA string.', '1107.4048-3-66-3': 'This phenomenon is similar to the Hagedorn transition in string theory [CITATION], where the associated instability is caused by temporal winding modes.', '1107.4048-3-66-4': 'Thus the GL transition in the IIB description might correspond to the Hagedorn transition in the IIA description.', '1107.4048-3-67-0': 'Note that from large [MATH] gauge-theoretic calculations also, the usual confinement/deconfinement transition is believed to be related to the Hagedorn transition.', '1107.4048-3-67-1': 'It has been explicitly shown in weakly coupled gauge theories [CITATION].', '1107.4048-3-67-2': 'This makes it plausible that the Hagedorn transition in the Yang-Mills theory continues to the Hagedorn transition in the IIA string, which, as we argued above, is possibly the dual of the GL transition in the IIB supergravity.', '1107.4048-3-68-0': '# Chiral symmetry restoration in Sakai-Sugimoto model', '1107.4048-3-69-0': 'In the previous sections, we have seen that the conventional holographic representation of the confinement/deconfinement transition as the SS transition is fraught with problems, to circumvent which we proposed in Section [REF] a different interpretation in terms of a GL transition.', '1107.4048-3-69-1': 'However, the SS transition has widely been employed in holographic QCD and purports to explain several phenomena in real QCD.', '1107.4048-3-69-2': 'In particular, the chiral symmetry restoration in the Sakai-Sugimoto model was neatly explained in [CITATION].', '1107.4048-3-69-3': 'In their scenario the black D4 brane plays a crucial role.', '1107.4048-3-69-4': 'However, since in our proposal the black branes do not appear anymore, we need to find an alternative idea for realizing chiral symmetry restoration.', '1107.4048-3-69-5': 'In this section, we discuss how chiral symmetry restoration can happen in the localized solitonic D3 background.', '1107.4048-3-70-0': '## Sakai-Sugimoto model and chiral symmetry breaking', '1107.4048-3-71-0': 'The Sakai-Sugimoto model [CITATION] was proposed to describe low energy hadron physics in holographic QCD and elegantly reproduces many aspects of the real QCD.', '1107.4048-3-71-1': 'We first briefly review this model and show how chiral symmetry breaking at low temperatures is realized in terms of the dual gravity.', '1107.4048-3-72-0': 'The Sakai-Sugimoto model is an extension of the holographic model of QCD discussed in section [REF].', '1107.4048-3-72-1': 'Sakai and Sugimoto added, to the [MATH] D4 brane system, [MATH] D8 and [MATH] branes which are localized on the [MATH] circle and fill all other directions as follows [EQUATION]', '1107.4048-3-72-2': 'Here [MATH] denotes the compactified directions.', '1107.4048-3-72-3': '(See Figure [REF] (a).)', '1107.4048-3-72-4': 'This model has a free parameter [MATH], which is the asymptotic ([MATH]) distance between the D8 and [MATH] brane on the [MATH] circle.', '1107.4048-3-72-5': 'In the original Sakai-Sugimoto model [CITATION] [MATH] was taken to be [MATH]; however, we will let it be general and take values in [MATH] (the other half, [MATH] is related by a reflection and need not be considered separately).', '1107.4048-3-73-0': 'This model has [MATH] gauge symmetry on the [MATH] D8 and [MATH] branes, which can be interpreted as a chiral [MATH] flavor symmetry in QCD.', '1107.4048-3-73-1': 'As we will see soon, in some situations, the D8 and [MATH] branes merge and the chiral symmetry is broken to a single [MATH].', '1107.4048-3-73-2': 'Sakai and Sugimoto proposed that this is the holographic realization of chiral symmetry breaking [CITATION].', '1107.4048-3-74-0': 'Let us take a large [MATH] limit a la Maldacena and, according to the principle of holography, replace the [MATH] D4 branes with a corresponding gravity solution.', '1107.4048-3-74-1': ""Here we will keep [MATH] such that we can ignore the back-reaction of the D8/[MATH] branes onto the background geometry (this is the so-called 'probe approximation')."", '1107.4048-3-74-2': 'In that case, the background (D4 brane) geometry is determined thermodynamically (as the dominant classical solution at a given temperature, as we have done in sections [REF] and [REF]), and the D8/[MATH] brane configuration coupled to this background, is determined dynamically, with the given distance [MATH] as a boundary condition.', '1107.4048-3-75-0': 'Chiral symmetry breaking in this model happens as follows.', '1107.4048-3-75-1': 'At sufficiently low temperatures, the favoured geometric background is that of solitonic D4 solution ([REF]), as we have seen in section [REF] and [REF].', '1107.4048-3-75-2': 'On this background, the D8 and [MATH] cannot extend separately and need to merge as shown in Figure [REF] (b).', '1107.4048-3-75-3': 'As a result, the [MATH] gauge symmetry on the D8 and [MATH] is broken to [MATH], representing chiral symmetry breaking ([MATH]SB) in the dual gauge theory.', '1107.4048-3-76-0': '## Chiral symmetry restoration in the black D4 brane background', '1107.4048-3-77-0': 'In the gauge theory, it is expected that chiral symmetry is restored at a sufficiently high temperature.', '1107.4048-3-77-1': 'Thus if holographic QCD is to work, there should be a corresponding phenomena in the dual gravity description.', '1107.4048-3-78-0': 'In [CITATION], a mechanism for chiral symmetry restoration was suggested by considering D8 and [MATH] in the black D4 brane background ([REF]) in the (AP,AP) case.', '1107.4048-3-78-1': ""Contrary to the solitonic D4 brane geometry ([REF]), the [MATH] plane of the black D4 background has a 'cigar' geometry."", '1107.4048-3-78-2': 'Thus the D8 and [MATH] branes can wrap the cigar separately as shown in Figure [REF] (c).', '1107.4048-3-78-3': 'If such a configuration is energetically favoured, [MATH] gauge symmetry is preserved and chiral symmetry is restored.', '1107.4048-3-79-0': 'In addition to this configuration, another configuration shown in Figure [REF] (d) is possible in the black D4 background.', '1107.4048-3-79-1': 'This configuration is similar to Figure [REF] (b) and chiral symmetry is broken.', '1107.4048-3-79-2': 'In [CITATION], energies of these two configurations are compared with the black D4 background.', '1107.4048-3-79-3': 'It was found (see Figure 7 of [CITATION]) that for [MATH], it is configuration (c) which is always favoured, indicating a concurrence of deconfinement and chiral symmetry restoration transitions at [MATH], while for [MATH], a new window [MATH] opens up where configuration (d) is the favoured one, indicating coexistence of deconfinement and broken chiral symmetry.', '1107.4048-3-80-0': '## Chiral symmetry restoration in the localized solitonic D3 brane.', '1107.4048-3-81-0': 'Although the chiral symmetry restoration was explained, as above, in the black D4 brane background, our main thesis in this paper is that the black D4 solution itself is fraught with problems, if we interpret this solution as the deconfinement phase of the four dimensional gauge theory (see Section [REF]).', '1107.4048-3-81-1': 'The alternative we proposed in Section [REF] is that the localized solitonic D3 brane solution ([REF]) in the (P,AP) b.c. should be taken as the correct representation of the deconfinement phase in the dual gauge theory.', '1107.4048-3-81-2': 'In view of this, we need to understand chiral symmetry restoration in this solution instead of in the black D4 solution.', '1107.4048-3-82-0': 'One subtlety in the application of our proposal to the Sakai-Sugimoto model is the existence of the fundamental quarks from the open string between the D4 and D8/[MATH] brane.', '1107.4048-3-82-1': 'These quarks are, of course, not decoupled in the 4 dimensional limit [MATH].', '1107.4048-3-82-2': ""Hence, unlike in case of the adjoint fermions, where the P and the AP b.c. on the thermal cycle reduce to the same quantity in the four dimensional limit (see ([REF])), for the fundamental quarks the P and the AP b.c's differ in the 4D limit; hence, we must choose the AP b.c. on the temporal cycle to investigate the thermodynamics of the Sakai-Sugimoto model."", '1107.4048-3-82-3': 'Since our proposal crucially uses P b.c. for the fermions along the temporal cycle, we must address this issue .', '1107.4048-3-82-4': 'It turns out that with a small addition to our model, viz. that of an imaginary chemical potential [CITATION], we can obtain the desired AP b.c. around the temporal cycle for the quarks retaining the P b.c. for the adjoint fermions.', '1107.4048-3-82-5': 'Details of this approach will appear in a forthcoming paper [CITATION].', '1107.4048-3-82-6': 'Introduction of such a chemical potential does not cause any essential modification to the analysis that follows below, as shown in detail in [CITATION].', '1107.4048-3-82-7': 'Consequently, we will proceed below with periodic (fundamental as well as adjoint) fermions.', '1107.4048-3-83-0': 'In our proposal, the gravity analysis has been done in IIB supergravity by performing a T-duality along the [MATH]-cycle: [MATH].', '1107.4048-3-83-1': 'Thus it is convenient to dualize the brane configuration of the Sakai-Sugimoto model to the IIB frame.', '1107.4048-3-83-2': 'Since all the D branes in the Sakai-Sugimoto model wrap the [MATH], the above T-duality maps the D4 branes and D8/[MATH] branes to D3 and D7/[MATH] branes spreading [EQUATION]', '1107.4048-3-83-3': 'See Figure [REF] (e) also.', '1107.4048-3-83-4': 'The distribution along [MATH] of the D branes in the IIB description (related to the gauge field [MATH] in the IIA description), is determined dynamically.', '1107.4048-3-83-5': 'In the probe approximation, according to the analyses in section [REF], at temperatures below the GL critical temperature ([REF]), the D3 branes are distributed uniformly and, above it, they are localized on the [MATH].', '1107.4048-3-83-6': 'The fate of the chiral gauge group in the gravity representation depends on the stable configurations of the probe D7/[MATH] branes in these backgrounds.', '1107.4048-3-84-0': 'In the uniformly smeared solitonic D3 brane geometry ([REF]) shown in Figure [REF] (f), the situation is similar to (b) in Figure [REF].', '1107.4048-3-84-1': 'Since the [MATH] direction is smoothly pinched off at [MATH], the D7 and [MATH] have to merge and chiral symmetry is broken.', '1107.4048-3-85-0': 'On the other hand, in the localized solitonic D3 brane geometry, the horizontal direction is not fully pinched off.', '1107.4048-3-85-1': 'Recall that the geometry ([REF]) is pinched off at [MATH], where [MATH] and [MATH].', '1107.4048-3-85-2': 'Thus [MATH] can reach zero.', '1107.4048-3-85-3': 'As a result, D7 and [MATH] can extend separately as shown in Figure [REF] (g).', '1107.4048-3-85-4': 'This configuration is similar to (e) in Figure [REF] and would restore chiral symmetry.', '1107.4048-3-85-5': 'In addition to this configuration, a chiral symmetry broken configuration is also possible as shown in Figure [REF] (h).', '1107.4048-3-86-0': 'The last task is the evaluation of the stability of these two configurations at temperatures higher than [MATH] ([REF]).', '1107.4048-3-86-1': 'In the flat space, the force between a single D3 and single D7 (or between a single D3 and a single [MATH]) in our configuration is repulsive, since the number of the Neumann-Dirichlet open strings between them is 6 [CITATION].', '1107.4048-3-86-2': 'Thus we expect that even after taking the large [MATH] and near horizon limits, the force may be repulsive.', '1107.4048-3-86-3': 'As a result, the D7/[MATH] branes, which are separated from the localized SD3 in the [MATH]-direction, would try to move away as far as possible in this direction.', '1107.4048-3-86-4': 'However, since the [MATH]-direction is compactified on a circle, the D7/[MATH] branes should then end as being fixed at the point on the [MATH]-circle which is antipodal to the localized SD3.', '1107.4048-3-86-5': 'E.g. if we put the localized SD3 brane at [MATH], the D7 as well as the [MATH] branes will both be at [MATH].', '1107.4048-3-86-6': 'See the left diagram of Figure [REF].', '1107.4048-3-86-7': 'In this case, we can effectively restrict the dynamics of these branes to ([MATH]) plane (given by [MATH]).', '1107.4048-3-86-8': 'Thus the problem reduces to finding classical solutions [MATH] with a boundary condition: [MATH]), where we put the D7 at [MATH] and [MATH] at [MATH].', '1107.4048-3-87-0': 'The problem stated above is difficult to solve precisely near the GL transition since the background metric ([REF]) is only an approximate description.', '1107.4048-3-87-1': 'The metric around the D7/[MATH] brane (at [MATH]) becomes more and more accurate, however, when [MATH].', '1107.4048-3-87-2': 'As a result, we solve for the stable configuration in this limit.', '1107.4048-3-87-3': 'Details of the calculation are presented in appendix [REF].', '1107.4048-3-87-4': 'We find three solutions, corresponding to the D brane configurations of Figures [REF] (g) and (h), as shown in Figure [REF].', '1107.4048-3-87-5': 'In the appendix, we compare the classical DBI actions of these solutions numerically, as depicted in Figure [REF] (a) (see also the phase diagram in Figure [REF]): as one increases temperature beyond a certain value, chiral symmetry is restored, the transition being of first order.', '1107.4048-3-87-6': 'Therefore, one can see that even in our proposal, similarly to the case of the black D4 [CITATION], we can explain chiral symmetry restoration at high enough temperatures.', '1107.4048-3-88-0': '# Conclusions', '1107.4048-3-89-0': 'In this paper, we showed that the conventional representation of the confinement/deconfinement transition in holographic QCD has several problems and proposed an alternative representation which resolves these problems.', '1107.4048-3-90-0': 'As mentioned earlier, problems similar to the above had also been encountered in the study of two dimensional bosonic gauge theory in [CITATION].', '1107.4048-3-90-1': 'This indicates that the issues addressed in this paper are rather general in the discussion of holography for non-supersymmetric gauge theories at finite temperatures.', '1107.4048-3-90-2': 'To elaborate, in the standard holographic procedure, a [MATH]-dimensional non-supersymmetric gauge theory is first constructed through the KK reduction of a [MATH]-dimensional super Yang-Mills theory on a Scherk-Schwarz circle.', '1107.4048-3-90-3': 'The [MATH]-dimensional SYM at large [MATH] can be mapped to a scaling limit of D[MATH] brane geometries [CITATION].', '1107.4048-3-90-4': 'At finite temperatures, because of the two compact cycles (temporal and Scherk-Schwarz), several distinct solutions (depending on boundary conditions) appear in gravity as shown in section [REF] and [REF]: solitonic D[MATH] (equivalently, uniformly smeared solitonic D[MATH]), localized solitonic D[MATH], and black D[MATH].', '1107.4048-3-90-5': 'The black D[MATH] brane solution appears at high temperatures in the (AP,AP) case, while the localized solitonic D[MATH] brane solution is the high temperature phase in the (P,AP) case.', '1107.4048-3-90-6': 'A table similar to Table [REF] can again be constructed, where the appropriate order parameters would be [MATH]; these would again appear to favour the localized solitonic D[MATH] phase as the more suitable representation of the deconfinement of [MATH]-dimensional YM theory (rather than the more conventional black D[MATH] phase which appears only in the (AP,AP) b.c.).', '1107.4048-3-90-7': 'Following this logic, a [MATH]-dimensional analogue of our proposal (see Section [REF]) would appear to give a better description of holographic QCD in [MATH] dimensions.', '1107.4048-3-90-8': 'In particular, we believe that the Gregory-Laflamme transition between the solitonic D[MATH] and localized solitonic D[MATH], would, as in this paper, be related to the confinement/deconfinement transition in the [MATH]-dimensional gauge theory.', '1107.4048-3-91-0': '## Further questions', '1107.4048-3-92-0': 'In order to further understand holographic QCD through the above proposal, it would be of interest to address the following questions.', '1107.4048-3-93-0': 'Transition temperature', '1107.4048-3-94-0': 'From ([REF]), the critical temperature of the confinement/deconfinement transition (the GL transition) would be [MATH].', '1107.4048-3-94-1': 'On the other hand, holographic QCD seems to predict that the square root of the QCD string tension is [MATH], as in ([REF]), whereas the glueball masses are [MATH] [CITATION].', '1107.4048-3-94-2': 'It would be important to understand the reason for the separation of these scales and how they evolve from strong coupling to weak coupling.', '1107.4048-3-95-0': 'Quantitative correspondence', '1107.4048-3-96-0': 'Although our new proposal reproduces the known qualitative features of 4 dimensional Yang-Mills theory, it does not automatically lead to a quantitative agreement.', '1107.4048-3-96-1': 'For example, the free energy of the deconfinement phase of the YM theory at a sufficiently high temperature must be proportional to [MATH], since the coupling becomes weak and the theory becomes approximately conformal.', '1107.4048-3-96-2': 'However the free energy of the localized solitonic D3 is proportional to [MATH] (as can be seen from ([REF])).', '1107.4048-3-96-3': 'This is not entirely surprising since the functional form of [MATH] can change as one evolves from weak coupling to strong coupling.', '1107.4048-3-96-4': 'Furthermore, one has to exercise caution in defining a high temperature limit of YM4 in the holographic context since the temperature must always remain much smaller than the KK scale.', '1107.4048-3-97-0': 'YM4 from SYM5 with (AP,AP) In this article, we have emphasized the correspondence between YM4 and SYM5 with (P,AP) b.c. However, as we pointed out in ([REF]), the 5 dimensional SYM with (AP,AP) b.c. should also be related to YM4, since the boundary condition becomes irrelevant in the limit [MATH] and [MATH].', '1107.4048-3-97-1': 'A possible way this correspondence may work is as follows.', '1107.4048-3-97-2': 'In the (AP,AP) case, if we treat the black D4 solution (the blue region in the bottom half of Figure [REF]), which is not related to YM4, as irrelevant, and focus on the solitonic D4 brane, its winding modes would appear to be light around the temperature ([REF]).', '1107.4048-3-97-3': 'A Hagedorn transition parallel to the (P,AP) case might occur above this temperature and might continue to the confinement/deconfinement transition in YM4.', '1107.4048-3-97-4': 'However, in order to investigate it further through gravity, we need to understand the gauge/gravity correspondence in the 0B frame as mentioned in footnote [REF].', '1107.4048-3-98-0': 'Real time It would be important to explore what geometry corresponds to the deconfinement phase in the real time formalism.', '1107.4048-3-98-1': 'An understanding of this would allow us to address dynamical properties of the deconfinement phase, e.g. transport properties.', '1107.4048-3-98-2': 'Since previous results in this area were based on the black brane solutions, it would be important to see how well-known results such as the viscosity bound [CITATION] can be derived in our proposal.', '1107.4048-3-99-0': 'AdS/CMT and chemical potential dependence', '1107.4048-3-100-0': 'The SS transition for D3 branes has been studied in the context of the AdS/CMT correspondence to investigate the superconductor/insulator transition in 2+1 dimension [CITATION].', '1107.4048-3-100-1': 'In these studies, a chemical potential for a [MATH] charge was introduced and the phase structure involving this chemical potential has been derived.', '1107.4048-3-100-2': 'It would be interesting to ask whether our proposal has any bearing on these studies, e.g., whether the GL transition analogous to the one discussed here can be a candidate for the superconductor/insulator or some other transition.', '1107.4048-3-100-3': 'A possible line of investigation could be to study the chemical potential dependence of [MATH] for the GL transition vis-a-vis that for the SS transition and see whether any qualitative differences appear.'}","{'1107.4048-4-0-0': 'We discuss the phase structure of [MATH] D4 branes wrapped on a temporal (Euclidean) and a spatial circle, in terms of the near-horizon geometries.', '1107.4048-4-0-1': 'This system has been studied previously to understand four dimensional pure [MATH] Yang-Mills theory (YM4) through holography.', '1107.4048-4-0-2': 'In the usual treatment of the subject, the phase transition between the solitonic D4 brane and the black D4 brane is interpreted as the strong coupling continuation of the confinement/deconfinement transition in YM4.', '1107.4048-4-0-3': 'We show that this interpretation is not valid, since the black D4 brane and the deconfinement phase of YM4 have different realizations of the [MATH] centre symmetry and cannot be identified.', '1107.4048-4-0-4': 'We propose an alternative gravity dual of the confinement/deconfinement transition in terms of a Gregory-Laflamme transition of the soliton in the IIB frame, where the strong coupling continuation of the deconfinement phase of YM4 is a localized D3 soliton.', '1107.4048-4-0-5': 'Our proposal offers a new explanation of several aspects of the thermodynamics of holographic QCD.', '1107.4048-4-0-6': 'As an example, we show a new mechanism of chiral symmetry restoration in the Sakai-Sugimoto model.', '1107.4048-4-0-7': 'The issues discussed in this paper pertain to gravity duals of non-supersymmetric gauge theories in general.', '1107.4048-4-1-0': '# Introduction', '1107.4048-4-2-0': 'Strongly coupled field theories such as QCD and various condensed matter systems are not amenable to perturbative calculations and require numerical or other non-perturbative tools.', '1107.4048-4-2-1': 'A powerful new tool in this context is holography, which maps gauge theories to gravity in a higher dimension [CITATION].', '1107.4048-4-2-2': 'Holography, however, is most reliable and quantitative for special supersymmetric gauge theories.', '1107.4048-4-2-3': 'In the generic case, including for pure Yang Mills theories, inferences from gravity tend to be less quantitative due to various degrees of extrapolations, although such exercises have proved to be rather valuable, e.g., in the context of holographic QCD [CITATION].', '1107.4048-4-3-0': 'Recently, it was pointed out in [CITATION] that there are problems with a naive application of holography to a two dimensional large [MATH] bosonic gauge theory at finite temperature.', '1107.4048-4-3-1': 'The dual of this system is based on gravity backgrounds involving [MATH] D2 branes on [MATH].', '1107.4048-4-3-2': 'The phase diagram of gravity, interpreted naively, does not admit a unique continuation to the regime of gauge theory, owing to its dependence on the boundary condition for fermions on the brane.', '1107.4048-4-3-3': 'However, it was shown in [CITATION] that the phase structures in gravity and gauge theory could be smoothly connected through an appropriate choice of fermion boundary conditions.', '1107.4048-4-4-0': 'In this paper, we will show that similar problems arise when we holographically analyse more general non-supersymmetric gauge theories at finite temperature.', '1107.4048-4-4-1': 'The particular example we will focus on is that of holographic QCD from D4 branes [CITATION].', '1107.4048-4-4-2': 'We will discuss some problems with the usual correspondence between the confinement/deconfinement transition in QCD and the Scherk-Schwarz transition between a solitonic D4 brane and a black D4 brane in the gravity dual.', '1107.4048-4-4-3': 'Some of these problems were first discussed in [CITATION] (see section [REF] for further details).', '1107.4048-4-4-4': 'Especially we will show that the black D4 brane cannot be identified with the (strong coupling continuation of the) deconfinement phase in QCD in four dimensions.', '1107.4048-4-4-5': 'As a resolution of these problems, we will propose an alternative scenario in which the confinement/deconfinement transition corresponds to a Gregory-Laflamme transition [CITATION] between a uniformly distributed D3 soliton and a localized D3 soliton in the IIB frame.', '1107.4048-4-5-0': 'The scenario we propose suggests that we need to reconsider several previous results in holographic QCD including the Sakai-Sugimoto model [CITATION].', '1107.4048-4-5-1': 'One important ingredient in the Sakai-Sugimoto model is the mechanism of chiral symmetry restoration at high temperatures [CITATION].', '1107.4048-4-5-2': 'We will propose a new mechanism for chiral symmetry restoration in our framework.', '1107.4048-4-6-0': 'The plan of the paper is as follows.', '1107.4048-4-7-0': 'Section [REF] contains a short review of finite temperature QCD and the holographic approach to four dimensional Yang Mills theory (YM4) using D4 branes.', '1107.4048-4-7-1': ""In Section [REF] we discuss the gravity theory at a finite temperature and a 'Scherk-Schwarz' transition between a solitonic D4 brane and a black D4 brane."", '1107.4048-4-7-2': 'In Section [REF] we recall the conventional correspondence between this transition and the confinement/deconfinement transition of the Yang Mills theory, and discuss problems with this correspondence, with special emphasis on the mismatch of [MATH] centre symmetry between the black D4 phase and the deconfined 4 dimensional YM theory.', '1107.4048-4-7-3': 'In Section [REF], we discuss the phase structure of the D4 branes using an unconventional (periodic) fermion boundary condition along the Euclidean time circle and show that the high temperature in gravity is described by a localized solitonic D3 brane (or its T-dual) whose centre symmetry precisely matches with the deconfinement phase of four dimensional YM theory.', '1107.4048-4-7-4': 'This leads us, in Section [REF], to propose a new representation of the confinement/deconfinement phase transition in terms of a GL transition between the solitonic D4 and the (T-dual of) the localized solitonic D3 (see Table [REF]).', '1107.4048-4-7-5': 'In Section [REF], we show some new correspondences between phenomena in QCD and their counterparts in gravity following our proposal.', '1107.4048-4-7-6': 'In Section [REF], we suggest a new mechanism of chiral symmetry restoration in the Sakai-Sugimoto model in keeping with our proposal.', '1107.4048-4-7-7': 'Section [REF] contains the concluding remarks and some open problems.', '1107.4048-4-8-0': '# Review of QCD and its gravity dual', '1107.4048-4-9-0': '## Finite temperature QCD', '1107.4048-4-10-0': 'In this section, we briefly review some salient properties of four dimensional [MATH] pure Yang-Mills theory at a finite temperature at large [MATH], a system which we will subsequently investigate through holography.', '1107.4048-4-11-0': 'An important symmetry in this theory is the [MATH] symmetry along the temporal cycle, which is the centre of the [MATH] symmetry.', '1107.4048-4-11-1': 'The order parameter of this symmetry is the temporal Polyakov loop operator [EQUATION]', '1107.4048-4-11-2': 'If this operator is zero, the [MATH] symmetry is preserved and, if not, it is broken.', '1107.4048-4-11-3': ""Especially, if the [MATH] symmetry is preserved, physical quantities do not depend on the temporal radius [MATH] at [MATH] order due to 'large [MATH] volume independence' [CITATION]."", '1107.4048-4-11-4': 'If the gauge theory is on a torus [MATH], the large [MATH] volume independence is generalized and physical quantities do not depend on [MATH], if the [MATH] symmetry along the [MATH]-th direction is preserved [CITATION].', '1107.4048-4-12-0': 'Let us consider the phase structure of the Yang-Mills theory.', '1107.4048-4-12-1': 'At low temperatures, the confinement phase is the dominant thermodynamic phase.', '1107.4048-4-12-2': 'In this phase, the [MATH] symmetry is preserved ([MATH]).', '1107.4048-4-12-3': 'At high temperatures, the deconfinement phase, in which the [MATH] symmetry is broken ([MATH]), dominates.', '1107.4048-4-12-4': 'There is a confinement/deconfinement transition at an intermediate temperature.', '1107.4048-4-12-5': 'From studies in large [MATH] lattice gauge theories, this is expected to be a first order phase transition [CITATION].', '1107.4048-4-13-0': ""As we remarked above, the free energy in the confinement phase does not depend on the temperature through the 'large [MATH] volume independence'."", '1107.4048-4-13-1': 'It implies that the entropy becomes zero at [MATH].', '1107.4048-4-13-2': 'On the other hand, the entropy in the deconfinement phase would be [MATH].', '1107.4048-4-13-3': 'This has a simple physical interpretation: in the confinement phase, the spectrum excludes gluon states, which have [MATH] degree of the freedom, and consists only of gauge singlet states like glueballs, leading to an [MATH] entropy.', '1107.4048-4-14-0': '## Holographic QCD', '1107.4048-4-15-0': 'In this section, we will review the construction of four dimensional [MATH] pure Yang-Mills theory from [MATH] D4 branes [CITATION].', '1107.4048-4-15-1': 'Let us first consider a 10 dimensional Euclidean spacetime with an [MATH] and consider D4 branes wrapping on the [MATH].', '1107.4048-4-15-2': 'We define the coordinate along this [MATH] as [MATH] and its periodicity as [MATH].', '1107.4048-4-15-3': 'The effective theory on this brane is a 5 dimensional supersymmetric Yang-Mills theory on the [MATH].', '1107.4048-4-15-4': 'For the fermions on the brane, the boundary condition along the circle can be AP (antiperiodic) or P (periodic); to specify the theory, we must pick one of these two boundary conditions.', '1107.4048-4-15-5': 'Let us take the AP boundary condition.', '1107.4048-4-15-6': 'This gives rise to fermion masses proportional to the Kaluza-Klein scale [MATH], leading to supersymmetry breaking (this is called the SS - Scherk-Schwarz- mechanism).', '1107.4048-4-15-7': 'This, in turn, induces masses for the adjoint scalars and for [MATH], which are proportional to [MATH] at one-loop.', '1107.4048-4-15-8': 'Therefore, if [MATH] is sufficiently small and the dynamical scale [MATH] is much less than both the above mass scales, then the fermions and adjoint scalars are decoupled and the 5 dimensional supersymmetric Yang-Mills theory is reduced to a four dimensional pure Yang-Mills theory.', '1107.4048-4-16-0': 'By taking the large [MATH] limit of this system a la Maldacena at low temperatures, we obtain the dual gravity description of the compactified 5 dimensional SYM theory [CITATION], which consists, at low temperatures, of a solitonic D[MATH] brane solution wrapping the [MATH].', '1107.4048-4-16-1': 'The explicit metric is given by [EQUATION]', '1107.4048-4-16-2': 'There is also a non-trivial value of the five form potential which we do not write explicitly.', '1107.4048-4-16-3': 'Here [MATH] is the YM coupling on the D[MATH] world-volume and [MATH] is found by putting [MATH] in the general formula [EQUATION]', '1107.4048-4-16-4': 'Since the [MATH]-cycle shrinks to zero at [MATH], in order to avoid possible conical singularities we must choose the asymptotic radius [MATH] as follows [EQUATION]', '1107.4048-4-16-5': ""With this choice, the contractible [MATH]-cycle, together with the radial direction [MATH], forms a so-called 'cigar' geometry which is topologically a disc."", '1107.4048-4-16-6': 'In order that the fermions are well-defined on this geometry, they must obey the AP boundary condition.', '1107.4048-4-16-7': 'The AP boundary condition along [MATH] is, of course, consistent with the choice of fermion b.c. in the boundary theory.', '1107.4048-4-17-0': 'The leading order gravity solution described above is not always valid.', '1107.4048-4-17-1': 'E.g. in order that the stringy modes can be ignored, we should ensure that the curvature in string units must be small.', '1107.4048-4-17-2': 'In other words, the typical length scale of this solution near [MATH], viz, [MATH], must satisfy [MATH] [CITATION].', '1107.4048-4-17-3': 'This condition turns out to be equivalent to [EQUATION]', '1107.4048-4-17-4': 'This is opposite to the condition which we found for the validity of the four-dimensional gauge theory description (see footnote [REF]).', '1107.4048-4-17-5': 'Thus, this gravity solution can describe the 5 dimensional SYM but cannot directly describe the 4 dimensional YM theory, inferences about which can only be obtained through extrapolation.', '1107.4048-4-17-6': 'This is a common problem in the construction of holographic duals of non-supersymmetric gauge theories.', '1107.4048-4-17-7': 'This has been discussed at length; in particular, the gravity description, which necessitates extrapolation to strong coupling, has been likened (cf. [CITATION] and [CITATION], p. 196-197) to strong coupling lattice gauge theory.', '1107.4048-4-17-8': 'Many interesting results, including the qualitative predictions in [CITATION], have been obtained using this prescription.', '1107.4048-4-18-0': 'Let us explore some properties of the soliton solution ([REF]).', '1107.4048-4-18-1': 'This solution is expected, on the basis of several arguments, to correspond to the confinement phase in the four dimensional gauge theory.', '1107.4048-4-18-2': 'For example, it can be shown that the classical action of this solution at a finite temperature [MATH] is [CITATION] [EQUATION] where [MATH] is the volume of the three spatial dimensions ([MATH]) and [MATH] is a constant which is obtainable from the formula below [CITATION] by putting [MATH] [EQUATION]', '1107.4048-4-18-3': 'Thus the free energy of this solution, [MATH], is independent of temperature, and hence the entropy is zero at [MATH] order.', '1107.4048-4-18-4': 'These facts are consistent with the interpretation of this solution as the confinement phase in the large [MATH] gauge theory, in which the temporal [MATH] symmetry is preserved.', '1107.4048-4-19-0': 'Note that the action ([REF]) divided by [MATH] does depend on [MATH], implying that the [MATH] symmetry along the [MATH]-cycle is broken.', '1107.4048-4-19-1': 'Although this observation might appear irrelevant for the 4 dimensional Yang-Mills theory, it is this symmetry breaking which allows a conventional KK reduction to four dimensions (see the next two sections for more details).', '1107.4048-4-20-0': '# Finite temperature holographic QCD', '1107.4048-4-21-0': '## Confinement/deconfinement transition in Holographic QCD in the standard scenario', '1107.4048-4-22-0': 'In existing holographic studies of the thermodynamics in the Yang-Mills theory, it is assumed that the black D4 brane solution corresponds to the deconfinement phase and the Scherk-Schwarz transition in the gravity is related to the confinement/deconfinement transition.', '1107.4048-4-22-1': 'In this section, we review this scenario and we will explain its problems in section [REF].', '1107.4048-4-23-0': 'To discuss holographic QCD at finite temperatures, we begin by compactifying Euclidean time in the boundary theory on a circle with periodicity [MATH].', '1107.4048-4-23-1': 'To permit description in terms of four dimensional gauge theory, the temperature should, of course, be kept far below the KK scale, [MATH] .', '1107.4048-4-23-2': 'In order to determine the gravitational theory, we need to fix the periodicity of fermions in the gauge theory along the time cycle.', '1107.4048-4-23-3': 'In the existing literature, the fermion boundary condition is chosen to be AP according to the usual practice for thermal fermions (we will see in section [REF] that this choice is not imperative here).', '1107.4048-4-23-4': ""The resulting fermionic bc's: (AP, AP) along [MATH] lead to a [MATH] symmetry of the system under [MATH]."", '1107.4048-4-23-5': 'In the gravity dual, we must therefore, include the [MATH]-transform of the solution ([REF]), which is a black D4 solution: [EQUATION]', '1107.4048-4-23-6': 'Here [MATH] in this metric is related to [MATH] as in ([REF]) [EQUATION]', '1107.4048-4-23-7': 'This solution has a contractible [MATH]-cycle and the fermions must obey the anti-periodic boundary condition (see footnote [REF]), consistent with the AP b.c. in the gauge theory.', '1107.4048-4-23-8': 'Analogously to ([REF]), the gravity description of this solution is reliable if [MATH].', '1107.4048-4-24-0': 'The classical action density can be evaluated, as before, yielding [EQUATION]', '1107.4048-4-24-1': 'In contrast with ([REF]), the free energy now is a function of [MATH] and hence the entropy is [MATH] which is appropriate for a description of the deconfinement phase in the gauge theory.', '1107.4048-4-25-0': 'By comparing ([REF]) and ([REF]), we see that at low temperatures (large [MATH]) the solitonic D4 solution dominates, while at high temperatures the black brane dominates.', '1107.4048-4-25-1': 'The transition between the two solutions, which we will call the ""Scherk-Schwarz (SS) transition"", occurs at [CITATION] [EQUATION]', '1107.4048-4-25-2': 'In the standard discussions of holographic QCD [CITATION], this transition is interpreted as the (continuation of the) confinement/deconfinement transition in the gauge theory.', '1107.4048-4-25-3': 'In the following sections, we will explain various problems with this interpretation and present a resolution by proposing an alternative scenario.', '1107.4048-4-26-0': '## Problems with the standard gravity analysis', '1107.4048-4-27-0': 'The previous subsection describes the usual relation between the confinement/deconfinement transition in 4 dimensional Yang-Mills theory and the SS transition in holographic QCD.', '1107.4048-4-27-1': 'However as we will now discuss, there are several problems associated with this correspondence.', '1107.4048-4-27-2': 'Some of these problems were mentioned earlier in [CITATION].', '1107.4048-4-28-0': 'In order to understand these problems, it is convenient to consider first the phase structure of the 5 dimensional SYM on the [MATH].', '1107.4048-4-28-1': 'See Figure [REF].', '1107.4048-4-28-2': 'As we have seen, the gravity description is valid in the strong coupling regime of this theory (the blue region in Figure [REF]) which is characterized by the solitonic D4 solution and the black D4 solution.', '1107.4048-4-28-3': 'On the other hand, the 4 dimensional Yang-Mills theory is realized at weak coupling ([MATH]) and low temperature ([MATH]) (the upper green region in Figure [REF]) which is characterized by the confinement phase and the deconfinement phase.', '1107.4048-4-28-4': 'The usual proposal is that the solitonic D4 and black D4 solutions correspond to the confinement and deconfinement phases respectively.', '1107.4048-4-29-0': 'To examine this proposal in detail, it is useful to consider the realization of the [MATH] centre symmetry in various phases of the five-dimensional SYM theory along [MATH] and [MATH].', '1107.4048-4-29-1': 'The [MATH] symmetry along [MATH] is characterized by the order parameter [MATH] which is the temporal Polyakov loop ([REF]).', '1107.4048-4-29-2': 'Similarly the [MATH] symmetry along [MATH] is characterized by the order parameter [MATH] which is the Polyakov loop around the [MATH]-cycle: [EQUATION]', '1107.4048-4-29-3': 'In Table [REF] we have collected values of [MATH] for the phases of 5D SYM discussed in this paper.', '1107.4048-4-30-0': 'Before testing the above proposal for holographic QCD in terms of these order parameters, let us see how to obtain this table.', '1107.4048-4-30-1': ""The 'confinement' phase denotes a phase of the 5D SYM whose KK compactification, at small enough [MATH], coincides with the usual confinement phase of YM4."", '1107.4048-4-30-2': 'The latter, clearly has [MATH].', '1107.4048-4-30-3': 'Now, the fact that we have obtained the 4D phase through a KK reduction, along [MATH], implies that the corresponding phase of the 5D theory must have [MATH].', '1107.4048-4-30-4': 'This follows from the fact [CITATION] that in a [MATH] phase, KK reduction does not work since the effective KK scale in a large [MATH] gauge theory becomes [MATH] in stead of the usual [MATH], and goes to zero at large [MATH] .', '1107.4048-4-30-5': ""Thus, the 'confinement' phase must have [MATH], as shown in the first row of Table [REF]."", '1107.4048-4-30-6': ""By similar arguments, the 'deconfinement' phase of 5D SYM which coincides with the deconfinement phase of YM4 must have [MATH] (for validity of the KK reduction) and [MATH] (to exhibit 4D deconfinement)."", '1107.4048-4-30-7': 'This is shown in the second row of Table [REF].', '1107.4048-4-31-0': 'The next two rows in Table [REF] describe two additional phases of SYM5, which appear only for the (AP,AP) b.c. via the [MATH] symmetry.', '1107.4048-4-31-1': 'The third row describes the [MATH] mirror of the confinement phase of YM4, which is characterized by [MATH] and [MATH].', '1107.4048-4-31-2': 'Since [MATH], this phase is a fully 5 dimensional object and is not related to the YM4 (indeed, so far as the Polyakov loop order parameters are concerned, this phase can be identified with a deconfinement phase of SYM5).', '1107.4048-4-31-3': 'Another phase is the mirror of the deconfinement phase in YM4, which is characterized by [MATH] and [MATH].', '1107.4048-4-31-4': 'Since this phase has the same order parameters as the original deconfinement phase, it is not obvious whether these two phases are distinct or smoothly connected through a cross over (we represent the possible phase boundary/crossover by the horizontal dotted line in Figure [REF]).', '1107.4048-4-32-0': 'To describe the next three rows in Table [REF], note that the order parameters [MATH] have a manifestation in gravity.', '1107.4048-4-32-1': ""Thus, e.g. if the [MATH] symmetry along [MATH] is preserved ([MATH]), then, due to the 'large [MATH] volume independence' mentioned in Section [REF], the free energy [MATH] should be independent of [MATH] (equivalently, the action should be linear in [MATH])."", '1107.4048-4-32-2': 'This is satisfied by the action in ([REF]).', '1107.4048-4-32-3': 'Hence for the parameter regime in which the solitonic D4 brane is the dominant phase of the theory, the 5 dim SYM dual exists in a phase characterized by [MATH].', '1107.4048-4-32-4': 'Similarly, the non-linear dependence of the action ([REF]) on [MATH] implies that the gauge theory dual has [MATH].', '1107.4048-4-32-5': 'These results are represented by the 5th row of Table [REF].', '1107.4048-4-32-6': 'One can similarly deduce by looking at the [MATH] dependence of ([REF]) that for the phase of the gauge theory represented by the black D4 solution, [MATH], which could also be inferred by the [MATH] symmetry.', '1107.4048-4-32-7': 'This is represented by the 6th row of Table [REF].', '1107.4048-4-32-8': 'In the last row, we have included a phase of SYM5 with (P, AP) fermion b.c. along ([MATH]) which corresponds to a localized D3 soliton, to be discussed in the next section.', '1107.4048-4-33-0': 'We now come back to the conventional proposal of holographic QCD discussed in the beginning of this section; let us define this proposal as a combination of three propositions:', '1107.4048-4-34-0': '(a) the solitonic D4 brane phase of gravity corresponds to the confinement phase of YM4,', '1107.4048-4-35-0': '(b) the black D4 corresponds to the deconfinement phase of YM4, and', '1107.4048-4-36-0': '(c) the SS transition between the two gravity solutions corresponds to the confinement/deconfinement transition of YM4.', '1107.4048-4-37-0': 'In terms of Table [REF], both the solitonic D4 and the confinement phase of YM4 (regarded as a phase of SYM) have [MATH]; hence both of these phases have the same [MATH] symmetry and (a) can in principle be valid.', '1107.4048-4-37-1': 'However, (b) cannot be valid, since the black D4 phase ([MATH]) and the deconfinement phase of YM4 ([MATH]) have different [MATH] symmetry.', '1107.4048-4-37-2': 'In particular, the black D4 corresponds to a phase that cannot be KK reduced to YM4 at all because of vanishing [MATH].', '1107.4048-4-37-3': 'In fact, it would correspond to the [MATH] mirror of the confinement phase of YM4 (which can be identified with a deconfinement phase of SYM5 which is intrinsically 5 dimensional).', '1107.4048-4-37-4': 'Thus, there must be at least one phase boundary between the deconfinement phase of YM4 and the black D4 phase (see Figure [REF] for one possibility).', '1107.4048-4-37-5': 'Regarding (c), if the SS transition persists even at weak coupling, then it clearly cannot coincide with the confinement/deconfinement transition.', '1107.4048-4-37-6': 'The reason is that, even at weak coupling the SS transition can only occur at [MATH] (since even stringy effects should satisfy the [MATH] symmetry), whereas it is known that the confinement/deconfinement phase transition temperature of YM4 goes down ([MATH] goes up) at smaller values of [MATH] (see Figure [REF]).', '1107.4048-4-38-0': 'Through the arguments in this section, it is obvious that the black D4 brane solution does not correspond to the deconfinement phase in the 4 dimensional Yang-Mills theory and the SS transition does not correspond to the confinement/deconfinement transition.', '1107.4048-4-38-1': 'In the next section, we propose an alternative correspondence, which resolves these problems.', '1107.4048-4-39-0': '# Our proposal for the confinement/deconfinement transition in holographic QCD', '1107.4048-4-40-0': 'In the previous section, we considered the conventional picture of holographic QCD at finite temperature and discussed some of the problems in describing phases of YM4.', '1107.4048-4-40-1': 'In this section, we propose an alternative picture to resolve these problems.', '1107.4048-4-41-0': 'In the above discussion, we described finite temperature gauge theory by choosing AP b.c. for fermions on the brane along the compactified Euclidean time.', '1107.4048-4-41-1': 'However, let us recall that the gauge theory of interest here is pure Yang Mills theory in four dimensions, which does not have fermions.', '1107.4048-4-41-2': 'Fermions reappear when the validity condition ([REF]) is enforced and [MATH] goes above the KK scales [MATH].', '1107.4048-4-41-3': 'In this sense, fermions are an artifact of the holographic method and in the region of validity of the pure YM theory, the periodicity of the fermion should not affect the gauge theory results.', '1107.4048-4-42-0': 'If we start with the 5 dimensional SYM on [MATH] with a periodic temporal circle, the partition function is the twisted one: Tr [MATH], and is not the standard thermal partition function.', '1107.4048-4-42-1': 'However, under the limit [MATH] and [MATH], in which the 4 dimensional YM theory appears, the fermions are effectively decoupled; hence [MATH].', '1107.4048-4-42-2': ""Thus in this limit the 'twisted' partition becomes the usual thermal partition function, even with the periodic b.c. for the fermions."", '1107.4048-4-42-3': 'More precisely, we have [EQUATION]', '1107.4048-4-42-4': 'Thus, independent of the boundary condition along the [MATH], we obtain the thermal partition function of YM4.', '1107.4048-4-43-0': 'As a consequence, it is pertinent to study the phase structure of the 5 dimensional SYM on [MATH] with (P,AP) boundary condition through holography.', '1107.4048-4-43-1': 'Indeed, in this sector, we will find a phase transition in gravity, which continues naturally, at weak coupling, to the confinement/deconfinement transition in the 4D Yang-Mills theory.', '1107.4048-4-44-0': '## Gregory-Laflamme transition in the gravity theory with (P,AP) boundary condition', '1107.4048-4-45-0': 'In this section, we reconsider the gravity theory described in Section [REF], now compactified on [MATH] with (P,AP) b.c. for fermions.', '1107.4048-4-45-1': 'In this case, contrary to the (AP,AP) case, the black D4 brane solution ([REF]) is not allowed (see the comment below ([REF])).', '1107.4048-4-45-2': 'However a non-trivial phase boundary still exists, as we will see shortly.', '1107.4048-4-45-3': 'The detailed phase structure for this boundary condition is summarized in [CITATION] (with [MATH] and [MATH] exchanged); see also [CITATION].', '1107.4048-4-45-4': 'We will now outline the salient features.', '1107.4048-4-46-0': 'Let us start from large [MATH].', '1107.4048-4-46-1': 'In this region the solitonic D4 brane solution ([REF]) is thermodynamically dominant.', '1107.4048-4-46-2': 'As we decrease [MATH], it reaches [MATH], which is the order of the effective string length at [MATH] [CITATION].', '1107.4048-4-46-3': 'Below this value, winding modes of the string wrapping on the [MATH]-cycle would be excited.', '1107.4048-4-46-4': 'Thus the gravity description given by ([REF]) would be valid only if [EQUATION]', '1107.4048-4-46-5': 'In order to avoid this problem, we perform a T-duality along the [MATH]-cycle and go to the IIB frame, where the solitonic D4 solution becomes a solitonic D3 brane uniformly smeared on the dual [MATH]-cycle.', '1107.4048-4-46-6': 'The metric describing this solution is given by [EQUATION]', '1107.4048-4-46-7': 'Here [MATH] is the dual of [MATH], hence [MATH] has a periodicity [MATH].', '1107.4048-4-46-8': 'By considering winding modes on this metric, we can see that the gravity description is now valid if [MATH].', '1107.4048-4-46-9': 'Thus even when [MATH] is below ([REF]), the gravity analysis in the IIB frame is possible.', '1107.4048-4-47-0': 'As [MATH] decreases (hence the dual radius in the IIB frame increases), the uniformly smeared solitonic D3 brane becomes thermodynamically unstable at a certain temperature.', '1107.4048-4-47-1': 'This temperature, called the Gregory-Laflamme (GL) instability point, is numerically given by [CITATION]: [EQUATION]', '1107.4048-4-47-2': 'It is expected that, even before [MATH] is lowered all the way to [MATH], the smeared solitonic D3 brane becomes meta-stable and undergoes a first order Gregory-Laflamme (GL) transition at an inverse temperature [MATH], leading to a more stable, solitonic D3 brane localized on the dual cycle.', '1107.4048-4-47-3': 'This localized solution spontaneously breaks translation symmetry along the temporal direction.', '1107.4048-4-47-4': 'Although it is difficult to derive the critical temperature [MATH] precisely, we can approximately evaluate it as follows [CITATION].', '1107.4048-4-47-5': 'The metric of the localized solitonic D3 brane is approximately given by that of the solitonic D3 on [MATH] for a sufficiently large radius [MATH] of the dual cycle [CITATION], [EQUATION]', '1107.4048-4-47-6': 'Here [MATH] (see Eq. ([REF])).', '1107.4048-4-47-7': 'The value of [MATH] is determined by the smoothness condition (as in ([REF])).', '1107.4048-4-47-8': '[MATH] is regarded as a radial component of a spherical coordinate and is related to the cylindrical coordinates [MATH] of ([REF]) by roughly [MATH].', '1107.4048-4-47-9': 'Here we implicitly assume that the soliton is localized at [MATH].', '1107.4048-4-47-10': 'This approximation is valid if [MATH] and [MATH] are sufficiently small such that we can ignore the finite size effect associated with the dual circle [CITATION] .', '1107.4048-4-48-0': 'The classical action of this solution turns out to be [EQUATION]', '1107.4048-4-48-1': 'By comparing this classical action with ([REF]) , we can estimate that the GL transition will happen around [EQUATION]', '1107.4048-4-48-2': 'Note that, near this critical point, [MATH] is similar in magnitude to, but less than, the dual radius [CITATION], so we expect ([REF])-([REF]) to receive some corrections from the finite size effect.', '1107.4048-4-49-0': 'In addition to these two solutions, ([REF]) and ([REF]), there is another solution: solitonic D3 brane non-uniformly smeared on the dual cycle.', '1107.4048-4-49-1': 'The metric of this solution is perturbatively derived in [CITATION].', '1107.4048-4-49-2': 'This non-uniform solution arises at the GL instability point and is always unstable.', '1107.4048-4-49-3': 'Although the complete behaviour of the non-uniform solution has not been explored, it is expected that it merges with the localized solitonic D3 as shown in Figure [REF] [CITATION].', '1107.4048-4-49-4': 'The fact that the GL instability temperature ([REF]) is higher than the approximately obtained GL critical temperature ([REF]) is consistent with this expectation.', '1107.4048-4-50-0': 'Let us make a few comments on the localized solitonic D3 solution ([REF]).', '1107.4048-4-50-1': 'Firstly, through a calculation similar to ([REF]), the gravity description can be shown to be valid (i.e. stringy effects can be ignored) if [MATH].', '1107.4048-4-50-2': 'Secondly, this solution ceases to exist if [MATH] is too large, as shown in Figure [REF]; intuitively, if [MATH] were too large, the dual cycle would become smaller than the size of the localized soliton, which is not possible.', '1107.4048-4-50-3': 'Thirdly, the free energy [MATH] of this solution, unlike that of ([REF]), is proportional to temperature; hence the entropy of this solution is non-zero at [MATH] and the temporal [MATH] symmetry is broken.', '1107.4048-4-50-4': 'In addition, because of the non-trivial [MATH]-dependence of the classical action ([REF]) we can see that the [MATH] symmetry along the [MATH]-cycle is also broken.', '1107.4048-4-50-5': 'Thus the Polyakov loop [MATH] and [MATH] in this solution are both non-zero as shown in Table [REF] and, contrary to the black D4 solution, the localized solitonic D3 solution is appropriate for a description of the deconfinement phase in the dual gauge theory (which also has [MATH]).', '1107.4048-4-50-6': 'We will build on this observation in the next subsection.', '1107.4048-4-51-0': '## Gregory-Laflamme transition as a confinement/deconfinement transition', '1107.4048-4-52-0': 'The phase structure of the 5 dimensional SYM on [MATH] with the (P,AP) boundary condition is shown in Figure [REF].', '1107.4048-4-52-1': 'The strong coupling region (the blue region in Figure [REF]) described by type II supergravity and is characterized by the GL phase transition which occurs at a temperature given by ([REF]).', '1107.4048-4-52-2': 'In the weak coupling region (the green region in Figure [REF]), the 4 dimensional Yang-Mills is realized at low temperatures ([MATH]).', '1107.4048-4-52-3': 'Although this region is common to the (P,AP) phase diagram in Figure [REF] and the (AP,AP) phase diagram in Figure [REF], the mirror of this region under the [MATH]) does not exist in Figure [REF] since the (P,AP) b.c is not [MATH]-symmetric.', '1107.4048-4-53-0': 'The main point to emphasize here it that: contrary to the previous phase structure in the (AP,AP) case, now the localized solitonic D3 phase has the same order parameters ([MATH]) as the deconfinement phase, thus making it plausible that these two phases are smoothly connected.', '1107.4048-4-53-1': 'In Figure [REF], we have indicated this by assuming the simplest extrapolation through the region of the intermediate coupling.', '1107.4048-4-54-0': 'Our proposal', '1107.4048-4-55-0': 'In view of the above observations, we propose a strong coupling continuation of weakly coupled 4-dimensional Yang Mills theory as shown in Table [REF].', '1107.4048-4-56-0': 'Of course, our proposal is based on a simple extrapolation between the intermediate coupling regime, and the real story there could be more involved.', '1107.4048-4-56-1': 'However the mere existence of such a simple extrapolation is a significant improvement over the previous proposal in the (AP,AP) case, where we are certain that there has to be at least one phase boundary between the deconfinement phase and black D4 solution (for the simple reason that their order parameters have different values).', '1107.4048-4-56-2': 'It is clearly important, therefore, to further investigate the nature of the deconfinement phase in terms of the localized solitonic D3 solution based on the correspondence outlined above.', '1107.4048-4-57-0': '# New correspondences in holographic QCD', '1107.4048-4-58-0': 'Using our proposed correspondence, we can explain anew several phenomena in the gauge theory from gravity.', '1107.4048-4-58-1': 'In this section, we list some of these phenomena.', '1107.4048-4-59-0': '## Polyakov loop and D3 brane distribution', '1107.4048-4-60-0': 'In our proposal, we identified the GL transition in the IIB frame to the confinement/deconfinement transition in the gauge theory.', '1107.4048-4-60-1': 'In the GL transition, the distribution of the D3 branes on the dual circle changes from a uniform distribution at low temperatures to a localized one at high temperatures.', '1107.4048-4-60-2': 'In this subsection, we explain what the corresponding phenomenon is in the gauge theory.', '1107.4048-4-61-0': 'Since the original five-dimensional gauge theory (of the D4 branes) appears in the IIA frame, in order to understand the role of the D3 branes, we need to consider a T-duality along the temporal direction.', '1107.4048-4-61-1': 'Under this T-duality, [MATH], which is transverse to the D3 brane, is mapped to the gauge potential [MATH] on the D4 brane.', '1107.4048-4-61-2': ""Thus the D3 brane distribution on the dual circle is related to a distribution of [MATH]'s, which are defined as the eigenvalues of the Polyakov loop operator [MATH]."", '1107.4048-4-62-0': 'If the D3 branes are uniformly distributed, the [MATH] are also uniformly distributed.', '1107.4048-4-62-1': 'In that case, by choosing an appropriate gauge, we can take [MATH].', '1107.4048-4-62-2': 'Then the temporal Polyakov loop operator ([REF]) becomes [EQUATION]', '1107.4048-4-62-3': 'Hence this distribution corresponds to the confinement phase.', '1107.4048-4-62-4': 'On the other hand, if the D3 branes are localized, then [MATH] are also localized and the Polyakov loop becomes non-zero, which characterizes the deconfinement phase.', '1107.4048-4-62-5': 'This observation is consistent with the entropy arguments in section [REF].', '1107.4048-4-63-0': 'The above discussion shows a direct relation between the D3 brane distribution and the eigenvalue distribution of the Polyakov loop operator.', '1107.4048-4-63-1': 'This latter quantity can sometimes be explicitly evaluated [CITATION].', '1107.4048-4-63-2': 'E.g. in [CITATION], for four dimensional Yang Mills theory on a small [MATH], we found that the uniform, non-uniform and localized distribution of [MATH], all appear, with obvious correspondence to similar gravitational solutions.', '1107.4048-4-63-3': 'Especially the free energies of these three solutions show a ""swallow tail"" relation similar to Figure [REF].', '1107.4048-4-63-4': '(See Figure 3 in [CITATION].)', '1107.4048-4-63-5': 'This correspondence strongly supports our proposal.', '1107.4048-4-64-0': '## Gregory-Laflamme transition as a Hagedorn transition', '1107.4048-4-65-0': 'Our proposal opens up the interesting possibility of a relation between the GL transition and the Hagedorn transition.', '1107.4048-4-66-0': 'It is known that the GL instability is an instability of the KK modes of the graviton along the compact circle [CITATION].', '1107.4048-4-66-1': 'In our case, the KK modes along the dual temporal circle, which are associated with the GL instability at ([REF]) in the IIB description, are mapped to winding modes around the temporal circle through the T-duality [CITATION].', '1107.4048-4-66-2': 'This then indicates that the type IIB GL transition is associated with the excitation of the winding modes of the IIA string.', '1107.4048-4-66-3': 'This phenomenon is similar to the Hagedorn transition in string theory [CITATION], where the associated instability is caused by temporal winding modes.', '1107.4048-4-66-4': 'Thus the GL transition in the IIB description might correspond to the Hagedorn transition in the IIA description.', '1107.4048-4-67-0': 'Note that from large [MATH] gauge-theoretic calculations also, the usual confinement/deconfinement transition is believed to be related to the Hagedorn transition.', '1107.4048-4-67-1': 'It has been explicitly shown in weakly coupled gauge theories [CITATION].', '1107.4048-4-67-2': 'This makes it plausible that the Hagedorn transition in the Yang-Mills theory continues to the Hagedorn transition in the IIA string, which, as we argued above, is possibly the dual of the GL transition in the IIB supergravity.', '1107.4048-4-68-0': '# Chiral symmetry restoration in Sakai-Sugimoto model', '1107.4048-4-69-0': 'In the previous sections, we have seen that the conventional holographic representation of the confinement/deconfinement transition as the SS transition is fraught with problems, to circumvent which we proposed in Section [REF] a different interpretation in terms of a GL transition.', '1107.4048-4-69-1': 'However, the SS transition has widely been employed in holographic QCD and purports to explain several phenomena in real QCD.', '1107.4048-4-69-2': 'In particular, the chiral symmetry restoration in the Sakai-Sugimoto model was neatly explained in [CITATION].', '1107.4048-4-69-3': 'In their scenario the black D4 brane plays a crucial role.', '1107.4048-4-69-4': 'However, since in our proposal the black branes do not appear anymore, we need to find an alternative idea for realizing chiral symmetry restoration.', '1107.4048-4-69-5': 'In this section, we discuss how chiral symmetry restoration can happen in the localized solitonic D3 background.', '1107.4048-4-70-0': '## Sakai-Sugimoto model and chiral symmetry breaking', '1107.4048-4-71-0': 'The Sakai-Sugimoto model [CITATION] was proposed to describe low energy hadron physics in holographic QCD and elegantly reproduces many aspects of the real QCD.', '1107.4048-4-71-1': 'We first briefly review this model and show how chiral symmetry breaking at low temperatures is realized in terms of the dual gravity.', '1107.4048-4-72-0': 'The Sakai-Sugimoto model is an extension of the holographic model of QCD discussed in section [REF].', '1107.4048-4-72-1': 'Sakai and Sugimoto added, to the [MATH] D4 brane system, [MATH] D8 and [MATH] branes which are localized on the [MATH] circle and fill all other directions as follows [EQUATION]', '1107.4048-4-72-2': 'Here [MATH] denotes the compactified directions.', '1107.4048-4-72-3': '(See Figure [REF] (a).)', '1107.4048-4-72-4': 'This model has a free parameter [MATH], which is the asymptotic ([MATH]) distance between the D8 and [MATH] brane on the [MATH] circle.', '1107.4048-4-72-5': 'In the original Sakai-Sugimoto model [CITATION] [MATH] was taken to be [MATH]; however, we will let it be general and take values in [MATH] (the other half, [MATH] is related by a reflection and need not be considered separately).', '1107.4048-4-73-0': 'This model has [MATH] gauge symmetry on the [MATH] D8 and [MATH] branes, which can be interpreted as a chiral [MATH] flavor symmetry in QCD.', '1107.4048-4-73-1': 'As we will see soon, in some situations, the D8 and [MATH] branes merge and the chiral symmetry is broken to a single [MATH].', '1107.4048-4-73-2': 'Sakai and Sugimoto proposed that this is the holographic realization of chiral symmetry breaking [CITATION].', '1107.4048-4-74-0': 'Let us take a large [MATH] limit a la Maldacena and, according to the principle of holography, replace the [MATH] D4 branes with a corresponding gravity solution.', '1107.4048-4-74-1': ""Here we will keep [MATH] such that we can ignore the back-reaction of the D8/[MATH] branes onto the background geometry (this is the so-called 'probe approximation')."", '1107.4048-4-74-2': 'In that case, the background (D4 brane) geometry is determined thermodynamically (as the dominant classical solution at a given temperature, as we have done in sections [REF] and [REF]), and the D8/[MATH] brane configuration coupled to this background, is determined dynamically, with the given distance [MATH] as a boundary condition.', '1107.4048-4-75-0': 'Chiral symmetry breaking in this model happens as follows.', '1107.4048-4-75-1': 'At sufficiently low temperatures, the favoured geometric background is that of solitonic D4 solution ([REF]), as we have seen in section [REF] and [REF].', '1107.4048-4-75-2': 'On this background, the D8 and [MATH] cannot extend separately and need to merge as shown in Figure [REF] (b).', '1107.4048-4-75-3': 'As a result, the [MATH] gauge symmetry on the D8 and [MATH] is broken to [MATH], representing chiral symmetry breaking ([MATH]SB) in the dual gauge theory.', '1107.4048-4-76-0': '## Chiral symmetry restoration in the black D4 brane background', '1107.4048-4-77-0': 'In the gauge theory, it is expected that chiral symmetry is restored at a sufficiently high temperature.', '1107.4048-4-77-1': 'Thus if holographic QCD is to work, there should be a corresponding phenomena in the dual gravity description.', '1107.4048-4-78-0': 'In [CITATION], a mechanism for chiral symmetry restoration was suggested by considering D8 and [MATH] in the black D4 brane background ([REF]) in the (AP,AP) case.', '1107.4048-4-78-1': ""Contrary to the solitonic D4 brane geometry ([REF]), the [MATH] plane of the black D4 background has a 'cigar' geometry."", '1107.4048-4-78-2': 'Thus the D8 and [MATH] branes can wrap the cigar separately as shown in Figure [REF] (c).', '1107.4048-4-78-3': 'If such a configuration is energetically favoured, [MATH] gauge symmetry is preserved and chiral symmetry is restored.', '1107.4048-4-79-0': 'In addition to this configuration, another configuration shown in Figure [REF] (d) is possible in the black D4 background.', '1107.4048-4-79-1': 'This configuration is similar to Figure [REF] (b) and chiral symmetry is broken.', '1107.4048-4-79-2': 'In [CITATION], energies of these two configurations are compared with the black D4 background.', '1107.4048-4-79-3': 'It was found (see Figure 7 of [CITATION]) that for [MATH], it is configuration (c) which is always favoured, indicating a concurrence of deconfinement and chiral symmetry restoration transitions at [MATH], while for [MATH], a new window [MATH] opens up where configuration (d) is the favoured one, indicating coexistence of deconfinement and broken chiral symmetry.', '1107.4048-4-80-0': '## Chiral symmetry restoration in the localized solitonic D3 brane.', '1107.4048-4-81-0': 'Although the chiral symmetry restoration was explained, as above, in the black D4 brane background, our main thesis in this paper is that the black D4 solution itself is fraught with problems, if we interpret this solution as the deconfinement phase of the four dimensional gauge theory (see Section [REF]).', '1107.4048-4-81-1': 'The alternative we proposed in Section [REF] is that the localized solitonic D3 brane solution ([REF]) in the (P,AP) b.c. should be taken as the correct representation of the deconfinement phase in the dual gauge theory.', '1107.4048-4-81-2': 'In view of this, we need to understand chiral symmetry restoration in this solution instead of in the black D4 solution.', '1107.4048-4-82-0': 'One subtlety in the application of our proposal to the Sakai-Sugimoto model is the existence of the fundamental quarks from the open string between the D4 and D8/[MATH] brane.', '1107.4048-4-82-1': 'These quarks are, of course, not decoupled in the 4 dimensional limit [MATH].', '1107.4048-4-82-2': ""Hence, unlike in case of the adjoint fermions, where the P and the AP b.c. on the thermal cycle reduce to the same quantity in the four dimensional limit (see ([REF])), for the fundamental quarks the P and the AP b.c's differ in the 4D limit; hence, we must choose the AP b.c. on the temporal cycle to investigate the thermodynamics of the Sakai-Sugimoto model."", '1107.4048-4-82-3': 'Since our proposal crucially uses P b.c. for the fermions along the temporal cycle, we must address this issue .', '1107.4048-4-82-4': 'It turns out that with a small addition to our model, viz. that of an imaginary chemical potential [CITATION], we can obtain the desired AP b.c. around the temporal cycle for the quarks retaining the P b.c. for the adjoint fermions.', '1107.4048-4-82-5': 'Details of this approach will appear in a forthcoming paper [CITATION].', '1107.4048-4-82-6': 'Introduction of such a chemical potential does not cause any essential modification to the analysis that follows below, as shown in detail in [CITATION].', '1107.4048-4-82-7': 'Consequently, we will proceed below with periodic (fundamental as well as adjoint) fermions.', '1107.4048-4-83-0': 'In our proposal, the gravity analysis has been done in IIB supergravity by performing a T-duality along the [MATH]-cycle: [MATH].', '1107.4048-4-83-1': 'Thus it is convenient to dualize the brane configuration of the Sakai-Sugimoto model to the IIB frame.', '1107.4048-4-83-2': 'Since all the D branes in the Sakai-Sugimoto model wrap the [MATH], the above T-duality maps the D4 branes and D8/[MATH] branes to D3 and D7/[MATH] branes spreading [EQUATION]', '1107.4048-4-83-3': 'See Figure [REF] (e) also.', '1107.4048-4-83-4': 'The distribution along [MATH] of the D branes in the IIB description (related to the gauge field [MATH] in the IIA description), is determined dynamically.', '1107.4048-4-83-5': 'In the probe approximation, according to the analyses in section [REF], at temperatures below the GL critical temperature ([REF]), the D3 branes are distributed uniformly and, above it, they are localized on the [MATH].', '1107.4048-4-83-6': 'The fate of the chiral gauge group in the gravity representation depends on the stable configurations of the probe D7/[MATH] branes in these backgrounds.', '1107.4048-4-84-0': 'In the uniformly smeared solitonic D3 brane geometry ([REF]) shown in Figure [REF] (f), the situation is similar to (b) in Figure [REF].', '1107.4048-4-84-1': 'Since the [MATH] direction is smoothly pinched off at [MATH], the D7 and [MATH] have to merge and chiral symmetry is broken.', '1107.4048-4-85-0': 'On the other hand, in the localized solitonic D3 brane geometry, the horizontal direction is not fully pinched off.', '1107.4048-4-85-1': 'Recall that the geometry ([REF]) is pinched off at [MATH], where [MATH] and [MATH].', '1107.4048-4-85-2': 'Thus [MATH] can reach zero.', '1107.4048-4-85-3': 'As a result, D7 and [MATH] can extend separately as shown in Figure [REF] (g).', '1107.4048-4-85-4': 'This configuration is similar to (e) in Figure [REF] and would restore chiral symmetry.', '1107.4048-4-85-5': 'In addition to this configuration, a chiral symmetry broken configuration is also possible as shown in Figure [REF] (h).', '1107.4048-4-86-0': 'The last task is the evaluation of the stability of these two configurations at temperatures higher than [MATH] ([REF]).', '1107.4048-4-86-1': 'In the flat space, the force between a single D3 and single D7 (or between a single D3 and a single [MATH]) in our configuration is repulsive, since the number of the Neumann-Dirichlet open strings between them is 6 [CITATION].', '1107.4048-4-86-2': 'Thus we expect that even after taking the large [MATH] and near horizon limits, the force may be repulsive.', '1107.4048-4-86-3': 'As a result, the D7/[MATH] branes, which are separated from the localized SD3 in the [MATH]-direction, would try to move away as far as possible in this direction.', '1107.4048-4-86-4': 'However, since the [MATH]-direction is compactified on a circle, the D7/[MATH] branes should then end as being fixed at the point on the [MATH]-circle which is antipodal to the localized SD3.', '1107.4048-4-86-5': 'E.g. if we put the localized SD3 brane at [MATH], the D7 as well as the [MATH] branes will both be at [MATH].', '1107.4048-4-86-6': 'See the left diagram of Figure [REF].', '1107.4048-4-86-7': 'In this case, we can effectively restrict the dynamics of these branes to ([MATH]) plane (given by [MATH]).', '1107.4048-4-86-8': 'Thus the problem reduces to finding classical solutions [MATH] with a boundary condition: [MATH]), where we put the D7 at [MATH] and [MATH] at [MATH].', '1107.4048-4-87-0': 'The problem stated above is difficult to solve precisely near the GL transition since the background metric ([REF]) is only an approximate description.', '1107.4048-4-87-1': 'The metric around the D7/[MATH] brane (at [MATH]) becomes more and more accurate, however, when [MATH].', '1107.4048-4-87-2': 'As a result, we solve for the stable configuration in this limit.', '1107.4048-4-87-3': 'Details of the calculation are presented in appendix [REF].', '1107.4048-4-87-4': 'We find three solutions, corresponding to the D brane configurations of Figures [REF] (g) and (h), as shown in Figure [REF].', '1107.4048-4-87-5': 'In the appendix, we compare the classical DBI actions of these solutions numerically, as depicted in Figure [REF] (a) (see also the phase diagram in Figure [REF]): as one increases temperature beyond a certain value, chiral symmetry is restored, the transition being of first order.', '1107.4048-4-87-6': 'Therefore, one can see that even in our proposal, similarly to the case of the black D4 [CITATION], we can explain chiral symmetry restoration at high enough temperatures.', '1107.4048-4-88-0': '# Conclusions', '1107.4048-4-89-0': 'In this paper, we showed that the conventional representation of the confinement/deconfinement transition in holographic QCD has several problems and proposed an alternative representation which resolves these problems.', '1107.4048-4-90-0': 'As mentioned earlier, problems similar to the above had also been encountered in the study of two dimensional bosonic gauge theory in [CITATION].', '1107.4048-4-90-1': 'This indicates that the issues addressed in this paper are rather general in the discussion of holography for non-supersymmetric gauge theories at finite temperatures.', '1107.4048-4-90-2': 'To elaborate, in the standard holographic procedure, a [MATH]-dimensional non-supersymmetric gauge theory is first constructed through the KK reduction of a [MATH]-dimensional super Yang-Mills theory on a Scherk-Schwarz circle.', '1107.4048-4-90-3': 'The [MATH]-dimensional SYM at large [MATH] can be mapped to a scaling limit of D[MATH] brane geometries [CITATION].', '1107.4048-4-90-4': 'At finite temperatures, because of the two compact cycles (temporal and Scherk-Schwarz), several distinct solutions (depending on boundary conditions) appear in gravity as shown in section [REF] and [REF]: solitonic D[MATH] (equivalently, uniformly smeared solitonic D[MATH]), localized solitonic D[MATH], and black D[MATH].', '1107.4048-4-90-5': 'The black D[MATH] brane solution appears at high temperatures in the (AP,AP) case, while the localized solitonic D[MATH] brane solution is the high temperature phase in the (P,AP) case.', '1107.4048-4-90-6': 'A table similar to Table [REF] can again be constructed, where the appropriate order parameters would be [MATH]; these would again appear to favour the localized solitonic D[MATH] phase as the more suitable representation of the deconfinement of [MATH]-dimensional YM theory (rather than the more conventional black D[MATH] phase which appears only in the (AP,AP) b.c.).', '1107.4048-4-90-7': 'Following this logic, a [MATH]-dimensional analogue of our proposal (see Section [REF]) would appear to give a better description of holographic QCD in [MATH] dimensions.', '1107.4048-4-90-8': 'In particular, we believe that the Gregory-Laflamme transition between the solitonic D[MATH] and localized solitonic D[MATH], would, as in this paper, be related to the confinement/deconfinement transition in the [MATH]-dimensional gauge theory.', '1107.4048-4-91-0': '## Further questions', '1107.4048-4-92-0': 'In order to further understand holographic QCD through the above proposal, it would be of interest to address the following questions.', '1107.4048-4-93-0': 'Transition temperature', '1107.4048-4-94-0': 'From ([REF]), the critical temperature of the confinement/deconfinement transition (the GL transition) would be [MATH].', '1107.4048-4-94-1': 'On the other hand, holographic QCD seems to predict that the square root of the QCD string tension is [MATH], as in ([REF]), whereas the glueball masses are [MATH] [CITATION].', '1107.4048-4-94-2': 'It would be important to understand the reason for the separation of these scales and how they evolve from strong coupling to weak coupling.', '1107.4048-4-95-0': 'Quantitative correspondence', '1107.4048-4-96-0': 'Although our new proposal reproduces the known qualitative features of 4 dimensional Yang-Mills theory, it does not automatically lead to a quantitative agreement.', '1107.4048-4-96-1': 'For example, the free energy of the deconfinement phase of the YM theory at a sufficiently high temperature must be proportional to [MATH], since the coupling becomes weak and the theory becomes approximately conformal.', '1107.4048-4-96-2': 'However the free energy of the localized solitonic D3 is proportional to [MATH] (as can be seen from ([REF])).', '1107.4048-4-96-3': 'This is not entirely surprising since the functional form of [MATH] can change as one evolves from weak coupling to strong coupling.', '1107.4048-4-96-4': 'Furthermore, one has to exercise caution in defining a high temperature limit of YM4 in the holographic context since the temperature must always remain much smaller than the KK scale.', '1107.4048-4-97-0': 'YM4 from SYM5 with (AP,AP) In this article, we have emphasized the correspondence between YM4 and SYM5 with (P,AP) b.c. However, as we pointed out in ([REF]), the 5 dimensional SYM with (AP,AP) b.c. should also be related to YM4, since the boundary condition becomes irrelevant in the limit [MATH] and [MATH].', '1107.4048-4-97-1': 'A possible way this correspondence may work is as follows.', '1107.4048-4-97-2': 'In the (AP,AP) case, if we treat the black D4 solution (the blue region in the bottom half of Figure [REF]), which is not related to YM4, as irrelevant, and focus on the solitonic D4 brane, its winding modes would appear to be light around the temperature ([REF]).', '1107.4048-4-97-3': 'A Hagedorn transition parallel to the (P,AP) case might occur above this temperature and might continue to the confinement/deconfinement transition in YM4.', '1107.4048-4-97-4': 'However, in order to investigate it further through gravity, we need to understand the gauge/gravity correspondence in the 0B frame as mentioned in footnote [REF].', '1107.4048-4-98-0': 'Real time It would be important to explore what geometry corresponds to the deconfinement phase in the real time formalism.', '1107.4048-4-98-1': 'An understanding of this would allow us to address dynamical properties of the deconfinement phase, e.g. transport properties.', '1107.4048-4-98-2': 'Since previous results in this area were based on the black brane solutions, it would be important to see how well-known results such as the viscosity bound [CITATION] can be derived in our proposal.', '1107.4048-4-99-0': 'AdS/CMT and chemical potential dependence', '1107.4048-4-100-0': 'The SS transition for D3 branes has been studied in the context of the AdS/CMT correspondence to investigate the superconductor/insulator transition in 2+1 dimension [CITATION].', '1107.4048-4-100-1': 'In these studies, a chemical potential for a [MATH] charge was introduced and the phase structure involving this chemical potential has been derived.', '1107.4048-4-100-2': 'It would be interesting to ask whether our proposal has any bearing on these studies, e.g., whether the GL transition analogous to the one discussed here can be a candidate for the superconductor/insulator or some other transition.', '1107.4048-4-100-3': 'A possible line of investigation could be to study the chemical potential dependence of [MATH] for the GL transition vis-a-vis that for the SS transition and see whether any qualitative differences appear.'}",
1407.7975,"{'1407.7975-1-0-0': 'We present a systematic investigation of the effect of H, B, C, and N interstitials on the electronic, lattice and magnetic properties of La(Fe,Si)[MATH] using density functional theory.', '1407.7975-1-0-1': 'The parent LaSiFe[MATH] alloy has a shallow, double-well free energy function that is the basis of first order itinerant electron metamagnetism.', '1407.7975-1-0-2': 'On increasing the dopant concentration, the resulting lattice expansion causes an initial increase in magnetisation for all interstitials that is only maintained at higher levels of doping in the case of hydrogen.', '1407.7975-1-0-3': 'Strong s-p band hybridisation occurs at high B,C and N concentrations.', '1407.7975-1-0-4': 'We thus find that the electronic effects of hydrogen doping are much less pronounced than those of other interstitials, and result in the double-well structure of the free energy function being least sensitive to the amount of hydrogen.', '1407.7975-1-0-5': 'This microscopic picture accounts for the change in the metamagnetic transition from first order to second order on doping with B,C, and N interstitials, as observed experimentally.', '1407.7975-1-1-0': '# Introduction', '1407.7975-1-2-0': 'The magnetocaloric effect (MCE) is the temperature change of a substance subjected to a change in applied magnetic field.', '1407.7975-1-2-1': ""The discovery of the effect can be attributed to Weiss and Piccard's observation of the magnetization of nickel close to its Curie point in 1917, [CITATION]after a recent re-examination of the original literature by Smith.["", '1407.7975-1-2-2': '[CITATION] The adiabatic demagnetization of paramagnetic salts was shown by Giauque and MacDougall in 1933[CITATION] following initial proposals by both Debye and Giauque in the previous decade.[', '1407.7975-1-2-3': ""[CITATION] The study of room temperature MCEs associated with a magnetic phase transition was revived in 1997 by Pecharsky and Gschneidner who observed a 'giant' entropy change of [MATH]14 JK[MATH]kg[MATH] in a 0-2 Tesla field change in Gd[MATH]Si[MATH]Ge[MATH].["", '1407.7975-1-2-4': '[CITATION] While experiments up to that point had postulated the possibility of room temperature refrigeration using, for example, a second order Curie transition such as that found in Gd, [CITATION] it was the effects associated with the first order transition seen in Gd[MATH]Si[MATH]Ge[MATH] that initiated widespread research interest in the MCE.', '1407.7975-1-2-5': ""Today, a large set of magnetic materials show 'large' or 'giant' magnetocaloric effects."", '1407.7975-1-2-6': '[CITATION] However, a good refrigerant material also needs to fulfil auxiliary requirements such as tuneable thermal conductivity, durability and elemental abundance and so the number of material systems that are close to commercialisation is relatively small.', '1407.7975-1-2-7': 'This situation provides motivation for the use of theoretical models that may aid the understanding and prediction of magnetocaloric effects.', '1407.7975-1-3-0': 'Density functional theory (DFT) is a valuable tool with which to describe the changes in matter at the electronic level that may lead to a large MCE.', '1407.7975-1-3-1': 'Elemental Gd has a ferromagnetic (FM) ordering temperature around room temperature that makes it an ideal candidate magnetocaloric material.', '1407.7975-1-3-2': 'DFT calculations based on thermally induced spin fluctuations in a disordered local moment picture showed that the magnetic order in Gd is linked to the [MATH] ratio and atomic unit cell volume.', '1407.7975-1-3-3': '[CITATION] Such magneto-elastic coupling is useful for generating a large MCE.', '1407.7975-1-3-4': 'However, the cost of heavy rare-earth Gd inhibits its use in everyday applications as refrigerant.', '1407.7975-1-3-5': 'In Gd[MATH](Si[MATH]Ge[MATH]), DFT calculations indicated breaking and reforming of Si-Ge bonds between layers within the unit cell, affecting both the location of the Fermi level and the effective magnetic exchange coupling, increasing the latter to the level where a first order magneto-structural transition is observed.', '1407.7975-1-3-6': '[CITATION]', '1407.7975-1-4-0': 'Manganites and manganese silicides have also been the subject of DFT studies.', '1407.7975-1-4-1': 'In manganites, the broad variety of crystallographic, magnetic and electronic phases are attributed to the strong interplay between spin, charge, orbital and lattice degrees of freedom that often couples to external magnetic fields and results in measurable MCE.', '1407.7975-1-4-2': 'For a qualitative description of these correlated physical quantities, state-of-the-art hybrid exchange density functionals such can be applied.', '1407.7975-1-4-3': '[CITATION] In the case of manganese based metallic silicides, ground state and finite temperature DFT models have been used to model and predict new Mn-based metamagnets.', '1407.7975-1-4-4': '[CITATION] Those calculations used accurate structural data obtained from high resolution neutron diffraction on CoMnSi, a noncollinear antiferromagnet (AFM) that exhibits giant magneto-elastic coupling.', '1407.7975-1-4-5': '[CITATION]', '1407.7975-1-5-0': 'Experimentally, the most intensively studied MCE materials are based on either Fe[MATH]P or La(Fe,Si)[MATH].', '1407.7975-1-5-1': 'Both have been the subject of some modelling studies.', '1407.7975-1-5-2': 'In Fe[MATH]P, iron has two inequivalent crystallographic sites and the low moment site (3[MATH]) has a metamagnetic transition[CITATION] at the Curie temperature, 212 K.', '1407.7975-1-5-3': 'The Curie point can be tuned through room temperature by partial replacement of Fe by Mn as well as P by, for example, Si.[', '1407.7975-1-5-4': '[CITATION] The so-called mixed magnetism of this material has been investigated by a number of DFT studies that have identified the mechanism of magneto-elastic coupling and the change of electron density across the Curie transition.[', '1407.7975-1-6-0': 'In this article, we perform a DFT study of compounds based on LaFe[MATH]M[MATH] (M=Si, Al), a cubic NaZn[MATH]-type material which was first synthesized by Kripyakevich et al.[CITATION] To date, much of the compositional tuning that is used to adjust the magnetocaloric effect and its temperature range is the result of empirical work rather than theory-led prediction.', '1407.7975-1-7-0': 'LaFe[MATH]M[MATH] exhibits a large MCE associated with a paramagnetic to ferromagnetic transition on cooling at a temperature, [MATH], between 180 and 250 K.', '1407.7975-1-7-1': 'A magnetic field-dependent itinerant-electron metamagnetic (IEM) transition above [MATH], can be shifted towards room temperature by Si addition.', '1407.7975-1-7-2': 'However, on increasing the Si content above [MATH], a change in the nature of the FM phase transition from first-order to second-order takes place that results in a considerable reduction of the useful MCE.', '1407.7975-1-7-3': 'A first-principles calculation by Wang et al.[CITATION] indicated that hybridization between the Fe-[MATH] and Si-[MATH] states is linked to the reduction of Fe magnetic moment as well as to the smearing of the first-order type transition for alloys with high Si-content.', '1407.7975-1-8-0': 'The partial replacement of the transition metal element Fe by Co or Mn has been explored in an attempt to preserve the first order nature of the transition around the Curie temperature, [MATH], although both elements cause significant weakening of the field-induced IEM transition[CITATION].', '1407.7975-1-8-1': 'Similarly, interstitial doping of [MATH]-block or [MATH]-block elements was also pursued experimentally in order to raise the IEM to room temperature.', '1407.7975-1-8-2': 'These empirical studies found that the preparation of single phase compositions is limited to low interstitial concentrations and that only hydrogen is capable of the increase of magnetic transition to room temperature without the diminution of useful isothermal entropy change.', '1407.7975-1-8-3': ""[CITATION] Theoretical calculations by Kuz'min and Richter[CITATION] on LaFe[MATH]Si, without interstitial substitution, found that the free energy, as a function of magnetization ([MATH]) has several shallow minima and maxima, to which they attributed the reduced hysteresis and improved magnetocaloric performance of La(Fe,Si)[MATH]."", '1407.7975-1-8-4': 'Recently, Fujita and Yako [CITATION] extended this approach and further detailed the dependence of such an energy plot on both the lattice size and the degree of Fe/Si substitution.', '1407.7975-1-9-0': 'We note that no systematic investigation of the effect of interstitial [MATH]-block or [MATH]-block elements on the electronic, lattice and magnetic properties of La(Fe,Si)[MATH] has been carried out to date.', '1407.7975-1-9-1': 'The work presented here, using a theoretical approach based on DFT, attempts to describe the effect of the size of four different dopants and their valence electrons to understand how interstitials can influence the magnetocaloric performance of these alloys.', '1407.7975-1-9-2': 'We describe our theoretical methods in section [REF] before presenting our results and discussion in section [REF].', '1407.7975-1-9-3': 'Conclusions are drawn in section [REF].', '1407.7975-1-10-0': '# Methods', '1407.7975-1-11-0': '## Computational models used', '1407.7975-1-12-0': 'Our computational approach is divided into two, complementary parts.', '1407.7975-1-12-1': 'In the first part, we investigate the effect of interstitial doping on the equilibrium unit cell volume using the projector augmented wave (PAW) method[CITATION] as implemented in the Vienna ab-initio simulation package (VASP).', '1407.7975-1-12-2': 'The VASP code with Perdew-Burke-Ernzerhof (PBE) parameterization[CITATION] is employed, where site-based magnetic moments were calculated using the Vosko-Wilk-Nusair interpolation[CITATION] within the general gradient approximation (GGA) for the exchange-correlation potential.', '1407.7975-1-13-0': 'La(Fe,Si)[MATH] has 8 formula units per conventional cell.', '1407.7975-1-13-1': 'The La atoms occupy the 8[MATH] sites ([MATH],[MATH],[MATH]), while diffraction studies show that Fe and Si atoms can occupy both 8[MATH] and the 96[MATH] crystallographic positions.', '1407.7975-1-13-2': ""[CITATION] In order to keep the computational requirements at a feasible level, we follow the approach previously adopted by Kuz'min and Richter,[CITATION] limiting our investigations to an atomically ordered version of LaSiFe[MATH]Z[MATH], where the 8[MATH] sites are occupied solely by silicon whilst iron is located exclusively on the 96[MATH] sites."", '1407.7975-1-13-3': 'In such a case, the cell that forms the basis of the calculations contains 2 La, 26 Fe and 2 Si atoms.', '1407.7975-1-13-4': 'Furthermore, interstitial elements (Z) H, B, C, and N were considered to occupy the 24[MATH] crystallographic site only.', '1407.7975-1-13-5': 'Using this model, we may vary the concentration of interstitials, [MATH] in a step size of 0.5 from [MATH] to 3 in LaSiFe[MATH]Z[MATH].', '1407.7975-1-14-0': 'Full structural relaxation was carried out for both collinear ferromagnetic (FM) and non-magnetic (NM) states in the case of parent LaFe[MATH]Si alloy, while only the lattice parameter [MATH] was relaxed (without relaxation of the internal atomic positions) for the materials doped with [MATH]- or [MATH]-block interstitials.', '1407.7975-1-14-1': 'A 7 [MATH]-point grid was used to discretize the first Brillouin zone and the energy convergence criterion was set to [MATH] eV during the energy minimization process.', '1407.7975-1-14-2': 'The spin-orbit interaction was turned off during the calculations.', '1407.7975-1-14-3': 'Finally, data presented in Fig. [REF] was calculated on a dense 19 [MATH] 19 grid of [MATH]-points for high accuracy.', '1407.7975-1-15-0': 'In the second part of this study, we have taken a fixed spin moment (FSM) approach within the tight-binding theorem using linear muffin tin orbitals (TB-LMTO) as implemented in Stuttgart TB-LMTO code.', '1407.7975-1-15-1': '[CITATION] This method requires carefully adjusted overlapping Wigner-Seitz (WS) atomic spheres included in the calculations to complete the basis and to provide an accurate description of the electron density throughout the entire unit cell.', '1407.7975-1-15-2': 'Consequently, the structural parameters of the relaxed lattice are inherently dependent on the volume occupied by the WS spheres and/or empty spheres.', '1407.7975-1-15-3': 'For this reason, we used VASP code (see above) for relaxation.', '1407.7975-1-15-4': 'Nevertheless, the TB-LMTO approach allows us to evaluate the total energy difference between FM and NM states, [MATH] as a function of fixed spin moment [MATH] as well as the corresponding density of states (DOS) and band dispersions.', '1407.7975-1-15-5': 'H, B, C, and N atoms were considered to fully occupy the 24[MATH] crystallographic site, LaSiFe[MATH]Z[MATH], for the study.', '1407.7975-1-15-6': 'A dense mesh with 48[MATH]-points (for the DOS calculations) or with 12[MATH]-points (for the FSM calculations) was used.', '1407.7975-1-16-0': '# Results and Disscussion', '1407.7975-1-17-0': '## Effect of dopants on the lattice expansion and magnetic properties', '1407.7975-1-18-0': 'Fig. [REF] shows the calculated lattice parameter in the FM state in LaSiFe[MATH]Z[MATH] as a function of interstitial doping.', '1407.7975-1-18-1': 'Our calculations obtain a relaxed structure that differs by only about 0.1% from the experimentally reported value.', '1407.7975-1-18-2': 'This remarkable agreement validates our choice of exchange correlation, GGA.', '1407.7975-1-18-3': 'The lattice expansion increases monotonically with dopant concentration at a rate that depends strongly on the size of the interstitial element.', '1407.7975-1-18-4': 'The empirical atomic radius of hydrogen (25 pm) is much smaller than that of the boron (85 pm), carbon (70 pm), or nitrogen (65 pm).', '1407.7975-1-18-5': 'The trend in calculated lattice expansion in Fig. [REF] correlates well with the relative atomic size of the interstitial, showing the predominant influence of the latter on the size of the unit cell.', '1407.7975-1-18-6': 'At full doping ([MATH]), we here find a relative lattice expansion of 0.4 for hydrogen and a considerably higher value of 1.7 for carbon, which match with the experimental values H and C, respectively.', '1407.7975-1-18-7': '[CITATION] The values of [MATH] are 1.8 and 1.25 for Z=B and N respectively at full doping, but these are yet to be confirmed experimentally.', '1407.7975-1-19-0': 'There are only limited experimental data available on the relative effects of lattice expansion of the dopants studied here, particularly as full occupation of the 24[MATH] site ([MATH]) by any of the dopants has not been achieved in practice.', '1407.7975-1-19-1': 'In terms of valence electron number, however, a different sequence exists: H[MATH] B[MATH] C[MATH] N[MATH]).', '1407.7975-1-19-2': 'A closer look at Fig. [REF] reveals a non-monotonic behaviour in the lattice expansion as a function of doping, especially in the case of nitrogen.', '1407.7975-1-19-3': 'Indeed, additional charges significantly alter the electronic structure (apart from H) in ways that go beyond the simple picture of chemical pressure effects, as we discuss later.', '1407.7975-1-19-4': 'The contribution of additional valence electrons is also reflected in the calculated magnetic moment ([MATH]).', '1407.7975-1-19-5': '[MATH] rises initially (Fig. [REF], top) for each dopant but a monotonic increase up to [MATH] is only seen in the case of hydrogen.', '1407.7975-1-19-6': 'These observations imply a mechanism for hybrid band formation and band broadening for any dopant with a larger atomic radius than that of hydrogen.', '1407.7975-1-20-0': 'In order to depict the changes in the electronic structure that are brought about by the interstitial elements, we next examine the partial electronic density of states (PDOS).', '1407.7975-1-20-1': 'Fig. [REF] shows the PDOS of the parent alloy together with the fully hydrogenated and fully nitrogenated materials ([MATH]).', '1407.7975-1-20-2': 'In the parent alloy (bottom), the PDOS is dominated by Fe (red line) around the Fermi level (E[MATH]) with typical spin-split states.', '1407.7975-1-20-3': 'The spin-up ([MATH]) states are mostly occupied, while the unoccupied states are dominated by spin-down ([MATH]) states separated by about 2.5eV in energy.', '1407.7975-1-20-4': 'Furthermore, the filled bands at the lower end of energy range (-9.5 eV) relate mostly to silicon 3[MATH] states which are overlapped with [MATH]-states of both La and Fe.', '1407.7975-1-20-5': 'A large energy gap appears from -9.5 eV up to about -6.5eV, where a high population of 3[MATH] states of Si (black) is located (-6.5 to -4.5 eV).', '1407.7975-1-20-6': 'In this latter energy range, there is negligible contribution from Fe [MATH]-states.', '1407.7975-1-20-7': 'Most of the aforementioned features in the electronic structure are preserved in fully hydrogenated LaSiFe[MATH]H[MATH] (middle of Fig. [REF]).', '1407.7975-1-20-8': 'The main difference in the PDOS compared to the parent alloy is the development of additional states in the gap around -7.5 eV related to the hydrogen interstitials.', '1407.7975-1-20-9': 'Small additional peaks also appear around -5 eV, where they overlap with the [MATH] states of Si.', '1407.7975-1-21-0': 'In strong contrast to hydrogenation, fully nitrogenated LaSiFe[MATH]N[MATH] exhibits a large overlap of N [MATH] states with Fe [MATH] states in the -7.5 to -4 eV energy interval (top of Fig. [REF]).', '1407.7975-1-21-1': 'These peaks indicate [MATH] hybridization, and as a result, increased covalency in bond formation.', '1407.7975-1-21-2': 'Another important consequence of nitrogenation is the appearance of states in the vicinity of the Fermi level.', '1407.7975-1-21-3': 'The existence of a ""double peak"" feature just below and above E[MATH] for LaSiFe[MATH], created by Fe [MATH] states in the minority DOS, is altered only a little by hydrogenation.', '1407.7975-1-21-4': 'On the other hand, nitrogenation fills this valley at E[MATH], which results in the strong alteration of the magnetic properties and ultimately leads to the disappearance of IEM transition.', '1407.7975-1-21-5': 'We address the latter behavior in detail in the next section.', '1407.7975-1-22-0': '## The free energy landscape', '1407.7975-1-23-0': 'We now turn our interest to the results of our second computational approach, fixed spin moment calculations using TB-LMTO.', '1407.7975-1-23-1': 'Our aim is to visualize the energy difference between FM and NM states in the parent alloy and the doped materials.', '1407.7975-1-23-2': 'The purpose of our analysis is to identify the main factors that lead to the field-induced isothermal entropy change of LaSiFe[MATH]Z[MATH] around the magnetic transition being lower than that of LaSiFe[MATH], as found experimentally for interstitials other than hydrogen.', '1407.7975-1-24-0': 'Fig. [REF] compares the free energy curves [MATH] calculated by the FSM method for LaSiFe[MATH]Z[MATH], where Z = H, N, B and empty sphere (Es), respectively, together with those of the parent alloy.', '1407.7975-1-24-1': 'For direct comparison, we set the non-magnetic energy state as F(0) for each individual composition.', '1407.7975-1-24-2': ""Fig. [REF](a) shows that the parent LaSiFe[MATH] has a very shallow magnetic energy landscape, in accordance with the predictions by Kuz'min and Richter, who used the full-potential local-orbital (FPLO) method."", '1407.7975-1-24-3': '[CITATION] They noted the advantage of such a potential energy landscape in permitting a low hysteresis, first order metamagnetic transition (IEM).', '1407.7975-1-24-4': 'The main difference in our calculation of the parent compound is that we find only two minima rather than the multiple minima that were predicted in their work.', '1407.7975-1-24-5': ""The field-induced magnetisation of hydrogenated La-Fe-Si under pressure previously was seen to exhibit multiple steps, confirming Kuz'min and Richter's predictions."", '1407.7975-1-24-6': 'It may be interesting to investigate this property in the undoped compound to eliminate the possibility of pressure-induced hydrogen segregation.', '1407.7975-1-24-7': 'Spontaneous hydrogen segregation is known to occur in materials with a smaller hydrogen content than the empirical maximum.', '1407.7975-1-24-8': '[CITATION]', '1407.7975-1-25-0': 'The derivative of [MATH] is also shown in Fig. [REF]; local minima in the free energy function are where [MATH].', '1407.7975-1-25-1': 'We may conclude that full doping of any of the four interstitials studied removes the shallow double-well potential, resulting in only a single well.', '1407.7975-1-25-2': 'This corresponds to the disappearance of the first order metamagnetic transition, as found experimentally for B, C, and N doping.', '1407.7975-1-25-3': 'We note that boron addition is detrimental to the total magnetisation as the minimum in [MATH] occurs at a lower value of [MATH] than for any other dopant.', '1407.7975-1-25-4': 'Of all the interstitials studied, hydrogen alters the double well picture the least.', '1407.7975-1-25-5': 'The shallow landscape of [MATH] takes on a concave curvature in the range of [MATH] at full doping.', '1407.7975-1-25-6': 'The intrinsically small energy barrier for the parent compound and the hydrogen-doped material makes these compositions particularly sensitive to external parameters such as magnetic field, pressure and temperature and renders the first order IEM transition quasi-reversible.', '1407.7975-1-26-0': 'Fig. [REF]b also reveals the sensitive nature of the metamagnetic states to the lattice parameters.', '1407.7975-1-26-1': 'An increase of about 0.5% in the lattice parameter upon H-doping generates a ground state with a high-spin configuration.', '1407.7975-1-26-2': 'In order to separate the changes in the magnetic state caused by the volume expansion (chemical pressure) due to the inclusion of large interstitials such as nitrogen from those caused by additional valence electrons, we also carried out calculations with empty spheres (Es) included at the 24[MATH] crystallographic site.', '1407.7975-1-26-3': 'For a direct comparison, the same lattice constant was adopted for both Z = N and Es.', '1407.7975-1-27-0': 'To further examine the appearance of a metastable low-spin state at around [MATH] and the emergence of an energy barrier near [MATH], DOS calculations were performed with fixed spin moments at these two values of magnetisation for each system.', '1407.7975-1-27-1': 'The results are shown in Fig. [REF].', '1407.7975-1-27-2': 'For both X=H and Es, the Fermi level is located in a deep valley for both the minority and majority-spin DOS at M=7[MATH], similar to that found in the parent alloy.', '1407.7975-1-27-3': 'This well-defined valley in the DOS is destroyed for X = B and N. By contrast, the electronic features found in the DOS for M=13[MATH] differ significantly; high peaks in the DOS appears at E[MATH] for both minority and majority-spin states for the undoped as well as for Z=H and Es alloys, whilst only a moderate peak in height for Z=N and B are observed.', '1407.7975-1-27-4': 'It is thus apparent that the valleys and peaks around E[MATH] can be attributed to the low-spin state and the energy barrier in [MATH].', '1407.7975-1-28-0': 'In Fig. [REF], we show the electronic band structure calculated for [MATH], in order to provide an explanation for the above peaks in the electronic DOS.', '1407.7975-1-28-1': 'We note that a 0.01 eV offset was added to the energy level of [MATH] bands of N for clarity in the figure.', '1407.7975-1-28-2': 'We may make several qualitative observations.', '1407.7975-1-28-3': 'First, the flat and quasi-degenerate t[MATH] and e[MATH] bands are widely observed toward representative [MATH]-directions along the Brillouin zone in the parent alloy.', '1407.7975-1-28-4': 'The additional charge supplied by the H atoms shifts the position of the Fermi level upwards but the character of these 3[MATH] bands is conserved.', '1407.7975-1-28-5': 'Second, in the empty sphere configuration (Z=Es), where the significantly larger lattice structure of the nitrogenated alloy is adopted without any addition of charge, strong narrowing of the bandwidth is found but the dispersion of each band is once again preserved (not shown).', '1407.7975-1-29-0': 'Third, the situation is very different for both Z=N and B, as the band structure is strongly altered at full doping.', '1407.7975-1-29-1': 'For Z=B, strong [MATH] band mixing occurs between -0.3 and -0.6 eV for the majority spin states in all [MATH]-directions.', '1407.7975-1-29-2': 'In addition, broad bands originating from the [MATH]-states of boron appear at the [MATH] point about -0.3 eV for the minority spins.', '1407.7975-1-29-3': 'The formation of these hybrid [MATH] bands ultimately results in the vanishing of the well-formed peak and valley structures of the DOS around E[MATH] as shown in Fig. [REF] and in Fig. [REF].', '1407.7975-1-29-4': 'Finally, for nitrogen doping most of the flat bands appear just above E[MATH], apart from some minor ones around -0.5 eV in the W direction for the minority spins.', '1407.7975-1-29-5': 'Here, the [MATH] mixing occurs mainly between bands in the K and W directions.', '1407.7975-1-29-6': 'Bands of [MATH] character show especially pronounced mixing with [MATH]-electrons.', '1407.7975-1-29-7': 'The number of flat bands around E[MATH] is decreased as compared to the undoped system and as a result the DOS has uneven features with small peaks that are detrimental to the IEM.', '1407.7975-1-29-8': 'transition.', '1407.7975-1-30-0': '# Conclusions', '1407.7975-1-31-0': 'We have investigated the effect of selected [MATH]- and [MATH]-block interstitial elements on the electronic, lattice and magnetic properties of La(Fe,Si)[MATH] using DFT.', '1407.7975-1-31-1': 'Our calculations find that a good correlation between the expansion of the unit cell and the size of the dopant.', '1407.7975-1-31-2': 'Fixed spin moment calculations yield a double well structure in the free energy of LaFe[MATH]Si as a function of magnetisation, which may be seen as the basis of the itinerant electron metamagnetic transition.', '1407.7975-1-31-3': 'Significantly, hydrogenation alters the electronic and magnetic structure of LaSiFe[MATH] to a much smaller degree than B, C and N dopants.', '1407.7975-1-31-4': 'This means that the first order IEM is much more robust to H insertion than to interstitial B, C, or N.', '1407.7975-1-32-0': 'An analysis of the projected electronic DOS reveals that the dominant electronic states related to hydrogen insertion appear at around -8 to -7 eV, where very little contribution from Fe, Si and La elements is present.', '1407.7975-1-32-1': 'The additional charge of the hydrogen atoms elevates the Fermi level but the character of the bands is unaltered.', '1407.7975-1-32-2': 'The latter feature is also evident in the empty sphere configuration, where the nitrogenated lattice parameters are simulated without the N inclusions; only a narrowing of the band-width is found but the dispersion of each band remains mostly unaffected.', '1407.7975-1-33-0': 'Consequently, hydrogen provides perhaps the only chemical pressure on the lattice that avoids significant alterations to the electronic structure of LaSiFe[MATH].', '1407.7975-1-33-1': 'In the case of the other dopants (B, C, N) broad bands originating from their [MATH]-states appear at energy levels, where the 3[MATH] states of Fe are also present.', '1407.7975-1-33-2': 'The formation of these hybrid [MATH] bands results in the disappearance of the peak and valley structures in the electronic DOS around E[MATH], thereby reshaping the shallow free energy landscape and ultimately destroying the first order IEM of LaSiFe[MATH].', '1407.7975-1-34-0': ""The research leading to these results has received funding from the European Community's 7th Framework Programme under Grant Agreement No. 310748 DRREAM."", '1407.7975-1-34-1': 'Computing resources provided by Darwin HPC and Camgrid facilities at The University of Cambridge and the HPC Service at Imperial College London are also gratefully acknowledged.'}","{'1407.7975-2-0-0': 'We present a systematic investigation of the effect of H, B, C, and N interstitials on the electronic, lattice and magnetic properties of La(Fe,Si)[MATH] using density functional theory.', '1407.7975-2-0-1': 'The parent LaSiFe[MATH] alloy has a shallow, double-well free energy function that is the basis of itinerant metamagnetism.', '1407.7975-2-0-2': 'On increasing the dopant concentration, the resulting lattice expansion causes an initial increase in magnetisation for all interstitials that is only maintained at higher levels of doping in the case of hydrogen.', '1407.7975-2-0-3': 'Strong s-p band hybridisation occurs at high B,C and N concentrations.', '1407.7975-2-0-4': 'We thus find that the electronic effects of hydrogen doping are much less pronounced than those of other interstitials, and result in the double-well structure of the free energy function being least sensitive to the amount of hydrogen.', '1407.7975-2-0-5': 'This microscopic picture accounts for the vanishing first order nature of the transition by B,C, and N dopants as observed experimentally.', '1407.7975-2-0-6': 'We use our calculated electronic density of states for LaSiFe[MATH] and the hydrogenated alloy to infer changes in magneto-elastic coupling and in phonon entropy on heating through [MATH] by calculating the fermionic entropy due to the itinerant electrons.', '1407.7975-2-0-7': 'Lastly, we predict the electron thermopower in a spin-mixing, high temperature limit and compare our findings to recent literature data.', '1407.7975-2-1-0': '# Introduction', '1407.7975-2-2-0': 'The magnetocaloric effect (MCE) is the temperature change of a substance subjected to a change in applied magnetic field.', '1407.7975-2-2-1': ""The discovery of the effect can be attributed to Weiss and Piccard's observation of the magnetization of nickel close to its Curie point in 1917 [CITATION], after a recent re-examination of the original literature by Smith [CITATION]."", '1407.7975-2-2-2': 'The adiabatic demagnetization of paramagnetic salts was shown by Giauque and MacDougall in 1933[CITATION] following initial proposals by both Debye and Giauque in the previous decade [CITATION].', '1407.7975-2-2-3': ""The study of room temperature MCEs associated with a magnetic phase transition was revived in 1997 by Pecharsky and Gschneidner who observed a 'giant' entropy change of [MATH]14 JK[MATH]kg[MATH] in a 0-2 Tesla field change in Gd[MATH]Si[MATH]Ge[MATH] [CITATION]."", '1407.7975-2-2-4': 'While experiments up to that point had postulated the possibility of room temperature refrigeration using, for example, a second order Curie transition such as that found in Gd [CITATION], it was the effects associated with the first order transition seen in Gd[MATH]Si[MATH]Ge[MATH] that initiated widespread research interest in the MCE.', '1407.7975-2-2-5': ""Today, a large set of magnetic materials show 'large' or 'giant' magnetocaloric effects [CITATION] and form one family in a more general set of ferroic refrigerants [CITATION]."", '1407.7975-2-2-6': 'However, a good refrigerant material also needs to fulfil auxiliary requirements such as tuneable thermal conductivity, durability and elemental abundance and so the number of material systems that are close to commercialisation is relatively small.', '1407.7975-2-2-7': 'This situation provides motivation for the use of theoretical models that may aid the understanding and prediction of caloric effects.', '1407.7975-2-3-0': 'Density functional theory (DFT) is a valuable tool with which to describe the changes in matter at the electronic level that may lead to a large MCE.', '1407.7975-2-3-1': 'Elemental Gd has a ferromagnetic (FM) ordering temperature around room temperature that makes it an ideal candidate magnetocaloric material.', '1407.7975-2-3-2': 'DFT calculations based on thermally induced spin fluctuations in a disordered local moment picture showed that the magnetic order in Gd is linked to the [MATH] ratio and atomic unit cell volume [CITATION].', '1407.7975-2-3-3': 'Such magneto-elastic coupling is useful for generating a large MCE since it increases the rate of change of magnetization with temperature.', '1407.7975-2-3-4': 'In Gd[MATH](Si[MATH]Ge[MATH]), DFT calculations indicated breaking and reforming of Si-Ge bonds between layers within the unit cell, affecting both the location of the Fermi level and the effective magnetic exchange coupling, increasing the latter to the level where a first order magneto-structural transition is observed [CITATION].', '1407.7975-2-3-5': 'However, the cost of heavy rare-earth Gd renders magnetocaloric alloys with high d-metal content preferable [CITATION].', '1407.7975-2-4-0': 'Manganites and manganese silicides have also been the subject of DFT studies.', '1407.7975-2-4-1': 'In manganites, the broad variety of crystallographic, magnetic and electronic phases are attributed to the strong interplay between spin, charge, orbital and lattice degrees of freedom that often couples to external magnetic fields and results in measurable MCE.', '1407.7975-2-4-2': 'For a qualitative description of these correlated physical quantities, state-of-the-art hybrid exchange density functionals can be applied [CITATION].', '1407.7975-2-4-3': 'In the case of manganese based metallic silicides, ground state and finite temperature DFT models have been used to model and predict new Mn-based metamagnets [CITATION].', '1407.7975-2-4-4': 'Those calculations used accurate structural data obtained from high resolution neutron diffraction on CoMnSi, a noncollinear antiferromagnet (AFM) that exhibits giant magneto-elastic coupling [CITATION].', '1407.7975-2-5-0': 'Experimentally, the most intensively studied MCE materials are based on either Fe[MATH]P or La(Fe,Si)[MATH].', '1407.7975-2-5-1': 'Both have been the subject of some modelling studies.', '1407.7975-2-5-2': 'In Fe[MATH]P, iron has two inequivalent crystallographic sites and the low moment site (3[MATH]) has a metamagnetic transition[CITATION] at the Curie temperature, 212 K.', '1407.7975-2-5-3': 'The Curie point can be tuned through room temperature by partial replacement of Fe by Mn as well as P by, for example, Si [CITATION].', '1407.7975-2-5-4': 'The so-called mixed magnetism of this material has been investigated by a number of DFT studies that have identified the mechanism of magneto-elastic coupling and the change of electron density across the Curie transition [CITATION].', '1407.7975-2-6-0': 'In this article, we perform a DFT study of compounds based on LaFe[MATH]M[MATH] (M=Si, Al), a cubic NaZn[MATH]-type material which was first synthesized by Kripyakevich et al. [CITATION].', '1407.7975-2-6-1': 'To date, much of the compositional tuning that is used to adjust the magnetocaloric effect and its temperature range is the result of empirical work rather than theory-led prediction.', '1407.7975-2-6-2': 'As a function of Si (Al) content, LaFe[MATH]M[MATH] exhibits an FM (or AFM) transition upon cooling at a temperature between 180 and 250 K with large MCE.', '1407.7975-2-6-3': 'Furthermore, the magnetic field-dependent itinerant-electron metamagnetic (IEM) transition[CITATION] can be shifted towards room temperature by Si addition.', '1407.7975-2-6-4': 'However, on increasing the Si content above [MATH], a change in the nature of the FM phase transition from first-order to second-order takes place that results in a considerable reduction of the useful MCE.', '1407.7975-2-6-5': 'A first-principles calculation by Wang et al. [CITATION] indicated that hybridization between the Fe-[MATH] and Si-[MATH] states is linked to the reduction of Fe magnetic moment as well as to the smearing of the first-order type transition for alloys with high Si-content.', '1407.7975-2-7-0': 'The partial replacement of the transition metal element Fe by Co or Mn has been explored in an attempt to preserve the first order nature of the transition around the Curie temperature, [MATH], although both elements cause significant weakening of the field-induced IEM transition [CITATION].', '1407.7975-2-7-1': 'Similarly, interstitial doping of [MATH]-block or [MATH]-block elements has also been pursued experimentally in order to raise the IEM to room temperature.', '1407.7975-2-7-2': 'These empirical studies found that the preparation of single phase compositions is limited to low interstitial concentrations and that only hydrogen is capable of the increase of magnetic transition to room temperature without the diminution of useful isothermal entropy change [CITATION].', '1407.7975-2-7-3': 'Theoretical calculations by Kuzmin and Richter [CITATION] on LaFe[MATH]Si, without interstitial substitution, found that the free energy, as a function of magnetization ([MATH]) has several shallow minima and maxima, to which they attributed the reduced hysteresis and improved magnetocaloric performance of La(Fe,Si)[MATH].', '1407.7975-2-7-4': 'Fujita and Yako [CITATION] extended this approach and further detailed the dependence of such an energy plot on both the lattice size and the degree of Fe/Si substitution.', '1407.7975-2-7-5': 'Most recently, Gruner et al. used a DFT approach to model the difference in phonon density of states between the paramagnetic and ferromagnetic states [CITATION].', '1407.7975-2-7-6': 'They found, consistent with nuclear resonant inelastic x-ray scattering (NRIXS), that phonon entropy appears on heating through [MATH], despite the decrease in unit cell volume.', '1407.7975-2-8-0': 'We note that no systematic investigation of the effect of interstitial [MATH]-block or [MATH]-block elements on the electronic, lattice and magnetic properties of La(Fe,Si)[MATH] had been carried out before this work.', '1407.7975-2-8-1': 'The work presented here, using a theoretical approach based on DFT, attempts to describe the effect of the size of four different dopants and their valence electrons to understand how interstitials can influence the magnetocaloric performance of these alloys.', '1407.7975-2-8-2': 'We use our calculated electronic density of states to further examine two experimental quantities linked to the entropy change present at [MATH].', '1407.7975-2-8-3': 'Firstly, we infer changes in magneto-elastic coupling and in phonon entropy on heating through [MATH] by calculating the fermionic entropy due to the itinerant electrons.', '1407.7975-2-8-4': 'Secondly, we predict the electron thermopower in a spin-mixing, high temperature limit and compare our findings to recent literature data.', '1407.7975-2-8-5': 'We describe our theoretical methods in section [REF] before presenting our results and discussion in section [REF].', '1407.7975-2-8-6': 'Conclusions are drawn in section [REF].', '1407.7975-2-9-0': '# Methods', '1407.7975-2-10-0': '## Computational models used', '1407.7975-2-11-0': 'Our computational approach is divided into two, complementary parts.', '1407.7975-2-11-1': 'In the first part, we investigate the effect of interstitial doping on the equilibrium unit cell volume using the projector augmented wave (PAW) method [CITATION] as implemented in the Vienna ab-initio simulation package (VASP).', '1407.7975-2-11-2': 'The VASP code with Perdew-Burke-Ernzerhof (PBE) parameterization [CITATION] is employed, where site-based magnetic moments were calculated using the Vosko-Wilk-Nusair interpolation [CITATION] within the general gradient approximation (GGA) for the exchange-correlation potential.', '1407.7975-2-12-0': 'La(Fe,Si)[MATH] has 8 formula units per conventional cell.', '1407.7975-2-12-1': 'The La atoms occupy the 8[MATH] sites ([MATH],[MATH],[MATH]), while diffraction studies show that Fe and Si atoms can occupy both 8[MATH] and the 96[MATH] crystallographic positions [CITATION].', '1407.7975-2-12-2': 'In order to keep the computational requirements at a feasible level, we follow the approach previously adopted by Kuzmin and Richter [CITATION], limiting our investigations to an atomically ordered version of LaSiFe[MATH]Z[MATH], where the 8[MATH] sites are occupied solely by silicon whilst iron is located exclusively on the 96[MATH] sites.', '1407.7975-2-12-3': 'In such a case, the cell that forms the basis of the calculations contains 2 La, 26 Fe and 2 Si atoms.', '1407.7975-2-12-4': 'Furthermore, interstitial elements H, B, C, and N were considered to occupy the 24[MATH] crystallographic site only.', '1407.7975-2-12-5': 'Using this model, we may vary the concentration of interstitials, [MATH] in steps of 0.5 from [MATH] to 3 in LaSiFe[MATH]Z[MATH].', '1407.7975-2-12-6': 'For a picture of the crystal structure including interstitial sites, we refer the reader to Figure 1 of Fujieda et al. [CITATION].', '1407.7975-2-13-0': 'Full structural relaxation was carried out for both collinear ferromagnetic (FM) and non-magnetic (NM) states in the case of parent LaFe[MATH]Si alloy, while only the lattice parameter [MATH] was relaxed (without relaxation of the internal atomic positions) for the materials doped with [MATH]- or [MATH]-block interstitials.', '1407.7975-2-13-1': 'A 7 [MATH]-point grid was used to discretize the first Brillouin zone and the energy convergence criterion was set to [MATH] eV during the energy minimization process.', '1407.7975-2-13-2': 'The effect of spin-orbit coupling was tested for the parent alloy, where we found negligible contributions to the magnetic moments ([MATH]) and total energies ([MATH] eV) and thus it was turned off for the calculations presented here.', '1407.7975-2-13-3': 'Finally, data presented in Fig. [REF] was calculated on a dense 19 [MATH] 19 grid of [MATH]-points for high accuracy.', '1407.7975-2-14-0': 'In the second part of this study, we have taken a fixed spin moment (FSM) approach within the tight-binding theorem using linear muffin tin orbitals (TB-LMTO) as implemented in the Stuttgart TB-LMTO code [CITATION].', '1407.7975-2-14-1': 'This method requires carefully adjusted overlapping Wigner-Seitz (WS) atomic spheres included in the calculations to complete the basis and to provide an accurate description of the electron density throughout the entire unit cell.', '1407.7975-2-14-2': 'Consequently, the structural parameters of the relaxed lattice are inherently dependent on the volume occupied by the WS spheres and/or empty spheres.', '1407.7975-2-14-3': 'For this reason, we used VASP code (see above) for relaxation.', '1407.7975-2-14-4': 'Nevertheless, the TB-LMTO approach allows us to evaluate the total energy difference between FM and NM states, [MATH] as a function of fixed spin moment [MATH] as well as the corresponding density of states (DOS) and band dispersions.', '1407.7975-2-14-5': 'A dense mesh with 48[MATH]-points (for the DOS calculations) or with 12[MATH]-points (for the FSM calculations) was used.', '1407.7975-2-15-0': '# Results and Discussion', '1407.7975-2-16-0': '## Effect of dopants on lattice expansion and magnetic properties', '1407.7975-2-17-0': 'Fig. [REF] shows the calculated lattice parameter in the FM state in LaSiFe[MATH]Z[MATH] as a function of interstitial doping.', '1407.7975-2-17-1': 'Our calculations obtain a relaxed structure that differs by only about 0.1% from the experimentally reported value.', '1407.7975-2-17-2': 'This remarkable agreement validates our choice of exchange correlation, GGA.', '1407.7975-2-17-3': 'The lattice expansion increases monotonically with dopant concentration at a rate that depends strongly on the size of the interstitial element.', '1407.7975-2-17-4': 'The empirical atomic radius of hydrogen (25 pm) is much smaller than that of the boron (85 pm), carbon (70 pm), or nitrogen (65 pm).', '1407.7975-2-17-5': 'The trend in calculated lattice expansion in Fig. [REF] correlates well with the relative atomic size of the interstitial, showing the predominant influence of the latter on the size of the unit cell.', '1407.7975-2-17-6': 'Our calculations are consistent with the experimental values for X=H and C interstitials respectively [CITATION].', '1407.7975-2-17-7': 'At full doping, we here find a relative lattice expansion of 0.4 for hydrogen and a considerably higher value of 1.7 for carbon.', '1407.7975-2-17-8': 'The values of [MATH] are 1.8 and 1.25 for Z=B and N respectively at full doping, but these are yet to be confirmed experimentally.', '1407.7975-2-17-9': 'There is only limited experimental data available on the relative effects of lattice expansion of the dopants studied here, particularly as full occupation of the 24[MATH] site ([MATH]) by any of the dopants has not been achieved in practice.', '1407.7975-2-18-0': 'In terms of valence electron number, however, a different sequence exists: H[MATH] B[MATH] C[MATH] N[MATH]).', '1407.7975-2-18-1': 'A closer look at Fig. [REF] reveals a non-monotonic behaviour in the magnetic moment per formula unit as a function of doping, especially in the case of nitrogen.', '1407.7975-2-18-2': 'Indeed, additional charges significantly alter the electronic structure (apart from H) in ways that go beyond the simple picture of chemical pressure effects, as we discuss later.', '1407.7975-2-18-3': 'The contribution of additional valence electrons is also reflected in the calculated magnetic moment ([MATH]).', '1407.7975-2-18-4': '[MATH] rises initially (Fig. [REF], top) for each dopant but a monotonic increase up to [MATH] is only seen in the case of hydrogen.', '1407.7975-2-18-5': 'These observations imply a mechanism for hybrid band formation and band broadening for any dopant with a larger atomic radius than that of hydrogen.', '1407.7975-2-18-6': 'In order to depict the changes in the electronic structure that are brought about by the interstitial elements, we next examine the partial electronic density of states (PDOS).', '1407.7975-2-19-0': 'Fig. [REF] shows the PDOS of the parent alloy together with that of the fully hydrogenated and fully nitrogenated materials ([MATH]).', '1407.7975-2-19-1': 'In the parent alloy (bottom), the PDOS is dominated by Fe (red line) around the Fermi level (E[MATH]) with typical spin-split states.', '1407.7975-2-19-2': 'The spin-up ([MATH]) states are mostly occupied, while the unoccupied states are dominated by spin-down ([MATH]) states separated by about 2.5eV in energy.', '1407.7975-2-19-3': 'Furthermore, the filled bands at the lower end of energy range (-9.5 eV) relate mostly to silicon 3[MATH] states which are overlapped with [MATH]-states of both La and Fe.', '1407.7975-2-19-4': 'A large energy gap appears from -9.5 eV up to about -6.5eV, where a high population of 3[MATH] states of Si (black) is located (-6.5 to -4.5 eV).', '1407.7975-2-19-5': 'In this latter energy range, there is negligible contribution from Fe [MATH]-states.', '1407.7975-2-20-0': 'Most of the aforementioned features in the electronic structure are preserved in fully hydrogenated LaSiFe[MATH]H[MATH] (middle of Fig. [REF]).', '1407.7975-2-20-1': 'The main difference in the PDOS compared to the parent alloy is the development of additional states in the gap around -7.5 eV related to the hydrogen interstitials.', '1407.7975-2-20-2': 'Small additional peaks also appear around -5 eV, where they overlap with the [MATH] states of Si.', '1407.7975-2-20-3': 'In strong contrast to hydrogenation, fully nitrogenated LaSiFe[MATH]N[MATH] exhibits a large overlap of N [MATH] states with Fe [MATH] states in the -7.5 to -4 eV energy interval (top of Fig. [REF]).', '1407.7975-2-20-4': 'These peaks indicate [MATH] hybridization, and as a result, increased covalency in bond formation.', '1407.7975-2-20-5': 'Such features also help to explain the non-monotonic moment as a function of doping (especially nitrogen).', '1407.7975-2-20-6': 'Another important consequence of nitrogenation is the appearance of states in the vicinity of the Fermi level.', '1407.7975-2-21-0': 'The existence of a ""double peak"" feature just below and above E[MATH] for LaSiFe[MATH], created by Fe [MATH] states in the minority DOS, is altered only a little by hydrogenation.', '1407.7975-2-21-1': 'On the other hand, nitrogenation fills this valley at E[MATH], which results in the strong alteration of the magnetic properties and ultimately leads to the disappearance of IEM transition.', '1407.7975-2-21-2': 'We address the latter behavior in detail in the next section.', '1407.7975-2-22-0': '## The free energy landscape', '1407.7975-2-23-0': 'We now turn our interest to the results of our second computational approach, fixed spin moment calculations using TB-LMTO.', '1407.7975-2-23-1': 'Our aim is to visualize the energy difference between FM and NM states in the parent alloy and the doped materials.', '1407.7975-2-23-2': 'The purpose of our analysis is to identify the main factors that lead to the field-induced isothermal entropy change of LaSiFe[MATH]Z[MATH] around the magnetic transition being lower than that of LaSiFe[MATH], as found experimentally for interstitials other than hydrogen [CITATION].', '1407.7975-2-24-0': 'Fig. [REF] compares the free energy curves [MATH] calculated by the FSM method for LaSiFe[MATH]Z[MATH], where Z = H, N, B and empty sphere (Es), respectively, together with those of the parent alloy.', '1407.7975-2-24-1': 'For direct comparison, we set the non-magnetic energy state as F(0) for each individual composition.', '1407.7975-2-24-2': 'We also employ a constant volume approximation, which is used here to examine trends between differently doped compositions.', '1407.7975-2-24-3': 'Fig. [REF](a) shows that the parent LaSiFe[MATH] has a very shallow magnetic energy landscape, in accordance with the predictions by Kuzmin and Richter, who used the full-potential local-orbital (FPLO) method, also in a constant volume approach [CITATION].', '1407.7975-2-24-4': 'They noted the applied advantage of such a potential energy landscape in permitting a low hysteresis, first order metamagnetic transition (IEM).', '1407.7975-2-24-5': 'The main difference in our calculation of the parent compound is that we find only two minima rather than the multiple minima that were predicted in their work.', '1407.7975-2-24-6': ""The field-induced magnetisation of hydrogenated La-Fe-Si under pressure previously was seen to exhibit multiple steps, confirming Kuzmin and Richter's predictions."", '1407.7975-2-24-7': 'It may be interesting to investigate this property in the undoped compound to eliminate the possibility of pressure-induced hydrogen segregation.', '1407.7975-2-24-8': 'Spontaneous hydrogen segregation is known to occur in materials with a smaller hydrogen content than the empirical maximum [CITATION].', '1407.7975-2-25-0': 'The derivative of [MATH] with respect to [MATH] is also shown in Fig. [REF]; local minima in the free energy function are where [MATH].', '1407.7975-2-25-1': 'We may conclude that full doping of any of the four interstitials studied removes the shallow double-well potential, resulting in only a single well.', '1407.7975-2-25-2': 'This corresponds to the disappearance of the first order metamagnetic transition, as found experimentally for B, C, and N doping.', '1407.7975-2-25-3': 'We note that boron addition is detrimental to the total magnetisation as the minimum in [MATH] occurs at a lower value of [MATH] than for any other dopant.', '1407.7975-2-25-4': 'Of all the interstitials studied, hydrogen alters the double well picture the least.', '1407.7975-2-25-5': 'The shallow landscape of [MATH] takes on a concave curvature in the range of [MATH] at full doping.', '1407.7975-2-25-6': 'The intrinsically small energy barrier for the parent compound and the hydrogen-doped material makes these compositions particularly sensitive to external parameters such as magnetic field, pressure and temperature and renders the first order IEM transition quasi-reversible.', '1407.7975-2-26-0': 'Fig. [REF]b also reveals the sensitive nature of the metamagnetic states to the lattice parameters.', '1407.7975-2-26-1': 'An increase of about 0.5% in the lattice parameter upon H-doping generates a ground state with a high-spin configuration.', '1407.7975-2-26-2': 'In order to separate the changes in the magnetic state caused by the volume expansion (chemical pressure) due to the inclusion of large interstitials such as nitrogen from those caused by additional valence electrons, we also carried out calculations with empty spheres (Es) included at the 24[MATH] crystallographic site.', '1407.7975-2-26-3': 'For a direct comparison, the same lattice constant was adopted for both Z = N and Es.', '1407.7975-2-27-0': 'To further examine the appearance of a metastable low-spin state at around [MATH] and the emergence of an energy barrier near [MATH], DOS calculations were performed with fixed spin moments at these two values of magnetisation for each system.', '1407.7975-2-27-1': 'The results are shown in Fig. [REF].', '1407.7975-2-27-2': 'For both Z = H and Es, the Fermi level is located in a deep valley for both the minority and majority-spin DOS at M=7[MATH], similar to that found in the parent alloy.', '1407.7975-2-27-3': 'This well-defined valley in the DOS is destroyed for Z = B and N. By contrast, the electronic features found in the DOS for M=13[MATH] differ significantly: high peaks in the DOS appears at E[MATH] for both minority and majority-spin states for the undoped compound as well as for Z=H and Es alloys, while only moderate peaks for Z = N and B are observed.', '1407.7975-2-27-4': 'It is thus apparent that the valleys and peaks around E[MATH] can be attributed to the low-spin state and the energy barrier in [MATH].', '1407.7975-2-28-0': 'In Fig. [REF], we show the electronic band structure calculated for [MATH], in order to provide an explanation for the above peaks in the electronic DOS.', '1407.7975-2-28-1': 'We note that a 0.01 eV offset was added to the energy level of p bands of the N-, B- and H-doped compounds for clarity in the figure.', '1407.7975-2-28-2': 'We may make several qualitative observations.', '1407.7975-2-28-3': 'First, the quasi-degenerate t[MATH] and e[MATH] bands are widely observed toward representative k-directions along the Brillouin zone in the parent alloy.', '1407.7975-2-28-4': 'The additional charge supplied by the H atoms shifts the position of the Fermi level upwards but the character of these 3[MATH] bands is conserved.', '1407.7975-2-28-5': 'Second, in the empty sphere configuration (Z = Es), where the significantly larger lattice structure of the nitrogenated alloy is adopted without any addition of charge, strong narrowing of the bandwidth is found but the dispersion of each band is once again preserved (not shown).', '1407.7975-2-29-0': 'Third, the situation is very different for both Z=N and B, as the band structure is strongly altered at full doping.', '1407.7975-2-29-1': 'For Z=B, strong [MATH] band mixing occurs between -0.3 and -0.6 eV for the majority spin states in all [MATH]-directions.', '1407.7975-2-29-2': 'In addition, broad bands originating from the [MATH]-states of boron appear at the [MATH] point about -0.3 eV for the minority spins.', '1407.7975-2-29-3': 'The formation of these hybrid [MATH] bands ultimately results in the vanishing of the well-formed peak and valley structures of the DOS around E[MATH] as shown in Fig. [REF] and in Fig. [REF].', '1407.7975-2-29-4': 'Finally, for nitrogen doping most of the [MATH]-states appear just above E[MATH], apart from some minor ones around -0.5 eV in the W direction for the minority spins.', '1407.7975-2-29-5': 'Here, the [MATH] mixing occurs mainly between bands in the K and W directions.', '1407.7975-2-29-6': 'Bands of [MATH] character show especially pronounced mixing with [MATH]-electrons.', '1407.7975-2-29-7': 'The number of flat bands around E[MATH] is decreased as compared to the undoped system and as a result the DOS has uneven features with small peaks that are detrimental to the IEM.', '1407.7975-2-29-8': 'transition.', '1407.7975-2-30-0': '## Electron coupling and inferred phonon entropy changes', '1407.7975-2-31-0': 'We may use the electronic DOS calculated in the previous sections to predict certain experimental quantities.', '1407.7975-2-31-1': 'In this section we investigate the first of two: the electronic entropy change (and thereby the phonon entropy change) expected at the Curie temperature.', '1407.7975-2-31-2': 'In the next section we will examine the variation of thermopower in the ferromagnetic state.', '1407.7975-2-31-3': 'Both have been the subject of recent experimental work.', '1407.7975-2-32-0': 'Several authors have attempted to decompose the entropy change at a first order magnetic phase transition into parts that can be ascribed to changes in the magnetic, phononic and electronic degrees of freedom.', '1407.7975-2-32-1': 'This has especially been of interest in magnetocaloric materials studies, with examples including antiferromagnetic metamagnets such as CeFe[MATH]Co[MATH] [CITATION].', '1407.7975-2-32-2': ', (Fe[MATH]Ni[MATH]Rh[MATH] with [MATH] [CITATION].', '1407.7975-2-32-3': ', and CoMnSi [CITATION] and the itinerant metamagnetic system considered in this article.', '1407.7975-2-32-4': 'A significant point of divergence in approach occurs between adopting an itinerant or a localised view of the principle magnetic moments that order at the phase transition and also in the treatment of the phonons as Debye-like or otherwise.', '1407.7975-2-32-5': 'Early work by Jia [CITATION] proposed that upon ferromagnetic ordering, the entropy associated with a localised Fe moments decreased.', '1407.7975-2-32-6': 'In La(Fe,Si)[MATH] compositions, there is a negative thermal expansion at the first order Curie point.', '1407.7975-2-32-7': 'Jia et al. proposed that the material behaved as a phonon gas and that the larger ferromagnetic volume resulted in a counteracting, positive phonon entropy change.', '1407.7975-2-32-8': 'However, a more recent inelastic resonant x-ray (INRXS) study by Gruner et al. found that the phonon entropy change on entering the ferromagnetic state was conventional (negative) despite the negative thermal expansion associated with the transition [CITATION].', '1407.7975-2-33-0': 'We have therefore set out to examine the first order phase transition from a partly itinerant electron standpoint.', '1407.7975-2-33-1': 'The itineracy of the 3[MATH] Fe electrons in La(Fe,Si)[MATH] is well established [CITATION].', '1407.7975-2-33-2': 'There is also mounting evidence that the magnetism of La(Fe,Si)[MATH] is intermediate between a full itineracy and full localisation.', '1407.7975-2-33-3': 'The presence of disordered local moments (DLMs) in the paramagnetic state has been supported by anomalous Hall effect measurements [CITATION], photoemission [CITATION], coherent potential approximation (CPA) calculations [CITATION] and recent fixed spin moment calculations [CITATION].', '1407.7975-2-33-4': 'Since the DLM moment may be of the order of 1 [MATH], we calculate the field-induced change of magnetic entropy as a fixed value, [MATH], where we set [MATH] rather than the value of [MATH] that Jia et al. used, to reduce the possibility of overestimating the magnetic contribution (since the expression [MATH] yields an overestimate for the true change in magnetic entropy at a finite temperature, first order phase transition, even if the appropriate value of [MATH] is used).', '1407.7975-2-33-5': 'The actual value of [MATH] or of [MATH] is not of primary importance here; as we will see in Table [REF], the trends between materials and between FM and PM states are retained by a constant shift in the value of [MATH] chosen.', '1407.7975-2-33-6': 'We calculate a ""bare"" electronic entropy as a function of temperature via the usual fermionic entropy relation: [EQUATION] where [MATH] is the Boltzmann constant, [MATH] is the fermi function and [MATH] is the density of states.', '1407.7975-2-33-7': 'The chemical potential is adjusted self-consistently at different temperatures by fixing the total electron number.', '1407.7975-2-33-8': 'We are thereby able to calculate bare electron Sommerfield coefficients, as the gradient of [MATH] curves where the non-magnetic (NM) state is taken as a proxy for the actual paramagnetic state.', '1407.7975-2-33-9': 'We find that there is a near-linear relationship between electronic entropy and temperature in the non-magnetic state while it is almost perfectly linear in the FM state.', '1407.7975-2-33-10': 'We note that the full Sommerfeld coefficient may be estimated on the basis of a modified free electron relation: [EQUATION] where [MATH] is the magnitude of all other couplings of the electrons to phonons, spin fluctuations and so on and [MATH] is the Fermi energy.', '1407.7975-2-33-11': 'We may use this relation to estimate the strength of the electron-coupling mechanisms in the FM and PM state.', '1407.7975-2-34-0': 'For LaSiFe[MATH], our estimate of the bare [MATH] (27-29 mJK[MATH]kg[MATH] in the FM state) depends on whether the density of states or gradient method is used (i.e. with [MATH]) and is fairly close to theoretical estimates made by Gruner at al. [CITATION].', '1407.7975-2-34-1': 'However, all such estimates are a magnitude of about ten lower than experimental values such as those found by Fang et al. [CITATION] (236 mJK[MATH]kg[MATH] for LaAl[MATH]Fe[MATH]) and [MATH] times lower than those found by Fujita et al., [CITATION] on La(Si[MATH]Fe[MATH] (LaSi[MATH]Fe[MATH]) and by Lovell et al. [CITATION]( 100-120 mJK[MATH]kg[MATH] in samples of LaSi[MATH]Fe[MATH] and LaSi[MATH]Fe[MATH]).', '1407.7975-2-34-2': 'Since there is strong variation in the experimental value of [MATH] between those samples containing Si and those containing Al, we take the early data of Fujita et al. [CITATION], and the recent data on LaSi[MATH]Fe[MATH] by Lovell et al. as our reference points for [MATH] of our Al-free material models.', '1407.7975-2-34-3': 'Both give [MATH] mJK[MATH]kg[MATH].', '1407.7975-2-35-0': 'Information about the paramagnetic Sommerfeld coefficient is lacking and thus is predominately guided by theoretical, rather than empirical, considerations.', '1407.7975-2-35-1': 'The Sommerfeld [MATH] values that we find for the PM state of the hydrogen-doped LaSiFe[MATH]H[MATH] compound vary somewhat, depending on whether the free electron formula in Equation [REF] (yielding 60.7 mJK[MATH]kg[MATH]) or the gradient method (yielding 78.2 mJK[MATH]kg[MATH]) is used.', '1407.7975-2-35-2': 'This can perhaps be attributed to the strong variation in the DOS in the immediate vicinity of the Fermi energy.', '1407.7975-2-35-3': '(The method-dependence of the Sommerfeld coefficient in the FM state of this compound and in the FM or PM states of LaSiFe[MATH] is considerably less.)', '1407.7975-2-36-0': ""To gain more insight, we infer possible phonon contributions to the total isothermal entropy change, [MATH], induced by a magnetic field, motivated by the contradiction between the recent observations by Gruner et al. and Jia and co-workers' early theoretical predictions."", '1407.7975-2-36-1': 'The fact that the Sommerfeld [MATH] coefficients are significantly lower than observed in low temperature experiments invites us to investigate equation (2) with [MATH] i.e. with finite coupling between the electrons and phonons, spin fluctuations, and so on.', '1407.7975-2-36-2': 'However, we will be unable to distinguish between such sources of mass enhancement of the electron.', '1407.7975-2-36-3': 'We thus consider that [MATH] is given principally by [EQUATION] where the subscripts indicate the magnetic, bare electron, electron-coupling and pure phonon terms, respectively.', '1407.7975-2-36-4': 'From Equation [REF] the [MATH] term above is given by [MATH].', '1407.7975-2-36-5': 'In Table [REF] we have compared the entropy changes that we find in LaSiFe[MATH] due to the bare electron and the coupled electron-phonon terms at [MATH] K.', '1407.7975-2-36-6': 'The factor of 3.5 discrepancy between the [MATH] values found here and those inferred from experimental data would imply that [MATH] in the FM state is about 2.5, which is in a similar range to the value for spin fluctuation enhancement of gamma (3.3) found by Michor and co-workers in paramagnetic LaCo[MATH]Si[MATH] [CITATION].', '1407.7975-2-36-7': 'To mimic the effect of magnetic field we calculate the difference in entropy between the FM and PM states as [MATH].', '1407.7975-2-37-0': 'If in the case of LaSiFe[MATH] we set [MATH] so that [MATH] is brought closer to the values found in experiment, we find that [MATH] Jkg[MATH]K[MATH] and we are forced to infer a very large negative (positive) phonon entropy change on entering (leaving) the FM state since [MATH] where [MATH] -20 Jkg[MATH]K[MATH].', '1407.7975-2-37-1': 'If however, we set [MATH], then it is possible to adjust the inferred phonon contribution to a more reasonable range, in agreement with the findings of Gruner et al., as demonstrated in Table [REF].', '1407.7975-2-37-2': 'For [MATH] and [MATH] we find [MATH]+[MATH] Jkg[MATH]K[MATH], and so [MATH] Jkg[MATH]K[MATH].', '1407.7975-2-37-3': 'It should be noted that this level of suppression of [MATH] is not the lowest possible (which is [MATH]); further suppression of [MATH] would result in more negative changes in phonon entropy while higher [MATH] values would (contrary to the recent experimental work by Gruner et al.) again infer a positive phonon entropy change.', '1407.7975-2-37-4': 'The parameter set [MATH] and [MATH] delivers an approximately equal division of entropy change between electronic and purely phononic effects.', '1407.7975-2-38-0': 'A similiar, if even more stark, conclusion can be drawn from an analysis of the fictitious LaSiFe[MATH]H[MATH] compound.', '1407.7975-2-38-1': 'The recent experimental data of Lovell et al. demonstrate that the low temperature, [MATH] values of La-Fe-Si compounds decrease with hydrogenation.', '1407.7975-2-38-2': 'However, the bare [MATH] value we find in DFT is slightly greater in the hydrogenated material.', '1407.7975-2-38-3': 'Therefore, we must lower the [MATH] value for the hydrogenated compound to around 1.5.', '1407.7975-2-38-4': 'Since [MATH] -20 Jkg[MATH]K[MATH] in this family of materials, if we consider [MATH], the resulting magnetic+electron entropy change is [MATH] Jkg[MATH]K[MATH] (see Table [REF]).', '1407.7975-2-38-5': 'Such a reduced value of [MATH] in the paramagnetic state yields only a very small phonon entropy change of around [MATH] Jkg[MATH]K[MATH].', '1407.7975-2-38-6': 'Given that this fictitious LaSiFe[MATH]H[MATH] compound is basically second order (see earlier), our analysis implies that hydrogenation decreases the mass enhancement in both the FM and PM states, and reduces the role in the total entropy change played by the purely phononic term.', '1407.7975-2-38-7': 'In Table [REF] we also show another parameter set for modelling the LaSiFe[MATH]H[MATH] compound.', '1407.7975-2-38-8': 'We may see that by lowering [MATH] slightly, the balance of electronic and phonon entropy can be maintained at the level modelled in the case of LaSiFe[MATH].', '1407.7975-2-38-9': 'The principal conclusion is therefore that the coupling of electrons to other degrees of freedom is reduced in the PM state and also by hydrogenation.', '1407.7975-2-38-10': 'We have been able to achieve this perspective through the use of DFT-derived Sommerfeld [MATH] factors, and a comparison with experimental data available for the La(Si,Fe)[MATH]H[MATH] material family.', '1407.7975-2-39-0': 'The division of entropy, as noted above, is done with the strict assumption of there being purely a (DLM) magnetic, itinerant electronic, phonon, and electron-coupled term.', '1407.7975-2-39-1': 'However, there are other considerations, some of which our model has not considered.', '1407.7975-2-39-2': 'Firstly, we have not considered the effects of spin waves at finite temperatures; we refer the reader to the recent analysis of spin waves from heat capacity data in Mn-containing La-Fe-Si compounds [CITATION].', '1407.7975-2-39-3': 'Further experiments are therefore warranted and will be important in determining a electronic coupling (and thereby the required phonon entropy change) more precisely as a function of hydrogenation.', '1407.7975-2-40-0': '## Thermopower', '1407.7975-2-41-0': 'We make a final comparison with experimental data using the electronic DOS calculated here.', '1407.7975-2-41-1': 'Hannemann et al. [CITATION] measured the thermopower in the ferromagnetic state of an unhydrogengated LaSi[MATH]Fe[MATH] and hydrogenated LaSi[MATH]Fe[MATH]H[MATH] and observed a broadening of the negative thermopower response in the latter material.', '1407.7975-2-41-2': 'They ascribed this to a broadened electronic density of states, applying a paramagnetic (single spin) electron model used by Burkov et al. [CITATION] to the thermopower in the ferromagnetic state.', '1407.7975-2-42-0': 'We may reverse the comparison by trying to predict the thermopower of LaSiFe[MATH] in the FM state while taking into account the effect of both spin channels.', '1407.7975-2-42-1': 'We do so to motivate further work in this area as the level of agreement we find with available experimental data is mixed.', '1407.7975-2-42-2': 'We employ a high temperature spin mixing approximation, such that the total thermopower is the average of that found in the two spin bands [CITATION].', '1407.7975-2-42-3': 'We invoke one approximation that Hannemann et al. and Burkov et al. also used in their analysis; namely that the electronic conductivity is a separable function of temperature and of the energy of an electron and that it can be considered as inversely proportional to the electronic density of [MATH]-states.', '1407.7975-2-42-4': 'Then the thermopower may be obtained from the linearised Boltzmann relation: [EQUATION]', '1407.7975-2-42-5': 'We also allow essentially no variation of the magnetisation (chemical potential) within the FM state with temperature, up to the first order Curie point at 195 K. For the non-hydrogenated compound, this is a reasonable approximation away from the Curie temperature.', '1407.7975-2-43-0': 'Our results for the non-hydrogenated compound are shown in Figure [REF], along with experimental data from Hannemann et al. extracted from the best fit curve to data on a bulk sample of LaSi[MATH]Fe[MATH].', '1407.7975-2-43-1': 'We clearly see that some aspects of the thermopower in the FM state are reproduced by our simplified model.', '1407.7975-2-43-2': 'The broad form and magnitude are similar, but the sign of the low temperature thermopower is incorrect and the theoretical curve contains an additional feature at around 50 K which is due to band structure features but which is not seen in experiment.', '1407.7975-2-43-3': 'Experimental thermopower data on the fictitious [MATH] hydrogenated material are not available.', '1407.7975-2-43-4': 'A comparison using a band structure calculation for [MATH] published elsewhere [CITATION] results in a poor agreement with experiment (not shown), even when temperature variation of the magnetisation (chemical potential) is included.', '1407.7975-2-43-5': 'The calculated thermopower for [MATH] increases from 100-300 K, in contrast to the experimental data.', '1407.7975-2-43-6': 'We nonetheless note three aspects of our comparisons of calculated thermopower with experiment: (i) the relative success of thermopower modelling in the non-hydrogenated compound; (ii) the observation of theoretical features at low temperatures that can arise solely from band structure effects; and (iii) that scattering effects beyond our model are likely to play a larger role in the accurate modelling of the thermopower in the hydrogenated La-Fe-Si samples.', '1407.7975-2-43-7': 'It has already been suggested that experimental features in the low temperature thermopower of the hydrogenated material are due to the presence of such effects [CITATION].', '1407.7975-2-43-8': 'Further work will be required to separate the effect of band structure from additional scattering mechanisms.', '1407.7975-2-44-0': '# Conclusions', '1407.7975-2-45-0': 'We have investigated the effect of select [MATH]- and [MATH]-block interstitial elements on the electronic, lattice and magnetic properties of La(Fe,Si)[MATH] using DFT.', '1407.7975-2-45-1': 'Our calculations find that a good correlation between the size of the unit cell and the size of the dopant.', '1407.7975-2-45-2': 'Fixed spin moment calculations yield a double well structure in the free energy of LaFe[MATH]Si as a function of magnetisation, which may be seen as the basis of the itinerant electron metamagnetic transition.', '1407.7975-2-45-3': 'Significantly, hydrogenation alters the electronic and magnetic structure of LaSiFe[MATH] to a much smaller degree than B, C and N dopants.', '1407.7975-2-45-4': 'This means that the first order IEM is much more robust to H insertion than to interstitial B, C, or N.', '1407.7975-2-46-0': 'An analysis of the projected electronic DOS reveals that the dominant electronic states related to hydrogen insertion appear at around -8 to -7 eV, where very little contribution from Fe, Si and La elements is present.', '1407.7975-2-46-1': 'The additional charge of the hydrogen atoms elevates the Fermi level but the character of the bands is unaltered.', '1407.7975-2-46-2': 'The latter feature is also evident in the empty sphere configuration, where the nitrogenated lattice parameters are simulated without the N inclusions; only a narrowing of the band-width is found but the dispersion of each band remains mostly unaffected.', '1407.7975-2-47-0': 'Consequently, hydrogen provides perhaps the only chemical pressure on the lattice that avoids significant alterations to the electronic structure of LaSiFe[MATH].', '1407.7975-2-47-1': 'In the case of the other dopants (B, C, N), broad bands originating from their [MATH]-states appear at energy levels where the 3[MATH] states of Fe are also present.', '1407.7975-2-47-2': 'The formation of these hybrid [MATH] bands results in the disappearance of the peak and valley structures in the electronic DOS around E[MATH], thereby reshaping the shallow free energy landscape and ultimately destroying the first order IEM of LaSiFe[MATH].', '1407.7975-2-47-3': 'Our theoretical findings are in good agreement with the experimentally-determined properties of interstitially doped La(Si,Fe)[MATH] compounds [CITATION].', '1407.7975-2-48-0': 'We have used our calculated band structure for the non-hydrogenated and hydrogenated materials to predict a range of Sommerfeld [MATH] enhancements that would achieve the same sign of phonon entropy change at the magnetic ordering transition that Gruner et al. have reported.', '1407.7975-2-48-1': 'We demonstrate that the [MATH] enhancement of the FM state is greater than that of the PM state, consistent with a picture that such an enhancement comes mainly from spin fluctuations.', '1407.7975-2-48-2': 'Hydrogenation lowers the relative enhancement of the electronic heat capacity (implying a reduction of spin fluctuation effects) and the bare electronic [MATH] is slightly increased.', '1407.7975-2-48-3': 'We have further calculated thermopower, in good agreement with experiments on non-hydrogenated material.', '1407.7975-2-48-4': 'Thermopower values in the hydrogenated material seem to require a model that includes other scattering effects, although these are unlikely to spin waves, which are thought to be heavily suppressed [CITATION].', '1407.7975-2-49-0': 'The authors thank L.F. Cohen for useful discussions.', '1407.7975-2-49-1': 'The research leading to these results has received funding from the European Community\'s 7th Framework Programme under Grant Agreement No. 310748 ""DRREAM"".', '1407.7975-2-49-2': 'Computing resources provided by Darwin HPC and Camgrid facilities at The University of Cambridge and the HPC Service at Imperial College London are also gratefully acknowledged.', '1407.7975-2-49-3': 'NF acknowledges the Brooklyn College LSAMP program and the NSF for funding.', '1407.7975-2-49-4': 'This material is based upon work supported by the National Science Foundation under Grant no. 1202520 (NYC Louis Stokes Alliance).', '1407.7975-2-50-0': '# References'}","[['1407.7975-1-9-1', '1407.7975-2-8-1'], ['1407.7975-1-9-2', '1407.7975-2-8-5'], ['1407.7975-1-9-3', '1407.7975-2-8-6'], ['1407.7975-1-5-0', '1407.7975-2-5-0'], ['1407.7975-1-5-1', '1407.7975-2-5-1'], ['1407.7975-1-5-2', '1407.7975-2-5-2'], ['1407.7975-1-31-2', '1407.7975-2-45-2'], ['1407.7975-1-31-3', '1407.7975-2-45-3'], ['1407.7975-1-31-4', '1407.7975-2-45-4'], ['1407.7975-1-33-0', '1407.7975-2-47-0'], ['1407.7975-1-33-2', '1407.7975-2-47-2'], ['1407.7975-1-15-2', '1407.7975-2-14-2'], ['1407.7975-1-15-3', '1407.7975-2-14-3'], ['1407.7975-1-15-4', '1407.7975-2-14-4'], ['1407.7975-1-15-6', '1407.7975-2-14-5'], ['1407.7975-1-27-0', '1407.7975-2-27-0'], ['1407.7975-1-27-1', '1407.7975-2-27-1'], ['1407.7975-1-27-4', '1407.7975-2-27-4'], ['1407.7975-1-2-0', '1407.7975-2-2-0'], ['1407.7975-1-12-0', '1407.7975-2-11-0'], ['1407.7975-1-3-0', '1407.7975-2-3-0'], ['1407.7975-1-3-1', '1407.7975-2-3-1'], ['1407.7975-1-14-0', '1407.7975-2-13-0'], ['1407.7975-1-14-1', '1407.7975-2-13-1'], ['1407.7975-1-14-3', '1407.7975-2-13-3'], ['1407.7975-1-21-3', '1407.7975-2-21-0'], ['1407.7975-1-21-4', '1407.7975-2-21-1'], ['1407.7975-1-21-5', '1407.7975-2-21-2'], ['1407.7975-1-4-0', '1407.7975-2-4-0'], ['1407.7975-1-4-1', '1407.7975-2-4-1'], ['1407.7975-1-26-0', '1407.7975-2-26-0'], ['1407.7975-1-26-1', '1407.7975-2-26-1'], ['1407.7975-1-26-2', '1407.7975-2-26-2'], ['1407.7975-1-26-3', '1407.7975-2-26-3'], ['1407.7975-1-13-0', '1407.7975-2-12-0'], ['1407.7975-1-13-3', '1407.7975-2-12-3'], ['1407.7975-1-28-0', '1407.7975-2-28-0'], ['1407.7975-1-28-2', '1407.7975-2-28-2'], ['1407.7975-1-28-4', '1407.7975-2-28-4'], ['1407.7975-1-0-0', '1407.7975-2-0-0'], ['1407.7975-1-0-2', '1407.7975-2-0-2'], ['1407.7975-1-0-3', '1407.7975-2-0-3'], ['1407.7975-1-0-4', '1407.7975-2-0-4'], ['1407.7975-1-34-1', '1407.7975-2-49-2'], ['1407.7975-1-20-2', '1407.7975-2-19-1'], ['1407.7975-1-20-3', '1407.7975-2-19-2'], ['1407.7975-1-20-4', '1407.7975-2-19-3'], ['1407.7975-1-20-5', '1407.7975-2-19-4'], ['1407.7975-1-20-6', '1407.7975-2-19-5'], ['1407.7975-1-23-0', '1407.7975-2-23-0'], ['1407.7975-1-23-1', '1407.7975-2-23-1'], ['1407.7975-1-32-0', '1407.7975-2-46-0'], ['1407.7975-1-32-1', '1407.7975-2-46-1'], ['1407.7975-1-32-2', '1407.7975-2-46-2'], ['1407.7975-1-24-0', '1407.7975-2-24-0'], 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1905.00734,"{'1905.00734-1-0-0': 'We present a series of SIR-network models, extended with a game-theoretic treatment of imitation dynamics which result from regular population mobility across residential and work areas and the ensuing interactions.', '1905.00734-1-0-1': 'Each considered SIR-network model captures a class of vaccination behaviors influenced by epidemic characteristics, interaction topology, and imitation dynamics.', '1905.00734-1-0-2': 'Our focus is the eventual vaccination coverage, produced under voluntary vaccination schemes, in response to these varying factors.', '1905.00734-1-0-3': 'Using the next generation matrix method, we analytically derive and compare expressions for the basic reproduction number [MATH] for the proposed SIR-network models.', '1905.00734-1-0-4': 'Furthermore, we simulate the epidemic dynamics over time for the considered models, and show that if individuals are sufficiently responsive towards the changes in the disease prevalence, then the more expansive travelling patterns encourage convergence to the endemic, mixed equilibria.', '1905.00734-1-0-5': 'On the contrary, if individuals are insensitive to changes in the disease prevalence, we find that they tend to remain unvaccinated in all the studied models.', '1905.00734-1-0-6': 'Our results concur with earlier studies in showing that residents from highly connected residential areas are more likely to get vaccinated.', '1905.00734-1-0-7': 'We also show that the existence of the individuals committed to receiving vaccination reduces [MATH] and delays the disease prevalence, and thus is essential to containing epidemics.', '1905.00734-1-1-0': '# Introduction', '1905.00734-1-2-0': 'Vaccination has long been established as a powerful tool in managing and controlling infectious diseases by providing protection to susceptible individuals [CITATION].', '1905.00734-1-2-1': 'With a sufficiently high vaccination coverage, the probability of the remaining unvaccinated individuals getting infected reduces significantly.', '1905.00734-1-2-2': 'However, such systematic programs by necessity may limit the freedom of choice of individuals.', '1905.00734-1-2-3': 'When vaccination programs are made voluntary, the vaccination uptake declines as a result of individuals choosing not to vaccinate, as seen in Britain in 2003 when the vaccination program for Measles-Mumps-Rubella (MMR) was made voluntary [CITATION].', '1905.00734-1-2-4': ""Parents feared possible complications from vaccination [CITATION] and hoped to exploit the 'herd immunity' by assuming other parents would choose to vaccinate their children."", '1905.00734-1-2-5': 'Such hopes did not materialize precisely because other parents also thought similarly.', '1905.00734-1-3-0': ""Under a voluntary vaccination policy an individual's decision depends on several factors: the social influence from one's social network, the risk perception of vaccination, and the risk perception of infection, in terms of both likelihood and impact."", '1905.00734-1-3-1': 'This decision-making is often modelled using game theory [CITATION], by allowing individuals to compare the cost of vaccination and the potential cost of non-vaccination (in terms of the likelihood and impact of infection), and adopting imitation dynamics in modelling the influence of social interactions.', '1905.00734-1-3-2': 'However, many of the earlier studies [CITATION] were based on the key assumption of a well-mixed homogeneous population where each individual is assumed to have an equal chance of making contact with any other individual in the population.', '1905.00734-1-3-3': 'This population assumption is rather unrealistic as large populations are often diverse with varying levels of interactions.', '1905.00734-1-3-4': 'To address this, more recent studies [CITATION] model populations as complex networks where each individual, represented by a node, has a finite set of contacts, represented by links.', '1905.00734-1-4-0': 'It has been shown that there is a critical cost threshold in the vaccination cost above which the likelihood of vaccination drops steeply [CITATION].', '1905.00734-1-4-1': 'In addition, highly connected individuals were shown to be more likely to choose to be vaccinated as they perceive themselves to be at a greater risk of being infected due to high exposure within the community.', '1905.00734-1-4-2': 'In modelling large-scale epidemics, the population size (i.e., number of nodes) can easily reach millions of individuals, interacting in a complex way.', '1905.00734-1-4-3': 'The challenge, therefore, is to extend epidemic modelling not only with the individual vaccination decision-making, but also capture diverse interaction patterns encoded within a network.', '1905.00734-1-4-4': 'Such an integration has not yet been formalized, motivating our study.', '1905.00734-1-4-5': 'Furthermore, once an integrated model is developed, a specific challenge is to consider how the vaccination imitation dynamics developing across a network affects the basic reproduction number, [MATH].', '1905.00734-1-4-6': 'This question forms our second main objective.', '1905.00734-1-5-0': 'One relevant approach partially addressing this objective is offered by the multi-suburb (or multi-city) SIR-network model [CITATION] where each node represents a neighbourhood (or city) with a certain number of residents.', '1905.00734-1-5-1': 'The daily commute of individuals between two neighbourhoods is modelled along the network link connecting the two nodes, which allows to quantify the disease spread between individuals from different neighbourhoods.', '1905.00734-1-5-2': 'Ultimately, this model captures meta-population dynamics in a multi-suburb setting affected by an epidemic spread at a greater scale.', '1905.00734-1-5-3': 'To date, these models have not yet considered intervention (e.g., vaccination) options, and the corresponding social interaction across the populations.', '1905.00734-1-6-0': 'To incorporate the imitation dynamics, modelled game-theoretically, within a multi-suburb model representing mobility, we propose a series of integrated vaccination-focused SIR-network models.', '1905.00734-1-6-1': 'This allows us to systematically analyze how travelling patterns affect the voluntary vaccination uptake due to adoption of different imitation choices, in a large distributed population.', '1905.00734-1-6-2': 'The developed models use an increasingly complex set of vaccination strategies.', '1905.00734-1-6-3': 'Thus, our specific contribution is the study of the vaccination uptake, driven by imitation dynamics under a voluntary vaccination scheme, using an SIR model on a complex network representing a multi-suburb environment, within which the individuals commute between residential and work areas.', '1905.00734-1-7-0': 'In section [REF], we present the models and methods associated with this study: in particular, we present analytical derivations of the basic reproduction number [MATH] for the proposed models using the Next Generation Operator Approach [CITATION], and carry out a comparative analysis of [MATH] across these models.', '1905.00734-1-7-1': 'In section [REF], we simulate the epidemic and vaccination dynamics over time using the proposed models in different network settings, including a pilot case of a 3-node network, and a 3000-node Erdos-Renyi random network.', '1905.00734-1-7-2': 'The comparison of the produced results across different models and settings is carried out for the larger network, with the focus on the emergent attractor dynamics, in terms of the proportion of vaccinated individuals.', '1905.00734-1-7-3': 'Particularly, we analyze the sensitivity of the individual strategies (whether to vaccinate or not) to the levels of disease prevalence produced by the different considered models.', '1905.00734-1-7-4': 'Section [REF] concludes the study with a brief discussion of the importance of these results.', '1905.00734-1-8-0': '# Technical background', '1905.00734-1-9-0': '## Basic reproduction number [MATH]', '1905.00734-1-10-0': 'The basic reproduction number [MATH] is defined as the number of secondary infections produced by an infected individual in an otherwise completely susceptible population [CITATION].', '1905.00734-1-10-1': 'It is well-known that [MATH] is an epidemic threshold, with the disease dying out as [MATH], or becoming endemic as [MATH] [CITATION].', '1905.00734-1-10-2': 'This finding strictly holds only in deterministic models with infinite population [CITATION].', '1905.00734-1-10-3': 'The topology of the underlying contact network is known to affect the epidemic threshold [CITATION].', '1905.00734-1-10-4': 'Many disease transmission models have shown important correlations between [MATH] and the key epidemic characteristics (e.g., disease prevalence, attack rates, etc.) [CITATION].', '1905.00734-1-10-5': 'In addition, [MATH] has been considered as a critical threshold for phase transitions studied with methods of statistical physics or information theory [CITATION].', '1905.00734-1-11-0': '## Vaccination model with imitation dynamics', '1905.00734-1-12-0': 'Imitation dynamics, a process by which individuals copy the strategy of other individuals, is widely used to model vaccinating behaviours incorporated with SIR models.', '1905.00734-1-12-1': ""The model proposed in [CITATION] applied game theory to represent parents' decision-making about whether to get their newborns vaccinated against childhood disease (e.g., measles, mumps, rubella, pertussis)."", '1905.00734-1-12-2': ""In this model, individuals are in a homogeneously mixing population, and susceptible individuals have two 'pure strategies' regarding vaccination: to vaccinate or not to vaccinate."", '1905.00734-1-12-3': 'The non-vaccination decision can change to the vaccination decision at a particular sampling rate, however the vaccination decision cannot be changed to non-vaccination.', '1905.00734-1-12-4': 'Individuals adopt one of these strategies by weighing up their perceived payoffs, measured by the probability of morbidity from vaccination, and the risk of infection respectively.', '1905.00734-1-12-5': 'The payoff for vaccination ([MATH]) is given as, [EQUATION] and the payoff for non-vaccination ([MATH]), measured as the risk of infection, is given as [EQUATION] where [MATH] is the perceived risk of morbidity from vaccination, [MATH] is the perceived risk of morbidity from non-vaccination (i.e., infection), [MATH] is the current disease prevalence in population fraction at time [MATH], and [MATH] is the sensitivity to disease prevalence [CITATION].', '1905.00734-1-13-0': 'From Equation [REF] and [REF], it can be seen that the payoff for vaccinated individuals is a simple constant, and the payoff for unvaccinated individuals is proportional to the extent of epidemic prevalence (i.e., the severity of the disease).', '1905.00734-1-14-0': 'It is assumed that life-long immunity is granted with effective vaccination.', '1905.00734-1-14-1': 'If one individual decides to vaccinate, s/he cannot revert back to the unvaccinated status.', '1905.00734-1-14-2': 'To make an individual switch to a vaccinating strategy, the payoff gain, [MATH] must be positive ([MATH]).', '1905.00734-1-14-3': ""Let [MATH] denote the relative proportion of vaccinated individuals, and assume that an unvaccinated individual can sample a strategy from a vaccinated individual at certain rate [MATH], and can switch to the 'vaccinate strategy' with probability [MATH]."", '1905.00734-1-14-4': 'Then the time evolution of [MATH] can be defined as: [EQUATION] where [MATH] can be interpreted as the combined imitation rate that individuals use to sample and imitate strategies of other individuals.', '1905.00734-1-14-5': 'Equation [REF] can be rewritten so that [MATH] only depends on two parameters: [EQUATION] where [MATH] and [MATH], assuming that the risk of morbidity from vaccination, [MATH], and the risk of morbidity from non-vaccination (i.e., morbidity from infection), [MATH], are constant during the course of epidemics.', '1905.00734-1-14-6': ""Hence, [MATH] is the adjusted imitation rate, and [MATH] measures individual's responsiveness towards the changes in disease prevalence."", '1905.00734-1-15-0': 'The disease prevalence [MATH] is determined from a simple SIR model [CITATION] with birth and death that divides the population into three health compartments: [MATH] (susceptible), [MATH] (infected) and [MATH] (recovered).', '1905.00734-1-15-1': 'If a susceptible individual encounters an infected individual, s/he contracts the infection with transmission rate [MATH], and thus progresses to the infected compartment, while an infected individual recovers at rate [MATH].', '1905.00734-1-15-2': 'It is assumed that the birth and death rate are equal, denoted by [MATH] [CITATION].', '1905.00734-1-15-3': 'Individuals in all compartments die at an equal rate, and all newborns are added to the susceptible compartment unless and until they are vaccinated.', '1905.00734-1-15-4': 'Vaccinated newborns are moved to the recovered class directly, with a rate [MATH].', '1905.00734-1-15-5': 'Therefore, the dynamics are modelled as follows [CITATION]: [EQUATION] where [MATH], [MATH], and [MATH] represent the proportion of susceptible, infected, and recovered individuals respectively (that is, [MATH]), and [MATH] represents the proportion of vaccinated individuals within the susceptible class at a given time.', '1905.00734-1-15-6': 'The variables [MATH], [MATH], and [MATH], as well as [MATH], are time-dependent, but for simplicity, we omit subscript [MATH] for these and other time-dependent variables.', '1905.00734-1-16-0': 'Model [REF] predicts oscillations in vaccine uptake in response to changes in disease prevalence.', '1905.00734-1-16-1': 'It is found that oscillations are more likely to occur when individuals imitate each other more quickly (i.e., higher [MATH]).', '1905.00734-1-16-2': 'The oscillations are observed to be more volatile when people alter their vaccinating behaviours promptly in response to changes in disease prevalence (i.e., higher [MATH]).', '1905.00734-1-16-3': 'Overall, higher [MATH] or [MATH] produce stable limit cycles at greater amplitude.', '1905.00734-1-16-4': 'Conversely, when individuals are insensitive to changes in disease prevalence (i.e., low [MATH]), or imitate at a slower rate (i.e., low [MATH]), the resultant vaccinating dynamics converge to equilibrium [CITATION].', '1905.00734-1-17-0': '## Multi-city epidemic model', '1905.00734-1-18-0': 'Several multi-city epidemic models [CITATION] consider a network of suburbs as [MATH] nodes, in which each node [MATH] represents a suburb, where [MATH].', '1905.00734-1-18-1': 'If two nodes [MATH] are linked, a fraction of population living in node [MATH] can travel to node [MATH] and back (commute from [MATH] to [MATH], for example for work) on a daily basis.', '1905.00734-1-18-2': 'The connectivity of suburbs and the fraction of people commuting between them are represented by the population flux matrix [MATH] whose entries represent the fraction of population daily commuting from [MATH] to [MATH], [MATH] (Equation [REF]).', '1905.00734-1-18-3': 'Note that [MATH], thus [MATH] is not a symmetric matrix.', '1905.00734-1-18-4': 'Figure [REF] shows an example of such travel dynamics.', '1905.00734-1-19-0': '[EQUATION]', '1905.00734-1-19-1': 'Trivially, each row in [MATH], representing proportions of the population of a suburb commuting to various destinations, has to sum up to unity: [EQUATION]', '1905.00734-1-19-2': ""However, the column sum measures the population influx to a suburb during a day, and therefore depends on the node's connectivity."", '1905.00734-1-19-3': 'Column summation in [MATH] does not necessarily equal to unity.', '1905.00734-1-20-0': 'It is also necessary to differentiate between present population [MATH] and native population [MATH] in node [MATH] on a particular day.', '1905.00734-1-20-1': '[MATH] is used as a normalizing factor to account for the differences in population flux for different nodes: [EQUATION]', '1905.00734-1-20-2': 'Assuming the disease transmission parameters are identical in all cities, the standard incidence can be expressed as [CITATION]: [EQUATION] where [MATH] and [MATH] denote respectively the number of infective and susceptible individuals in city [MATH], and [MATH] is the average number of contacts in city [MATH] per unit time.', '1905.00734-1-20-3': 'Incorporating travelling pattern defined by [MATH], yields [CITATION]: [EQUATION]', '1905.00734-1-20-4': 'The double summation term in this expression captures the infection at suburb [MATH] due to the encounters between the residents from suburb [MATH] and the residents from suburb [MATH] could be any suburb including [MATH] or [MATH]) occurring at suburb [MATH], provided that suburbs [MATH] are connected with non-zero population flux entries in [MATH].', '1905.00734-1-21-0': '[MATH] can be derived using the Next Generation Approach (see Appendix [REF] for more details).', '1905.00734-1-21-1': 'For example, for a multi-city SEIRS model with 4 compartments (susceptible, exposed, infective, recovered), introduced in [CITATION], and a special case where the contact rate [MATH] is set to 1, while [MATH] is identical across all cities, [MATH] has the following analytical solution: [EQUATION] where [MATH] and [MATH] denote the average lifetime, exposed period and infective period.', '1905.00734-1-21-2': 'It can be seen that this solution concurs with a classical SEIRS model with no mobility.', '1905.00734-1-21-3': 'When [MATH], Equation [REF] reduces to [MATH], being a solution for a canonical SIR model.', '1905.00734-1-21-4': 'Furthermore, if the population dynamics is not considered (i.e., [MATH]), Equation [REF] can be further reduced to [MATH], agreeing with [CITATION].', '1905.00734-1-22-0': '# Methods', '1905.00734-1-23-0': '## Integrated model', '1905.00734-1-24-0': 'Expanding on models described in sections [REF] and [REF], we bring together population mobility and vaccinating behaviours in a network setting, and propose three extensions within an integrated vaccination-focused SIR-network model:', '1905.00734-1-25-0': 'In this study, we only consider vaccinations that confer lifelong immunity, mostly related to childhood diseases such as measles, mumps, rubella and pertussis.', '1905.00734-1-25-1': 'Vaccinations against such diseases are often administered for individuals at a young age, implying that newborns (more precisely, their parents) often face the vaccination decision.', '1905.00734-1-25-2': 'These are captured by the first extension above.', '1905.00734-1-25-3': 'However, during an outbreak, adults may face the vaccination decision themselves if they were not previously vaccinated.', '1905.00734-1-25-4': 'For example, diseases which are not included in formal childhood vaccination programs, such as smallpox [CITATION], may present vaccination decisions to the entire susceptible population.', '1905.00734-1-25-5': 'These are captured by the second extension.', '1905.00734-1-25-6': 'The third extension introduces a small fraction of susceptible population as committed vaccine recipients, i.e., those who would always choose to vaccinate regardless.', '1905.00734-1-25-7': 'The purpose of these committed vaccine recipients is to demonstrate how successful immunization education campaigns could affect vaccination dynamics.', '1905.00734-1-26-0': 'Our models divide the population into many homogeneous groups [CITATION], based on their residential suburbs.', '1905.00734-1-26-1': 'Within each suburb (i.e., node), residents are treated as a homogeneous population.', '1905.00734-1-26-2': 'It is assumed that the total population within each node is conserved over time.', '1905.00734-1-26-3': ""The dynamics of epidemic and vaccination at each node, are referred to as 'local dynamics'."", '1905.00734-1-26-4': ""The aggregate epidemic dynamics of the entire network can be obtained by summing over all nodes, producing 'global dynamics'."", '1905.00734-1-27-0': ""We model the 'imitation dynamics' based on the individual's travelling pattern and the connectivity of their node defined by Equation [REF]."", '1905.00734-1-27-1': 'For any node [MATH], let [MATH] denote the fraction of vaccinated individuals in the susceptible class in suburb [MATH].', '1905.00734-1-27-2': 'On a particular day, unvaccinated susceptible individuals [MATH] commute to suburb [MATH] and encounter vaccinated individuals from node [MATH] (where [MATH] may be [MATH] itself, [MATH] or any other nodes) and imitate their strategy.', '1905.00734-1-27-3': ""However, this 'imitation' is only applicable in the case of a non-vaccinated individual imitating the strategy of a vaccinated individual (that is, deciding to vaccinate), since the opposite 'imitation' cannot occur."", '1905.00734-1-27-4': ""Therefore, in our model, every time a non-vaccinated person from [MATH] comes in contact with a vaccinated person from [MATH], they imitate the 'vaccinate' strategy if the perceived payoff outweighs the non-vaccination strategy."", '1905.00734-1-28-0': 'Following Equation [REF] proposed in [CITATION], the rate of change of the proportion of vaccinated individuals in [MATH], that is, [MATH], over time can be expressed by: [EQUATION]', '1905.00734-1-28-1': 'The players of the vaccination game are the parents, deciding whether or not to vaccinate their children using the information of the disease prevalence collected from their daily commute.', '1905.00734-1-28-2': 'For example, a susceptible individual residing at node [MATH] and working at node [MATH] uses the local disease prevalence at node [MATH] to decide whether to vaccinate or not.', '1905.00734-1-28-3': 'If such a susceptible individual is infected, that individual will be counted towards the local epidemic prevalence at node [MATH].', '1905.00734-1-29-0': 'We measure each health compartment as a proportion of the population.', '1905.00734-1-29-1': ""Hence, we define a ratio, [MATH], as the ratio between present population [MATH] and the 'native' population [MATH] in node [MATH] on a particular day as: [EQUATION]"", '1905.00734-1-29-2': 'Our main focus is to investigate the effects of vaccinating behaviours on the global epidemic dynamics.', '1905.00734-1-29-3': 'To do so, we vary three parameters:', '1905.00734-1-30-0': 'While [MATH] and [MATH] have had been considered in [CITATION], we introduce a new parameter [MATH] as the vaccination failure rate, [MATH] to consider the cases where the vaccination may not be fully effective.', '1905.00734-1-30-1': 'The imitation component (Equation [REF]) is common in all model extensions.', '1905.00734-1-30-2': 'While the epidemic compartments vary depending on the specific extension, Equation [REF] is used consistently to model the relative rate of change in vaccination behaviours.', '1905.00734-1-31-0': '## Vaccination available to newborns only', '1905.00734-1-32-0': 'Model ([REF]) captures the scenario when vaccination opportunities are provided to newborns only: [EQUATION]', '1905.00734-1-32-1': 'Unvaccinated newborns and newborns with unsuccessful vaccination [MATH] stay in the susceptible class.', '1905.00734-1-32-2': 'Successfully vaccinated newborns, on the other hand, move to the recovered class [MATH].', '1905.00734-1-32-3': 'Other population dynamics across health compartments follow the model ([REF]).', '1905.00734-1-33-0': 'Since [MATH], model ([REF]) can be reduced to: [EQUATION]', '1905.00734-1-34-0': 'Model ([REF]) has a disease-free equilibrium (disease-free initial condition) [MATH] for which [EQUATION]', '1905.00734-1-34-1': 'Proposition 1.', '1905.00734-1-34-2': 'At the disease free equilibrium, [MATH] has two solutions ([MATH] or [MATH]) if [MATH] and [MATH] are uniform across all nodes.', '1905.00734-1-35-0': 'Proof: At the disease free equilibrium, assuming [MATH] are uniform across all nodes, and therefore: [EQUATION]', '1905.00734-1-35-1': 'Hence, under disease-free condition where [MATH] and [MATH], Equation [REF] becomes: [EQUATION]', '1905.00734-1-35-2': 'Equation ([REF]) leads to two solutions: [MATH] or [MATH].', '1905.00734-1-36-0': ""We can now obtain [MATH] for the global dynamics in this model by using the Next Generation Approach, where [MATH] is given by the most dominant eigenvalue (or 'spatial radius' [MATH]) of [MATH], where [MATH] and [MATH] are [MATH] matrices, representing the 'new infections' and 'cases removed or transferred from the infected class', respectively in the disease free condition [CITATION]."", '1905.00734-1-36-1': 'As a result, [MATH] is determined as follows (see Appendix [REF] for detailed derivation): [EQUATION] where [EQUATION] while [EQUATION]', '1905.00734-1-36-2': 'Assuming [MATH] for all nodes, we can derive [MATH] and [MATH] as follows: [EQUATION] where [MATH] is a [MATH] identity matrix.', '1905.00734-1-37-0': 'Using Proposition 1, [MATH] can be simplified as: [EQUATION]', '1905.00734-1-37-1': 'We can now prove the following simple but useful proposition.', '1905.00734-1-38-0': 'Proposition 2.', '1905.00734-1-38-1': 'Matrix [MATH], defined as follows: [EQUATION] is a Markov matrix.', '1905.00734-1-39-0': 'Proof: We assume all nodes have the same population [MATH].', '1905.00734-1-39-1': 'Then Equation ([REF]) can be reduced to: [EQUATION]', '1905.00734-1-39-2': 'Without loss of generality, substituting Equation ([REF]) into Equation ([REF]) and expanding entries in the first column yields the first column [MATH] of matrix [MATH]: [EQUATION]', '1905.00734-1-39-3': 'Continuing with column 1, the column sum is: [EQUATION]', '1905.00734-1-39-4': 'Similarly, the sums of each column are equal to 1, with all entries being non-negative population fractions.', '1905.00734-1-39-5': 'Hence, [MATH] is a Markov matrix.', '1905.00734-1-40-0': 'As a Markov matrix, [MATH] always has the most dominant eigenvalue of unity.', '1905.00734-1-40-1': 'All other eigenvalues are smaller than unity in absolute value [CITATION].', '1905.00734-1-41-0': 'Now the next generation matrix [MATH] can be obtained as follows: [EQUATION]', '1905.00734-1-41-1': 'From Proposition 2 and Equation ([REF]), [MATH] can be obtained as: [EQUATION]', '1905.00734-1-41-2': 'Noting Proposition 1, there are two cases: [MATH] and [MATH].', '1905.00734-1-41-3': 'When nobody vaccinates ([MATH]), the entire population remains susceptible, which reduces model ([REF]) to a canonical SIR model without vaccination intervention and [MATH] returns to [MATH].', '1905.00734-1-41-4': 'On the other hand, if the whole population is vaccinated ([MATH]), the fraction of susceptible population only depends on the vaccine failure rate [MATH], resulting in: [EQUATION]', '1905.00734-1-41-5': 'Clearly, fully effective vaccination ([MATH]) would prohibit disease spread ([MATH]).', '1905.00734-1-41-6': 'Partially effective vaccination could potentially suppress disease transmission, or even eradicate disease spread if [MATH].', '1905.00734-1-41-7': 'If all vaccinations fail ([MATH]), model [REF] concurs with a canonical SIR model without vaccination, which also corresponds to a special case in Equation [REF] where [MATH].', '1905.00734-1-42-0': '## Vaccination available to the entire susceptible class', '1905.00734-1-43-0': 'If the vaccination opportunity is expanded to the entire susceptible class, including newborns and adults, the following model is proposed based on model ([REF]): [EQUATION]', '1905.00734-1-44-0': 'Model ([REF]) has a disease-free equilibrium [MATH]: [EQUATION] while [EQUATION]', '1905.00734-1-44-1': 'Substituting [MATH] using Equation ([REF]) yields [EQUATION] where [MATH] is a [MATH] identity matrix.', '1905.00734-1-45-0': 'The next generation matrix [MATH] can then be obtained as: [EQUATION]', '1905.00734-1-45-1': 'Using Proposition 2, [MATH] can be obtained as: [EQUATION]', '1905.00734-1-45-2': 'When nobody vaccinates ([MATH]), model ([REF]) concurs with a canonical SIR model with [MATH] reducing to [MATH].', '1905.00734-1-45-3': 'When the vaccination attains the full coverage in the population ([MATH]), the magnitude of [MATH] depends on the vaccine failure rate [MATH]: [EQUATION]', '1905.00734-1-45-4': 'Equation [REF] reduces to [MATH] if all vaccines fail ([MATH]).', '1905.00734-1-45-5': 'Conversely, if all vaccines are effective [MATH], Equation [REF] yields [EQUATION]', '1905.00734-1-45-6': 'Given that [MATH], [MATH] is well below the critical threshold: [EQUATION]', '1905.00734-1-45-7': 'If [MATH], by comparing Equations ([REF]) and ([REF]), we note that [MATH] becomes smaller, provided [MATH], that is: [EQUATION]', '1905.00734-1-46-0': '## Vaccination available to the entire susceptible class with committed vaccine recipients', '1905.00734-1-47-0': 'We now consider the existence of committed vaccine recipients, [MATH], as a fraction of individuals who would choose to vaccinate regardless of payoff assessment [CITATION] ([MATH]).', '1905.00734-1-47-1': 'We assume that committed vaccine recipients are also exposed to vaccination failure rate [MATH] and are distributed uniformly across all nodes.', '1905.00734-1-47-2': 'It is also important to point out that the fraction of committed vaccine recipients is constant over time.', '1905.00734-1-47-3': 'However, they can still affect vaccination decision for those who are not vaccinated, and consequently, contribute to the rate of change of the vaccinated fraction [MATH].', '1905.00734-1-47-4': 'Model ([REF]) can be further extended to reflect these considerations, as follows: [EQUATION]', '1905.00734-1-48-0': 'Model ([REF]) has a disease-free equilibrium: [EQUATION] while [EQUATION]', '1905.00734-1-48-1': 'Substituting [MATH], using Equation [REF], yields: [EQUATION] where [MATH] is a [MATH] identity matrix.', '1905.00734-1-49-0': 'The next generation matrix [MATH] can now be obtained as: [EQUATION]', '1905.00734-1-49-1': 'Proposition 3 At the disease free equilibrium, [MATH] has two solutions [MATH] or [MATH] if [MATH], [MATH] and [MATH] are uniform across all nodes.', '1905.00734-1-50-0': 'Proof: In analogy with Proposition 1, with committed vaccine recipients, at the disease free equilibrium, [MATH] and [MATH] are uniform across all nodes, and therefore: [EQUATION]', '1905.00734-1-50-1': 'Hence, under disease-free condition where [MATH] and [MATH], Equation [REF] becomes: [EQUATION]', '1905.00734-1-50-2': 'Equation [REF] leads to two solutions: [MATH] or [MATH].', '1905.00734-1-51-0': '[MATH] can be obtained for this case as: [EQUATION]', '1905.00734-1-51-1': 'If nobody chooses to vaccinate ([MATH]), [MATH] can be reduced to [MATH].', '1905.00734-1-51-2': 'Conversely, if the entire population is vaccinated [MATH], Equation ([REF]) reduces to equation ([REF]), and [MATH] is purely dependent on the vaccine failure rate [MATH].', '1905.00734-1-52-0': '## Model parametrization', '1905.00734-1-53-0': 'The proposed models aim to simulate a scenario of a generic childhood disease (e.g., measles) where life-long full immunity is acquired after effective vaccination.', '1905.00734-1-53-1': 'The vaccination failure rate, [MATH], is set as the probability of ineffective vaccination, to showcase the influence of unsuccessful vaccination on the global epidemic dynamics.', '1905.00734-1-53-2': 'In reality, the vaccination for Measles-Mumps-Rubella (MMR) is highly effective: for example, in Australia, an estimated 96% is successful in conferring immunity [CITATION].', '1905.00734-1-54-0': 'The population flux matrix [MATH] is derived from the network topology, in which each entry represents the connectivity between two nodes: if two nodes are not connected, [MATH]; if two nodes are connected, [MATH] is randomly assigned in the range of [MATH].', '1905.00734-1-54-1': 'The population influx into a node [MATH] within a day is represented by the column sum [MATH] in flux matrix [MATH].', '1905.00734-1-55-0': 'The parameters used for all simulations are summarized in Table [REF].', '1905.00734-1-55-1': 'Same initial conditions are applied to all nodes.', '1905.00734-1-55-2': 'Parameters [MATH] and [MATH] are calibrated based on values used in [CITATION].', '1905.00734-1-56-0': 'Our simulations were carried out on two networks:', '1905.00734-1-57-0': 'Each of these networks was used in conjunction with the following three models of vaccination behaviors:', '1905.00734-1-58-0': 'In the pilot case of 3-node network, two network topologies are studied: an isolated 3-node network where all residents remain in their residential nodes without travelling to other nodes (equivalent to models proposed in [CITATION]), and a fully connected network where the residents at each node commute to the other two nodes, and the population fractions commuting are symmetric and uniformly distributed (Figure [REF]).', '1905.00734-1-59-0': 'We then consider a suburb-network modelled as an Erdos-Renyi random graph (with number of nodes [MATH]) with average degree [MATH], in order to study how an expansive travelling pattern affects the global epidemic dynamics.', '1905.00734-1-59-1': 'Other topologies, such as lattice, scale-free and small-world networks, can easily be substituted here, though in this study our focus is on an Erdos-Renyi random graphs.', '1905.00734-1-60-0': '# Results', '1905.00734-1-61-0': '## 3-node network', '1905.00734-1-62-0': '### Vaccinating newborns only', '1905.00734-1-63-0': 'When there is no population mobility, we observe three distinctive equilibria (Figure [REF]; dotted lines): a pure non-vaccinating equilibrium (where [MATH], representing the final condition, and [MATH]), a mixed equilibrium (where [MATH], [MATH]), and stable limit cycles (where [MATH], [MATH]).', '1905.00734-1-63-1': 'This observation is in qualitative agreement with the vaccinating dynamics reported by [CITATION].', '1905.00734-1-64-0': 'When commuting is allowed, individuals commute to different nodes, and their decision will no longer rely on the single source of information (i.e., the disease prevalence in their residential node) but will also depend on the disease prevalence at their destination.', '1905.00734-1-64-1': 'As a consequence, the three distinctive equilibria are affected in different ways (Figure [REF]; solid lines).', '1905.00734-1-64-2': 'The amplitude of the stable limit cycles at [MATH] is reduced as a result of the reduced disease prevalence.', '1905.00734-1-64-3': 'It takes comparatively longer (compare the dotted lines and solid lines in Figure [REF]) to converge to the pure non-vaccinating equilibrium at [MATH], and to the endemic, mixed equilibrium at [MATH] with high amplitude of oscillation at the start of the epidemic spread.', '1905.00734-1-64-4': 'As [MATH] can be interpreted as the responsiveness of vaccinating behavior to the disease prevalence [CITATION], if individuals are sufficiently responsive (i.e., [MATH] is high), the overall epidemic is suppressed more due to population mobility (and the ensuing imitation), as evidenced by smaller prevalence peaks.', '1905.00734-1-64-5': 'Conversely, if individuals are insensitive to the prevalence change (i.e., [MATH] is low), epidemic dynamics with equal population mobility may appear to be more volatile at the start, but the converged levels of both prevalence peak and vaccine uptake remain unchanged in comparison to the case where there is no population mobility.', '1905.00734-1-65-0': 'We further investigated how the vaccination failure rate [MATH] affects the global epidemic dynamics.', '1905.00734-1-65-1': 'If, for example, only a half of the vaccine administered is effective ([MATH]), as shown in Figure [REF] (b-d), the infection peaks arrive sooner, for all values of [MATH].', '1905.00734-1-65-2': 'As a result of having the earlier infection peaks, the individuals respond to the breakout and choose to vaccinate sooner, causing the vaccine uptake to rise.', '1905.00734-1-65-3': ""When [MATH], the prevalence peaks take longer to develop and the extended period gives individuals an illusion that there may not be an epidemic breakout, and consequently encourages 'free-riding' behavior."", '1905.00734-1-65-4': 'In the case where half of the vaccinations fail, epidemic breaks out significantly earlier at lower peaks, an observation that is beneficial to encourage responsive individuals to choose to vaccinate.', '1905.00734-1-65-5': 'Although some (in this case half) of the vaccines fail, a sufficiently high vaccine uptake still curbs prevalence peaks and shortens the length of breakout period.', '1905.00734-1-65-6': 'In terms of the final vaccination uptake, the vaccine failure rate predominantly affects behaviors of responsive individuals (when [MATH] is high, such as [MATH]).', '1905.00734-1-65-7': 'In this case, instead of the stable limit cycles observed when [MATH], an endemic, mixed equilibrium is reached when the vaccination failure rate is significant.', '1905.00734-1-65-8': 'Final vaccination uptake is impacted little by vaccination failure rate when individuals are insensitive to changes in disease prevalence (when [MATH] is lower in value, such as [MATH] or [MATH]).', '1905.00734-1-65-9': 'These observations are illustrated in Figure [REF].', '1905.00734-1-66-0': '### Vaccinating the entire susceptible class', '1905.00734-1-67-0': ""If (voluntary) vaccination is offered to the susceptible class regardless of the age, the initial condition of [MATH] represents the scenario that the vast majority of population are immunized to begin with, and the epidemic would not breakout until the false sense of security provided by the temporary 'herd immunity' settles in."", '1905.00734-1-67-1': 'This feature is observed in Figure [REF] (a)-(b): the vaccination coverage continues to drop at the start of the epidemic breakout, indicating that individuals, regardless of their responsiveness towards the prevalence change, exploit the temporary herd immunity until an infection peak emerges.', '1905.00734-1-67-2': 'Since vaccination is available to the entire susceptible class, a small increase in [MATH] could help suppresses disease prevalence.', '1905.00734-1-67-3': 'However, when vaccination is partially effective (i.e., [MATH]), the peaks in vaccine uptake, [MATH], are no longer an accurate reflection of the actual vaccination coverage, and such peaks therefore may not be sufficient to adequately suppress infection peaks.', '1905.00734-1-67-4': 'It can also be observed that vaccinating the susceptible class encourages non-vaccinating behaviors due to the perception of herd immunity, and therefore pushes the epidemic towards an endemic equilibrium, particularly when sensitivity to prevalence is relatively high ( mid and high [MATH]), as the responsive individuals would react promptly to the level of disease prevalence, altering their vaccination decisions.', '1905.00734-1-67-5': 'However, no substantial impact is observed on those individuals who are insensitive to changes in the disease prevalence ( low [MATH]).', '1905.00734-1-68-0': '## Erdos-Renyi random network of 3000 nodes', '1905.00734-1-69-0': 'We now present the simulation results on a much larger Erdo-Renyi random network of [MATH] nodes, which more realistically reflects the size of a modern city and its commuting patterns.', '1905.00734-1-69-1': ""It was observed by previous studies [CITATION] that such a larger system requires a higher vaccination coverage to achieve herd immunity, and thus curbs 'free-riding' behaviors more effectively."", '1905.00734-1-70-0': '### Vaccinating newborns only', '1905.00734-1-71-0': 'In this case, three distinct equilibria are observed (as shown in Figure [REF] (a)) for three values of [MATH]: a pure non-vaccinating equilibrium at [MATH] and two endemic mixed equilibria at [MATH] and [MATH] respectively, replacing the stable limit cycles at high [MATH] previously observed in the 3-node case.', '1905.00734-1-71-1': 'As expected, when individuals are more responsive to disease prevalence (higher [MATH]), they are more likely to get vaccinated.', '1905.00734-1-71-2': 'The expansive travelling pattern also somewhat elevates the global vaccination coverage level (compared to the 3-node case), particularly in the case that the individuals show a moderate level of responsiveness to prevalence ([MATH]), and shortens the convergence time to reach the equilibrium.', '1905.00734-1-71-3': 'However, for those who are insensitive to the changes in disease prevalence ([MATH]), the more expansive commuting presented by the larger network does not affect either the level of voluntary vaccination, or the convergence time in global dynamics - these individuals remain unvaccinated as in the case of the smaller network.', '1905.00734-1-71-4': 'It is also found that the vaccination coverage is very sensitive to the disease prevalence change at the larger network since the disease prevalence peaks are notably lower than the 3-node case counterparts.', '1905.00734-1-71-5': 'If half of the vaccines administered are unsuccessful ([MATH]), the impact of early peaks on global dynamics is magnified in a larger network (Figure [REF] (b-d)), leading to a shorter convergence time, although the final equilibria are hardly affected compared to the 3-node case as shown in figure [REF].', '1905.00734-1-72-0': 'We also compared the epidemic dynamics in terms of the adjusted imitation rate, represented by the parameter [MATH], as shown in figure [REF].', '1905.00734-1-72-1': 'Recall that the value of imitation rate used in our simulations, unless otherwise stated, is [MATH] as reported in Table 1.', '1905.00734-1-72-2': 'Here we presents results where this value of imitation rate is compared with a much smaller value of [MATH].', '1905.00734-1-72-3': 'In populations where the individuals imitate more quickly (i.e., higher [MATH]), the oscillations in prevalence and vaccination dynamics arrive quicker with larger amplitude, although the converged vaccine uptake and disease prevalence are similar for higher and lower [MATH].', '1905.00734-1-72-4': 'For a higher [MATH], the convergence to equilibria is quicker.', '1905.00734-1-72-5': 'These observations are consistent with results reported by earlier studies [CITATION].', '1905.00734-1-72-6': 'A smaller value of [MATH]) also altered behaviors of those individuals who are insensitive to prevalence change.', '1905.00734-1-72-7': 'Instead of converging to the pure non-vaccination equilibrium, the dynamics converge to a mixed endemic state, meaning that when imitation rate is very low, some individuals would still choose to vaccinate even when they are not very sensitive to disease prevalence.', '1905.00734-1-73-0': 'For such a large network, vaccinating behavior of individuals also depends on the weighted degree (number and weight of connections) of the node (suburb) in which they live.', '1905.00734-1-73-1': 'This is illustrated in Figure [REF].', '1905.00734-1-73-2': 'For individuals living in highly connected nodes, there are many commuting destinations, allowing access to a broad spectrum of information on the local disease prevalence.', '1905.00734-1-73-3': ""Therefore, it is not surprising that we found that individuals living in 'hubs' are more likely to get vaccinated, particularly when the population is sensitive to prevalence ([MATH] is high), as shown in Figure [REF] (a)."", '1905.00734-1-73-4': 'This observation is in accordance with previous studies [CITATION].', '1905.00734-1-73-5': 'Note that in in Figure [REF] (a), nodes with degree [MATH] are grouped into one bin to represent hubs, as the frequency count of these nodes is extremely low.', '1905.00734-1-73-6': 'Note also that there is a positive correlation between the number of degrees and the volume of population influx as measured by a sum of proportions from the source nodes (Figure [REF] (b)), indicating that a highly connected suburb has greater population influx on a daily basis.', '1905.00734-1-74-0': '### Vaccinating entire susceptible class', '1905.00734-1-75-0': 'If (voluntary) vaccination is offered to the susceptible class regardless of age, we found that the global epidemic dynamics converge quicker compared to the similar scenario in the 3-node case.', '1905.00734-1-75-1': 'Only one predominant infection peak is observed, corresponding to the high infection prevalence around year 25, as shown in figure [REF].', '1905.00734-1-75-2': 'Oscillations of small magnitude are observed for both vaccine uptake and disease prevalence at later time-steps for middle or high values of [MATH] (as shown in insets of figure [REF]).', '1905.00734-1-75-3': 'These findings also largely hold if half of the vaccines administered are ineffective (i.e., [MATH]), although the predominant prevalence peak and the corresponding vaccination peak around year 25 are both higher than their counterparts observed for the case where [MATH].', '1905.00734-1-76-0': 'We also found that employing a small fraction of committed vaccine recipients prevents major epidemic by curbing disease prevalence.', '1905.00734-1-76-1': ""Such a finding holds for all [MATH] (i.e., regardless population's responsiveness towards disease prevalence) as the magnitude of prevalence is too small to make a non-vaccinated individual switch to the vaccination strategy (Figure [REF])."", '1905.00734-1-77-0': '# Discussion and conclusion', '1905.00734-1-78-0': 'We presented a series of SIR-network models with imitation dynamics, aiming to model scenarios where individuals commute between their residence and work across a network (e.g., where each node represents a suburb).', '1905.00734-1-78-1': 'These models are able to capture diverse travelling patterns (i.e., reflecting local connectivity of suburbs), and different vaccinating behaviors affecting the global vaccination uptake and epidemic dynamics.', '1905.00734-1-78-2': 'We also analytically derived expressions for the reproductive number [MATH] for the considered SIR-network models, and demonstrated how epidemics may evolve over time in these models.', '1905.00734-1-79-0': 'We showed that the stable oscillations in the vaccinating dynamics are only likely to occur either when there is no population mobility across nodes, or only with limited commuting destinations.', '1905.00734-1-79-1': 'We observed that, compared to the case where vaccination is only provided to newborns, if vaccination is provided to the entire susceptible class, higher disease prevalence and more volatile oscillations in vaccination uptake are observed (particularly in populations which are relatively responsive to the changes in disease prevalence).', '1905.00734-1-79-2': 'A more expansive travelling pattern simulated in a larger network encourages the attractor dynamics and the convergence to the endemic, mixed equilibria, again if individuals are sufficiently responsive towards the changes in the disease prevalence.', '1905.00734-1-79-3': 'If individuals are insensitive to the prevalence, they are hardly affected by different vaccinating models and remain as unvaccinated individuals, although the existence of committed vaccine recipients noticeably delays the convergence to the non-vaccinating equilibrium.', '1905.00734-1-79-4': 'The presented models highlight the important role of committed vaccine recipients in actively reducing [MATH] and disease prevalence, strongly contributing to eradicating an epidemic spread.', '1905.00734-1-79-5': 'Similarly conclusions have been reached previously [CITATION], and our results extend these to imitation dynamics in SIR-network models.', '1905.00734-1-80-0': 'Previous studies drew an important conclusion that highly connected hubs play a key role in containing infections as they are more likely to get vaccinated due to the higher risk of infection in social networks [CITATION].', '1905.00734-1-80-1': 'Our results complement this finding by showing that a higher fraction of individuals who reside in highly connected suburbs choose to vaccinate compared to those living in relatively less connected suburbs.', '1905.00734-1-80-2': 'These hubs, often recognized as business districts, also have significantly higher population influx as the destination for many commuters from other suburbs.', '1905.00734-1-80-3': 'Therefore, it is important for policy makers to leverage these job hubs in promoting vaccination campaigns and public health programs.', '1905.00734-1-81-0': 'Overall, our results demonstrate that, in order to encourage vaccination behavior and shorten the course of epidemic, policy makers of vaccination campaigns need to carefully orchestrate the following three factors: ensuring a number of committed vaccine recipients in each suburb, utilizing the vaccinating tendency of well-connected suburbs, and increasing individual awareness towards the prevalence change.', '1905.00734-1-82-0': 'There are several avenues to extend this work further.', '1905.00734-1-82-1': 'This work assumes that the individuals from different nodes (suburbs) only differ in their travelling patterns, using the same epidemic and behavior parameters for individuals from all nodes.', '1905.00734-1-82-2': 'Also, the same [MATH] and [MATH] are used for all nodes, by assuming that individuals living in all nodes are equally responsive towards disease prevalence and imitation.', '1905.00734-1-82-3': ""Realistically, greater heterogeneity can be implemented by establishing context-specific [MATH] and imitating parameters for factors such as the local population density, community size [CITATION], and suburbs' level of connectivity."", '1905.00734-1-82-4': 'For example, residents living in highly connected suburbs may be more alert to changes in disease prevalence, and adopt imitation behaviors more quickly.', '1905.00734-1-82-5': 'Different network topologies can also be used, particularly scale-free networks [CITATION] where a small number of nodes have a large number of links each.', '1905.00734-1-82-6': 'These highly connected nodes are a better representation of suburbs with extremely high population influx (e.g., central business districts and job hubs).', '1905.00734-1-82-7': 'It may also be instructive to translate the risk perception of vaccination and infection into tangible measures to demonstrate the aggregate social cost of an epidemic breakout, and help policy makers to visualize the cost effectiveness of different vaccinating strategies and estimate the financial burden for public health care.', '1905.00734-1-82-8': 'This study can also be extended by calibrating the network to real-life connectivity and demographic to mimic a breakout in a targeted region [CITATION], so that a model of vaccination campaign can demonstrate a reduction in the severity of epidemic.', '1905.00734-1-82-9': 'These considerations could advance this study to more accurately reflect contagion dynamics in urban environment, and provide further insights to public health planning.', '1905.00734-1-83-0': '# Appendix', '1905.00734-1-84-0': '## Next Generation Operator Approach', '1905.00734-1-85-0': 'The proposed model ([REF]) can be seen as a finely categorized SIR deterministic model, so that each health compartment (Susceptible, Infected and Recovered) has [MATH] sub-classes.', '1905.00734-1-85-1': 'Let us denote [MATH] to represent [MATH] infected host compartments, and [MATH] to represent [MATH] other host compartments consisting of susceptible compartments [MATH] and recovered compartments [MATH].', '1905.00734-1-85-2': '[EQUATION] where [MATH] and [MATH].', '1905.00734-1-86-0': '[MATH] is the rate at which new infection enters infected compartments and [MATH] is the transfer of individuals out of or into the infected compartments.', '1905.00734-1-87-0': 'When close to disease-free equilibrium where [MATH], the model can be linearized to: [EQUATION] where [MATH] and [MATH]', '1905.00734-1-88-0': 'The next generation matrix, [MATH], is then given by: [EQUATION] where each entry [MATH] represents the expected number of secondary cases which an infected individual imposes on the rest of the compartments.', '1905.00734-1-88-1': '[MATH] and [MATH] are given as follows: [EQUATION]', '1905.00734-1-88-2': ""The basic reproduction number [MATH] is given by the most dominant eigenvalue (or 'spatial radius' [MATH]) of [MATH] [CITATION], and therefore: [EQUATION]"", '1905.00734-1-89-0': '## Declaration of interest', '1905.00734-1-90-0': 'The authors declare that they have no competing interests.'}","{'1905.00734-2-0-0': 'We present a series of SIR-network models, extended with a game-theoretic treatment of imitation dynamics which result from regular population mobility across residential and work areas and the ensuing interactions.', '1905.00734-2-0-1': 'Each considered SIR-network model captures a class of vaccination behaviours influenced by epidemic characteristics, interaction topology, and imitation dynamics.', '1905.00734-2-0-2': 'Our focus is the eventual vaccination coverage, produced under voluntary vaccination schemes, in response to these varying factors.', '1905.00734-2-0-3': 'Using the next generation matrix method, we analytically derive and compare expressions for the basic reproduction number [MATH] for the proposed SIR-network models.', '1905.00734-2-0-4': 'Furthermore, we simulate the epidemic dynamics over time for the considered models, and show that if individuals are sufficiently responsive towards the changes in the disease prevalence, then the more expansive travelling patterns encourage convergence to the endemic, mixed equilibria.', '1905.00734-2-0-5': 'On the contrary, if individuals are insensitive to changes in the disease prevalence, we find that they tend to remain unvaccinated in all the studied models.', '1905.00734-2-0-6': 'Our results concur with earlier studies in showing that residents from highly connected residential areas are more likely to get vaccinated.', '1905.00734-2-0-7': 'We also show that the existence of the individuals committed to receiving vaccination reduces [MATH] and delays the disease prevalence, and thus is essential to containing epidemics.', '1905.00734-2-1-0': '# Introduction', '1905.00734-2-2-0': 'Vaccination has long been established as a powerful tool in managing and controlling infectious diseases by providing protection to susceptible individuals [CITATION].', '1905.00734-2-2-1': 'With a sufficiently high vaccination coverage, the probability of the remaining unvaccinated individuals getting infected reduces significantly.', '1905.00734-2-2-2': 'However, such systematic programs by necessity may limit the freedom of choice of individuals.', '1905.00734-2-2-3': 'When vaccination programs are made voluntary, the vaccination uptake declines as a result of individuals choosing not to vaccinate, as seen in Britain in 2003 when the vaccination program for Measles-Mumps-Rubella (MMR) was made voluntary [CITATION].', '1905.00734-2-2-4': ""Parents feared possible complications from vaccination [CITATION] and hoped to exploit the 'herd immunity' by assuming other parents would choose to vaccinate their children."", '1905.00734-2-2-5': 'Such hopes did not materialize precisely because other parents also thought similarly.', '1905.00734-2-3-0': ""Under a voluntary vaccination policy an individual's decision depends on several factors: the social influence from one's social network, the risk perception of vaccination, and the risk perception of infection, in terms of both likelihood and impact [CITATION]."", '1905.00734-2-3-1': 'This decision-making is often modelled using game theory [CITATION], by allowing individuals to compare the cost of vaccination and the potential cost of non-vaccination (in terms of the likelihood and impact of infection), and adopting imitation dynamics in modelling the influence of social interactions.', '1905.00734-2-3-2': 'However, many of the earlier studies [CITATION] were based on the key assumption of a well-mixed homogeneous population where each individual is assumed to have an equal chance of making contact with any other individual in the population.', '1905.00734-2-3-3': 'This population assumption is rather unrealistic as large populations are often diverse with varying levels of interactions.', '1905.00734-2-3-4': 'To address this, more recent studies [CITATION] model populations as complex networks where each individual, represented by a node, has a finite set of contacts, represented by links [CITATION].', '1905.00734-2-4-0': 'It has been shown that there is a critical cost threshold in the vaccination cost above which the likelihood of vaccination drops steeply [CITATION].', '1905.00734-2-4-1': 'In addition, highly connected individuals were shown to be more likely to choose to be vaccinated as they perceive themselves to be at a greater risk of being infected due to high exposure within the community.', '1905.00734-2-4-2': 'In modelling large-scale epidemics, the population size (i.e., number of nodes) can easily reach millions of individuals, interacting in a complex way.', '1905.00734-2-4-3': 'The challenge, therefore, is to extend epidemic modelling not only with the individual vaccination decision-making, but also capture diverse interaction patterns encoded within a network.', '1905.00734-2-4-4': 'Such an integration has not yet been formalized, motivating our study.', '1905.00734-2-4-5': 'Furthermore, once an integrated model is developed, a specific challenge is to consider how the vaccination imitation dynamics developing across a network affects the basic reproduction number, [MATH].', '1905.00734-2-4-6': 'This question forms our second main objective.', '1905.00734-2-5-0': 'One relevant approach partially addressing this objective is offered by the multi-suburb (or multi-city) SIR-network model [CITATION] where each node represents a neighbourhood (or city) with a certain number of residents.', '1905.00734-2-5-1': 'The daily commute of individuals between two neighbourhoods is modelled along the network link connecting the two nodes, which allows to quantify the disease spread between individuals from different neighbourhoods.', '1905.00734-2-5-2': 'Ultimately, this model captures meta-population dynamics in a multi-suburb setting affected by an epidemic spread at a greater scale.', '1905.00734-2-5-3': 'To date, these models have not yet considered intervention (e.g., vaccination) options, and the corresponding social interaction across the populations.', '1905.00734-2-6-0': 'To incorporate the imitation dynamics, modelled game-theoretically, within a multi-suburb model representing mobility, we propose a series of integrated vaccination-focused SIR-network models.', '1905.00734-2-6-1': 'This allows us to systematically analyze how travelling patterns affect the voluntary vaccination uptake due to adoption of different imitation choices, in a large distributed population.', '1905.00734-2-6-2': 'The developed models use an increasingly complex set of vaccination strategies.', '1905.00734-2-6-3': 'Thus, our specific contribution is the study of the vaccination uptake, driven by imitation dynamics under a voluntary vaccination scheme, using an SIR model on a complex network representing a multi-suburb environment, within which the individuals commute between residential and work areas.', '1905.00734-2-7-0': 'In section [REF], we present the models and methods associated with this study: in particular, we present analytical derivations of the basic reproduction number [MATH] for the proposed models using the Next Generation Operator Approach [CITATION], and carry out a comparative analysis of [MATH] across these models.', '1905.00734-2-7-1': 'In section [REF], we simulate the epidemic and vaccination dynamics over time using the proposed models in different network settings, including a pilot case of a 3-node network, and a 3000-node Erdos-Renyi random network.', '1905.00734-2-7-2': 'The comparison of the produced results across different models and settings is carried out for the larger network, with the focus on the emergent attractor dynamics, in terms of the proportion of vaccinated individuals.', '1905.00734-2-7-3': 'Particularly, we analyze the sensitivity of the individual strategies (whether to vaccinate or not) to the levels of disease prevalence produced by the different considered models.', '1905.00734-2-7-4': 'Section [REF] concludes the study with a brief discussion of the importance of these results.', '1905.00734-2-8-0': '# Technical background', '1905.00734-2-9-0': '## Basic reproduction number [MATH]', '1905.00734-2-10-0': 'The basic reproduction number [MATH] is defined as the number of secondary infections produced by an infected individual in an otherwise completely susceptible population [CITATION].', '1905.00734-2-10-1': 'It is well-known that [MATH] is an epidemic threshold, with the disease dying out as [MATH], or becoming endemic as [MATH] [CITATION].', '1905.00734-2-10-2': 'This finding strictly holds only in deterministic models with infinite population [CITATION].', '1905.00734-2-10-3': 'The topology of the underlying contact network is known to affect the epidemic threshold [CITATION].', '1905.00734-2-10-4': 'Many disease transmission models have shown important correlations between [MATH] and the key epidemic characteristics (e.g., disease prevalence, attack rates, etc.) [CITATION].', '1905.00734-2-10-5': 'In addition, [MATH] has been considered as a critical threshold for phase transitions studied with methods of statistical physics or information theory [CITATION].', '1905.00734-2-11-0': '## Vaccination model with imitation dynamics', '1905.00734-2-12-0': 'Imitation dynamics, a process by which individuals copy the strategy of other individuals, is widely used to model vaccinating behaviours incorporated with SIR models.', '1905.00734-2-12-1': ""The model proposed in [CITATION] applied game theory to represent parents' decision-making about whether to get their newborns vaccinated against childhood disease (e.g., measles, mumps, rubella, pertussis)."", '1905.00734-2-12-2': ""In this model, individuals are in a homogeneously mixing population, and susceptible individuals have two 'pure strategies' regarding vaccination: to vaccinate or not to vaccinate."", '1905.00734-2-12-3': 'The non-vaccination decision can change to the vaccination decision at a particular sampling rate, however the vaccination decision cannot be changed to non-vaccination.', '1905.00734-2-12-4': 'Individuals adopt one of these strategies by weighing up their perceived payoffs, measured by the probability of morbidity from vaccination, and the risk of infection respectively.', '1905.00734-2-12-5': 'The payoff for vaccination ([MATH]) is given as, [EQUATION] and the payoff for non-vaccination ([MATH]), measured as the risk of infection, is given as [EQUATION] where [MATH] is the perceived risk of morbidity from vaccination, [MATH] is the perceived risk of morbidity from non-vaccination (i.e., infection), [MATH] is the current disease prevalence in population fraction at time [MATH], and [MATH] is the sensitivity to disease prevalence [CITATION].', '1905.00734-2-13-0': 'From Equation [REF] and [REF], it can be seen that the payoff for vaccinated individuals is a simple constant, and the payoff for unvaccinated individuals is proportional to the extent of epidemic prevalence (i.e., the severity of the disease).', '1905.00734-2-14-0': 'It is assumed that life-long immunity is granted with effective vaccination.', '1905.00734-2-14-1': 'If one individual decides to vaccinate, s/he cannot revert back to the unvaccinated status.', '1905.00734-2-14-2': 'To make an individual switch to a vaccinating strategy, the payoff gain, [MATH] must be positive ([MATH]).', '1905.00734-2-14-3': ""Let [MATH] denote the relative proportion of vaccinated individuals, and assume that an unvaccinated individual can sample a strategy from a vaccinated individual at certain rate [MATH], and can switch to the 'vaccinate strategy' with probability [MATH]."", '1905.00734-2-14-4': 'Then the time evolution of [MATH] can be defined as: [EQUATION] where [MATH] can be interpreted as the combined imitation rate that individuals use to sample and imitate strategies of other individuals.', '1905.00734-2-14-5': 'Equation [REF] can be rewritten so that [MATH] only depends on two parameters: [EQUATION] where [MATH] and [MATH], assuming that the risk of morbidity from vaccination, [MATH], and the risk of morbidity from non-vaccination (i.e., morbidity from infection), [MATH], are constant during the course of epidemics.', '1905.00734-2-14-6': ""Hence, [MATH] is the adjusted imitation rate, and [MATH] measures individual's responsiveness towards the changes in disease prevalence."", '1905.00734-2-15-0': 'The disease prevalence [MATH] is determined from a simple SIR model [CITATION] with birth and death that divides the population into three health compartments: [MATH] (susceptible), [MATH] (infected) and [MATH] (recovered).', '1905.00734-2-15-1': 'If a susceptible individual encounters an infected individual, s/he contracts the infection with transmission rate [MATH], and thus progresses to the infected compartment, while an infected individual recovers at rate [MATH].', '1905.00734-2-15-2': 'It is assumed that the birth and death rate are equal, denoted by [MATH] [CITATION].', '1905.00734-2-15-3': 'Individuals in all compartments die at an equal rate, and all newborns are added to the susceptible compartment unless and until they are vaccinated.', '1905.00734-2-15-4': 'Vaccinated newborns are moved to the recovered class directly, with a rate [MATH].', '1905.00734-2-15-5': 'Therefore, the dynamics are modelled as follows [CITATION]: [EQUATION] where [MATH], [MATH], and [MATH] represent the proportion of susceptible, infected, and recovered individuals respectively (that is, [MATH]), and [MATH] represents the proportion of vaccinated individuals within the susceptible class at a given time.', '1905.00734-2-15-6': 'The variables [MATH], [MATH], and [MATH], as well as [MATH], are time-dependent, but for simplicity, we omit subscript [MATH] for these and other time-dependent variables.', '1905.00734-2-16-0': 'Model [REF] predicts oscillations in vaccine uptake in response to changes in disease prevalence.', '1905.00734-2-16-1': 'It is found that oscillations are more likely to occur when individuals imitate each other more quickly (i.e., higher [MATH]).', '1905.00734-2-16-2': 'The oscillations are observed to be more volatile when people alter their vaccinating behaviours promptly in response to changes in disease prevalence (i.e., higher [MATH]).', '1905.00734-2-16-3': 'Overall, higher [MATH] or [MATH] produce stable limit cycles at greater amplitude.', '1905.00734-2-16-4': 'Conversely, when individuals are insensitive to changes in disease prevalence (i.e., low [MATH]), or imitate at a slower rate (i.e., low [MATH]), the resultant vaccinating dynamics converge to equilibrium [CITATION].', '1905.00734-2-17-0': '## Multi-city epidemic model', '1905.00734-2-18-0': 'Several multi-city epidemic models [CITATION] consider a network of suburbs as [MATH] nodes, in which each node [MATH] represents a suburb, where [MATH].', '1905.00734-2-18-1': 'If two nodes [MATH] are linked, a fraction of population living in node [MATH] can travel to node [MATH] and back (commute from [MATH] to [MATH], for example for work) on a daily basis.', '1905.00734-2-18-2': 'The connectivity of suburbs and the fraction of people commuting between them are represented by the population flux matrix [MATH] whose entries represent the fraction of population daily commuting from [MATH] to [MATH], [MATH] (Equation [REF]).', '1905.00734-2-18-3': 'Note that [MATH], thus [MATH] is not a symmetric matrix.', '1905.00734-2-18-4': 'Figure [REF] shows an example of such travel dynamics.', '1905.00734-2-19-0': '[EQUATION]', '1905.00734-2-19-1': 'Trivially, each row in [MATH], representing proportions of the population of a suburb commuting to various destinations, has to sum up to unity: [EQUATION]', '1905.00734-2-19-2': ""However, the column sum measures the population influx to a suburb during a day, and therefore depends on the node's connectivity."", '1905.00734-2-19-3': 'Column summation in [MATH] does not necessarily equal to unity.', '1905.00734-2-20-0': 'It is also necessary to differentiate between present population [MATH] and native population [MATH] in node [MATH] on a particular day.', '1905.00734-2-20-1': '[MATH] is used as a normalizing factor to account for the differences in population flux for different nodes: [EQUATION]', '1905.00734-2-20-2': 'Assuming the disease transmission parameters are identical in all cities, the standard incidence can be expressed as [CITATION]: [EQUATION] where [MATH] and [MATH] denote respectively the number of infective and susceptible individuals in city [MATH], and [MATH] is the average number of contacts in city [MATH] per unit time.', '1905.00734-2-20-3': 'Incorporating travelling pattern defined by [MATH], yields [CITATION]: [EQUATION]', '1905.00734-2-20-4': 'The double summation term in this expression captures the infection at suburb [MATH] due to the encounters between the residents from suburb [MATH] and the residents from suburb [MATH] could be any suburb including [MATH] or [MATH]) occurring at suburb [MATH], provided that suburbs [MATH] are connected with non-zero population flux entries in [MATH].', '1905.00734-2-21-0': '[MATH] can be derived using the Next Generation Approach (see Appendix [REF] for more details).', '1905.00734-2-21-1': 'For example, for a multi-city SEIRS model with 4 compartments (susceptible, exposed, infective, recovered), introduced in [CITATION], and a special case where the contact rate [MATH] is set to 1, while [MATH] is identical across all cities, [MATH] has the following analytical solution: [EQUATION] where [MATH] and [MATH] denote the average lifetime, exposed period and infective period.', '1905.00734-2-21-2': 'It can be seen that this solution concurs with a classical SEIRS model with no mobility.', '1905.00734-2-21-3': 'When [MATH], Equation [REF] reduces to [MATH], being a solution for a canonical SIR model.', '1905.00734-2-21-4': 'Furthermore, if the population dynamics is not considered (i.e., [MATH]), Equation [REF] can be further reduced to [MATH], agreeing with [CITATION].', '1905.00734-2-22-0': '# Methods', '1905.00734-2-23-0': '## Integrated model', '1905.00734-2-24-0': 'Expanding on models described in sections [REF] and [REF], we bring together population mobility and vaccinating behaviours in a network setting, and propose three extensions within an integrated vaccination-focused SIR-network model:', '1905.00734-2-25-0': 'In this study, we only consider vaccinations that confer lifelong immunity, mostly related to childhood diseases such as measles, mumps, rubella and pertussis.', '1905.00734-2-25-1': 'Vaccinations against such diseases are often administered for individuals at a young age, implying that newborns (more precisely, their parents) often face the vaccination decision.', '1905.00734-2-25-2': 'These are captured by the first extension above.', '1905.00734-2-25-3': 'However, during an outbreak, adults may face the vaccination decision themselves if they were not previously vaccinated.', '1905.00734-2-25-4': 'For example, diseases which are not included in formal childhood vaccination programs, such as smallpox [CITATION], may present vaccination decisions to the entire susceptible population.', '1905.00734-2-25-5': 'These are captured by the second extension.', '1905.00734-2-25-6': 'The third extension introduces a small fraction of susceptible population as committed vaccine recipients, i.e., those who would always choose to vaccinate regardless.', '1905.00734-2-25-7': 'The purpose of these committed vaccine recipients is to demonstrate how successful immunization education campaigns could affect vaccination dynamics.', '1905.00734-2-26-0': 'Our models divide the population into many homogeneous groups [CITATION], based on their residential suburbs.', '1905.00734-2-26-1': 'Within each suburb (i.e., node), residents are treated as a homogeneous population.', '1905.00734-2-26-2': 'It is assumed that the total population within each node is conserved over time.', '1905.00734-2-26-3': ""The dynamics of epidemic and vaccination at each node, are referred to as 'local dynamics'."", '1905.00734-2-26-4': ""The aggregate epidemic dynamics of the entire network can be obtained by summing over all nodes, producing 'global dynamics'."", '1905.00734-2-27-0': ""We model the 'imitation dynamics' based on the individual's travelling pattern and the connectivity of their node defined by Equation [REF]."", '1905.00734-2-27-1': 'For any node [MATH], let [MATH] denote the fraction of vaccinated individuals in the susceptible class in suburb [MATH].', '1905.00734-2-27-2': 'On a particular day, unvaccinated susceptible individuals [MATH] commute to suburb [MATH] and encounter vaccinated individuals from node [MATH] (where [MATH] may be [MATH] itself, [MATH] or any other nodes) and imitate their strategy.', '1905.00734-2-27-3': ""However, this 'imitation' is only applicable in the case of a non-vaccinated individual imitating the strategy of a vaccinated individual (that is, deciding to vaccinate), since the opposite 'imitation' cannot occur."", '1905.00734-2-27-4': ""Therefore, in our model, every time a non-vaccinated person from [MATH] comes in contact with a vaccinated person from [MATH], they imitate the 'vaccinate' strategy if the perceived payoff outweighs the non-vaccination strategy."", '1905.00734-2-28-0': 'Following Equation [REF] proposed in [CITATION], the rate of change of the proportion of vaccinated individuals in [MATH], that is, [MATH], over time can be expressed by: [EQUATION]', '1905.00734-2-28-1': 'The players of the vaccination game are the parents, deciding whether or not to vaccinate their children using the information of the disease prevalence collected from their daily commute.', '1905.00734-2-28-2': 'For example, a susceptible individual residing at node [MATH] and working at node [MATH] uses the local disease prevalence at node [MATH] to decide whether to vaccinate or not.', '1905.00734-2-28-3': 'If such a susceptible individual is infected, that individual will be counted towards the local epidemic prevalence at node [MATH].', '1905.00734-2-29-0': 'We measure each health compartment as a proportion of the population.', '1905.00734-2-29-1': ""Hence, we define a ratio, [MATH], as the ratio between present population [MATH] and the 'native' population [MATH] in node [MATH] on a particular day as: [EQUATION]"", '1905.00734-2-29-2': 'Our main focus is to investigate the effects of vaccinating behaviours on the global epidemic dynamics.', '1905.00734-2-29-3': 'To do so, we vary three parameters:', '1905.00734-2-30-0': 'While [MATH] and [MATH] have had been considered in [CITATION], we introduce a new parameter [MATH] as the vaccination failure rate, [MATH] to consider the cases where the vaccination may not be fully effective.', '1905.00734-2-30-1': 'The imitation component (Equation [REF]) is common in all model extensions.', '1905.00734-2-30-2': 'While the epidemic compartments vary depending on the specific extension, Equation [REF] is used consistently to model the relative rate of change in vaccination behaviours.', '1905.00734-2-31-0': '## Vaccination available to newborns only', '1905.00734-2-32-0': 'Model ([REF]) captures the scenario when vaccination opportunities are provided to newborns only: [EQUATION]', '1905.00734-2-32-1': 'Unvaccinated newborns and newborns with unsuccessful vaccination [MATH] stay in the susceptible class.', '1905.00734-2-32-2': 'Successfully vaccinated newborns, on the other hand, move to the recovered class [MATH].', '1905.00734-2-32-3': 'Other population dynamics across health compartments follow the model ([REF]).', '1905.00734-2-33-0': 'Since [MATH], model ([REF]) can be reduced to: [EQUATION]', '1905.00734-2-34-0': 'Model ([REF]) has a disease-free equilibrium (disease-free initial condition) [MATH] for which [EQUATION]', '1905.00734-2-34-1': 'Proposition 1.', '1905.00734-2-34-2': 'At the disease free equilibrium, [MATH] has two solutions ([MATH] or [MATH]) if [MATH] and [MATH] are uniform across all nodes.', '1905.00734-2-35-0': 'Proof: At the disease free equilibrium, assuming [MATH] are uniform across all nodes, and therefore: [EQUATION]', '1905.00734-2-35-1': 'Hence, under disease-free condition where [MATH] and [MATH], Equation [REF] becomes: [EQUATION]', '1905.00734-2-35-2': 'Equation ([REF]) leads to two solutions: [MATH] or [MATH].', '1905.00734-2-36-0': ""We can now obtain [MATH] for the global dynamics in this model by using the Next Generation Approach, where [MATH] is given by the most dominant eigenvalue (or 'spatial radius' [MATH]) of [MATH], where [MATH] and [MATH] are [MATH] matrices, representing the 'new infections' and 'cases removed or transferred from the infected class', respectively in the disease free condition [CITATION]."", '1905.00734-2-36-1': 'As a result, [MATH] is determined as follows (see Appendix [REF] for detailed derivation): [EQUATION] where [EQUATION] while [EQUATION]', '1905.00734-2-36-2': 'Assuming [MATH] for all nodes, we can derive [MATH] and [MATH] as follows: [EQUATION] where [MATH] is a [MATH] identity matrix.', '1905.00734-2-37-0': 'Using Proposition 1, [MATH] can be simplified as: [EQUATION]', '1905.00734-2-37-1': 'We can now prove the following simple but useful proposition.', '1905.00734-2-38-0': 'Proposition 2.', '1905.00734-2-38-1': 'Matrix [MATH], defined as follows: [EQUATION] is a Markov matrix.', '1905.00734-2-39-0': 'Proof: We assume all nodes have the same population [MATH].', '1905.00734-2-39-1': 'Then Equation ([REF]) can be reduced to: [EQUATION]', '1905.00734-2-39-2': 'Without loss of generality, substituting Equation ([REF]) into Equation ([REF]) and expanding entries in the first column yields the first column [MATH] of matrix [MATH]: [EQUATION]', '1905.00734-2-39-3': 'Continuing with column 1, the column sum is: [EQUATION]', '1905.00734-2-39-4': 'Similarly, the sums of each column are equal to 1, with all entries being non-negative population fractions.', '1905.00734-2-39-5': 'Hence, [MATH] is a Markov matrix.', '1905.00734-2-40-0': 'As a Markov matrix, [MATH] always has the most dominant eigenvalue of unity.', '1905.00734-2-40-1': 'All other eigenvalues are smaller than unity in absolute value [CITATION].', '1905.00734-2-41-0': 'Now the next generation matrix [MATH] can be obtained as follows: [EQUATION]', '1905.00734-2-41-1': 'From Proposition 2 and Equation ([REF]), [MATH] can be obtained as: [EQUATION]', '1905.00734-2-41-2': 'Noting Proposition 1, there are two cases: [MATH] and [MATH].', '1905.00734-2-41-3': 'When nobody vaccinates ([MATH]), the entire population remains susceptible, which reduces model ([REF]) to a canonical SIR model without vaccination intervention and [MATH] returns to [MATH].', '1905.00734-2-41-4': 'On the other hand, if the whole population is vaccinated ([MATH]), the fraction of susceptible population only depends on the vaccine failure rate [MATH], resulting in: [EQUATION]', '1905.00734-2-41-5': 'Clearly, fully effective vaccination ([MATH]) would prohibit disease spread ([MATH]).', '1905.00734-2-41-6': 'Partially effective vaccination could potentially suppress disease transmission, or even eradicate disease spread if [MATH].', '1905.00734-2-41-7': 'If all vaccinations fail ([MATH]), model [REF] concurs with a canonical SIR model without vaccination, which also corresponds to a special case in Equation [REF] where [MATH].', '1905.00734-2-42-0': '## Vaccination available to the entire susceptible class', '1905.00734-2-43-0': 'If the vaccination opportunity is expanded to the entire susceptible class, including newborns and adults, the following model is proposed based on model ([REF]): [EQUATION]', '1905.00734-2-44-0': 'Model ([REF]) has a disease-free equilibrium [MATH]: [EQUATION] while [EQUATION]', '1905.00734-2-44-1': 'Substituting [MATH] using Equation ([REF]) yields [EQUATION] where [MATH] is a [MATH] identity matrix.', '1905.00734-2-45-0': 'The next generation matrix [MATH] can then be obtained as: [EQUATION]', '1905.00734-2-45-1': 'Using Proposition 2, [MATH] can be obtained as: [EQUATION]', '1905.00734-2-45-2': 'When nobody vaccinates ([MATH]), model ([REF]) concurs with a canonical SIR model with [MATH] reducing to [MATH].', '1905.00734-2-45-3': 'When the vaccination attains the full coverage in the population ([MATH]), the magnitude of [MATH] depends on the vaccine failure rate [MATH]: [EQUATION]', '1905.00734-2-45-4': 'Equation [REF] reduces to [MATH] if all vaccines fail ([MATH]).', '1905.00734-2-45-5': 'Conversely, if all vaccines are effective [MATH], Equation [REF] yields [EQUATION]', '1905.00734-2-45-6': 'Given that [MATH], [MATH] is well below the critical threshold: [EQUATION]', '1905.00734-2-45-7': 'If [MATH], by comparing Equations ([REF]) and ([REF]), we note that [MATH] becomes smaller, provided [MATH], that is: [EQUATION]', '1905.00734-2-46-0': '## Vaccination available to the entire susceptible class with committed vaccine recipients', '1905.00734-2-47-0': 'We now consider the existence of committed vaccine recipients, [MATH], as a fraction of individuals who would choose to vaccinate regardless of payoff assessment [CITATION] ([MATH]).', '1905.00734-2-47-1': 'We assume that committed vaccine recipients are also exposed to vaccination failure rate [MATH] and are distributed uniformly across all nodes.', '1905.00734-2-47-2': 'It is also important to point out that the fraction of committed vaccine recipients is constant over time.', '1905.00734-2-47-3': 'However, they can still affect vaccination decision for those who are not vaccinated, and consequently, contribute to the rate of change of the vaccinated fraction [MATH].', '1905.00734-2-47-4': 'Model ([REF]) can be further extended to reflect these considerations, as follows: [EQUATION]', '1905.00734-2-48-0': 'Model ([REF]) has a disease-free equilibrium: [EQUATION] while [EQUATION]', '1905.00734-2-48-1': 'Substituting [MATH], using Equation [REF], yields: [EQUATION] where [MATH] is a [MATH] identity matrix.', '1905.00734-2-49-0': 'The next generation matrix [MATH] can now be obtained as: [EQUATION]', '1905.00734-2-49-1': 'Proposition 3 At the disease free equilibrium, [MATH] has two solutions [MATH] or [MATH] if [MATH], [MATH] and [MATH] are uniform across all nodes.', '1905.00734-2-50-0': 'Proof: In analogy with Proposition 1, with committed vaccine recipients, at the disease free equilibrium, [MATH] and [MATH] are uniform across all nodes, and therefore: [EQUATION]', '1905.00734-2-50-1': 'Hence, under disease-free condition where [MATH] and [MATH], Equation [REF] becomes: [EQUATION]', '1905.00734-2-50-2': 'Equation [REF] leads to two solutions: [MATH] or [MATH].', '1905.00734-2-51-0': '[MATH] can be obtained for this case as: [EQUATION]', '1905.00734-2-51-1': 'If nobody chooses to vaccinate ([MATH]), [MATH] can be reduced to [MATH].', '1905.00734-2-51-2': 'Conversely, if the entire population is vaccinated [MATH], Equation ([REF]) reduces to equation ([REF]), and [MATH] is purely dependent on the vaccine failure rate [MATH].', '1905.00734-2-52-0': '## Model parametrization', '1905.00734-2-53-0': 'The proposed models aim to simulate a scenario of a generic childhood disease (e.g., measles) where life-long full immunity is acquired after effective vaccination.', '1905.00734-2-53-1': 'The vaccination failure rate, [MATH], is set as the probability of ineffective vaccination, to showcase the influence of unsuccessful vaccination on the global epidemic dynamics.', '1905.00734-2-53-2': 'In reality, the vaccination for Measles-Mumps-Rubella (MMR) is highly effective: for example, in Australia, an estimated 96% is successful in conferring immunity [CITATION].', '1905.00734-2-54-0': 'The population flux matrix [MATH] is derived from the network topology, in which each entry represents the connectivity between two nodes: if two nodes are not connected, [MATH]; if two nodes are connected, [MATH] is randomly assigned in the range of [MATH].', '1905.00734-2-54-1': 'The population influx into a node [MATH] within a day is represented by the column sum [MATH] in flux matrix [MATH].', '1905.00734-2-55-0': 'The parameters used for all simulations are summarized in Table [REF].', '1905.00734-2-55-1': 'Same initial conditions are applied to all nodes.', '1905.00734-2-55-2': 'Parameters [MATH] and [MATH] are calibrated based on values used in [CITATION].', '1905.00734-2-56-0': 'Our simulations were carried out on two networks:', '1905.00734-2-57-0': 'Each of these networks was used in conjunction with the following three models of vaccination behaviours:', '1905.00734-2-58-0': 'In the pilot case of 3-node network, two network topologies are studied: an isolated 3-node network where all residents remain in their residential nodes without travelling to other nodes (equivalent to models proposed in [CITATION]), and a fully connected network where the residents at each node commute to the other two nodes, and the population fractions commuting are symmetric and uniformly distributed (Figure [REF]).', '1905.00734-2-59-0': 'We then consider a suburb-network modelled as an Erdos-Renyi random graph (with number of nodes [MATH]) with average degree [MATH], in order to study how an expansive travelling pattern affects the global epidemic dynamics.', '1905.00734-2-59-1': 'Other topologies, such as lattice, scale-free and small-world networks, can easily be substituted here, though in this study our focus is on an Erdos-Renyi random graphs.', '1905.00734-2-60-0': '# Results', '1905.00734-2-61-0': '## 3-node network', '1905.00734-2-62-0': '### Vaccinating newborns only', '1905.00734-2-63-0': 'When there is no population mobility, we observe three distinctive equilibria (Figure [REF]; dotted lines): a pure non-vaccinating equilibrium (where [MATH], representing the final condition, and [MATH]), a mixed equilibrium (where [MATH], [MATH]), and stable limit cycles (where [MATH], [MATH]).', '1905.00734-2-63-1': 'This observation is in qualitative agreement with the vaccinating dynamics reported by [CITATION].', '1905.00734-2-64-0': 'When commuting is allowed, individuals commute to different nodes, and their decision will no longer rely on the single source of information (i.e., the disease prevalence in their residential node) but will also depend on the disease prevalence at their destination.', '1905.00734-2-64-1': 'As a consequence, the three distinctive equilibria are affected in different ways (Figure [REF]; solid lines).', '1905.00734-2-64-2': 'The amplitude of the stable limit cycles at [MATH] is reduced as a result of the reduced disease prevalence.', '1905.00734-2-64-3': 'It takes comparatively longer (compare the dotted lines and solid lines in Figure [REF]) to converge to the pure non-vaccinating equilibrium at [MATH], and to the endemic, mixed equilibrium at [MATH] with high amplitude of oscillation at the start of the epidemic spread.', '1905.00734-2-64-4': 'As [MATH] can be interpreted as the responsiveness of vaccinating behaviour to the disease prevalence [CITATION], if individuals are sufficiently responsive (i.e., [MATH] is high), the overall epidemic is suppressed more due to population mobility (and the ensuing imitation), as evidenced by smaller prevalence peaks.', '1905.00734-2-64-5': 'Conversely, if individuals are insensitive to the prevalence change (i.e., [MATH] is low), epidemic dynamics with equal population mobility may appear to be more volatile at the start, but the converged levels of both prevalence peak and vaccine uptake remain unchanged in comparison to the case where there is no population mobility.', '1905.00734-2-65-0': 'We further investigated how the vaccination failure rate [MATH] affects the global epidemic dynamics.', '1905.00734-2-65-1': 'If, for example, only a half of the vaccine administered is effective ([MATH]), as shown in Figure [REF] (b-d), the infection peaks arrive sooner, for all values of [MATH].', '1905.00734-2-65-2': 'As a result of having the earlier infection peaks, the individuals respond to the breakout and choose to vaccinate sooner, causing the vaccine uptake to rise.', '1905.00734-2-65-3': 'This seemingly counter-intuitive behaviour has also been reported in [CITATION].', '1905.00734-2-65-4': ""When [MATH], the prevalence peaks take longer to develop and the extended period gives individuals an illusion that there may not be an epidemic breakout, and consequently encourages 'free-riding' behaviour."", '1905.00734-2-65-5': 'In the case where half of the vaccinations fail, epidemic breaks out significantly earlier at lower peaks, an observation that is beneficial to encourage responsive individuals to choose to vaccinate.', '1905.00734-2-65-6': 'Although some (in this case half) of the vaccines fail, a sufficiently high vaccine uptake still curbs prevalence peaks and shortens the length of breakout period.', '1905.00734-2-65-7': 'In terms of the final vaccination uptake, the vaccine failure rate predominantly affects behaviours of responsive individuals (when [MATH] is high, such as [MATH]).', '1905.00734-2-65-8': 'In this case, instead of the stable limit cycles observed when [MATH], an endemic, mixed equilibrium is reached when the vaccination failure rate is significant.', '1905.00734-2-65-9': 'Final vaccination uptake is impacted little by vaccination failure rate when individuals are insensitive to changes in disease prevalence (when [MATH] is lower in value, such as [MATH] or [MATH]).', '1905.00734-2-65-10': 'These observations are illustrated in Figure [REF].', '1905.00734-2-66-0': '### Vaccinating the entire susceptible class', '1905.00734-2-67-0': ""If (voluntary) vaccination is offered to the susceptible class regardless of the age, the initial condition of [MATH] represents the scenario that the vast majority of population are immunized to begin with, and the epidemic would not breakout until the false sense of security provided by the temporary 'herd immunity' settles in."", '1905.00734-2-67-1': 'This feature is observed in Figure [REF] (a)-(b): the vaccination coverage continues to drop at the start of the epidemic breakout, indicating that individuals, regardless of their responsiveness towards the prevalence change, exploit the temporary herd immunity until an infection peak emerges.', '1905.00734-2-67-2': 'Since vaccination is available to the entire susceptible class, a small increase in [MATH] could help suppresses disease prevalence.', '1905.00734-2-67-3': 'However, when vaccination is partially effective (i.e., [MATH]), the peaks in vaccine uptake, [MATH], are no longer an accurate reflection of the actual vaccination coverage, and such peaks therefore may not be sufficient to adequately suppress infection peaks.', '1905.00734-2-67-4': 'It can also be observed that vaccinating the susceptible class encourages non-vaccinating behaviours due to the perception of herd immunity, and therefore pushes the epidemic towards an endemic equilibrium, particularly when sensitivity to prevalence is relatively high ( mid and high [MATH]), as the responsive individuals would react promptly to the level of disease prevalence, altering their vaccination decisions.', '1905.00734-2-67-5': 'However, no substantial impact is observed on those individuals who are insensitive to changes in the disease prevalence ( low [MATH]).', '1905.00734-2-68-0': '## Erdos-Renyi random network of 3000 nodes', '1905.00734-2-69-0': 'We now present the simulation results on a much larger Erdo-Renyi random network of [MATH] nodes, which more realistically reflects the size of a modern city and its commuting patterns.', '1905.00734-2-69-1': ""It was observed by previous studies [CITATION] that such a larger system requires a higher vaccination coverage to achieve herd immunity, and thus curbs 'free-riding' behaviours more effectively."", '1905.00734-2-70-0': '### Vaccinating newborns only', '1905.00734-2-71-0': 'In this case, three distinct equilibria are observed (as shown in Figure [REF] (a)) for three values of [MATH]: a pure non-vaccinating equilibrium at [MATH] and two endemic mixed equilibria at [MATH] and [MATH] respectively, replacing the stable limit cycles at high [MATH] previously observed in the 3-node case.', '1905.00734-2-71-1': 'As expected, when individuals are more responsive to disease prevalence (higher [MATH]), they are more likely to get vaccinated.', '1905.00734-2-71-2': 'The expansive travelling pattern also somewhat elevates the global vaccination coverage level (compared to the 3-node case), particularly in the case that the individuals show a moderate level of responsiveness to prevalence ([MATH]), and shortens the convergence time to reach the equilibrium.', '1905.00734-2-71-3': 'However, for those who are insensitive to the changes in disease prevalence ([MATH]), the more expansive commuting presented by the larger network does not affect either the level of voluntary vaccination, or the convergence time in global dynamics - these individuals remain unvaccinated as in the case of the smaller network.', '1905.00734-2-71-4': 'It is also found that the vaccination coverage is very sensitive to the disease prevalence change at the larger network since the disease prevalence peaks are notably lower than the 3-node case counterparts.', '1905.00734-2-71-5': 'If half of the vaccines administered are unsuccessful ([MATH]), the impact of early peaks on global dynamics is magnified in a larger network (Figure [REF] (b-d)), leading to a shorter convergence time, although the final equilibria are hardly affected compared to the 3-node case as shown in figure [REF].', '1905.00734-2-72-0': 'We also compared the epidemic dynamics in terms of the adjusted imitation rate, represented by the parameter [MATH], as shown in figure [REF].', '1905.00734-2-72-1': 'Recall that the value of imitation rate used in our simulations, unless otherwise stated, is [MATH] as reported in Table 1.', '1905.00734-2-72-2': 'Here we presents results where this value of imitation rate is compared with a much smaller value of [MATH].', '1905.00734-2-72-3': 'In populations where the individuals imitate more quickly (i.e., higher [MATH]), the oscillations in prevalence and vaccination dynamics arrive quicker with larger amplitude, although the converged vaccine uptake and disease prevalence are similar for higher and lower [MATH].', '1905.00734-2-72-4': 'For a higher [MATH], the convergence to equilibria is quicker.', '1905.00734-2-72-5': 'These observations are consistent with results reported by earlier studies [CITATION].', '1905.00734-2-72-6': 'A smaller value of [MATH]) also altered behaviours of those individuals who are insensitive to prevalence change.', '1905.00734-2-72-7': 'Instead of converging to the pure non-vaccination equilibrium, the dynamics converge to a mixed endemic state, meaning that when imitation rate is very low, some individuals would still choose to vaccinate even when they are not very sensitive to disease prevalence.', '1905.00734-2-73-0': 'For such a large network, vaccinating behaviour of individuals also depends on the weighted degree (number and weight of connections) of the node (suburb) in which they live.', '1905.00734-2-73-1': 'This is illustrated in Figure [REF].', '1905.00734-2-73-2': 'For individuals living in highly connected nodes, there are many commuting destinations, allowing access to a broad spectrum of information on the local disease prevalence.', '1905.00734-2-73-3': ""Therefore, it is not surprising that we found that individuals living in 'hubs' are more likely to get vaccinated, particularly when the population is sensitive to prevalence ([MATH] is high), as shown in Figure [REF] (a)."", '1905.00734-2-73-4': 'This observation is in accordance with previous studies [CITATION].', '1905.00734-2-73-5': 'Note that in in Figure [REF] (a), nodes with degree [MATH] are grouped into one bin to represent hubs, as the frequency count of these nodes is extremely low.', '1905.00734-2-73-6': 'Note also that there is a positive correlation between the number of degrees and the volume of population influx as measured by a sum of proportions from the source nodes (Figure [REF] (b)), indicating that a highly connected suburb has greater population influx on a daily basis.', '1905.00734-2-74-0': '### Vaccinating entire susceptible class', '1905.00734-2-75-0': 'If (voluntary) vaccination is offered to the susceptible class regardless of age, we found that the global epidemic dynamics converge quicker compared to the similar scenario in the 3-node case.', '1905.00734-2-75-1': 'Only one predominant infection peak is observed, corresponding to the high infection prevalence around year 25, as shown in figure [REF].', '1905.00734-2-75-2': 'Oscillations of small magnitude are observed for both vaccine uptake and disease prevalence at later time-steps for middle or high values of [MATH] (as shown in insets of figure [REF]).', '1905.00734-2-75-3': 'These findings also largely hold if half of the vaccines administered are ineffective (i.e., [MATH]), although the predominant prevalence peak and the corresponding vaccination peak around year 25 are both higher than their counterparts observed for the case where [MATH].', '1905.00734-2-76-0': 'We also found that employing a small fraction of committed vaccine recipients prevents major epidemic by curbing disease prevalence.', '1905.00734-2-76-1': ""Such a finding holds for all [MATH] (i.e., regardless population's responsiveness towards disease prevalence) as the magnitude of prevalence is too small to make a non-vaccinated individual switch to the vaccination strategy (Figure [REF])."", '1905.00734-2-77-0': '# Discussion and conclusion', '1905.00734-2-78-0': 'We presented a series of SIR-network models with imitation dynamics, aiming to model scenarios where individuals commute between their residence and work across a network (e.g., where each node represents a suburb).', '1905.00734-2-78-1': 'These models are able to capture diverse travelling patterns (i.e., reflecting local connectivity of suburbs), and different vaccinating behaviours affecting the global vaccination uptake and epidemic dynamics.', '1905.00734-2-78-2': 'We also analytically derived expressions for the reproductive number [MATH] for the considered SIR-network models, and demonstrated how epidemics may evolve over time in these models.', '1905.00734-2-79-0': 'We showed that the stable oscillations in the vaccinating dynamics are only likely to occur either when there is no population mobility across nodes, or only with limited commuting destinations.', '1905.00734-2-79-1': 'We observed that, compared to the case where vaccination is only provided to newborns, if vaccination is provided to the entire susceptible class, higher disease prevalence and more volatile oscillations in vaccination uptake are observed (particularly in populations which are relatively responsive to the changes in disease prevalence).', '1905.00734-2-79-2': 'A more expansive travelling pattern simulated in a larger network encourages the attractor dynamics and the convergence to the endemic, mixed equilibria, again if individuals are sufficiently responsive towards the changes in the disease prevalence.', '1905.00734-2-79-3': 'If individuals are insensitive to the prevalence, they are hardly affected by different vaccinating models and remain as unvaccinated individuals, although the existence of committed vaccine recipients noticeably delays the convergence to the non-vaccinating equilibrium.', '1905.00734-2-79-4': 'The presented models highlight the important role of committed vaccine recipients in actively reducing [MATH] and disease prevalence, strongly contributing to eradicating an epidemic spread.', '1905.00734-2-79-5': 'Similarly conclusions have been reached previously [CITATION], and our results extend these to imitation dynamics in SIR-network models.', '1905.00734-2-80-0': 'Previous studies drew an important conclusion that highly connected hubs play a key role in containing infections as they are more likely to get vaccinated due to the higher risk of infection in social networks [CITATION].', '1905.00734-2-80-1': 'Our results complement this finding by showing that a higher fraction of individuals who reside in highly connected suburbs choose to vaccinate compared to those living in relatively less connected suburbs.', '1905.00734-2-80-2': 'These hubs, often recognized as business districts, also have significantly higher population influx as the destination for many commuters from other suburbs.', '1905.00734-2-80-3': 'Therefore, it is important for policy makers to leverage these job hubs in promoting vaccination campaigns and public health programs.', '1905.00734-2-81-0': 'Overall, our results demonstrate that, in order to encourage vaccination behaviour and shorten the course of epidemic, policy makers of vaccination campaigns need to carefully orchestrate the following three factors: ensuring a number of committed vaccine recipients in each suburb, utilizing the vaccinating tendency of well-connected suburbs, and increasing individual awareness towards the prevalence change.', '1905.00734-2-82-0': 'There are several avenues to extend this work further.', '1905.00734-2-82-1': 'This work assumes that the individuals from different nodes (suburbs) only differ in their travelling patterns, using the same epidemic and behaviour parameters for individuals from all nodes.', '1905.00734-2-82-2': 'Also, the same [MATH] and [MATH] are used for all nodes, by assuming that individuals living in all nodes are equally responsive towards disease prevalence and imitation.', '1905.00734-2-82-3': ""Realistically, greater heterogeneity can be implemented by establishing context-specific [MATH] and imitating parameters for factors such as the local population density, community size [CITATION], and suburbs' level of connectivity."", '1905.00734-2-82-4': 'For example, residents living in highly connected suburbs may be more alert to changes in disease prevalence, and adopt imitation behaviours more quickly.', '1905.00734-2-82-5': 'Different network topologies can also be used, particularly scale-free networks [CITATION] where a small number of nodes have a large number of links each.', '1905.00734-2-82-6': 'These highly connected nodes are a better representation of suburbs with extremely high population influx (e.g., central business districts and job hubs).', '1905.00734-2-82-7': 'It may also be instructive to translate the risk perception of vaccination and infection into tangible measures to demonstrate the aggregate social cost of an epidemic breakout, and help policy makers to visualize the cost effectiveness of different vaccinating strategies and estimate the financial burden for public health care.', '1905.00734-2-82-8': 'This study can also be extended by calibrating the network to scale-free networks [CITATION] while incorporating information theory [CITATION].', '1905.00734-2-82-9': 'Networks from real-life connectivity and demographic are also of strong interest to mimic a breakout in a targeted region [CITATION], so that a model of vaccination campaign can demonstrate a reduction in the severity of epidemic.', '1905.00734-2-82-10': 'These considerations could advance this study to more accurately reflect contagion dynamics in urban environment, and provide further insights to public health planning.', '1905.00734-2-83-0': '# Appendix', '1905.00734-2-84-0': '## Next Generation Operator Approach', '1905.00734-2-85-0': 'The proposed model ([REF]) can be seen as a finely categorized SIR deterministic model, so that each health compartment (Susceptible, Infected and Recovered) has [MATH] sub-classes.', '1905.00734-2-85-1': 'Let us denote [MATH] to represent [MATH] infected host compartments, and [MATH] to represent [MATH] other host compartments consisting of susceptible compartments [MATH] and recovered compartments [MATH].', '1905.00734-2-85-2': '[EQUATION] where [MATH] and [MATH].', '1905.00734-2-86-0': '[MATH] is the rate at which new infection enters infected compartments and [MATH] is the transfer of individuals out of or into the infected compartments.', '1905.00734-2-87-0': 'When close to disease-free equilibrium where [MATH], the model can be linearized to: [EQUATION] where [MATH] and [MATH]', '1905.00734-2-88-0': 'The next generation matrix, [MATH], is then given by: [EQUATION] where each entry [MATH] represents the expected number of secondary cases which an infected individual imposes on the rest of the compartments.', '1905.00734-2-88-1': '[MATH] and [MATH] are given as follows: [EQUATION]', '1905.00734-2-88-2': ""The basic reproduction number [MATH] is given by the most dominant eigenvalue (or 'spatial radius' [MATH]) of [MATH] [CITATION], and therefore: [EQUATION]"", '1905.00734-2-89-0': '## Declaration of interest', '1905.00734-2-90-0': 'The authors declare that they have no competing interests.'}","[['1905.00734-1-20-0', '1905.00734-2-20-0'], ['1905.00734-1-20-1', '1905.00734-2-20-1'], ['1905.00734-1-20-2', 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'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1905.00734,"{'1905.00734-3-0-0': 'We present a series of SIR-network models, extended with a game-theoretic treatment of imitation dynamics which result from regular population mobility across residential and work areas and the ensuing interactions.', '1905.00734-3-0-1': 'Each considered SIR-network model captures a class of vaccination behaviours influenced by epidemic characteristics, interaction topology, and imitation dynamics.', '1905.00734-3-0-2': 'Our focus is the eventual vaccination coverage, produced under voluntary vaccination schemes, in response to these varying factors.', '1905.00734-3-0-3': 'Using the next generation matrix method, we analytically derive and compare expressions for the basic reproduction number [MATH] for the proposed SIR-network models.', '1905.00734-3-0-4': 'Furthermore, we simulate the epidemic dynamics over time for the considered models, and show that if individuals are sufficiently responsive towards the changes in the disease prevalence, then the more expansive travelling patterns encourage convergence to the endemic, mixed equilibria.', '1905.00734-3-0-5': 'On the contrary, if individuals are insensitive to changes in the disease prevalence, we find that they tend to remain unvaccinated in all the studied models.', '1905.00734-3-0-6': 'Our results concur with earlier studies in showing that residents from highly connected residential areas are more likely to get vaccinated.', '1905.00734-3-0-7': 'We also show that the existence of the individuals committed to receiving vaccination reduces [MATH] and delays the disease prevalence, and thus is essential to containing epidemics.', '1905.00734-3-1-0': '# Introduction', '1905.00734-3-2-0': 'Vaccination has long been established as a powerful tool in managing and controlling infectious diseases by providing protection to susceptible individuals [CITATION].', '1905.00734-3-2-1': 'With a sufficiently high vaccination coverage, the probability of the remaining unvaccinated individuals getting infected reduces significantly.', '1905.00734-3-2-2': 'However, such systematic programs by necessity may limit the freedom of choice of individuals.', '1905.00734-3-2-3': 'When vaccination programs are made voluntary, the vaccination uptake declines as a result of individuals choosing not to vaccinate, as seen in Britain in 2003 when the vaccination program for Measles-Mumps-Rubella (MMR) was made voluntary [CITATION].', '1905.00734-3-2-4': ""Parents feared possible complications from vaccination [CITATION] and hoped to exploit the 'herd immunity' by assuming other parents would choose to vaccinate their children."", '1905.00734-3-2-5': 'Such hopes did not materialize precisely because other parents also thought similarly.', '1905.00734-3-3-0': ""Under a voluntary vaccination policy an individual's decision depends on several factors: the social influence from one's social network, the risk perception of vaccination, and the risk perception of infection, in terms of both likelihood and impact [CITATION]."", '1905.00734-3-3-1': 'This decision-making is often modelled using game theory [CITATION], by allowing individuals to compare the cost of vaccination and the potential cost of non-vaccination (in terms of the likelihood and impact of infection), and adopting imitation dynamics in modelling the influence of social interactions.', '1905.00734-3-3-2': 'However, many of the earlier studies [CITATION] were based on the key assumption of a well-mixed homogeneous population where each individual is assumed to have an equal chance of making contact with any other individual in the population.', '1905.00734-3-3-3': 'This population assumption is rather unrealistic as large populations are often diverse with varying levels of interactions.', '1905.00734-3-3-4': 'To address this, more recent studies [CITATION] model populations as complex networks where each individual, represented by a node, has a finite set of contacts, represented by links [CITATION].', '1905.00734-3-4-0': 'It has been shown that there is a critical cost threshold in the vaccination cost above which the likelihood of vaccination drops steeply [CITATION].', '1905.00734-3-4-1': 'In addition, highly connected individuals were shown to be more likely to choose to be vaccinated as they perceive themselves to be at a greater risk of being infected due to high exposure within the community.', '1905.00734-3-4-2': 'In modelling large-scale epidemics, the population size (i.e., number of nodes) can easily reach millions of individuals, interacting in a complex way.', '1905.00734-3-4-3': 'The challenge, therefore, is to extend epidemic modelling not only with the individual vaccination decision-making, but also capture diverse interaction patterns encoded within a network.', '1905.00734-3-4-4': 'Such an integration has not yet been formalized, motivating our study.', '1905.00734-3-4-5': 'Furthermore, once an integrated model is developed, a specific challenge is to consider how the vaccination imitation dynamics developing across a network affects the basic reproduction number, [MATH].', '1905.00734-3-4-6': 'This question forms our second main objective.', '1905.00734-3-5-0': 'One relevant approach partially addressing this objective is offered by the multi-suburb (or multi-city) SIR-network model [CITATION] where each node represents a neighbourhood (or city) with a certain number of residents.', '1905.00734-3-5-1': 'The daily commute of individuals between two neighbourhoods is modelled along the network link connecting the two nodes, which allows to quantify the disease spread between individuals from different neighbourhoods.', '1905.00734-3-5-2': 'Ultimately, this model captures meta-population dynamics in a multi-suburb setting affected by an epidemic spread at a greater scale.', '1905.00734-3-5-3': 'To date, these models have not yet considered intervention (e.g., vaccination) options, and the corresponding social interaction across the populations.', '1905.00734-3-6-0': 'To incorporate the imitation dynamics, modelled game-theoretically, within a multi-suburb model representing mobility, we propose a series of integrated vaccination-focused SIR-network models.', '1905.00734-3-6-1': 'This allows us to systematically analyze how travelling patterns affect the voluntary vaccination uptake due to adoption of different imitation choices, in a large distributed population.', '1905.00734-3-6-2': 'The developed models use an increasingly complex set of vaccination strategies.', '1905.00734-3-6-3': 'Thus, our specific contribution is the study of the vaccination uptake, driven by imitation dynamics under a voluntary vaccination scheme, using an SIR model on a complex network representing a multi-suburb environment, within which the individuals commute between residential and work areas.', '1905.00734-3-7-0': 'In section [REF], we present the models and methods associated with this study: in particular, we present analytical derivations of the basic reproduction number [MATH] for the proposed models using the Next Generation Operator Approach [CITATION], and carry out a comparative analysis of [MATH] across these models.', '1905.00734-3-7-1': 'In section [REF], we simulate the epidemic and vaccination dynamics over time using the proposed models in different network settings, including a pilot case of a 3-node network, and a 3000-node Erdos-Renyi random network.', '1905.00734-3-7-2': 'The comparison of the produced results across different models and settings is carried out for the larger network, with the focus on the emergent attractor dynamics, in terms of the proportion of vaccinated individuals.', '1905.00734-3-7-3': 'Particularly, we analyze the sensitivity of the individual strategies (whether to vaccinate or not) to the levels of disease prevalence produced by the different considered models.', '1905.00734-3-7-4': 'Section [REF] concludes the study with a brief discussion of the importance of these results.', '1905.00734-3-8-0': '# Technical background', '1905.00734-3-9-0': '## Basic reproduction number [MATH]', '1905.00734-3-10-0': 'The basic reproduction number [MATH] is defined as the number of secondary infections produced by an infected individual in an otherwise completely susceptible population [CITATION].', '1905.00734-3-10-1': 'It is well-known that [MATH] is an epidemic threshold, with the disease dying out as [MATH], or becoming endemic as [MATH] [CITATION].', '1905.00734-3-10-2': 'This finding strictly holds only in deterministic models with infinite population [CITATION].', '1905.00734-3-10-3': 'The topology of the underlying contact network is known to affect the epidemic threshold [CITATION].', '1905.00734-3-10-4': 'Many disease transmission models have shown important correlations between [MATH] and the key epidemic characteristics (e.g., disease prevalence, attack rates, etc.) [CITATION].', '1905.00734-3-10-5': 'In addition, [MATH] has been considered as a critical threshold for phase transitions studied with methods of statistical physics or information theory [CITATION].', '1905.00734-3-11-0': '## Vaccination model with imitation dynamics', '1905.00734-3-12-0': 'Imitation dynamics, a process by which individuals copy the strategy of other individuals, is widely used to model vaccinating behaviours incorporated with SIR models.', '1905.00734-3-12-1': ""The model proposed in [CITATION] applied game theory to represent parents' decision-making about whether to get their newborns vaccinated against childhood disease (e.g., measles, mumps, rubella, pertussis)."", '1905.00734-3-12-2': ""In this model, individuals are in a homogeneously mixing population, and susceptible individuals have two 'pure strategies' regarding vaccination: to vaccinate or not to vaccinate."", '1905.00734-3-12-3': 'The non-vaccination decision can change to the vaccination decision at a particular sampling rate, however the vaccination decision cannot be changed to non-vaccination.', '1905.00734-3-12-4': 'Individuals adopt one of these strategies by weighing up their perceived payoffs, measured by the probability of morbidity from vaccination, and the risk of infection respectively.', '1905.00734-3-12-5': 'The payoff for vaccination ([MATH]) is given as, [EQUATION] and the payoff for non-vaccination ([MATH]), measured as the risk of infection, is given as [EQUATION] where [MATH] is the perceived risk of morbidity from vaccination, [MATH] is the perceived risk of morbidity from non-vaccination (i.e., infection), [MATH] is the current disease prevalence in population fraction at time [MATH], and [MATH] is the sensitivity to disease prevalence [CITATION].', '1905.00734-3-13-0': 'From Equation [REF] and [REF], it can be seen that the payoff for vaccinated individuals is a simple constant, and the payoff for unvaccinated individuals is proportional to the extent of epidemic prevalence (i.e., the severity of the disease).', '1905.00734-3-14-0': 'It is assumed that life-long immunity is granted with effective vaccination.', '1905.00734-3-14-1': 'If one individual decides to vaccinate, s/he cannot revert back to the unvaccinated status.', '1905.00734-3-14-2': 'To make an individual switch to a vaccinating strategy, the payoff gain, [MATH] must be positive ([MATH]).', '1905.00734-3-14-3': ""Let [MATH] denote the relative proportion of vaccinated individuals, and assume that an unvaccinated individual can sample a strategy from a vaccinated individual at certain rate [MATH], and can switch to the 'vaccinate strategy' with probability [MATH]."", '1905.00734-3-14-4': 'Then the time evolution of [MATH] can be defined as: [EQUATION] where [MATH] can be interpreted as the combined imitation rate that individuals use to sample and imitate strategies of other individuals.', '1905.00734-3-14-5': 'Equation [REF] can be rewritten so that [MATH] only depends on two parameters: [EQUATION] where [MATH] and [MATH], assuming that the risk of morbidity from vaccination, [MATH], and the risk of morbidity from non-vaccination (i.e., morbidity from infection), [MATH], are constant during the course of epidemics.', '1905.00734-3-14-6': ""Hence, [MATH] is the adjusted imitation rate, and [MATH] measures individual's responsiveness towards the changes in disease prevalence."", '1905.00734-3-15-0': 'The disease prevalence [MATH] is determined from a simple SIR model [CITATION] with birth and death that divides the population into three health compartments: [MATH] (susceptible), [MATH] (infected) and [MATH] (recovered).', '1905.00734-3-15-1': 'If a susceptible individual encounters an infected individual, s/he contracts the infection with transmission rate [MATH], and thus progresses to the infected compartment, while an infected individual recovers at rate [MATH].', '1905.00734-3-15-2': 'It is assumed that the birth and death rate are equal, denoted by [MATH] [CITATION].', '1905.00734-3-15-3': 'Individuals in all compartments die at an equal rate, and all newborns are added to the susceptible compartment unless and until they are vaccinated.', '1905.00734-3-15-4': 'Vaccinated newborns are moved to the recovered class directly, with a rate [MATH].', '1905.00734-3-15-5': 'Therefore, the dynamics are modelled as follows [CITATION]: [EQUATION] where [MATH], [MATH], and [MATH] represent the proportion of susceptible, infected, and recovered individuals respectively (that is, [MATH]), and [MATH] represents the proportion of vaccinated individuals within the susceptible class at a given time.', '1905.00734-3-15-6': 'The variables [MATH], [MATH], and [MATH], as well as [MATH], are time-dependent, but for simplicity, we omit subscript [MATH] for these and other time-dependent variables.', '1905.00734-3-16-0': 'Model [REF] predicts oscillations in vaccine uptake in response to changes in disease prevalence.', '1905.00734-3-16-1': 'It is found that oscillations are more likely to occur when individuals imitate each other more quickly (i.e., higher [MATH]).', '1905.00734-3-16-2': 'The oscillations are observed to be more volatile when people alter their vaccinating behaviours promptly in response to changes in disease prevalence (i.e., higher [MATH]).', '1905.00734-3-16-3': 'Overall, higher [MATH] or [MATH] produce stable limit cycles at greater amplitude.', '1905.00734-3-16-4': 'Conversely, when individuals are insensitive to changes in disease prevalence (i.e., low [MATH]), or imitate at a slower rate (i.e., low [MATH]), the resultant vaccinating dynamics converge to equilibrium [CITATION].', '1905.00734-3-17-0': '## Multi-city epidemic model', '1905.00734-3-18-0': 'Several multi-city epidemic models [CITATION] consider a network of suburbs as [MATH] nodes, in which each node [MATH] represents a suburb, where [MATH].', '1905.00734-3-18-1': 'If two nodes [MATH] are linked, a fraction of population living in node [MATH] can travel to node [MATH] and back (commute from [MATH] to [MATH], for example for work) on a daily basis.', '1905.00734-3-18-2': 'The connectivity of suburbs and the fraction of people commuting between them are represented by the population flux matrix [MATH] whose entries represent the fraction of population daily commuting from [MATH] to [MATH], [MATH] (Equation [REF]).', '1905.00734-3-18-3': 'Note that [MATH], thus [MATH] is not a symmetric matrix.', '1905.00734-3-18-4': 'Figure [REF] shows an example of such travel dynamics.', '1905.00734-3-19-0': '[EQUATION]', '1905.00734-3-19-1': 'Trivially, each row in [MATH], representing proportions of the population of a suburb commuting to various destinations, has to sum up to unity: [EQUATION]', '1905.00734-3-19-2': ""However, the column sum measures the population influx to a suburb during a day, and therefore depends on the node's connectivity."", '1905.00734-3-19-3': 'Column summation in [MATH] does not necessarily equal to unity.', '1905.00734-3-20-0': 'It is also necessary to differentiate between present population [MATH] and native population [MATH] in node [MATH] on a particular day.', '1905.00734-3-20-1': '[MATH] is used as a normalizing factor to account for the differences in population flux for different nodes: [EQUATION]', '1905.00734-3-20-2': 'Assuming the disease transmission parameters are identical in all cities, the standard incidence can be expressed as [CITATION]: [EQUATION] where [MATH] and [MATH] denote respectively the number of infective and susceptible individuals in city [MATH], and [MATH] is the average number of contacts in city [MATH] per unit time.', '1905.00734-3-20-3': 'Incorporating travelling pattern defined by [MATH], yields [CITATION]: [EQUATION]', '1905.00734-3-20-4': 'The double summation term in this expression captures the infection at suburb [MATH] due to the encounters between the residents from suburb [MATH] and the residents from suburb [MATH] could be any suburb including [MATH] or [MATH]) occurring at suburb [MATH], provided that suburbs [MATH] are connected with non-zero population flux entries in [MATH].', '1905.00734-3-21-0': '[MATH] can be derived using the Next Generation Approach (see Appendix [REF] for more details).', '1905.00734-3-21-1': 'For example, for a multi-city SEIRS model with 4 compartments (susceptible, exposed, infective, recovered), introduced in [CITATION], and a special case where the contact rate [MATH] is set to 1, while [MATH] is identical across all cities, [MATH] has the following analytical solution: [EQUATION] where [MATH] and [MATH] denote the average lifetime, exposed period and infective period.', '1905.00734-3-21-2': 'It can be seen that this solution concurs with a classical SEIRS model with no mobility.', '1905.00734-3-21-3': 'When [MATH], Equation [REF] reduces to [MATH], being a solution for a canonical SIR model.', '1905.00734-3-21-4': 'Furthermore, if the population dynamics is not considered (i.e., [MATH]), Equation [REF] can be further reduced to [MATH], agreeing with [CITATION].', '1905.00734-3-22-0': '# Methods', '1905.00734-3-23-0': '## Integrated model', '1905.00734-3-24-0': 'Expanding on models described in sections [REF] and [REF], we bring together population mobility and vaccinating behaviours in a network setting, and propose three extensions within an integrated vaccination-focused SIR-network model:', '1905.00734-3-25-0': 'In this study, we only consider vaccinations that confer lifelong immunity, mostly related to childhood diseases such as measles, mumps, rubella and pertussis.', '1905.00734-3-25-1': 'Vaccinations against such diseases are often administered for individuals at a young age, implying that newborns (more precisely, their parents) often face the vaccination decision.', '1905.00734-3-25-2': 'These are captured by the first extension above.', '1905.00734-3-25-3': 'However, during an outbreak, adults may face the vaccination decision themselves if they were not previously vaccinated.', '1905.00734-3-25-4': 'For example, diseases which are not included in formal childhood vaccination programs, such as smallpox [CITATION], may present vaccination decisions to the entire susceptible population.', '1905.00734-3-25-5': 'These are captured by the second extension.', '1905.00734-3-25-6': 'The third extension introduces a small fraction of susceptible population as committed vaccine recipients, i.e., those who would always choose to vaccinate regardless.', '1905.00734-3-25-7': 'The purpose of these committed vaccine recipients is to demonstrate how successful immunization education campaigns could affect vaccination dynamics.', '1905.00734-3-26-0': 'Our models divide the population into many homogeneous groups [CITATION], based on their residential suburbs.', '1905.00734-3-26-1': 'Within each suburb (i.e., node), residents are treated as a homogeneous population.', '1905.00734-3-26-2': 'It is assumed that the total population within each node is conserved over time.', '1905.00734-3-26-3': ""The dynamics of epidemic and vaccination at each node, are referred to as 'local dynamics'."", '1905.00734-3-26-4': ""The aggregate epidemic dynamics of the entire network can be obtained by summing over all nodes, producing 'global dynamics'."", '1905.00734-3-27-0': ""We model the 'imitation dynamics' based on the individual's travelling pattern and the connectivity of their node defined by Equation [REF]."", '1905.00734-3-27-1': 'For any node [MATH], let [MATH] denote the fraction of vaccinated individuals in the susceptible class in suburb [MATH].', '1905.00734-3-27-2': 'On a particular day, unvaccinated susceptible individuals [MATH] commute to suburb [MATH] and encounter vaccinated individuals from node [MATH] (where [MATH] may be [MATH] itself, [MATH] or any other nodes) and imitate their strategy.', '1905.00734-3-27-3': ""However, this 'imitation' is only applicable in the case of a non-vaccinated individual imitating the strategy of a vaccinated individual (that is, deciding to vaccinate), since the opposite 'imitation' cannot occur."", '1905.00734-3-27-4': ""Therefore, in our model, every time a non-vaccinated person from [MATH] comes in contact with a vaccinated person from [MATH], they imitate the 'vaccinate' strategy if the perceived payoff outweighs the non-vaccination strategy."", '1905.00734-3-28-0': 'Following Equation [REF] proposed in [CITATION], the rate of change of the proportion of vaccinated individuals in [MATH], that is, [MATH], over time can be expressed by: [EQUATION]', '1905.00734-3-28-1': 'The players of the vaccination game are the parents, deciding whether or not to vaccinate their children using the information of the disease prevalence collected from their daily commute.', '1905.00734-3-28-2': 'For example, a susceptible individual residing at node [MATH] and working at node [MATH] uses the local disease prevalence at node [MATH] to decide whether to vaccinate or not.', '1905.00734-3-28-3': 'If such a susceptible individual is infected, that individual will be counted towards the local epidemic prevalence at node [MATH].', '1905.00734-3-29-0': 'We measure each health compartment as a proportion of the population.', '1905.00734-3-29-1': ""Hence, we define a ratio, [MATH], as the ratio between present population [MATH] and the 'native' population [MATH] in node [MATH] on a particular day as: [EQUATION]"", '1905.00734-3-29-2': 'Our main focus is to investigate the effects of vaccinating behaviours on the global epidemic dynamics.', '1905.00734-3-29-3': 'To do so, we vary three parameters:', '1905.00734-3-30-0': 'While [MATH] and [MATH] have had been considered in [CITATION], we introduce a new parameter [MATH] as the vaccination failure rate, [MATH] to consider the cases where the vaccination may not be fully effective.', '1905.00734-3-30-1': 'The imitation component (Equation [REF]) is common in all model extensions.', '1905.00734-3-30-2': 'While the epidemic compartments vary depending on the specific extension, Equation [REF] is used consistently to model the relative rate of change in vaccination behaviours.', '1905.00734-3-31-0': '## Vaccination available to newborns only', '1905.00734-3-32-0': 'Model ([REF]) captures the scenario when vaccination opportunities are provided to newborns only: [EQUATION]', '1905.00734-3-32-1': 'Unvaccinated newborns and newborns with unsuccessful vaccination [MATH] stay in the susceptible class.', '1905.00734-3-32-2': 'Successfully vaccinated newborns, on the other hand, move to the recovered class [MATH].', '1905.00734-3-32-3': 'Other population dynamics across health compartments follow the model ([REF]).', '1905.00734-3-33-0': 'Since [MATH], model ([REF]) can be reduced to: [EQUATION]', '1905.00734-3-34-0': 'Model ([REF]) has a disease-free equilibrium (disease-free initial condition) [MATH] for which [EQUATION]', '1905.00734-3-34-1': 'Proposition 1.', '1905.00734-3-34-2': 'At the disease free equilibrium, [MATH] has two solutions ([MATH] or [MATH]) if [MATH] and [MATH] are uniform across all nodes.', '1905.00734-3-35-0': 'Proof: At the disease free equilibrium, assuming [MATH] are uniform across all nodes, and therefore: [EQUATION]', '1905.00734-3-35-1': 'Hence, under disease-free condition where [MATH] and [MATH], Equation [REF] becomes: [EQUATION]', '1905.00734-3-35-2': 'Equation ([REF]) leads to two solutions: [MATH] or [MATH].', '1905.00734-3-36-0': ""We can now obtain [MATH] for the global dynamics in this model by using the Next Generation Approach, where [MATH] is given by the most dominant eigenvalue (or 'spatial radius' [MATH]) of [MATH], where [MATH] and [MATH] are [MATH] matrices, representing the 'new infections' and 'cases removed or transferred from the infected class', respectively in the disease free condition [CITATION]."", '1905.00734-3-36-1': 'As a result, [MATH] is determined as follows (see Appendix [REF] for detailed derivation): [EQUATION] where [EQUATION] while [EQUATION]', '1905.00734-3-36-2': 'Assuming [MATH] for all nodes, we can derive [MATH] and [MATH] as follows: [EQUATION] where [MATH] is a [MATH] identity matrix.', '1905.00734-3-37-0': 'Using Proposition 1, [MATH] can be simplified as: [EQUATION]', '1905.00734-3-37-1': 'We can now prove the following simple but useful proposition.', '1905.00734-3-38-0': 'Proposition 2.', '1905.00734-3-38-1': 'Matrix [MATH], defined as follows: [EQUATION] is a Markov matrix.', '1905.00734-3-39-0': 'Proof: We assume all nodes have the same population [MATH].', '1905.00734-3-39-1': 'Then Equation ([REF]) can be reduced to: [EQUATION]', '1905.00734-3-39-2': 'Without loss of generality, substituting Equation ([REF]) into Equation ([REF]) and expanding entries in the first column yields the first column [MATH] of matrix [MATH]: [EQUATION]', '1905.00734-3-39-3': 'Continuing with column 1, the column sum is: [EQUATION]', '1905.00734-3-39-4': 'Similarly, the sums of each column are equal to 1, with all entries being non-negative population fractions.', '1905.00734-3-39-5': 'Hence, [MATH] is a Markov matrix.', '1905.00734-3-40-0': 'As a Markov matrix, [MATH] always has the most dominant eigenvalue of unity.', '1905.00734-3-40-1': 'All other eigenvalues are smaller than unity in absolute value [CITATION].', '1905.00734-3-41-0': 'Now the next generation matrix [MATH] can be obtained as follows: [EQUATION]', '1905.00734-3-41-1': 'From Proposition 2 and Equation ([REF]), [MATH] can be obtained as: [EQUATION]', '1905.00734-3-41-2': 'Noting Proposition 1, there are two cases: [MATH] and [MATH].', '1905.00734-3-41-3': 'When nobody vaccinates ([MATH]), the entire population remains susceptible, which reduces model ([REF]) to a canonical SIR model without vaccination intervention and [MATH] returns to [MATH].', '1905.00734-3-41-4': 'On the other hand, if the whole population is vaccinated ([MATH]), the fraction of susceptible population only depends on the vaccine failure rate [MATH], resulting in: [EQUATION]', '1905.00734-3-41-5': 'Clearly, fully effective vaccination ([MATH]) would prohibit disease spread ([MATH]).', '1905.00734-3-41-6': 'Partially effective vaccination could potentially suppress disease transmission, or even eradicate disease spread if [MATH].', '1905.00734-3-41-7': 'If all vaccinations fail ([MATH]), model [REF] concurs with a canonical SIR model without vaccination, which also corresponds to a special case in Equation [REF] where [MATH].', '1905.00734-3-42-0': '## Vaccination available to the entire susceptible class', '1905.00734-3-43-0': 'If the vaccination opportunity is expanded to the entire susceptible class, including newborns and adults, the following model is proposed based on model ([REF]): [EQUATION]', '1905.00734-3-44-0': 'Model ([REF]) has a disease-free equilibrium [MATH]: [EQUATION] while [EQUATION]', '1905.00734-3-44-1': 'Substituting [MATH] using Equation ([REF]) yields [EQUATION] where [MATH] is a [MATH] identity matrix.', '1905.00734-3-45-0': 'The next generation matrix [MATH] can then be obtained as: [EQUATION]', '1905.00734-3-45-1': 'Using Proposition 2, [MATH] can be obtained as: [EQUATION]', '1905.00734-3-45-2': 'When nobody vaccinates ([MATH]), model ([REF]) concurs with a canonical SIR model with [MATH] reducing to [MATH].', '1905.00734-3-45-3': 'When the vaccination attains the full coverage in the population ([MATH]), the magnitude of [MATH] depends on the vaccine failure rate [MATH]: [EQUATION]', '1905.00734-3-45-4': 'Equation [REF] reduces to [MATH] if all vaccines fail ([MATH]).', '1905.00734-3-45-5': 'Conversely, if all vaccines are effective [MATH], Equation [REF] yields [EQUATION]', '1905.00734-3-45-6': 'Given that [MATH], [MATH] is well below the critical threshold: [EQUATION]', '1905.00734-3-45-7': 'If [MATH], by comparing Equations ([REF]) and ([REF]), we note that [MATH] becomes smaller, provided [MATH], that is: [EQUATION]', '1905.00734-3-46-0': '## Vaccination available to the entire susceptible class with committed vaccine recipients', '1905.00734-3-47-0': 'We now consider the existence of committed vaccine recipients, [MATH], as a fraction of individuals who would choose to vaccinate regardless of payoff assessment [CITATION] ([MATH]).', '1905.00734-3-47-1': 'We assume that committed vaccine recipients are also exposed to vaccination failure rate [MATH] and are distributed uniformly across all nodes.', '1905.00734-3-47-2': 'It is also important to point out that the fraction of committed vaccine recipients is constant over time.', '1905.00734-3-47-3': 'However, they can still affect vaccination decision for those who are not vaccinated, and consequently, contribute to the rate of change of the vaccinated fraction [MATH].', '1905.00734-3-47-4': 'Model ([REF]) can be further extended to reflect these considerations, as follows: [EQUATION]', '1905.00734-3-48-0': 'Model ([REF]) has a disease-free equilibrium: [EQUATION] while [EQUATION]', '1905.00734-3-48-1': 'Substituting [MATH], using Equation [REF], yields: [EQUATION] where [MATH] is a [MATH] identity matrix.', '1905.00734-3-49-0': 'The next generation matrix [MATH] can now be obtained as: [EQUATION]', '1905.00734-3-49-1': 'Proposition 3 At the disease free equilibrium, [MATH] has two solutions [MATH] or [MATH] if [MATH], [MATH] and [MATH] are uniform across all nodes.', '1905.00734-3-50-0': 'Proof: In analogy with Proposition 1, with committed vaccine recipients, at the disease free equilibrium, [MATH] and [MATH] are uniform across all nodes, and therefore: [EQUATION]', '1905.00734-3-50-1': 'Hence, under disease-free condition where [MATH] and [MATH], Equation [REF] becomes: [EQUATION]', '1905.00734-3-50-2': 'Equation [REF] leads to two solutions: [MATH] or [MATH].', '1905.00734-3-51-0': '[MATH] can be obtained for this case as: [EQUATION]', '1905.00734-3-51-1': 'If nobody chooses to vaccinate ([MATH]), [MATH] can be reduced to [MATH].', '1905.00734-3-51-2': 'Conversely, if the entire population is vaccinated [MATH], Equation ([REF]) reduces to equation ([REF]), and [MATH] is purely dependent on the vaccine failure rate [MATH].', '1905.00734-3-52-0': '## Model parametrization', '1905.00734-3-53-0': 'The proposed models aim to simulate a scenario of a generic childhood disease (e.g., measles) where life-long full immunity is acquired after effective vaccination.', '1905.00734-3-53-1': 'The vaccination failure rate, [MATH], is set as the probability of ineffective vaccination, to showcase the influence of unsuccessful vaccination on the global epidemic dynamics.', '1905.00734-3-53-2': 'In reality, the vaccination for Measles-Mumps-Rubella (MMR) is highly effective: for example, in Australia, an estimated 96% is successful in conferring immunity [CITATION].', '1905.00734-3-54-0': 'The population flux matrix [MATH] is derived from the network topology, in which each entry represents the connectivity between two nodes: if two nodes are not connected, [MATH]; if two nodes are connected, [MATH] is randomly assigned in the range of [MATH].', '1905.00734-3-54-1': 'The population influx into a node [MATH] within a day is represented by the column sum [MATH] in flux matrix [MATH].', '1905.00734-3-55-0': 'The parameters used for all simulations are summarized in Table [REF].', '1905.00734-3-55-1': 'Same initial conditions are applied to all nodes.', '1905.00734-3-55-2': 'Parameters [MATH] and [MATH] are calibrated based on values used in [CITATION].', '1905.00734-3-56-0': 'Our simulations were carried out on two networks:', '1905.00734-3-57-0': 'Each of these networks was used in conjunction with the following three models of vaccination behaviours:', '1905.00734-3-58-0': 'In the pilot case of 3-node network, two network topologies are studied: an isolated 3-node network where all residents remain in their residential nodes without travelling to other nodes (equivalent to models proposed in [CITATION]), and a fully connected network where the residents at each node commute to the other two nodes, and the population fractions commuting are symmetric and uniformly distributed (Figure [REF]).', '1905.00734-3-59-0': 'We then consider a suburb-network modelled as an Erdos-Renyi random graph (with number of nodes [MATH]) with average degree [MATH], in order to study how an expansive travelling pattern affects the global epidemic dynamics.', '1905.00734-3-59-1': 'Other topologies, such as lattice, scale-free and small-world networks, can easily be substituted here, though in this study our focus is on an Erdos-Renyi random graphs.', '1905.00734-3-60-0': '# Results', '1905.00734-3-61-0': '## 3-node network', '1905.00734-3-62-0': '### Vaccinating newborns only', '1905.00734-3-63-0': 'When there is no population mobility, we observe three distinctive equilibria (Figure [REF]; dotted lines): a pure non-vaccinating equilibrium (where [MATH], representing the final condition, and [MATH]), a mixed equilibrium (where [MATH], [MATH]), and stable limit cycles (where [MATH], [MATH]).', '1905.00734-3-63-1': 'This observation is in qualitative agreement with the vaccinating dynamics reported by [CITATION].', '1905.00734-3-64-0': 'When commuting is allowed, individuals commute to different nodes, and their decision will no longer rely on the single source of information (i.e., the disease prevalence in their residential node) but will also depend on the disease prevalence at their destination.', '1905.00734-3-64-1': 'As a consequence, the three distinctive equilibria are affected in different ways (Figure [REF]; solid lines).', '1905.00734-3-64-2': 'The amplitude of the stable limit cycles at [MATH] is reduced as a result of the reduced disease prevalence.', '1905.00734-3-64-3': 'It takes comparatively longer (compare the dotted lines and solid lines in Figure [REF]) to converge to the pure non-vaccinating equilibrium at [MATH], and to the endemic, mixed equilibrium at [MATH] with high amplitude of oscillation at the start of the epidemic spread.', '1905.00734-3-64-4': 'As [MATH] can be interpreted as the responsiveness of vaccinating behaviour to the disease prevalence [CITATION], if individuals are sufficiently responsive (i.e., [MATH] is high), the overall epidemic is suppressed more due to population mobility (and the ensuing imitation), as evidenced by smaller prevalence peaks.', '1905.00734-3-64-5': 'Conversely, if individuals are insensitive to the prevalence change (i.e., [MATH] is low), epidemic dynamics with equal population mobility may appear to be more volatile at the start, but the converged levels of both prevalence peak and vaccine uptake remain unchanged in comparison to the case where there is no population mobility.', '1905.00734-3-65-0': 'We further investigated how the vaccination failure rate [MATH] affects the global epidemic dynamics.', '1905.00734-3-65-1': 'If, for example, only a half of the vaccine administered is effective ([MATH]), as shown in Figure [REF] (b-d), the infection peaks arrive sooner, for all values of [MATH].', '1905.00734-3-65-2': 'As a result of having the earlier infection peaks, the individuals respond to the breakout and choose to vaccinate sooner, causing the vaccine uptake to rise.', '1905.00734-3-65-3': 'This seemingly counter-intuitive behaviour has also been reported in [CITATION].', '1905.00734-3-65-4': ""When [MATH], the prevalence peaks take longer to develop and the extended period gives individuals an illusion that there may not be an epidemic breakout, and consequently encourages 'free-riding' behaviour."", '1905.00734-3-65-5': 'In the case where half of the vaccinations fail, epidemic breaks out significantly earlier at lower peaks, an observation that is beneficial to encourage responsive individuals to choose to vaccinate.', '1905.00734-3-65-6': 'Although some (in this case half) of the vaccines fail, a sufficiently high vaccine uptake still curbs prevalence peaks and shortens the length of breakout period.', '1905.00734-3-65-7': 'In terms of the final vaccination uptake, the vaccine failure rate predominantly affects behaviours of responsive individuals (when [MATH] is high, such as [MATH]).', '1905.00734-3-65-8': 'In this case, instead of the stable limit cycles observed when [MATH], an endemic, mixed equilibrium is reached when the vaccination failure rate is significant.', '1905.00734-3-65-9': 'Final vaccination uptake is impacted little by vaccination failure rate when individuals are insensitive to changes in disease prevalence (when [MATH] is lower in value, such as [MATH] or [MATH]).', '1905.00734-3-65-10': 'These observations are illustrated in Figure [REF].', '1905.00734-3-66-0': '### Vaccinating the entire susceptible class', '1905.00734-3-67-0': ""If (voluntary) vaccination is offered to the susceptible class regardless of the age, the initial condition of [MATH] represents the scenario that the vast majority of population are immunized to begin with, and the epidemic would not breakout until the false sense of security provided by the temporary 'herd immunity' settles in."", '1905.00734-3-67-1': 'This feature is observed in Figure [REF] (a)-(b): the vaccination coverage continues to drop at the start of the epidemic breakout, indicating that individuals, regardless of their responsiveness towards the prevalence change, exploit the temporary herd immunity until an infection peak emerges.', '1905.00734-3-67-2': 'Since vaccination is available to the entire susceptible class, a small increase in [MATH] could help suppresses disease prevalence.', '1905.00734-3-67-3': 'However, when vaccination is partially effective (i.e., [MATH]), the peaks in vaccine uptake, [MATH], are no longer an accurate reflection of the actual vaccination coverage, and such peaks therefore may not be sufficient to adequately suppress infection peaks.', '1905.00734-3-67-4': 'It can also be observed that vaccinating the susceptible class encourages non-vaccinating behaviours due to the perception of herd immunity, and therefore pushes the epidemic towards an endemic equilibrium, particularly when sensitivity to prevalence is relatively high ( mid and high [MATH]), as the responsive individuals would react promptly to the level of disease prevalence, altering their vaccination decisions.', '1905.00734-3-67-5': 'However, no substantial impact is observed on those individuals who are insensitive to changes in the disease prevalence ( low [MATH]).', '1905.00734-3-68-0': '## Erdos-Renyi random network of 3000 nodes', '1905.00734-3-69-0': 'We now present the simulation results on a much larger Erdo-Renyi random network of [MATH] nodes, which more realistically reflects the size of a modern city and its commuting patterns.', '1905.00734-3-69-1': ""It was observed by previous studies [CITATION] that such a larger system requires a higher vaccination coverage to achieve herd immunity, and thus curbs 'free-riding' behaviours more effectively."", '1905.00734-3-70-0': '### Vaccinating newborns only', '1905.00734-3-71-0': 'In this case, three distinct equilibria are observed (as shown in Figure [REF] (a)) for three values of [MATH]: a pure non-vaccinating equilibrium at [MATH] and two endemic mixed equilibria at [MATH] and [MATH] respectively, replacing the stable limit cycles at high [MATH] previously observed in the 3-node case.', '1905.00734-3-71-1': 'As expected, when individuals are more responsive to disease prevalence (higher [MATH]), they are more likely to get vaccinated.', '1905.00734-3-71-2': 'The expansive travelling pattern also somewhat elevates the global vaccination coverage level (compared to the 3-node case), particularly in the case that the individuals show a moderate level of responsiveness to prevalence ([MATH]), and shortens the convergence time to reach the equilibrium.', '1905.00734-3-71-3': 'However, for those who are insensitive to the changes in disease prevalence ([MATH]), the more expansive commuting presented by the larger network does not affect either the level of voluntary vaccination, or the convergence time in global dynamics - these individuals remain unvaccinated as in the case of the smaller network.', '1905.00734-3-71-4': 'It is also found that the vaccination coverage is very sensitive to the disease prevalence change at the larger network since the disease prevalence peaks are notably lower than the 3-node case counterparts.', '1905.00734-3-71-5': 'If half of the vaccines administered are unsuccessful ([MATH]), the impact of early peaks on global dynamics is magnified in a larger network (Figure [REF] (b-d)), leading to a shorter convergence time, although the final equilibria are hardly affected compared to the 3-node case as shown in figure [REF].', '1905.00734-3-72-0': 'We also compared the epidemic dynamics in terms of the adjusted imitation rate, represented by the parameter [MATH], as shown in figure [REF].', '1905.00734-3-72-1': 'Recall that the value of imitation rate used in our simulations, unless otherwise stated, is [MATH] as reported in Table 1.', '1905.00734-3-72-2': 'Here we presents results where this value of imitation rate is compared with a much smaller value of [MATH].', '1905.00734-3-72-3': 'In populations where the individuals imitate more quickly (i.e., higher [MATH]), the oscillations in prevalence and vaccination dynamics arrive quicker with larger amplitude, although the converged vaccine uptake and disease prevalence are similar for higher and lower [MATH].', '1905.00734-3-72-4': 'For a higher [MATH], the convergence to equilibria is quicker.', '1905.00734-3-72-5': 'These observations are consistent with results reported by earlier studies [CITATION].', '1905.00734-3-72-6': 'A smaller value of [MATH]) also altered behaviours of those individuals who are insensitive to prevalence change.', '1905.00734-3-72-7': 'Instead of converging to the pure non-vaccination equilibrium, the dynamics converge to a mixed endemic state, meaning that when imitation rate is very low, some individuals would still choose to vaccinate even when they are not very sensitive to disease prevalence.', '1905.00734-3-73-0': 'For such a large network, vaccinating behaviour of individuals also depends on the weighted degree (number and weight of connections) of the node (suburb) in which they live.', '1905.00734-3-73-1': 'This is illustrated in Figure [REF].', '1905.00734-3-73-2': 'For individuals living in highly connected nodes, there are many commuting destinations, allowing access to a broad spectrum of information on the local disease prevalence.', '1905.00734-3-73-3': ""Therefore, it is not surprising that we found that individuals living in 'hubs' are more likely to get vaccinated, particularly when the population is sensitive to prevalence ([MATH] is high), as shown in Figure [REF] (a)."", '1905.00734-3-73-4': 'This observation is in accordance with previous studies [CITATION].', '1905.00734-3-73-5': 'Note that in in Figure [REF] (a), nodes with degree [MATH] are grouped into one bin to represent hubs, as the frequency count of these nodes is extremely low.', '1905.00734-3-73-6': 'Note also that there is a positive correlation between the number of degrees and the volume of population influx as measured by a sum of proportions from the source nodes (Figure [REF] (b)), indicating that a highly connected suburb has greater population influx on a daily basis.', '1905.00734-3-74-0': '### Vaccinating entire susceptible class', '1905.00734-3-75-0': 'If (voluntary) vaccination is offered to the susceptible class regardless of age, we found that the global epidemic dynamics converge quicker compared to the similar scenario in the 3-node case.', '1905.00734-3-75-1': 'Only one predominant infection peak is observed, corresponding to the high infection prevalence around year 25, as shown in figure [REF].', '1905.00734-3-75-2': 'Oscillations of small magnitude are observed for both vaccine uptake and disease prevalence at later time-steps for middle or high values of [MATH] (as shown in insets of figure [REF]).', '1905.00734-3-75-3': 'These findings also largely hold if half of the vaccines administered are ineffective (i.e., [MATH]), although the predominant prevalence peak and the corresponding vaccination peak around year 25 are both higher than their counterparts observed for the case where [MATH].', '1905.00734-3-76-0': 'We also found that employing a small fraction of committed vaccine recipients prevents major epidemic by curbing disease prevalence.', '1905.00734-3-76-1': ""Such a finding holds for all [MATH] (i.e., regardless population's responsiveness towards disease prevalence) as the magnitude of prevalence is too small to make a non-vaccinated individual switch to the vaccination strategy (Figure [REF])."", '1905.00734-3-77-0': '# Discussion and conclusion', '1905.00734-3-78-0': 'We presented a series of SIR-network models with imitation dynamics, aiming to model scenarios where individuals commute between their residence and work across a network (e.g., where each node represents a suburb).', '1905.00734-3-78-1': 'These models are able to capture diverse travelling patterns (i.e., reflecting local connectivity of suburbs), and different vaccinating behaviours affecting the global vaccination uptake and epidemic dynamics.', '1905.00734-3-78-2': 'We also analytically derived expressions for the reproductive number [MATH] for the considered SIR-network models, and demonstrated how epidemics may evolve over time in these models.', '1905.00734-3-79-0': 'We showed that the stable oscillations in the vaccinating dynamics are only likely to occur either when there is no population mobility across nodes, or only with limited commuting destinations.', '1905.00734-3-79-1': 'We observed that, compared to the case where vaccination is only provided to newborns, if vaccination is provided to the entire susceptible class, higher disease prevalence and more volatile oscillations in vaccination uptake are observed (particularly in populations which are relatively responsive to the changes in disease prevalence).', '1905.00734-3-79-2': 'A more expansive travelling pattern simulated in a larger network encourages the attractor dynamics and the convergence to the endemic, mixed equilibria, again if individuals are sufficiently responsive towards the changes in the disease prevalence.', '1905.00734-3-79-3': 'If individuals are insensitive to the prevalence, they are hardly affected by different vaccinating models and remain as unvaccinated individuals, although the existence of committed vaccine recipients noticeably delays the convergence to the non-vaccinating equilibrium.', '1905.00734-3-79-4': 'The presented models highlight the important role of committed vaccine recipients in actively reducing [MATH] and disease prevalence, strongly contributing to eradicating an epidemic spread.', '1905.00734-3-79-5': 'Similarly conclusions have been reached previously [CITATION], and our results extend these to imitation dynamics in SIR-network models.', '1905.00734-3-80-0': 'Previous studies drew an important conclusion that highly connected hubs play a key role in containing infections as they are more likely to get vaccinated due to the higher risk of infection in social networks [CITATION].', '1905.00734-3-80-1': 'Our results complement this finding by showing that a higher fraction of individuals who reside in highly connected suburbs choose to vaccinate compared to those living in relatively less connected suburbs.', '1905.00734-3-80-2': 'These hubs, often recognized as business districts, also have significantly higher population influx as the destination for many commuters from other suburbs.', '1905.00734-3-80-3': 'Therefore, it is important for policy makers to leverage these job hubs in promoting vaccination campaigns and public health programs.', '1905.00734-3-81-0': 'Overall, our results demonstrate that, in order to encourage vaccination behaviour and shorten the course of epidemic, policy makers of vaccination campaigns need to carefully orchestrate the following three factors: ensuring a number of committed vaccine recipients in each suburb, utilizing the vaccinating tendency of well-connected suburbs, and increasing individual awareness towards the prevalence change.', '1905.00734-3-82-0': 'There are several avenues to extend this work further.', '1905.00734-3-82-1': 'This work assumes that the individuals from different nodes (suburbs) only differ in their travelling patterns, using the same epidemic and behaviour parameters for individuals from all nodes.', '1905.00734-3-82-2': 'Also, the same [MATH] and [MATH] are used for all nodes, by assuming that individuals living in all nodes are equally responsive towards disease prevalence and imitation.', '1905.00734-3-82-3': ""Realistically, greater heterogeneity can be implemented by establishing context-specific [MATH] and imitating parameters for factors such as the local population density, community size [CITATION], and suburbs' level of connectivity."", '1905.00734-3-82-4': 'For example, residents living in highly connected suburbs may be more alert to changes in disease prevalence, and adopt imitation behaviours more quickly.', '1905.00734-3-82-5': 'Different network topologies can also be used, particularly scale-free networks [CITATION] where a small number of nodes have a large number of links each.', '1905.00734-3-82-6': 'These highly connected nodes are a better representation of suburbs with extremely high population influx (e.g., central business districts and job hubs).', '1905.00734-3-82-7': 'It may also be instructive to translate the risk perception of vaccination and infection into tangible measures to demonstrate the aggregate social cost of an epidemic breakout, and help policy makers to visualize the cost effectiveness of different vaccinating strategies and estimate the financial burden for public health care.', '1905.00734-3-82-8': 'This study can also be extended by calibrating the network to scale-free networks [CITATION] while incorporating information theory [CITATION].', '1905.00734-3-82-9': 'Networks from real-life connectivity and demographic are also of strong interest to mimic a breakout in a targeted region [CITATION], so that a model of vaccination campaign can demonstrate a reduction in the severity of epidemic.', '1905.00734-3-82-10': 'These considerations could advance this study to more accurately reflect contagion dynamics in urban environment, and provide further insights to public health planning.', '1905.00734-3-83-0': '# Appendix', '1905.00734-3-84-0': '## Next Generation Operator Approach', '1905.00734-3-85-0': 'The proposed model ([REF]) can be seen as a finely categorized SIR deterministic model, so that each health compartment (Susceptible, Infected and Recovered) has [MATH] sub-classes.', '1905.00734-3-85-1': 'Let us denote [MATH] to represent [MATH] infected host compartments, and [MATH] to represent [MATH] other host compartments consisting of susceptible compartments [MATH] and recovered compartments [MATH].', '1905.00734-3-85-2': '[EQUATION] where [MATH] and [MATH].', '1905.00734-3-86-0': '[MATH] is the rate at which new infection enters infected compartments and [MATH] is the transfer of individuals out of or into the infected compartments.', '1905.00734-3-87-0': 'When close to disease-free equilibrium where [MATH], the model can be linearized to: [EQUATION] where [MATH] and [MATH]', '1905.00734-3-88-0': 'The next generation matrix, [MATH], is then given by: [EQUATION] where each entry [MATH] represents the expected number of secondary cases which an infected individual imposes on the rest of the compartments.', '1905.00734-3-88-1': '[MATH] and [MATH] are given as follows: [EQUATION]', '1905.00734-3-88-2': ""The basic reproduction number [MATH] is given by the most dominant eigenvalue (or 'spatial radius' [MATH]) of [MATH] [CITATION], and therefore: [EQUATION]"", '1905.00734-3-89-0': '## Declaration of interest', '1905.00734-3-90-0': 'The authors declare that they have no competing interests.'}",,
1805.06723,"{'1805.06723-1-0-0': 'Motivated by the randomized generation of slowly synchronizing automata, we study automata made of permutation letters and a merging letter of rank [MATH].', '1805.06723-1-0-1': 'We present a constructive randomized procedure to generate minimal synchronizing automata of that kind with (potentially) large alphabet size based on recent results on primitive sets of matrices.', '1805.06723-1-0-2': 'We report numerical results showing that our algorithm finds automata with much larger reset threshold than a mere uniform random generation and we present new families of automata with reset threshold of [MATH].', '1805.06723-1-0-3': 'We finally report theoretical results on randomized generation of primitive sets of matrices: a set of permutation matrices with a [MATH] entry changed into a [MATH] is primitive and has exponent of [MATH] with high probability in case of uniform random distribution and the same holds for a random set of binary matrices where each entry is set, independently, equal to [MATH] with probability [MATH] and equal to [MATH] with probability [MATH], when [MATH] as [MATH].', '1805.06723-1-1-0': '# Introduction', '1805.06723-1-2-0': 'An automaton [MATH] (see Section [REF] for definitions) is synchronizing if there exists a word that brings the automaton into a particular state, regardless of the initial one.', '1805.06723-1-2-1': 'The length of the shortest word of that kind is the reset threshold ([MATH]) of the automaton.', '1805.06723-1-2-2': 'Synchronizing automata appear in different research fields such as biocomputing [CITATION] or symbolic dynamics [CITATION], and they are often used as models of error-resistant systems [CITATION]; for a brief account on synchronizing automata we refer the reader to [CITATION].', '1805.06723-1-2-3': 'The importance of synchronizing automata also arises from one of the most longstanding open problems in this field, the Cerny conjecture, which states that any synchronizing automaton on [MATH] states has reset threshold of at most of [MATH].', '1805.06723-1-2-4': 'If it is true, the bound is sharp due to the existence of a family of [MATH]-letter automata attaining this value, family discovered by Cerny in [CITATION].', '1805.06723-1-2-5': 'Despite great effort, the best upper bound for the [MATH] known so far is [MATH], recently obtained by Szykula in [CITATION] and thereby beating the 30 years-standing upper bound of [MATH] found by Pin and Frankl in [CITATION].', '1805.06723-1-2-6': 'Better upper bounds have been obtained for certain families of automata and the search for automata attaining quadratic rt within these families have been the subject of several contributions in recent years.', '1805.06723-1-2-7': 'These results are (partly) summarized in Table [REF].', '1805.06723-1-2-8': 'Exhaustive search confirmed the conjecture for small values of [MATH] (see [CITATION], [CITATION]).', '1805.06723-1-3-0': 'The hunt for a possible counterexample to the conjecture turned out not to be an easy task as well; the search space is wide and calculating the [MATH] is computationally hard (see [CITATION]).', '1805.06723-1-3-1': 'Automata with reset thresholds close to [MATH], called extremal or slowly synchronizing automata, are also hard to detect and not so many families are known; Bondt et.', '1805.06723-1-3-2': 'al. [CITATION] make a thorough analysis of automata with small number of states and we recall, among others, the families found by Ananichev et al. [CITATION], by Gusev and Pribavkina [CITATION], by Kisielewicz and Szykula [CITATION] and by Dzyga et.', '1805.06723-1-3-3': 'al. [CITATION].', '1805.06723-1-3-4': 'These last two examples in particular, are some of the few examples of slowly synchronizing automata with more than two letters that can be found in the literature.', '1805.06723-1-3-5': 'Almost all the families of slowly synchronizing automata listed above are closely related to the Cerny automaton [MATH], where [MATH] is a cycle over [MATH] vertices and [MATH] fixes all the vertices but the [MATH]-th one, which is mapped to vertex [MATH]; indeed all these families present a letter that is a cycle over [MATH] vertices and the other letters have an action similar to the one of letter [MATH].', '1805.06723-1-3-6': 'As these examples seem to have a quite regular structure, it is natural to wonder whether a randomized procedure to generate automata could obtain less structured automata with possibly larger reset thresholds.', '1805.06723-1-3-7': ""This probabilistic approach can be rooted back to the work of Erdos in the 60's, where he developed a tool, called The Probabilistic Method, that permits to prove the existence of a structure with certain desired properties by defining a suitable probabilistic space in which to embed the problem; for an account on the probabilistic method we refer the reader to [CITATION]."", '1805.06723-1-3-8': 'The simplest way to randomly generate automata is to uniformly and independently sample [MATH] binary row-stochastic matrices: unfortunately, Berlinkov first proved in [CITATION] that two uniformly sampled random binary row-stochastic matrices synchronizes with high probability (i.e. the probability they form a synchronizing automaton tends to [MATH] as the matrix dimension tends to infinity), then Nicaud showed in [CITATION] that they also have reset threshold of order [MATH] with high probability; this overall implies that an uniformly sampled random automata of [MATH] letters has low reset threshold with high probability.', '1805.06723-1-3-9': 'This is also probably a reason why the majority of the literature focuses just on two-letter automata.', '1805.06723-1-3-10': 'We call minimal automaton an automaton that does not contain any proper synchronizing sub-automaton.', '1805.06723-1-3-11': 'Summarizing:', '1805.06723-1-4-0': 'With this motivation in place, our paper tackles the following questions:', '1805.06723-1-5-0': '[Q1] Is there a way to randomly generate minimal (slowly) synchronizing automata with more than two letters?', '1805.06723-1-5-1': '[Q2] Can we find some minimal automata families with more than two letters, quadratic reset threshold and that do not resemble the Cerny family?', '1805.06723-1-6-0': 'Our results.', '1805.06723-1-6-1': 'In this paper we give positive answers to both questions Q1 and Q2.', '1805.06723-1-6-2': 'For the first one, we rely on the concept of primitive set of matrices, introduced by Protasov and Voynov in [CITATION]: a finite set of matrices with nonnegative entries is said to be primitive if there exists a product of these matrices, with repetitions allowed, with all positive entries.', '1805.06723-1-6-3': 'The length of the shortest positive product of a primitive set [MATH] is called the exponent ([MATH]) of the set.', '1805.06723-1-6-4': 'Although the Protasov-Voynov primitivity has gained a lot of attention in different fields as in stochastic switching systems [CITATION] and consensus for discrete-time multi-agent systems [CITATION], we are interested in its connection with automata theory.', '1805.06723-1-6-5': 'In the following, we say that a matrix set is NZ if all its matrices have neither zero-rows nor zero-columns.', '1805.06723-1-7-0': 'Let [MATH] be a NZ-matrix set with nonnegative entries and, for any [MATH], let [MATH] be the binary matrix such that [MATH] if and only if [MATH], for every [MATH].', '1805.06723-1-7-1': 'The automaton associated to the set [MATH] is the set of all the binary row-stochastic matrices that are entrywise smaller than at least one matrix in [MATH].', '1805.06723-1-8-0': 'The following theorem summarizes two results proved by Blondel et.', '1805.06723-1-8-1': 'al. ([CITATION], Theorems 16-17) and a result proved by Gerencser et al. ([CITATION], Theorem 8):', '1805.06723-1-9-0': 'For any set [MATH] of NZ-matrices of size [MATH], if [MATH] is the automaton associated to [MATH] and [MATH] is the automaton associated to [MATH] then it holds that [MATH] is primitive if and only if [MATH] (equiv.', '1805.06723-1-9-1': '[MATH]) is synchronizing.', '1805.06723-1-9-2': 'If [MATH] is primitive, then it also holds that: [EQUATION]', '1805.06723-1-9-3': 'The above theorem will be extensively used throughout the paper.', '1805.06723-1-10-0': 'Here below a primitive set [MATH] and the automata [MATH] and [MATH] associated, respectively, to [MATH] and [MATH] in both their matrix and graph representations (Figure [REF]).', '1805.06723-1-11-0': '[MATH], [MATH], [MATH].', '1805.06723-1-11-1': '[MATH], [MATH], [MATH].', '1805.06723-1-12-0': 'Primitive sets can hence be used for generating automata and Equation ([REF]) tells us that the presence of a primitive set with large exponent implies the existence of an automata with large reset threshold; in particular the discovery of a primitive set [MATH] with [MATH] would disprove the Cerny conjecture.', '1805.06723-1-12-1': 'One advantage of using primitive sets is the Protasov-Voynov characterization theorem (see Theorem [REF] in Section [REF]) that describes a combinatorial property that a NZ-matrix set must have in order not to be primitive: by constructing a primitive set such that each of its proper subsets has this property, we can make it minimal.', '1805.06723-1-12-2': 'We decided to focus our attention on what we call perturbed permutation sets, i.e. sets made of permutation matrices (binary matrices having one and only one [MATH] in every row and every column) where a [MATH]-entry of one of these matrices is changed into a [MATH].', '1805.06723-1-12-3': 'These sets have many interesting properties:', '1805.06723-1-13-0': 'The characterization theorem for primitive sets and the above properties are the main ingredients of our randomized algorithm that finds minimal automata of [MATH] letters (and can eventually be generalized to [MATH] letters); to the best of our knowledge, this is the first time where a constructive procedure for finding minimal automata is presented.', '1805.06723-1-13-1': 'This is described in Section [REF] where numerical results are reported.', '1805.06723-1-13-2': 'The random construction let us also find new families of minimal [MATH]-letters automata of reset threshold [MATH], presented in section [REF], thus answering question Q2.', '1805.06723-1-13-3': 'Finally, in Section [REF] we extend a result on perturbed permutation sets obtained by Gonze et al. in [CITATION]: we show that a random perturbed permutation set is primitive with high probability for any matrix size [MATH] (and not just when [MATH] is a prime number as in [CITATION]) and that its exponent is of order [MATH] still with high probability.', '1805.06723-1-13-4': 'A further generalization is then presented for sets of random binary matrices: if each entry of each matrix is set to [MATH] with probability [MATH] and to [MATH] with probability [MATH], independently from each other, then the set is primitive and has exponent of order [MATH] with high probability for any [MATH] such that [MATH].', '1805.06723-1-14-0': '# Definitions and notation', '1805.06723-1-15-0': 'In this section we briefly go through some definitions and results that will be needed in the rest of the paper.', '1805.06723-1-16-0': 'We indicate with [MATH] the set [MATH] and with [MATH] the set of permutations over [MATH] elements; with a slight abuse of notation [MATH] will also denote the set of the [MATH] permutation matrices.', '1805.06723-1-17-0': 'A (complete deterministic finite) automaton [MATH] on [MATH] states can be defined as a set of [MATH] binary row-stochastic matrices [MATH] that are called the letters of the automaton.', '1805.06723-1-17-1': 'We say that [MATH] is synchronizing if there exist a product of the [MATH]s, with repetitions allowed, that has a all-ones column.', '1805.06723-1-17-2': 'The length of the shortest of these products is called the reset threshold ([MATH]) of the automaton.', '1805.06723-1-17-3': 'An automaton can be represented by a labelled digraph on [MATH] vertices where there is a direct edge from vertex [MATH] to vertex [MATH] labelled by [MATH] if [MATH]; in this case we also use the notation [MATH].', '1805.06723-1-18-0': 'A primitive set [MATH] is a set of [MATH] matrices [MATH] with nonnegative entries where there exists a product of the [MATH]s, with repetitions allowed, that has all positive entries.', '1805.06723-1-18-1': 'The length of the shortest of these products is called the exponent ([MATH]) of the set.', '1805.06723-1-18-2': 'A matrix is said to be NZ if it has no zero-rows nor zero-columns.', '1805.06723-1-18-3': 'Primitive sets of NZ matrices can be characterized as follows:', '1805.06723-1-19-0': 'Let [MATH] a partition of [MATH] with [MATH].', '1805.06723-1-19-1': 'We say that a [MATH] matrix [MATH] has a block-permutation structure on the partition [MATH] if there exists a permutation [MATH] such that for all [MATH] and for all [MATH], if [MATH] then [MATH].', '1805.06723-1-20-0': '[[CITATION], Theorem 1] An irreducible set of NZ matrices of size [MATH] is not primitive if and only if there exist a partition [MATH] of [MATH] on which every matrix of the set has a block-permutation structure.', '1805.06723-1-21-0': 'We remind that a set [MATH] is irreducible iff the matrix [MATH] is irreducible, where a matrix [MATH] is irreducible if there does not exist a permutation matrix [MATH] such that [MATH] is block-triangular.', '1805.06723-1-21-1': 'We say that a set of matrices has a block-permutation structure if there exists a partition on which all the matrices of the set have a block-permutation structure.', '1805.06723-1-21-2': 'The directed graph associated to a [MATH] matrix [MATH] is the digraph [MATH] on [MATH] vertices with a direct edge from [MATH] to [MATH] if [MATH].', '1805.06723-1-21-3': 'A matrix [MATH] is irreducible if and only if [MATH] is strongly connected, i.e. if and only if there exists a direct path between any two given vertices.', '1805.06723-1-21-4': 'A reducible set cannot be primitive.', '1805.06723-1-21-5': 'We end this section with the last definition and our first observation (Proposition [REF]).', '1805.06723-1-22-0': 'A matrix [MATH] dominates a matrix [MATH] if [MATH].', '1805.06723-1-23-0': 'Consider an irreducible set [MATH] in which every matrix dominates a permutation matrix.', '1805.06723-1-23-1': 'If the set has a block-permutation structure, then all the blocks of the partition must have the same size.', '1805.06723-1-24-0': 'See Appendix.', '1805.06723-1-25-0': '# Minimal primitive sets and minimal automata', '1805.06723-1-26-0': 'In this section we focus on perturbed permutation sets, i.e. matrix sets made of permutation matrices where a [MATH]-entry of one matrix is changed into a [MATH].', '1805.06723-1-26-1': 'We represent a set of this kind as [MATH], where [MATH] are permutation matrices, [MATH] is a matrix whose [MATH]-th entry is equal to [MATH] and all the others entries are equal to [MATH] and [MATH] for a [MATH].', '1805.06723-1-26-2': 'A primitive set is said to be minimal if it does not have any primitive proper subset; same definition can be applied to automata.', '1805.06723-1-26-3': 'The first result states that we can easily generate minimal automata starting from minimal perturbed permutation sets:', '1805.06723-1-27-0': 'Let [MATH] be a minimal primitive perturbed permutation set with [MATH] for a [MATH] and let [MATH] with [MATH] be the synchronizing automaton associated to [MATH] as in Definition [REF].', '1805.06723-1-27-1': 'If [MATH] is not minimal, then [MATH] is synchronizing and minimal.', '1805.06723-1-28-0': 'See Appendix.', '1805.06723-1-29-0': '## A randomized algorithm for constructing minimal primitive sets', '1805.06723-1-30-0': 'If we want to find minimal automata, Proposition [REF] tells us that we just need to generate minimal primitive perturbed permutation sets; in this section we implement a randomized procedure to build them.', '1805.06723-1-30-1': 'Theorem [REF] says that a matrix set is not primitive if all the matrices share the same block-permutation structure, therefore a primitive set of [MATH] matrices is minimal iff every subset of cardinality [MATH] has a block-permutation structure on a certain partition; this is the condition we will enforce.', '1805.06723-1-30-2': 'As we are dealing with perturbed permutation sets, Proposition [REF] tells us that we just need to consider partitions with blocks of the same size; if the blocks of the partition have size [MATH], we call it a [MATH]-partition and we say that the set has a [MATH]-permutation structure.', '1805.06723-1-30-3': 'Given [MATH] and a matrix [MATH], we indicate with [MATH] the submatrix of [MATH] with rows indexed by [MATH] and columns indexed by [MATH].', '1805.06723-1-30-4': 'The algorithm first generates a set of permutation matrices satisfying the requested block-permutation structures and then a [MATH]-entry of one of the obtained matrices is changed into a [MATH]; while doing this last step, we will make sure that such change will keep inalterated all the block-permutation structures of the matrix.', '1805.06723-1-30-5': 'We underline that our algorithm finds perturbed permutation sets that, if they are primitive, they are also minimal.', '1805.06723-1-30-6': 'Indeed, the construction itself only ensures minimality and not primitivity: this latter property has to be verified at the end.', '1805.06723-1-31-0': '### The algorithm', '1805.06723-1-32-0': 'For generating a set of [MATH] matrices [MATH] we choose [MATH] prime numbers [MATH] and we set [MATH].', '1805.06723-1-32-1': 'For [MATH], we require the set [MATH] (the set obtained from [MATH] by erasing matrix [MATH]) to have a [MATH]-permutation structure.', '1805.06723-1-32-2': 'This construction will ensure the minimality of the set, since a set that has a block-permutation structure cannot be primitive (Theorem [REF]).', '1805.06723-1-32-3': 'In particular, for all [MATH] we will enforce the existence of a [MATH]-partition [MATH] of [MATH] on which, for all [MATH], the matrix [MATH] has to have a block-permutation structure.', '1805.06723-1-32-4': 'As stated by Definition [REF], requiring a matrix to have a block-permutation structure on a partition means, in our case, that for every [MATH] and for every [MATH] there must exist a permutation [MATH] such that for all [MATH] and [MATH], [MATH] is a zero matrix.', '1805.06723-1-32-5': 'The main idea of the algorithm is to initialize every entry of each matrix to [MATH] and then, step by step, to set to [MATH] the entries that are not compatible with the conditions that we are requiring.', '1805.06723-1-32-6': 'As our final goal is to have a set of permutation matrices, at every step we need to make sure that each matrix dominates at least one permutation matrix, despite the increasing number of zeros among its entries.', '1805.06723-1-32-7': 'We say that a matrix [MATH] is compatible with a [MATH]-partition [MATH] and a permutation [MATH] if for all [MATH], there exists a permutation matrix [MATH] such that [EQUATION]', '1805.06723-1-32-8': 'The algorithm itself is formally presented in Listing 1; we here describe in words how it operates.', '1805.06723-1-32-9': 'Each entry of each matrix is initialized to [MATH].', '1805.06723-1-32-10': 'The algorithm has two for-loops: the outer one on [MATH], where a [MATH]-partition [MATH] of [MATH] is uniformly randomly sampled and then the inner one on [MATH] and [MATH] where a permutation [MATH] is uniformly randomly sampled and it is verified if [MATH] is compatible with [MATH] and [MATH].', '1805.06723-1-32-11': 'If it is compatible, the algorithm moves to the next step [MATH].', '1805.06723-1-32-12': 'If it is not compatible, the algorithm continues to randomly select another permutation [MATH] and to check the compatibility of [MATH] with it and [MATH].', '1805.06723-1-32-13': 'If after [MATH] steps this compatibility has not been established, the algorithm exits the inner for-loop and it randomly selects another [MATH]-partition [MATH] of [MATH]; it then repeats the inner loop for [MATH] and [MATH] with this new partition.', '1805.06723-1-32-14': 'Again, if after [MATH] steps it is chosing a different [MATH]-partition the compatibility of [MATH] with [MATH] and [MATH] is not established, we stop the algorithm and we say that it did not converge.', '1805.06723-1-32-15': 'If the inner for-loop is completed, then for each [MATH] the algorithm modifies the matrix [MATH] by keeping unchanged all the blocks [MATH] and by setting to zero all the other entries of [MATH]; the matrix [MATH] has now a block-permutation structure over the sampled partition [MATH].', '1805.06723-1-32-16': 'The algorithm then moves to the next step [MATH].', '1805.06723-1-32-17': 'If it manages to finish the outer for-loop, we have a set of binary matrices with the desired block-permutation structures.', '1805.06723-1-32-18': 'We then just need to select a permutation matrix [MATH] for every [MATH] and then to randomly change a [MATH]-entry into a [MATH] in one of the matrices without modifying the block-permutation structures: this is always possible as the blocks of the partitions are non trivial and a permutation matrix has just [MATH] positive entries.', '1805.06723-1-32-19': 'We finally check whether the set is primitive.', '1805.06723-1-32-20': 'Here below we describe the subprocedures that the algorithm uses.', '1805.06723-1-33-0': 'if the matrix !M', '1805.06723-1-33-1': 'is compatible with the partition !O', '1805.06723-1-33-2': 'and the permutation !s', '1805.06723-1-33-3': 'and in this case it returns the matrix !A', '1805.06723-1-33-4': 'but the fact that the entries not in the blocks defined by ([REF]) are set to zero; it returns !a', '1805.06723-1-33-5': 'returns a permutation matrix dominated by matrix !M', '1805.06723-1-33-6': '; there will be always at least one due to the use of subprocedure !D', '1805.06723-1-33-7': 'For this procedure we implemented an algorithm where we can change a parameter, represented by !m', '1805.06723-1-33-8': 'uniform among all the permutation matrices that are dominated by !M', '1805.06723-1-33-9': 'or to make it deterministic (see Appendix).', '1805.06723-1-33-10': 'This deterministic procedure will play an important role in our numerical experiments in Section [REF] and in the discovery of new families of automata with quadratic reset threshold in Section [REF].', '1805.06723-1-33-11': '!', '1805.06723-1-33-12': 'M=Addone(P1,...,Pm)!', '1805.06723-1-33-13': 'changes a [MATH]-entry of one of the matrices into a [MATH] keeping inalterated all the block-permutation structures.', '1805.06723-1-33-14': 'The matrix and the entry are chosen at random and the procedure iterates the choice till it finds a compatible entry (which always exists); it then returns the final perturbed permutation set !M', '1805.06723-1-33-15': 'if the set !M', '1805.06723-1-33-16': '[MATH] is primitive and !p', '1805.06723-1-33-17': 'It first verifies if the set is irreducible by checking the strong connectivity of the digraph [MATH] where [MATH] (see Section [REF]) via a breadth-first search on every node, then if the set is irreducible, primitivity is checked by the Protasov-Voynov algorithm ([CITATION], Section 4).', '1805.06723-1-34-0': 'All the above routines have polynomial time complexity in [MATH], apart from routine !', '1805.06723-1-34-1': 'Primitive!', '1805.06723-1-34-2': 'that has time complexity [MATH].', '1805.06723-1-35-0': 'In all our numerical experiments the algorithm always converged, i.e. it always ended in finite time for [MATH] and [MATH] large enough.', '1805.06723-1-35-1': 'This is probably due to the fact that the matrix dimension [MATH] grows exponentially as the number of matrices [MATH] increases, which produces enough degrees of freedom.', '1805.06723-1-35-2': 'We leave the proof of this fact for future work.', '1805.06723-1-36-0': 'A recent work of Alpin and Alpina [CITATION] generalizes Theorem [REF] for the characterization of primitive sets to sets that are allowed to be reducible and the matrices to have zero columns but not zero rows.', '1805.06723-1-36-1': 'Clearly, automata fall within this category.', '1805.06723-1-36-2': 'Without going into many details (for which we refer the reader to [CITATION], Theorem 3), Alpin and Alpina show that an automaton on [MATH] states is not synchronizing if and only if there exist a partition [MATH] of [MATH] such that it has a block-permutation structure on a subset of that partition.', '1805.06723-1-36-3': 'This characterization is clearly less restrictive: it just suffices to find a subset [MATH] such that for each letter [MATH] of the automaton there exist a permutation [MATH] such that for all [MATH], if [MATH] then [MATH].', '1805.06723-1-36-4': 'Our algorithm could leverage these recent results in order to directly construct minimal synchronizing automata.', '1805.06723-1-36-5': 'We also leave this for future work.', '1805.06723-1-37-0': '## Numerical results', '1805.06723-1-38-0': 'We have seen that once we have a minimal perturbed permutation set, it is easy to generate a minimal automaton from it, as stated by Proposition [REF].', '1805.06723-1-38-1': 'Our goal is to generate automata with large research threshold, but checking this property on our many randomly generated instances is prohibitive.', '1805.06723-1-38-2': 'Indeed, we recall that computing the reset threshold of an automaton is in general NP-hard [CITATION], it is thus unrealistic thinking of computing the [MATH] of thousands of sets.', '1805.06723-1-38-3': 'Instead, as a proxy for the reset threshold, we compute the diameter of the square graph, which we now introduce:', '1805.06723-1-39-0': 'The square graph [MATH] of a [MATH]-state automaton [MATH] is the directed graph with vertex set [MATH] and edge set [MATH] such that [MATH] iff there exist a letter [MATH] such that [MATH] and [MATH] or [MATH] and [MATH].', '1805.06723-1-39-1': 'A vertex of type [MATH] is called a singleton.', '1805.06723-1-40-0': 'We recall that an automaton is synchronizing if and only if in its square graph there is a path from any non-singleton vertex to a singleton one; the proof of this well-known fact implies that [MATH].', '1805.06723-1-40-1': 'The diameter [MATH] can be computed in polynomial time, namely [MATH].', '1805.06723-1-41-0': 'Definition [REF] can be used for NZ-matrix sets as well.', '1805.06723-1-41-1': 'Also in this case it holds that a set is primitive if and only if in its square graph there is a path from any non-singleton vertex to a singleton one; this relies on the fact that for a NZ-matrix set primitivity is equivalent to admitting a product with a all-ones column ([CITATION], Lemma 4).', '1805.06723-1-41-2': 'Notice that a primitive set and its corresponding automaton (Definition [REF]) share the same square graph.', '1805.06723-1-42-0': 'We now report our numerical results based on the diameter of the square graph.', '1805.06723-1-42-1': 'We compare three methods of generating minimal automata: we call method 1 the uniform random generation of a 2-letter automata made of one permutation matrix and a matrix of rank [MATH], method 2 refers to the automata generated by our minimal construction where the permutation matrices are selected by !E', '1805.06723-1-42-2': 'uniformly at random while method 3 refers to the same construction but !E', '1805.06723-1-42-3': 'behaving deterministically (see Appendix).', '1805.06723-1-42-4': 'We set [MATH].', '1805.06723-1-42-5': 'For each method and each choice of [MATH] we run the algorithm [MATH] times, thus producing each time [MATH] perturbed permutation sets.', '1805.06723-1-42-6': 'Among these sets, we select the primitive ones and we generate their associated automata (Definition [REF]); we then extract from each automata its minimal subautomaton (Proposition [REF]) and we compute its square graph diameter.', '1805.06723-1-42-7': 'Figure [REF] reports on the [MATH] axis the maximal square graph diameter found for each method and for each matrix dimension [MATH] when [MATH] is a product of three primes (left picture) and when it is a product of four primes (right picture).', '1805.06723-1-42-8': 'We can see that our minimal random construction manages to reach higher values of the square graph diameter than the mere random generation; in particular, method 3 reaches quadratic diameters in case of three matrices.', '1805.06723-1-43-0': '# New families of automata with quadratic reset threshold', '1805.06723-1-44-0': 'We present here four new families of (minimal) [MATH]-letter automata with square graph diameter of order [MATH], which represents a lower bound for their reset threshold.', '1805.06723-1-44-1': 'These families are all made of two symmetric permutation matrices and a matrix of rank [MATH] that merges two states and fixes all the others (a perturbed identity matrix): they thus lie within the class of automata with simple idempotents, class introduced by Rystov in [CITATION] in which every letter [MATH] of the automaton is requested either to be a permutation or to satisfy [MATH].', '1805.06723-1-44-2': 'These families have been found via the randomized algorithm described in Section [REF] using a deterministic procedure (method 3, see Appendix for details) to extract a permutation matrix from a binary one.', '1805.06723-1-44-3': 'The following proposition shows that primitive sets made of a perturbed identity matrix and two symmetric permutations must have a very specific shape; we then present our families, prove closed formulas for their square graph diameter and finally state a conjecture on their reset thresholds.', '1805.06723-1-44-4': 'With a slight abuse of notation we identify a permutation matrix [MATH] with its underlying permutation, that is we say that [MATH] if and only if [MATH]; the identity matrix is denoted by [MATH].', '1805.06723-1-44-5': 'Note that a permutation matrix is symmetric if and only if its cycle decomposition is made of fixed points and cycles of length [MATH].', '1805.06723-1-45-0': 'Let [MATH] be a matrix set of [MATH] matrices where [MATH], [MATH], is a perturbed identity and [MATH] and [MATH] are two symmetric permutations.', '1805.06723-1-45-1': 'If [MATH] is irreducible then, up to a relabelling of the vertices, [MATH] and [MATH] have the following form:', '1805.06723-1-46-0': '- if [MATH] is even [EQUATION] or [EQUATION]- if [MATH] is odd [EQUATION]', '1805.06723-1-46-1': 'See Appendix.', '1805.06723-1-47-0': 'A matrix set [MATH] of type ([REF]) is never primitive.', '1805.06723-1-48-0': 'See Appendix.', '1805.06723-1-49-0': 'We define [MATH] to be the automaton associated to [MATH] (see Definition [REF]), where [MATH].', '1805.06723-1-50-0': 'It is clear that [MATH] is with simple idempotents.', '1805.06723-1-50-1': 'Figure [REF] represents [MATH] with [MATH].', '1805.06723-1-50-2': 'We set now [MATH] for [MATH] and [MATH], [MATH] for [MATH] and [MATH], [MATH] for [MATH] and [MATH], [MATH] for [MATH] and [MATH].', '1805.06723-1-50-3': 'The following theorem holds:', '1805.06723-1-51-0': 'The automaton [MATH] has square graph diameter (SGD) of [MATH], [MATH] has SGD of [MATH], [MATH] has SGD of [MATH] and [MATH] has SGD of [MATH].', '1805.06723-1-52-0': 'See Appendix.', '1805.06723-1-53-0': 'Figure [REF] represents the square graph of the automaton [MATH], where its diameter is colored in red.', '1805.06723-1-53-1': 'All the singletons but the one that belongs to the diameter have been omitted.', '1805.06723-1-54-0': 'The automaton [MATH] has reset threshold of [MATH], [MATH] has reset threshold of [MATH] and [MATH] and [MATH] have reset threshold of [MATH].', '1805.06723-1-54-1': 'Furthermore, they represent the automata with the largest possible reset threshold among the family [MATH] for respectively [MATH], [MATH], [MATH] and [MATH].', '1805.06723-1-55-0': 'We end this section by remarking that, despite the fact that the randomized construction for minimal primitive sets presented in Section [REF] works just when the matrix size [MATH] is the product of at least three prime numbers, here we found an extremal automaton of quadratic reset threshold for any value of [MATH].', '1805.06723-1-56-0': '# Primitivity with high probability', '1805.06723-1-57-0': 'We call random perturbed permutation set a perturbed permutation set of [MATH] matrices constructed with the following randomized procedure:', '1805.06723-1-58-0': 'We sample [MATH] permutation matrices [MATH] independently and uniformly at random from the set [MATH] of all the permutations over [MATH] elements; A matrix [MATH] is uniformly randomly chosen from the set [MATH].', '1805.06723-1-58-1': 'Then, one of its [MATH]-entry is uniformly randomly selected among its [MATH]-entries and changed into a [MATH].', '1805.06723-1-58-2': 'It becomes then a perturbed permutation matrix [MATH]; The final random perturbed permutation set is the set [MATH].', '1805.06723-1-59-0': 'This procedure is also equivalent to choose independently and uniformly at random [MATH] permutation matrices from [MATH] and then one perturbed permutation matrix from [MATH] with [MATH].', '1805.06723-1-59-1': 'We say that a property [MATH] holds for a random matrix set with high probability if the probability that property [MATH] holds tends to [MATH] as the matrix dimension [MATH] tends to infinity.', '1805.06723-1-60-0': 'A random perturbed permutation set constructed via Procedure [REF] is primitive and has exponent of order [MATH] with high probability.', '1805.06723-1-61-0': 'See Appendix.', '1805.06723-1-62-0': 'Theorem [REF] can be extended to random sets of binary matrices.', '1805.06723-1-62-1': 'It is clear that focusing just on binary matrices is not restrictive as in the definition of primitivity what counts is just the position of the positive entries within the matrices and not their actual values.', '1805.06723-1-62-2': 'Let [MATH] denote a random binary matrix where each entry is independently set to [MATH] with probability [MATH] and to [MATH] with probability [MATH] and let [MATH] denote a set of [MATH] matrices obtained independently in this way.', '1805.06723-1-62-3': 'Under some mild assumptions over [MATH], we still have primitivity with high probability:', '1805.06723-1-63-0': 'If [MATH] as [MATH], then [MATH] is primitive and [MATH] with high probability.', '1805.06723-1-64-0': 'See Appendix.', '1805.06723-1-65-0': 'It is interesting to compare this result with the one obtained by Gerencser et al. in [CITATION]: they prove that, if [MATH] is the maximal value of the exponent among all the binary primitive sets of [MATH] matrices, then [MATH].', '1805.06723-1-65-1': 'This implies that, for [MATH] big enough, there must exist some primitive sets whose exponent is close to [MATH], but these sets must be very few as Theorem [REF] states that they are almost impossible to be detected by a mere random generation.', '1805.06723-1-65-2': 'We conclude with a result on when a set of two random binary matrices is not primitive with high probability:', '1805.06723-1-66-0': 'If [MATH] as [MATH], then [MATH] is reducible with high probability.', '1805.06723-1-66-1': 'This implies that [MATH] is not primitive with high probability.', '1805.06723-1-67-0': 'See Appendix.', '1805.06723-1-68-0': '# Conclusion', '1805.06723-1-69-0': 'In this paper we have proposed a randomized construction for generating slowly minimal synchronizing automata.', '1805.06723-1-69-1': 'Our strategy relies on a recent characterization of primitive sets (Theorem [REF]), together with a construction (Definition [REF] and Theorem [REF]) allowing to build (slowly minimal) synchronizing automata from (slowly minimal) primitive sets.', '1805.06723-1-69-2': 'We have obtained four new families of automata with simple idempotent with reset threshold of order [MATH].', '1805.06723-1-69-3': 'The primitive sets approach to synchronizing automata seems promising and we believe that out randomized construction could be further refined and developed in order to produce other interesting automata with large reset threshold.', '1805.06723-1-69-4': 'As mentioned at the end of Section [REF], it would be also of interest to apply the minimal construction directly to automata by leveraging the recent result of Alpin and Alpina ([CITATION], Theorem 3).'}","{'1805.06723-2-0-0': 'Motivated by the randomized generation of slowly synchronizing automata, we study automata made of permutation letters and a merging letter of rank [MATH].', '1805.06723-2-0-1': 'We present a constructive randomized procedure to generate synchronizing automata of that kind with (potentially) large alphabet size based on recent results on primitive sets of matrices.', '1805.06723-2-0-2': 'We report numerical results showing that our algorithm finds automata with much larger reset threshold than a mere uniform random generation and we present new families of automata with reset threshold of [MATH].', '1805.06723-2-0-3': 'We finally report theoretical results on randomized generation of primitive sets of matrices: a set of permutation matrices with a [MATH] entry changed into a [MATH] is primitive and has exponent of [MATH] with high probability in case of uniform random distribution and the same holds for a random set of binary matrices where each entry is set, independently, equal to [MATH] with probability [MATH] and equal to [MATH] with probability [MATH], when [MATH] as [MATH].', '1805.06723-2-1-0': '# Introduction', '1805.06723-2-2-0': 'A (complete deterministic finite) automaton [MATH] on [MATH] states can be defined as a set of [MATH] binary row-stochastic matrices [MATH] that are called the letters of the automaton.', '1805.06723-2-2-1': 'We say that [MATH] is synchronizing if there exists a product of its letters, with repetitions allowed, that has an all-ones column and the length of the shortest of these products is called the reset threshold ([MATH]) of the automaton.', '1805.06723-2-2-2': 'In other words, an automaton is synchronizing if there exists a word that brings the automaton into a particular state, regardless of the initial one.', '1805.06723-2-2-3': 'Synchronizing automata appear in different research fields; for example they are often used as models of error-resistant systems [CITATION] and in symbolic dynamics [CITATION].', '1805.06723-2-2-4': 'For a brief account on synchronizing automata and their other applications we refer the reader to [CITATION].', '1805.06723-2-2-5': 'The importance of synchronizing automata also arises from one of the most longstanding open problems in this field, the Cerny conjecture, which affirms that any synchronizing automaton on [MATH] states has reset threshold at most [MATH].', '1805.06723-2-2-6': 'If it is true, the bound is sharp due to the existence of a family of [MATH]-letter automata attaining this value, family discovered by Cerny in [CITATION].', '1805.06723-2-2-7': 'Despite great effort, the best upper bound for the reset threshold known so far is [MATH], recently obtained by Szykula in [CITATION] and thereby beating the 30 years-standing upper bound of [MATH] found by Pin and Frankl in [CITATION].', '1805.06723-2-2-8': 'Better upper bounds have been obtained for certain families of automata and the search for automata attaining quadratic reset threshold within these families have been the subject of several contributions in recent years.', '1805.06723-2-2-9': 'These results are (partly) summarized in Table [REF].', '1805.06723-2-2-10': 'Exhaustive search confirmed the conjecture for small values of [MATH] (see [CITATION]).', '1805.06723-2-3-0': 'The hunt for a possible counterexample to the conjecture turned out not to be an easy task as well; the search space is wide and calculating the reset threshold is computationally hard (see [CITATION]).', '1805.06723-2-3-1': 'Automata with reset thresholds close to [MATH], called extremal or slowly synchronizing automata, are also hard to detect and not so many families are known; Bondt et.', '1805.06723-2-3-2': 'al. [CITATION] make a thorough analysis of automata with small number of states and we recall, among others, the families found by Ananichev et al. [CITATION], by Gusev and Pribavkina [CITATION], by Kisielewicz and Szykula [CITATION] and by Dzyga et.', '1805.06723-2-3-3': 'al. [CITATION].', '1805.06723-2-3-4': 'These last two examples are, in particular, some of the few examples of slowly synchronizing automata with more than two letters that can be found in the literature.', '1805.06723-2-3-5': 'Almost all the families of slowly synchronizing automata listed above are closely related to the Cerny automaton [MATH], where [MATH] is the cycle over [MATH] vertices and [MATH] the letter that fixes all the vertices but one, which is mapped to the same vertex as done by [MATH]; indeed all these families present a letter that is a cycle over [MATH] vertices and the other letters have an action similar to the one of letter [MATH].', '1805.06723-2-3-6': 'As these examples seem to have a quite regular structure, it is natural to wonder whether a randomized procedure to generate automata could obtain less structured automata with possibly larger reset thresholds.', '1805.06723-2-3-7': ""This probabilistic approach can be rooted back to the work of Erdos in the 60's, where he developed the so-called Probabilistic Method, a tool that permits to prove the existence of a structure with certain desired properties by defining a suitable probabilistic space in which to embed the problem; for an account on the probabilistic method we refer the reader to [CITATION]."", '1805.06723-2-3-8': 'The simplest way to randomly generate an automaton of [MATH] letters is to uniformly and independently sample [MATH] binary row-stochastic matrices: unfortunately, Berlinkov first proved in [CITATION] that two uniformly sampled random binary row-stochastic matrices synchronize with high probability (i.e. the probability that they form a synchronizing automaton tends to [MATH] as the matrix dimension tends to infinity), then Nicaud showed in [CITATION] that they also have reset threshold of order [MATH] with high probability.', '1805.06723-2-3-9': 'We say that an automaton is minimally synchronizing if any proper subset of its letters is not synchronizing; what just presented before implies that a uniformly sampled random automaton of [MATH] letters has low reset threshold and is not minimally synchronizing with high probability.', '1805.06723-2-3-10': 'Summarizing:', '1805.06723-2-4-0': 'With this motivation in place, our paper tackles the following questions:', '1805.06723-2-5-0': '[Q1] Is there a way to randomly generate (minimally) slowly synchronizing automata (with more than two letters)?', '1805.06723-2-5-1': '[Q2] Can we find some automata families with more than two letters, quadratic reset threshold and that do not resemble the Cerny family?', '1805.06723-2-6-0': 'Our Contribution.', '1805.06723-2-6-1': 'In this paper we give positive answers to both questions Q1 and Q2.', '1805.06723-2-6-2': 'For the first one, we rely on the concept of primitive set of matrices, introduced by Protasov and Voynov in [CITATION]: a finite set of matrices with nonnegative entries is said to be primitive if there exists a product of these matrices, with repetitions allowed, with all positive entries.', '1805.06723-2-6-3': 'A product of this kind is called positive and the length of the shortest positive product of a primitive set [MATH] is called the exponent ([MATH]) of the set.', '1805.06723-2-6-4': 'Although the Protasov-Voynov primitivity has gained a lot of attention in different fields as in stochastic switching systems [CITATION] and consensus for discrete-time multi-agent systems [CITATION], we are interested in its connection with automata theory.', '1805.06723-2-6-5': 'In the following, we say that a matrix is NZ if it has neither zero-rows nor zero-columns; a matrix set is said to be NZ if all its matrices are NZ.', '1805.06723-2-7-0': 'Let [MATH] be a binary NZ-matrix set.', '1805.06723-2-7-1': 'The automaton associated to the set [MATH] is the automaton [MATH] whose letters are all the binary row-stochastic matrices that are entrywise not greater than at least one matrix in [MATH].', '1805.06723-2-8-0': 'We here provide an example of a primitive set [MATH] and the associated automata [MATH] and [MATH] in both their matrix and graph representations (Figure [REF]), where [MATH].', '1805.06723-2-9-0': '[MATH], [MATH], [MATH].', '1805.06723-2-10-0': 'The following theorem summarizes two results proved by Blondel et.', '1805.06723-2-10-1': 'al. ([CITATION], Theorems 16-17) and a result proved by Gerencser et al. ([CITATION], Theorem 8).', '1805.06723-2-10-2': 'Note that we state it for sets of binary NZ-matrices but it more generally holds for any set of NZ-matrices with nonnegative entries; this relies on the fact that in the notion of primitivity what counts is the position of the nonnegative entries within the matrices of the set and not their the actual values.', '1805.06723-2-10-3': 'In this case we should add to Definition [REF] the request of setting to [MATH] all the positive entries of the matrices of [MATH] before building [MATH].', '1805.06723-2-11-0': 'Let [MATH] a set of binary NZ-matrices of size [MATH] and [MATH].', '1805.06723-2-11-1': 'It holds that [MATH] is primitive if and only if [MATH] (equiv.', '1805.06723-2-11-2': '[MATH]) is synchronizing.', '1805.06723-2-11-3': 'If [MATH] is primitive, then it also holds that: [EQUATION]', '1805.06723-2-11-4': 'For the matrix set [MATH] defined in Example [REF], it holds that [MATH], [MATH] and [MATH].', '1805.06723-2-12-0': 'Theorem [REF] will be extensively used throughout the paper.', '1805.06723-2-12-1': 'It shows that primitive sets can be used for generating synchronizing automata and Equation ([REF]) tells us that the presence of a primitive set with large exponent implies the existence of an automaton with large reset threshold; in particular the discovery of a primitive set [MATH] with [MATH] would disprove the Cerny conjecture.', '1805.06723-2-12-2': 'On the other hand, the upper bounds on the automata reset threshold mentioned before imply that [MATH].', '1805.06723-2-13-0': 'One advantage of using primitive sets is the Protasov-Voynov characterization theorem (see Theorem [REF] in Section [REF]) that describes a combinatorial property that a NZ-matrix set must have in order not to be primitive: by constructing a primitive set such that each of its proper subsets has this property, we can make it minimally primitive.', '1805.06723-2-14-0': 'We decided to focus our attention on what we call perturbed permutation sets, i.e. sets made of permutation matrices (binary matrices having exactly one [MATH] in every row and in every column) where a [MATH]-entry of one of these matrices is changed into a [MATH].', '1805.06723-2-14-1': 'These sets have many interesting properties:', '1805.06723-2-15-0': 'The characterization theorem for primitive sets and the above properties are the main ingredients of our randomized algorithm that finds minimally synchronizing automata of [MATH] and [MATH] letters (and can eventually be generalized to [MATH] letters); to the best of our knowledge, this is the first time where a constructive procedure for finding minimally synchronizing automata is presented.', '1805.06723-2-15-1': 'This is described in Section [REF] where numerical results are reported.', '1805.06723-2-15-2': 'The random construction let us also find new families of [MATH]-letters automata, presented in Section [REF], with reset threshold [MATH] and that do not resemble the Cerny automaton, thus answering question Q2.', '1805.06723-2-15-3': 'Finally, in Section [REF] we extend a result on perturbed permutation sets obtained by Gonze et al. in [CITATION]: we show that a random perturbed permutation set is primitive with high probability for any matrix size [MATH] (and not just when [MATH] is a prime number as in [CITATION]) and that its exponent is of order [MATH] still with high probability.', '1805.06723-2-15-4': 'A further generalization is then presented for sets of random binary matrices: if each entry of each matrix is set to [MATH] with probability [MATH] and to [MATH] with probability [MATH], independently from each other, then the set is primitive and has exponent of order [MATH] with high probability for any [MATH] such that [MATH] as [MATH].', '1805.06723-2-15-5': 'The proofs of the results presented in this paper have been omitted due to length restrictions.', '1805.06723-2-16-0': '# Definitions and notation', '1805.06723-2-17-0': 'In this section we briefly go through some definitions and results that will be needed in the rest of the paper.', '1805.06723-2-18-0': 'We indicate with [MATH] the set [MATH] and with [MATH] the set of permutations over [MATH] elements; with a slight abuse of notation [MATH] will also denote the set of the [MATH] permutation matrices.', '1805.06723-2-19-0': 'An [MATH]-state automaton [MATH] can be represented by a labelled digraph on [MATH] vertices with a directed edge from vertex [MATH] to vertex [MATH] labelled by [MATH] if [MATH]; in this case we also use the notation [MATH].', '1805.06723-2-19-1': 'We remind that a matrix [MATH] is irreducible if there does not exist a permutation matrix [MATH] such that [MATH] is block-triangular; a set [MATH] is said to be irreducible iff the matrix [MATH] is irreducible.', '1805.06723-2-19-2': 'The directed graph associated to an [MATH] matrix [MATH] is the digraph [MATH] on [MATH] vertices with a directed edge from [MATH] to [MATH] if [MATH].', '1805.06723-2-19-3': 'A matrix [MATH] is irreducible if and only if [MATH] is strongly connected, i.e. if and only if there exists a directed path between any two given vertices.', '1805.06723-2-19-4': 'A primitive set [MATH] is a set of [MATH] matrices [MATH] with nonnegative entries where there exists a product [MATH] entrywise, for [MATH].', '1805.06723-2-19-5': 'The length of the shortest of these products is called the exponent ([MATH]) of the set.', '1805.06723-2-19-6': 'Irreducibility is a necessary (but not sufficient) condition for a matrix set to be primitive (see [CITATION], Section 1).', '1805.06723-2-19-7': 'Primitive sets of NZ-matrices can be characterized as follows:', '1805.06723-2-20-0': 'Let [MATH] be a partition of [MATH] with [MATH].', '1805.06723-2-20-1': 'We say that an [MATH] matrix [MATH] has a block-permutation structure on the partition [MATH] if there exists a permutation [MATH] such that [MATH] and [MATH], if [MATH] then [MATH].', '1805.06723-2-20-2': 'We say that a set of matrices has a block-permutation structure if there exists a partition on which all the matrices of the set have a block-permutation structure.', '1805.06723-2-21-0': '[[CITATION], Theorem 1] An irreducible set of NZ matrices of size [MATH] is not primitive if and only if the set has a block-permutation structure.', '1805.06723-2-22-0': 'We end this section with the last definition and our first observation (Proposition [REF]).', '1805.06723-2-23-0': 'A matrix [MATH] dominates a matrix [MATH] if [MATH].', '1805.06723-2-24-0': 'Consider an irreducible set [MATH] in which every matrix dominates a permutation matrix.', '1805.06723-2-24-1': 'If the set has a block-permutation structure, then all the blocks of the partition must have the same size.', '1805.06723-2-25-0': '# Minimally primitive sets and minimally synchronizing automata', '1805.06723-2-26-0': 'In this section we focus on perturbed permutation sets, i.e. matrix sets made of permutation matrices where a [MATH]-entry of one matrix is changed into a [MATH].', '1805.06723-2-26-1': 'We represent a set of this kind as [MATH], where [MATH] are permutation matrices, [MATH] is a matrix whose [MATH]-th entry is equal to [MATH] and all the others entries are equal to [MATH] and [MATH] for a [MATH].', '1805.06723-2-26-2': 'The first result states that we can easily generate minimally synchronizing automata starting from minimally primitive perturbed permutation sets:', '1805.06723-2-27-0': 'Let [MATH] be a minimally primitive perturbed permutation set and let [MATH] be the integer such that [MATH].', '1805.06723-2-27-1': 'The automaton [MATH] (see Definition [REF]) can be written as [MATH] with [MATH].', '1805.06723-2-27-2': 'If [MATH] is not minimally sychronizing, then [MATH] is.', '1805.06723-2-28-0': '## A randomized algorithm for constructing minimally primitive sets', '1805.06723-2-29-0': 'If we want to find minimally synchronizing automata, Proposition [REF] tells us that we just need to generate minimally primitive perturbed permutation sets; in this section we implement a randomized procedure to build them.', '1805.06723-2-30-0': 'Theorem [REF] says that a matrix set is not primitive if all the matrices share the same block-permutation structure, therefore a set of [MATH] matrices is minimally primitive iff every subset of cardinality [MATH] has a block-permutation structure on a certain partition; this is the condition we will enforce.', '1805.06723-2-30-1': 'As we are dealing with perturbed permutation sets, Proposition [REF] tells us that we just need to consider partitions with blocks of the same size; if the blocks of the partition have size [MATH], we call it a [MATH]-partition and we say that the set has a [MATH]-permutation structure.', '1805.06723-2-30-2': 'Given [MATH] and a matrix [MATH], we indicate with [MATH] the submatrix of [MATH] with rows indexed by [MATH] and columns indexed by [MATH].', '1805.06723-2-30-3': 'The algorithm first generates a set of permutation matrices satisfying the requested block-permutation structures and then a [MATH]-entry of one of the obtained matrices is changed into a [MATH]; while doing this last step, we will make sure that such change will preserve all the block-permutation structures of the matrix.', '1805.06723-2-30-4': 'We underline that our algorithm finds perturbed permutation sets that, if are primitive, are minimally primitive.', '1805.06723-2-30-5': 'Indeed, the construction itself only ensures minimality and not primitivity: this latter property has to be verified at the end.', '1805.06723-2-31-0': '### The algorithm', '1805.06723-2-32-0': 'For generating a set of [MATH] matrices [MATH] we choose [MATH] prime numbers [MATH] and we set [MATH].', '1805.06723-2-32-1': 'For [MATH], we require the set [MATH] (the set obtained from [MATH] by erasing matrix [MATH]) to have a [MATH]-permutation structure; this construction will ensure the minimality of the set.', '1805.06723-2-32-2': 'More in detail, for all [MATH] we will enforce the existence of a [MATH]-partition [MATH] of [MATH] on which, for all [MATH], the matrix [MATH] has to have a block-permutation structure.', '1805.06723-2-32-3': 'As stated by Definition [REF], this request means that for every [MATH] and for every [MATH] there must exist a permutation [MATH] such that for all [MATH] and [MATH], [MATH] is a zero matrix.', '1805.06723-2-33-0': 'The main idea of the algorithm is to initialize every entry of each matrix to [MATH] and then, step by step, to set to [MATH] the entries that are not compatible with the conditions that we are requiring.', '1805.06723-2-33-1': 'As our final goal is to have a set of permutation matrices, at every step we need to make sure that each matrix dominates at least one permutation matrix, despite the increasing number of zeros among its entries.', '1805.06723-2-34-0': 'Given a matrix [MATH] and a [MATH]-partition [MATH], we say that a permutation [MATH] is compatible with [MATH] and [MATH] if for all [MATH], there exists a permutation matrix [MATH] such that [EQUATION]', '1805.06723-2-34-1': 'The algorithm itself is formally presented in Listing 1; we here describe in words how it operates.', '1805.06723-2-34-2': 'Each entry of each matrix is initialized to [MATH].', '1805.06723-2-34-3': 'The algorithm has two for-loops: the outer one on [MATH], where a [MATH]-partition [MATH] of [MATH] is uniformly randomly sampled and then the inner one on [MATH] with [MATH] where we verify whether there exists a permutation [MATH] that is compatible with [MATH] and [MATH].', '1805.06723-2-34-4': 'If it does exist, then we choose one among all the compatible permutations and the algorithm moves to the next step [MATH].', '1805.06723-2-34-5': 'If such permutation does not exist, then the algorithm exits the inner for-loop and it randomly selects another [MATH]-partition [MATH] of [MATH]; it then repeats the inner loop for [MATH] with [MATH] with this new partition.', '1805.06723-2-34-6': 'If after [MATH] steps it is choosing a different [MATH]-partition [MATH] the existence, for each [MATH], of a permutation [MATH] that is compatible with [MATH] and [MATH] is not established, we stop the algorithm and we say that it did not converge.', '1805.06723-2-34-7': 'If the inner for-loop is completed, then for each [MATH] the algorithm modifies the matrix [MATH] by keeping unchanged each block [MATH] for [MATH] and by setting to zero all the other entries of [MATH], where [MATH] is the selected compatible permutation; the matrix [MATH] has now a block-permutation structure over the sampled partition [MATH].', '1805.06723-2-34-8': 'The algorithm then moves to the next step [MATH].', '1805.06723-2-34-9': 'If it manages to finish the outer for-loop, we have a set of binary matrices with the desired block-permutation structures.', '1805.06723-2-34-10': 'We then just need to select a permutation matrix [MATH] for every [MATH] and then to randomly change a [MATH]-entry into a [MATH] in one of the matrices without modifying its block-permutation structures: this is always possible as the blocks of the partitions are non trivial and a permutation matrix has just [MATH] positive entries.', '1805.06723-2-34-11': 'We finally check whether the set is primitive.', '1805.06723-2-35-0': 'Here below we present the procedures that the algorithm uses:', '1805.06723-2-36-0': '[MATH]', '1805.06723-2-37-0': 'This is the key function of the algorithm.', '1805.06723-2-37-1': 'It returns [MATH] if the matrix [MATH] dominates a permutation matrix, it returns [MATH] and [MATH] otherwise.', '1805.06723-2-37-2': 'In the former case it also returns a permutation matrix [MATH] selected among the ones dominated by [MATH] according to [MATH]; if [MATH] the matrix [MATH] is sampled uniformly at random, while if [MATH] we make the choice of [MATH] deterministic.', '1805.06723-2-37-3': 'More in detail, the procedure works as follows: we first count the numbers of [MATH]s in each column and in each row of the matrix [MATH].', '1805.06723-2-37-4': 'We then consider the row or the column with the least number of [MATH]s; if this number is zero we stop the procedure and we set [MATH], as in this case [MATH] does not dominate a permutation matrix.', '1805.06723-2-37-5': 'If this number is strictly greater than zero, we choose one of the [MATH]-entries of the row or the column attaining this minimum: if [MATH] (method 2) the entry is chosen uniformly at random while if [MATH] (method 3) we take the first [MATH]-entry in the lexicographic order.', '1805.06723-2-37-6': 'Suppose that such [MATH]-entry is in position [MATH]: we set to zero all the other entries in row [MATH] and column [MATH] and we iterate the procedure on the submatrix obtained from [MATH] by erasing row [MATH] and column [MATH].', '1805.06723-2-37-7': 'We can prove that this procedure is well-defined and in at most [MATH] steps it produces the desired output: [MATH] if and only if [MATH] does not dominate a permutation matrix and, in case [MATH], method 2 indeed sample uniformly one of the permutations dominated by [MATH], while method 3 is deterministic and the permutation obtained usually has its [MATH]s distributed around the main diagonal.', '1805.06723-2-37-8': 'Method 3 will play an important role in our numerical experiments in Section [REF] and in the discovery of new families of automata with quadratic reset threshold in Section [REF].', '1805.06723-2-38-0': '[MATH]', '1805.06723-2-39-0': 'It returns [MATH] if there exists a permutation compatible with the matrix [MATH] and the partition [MATH], it returns [MATH] and [MATH] otherwise.', '1805.06723-2-39-1': 'In the former case it chooses one of the compatible permutations, say [MATH], according to [MATH] and returns the matrix [MATH] equal to [MATH] but the entries not in the blocks defined by ([REF]) that are set to zero; [MATH] has then a block-permutation structure on [MATH].', '1805.06723-2-39-2': 'More precisely, [MATH] acts in two steps: it first defines a [MATH] matrix [MATH] such that, for all [MATH],', '1805.06723-2-40-0': '[MATH] this can be done by calling [MATH] with input [MATH] and [MATH] for all [MATH].', '1805.06723-2-40-1': 'At this point, asking if there exists a permutation compatible with [MATH] and [MATH] is equivalent of asking if [MATH] dominates a permutation matrix.', '1805.06723-2-40-2': 'Therefore, the second step is to call again [MATH]: if [MATH] we set [MATH] and [MATH], while if [MATH] we set [MATH] and [MATH] as described before with [MATH] (i.e. [MATH] iff [MATH]); indeed the permutation [MATH] is one of the permutations compatible with [MATH] and [MATH].', '1805.06723-2-41-0': '[MATH]', '1805.06723-2-42-0': 'It changes a [MATH]-entry of one of the matrices [MATH] into a [MATH] preserving all its block-permutation structures.', '1805.06723-2-42-1': 'The matrix and the entry are chosen uniformly at random and the procedure iterates the choice till it finds a compatible entry (which always exists); it then returns the final perturbed permutation set [MATH].', '1805.06723-2-42-2': '[MATH]', '1805.06723-2-43-0': 'It returns [MATH] if the matrix set [MATH] is primitive and [MATH] otherwise.', '1805.06723-2-43-1': 'It first verifies if the set is irreducible by checking the strong connectivity of the digraph [MATH] where [MATH] (see Section [REF]) via breadth-first search on every node, then if the set is irreducible, primitivity is checked by the Protasov-Voynov algorithm ([CITATION], Section 4).', '1805.06723-2-44-0': 'All the above routines have polynomial time complexity in [MATH], apart from routine [MATH] that has time complexity [MATH].', '1805.06723-2-45-0': 'In all our numerical experiments the algorithm always converged, i.e. it always ended before reaching the stopping value [MATH], for [MATH] large enough.', '1805.06723-2-45-1': 'This is probably due to the fact that the matrix dimension [MATH] grows exponentially as the number of matrices [MATH] increases, which produces enough degrees of freedom.', '1805.06723-2-45-2': 'We leave the proof of this fact for future work.', '1805.06723-2-46-0': 'A recent work of Alpin and Alpina [CITATION] generalizes Theorem [REF] for the characterization of primitive sets to sets that are allowed to be reducible and the matrices to have zero columns but not zero rows.', '1805.06723-2-46-1': 'Clearly, automata fall within this category.', '1805.06723-2-46-2': 'Without going into many details (for which we refer the reader to [CITATION], Theorem 3), Alpin and Alpina show that an [MATH]-state automaton is not synchronizing if and only if there exist a partition [MATH] of [MATH] such that it has a block-permutation structure on a subset of that partition.', '1805.06723-2-46-3': 'This characterization is clearly less restrictive: it just suffices to find a subset [MATH] such that for each letter [MATH] of the automaton there exists a permutation [MATH] such that for all [MATH], if [MATH] then [MATH].', '1805.06723-2-46-4': 'Our algorithm could leverage this recent result in order to directly construct minimal synchronizing automata.', '1805.06723-2-46-5': 'We also leave this for future work.', '1805.06723-2-47-0': '## Numerical results', '1805.06723-2-48-0': 'We have seen that once we have a minimally primitive perturbed permutation set, it is easy to generate a minimally synchronizing automaton from it, as stated by Proposition [REF].', '1805.06723-2-48-1': 'Our goal is to generate automata with large research threshold, but checking this property on many randomly generated instances is prohibitive.', '1805.06723-2-48-2': 'Indeed, we recall that computing the reset threshold of an automaton is in general NP-hard [CITATION].', '1805.06723-2-48-3': 'Instead, as a proxy for the reset threshold, we compute the diameter of the square graph, which we now introduce:', '1805.06723-2-49-0': 'The square graph [MATH] of an [MATH]-state automaton [MATH] is the labelled directed graph with vertex set [MATH] and edge set [MATH] such that [MATH] if there exists a letter [MATH] such that [MATH] and [MATH], or [MATH] and [MATH].', '1805.06723-2-49-1': 'In this case, we label the directed edge [MATH] by [MATH] (multiple labels are allowed).', '1805.06723-2-49-2': 'A vertex of type [MATH] is called a singleton.', '1805.06723-2-50-0': 'A well-known result ([CITATION], Proposition 1) states that an automaton is synchronizing if and only if in its square graph there exists a path from any non-singleton vertex to a singleton one; the proof of this fact also implies that [EQUATION] where [MATH] denotes the diameter of [MATH] i.e. the maximum length of the shortest path between any two given vertices, taken over all the pairs of vertices.', '1805.06723-2-50-1': 'The diameter can be computed in polynomial time, namely [MATH] with [MATH] the number of letters of the automaton.', '1805.06723-2-51-0': 'We now report our numerical results based on the diameter of the square graph.', '1805.06723-2-51-1': 'We compare three methods of generating automata: we call method 1 the uniform random generation of 2-letter automata made of one permutation matrix and a matrix of rank [MATH], while method 2 and method 3, already introduced in the previous paragraph, refer to the different ways of extracting a permutation matrix from a binary one in our randomized construction.', '1805.06723-2-51-2': 'We set [MATH] and for each method and each choice of [MATH] we run the algorithm [MATH] times, thus producing each time [MATH] sets.', '1805.06723-2-51-3': 'This choice for [MATH] has been made by taking into account two facts: on one hand, it is desirable to keep constant the rate [MATH] between the number of sampled sets [MATH] and the cardinality [MATH] of the set of the perturbed permutation sets made of [MATH] matrices.', '1805.06723-2-51-4': 'Unfortunately, [MATH] grows approximately as [MATH] and so [MATH] explodes very fast.', '1805.06723-2-51-5': 'On the other hand, we have to deal with the limited computational speed of our computers.', '1805.06723-2-51-6': 'The choice of [MATH] comes as a compromise between these two issues, at least when [MATH].', '1805.06723-2-52-0': 'Among the [MATH] generated sets, we select the primitive ones and we generate their associated automata (Definition [REF]); we then check which ones are not minimally synchronizing and we make them minimally synchronizing by using Proposition [REF].', '1805.06723-2-52-1': 'Finally, we compute the square graph diameter of all the minimally synchronizing automata obtained.', '1805.06723-2-52-2': 'Figure [REF] reports on the [MATH] axis the maximal square graph diameter found for each method and for each matrix dimension [MATH] when [MATH] is the product of three prime numbers (left picture) and when it is a product of four prime numbers (right picture).', '1805.06723-2-52-3': 'We can see that our randomized construction manages to reach higher values of the square graph diameter than the mere random generation; in particular, method 3 reaches quadratic diameters in case of three matrices.', '1805.06723-2-53-0': 'We also report in Figure [REF] (left) the behavior of the average diameter of the minimally synchronizing automata generated on [MATH] iterations when [MATH] is the product of three prime numbers: we can see that in this case method [MATH] does not perform better than method [MATH], while method [MATH] performs just slightly better.', '1805.06723-2-53-1': 'This behavior could have been expected since our primary goal was to randomly generate at least one slowly synchronizing automata; this is indeed what happens with method [MATH], that manages to reach quadratic reset thresholds most of the times.', '1805.06723-2-54-0': 'A remark can be done on the percentage of the generated sets that are not primitive; this is reported in Figure [REF] (right), where we divide nonprimitive sets into two categories: reducible sets and imprimitive sets, i.e. irreducible sets that are not primitive.', '1805.06723-2-54-1': 'We can see that the percentage of nonprimitive sets generated by method 1 goes to [MATH] as [MATH] increases, behavior that we expected (see Section [REF], Theorem [REF]), while method 2 seems to always produce a not negligible percentage of nonprimitive sets, although quite small.', '1805.06723-2-54-2': 'The behavior is reversed for method [MATH]: most of the generated sets are not primitive.', '1805.06723-2-54-3': 'This can be interpreted as a good sign.', '1805.06723-2-54-4': 'Indeed, nonprimitive sets can be seen as sets with infinite exponent; as we are generating a lot of them with method [MATH], we intuitively should expect that, when a primitive set is generated, it has high chances to have large diameter.', '1805.06723-2-55-0': 'The slowly synchronizing automata found by our randomized construction are presented in the following section.', '1805.06723-2-55-1': 'We believe that some parameters of our construction, as the way a permutation matrix is extracted from a binary one or the way the partitions of [MATH] are selected, could be further tuned or changed in order to generate new families of slowly synchronizing automata; for example, we could think about selecting the ones in the procedure [MATH] according to a given distribution.', '1805.06723-2-55-2': 'We leave this for future work.', '1805.06723-2-56-0': '# New families of automata with quadratic reset threshold', '1805.06723-2-57-0': 'We present here four new families of (minimally synchronizing) [MATH]-letter automata with square graph diameter of order [MATH], which represents a lower bound for their reset threshold.', '1805.06723-2-57-1': 'These families are all made of two symmetric permutation matrices and a matrix of rank [MATH] that merges two states and fixes all the others (a perturbed identity matrix): they thus lie within the class of automata with simple idempotents, class introduced by Rystsov in [CITATION] in which every letter [MATH] of the automaton is requested either to be a permutation or to satisfy [MATH].', '1805.06723-2-57-2': 'These families have been found via the randomized algorithm described in Section [REF] using the deterministic procedure to extract a permutation matrix from a binary one (method 3).', '1805.06723-2-57-3': 'The following proposition shows that primitive sets made of a perturbed identity matrix and two symmetric permutations must have a very specific shape; we then present our families, prove closed formulas for their square graph diameter and finally state a conjecture on their reset thresholds.', '1805.06723-2-57-4': 'With a slight abuse of notation we identify a permutation matrix [MATH] with its underlying permutation, that is we say that [MATH] if and only if [MATH]; the identity matrix is denoted by [MATH].', '1805.06723-2-57-5': 'Note that a permutation matrix is symmetric if and only if its cycle decomposition is made of fixed points and cycles of length [MATH].', '1805.06723-2-58-0': 'Let [MATH] be a matrix set of [MATH] matrices where [MATH], [MATH], is a perturbed identity and [MATH] and [MATH] are two symmetric permutations.', '1805.06723-2-58-1': 'If [MATH] is irreducible then, up to a relabelling of the vertices, [MATH] and [MATH] have the following form:', '1805.06723-2-59-0': '- if [MATH] is even [EQUATION] or [EQUATION]- if [MATH] is odd [EQUATION]', '1805.06723-2-59-1': 'A matrix set [MATH] of type ([REF]) is never primitive.', '1805.06723-2-60-0': 'We define [MATH] to be the associated automaton [MATH] of [MATH] (see Definition [REF]), where [MATH].', '1805.06723-2-61-0': 'It is clear that [MATH] is with simple idempotents.', '1805.06723-2-61-1': 'Figure [REF] represents [MATH] with [MATH].', '1805.06723-2-61-2': 'We set now [MATH] for [MATH] and [MATH], [MATH] for [MATH] and [MATH], [MATH] for [MATH] and [MATH], [MATH] for [MATH] and [MATH].', '1805.06723-2-61-3': 'The following theorem holds:', '1805.06723-2-62-0': 'The automaton [MATH] has square graph diameter (SGD) of [MATH], [MATH] has SGD of [MATH], [MATH] has SGD of [MATH] and [MATH] has SGD of [MATH].', '1805.06723-2-62-1': 'Therefore all the families [MATH], [MATH], [MATH] and [MATH] have reset threshold of [MATH].', '1805.06723-2-63-0': 'Figure [REF] represents the square graph of the automaton [MATH], where its diameter is colored in red.', '1805.06723-2-63-1': 'All the singletons but the one that belongs to the diameter have been omitted.', '1805.06723-2-64-0': 'The automaton [MATH] has reset threshold of [MATH], [MATH] has reset threshold of [MATH] and [MATH] and [MATH] have reset threshold of [MATH].', '1805.06723-2-64-1': 'Furthermore, they represent the automata with the largest possible reset threshold among the family [MATH] for respectively [MATH], [MATH], [MATH] and [MATH].', '1805.06723-2-65-0': 'We end this section by remarking that, despite the fact that the randomized construction for minimally primitive sets presented in Section [REF] works just when the matrix size [MATH] is the product of at least three prime numbers, here we found an extremal automaton of quadratic reset threshold for any value of [MATH].', '1805.06723-2-66-0': '# Primitivity with high probability', '1805.06723-2-67-0': 'We call random perturbed permutation set a perturbed permutation set of [MATH] matrices constructed with the following randomized procedure:', '1805.06723-2-68-0': 'We sample [MATH] permutation matrices [MATH] independently and uniformly at random from the set [MATH] of all the permutations over [MATH] elements; A matrix [MATH] is uniformly randomly chosen from the set [MATH].', '1805.06723-2-68-1': 'Then, one of its [MATH]-entry is uniformly randomly selected among its [MATH]-entries and changed into a [MATH].', '1805.06723-2-68-2': 'It becomes then a perturbed permutation matrix [MATH]; The final random perturbed permutation set is the set [MATH].', '1805.06723-2-69-0': 'This procedure is also equivalent to choosing independently and uniformly at random [MATH] permutation matrices from [MATH] and one perturbed permutation matrix from [MATH] with [MATH].', '1805.06723-2-69-1': 'We say that a property [MATH] holds for a random matrix set with high probability if the probability that property [MATH] holds tends to [MATH] as the matrix dimension [MATH] tends to infinity.', '1805.06723-2-70-0': 'A random perturbed permutation set constructed via Procedure [REF] is primitive and has exponent of order [MATH] with high probability.', '1805.06723-2-71-0': 'Theorem [REF] can be extended to random sets of binary matrices.', '1805.06723-2-71-1': 'It is clear that focusing just on binary matrices is not restrictive as in the definition of primitivity what counts is just the position of the positive entries within the matrices and not their actual values.', '1805.06723-2-71-2': 'Let [MATH] denote a random binary matrix where each entry is independently set to [MATH] with probability [MATH] and to [MATH] with probability [MATH] and let [MATH] denote a set of [MATH] matrices obtained independently in this way.', '1805.06723-2-71-3': 'Under some mild assumptions over [MATH], we still have primitivity with high probability:', '1805.06723-2-72-0': 'For any fixed integer [MATH], if [MATH] as [MATH], then [MATH] is primitive and [MATH] with high probability.', '1805.06723-2-73-0': 'It is interesting to compare this result with the one obtained by Gerencser et al. in [CITATION]: they prove that, if [MATH] is the maximal value of the exponent among all the binary primitive sets of [MATH] matrices, then [MATH].', '1805.06723-2-73-1': 'This implies that, for [MATH] big enough, there must exist some primitive sets whose exponent is close to [MATH], but these sets must be very few as Theorem [REF] states that they are almost impossible to be detected by a mere random generation.', '1805.06723-2-73-2': 'We conclude with a result on when a set of two random binary matrices is not primitive with high probability:', '1805.06723-2-74-0': 'For any fixed integer [MATH], if [MATH] as [MATH], then [MATH] is reducible with high probability.', '1805.06723-2-74-1': 'This implies that [MATH] is not primitive with high probability.', '1805.06723-2-75-0': '# Conclusion', '1805.06723-2-76-0': 'In this paper we have proposed a randomized construction for generating slowly minimally synchronizing automata.', '1805.06723-2-76-1': 'Our strategy relies on a recent characterization of primitive sets (Theorem [REF]), together with a construction (Definition [REF] and Theorem [REF]) allowing to build (slowly minimally) synchronizing automata from (slowly minimally) primitive sets.', '1805.06723-2-76-2': 'We have obtained four new families of automata with simple idempotents with reset threshold of order [MATH].', '1805.06723-2-76-3': 'The primitive sets approach to synchronizing automata seems promising and we believe that out randomized construction could be further refined and tweaked in order to produce other interesting automata with large reset threshold, for example by changing the way a permutation matrix is extracted by a binary one.', '1805.06723-2-76-4': 'As mentioned at the end of Section [REF], it would be also of interest to apply the minimal construction directly to automata by leveraging the recent result of Alpin and Alpina ([CITATION], Theorem 3).'}","[['1805.06723-1-44-3', '1805.06723-2-57-3'], ['1805.06723-1-44-4', '1805.06723-2-57-4'], ['1805.06723-1-44-5', '1805.06723-2-57-5'], ['1805.06723-1-35-1', '1805.06723-2-45-1'], ['1805.06723-1-35-2', '1805.06723-2-45-2'], ['1805.06723-1-50-0', '1805.06723-2-61-0'], ['1805.06723-1-50-1', '1805.06723-2-61-1'], ['1805.06723-1-16-0', '1805.06723-2-18-0'], ['1805.06723-1-3-1', '1805.06723-2-3-1'], ['1805.06723-1-3-2', '1805.06723-2-3-2'], ['1805.06723-1-3-6', '1805.06723-2-3-6'], ['1805.06723-1-58-0', '1805.06723-2-68-0'], ['1805.06723-1-58-1', '1805.06723-2-68-1'], ['1805.06723-1-58-2', '1805.06723-2-68-2'], 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['1805.06723-1-23-0', '1805.06723-2-24-0'], ['1805.06723-1-23-1', '1805.06723-2-24-1'], ['1805.06723-1-13-1', '1805.06723-2-15-1'], ['1805.06723-1-13-3', '1805.06723-2-15-3'], ['1805.06723-1-26-0', '1805.06723-2-26-0'], ['1805.06723-1-26-1', '1805.06723-2-26-1'], ['1805.06723-1-45-0', '1805.06723-2-58-0'], ['1805.06723-1-2-4', '1805.06723-2-2-6'], ['1805.06723-1-2-7', '1805.06723-2-2-9'], ['1805.06723-1-69-4', '1805.06723-2-76-4'], ['1805.06723-1-32-5', '1805.06723-2-33-0'], ['1805.06723-1-32-6', '1805.06723-2-33-1'], ['1805.06723-1-32-8', '1805.06723-2-34-1'], ['1805.06723-1-32-9', '1805.06723-2-34-2'], ['1805.06723-1-32-16', '1805.06723-2-34-8'], ['1805.06723-1-32-17', '1805.06723-2-34-9'], ['1805.06723-1-32-19', '1805.06723-2-34-11'], ['1805.06723-1-66-1', '1805.06723-2-74-1'], ['1805.06723-1-42-0', '1805.06723-2-51-0'], ['1805.06723-1-30-2', '1805.06723-2-30-1'], ['1805.06723-1-30-3', '1805.06723-2-30-2'], ['1805.06723-1-30-6', '1805.06723-2-30-5'], ['1805.06723-1-18-1', '1805.06723-2-19-5'], ['1805.06723-1-21-5', '1805.06723-2-22-0'], ['1805.06723-1-44-0', '1805.06723-2-57-0'], ['1805.06723-1-44-1', '1805.06723-2-57-1'], ['1805.06723-1-44-2', '1805.06723-2-57-2'], ['1805.06723-1-3-0', '1805.06723-2-3-0'], ['1805.06723-1-3-4', '1805.06723-2-3-4'], ['1805.06723-1-3-5', '1805.06723-2-3-5'], ['1805.06723-1-3-7', '1805.06723-2-3-7'], ['1805.06723-1-3-8', '1805.06723-2-3-8'], ['1805.06723-1-59-0', '1805.06723-2-69-0'], ['1805.06723-1-55-0', '1805.06723-2-65-0'], ['1805.06723-1-7-1', '1805.06723-2-7-1'], ['1805.06723-1-36-2', '1805.06723-2-46-2'], ['1805.06723-1-36-3', '1805.06723-2-46-3'], ['1805.06723-1-36-4', '1805.06723-2-46-4'], ['1805.06723-1-0-1', '1805.06723-2-0-1'], ['1805.06723-1-39-0', '1805.06723-2-49-0'], ['1805.06723-1-6-3', '1805.06723-2-6-3'], ['1805.06723-1-6-5', '1805.06723-2-6-5'], ['1805.06723-1-34-0', '1805.06723-2-44-0'], ['1805.06723-1-38-0', '1805.06723-2-48-0'], ['1805.06723-1-38-1', '1805.06723-2-48-1'], ['1805.06723-1-13-0', '1805.06723-2-15-0'], ['1805.06723-1-13-4', '1805.06723-2-15-4'], ['1805.06723-1-20-0', '1805.06723-2-21-0'], ['1805.06723-1-2-3', '1805.06723-2-2-5'], ['1805.06723-1-2-5', '1805.06723-2-2-7'], ['1805.06723-1-2-6', '1805.06723-2-2-8'], ['1805.06723-1-2-8', '1805.06723-2-2-10'], ['1805.06723-1-19-0', '1805.06723-2-20-0'], ['1805.06723-1-19-1', '1805.06723-2-20-1'], ['1805.06723-1-69-0', '1805.06723-2-76-0'], ['1805.06723-1-69-1', '1805.06723-2-76-1'], ['1805.06723-1-69-2', '1805.06723-2-76-2'], ['1805.06723-1-5-1', '1805.06723-2-5-1'], ['1805.06723-1-12-0', '1805.06723-2-12-1'], ['1805.06723-1-12-1', '1805.06723-2-13-0'], ['1805.06723-1-12-2', '1805.06723-2-14-0'], ['1805.06723-1-32-7', '1805.06723-2-34-0'], ['1805.06723-1-32-10', '1805.06723-2-34-3'], ['1805.06723-1-32-15', '1805.06723-2-34-7'], ['1805.06723-1-32-18', '1805.06723-2-34-10'], ['1805.06723-1-66-0', '1805.06723-2-74-0'], ['1805.06723-1-42-5', '1805.06723-2-51-2'], ['1805.06723-1-42-7', '1805.06723-2-52-2'], 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'1805.06723-1-67-0', '1805.06723-2-3-3', '1805.06723-2-3-10', '1805.06723-2-4-0', '1805.06723-2-6-0', '1805.06723-2-9-0', '1805.06723-2-11-2', '1805.06723-2-14-1', '1805.06723-2-19-7', '1805.06723-2-23-0', '1805.06723-2-26-2', '1805.06723-2-35-0', '1805.06723-2-36-0', '1805.06723-2-38-0', '1805.06723-2-39-2', '1805.06723-2-41-0', '1805.06723-2-42-2', '1805.06723-2-48-3', '1805.06723-2-58-1', '1805.06723-2-59-0', '1805.06723-2-59-1', '1805.06723-2-60-0', '1805.06723-2-61-2', '1805.06723-2-61-3', '1805.06723-2-62-0', '1805.06723-2-62-1', '1805.06723-2-67-0', '1805.06723-2-71-3', '1805.06723-2-73-2']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1805.06723,,,
1603.06312,"{'1603.06312-1-0-0': ""We discuss a natural game of competition and solve the corresponding mean field game with common noise when agents' rewards are rank dependent."", '1603.06312-1-0-1': 'We use this solution to provide an approximate Nash equilibrium for the finite player game and obtain the rate of convergence.', '1603.06312-1-1-0': 'headings', '1603.06312-1-2-0': '# Introduction', '1603.06312-1-3-0': 'Mean field games (MFGs), introduced independently by [CITATION] and [CITATION], provide a useful approximation for the finite player Nash equilibrium problems in which the players are coupled through their empirical distribution.', '1603.06312-1-3-1': ""In particular, the mean field game limit gives an approximate Nash equilibrium, in which the agents' decision making is decoupled."", '1603.06312-1-3-2': 'In this paper we will consider a particular game in which the interaction of the players is through their ranks.', '1603.06312-1-3-3': ""Our main goal is to construct an approximate Nash equilibrium for a finite player game when the agents' dynamics are modulated by common noise."", '1603.06312-1-4-0': 'Rank based mean field games, which have non-local mean field interactions, have been suggested as an extension of their model of oil production in [CITATION] and analyzed more generally by the recent paper by Carmona and Lacker [CITATION] using the weak formulation, when there is no common noise.', '1603.06312-1-4-1': 'There are currently no results on the rank dependent mean field games with common noise.', '1603.06312-1-4-2': 'In order to solve the problem with common noise, we will make use of the mechanism in [CITATION] by solving the strong formulation of the problem rank dependent mean field game without common noise and then by observing that purely rank dependent reward functions are translation invariant.', '1603.06312-1-5-0': 'The rest of the paper is organized as follows: In Section [REF] we introduce the N-player game in which the players are coupled through the reward function which is rank based.', '1603.06312-1-5-1': 'In Section [REF] we consider the case without common noise.', '1603.06312-1-5-2': 'We first find the mean field limit, discuss the uniqueness of the Nash equilibrium, and construct an approximate Nash equilibrium using the mean field limit.', '1603.06312-1-5-3': 'Using these results, in Section [REF] we use the mechanism in [CITATION] and obtain respective results for the common noise.', '1603.06312-1-6-0': '# The [MATH]-player game', '1603.06312-1-7-0': 'We consider [MATH] players each of whom control her own state variable and are rewarded based on their ranking.', '1603.06312-1-7-1': ""We will denote by [MATH], [MATH]-th player's state variable and assume that it satisfies the following stochastic differential equation (SDE) [EQUATION] where [MATH] represents the control by agent [MATH], and [MATH] and [MATH] are independent standard Brownian motions defined on some filtered probability space [MATH], representing the idiosyncratic noises and common noise, respectively."", '1603.06312-1-7-2': 'The game ends at time [MATH], when each player receives a rank-based reward minus the running cost of effort, which we will assumed to be quadratic [MATH] for some constant [MATH].', '1603.06312-1-7-3': 'In order to precisely define the rank based [EQUATION] denote the empirical measure of the terminal state of the [MATH]-player system.', '1603.06312-1-7-4': 'Then [MATH] gives the fraction of players that finishes the same or worse than player [MATH].', '1603.06312-1-7-5': 'Let [MATH] be a bounded continuous function that is non-decreasing in both arguments.', '1603.06312-1-7-6': 'For any probability measure [MATH] on [MATH], write [MATH] where [MATH] denotes the cumulative distribution function of [MATH].', '1603.06312-1-7-7': 'The reward player [MATH] receives is given by [EQUATION].', '1603.06312-1-7-8': 'When [MATH] is independent of [MATH], the compensation scheme is purely rank-based.', '1603.06312-1-7-9': 'In general, we could have a mixture of absolute performance compensation and relative performance compensation.', '1603.06312-1-7-10': ""The objective of each player is to observe the progress of all players and choose his effort level to maximize the expected payoff, while anticipating the other players' strategies."", '1603.06312-1-8-0': ""The players' equilibrium expected payoffs, as functions of time and state variables, satisfy a system of [MATH] coupled nonlinear partial differential equations subject to discontinuous boundary conditions, which appears to be analytically intractable."", '1603.06312-1-8-1': 'Fortunately, in a large-population game, the impact of any individual on the whole population is very small.', '1603.06312-1-8-2': 'So it is often good enough for each player to ignore the private state of any other individual and simply optimize against the aggregate distribution of the population.', '1603.06312-1-8-3': 'As a consequence, the equilibrium strategies decentralize in the limiting game as [MATH].', '1603.06312-1-8-4': 'We shall use the MFG solution to construct approximate Nash equilibrium for the [MATH]-player game, both in the case with and without common noise.', '1603.06312-1-9-0': '# Mean Field Approximation when there is no common noise', '1603.06312-1-10-0': 'In this section, we assume [MATH].', '1603.06312-1-10-1': 'Solving the mean field game consists of two sub-problems: a stochastic control problem and a fixed-point problem (also called the consistency condition).', '1603.06312-1-10-2': 'For any Polish space [MATH], denote by [MATH] the space of probability measures on [MATH], and [MATH].', '1603.06312-1-11-0': ""We first fix a distribution [MATH] of the terminal state of the population, and consider a single player's optimization problem: [EQUATION] where [EQUATION] and [MATH] ranges over the set of progressively measurable processes satisfying [MATH]."", '1603.06312-1-11-1': 'The associated dynamic programming equation is [EQUATION] with terminal condition [MATH].', '1603.06312-1-11-2': 'Using the first-order condition, we obtain that the candidate optimizer is [MATH], and the Hamilton-Jacobi-Bellman (HJB) equation can be written as [EQUATION].', '1603.06312-1-11-3': 'The above equation can be linearized using the transformation [MATH], giving [EQUATION].', '1603.06312-1-11-4': 'Together with the boundary condition [MATH], we can easily write down the solution: [EQUATION] where [MATH] is a standard normal random variable, possibly defined on another probability space with measure [MATH].', '1603.06312-1-11-5': 'Let us further write [MATH] as an integral: [EQUATION]', '1603.06312-1-11-6': 'Using dominated convergence theorem, we can differentiate under the integral sign and get [EQUATION]', '1603.06312-1-11-7': 'Similarly, we obtain [EQUATION]', '1603.06312-1-11-8': 'Using [REF]-[REF], together with the boundedness and monotonicity of [MATH], we easily get the following estimates.', '1603.06312-1-11-9': 'Note that all bounds are independent of [MATH].', '1603.06312-1-12-0': 'The functions [MATH] and [MATH] satisfy [EQUATION] where [MATH].', '1603.06312-1-13-0': 'Since [MATH] is bounded, the drift coefficient [MATH] is Lipschitz continuous in [MATH].', '1603.06312-1-13-1': 'It follows that the optimally controlled state process, denoted by [MATH], has a strong solution on [MATH].', '1603.06312-1-13-2': 'Observe that [EQUATION].', '1603.06312-1-13-3': 'So the optimal cumulative effort is bounded by some constant independent of [MATH].', '1603.06312-1-13-4': 'It also implies that [MATH] has a well-defined limit as [MATH].', '1603.06312-1-13-5': 'Standard verification theorem yields that the solution to the HJB equation is the value function of the problem [REF]-[REF], and that [MATH] is the optimal Markovian feedback control.', '1603.06312-1-13-6': 'Finally, using dominated convergence theorem again, we can show that for [MATH], [EQUATION].', '1603.06312-1-13-7': 'The same limit also holds for [MATH] since [MATH] is bounded away from zero.', '1603.06312-1-13-8': 'In other words, the optimal effort level is small when the progress is very large in absolute value.', '1603.06312-1-13-9': 'This agrees with many real life observations that when a player has a very big lead, it is easy for him to show slackness; and when one is too far behind, he often gives up on the game instead of trying to catch up.', '1603.06312-1-14-0': '## Existence of a Nash equilibrium', '1603.06312-1-15-0': 'For each fixed [MATH], solving the stochastic control problem [REF]-[REF] yields a value function [MATH] and a best response [MATH].', '1603.06312-1-15-1': 'Suppose the game is started at time zero, with zero initial progress, the optimally controlled state process [MATH] of the generic player satisfies the SDE [EQUATION]', '1603.06312-1-15-2': 'Finding a Nash equilibrium for the limiting game is equivalent to finding a fixed point of the mapping [MATH], where [MATH] denotes the law of its argument.', '1603.06312-1-15-3': 'Such a fixed point is sometimes referred to as an equilibrium measure.', '1603.06312-1-16-0': ""Similar to [CITATION], we will use Schauder's fixed point theorem."", '1603.06312-1-16-1': 'Observe that for any [MATH], we have [EQUATION].', '1603.06312-1-16-2': 'This implies the set of [MATH] is tight in [MATH], hence relatively compact for the topology of weak convergence by Prokhorov theorem.', '1603.06312-1-16-3': 'Recall that [MATH].', '1603.06312-1-16-4': 'Equip [MATH] with the topology induced by the 1-Wasserstein metric: [EQUATION]', '1603.06312-1-16-5': 'Here [MATH] denotes the space of Lipschitz continuous functions on [MATH] whose Lipschitz constant is bounded by one.', '1603.06312-1-16-6': 'It is known that [MATH] is a complete separable metric space (see e.g. [CITATION]).', '1603.06312-1-16-7': 'We shall work with a subset of [MATH] defined by [EQUATION].', '1603.06312-1-16-8': 'It is easy to check that [MATH] is non-empty, convex and closed (for the topology induced by the [MATH] metric).', '1603.06312-1-16-9': 'Moreover, one can show using [CITATION] that any weakly convergent sequence [MATH] is also [MATH]-convergent.', '1603.06312-1-16-10': 'Therefore, [MATH] is also relatively compact for the topology induced by the [MATH] metric.', '1603.06312-1-16-11': 'So we have found a non-empty, convex and compact set [MATH] such that [MATH] maps [MATH] into itself.', '1603.06312-1-16-12': 'It remains to show [MATH] is continuous on [MATH].', '1603.06312-1-16-13': 'In the rest of the proof, the constant [MATH] may change from line to line.', '1603.06312-1-17-0': 'Let [MATH] such that [MATH] as [MATH].', '1603.06312-1-17-1': 'We wish to show [MATH].', '1603.06312-1-17-2': 'Note that [EQUATION].', '1603.06312-1-17-3': 'From Lemma [REF], we know that [MATH].', '1603.06312-1-17-4': 'Since [MATH], thanks to the dominated convergence theorem, it suffices to show for [MATH], [EQUATION].', '1603.06312-1-17-5': 'By Lemma [REF] and mean value theorem, we have that [EQUATION]', '1603.06312-1-17-6': 'So to show [MATH] it suffices to demonstrate for each fixed [MATH], [EQUATION] and [EQUATION]', '1603.06312-1-17-7': 'We first show [REF].', '1603.06312-1-17-8': 'Using the estimates in Lemma [REF], we get [EQUATION]', '1603.06312-1-17-9': 'Since all integrands are bounded, to show the expectations converge to zero, it suffice to check that the integrands converge to zero a.s. Fix [MATH], we know from [REF] that [EQUATION]', '1603.06312-1-17-10': 'Since [MATH], [MATH] also converges to [MATH] weakly, and the cumulative distribution function [MATH] converges to [MATH] at every point [MATH] at which [MATH] is continuous.', '1603.06312-1-17-11': 'It follows from the continuity of [MATH] that [MATH] converges to [MATH] at every point [MATH] at which [MATH] is continuous.', '1603.06312-1-17-12': 'Since [MATH] has at most countably many points of discontinuity, the random variable (the randomness comes from [MATH]) inside the expectation converges to zero [MATH]-a.s. Dominated convergence theorem then allows us to interchange the limit and the expectation, giving that [EQUATION].', '1603.06312-1-17-13': 'Similarly, from [REF] we obtain [EQUATION]', '1603.06312-1-17-14': 'Again, using that [MATH] has countably many points of discontinuity, one can show that [EQUATION].', '1603.06312-1-17-15': 'Putting everything together, we have proved [REF].', '1603.06312-1-18-0': ""Next, we show [REF] by Gronwall's inequality."", '1603.06312-1-18-1': 'Let [MATH] be given.', '1603.06312-1-18-2': 'We have for any [MATH], [EQUATION]', '1603.06312-1-18-3': 'By [REF] and bounded convergence theorem, we obtain [EQUATION].', '1603.06312-1-18-4': 'So for [MATH] large enough, we have [EQUATION]', '1603.06312-1-18-5': ""By Gronwall's inequality, [EQUATION]."", '1603.06312-1-18-6': 'This completes the proof of [REF], and thus the continuity of [MATH].', '1603.06312-1-18-7': ""By Schauder's fixed point theorem, there exists a fixed point [MATH] of [MATH] in the set [MATH]."", '1603.06312-1-19-0': '## Uniqueness of Nash equilibrium.', '1603.06312-1-20-0': 'Nash equilibrium is rarely unique except under the following monotonicity assumption which is in the spirit of [CITATION]:', '1603.06312-1-21-0': 'For any two fixed points [MATH] and [MATH] of [MATH], we have [EQUATION].', '1603.06312-1-22-0': 'Assumption [REF] is satisfied when [MATH] and all fixed points of [MATH] have continuous cumulative distribution function.', '1603.06312-1-22-1': 'To see this, note that if [MATH] and [MATH] are continuous functions, then [EQUATION].', '1603.06312-1-22-2': 'If the fixed points of [MATH] do not necessarily have continuous cumulative distribution function, then Assumption [REF] is still satisfied if [MATH] is replaced by its ""regular"" version [MATH] (see [CITATION]).', '1603.06312-1-23-0': 'Suppose [MATH] and [MATH] are two fixed points of [MATH].', '1603.06312-1-23-1': 'To simplify notation, write [MATH] and [MATH].', '1603.06312-1-23-2': 'Let [MATH] and [MATH] be the optimally controlled state processes (starting at zero) in response to [MATH] and [MATH], respectively.', '1603.06312-1-23-3': 'Let [MATH].', '1603.06312-1-23-4': ""Using Ito's lemma and the PDE satisfied by [MATH] and [MATH], it is easy to show that [EQUATION] and [EQUATION]"", '1603.06312-1-23-5': 'Write [MATH], we obtain by subtracting [REF] from [REF] that [EQUATION]', '1603.06312-1-23-6': 'Letting [MATH] and using the continuity of [MATH] and [MATH] at the terminal time, we get [EQUATION]', '1603.06312-1-23-7': 'Now, exchange the role of [MATH] and [MATH].', '1603.06312-1-23-8': 'We also have [EQUATION]', '1603.06312-1-23-9': 'Adding [REF] and [REF], and using that [MATH], we get [EQUATION] where the last inequality follows from Assumption [REF].', '1603.06312-1-23-10': 'This implies [EQUATION]', '1603.06312-1-23-11': 'By the uniqueness of the solution of the SDE [REF], we must have [MATH] a.s. and [MATH].', '1603.06312-1-24-0': '## Approximate Nash equilibrium of the [MATH]-player game', '1603.06312-1-25-0': 'The MFG solution allows us to construct, using decentralized strategies, an approximate Nash equilibrium of the [MATH]-player game when [MATH] is large.', '1603.06312-1-25-1': 'In MFG literature, this is typically done using results from the propagation of chaos.', '1603.06312-1-25-2': 'Here we in some sense have a simpler problem since the mean-field interaction does not enter the dynamics of the state process.', '1603.06312-1-25-3': 'And it is this special structure that allows us to handle rank-based terminal payoff which fails to be Lipschitz continuous in general.', '1603.06312-1-26-0': 'A progressively measurable vector [MATH] is called an [MATH]-Nash equilibrium of the [MATH]-player game if', '1603.06312-1-27-0': 'We now state an additional Holder condition on [MATH] which allows us to get the convergence rate.', '1603.06312-1-27-1': 'It holds, for example, when [MATH] where [MATH] and [MATH].', '1603.06312-1-28-0': 'There exist constants [MATH] and [MATH] such that [MATH] for any [MATH] and [MATH].', '1603.06312-1-28-1': '[EQUATION]', '1603.06312-1-28-2': 'Let [MATH] be a fixed point of [MATH], and let [MATH] be defined as in the theorem statement.', '1603.06312-1-28-3': 'To keep the notation simple, we omit the superscript of any state process if it is controlled by the optimal Markovian feedback strategy [MATH].', '1603.06312-1-28-4': 'Let [EQUATION] be the value of the limiting game where [MATH] satisfies [REF], and [EQUATION] be the net gain of player [MATH] in an [MATH]-player game, if everybody use the candidate approximate Nash equilibrium [MATH].', '1603.06312-1-28-5': 'Here [MATH].', '1603.06312-1-28-6': 'Since our state processes do not depend on the empirical measure (the interaction is only through the terminal payoff), each [MATH] is simply an independent, identical copy of [MATH].', '1603.06312-1-28-7': 'Hence [EQUATION].', '1603.06312-1-28-8': 'Let us first show that [MATH] and [MATH] are close.', '1603.06312-1-28-9': 'We have [EQUATION].', '1603.06312-1-28-10': 'It follows from the [MATH]-Holder continuity of [MATH] that [EQUATION] where for [MATH], [MATH] denotes the empirical cumulative distribution function of [MATH] i.i.d. random variables with cumulative distribution function [MATH].', '1603.06312-1-28-11': 'By Dvoretzky-Kiefer-Wolfowitz inequality, we have [EQUATION].', '1603.06312-1-28-12': 'It follows that [EQUATION]', '1603.06312-1-28-13': 'Next, consider the system where player [MATH] makes a unilateral deviation from the candidate approximate NE equilibrium [MATH], say, he chooses an admissible control [MATH].', '1603.06312-1-28-14': 'Denote his controlled state process by [MATH], and the state processes of all other players by [MATH] as before for [MATH].', '1603.06312-1-28-15': 'Let [MATH] be the corresponding empirical measure of the terminal states, and [EQUATION] be the corresponding net gain for player [MATH].', '1603.06312-1-28-16': ""We have [EQUATION] where the inequality follows from the optimality of [MATH] for the [MATH]-th player's problem."", '1603.06312-1-28-17': ""Similar to how we estimate [MATH], we have [EQUATION] where we used Jensen's inequality in the fourth step."", '1603.06312-1-28-18': 'Combining the two estimates, we obtain [EQUATION].', '1603.06312-1-28-19': 'This shows [MATH] is an [MATH]-approximate Nash equilibrium.', '1603.06312-1-29-0': 'Without Assumption [REF], one can still use the continuity and boundedness of [MATH] to get convergence; that is, the MFG solution still provides an approximate Nash equilibrium of the [MATH]-player game.', '1603.06312-1-29-1': 'However, the convergence rate is no longer valid.', '1603.06312-1-30-0': '# Mean Field Approximation when there is common noise', '1603.06312-1-31-0': 'In this section, we assume [MATH] and [MATH] is independent of [MATH]; the latter means the reward is purely rank-dependent.', '1603.06312-1-31-1': 'Unlike the case with only idiosyncratic noises, since the common noise does not average out as [MATH], the limiting environment becomes a random measure instead of a deterministic one.', '1603.06312-1-31-2': 'So the MFG problem now reads:', '1603.06312-1-32-0': 'For [MATH], denote by [MATH] the probably measure obtained by shifting [MATH] to the left by [MATH].', '1603.06312-1-32-1': 'Observe that when [MATH] is independent of [MATH], we have [MATH].', '1603.06312-1-32-2': 'So we are precisely in the framework of translation invariant MFGs.', '1603.06312-1-32-3': 'In fact, purely rank-based functions should be another important example of translation invariant functions besides the convolution and local interaction given in [CITATION].', '1603.06312-1-32-4': 'In the general case without translation invariance, results have only been obtained in the weak formulation, see [CITATION].', '1603.06312-1-33-0': 'In the remaining discussion, let us refer to the problems [REF]-[REF] and [REF]-[REF] together with their respective fixed point problems as MFG[MATH] and MFG[MATH], respectively.', '1603.06312-1-33-1': 'A direct application of [CITATION] yields the following existence result.', '1603.06312-1-34-0': 'The intuition is that the whole population is affected in parallel by the common noise.', '1603.06312-1-34-1': 'Thus, the effect of common noise is essentially cancelled out in the optimization problem due to translation invariance.', '1603.06312-1-34-2': 'Such a random equilibrium measure is clearly [MATH]-measurable, hence is a strong MFG solution in the language of [CITATION].', '1603.06312-1-35-0': 'Here we are not taking the optimal feedback form [MATH] and plugging in the state process [MATH] with common noise, but directly using [MATH] where [MATH] is the solution to [REF] as an open loop control for MFG[MATH].', '1603.06312-1-35-1': 'Such a control is independent of the common noise.', '1603.06312-1-35-2': 'Here it is reasonable to use open loop controls because for an [MATH]-player game, the individual can observe all state processes.', '1603.06312-1-35-3': 'When [MATH] is large, the individual noises average out.', '1603.06312-1-35-4': 'Thus, observing the entire system should give some information about the common noise.', '1603.06312-1-35-5': 'Passing to the MFG limit, the agent should be allowed more information than the one generated by his own state process.', '1603.06312-1-36-0': 'Let Assumption [REF] hold.', '1603.06312-1-36-1': 'Let [MATH] be an equilibrium measure of MFG[MATH], and let [MATH] where [MATH] is the solution to [REF] with [MATH] replaced by [MATH].', '1603.06312-1-36-2': 'Then [MATH] form an [MATH]-Nash equilibrium of the [MATH]-player game with common noise.', '1603.06312-1-37-0': 'Let [MATH] and [MATH] be defined as in Proposition [REF] and Remark [REF].', '1603.06312-1-37-1': 'Also let [MATH] and [MATH].', '1603.06312-1-37-2': 'By Proposition [REF], we have [EQUATION].', '1603.06312-1-37-3': 'Also let [EQUATION] be the net gain of player [MATH] in an [MATH]-player game, if everybody use the candidate approximate Nash equilibrium [MATH].', '1603.06312-1-37-4': 'Here [MATH] denotes the empirical measure of the terminal state of the system.', '1603.06312-1-37-5': 'Translation invariance and the definition of [MATH] imply [EQUATION].', '1603.06312-1-37-6': 'Similarly, since [EQUATION] we also have [EQUATION].', '1603.06312-1-37-7': 'Hence we can rewrite [MATH] and [MATH] as [EQUATION] where we also used that [MATH] has the same distribution as [MATH].', '1603.06312-1-37-8': 'From the proof of Theorem [REF], we know [EQUATION].', '1603.06312-1-38-0': 'Next, suppose player [MATH] makes a unilateral deviation to some admissible control [MATH].', '1603.06312-1-38-1': 'Denote his controlled state process with and without common noise by [MATH] and [MATH], respectively.', '1603.06312-1-38-2': 'We have [MATH].', '1603.06312-1-38-3': 'Let [MATH] and [EQUATION].', '1603.06312-1-38-4': 'We have by the definition of [MATH] and the Holder continuity of [MATH] that [EQUATION]', '1603.06312-1-38-5': 'By translation invariance, [MATH].', '1603.06312-1-38-6': 'By definition of [MATH], [MATH].', '1603.06312-1-38-7': 'Also note that for [MATH], [MATH] since [MATH] is an equilibrium measure for MFG[MATH].', '1603.06312-1-38-8': 'Therefore, by expressing everything in terms of [MATH], [MATH] and [MATH], we are again back in the framework without common noise.', '1603.06312-1-38-9': 'The proof of Theorem [REF] implies that [EQUATION] where [MATH] denotes the empirical cumulative distribution function of [MATH] i.i.d. random variables with cumulative distribution function [MATH].', '1603.06312-1-38-10': 'We conclude that [MATH] and that [MATH] is an [MATH]-Nash equilibrium of the [MATH]-player game with common noise.', '1603.06312-1-39-0': 'The arbitrary control [MATH] in the above proof may depend on the common noise.', '1603.06312-1-39-1': ""However, the additional information of the common noise gives each player very little advantage when everyone else use their respective [MATH]'s which are independent of the common noise.""}","{'1603.06312-2-0-0': ""We discuss a natural game of competition and solve the corresponding mean field game with common noise when agents' rewards are rank-dependent."", '1603.06312-2-0-1': 'We use this solution to provide an approximate Nash equilibrium for the finite player game and obtain the rate of convergence.', '1603.06312-2-1-0': 'headings', '1603.06312-2-2-0': '# Introduction', '1603.06312-2-3-0': 'Mean field games (MFGs), introduced independently by [CITATION] and [CITATION], provide a useful approximation for the finite player Nash equilibrium problems in which the players are coupled through their empirical distribution.', '1603.06312-2-3-1': ""In particular, the mean field game limit gives an approximate Nash equilibrium, in which the agents' decision making is decoupled."", '1603.06312-2-3-2': 'In this paper we will consider a particular game in which the interaction of the players is through their ranks.', '1603.06312-2-3-3': ""Our main goal is to construct an approximate Nash equilibrium for a finite player game when the agents' dynamics are modulated by common noise."", '1603.06312-2-4-0': 'Rank-based mean field games, which have non-local mean field interactions, have been suggested as an extension of their model of oil production in [CITATION] and analyzed more generally by the recent paper by Carmona and Lacker [CITATION] using the weak formulation, when there is no common noise.', '1603.06312-2-4-1': 'There are currently no results on the rank-dependent mean field games with common noise.', '1603.06312-2-4-2': 'In order to solve the problem with common noise, we will make use of the mechanism in [CITATION] by solving the strong formulation of the rank-dependent mean field game without common noise and then by observing that purely rank-dependent reward functions are translation invariant.', '1603.06312-2-5-0': 'The rest of the paper is organized as follows: In Section [REF] we introduce the N-player game in which the players are coupled through the reward function which is rank-based.', '1603.06312-2-5-1': 'In Section [REF] we consider the case without common noise.', '1603.06312-2-5-2': 'We first find the mean field limit, discuss the uniqueness of the Nash equilibrium, and construct an approximate Nash equilibrium using the mean field limit.', '1603.06312-2-5-3': 'Using these results, in Section [REF] we use the mechanism in [CITATION] and obtain respective results for the common noise.', '1603.06312-2-6-0': '# The [MATH]-player game', '1603.06312-2-7-0': 'We consider [MATH] players each of whom control her own state variable and are rewarded based on their ranking.', '1603.06312-2-7-1': ""We will denote by [MATH] the [MATH]-th player's state variable, and assume that it satisfies the following stochastic differential equation (SDE) [EQUATION] where [MATH] represents the control by agent [MATH], and [MATH] and [MATH] are independent standard Brownian motions defined on some filtered probability space [MATH], representing the idiosyncratic noises and common noise, respectively."", '1603.06312-2-7-2': 'The game ends at time [MATH], when each player receives a rank-based reward minus the running cost of effort, which we will assumed to be quadratic [MATH] for some constant [MATH].', '1603.06312-2-7-3': 'In order to precisely define the rank-based reward [EQUATION] denote the empirical measure of the terminal state of the [MATH]-player system.', '1603.06312-2-7-4': 'Then [MATH] gives the fraction of players that finishes the same or worse than player [MATH].', '1603.06312-2-7-5': 'Let [MATH] be a bounded continuous function that is non-decreasing in both arguments.', '1603.06312-2-7-6': 'For any probability measure [MATH] on [MATH], write [MATH] where [MATH] denotes the cumulative distribution function of [MATH].', '1603.06312-2-7-7': 'The reward player [MATH] receives is given by [EQUATION].', '1603.06312-2-7-8': 'When [MATH] is independent of [MATH], the compensation scheme is purely rank-based.', '1603.06312-2-7-9': 'In general, we could have a mixture of absolute performance compensation and relative performance compensation.', '1603.06312-2-7-10': ""The objective of each player is to observe the progress of all players and choose his effort level to maximize the expected payoff, while anticipating the other players' strategies."", '1603.06312-2-8-0': ""The players' equilibrium expected payoffs, as functions of time and state variables, satisfy a system of [MATH] coupled nonlinear partial differential equations subject to discontinuous boundary conditions, which appears to be analytically intractable."", '1603.06312-2-8-1': 'Fortunately, in a large-population game, the impact of any individual on the whole population is very small.', '1603.06312-2-8-2': 'So it is often good enough for each player to ignore the private state of any other individual and simply optimize against the aggregate distribution of the population.', '1603.06312-2-8-3': 'As a consequence, the equilibrium strategies decentralize in the limiting game as [MATH].', '1603.06312-2-8-4': 'We shall use the mean field limit to construct approximate Nash equilibrium for the [MATH]-player game, both in the case with and without common noise.', '1603.06312-2-9-0': '# Mean Field Approximation when there is no common noise', '1603.06312-2-10-0': 'In this section, we assume [MATH].', '1603.06312-2-10-1': 'Solving the mean field game consists of two sub-problems: a stochastic control problem and a fixed-point problem (also called the consistency condition).', '1603.06312-2-10-2': 'For any Polish space [MATH], denote by [MATH] the space of probability measures on [MATH], and [MATH].', '1603.06312-2-11-0': ""We first fix a distribution [MATH] of the terminal state of the population, and consider a single player's optimization problem: [EQUATION] where [EQUATION] [MATH] is a Brownian motion, and [MATH] ranges over the set of progressively measurable processes satisfying [MATH]."", '1603.06312-2-11-1': 'The associated dynamic programming equation is [EQUATION] with terminal condition [MATH].', '1603.06312-2-11-2': 'Using the first-order condition, we obtain that the candidate optimizer is [MATH], and the Hamilton-Jacobi-Bellman (HJB) equation can be written as [EQUATION].', '1603.06312-2-11-3': 'The above equation can be linearized using the transformation [MATH], giving [EQUATION].', '1603.06312-2-11-4': 'Together with the boundary condition [MATH], we can easily write down the solution: [EQUATION] where [MATH] is a standard normal random variable, possibly defined on another probability space with measure [MATH].', '1603.06312-2-11-5': 'Let us further write [MATH] as an integral: [EQUATION]', '1603.06312-2-11-6': 'Using dominated convergence theorem, we can differentiate under the integral sign and get [EQUATION]', '1603.06312-2-11-7': 'Similarly, we obtain [EQUATION]', '1603.06312-2-11-8': 'Using [REF]-[REF], together with the boundedness and monotonicity of [MATH], we easily get the following estimates.', '1603.06312-2-11-9': 'Note that all bounds are independent of [MATH].', '1603.06312-2-12-0': 'The functions [MATH] and [MATH] satisfy [EQUATION] where [MATH].', '1603.06312-2-13-0': 'Since [MATH] is bounded, the drift coefficient [MATH] is Lipschitz continuous in [MATH].', '1603.06312-2-13-1': 'It follows that the optimally controlled state process, denoted by [MATH], has a strong solution on [MATH].', '1603.06312-2-13-2': 'Observe that [EQUATION].', '1603.06312-2-13-3': 'So the optimal cumulative effort is bounded by some constant independent of [MATH].', '1603.06312-2-13-4': 'It also implies that [MATH] has a well-defined limit as [MATH].', '1603.06312-2-13-5': 'Standard verification theorem yields that the solution to the HJB equation is the value function of the problem [REF]-[REF], and that [MATH] is the optimal Markovian feedback control.', '1603.06312-2-13-6': 'Finally, using dominated convergence theorem again, we can show that for [MATH], [EQUATION].', '1603.06312-2-13-7': 'The same limit also holds for [MATH] since [MATH] is bounded away from zero.', '1603.06312-2-13-8': 'In other words, the optimal effort level is small when the progress is very large in absolute value.', '1603.06312-2-13-9': 'This agrees with many real life observations that when a player has a very big lead, it is easy for him to show slackness; and when one is too far behind, he often gives up on the game instead of trying to catch up.', '1603.06312-2-14-0': '## Existence of a Nash equilibrium', '1603.06312-2-15-0': 'For each fixed [MATH], solving the stochastic control problem [REF]-[REF] yields a value function [MATH] and a best response [MATH].', '1603.06312-2-15-1': 'Suppose the game is started at time zero, with zero initial progress, the optimally controlled state process [MATH] of the generic player satisfies the SDE [EQUATION]', '1603.06312-2-15-2': 'Finding a Nash equilibrium for the limiting game is equivalent to finding a fixed point of the mapping [MATH], where [MATH] denotes the law of its argument.', '1603.06312-2-15-3': 'We shall sometimes refer to such a fixed point an equilibrium measure.', '1603.06312-2-16-0': ""Similar to [CITATION], we will use Schauder's fixed point theorem."", '1603.06312-2-16-1': 'Observe that for any [MATH], we have [EQUATION].', '1603.06312-2-16-2': 'This implies the set of [MATH] is tight in [MATH], hence relatively compact for the topology of weak convergence by Prokhorov theorem.', '1603.06312-2-16-3': 'Recall that [MATH].', '1603.06312-2-16-4': 'Equip [MATH] with the topology induced by the 1-Wasserstein metric: [EQUATION]', '1603.06312-2-16-5': 'Here [MATH] denotes the space of Lipschitz continuous functions on [MATH] whose Lipschitz constant is bounded by one.', '1603.06312-2-16-6': 'It is known that [MATH] is a complete separable metric space (see e.g. [CITATION]).', '1603.06312-2-16-7': 'We shall work with a subset of [MATH] defined by [EQUATION].', '1603.06312-2-16-8': 'It is easy to check that [MATH] is non-empty, convex and closed (for the topology induced by the [MATH] metric).', '1603.06312-2-16-9': 'Moreover, one can show using [CITATION] that any weakly convergent sequence [MATH] is also [MATH]-convergent.', '1603.06312-2-16-10': 'Therefore, [MATH] is also relatively compact for the topology induced by the [MATH] metric.', '1603.06312-2-16-11': 'So we have found a non-empty, convex and compact set [MATH] such that [MATH] maps [MATH] into itself.', '1603.06312-2-16-12': 'It remains to show [MATH] is continuous on [MATH].', '1603.06312-2-16-13': 'In the rest of the proof, the constant [MATH] may change from line to line.', '1603.06312-2-17-0': 'Let [MATH] such that [MATH] as [MATH].', '1603.06312-2-17-1': 'We wish to show [MATH].', '1603.06312-2-17-2': 'Note that [EQUATION].', '1603.06312-2-17-3': 'From Lemma [REF], we know that [MATH].', '1603.06312-2-17-4': 'Since [MATH], thanks to the dominated convergence theorem, it suffices to show for [MATH], [EQUATION].', '1603.06312-2-17-5': 'By Lemma [REF] and mean value theorem, we have that [EQUATION]', '1603.06312-2-17-6': 'So to show [MATH] it suffices to demonstrate for each fixed [MATH], [EQUATION] and [EQUATION]', '1603.06312-2-17-7': 'We first show [REF].', '1603.06312-2-17-8': 'Using the estimates in Lemma [REF], we get [EQUATION]', '1603.06312-2-17-9': 'Since all integrands are bounded, to show the expectations converge to zero, it suffice to check that the integrands converge to zero a.s. Fix [MATH], we know from [REF] that [EQUATION]', '1603.06312-2-17-10': 'Since [MATH], [MATH] also converges to [MATH] weakly, and the cumulative distribution function [MATH] converges to [MATH] at every point [MATH] at which [MATH] is continuous.', '1603.06312-2-17-11': 'It follows from the continuity of [MATH] that [MATH] converges to [MATH] at every point [MATH] at which [MATH] is continuous.', '1603.06312-2-17-12': 'Since [MATH] has at most countably many points of discontinuity, the random variable (the randomness comes from [MATH]) inside the expectation converges to zero [MATH]-a.s. Dominated convergence theorem then allows us to interchange the limit and the expectation, giving that [EQUATION].', '1603.06312-2-17-13': 'Similarly, from [REF] we obtain [EQUATION]', '1603.06312-2-17-14': 'Again, using that [MATH] has countably many points of discontinuity, one can show that [EQUATION].', '1603.06312-2-17-15': 'Putting everything together, we have proved [REF].', '1603.06312-2-18-0': ""Next, we show [REF] by Gronwall's inequality."", '1603.06312-2-18-1': 'Let [MATH] be given.', '1603.06312-2-18-2': 'We have for any [MATH], [EQUATION]', '1603.06312-2-18-3': 'By [REF] and bounded convergence theorem, we obtain [EQUATION].', '1603.06312-2-18-4': 'So for [MATH] large enough, we have [EQUATION]', '1603.06312-2-18-5': ""By Gronwall's inequality, [EQUATION]."", '1603.06312-2-18-6': 'This completes the proof of [REF], and thus the continuity of [MATH].', '1603.06312-2-18-7': ""By Schauder's fixed point theorem, there exists a fixed point of [MATH] in the set [MATH]."", '1603.06312-2-19-0': '## Uniqueness of Nash equilibrium.', '1603.06312-2-20-0': 'Nash equilibrium is rarely unique except under the following monotonicity assumption which is in the spirit of [CITATION]:', '1603.06312-2-21-0': 'For any two fixed points [MATH] and [MATH] of [MATH], we have [EQUATION].', '1603.06312-2-22-0': 'Assumption [REF] is satisfied when [MATH] and all fixed points of [MATH] have continuous cumulative distribution function.', '1603.06312-2-22-1': 'To see this, note that if [MATH] and [MATH] are continuous functions, then [EQUATION].', '1603.06312-2-22-2': 'If the fixed points of [MATH] do not necessarily have continuous cumulative distribution function, then Assumption [REF] is still satisfied if [MATH] is replaced by its ""regular"" version [MATH] (see [CITATION]).', '1603.06312-2-23-0': 'Suppose [MATH] and [MATH] are two fixed points of [MATH].', '1603.06312-2-23-1': 'To simplify notation, write [MATH] and [MATH].', '1603.06312-2-23-2': 'Let [MATH] and [MATH] be the optimally controlled state processes (starting at zero) in response to [MATH] and [MATH], respectively.', '1603.06312-2-23-3': 'Let [MATH].', '1603.06312-2-23-4': ""Using Ito's lemma and the PDE satisfied by [MATH] and [MATH], it is easy to show that [EQUATION] and [EQUATION]"", '1603.06312-2-23-5': 'Write [MATH], we obtain by subtracting [REF] from [REF] that [EQUATION]', '1603.06312-2-23-6': 'Letting [MATH] and using the continuity of [MATH] and [MATH] at the terminal time, we get [EQUATION]', '1603.06312-2-23-7': 'Now, exchange the role of [MATH] and [MATH].', '1603.06312-2-23-8': 'We also have [EQUATION]', '1603.06312-2-23-9': 'Adding [REF] and [REF], and using that [MATH], we get [EQUATION] where the last inequality follows from Assumption [REF].', '1603.06312-2-23-10': 'This implies [EQUATION]', '1603.06312-2-23-11': 'By the uniqueness of the solution of the SDE [REF], we must have [MATH] a.s. and [MATH].', '1603.06312-2-24-0': '## Approximate Nash equilibrium of the [MATH]-player game', '1603.06312-2-25-0': 'The MFG solution allows us to construct, using decentralized strategies, an approximate Nash equilibrium of the [MATH]-player game when [MATH] is large.', '1603.06312-2-25-1': 'In MFG literature, this is typically done using results from the propagation of chaos.', '1603.06312-2-25-2': 'Here we in some sense have a simpler problem since the mean-field interaction does not enter the dynamics of the state process.', '1603.06312-2-25-3': 'And it is this special structure that allows us to handle rank-based terminal payoff which fails to be Lipschitz continuous in general.', '1603.06312-2-26-0': 'A progressively measurable vector [MATH] is called an [MATH]-Nash equilibrium of the [MATH]-player game if', '1603.06312-2-27-0': 'We now state an additional Holder condition on [MATH] which allows us to get the convergence rate.', '1603.06312-2-27-1': 'It holds, for example, when [MATH] where [MATH] and [MATH].', '1603.06312-2-28-0': 'There exist constants [MATH] and [MATH] such that [MATH] for any [MATH] and [MATH].', '1603.06312-2-28-1': '[EQUATION]', '1603.06312-2-28-2': 'Let [MATH] be a fixed point of [MATH], and let [MATH] be defined as in the theorem statement.', '1603.06312-2-28-3': 'To keep the notation simple, we omit the superscript of any state process if it is controlled by the optimal Markovian feedback strategy [MATH].', '1603.06312-2-28-4': 'Let [EQUATION] be the value of the limiting game where [MATH] satisfies [REF], and [EQUATION] be the net gain of player [MATH] in an [MATH]-player game, if everybody use the candidate approximate Nash equilibrium [MATH].', '1603.06312-2-28-5': 'Here [MATH].', '1603.06312-2-28-6': 'Since our state processes do not depend on the empirical measure (the interaction is only through the terminal payoff), each [MATH] is simply an independent, identical copy of [MATH].', '1603.06312-2-28-7': 'Hence [EQUATION].', '1603.06312-2-28-8': 'Let us first show that [MATH] and [MATH] are close.', '1603.06312-2-28-9': 'We have [EQUATION].', '1603.06312-2-28-10': 'It follows from the [MATH]-Holder continuity of [MATH] that [EQUATION] where for [MATH], [MATH] denotes the empirical cumulative distribution function of [MATH] i.i.d. random variables with cumulative distribution function [MATH].', '1603.06312-2-28-11': 'By Dvoretzky-Kiefer-Wolfowitz inequality, we have [EQUATION].', '1603.06312-2-28-12': 'It follows that [EQUATION]', '1603.06312-2-28-13': 'Next, consider the system where player [MATH] makes a unilateral deviation from the candidate approximate Nash equilibrium [MATH]; say, he chooses an admissible control [MATH].', '1603.06312-2-28-14': 'Denote his controlled state process by [MATH], and the state processes of all other players by [MATH] as before for [MATH].', '1603.06312-2-28-15': 'Let [MATH] be the corresponding empirical measure of the terminal states, and [EQUATION] be the corresponding net gain for player [MATH].', '1603.06312-2-28-16': ""We have [EQUATION] where the inequality follows from the optimality of [MATH] for the [MATH]-th player's problem."", '1603.06312-2-28-17': ""Similar to how we estimate [MATH], we have [EQUATION] where we used Jensen's inequality in the fourth step."", '1603.06312-2-28-18': 'Combining the two estimates, we obtain [EQUATION].', '1603.06312-2-28-19': 'This shows [MATH] is an [MATH]-approximate Nash equilibrium.', '1603.06312-2-29-0': 'Without Assumption [REF], one can still use the continuity and boundedness of [MATH] to get convergence; that is, the MFG solution still provides an approximate Nash equilibrium of the [MATH]-player game.', '1603.06312-2-29-1': 'However, the convergence rate is no longer valid.', '1603.06312-2-30-0': '# Mean Field Approximation when there is common noise', '1603.06312-2-31-0': 'In this section, we assume [MATH] and [MATH] is independent of [MATH]; the latter means the reward is purely rank-dependent.', '1603.06312-2-31-1': 'Unlike the case with only idiosyncratic noises, since the common noise does not average out as [MATH], the limiting environment becomes a random measure instead of a deterministic one.', '1603.06312-2-31-2': 'So the MFG problem now reads:', '1603.06312-2-32-0': 'For [MATH], denote by [MATH] the probably measure obtained by shifting [MATH] to the left by [MATH].', '1603.06312-2-32-1': 'Observe that when [MATH] is independent of [MATH], we have [MATH].', '1603.06312-2-32-2': 'So we are precisely in the framework of translation invariant MFGs.', '1603.06312-2-32-3': 'In fact, purely rank-based functions should be another important example of translation invariant functions besides the convolution and local interaction given in [CITATION].', '1603.06312-2-32-4': 'In the general case without translation invariance, results have only been obtained in the weak formulation, see [CITATION].', '1603.06312-2-33-0': 'In the remaining discussion, let us refer to the problems [REF]-[REF] and [REF]-[REF] together with their respective fixed point problems as MFG[MATH] and MFG[MATH], respectively.', '1603.06312-2-33-1': 'A direct application of [CITATION] yields the following existence result.', '1603.06312-2-34-0': 'The intuition is that the whole population is affected in parallel by the common noise.', '1603.06312-2-34-1': 'Thus, the effect of common noise is essentially cancelled out in the optimization problem due to translation invariance.', '1603.06312-2-34-2': 'Such a random equilibrium measure is clearly [MATH]-measurable, hence is a strong MFG solution in the language of [CITATION].', '1603.06312-2-35-0': 'Here we are not taking the optimal feedback form [MATH] and plugging in the state process [MATH] with common noise, but directly using [MATH] where [MATH] is the solution to [REF] as an open loop control for MFG[MATH].', '1603.06312-2-35-1': 'Such a control is independent of the common noise.', '1603.06312-2-35-2': 'Here it is reasonable to use open loop controls because for an [MATH]-player game, the individual can observe all state processes.', '1603.06312-2-35-3': 'When [MATH] is large, the individual noises average out.', '1603.06312-2-35-4': 'Thus, observing the entire system should give some information about the common noise.', '1603.06312-2-35-5': 'Passing to the MFG limit, the agent should be allowed more information than the one generated by his own state process.', '1603.06312-2-36-0': 'Let Assumption [REF] hold.', '1603.06312-2-36-1': 'Let [MATH] be an equilibrium measure of MFG[MATH], and let [MATH] where [MATH] is the solution to [REF] with [MATH] replaced by [MATH].', '1603.06312-2-36-2': 'Then [MATH] form an [MATH]-Nash equilibrium of the [MATH]-player game with common noise.', '1603.06312-2-37-0': 'Let [MATH] and [MATH] be defined as in Proposition [REF] and Remark [REF].', '1603.06312-2-37-1': 'Also let [MATH] and [MATH].', '1603.06312-2-37-2': 'By Proposition [REF], we have [EQUATION].', '1603.06312-2-37-3': 'Also let [EQUATION] be the net gain of player [MATH] in an [MATH]-player game, if everybody use the candidate approximate Nash equilibrium [MATH].', '1603.06312-2-37-4': 'Here [MATH] denotes the empirical measure of the terminal state of the system.', '1603.06312-2-37-5': 'Translation invariance and the definition of [MATH] imply [EQUATION].', '1603.06312-2-37-6': 'Similarly, since [EQUATION] we also have [EQUATION].', '1603.06312-2-37-7': 'Hence we can rewrite [MATH] and [MATH] as [EQUATION] where we also used that [MATH] has the same distribution as [MATH].', '1603.06312-2-37-8': 'From the proof of Theorem [REF], we know [EQUATION].', '1603.06312-2-38-0': 'Next, suppose player [MATH] makes a unilateral deviation to some admissible control [MATH].', '1603.06312-2-38-1': 'Denote his controlled state process with and without common noise by [MATH] and [MATH], respectively.', '1603.06312-2-38-2': 'We have [MATH].', '1603.06312-2-38-3': 'Let [MATH] and [EQUATION].', '1603.06312-2-38-4': 'We have by the definition of [MATH] and the Holder continuity of [MATH] that [EQUATION]', '1603.06312-2-38-5': 'By translation invariance, [MATH].', '1603.06312-2-38-6': 'By definition of [MATH], [MATH].', '1603.06312-2-38-7': 'Also note that for [MATH], [MATH] since [MATH] is an equilibrium measure for MFG[MATH].', '1603.06312-2-38-8': 'Therefore, by expressing everything in terms of [MATH], [MATH] and [MATH], we are again back in the framework without common noise.', '1603.06312-2-38-9': 'The proof of Theorem [REF] implies that [EQUATION] where [MATH] denotes the empirical cumulative distribution function of [MATH] i.i.d. random variables with cumulative distribution function [MATH].', '1603.06312-2-38-10': 'We conclude that [MATH] and that [MATH] is an [MATH]-Nash equilibrium of the [MATH]-player game with common noise.', '1603.06312-2-39-0': 'The arbitrary control [MATH] in the above proof may depend on the common noise.', '1603.06312-2-39-1': ""However, the additional information of the common noise gives each player very little advantage when everyone else use their respective [MATH]'s which are independent of the common noise.""}","[['1603.06312-1-29-0', '1603.06312-2-29-0'], ['1603.06312-1-29-1', '1603.06312-2-29-1'], ['1603.06312-1-31-0', '1603.06312-2-31-0'], ['1603.06312-1-31-1', '1603.06312-2-31-1'], ['1603.06312-1-15-0', '1603.06312-2-15-0'], ['1603.06312-1-15-1', '1603.06312-2-15-1'], ['1603.06312-1-15-2', '1603.06312-2-15-2'], ['1603.06312-1-33-0', '1603.06312-2-33-0'], ['1603.06312-1-33-1', '1603.06312-2-33-1'], ['1603.06312-1-18-0', '1603.06312-2-18-0'], ['1603.06312-1-18-1', '1603.06312-2-18-1'], ['1603.06312-1-18-2', '1603.06312-2-18-2'], ['1603.06312-1-18-3', '1603.06312-2-18-3'], ['1603.06312-1-18-4', '1603.06312-2-18-4'], ['1603.06312-1-18-5', '1603.06312-2-18-5'], ['1603.06312-1-18-6', '1603.06312-2-18-6'], ['1603.06312-1-28-2', '1603.06312-2-28-2'], ['1603.06312-1-28-3', '1603.06312-2-28-3'], ['1603.06312-1-28-4', '1603.06312-2-28-4'], ['1603.06312-1-28-6', '1603.06312-2-28-6'], ['1603.06312-1-28-8', '1603.06312-2-28-8'], ['1603.06312-1-28-10', '1603.06312-2-28-10'], ['1603.06312-1-28-11', '1603.06312-2-28-11'], ['1603.06312-1-28-12', '1603.06312-2-28-12'], ['1603.06312-1-28-14', '1603.06312-2-28-14'], ['1603.06312-1-28-15', '1603.06312-2-28-15'], ['1603.06312-1-28-16', '1603.06312-2-28-16'], ['1603.06312-1-28-17', '1603.06312-2-28-17'], ['1603.06312-1-28-18', '1603.06312-2-28-18'], ['1603.06312-1-28-19', '1603.06312-2-28-19'], ['1603.06312-1-7-0', '1603.06312-2-7-0'], ['1603.06312-1-7-2', '1603.06312-2-7-2'], ['1603.06312-1-7-4', '1603.06312-2-7-4'], ['1603.06312-1-7-5', '1603.06312-2-7-5'], ['1603.06312-1-7-6', '1603.06312-2-7-6'], ['1603.06312-1-7-7', '1603.06312-2-7-7'], ['1603.06312-1-7-8', '1603.06312-2-7-8'], ['1603.06312-1-7-9', '1603.06312-2-7-9'], ['1603.06312-1-7-10', '1603.06312-2-7-10'], ['1603.06312-1-27-0', '1603.06312-2-27-0'], ['1603.06312-1-27-1', '1603.06312-2-27-1'], ['1603.06312-1-17-0', '1603.06312-2-17-0'], ['1603.06312-1-17-1', '1603.06312-2-17-1'], ['1603.06312-1-17-3', '1603.06312-2-17-3'], ['1603.06312-1-17-4', '1603.06312-2-17-4'], ['1603.06312-1-17-5', '1603.06312-2-17-5'], ['1603.06312-1-17-6', '1603.06312-2-17-6'], ['1603.06312-1-17-7', '1603.06312-2-17-7'], ['1603.06312-1-17-8', '1603.06312-2-17-8'], ['1603.06312-1-17-9', '1603.06312-2-17-9'], ['1603.06312-1-17-10', '1603.06312-2-17-10'], ['1603.06312-1-17-11', '1603.06312-2-17-11'], ['1603.06312-1-17-12', '1603.06312-2-17-12'], ['1603.06312-1-17-13', '1603.06312-2-17-13'], ['1603.06312-1-17-14', '1603.06312-2-17-14'], ['1603.06312-1-17-15', '1603.06312-2-17-15'], ['1603.06312-1-25-0', '1603.06312-2-25-0'], ['1603.06312-1-25-1', '1603.06312-2-25-1'], ['1603.06312-1-25-2', '1603.06312-2-25-2'], ['1603.06312-1-25-3', '1603.06312-2-25-3'], ['1603.06312-1-8-0', '1603.06312-2-8-0'], ['1603.06312-1-8-1', '1603.06312-2-8-1'], ['1603.06312-1-8-2', '1603.06312-2-8-2'], ['1603.06312-1-8-3', '1603.06312-2-8-3'], ['1603.06312-1-3-0', '1603.06312-2-3-0'], 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'1603.06312-3-10-2'], ['1603.06312-2-11-5', '1603.06312-3-10-5'], ['1603.06312-2-11-7', '1603.06312-3-10-7'], ['1603.06312-2-11-8', '1603.06312-3-10-8'], ['1603.06312-2-11-9', '1603.06312-3-10-9'], ['1603.06312-2-18-0', '1603.06312-3-19-0'], ['1603.06312-2-18-1', '1603.06312-3-19-1'], ['1603.06312-2-18-4', '1603.06312-3-19-4'], ['1603.06312-2-18-5', '1603.06312-3-19-5'], ['1603.06312-2-18-6', '1603.06312-3-19-6'], ['1603.06312-2-18-7', '1603.06312-3-19-7'], ['1603.06312-2-26-0', '1603.06312-3-28-0'], ['1603.06312-2-35-3', '1603.06312-3-40-3'], ['1603.06312-2-13-0', '1603.06312-3-12-0'], ['1603.06312-2-13-1', '1603.06312-3-12-1'], ['1603.06312-2-13-3', '1603.06312-3-12-3'], ['1603.06312-2-13-4', '1603.06312-3-12-4'], ['1603.06312-2-13-5', '1603.06312-3-12-5'], ['1603.06312-2-13-8', '1603.06312-3-12-8'], ['1603.06312-2-33-0', '1603.06312-3-37-0'], ['1603.06312-2-33-1', '1603.06312-3-37-1'], ['1603.06312-2-36-0', '1603.06312-3-41-0'], ['1603.06312-2-36-2', '1603.06312-3-41-2'], ['1603.06312-2-25-0', '1603.06312-3-27-0'], ['1603.06312-2-25-3', '1603.06312-3-27-3'], ['1603.06312-2-10-0', '1603.06312-3-9-0'], ['1603.06312-2-10-1', '1603.06312-3-9-1'], ['1603.06312-2-10-2', '1603.06312-3-9-2'], ['1603.06312-2-3-0', '1603.06312-3-1-0'], ['1603.06312-2-3-1', '1603.06312-3-1-1'], ['1603.06312-2-3-2', '1603.06312-3-1-2'], ['1603.06312-2-3-3', '1603.06312-3-1-3'], ['1603.06312-2-4-1', '1603.06312-3-2-1'], ['1603.06312-2-4-2', '1603.06312-3-2-2'], ['1603.06312-2-32-0', '1603.06312-3-36-0'], ['1603.06312-2-32-1', '1603.06312-3-36-1'], ['1603.06312-2-32-2', '1603.06312-3-36-2'], ['1603.06312-2-32-3', '1603.06312-3-36-3'], ['1603.06312-2-32-4', '1603.06312-3-36-4'], ['1603.06312-2-37-0', '1603.06312-3-42-0'], ['1603.06312-2-37-1', '1603.06312-3-42-1'], ['1603.06312-2-37-2', '1603.06312-3-42-2'], ['1603.06312-2-37-3', '1603.06312-3-42-3'], ['1603.06312-2-37-4', '1603.06312-3-42-4'], ['1603.06312-2-37-5', '1603.06312-3-42-5'], ['1603.06312-2-37-6', '1603.06312-3-42-6'], ['1603.06312-2-37-7', '1603.06312-3-42-7'], ['1603.06312-2-37-8', '1603.06312-3-42-8'], ['1603.06312-2-28-2', '1603.06312-3-32-0'], ['1603.06312-2-28-3', '1603.06312-3-32-1'], ['1603.06312-2-28-4', '1603.06312-3-32-2'], ['1603.06312-2-28-6', '1603.06312-3-32-4'], ['1603.06312-2-28-8', '1603.06312-3-32-6'], ['1603.06312-2-28-10', '1603.06312-3-32-8'], ['1603.06312-2-28-11', '1603.06312-3-32-9'], ['1603.06312-2-28-12', '1603.06312-3-32-10'], ['1603.06312-2-28-15', '1603.06312-3-32-13'], ['1603.06312-2-28-16', '1603.06312-3-32-14'], ['1603.06312-2-28-17', '1603.06312-3-32-15'], ['1603.06312-2-28-18', '1603.06312-3-32-16'], ['1603.06312-2-28-19', '1603.06312-3-32-17'], ['1603.06312-2-15-0', '1603.06312-3-14-0'], ['1603.06312-2-15-1', '1603.06312-3-14-1'], ['1603.06312-2-15-2', '1603.06312-3-14-2'], ['1603.06312-2-27-0', '1603.06312-3-29-0'], ['1603.06312-2-27-1', '1603.06312-3-29-1'], ['1603.06312-2-7-2', '1603.06312-3-5-2'], ['1603.06312-2-7-5', '1603.06312-3-6-2'], ['1603.06312-2-7-6', '1603.06312-3-6-3'], ['1603.06312-2-7-7', '1603.06312-3-6-4'], ['1603.06312-2-7-8', '1603.06312-3-6-5'], ['1603.06312-2-7-9', '1603.06312-3-6-6'], ['1603.06312-2-17-0', '1603.06312-3-17-0'], ['1603.06312-2-17-1', '1603.06312-3-17-1'], ['1603.06312-2-17-3', '1603.06312-3-17-3'], ['1603.06312-2-17-4', '1603.06312-3-17-4'], ['1603.06312-2-17-7', '1603.06312-3-18-2'], ['1603.06312-2-17-8', '1603.06312-3-18-3'], ['1603.06312-2-17-10', '1603.06312-3-18-5'], ['1603.06312-2-17-11', '1603.06312-3-18-6'], ['1603.06312-2-17-13', '1603.06312-3-18-9'], ['1603.06312-2-17-14', '1603.06312-3-18-10'], ['1603.06312-2-17-15', '1603.06312-3-18-11']]","[['1603.06312-1-4-0', '1603.06312-2-4-0'], ['1603.06312-1-4-1', '1603.06312-2-4-1'], ['1603.06312-1-4-2', '1603.06312-2-4-2'], ['1603.06312-1-15-3', '1603.06312-2-15-3'], ['1603.06312-1-18-7', '1603.06312-2-18-7'], ['1603.06312-1-28-13', '1603.06312-2-28-13'], ['1603.06312-1-7-1', '1603.06312-2-7-1'], ['1603.06312-1-7-3', '1603.06312-2-7-3'], ['1603.06312-1-8-4', '1603.06312-2-8-4'], ['1603.06312-1-5-0', '1603.06312-2-5-0'], ['1603.06312-1-0-0', '1603.06312-2-0-0'], ['1603.06312-1-11-0', '1603.06312-2-11-0'], ['1603.06312-2-38-1', '1603.06312-3-43-1'], ['1603.06312-2-23-0', '1603.06312-3-25-0'], ['1603.06312-2-23-5', '1603.06312-3-25-5'], ['1603.06312-2-11-3', '1603.06312-3-10-3'], ['1603.06312-2-11-6', '1603.06312-3-10-6'], ['1603.06312-2-18-3', '1603.06312-3-19-3'], ['1603.06312-2-35-2', '1603.06312-3-40-2'], ['1603.06312-2-35-4', '1603.06312-3-40-4'], ['1603.06312-2-35-5', '1603.06312-3-40-5'], ['1603.06312-2-13-6', '1603.06312-3-12-6'], ['1603.06312-2-13-7', '1603.06312-3-12-7'], ['1603.06312-2-13-9', '1603.06312-3-12-9'], ['1603.06312-2-36-1', '1603.06312-3-41-1'], ['1603.06312-2-25-1', '1603.06312-3-27-1'], ['1603.06312-2-25-2', '1603.06312-3-27-2'], ['1603.06312-2-4-0', '1603.06312-3-2-0'], ['1603.06312-2-28-13', '1603.06312-3-32-11'], ['1603.06312-2-28-14', '1603.06312-3-32-12'], ['1603.06312-2-15-3', '1603.06312-3-14-3'], ['1603.06312-2-7-0', '1603.06312-3-5-0'], ['1603.06312-2-7-1', '1603.06312-3-5-1'], ['1603.06312-2-7-3', '1603.06312-3-6-0'], ['1603.06312-2-7-4', '1603.06312-3-6-1'], ['1603.06312-2-7-10', '1603.06312-3-6-7'], ['1603.06312-2-17-5', '1603.06312-3-18-0'], ['1603.06312-2-17-6', '1603.06312-3-18-1'], ['1603.06312-2-17-9', '1603.06312-3-18-4']]",[],"[['1603.06312-2-11-4', '1603.06312-3-10-4'], ['1603.06312-2-22-2', '1603.06312-3-23-5'], ['1603.06312-2-17-12', '1603.06312-3-18-7']]",[],"['1603.06312-1-1-0', '1603.06312-1-12-0', '1603.06312-1-13-2', '1603.06312-1-16-3', '1603.06312-1-17-2', '1603.06312-1-20-0', '1603.06312-1-21-0', '1603.06312-1-23-3', '1603.06312-1-23-10', '1603.06312-1-28-0', '1603.06312-1-28-1', '1603.06312-1-28-5', '1603.06312-1-28-7', '1603.06312-1-28-9', '1603.06312-1-31-2', '1603.06312-1-38-2', '1603.06312-1-38-3', '1603.06312-2-1-0', '1603.06312-2-12-0', '1603.06312-2-13-2', '1603.06312-2-16-3', '1603.06312-2-17-2', '1603.06312-2-20-0', '1603.06312-2-21-0', '1603.06312-2-23-3', '1603.06312-2-23-10', '1603.06312-2-28-0', '1603.06312-2-28-1', '1603.06312-2-28-5', '1603.06312-2-28-7', '1603.06312-2-28-9', '1603.06312-2-31-2', '1603.06312-2-38-2', '1603.06312-2-38-3', '1603.06312-3-11-0', '1603.06312-3-12-2', '1603.06312-3-15-0', '1603.06312-3-16-3', '1603.06312-3-17-2', '1603.06312-3-19-2', '1603.06312-3-22-0', '1603.06312-3-25-3', '1603.06312-3-25-10', '1603.06312-3-30-0', '1603.06312-3-31-0', '1603.06312-3-31-1', '1603.06312-3-32-3', '1603.06312-3-32-5', '1603.06312-3-32-7', '1603.06312-3-35-2', '1603.06312-3-43-2', '1603.06312-3-43-3']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1603.06312,"{'1603.06312-3-0-0': '# Introduction', '1603.06312-3-1-0': 'Mean field games (MFGs), introduced independently by [CITATION] and [CITATION], provide a useful approximation for the finite player Nash equilibrium problems in which the players are coupled through their empirical distribution.', '1603.06312-3-1-1': ""In particular, the mean field game limit gives an approximate Nash equilibrium, in which the agents' decision making is decoupled."", '1603.06312-3-1-2': 'In this paper we will consider a particular game in which the interaction of the players is through their ranks.', '1603.06312-3-1-3': ""Our main goal is to construct an approximate Nash equilibrium for a finite player game when the agents' dynamics are modulated by common noise."", '1603.06312-3-2-0': 'Rank-based mean field games, which have non-local mean field interactions, have been suggested in [CITATION] and analyzed more generally by the recent paper by Carmona and Lacker [CITATION] using the weak formulation, when there is no common noise.', '1603.06312-3-2-1': 'There are currently no results on the rank-dependent mean field games with common noise.', '1603.06312-3-2-2': 'In order to solve the problem with common noise, we will make use of the mechanism in [CITATION] by solving the strong formulation of the rank-dependent mean field game without common noise and then by observing that purely rank-dependent reward functions are translation invariant.', '1603.06312-3-3-0': 'The rest of the paper is organized as follows: In Section [REF] we introduce the N-player game in which the players are coupled through the reward function which is rank-based.', '1603.06312-3-3-1': 'In Section [REF] we consider the case without common noise.', '1603.06312-3-3-2': 'We first find the mean field limit, discuss the uniqueness of the Nash equilibrium, and construct an approximate Nash equilibrium using the mean field limit.', '1603.06312-3-3-3': 'Using these results, in Section [REF] we use the mechanism in [CITATION] and obtain respective results for the common noise.', '1603.06312-3-4-0': '# The [MATH]-player game', '1603.06312-3-5-0': 'We consider [MATH] players each of whom controls her own state variable and is rewarded based on her rank.', '1603.06312-3-5-1': ""We will denote by [MATH] the [MATH]-th player's state variable, and assume that it satisfies the following stochastic differential equation (SDE) [EQUATION] where [MATH] is the control by agent [MATH], and [MATH] and [MATH] are independent standard Brownian motions defined on some filtered probability space [MATH], representing the idiosyncratic noises and common noise, respectively."", '1603.06312-3-5-2': 'The game ends at time [MATH], when each player receives a rank-based reward minus the running cost of effort, which we will assumed to be quadratic [MATH] for some constant [MATH].', '1603.06312-3-6-0': 'In order to precisely define the rank-based reward, let [EQUATION] denote the empirical measure of the terminal state of the [MATH]-player system.', '1603.06312-3-6-1': 'Then [MATH] gives the fraction of players that finish the same or worse than player [MATH].', '1603.06312-3-6-2': 'Let [MATH] be a bounded continuous function that is non-decreasing in both arguments.', '1603.06312-3-6-3': 'For any probability measure [MATH] on [MATH], write [MATH] where [MATH] denotes the cumulative distribution function of [MATH].', '1603.06312-3-6-4': 'The reward player [MATH] receives is given by [EQUATION].', '1603.06312-3-6-5': 'When [MATH] is independent of [MATH], the compensation scheme is purely rank-based.', '1603.06312-3-6-6': 'In general, we could have a mixture of absolute performance compensation and relative performance compensation.', '1603.06312-3-6-7': ""The objective of each player is to observe the progress of all players and choose her effort level to maximize the expected payoff, while anticipating the other players' strategies."", '1603.06312-3-7-0': ""The players' equilibrium expected payoffs, as functions of time and state variables, satisfy a system of [MATH] coupled nonlinear partial differential equations subject to discontinuous boundary conditions, which appears to be analytically intractable."", '1603.06312-3-7-1': 'Fortunately, in a large-population game, the impact of any individual on the whole population is very small.', '1603.06312-3-7-2': 'So it is often good enough for each player to ignore the private state of any other individual and simply optimize against the aggregate distribution of the population.', '1603.06312-3-7-3': 'As a consequence, the equilibrium strategies decentralize in the limiting game as [MATH].', '1603.06312-3-7-4': 'We shall use the mean field limit to construct approximate Nash equilibrium for the [MATH]-player game, both in the case with and without common noise.', '1603.06312-3-8-0': '# Mean field approximation when there is no common noise', '1603.06312-3-9-0': 'In this section, we assume [MATH].', '1603.06312-3-9-1': 'Solving the mean field game consists of two sub-problems: a stochastic control problem and a fixed-point problem (also called the consistency condition).', '1603.06312-3-9-2': 'For any Polish space [MATH], denote by [MATH] the space of probability measures on [MATH], and [MATH].', '1603.06312-3-10-0': ""We first fix a distribution [MATH] of the terminal state of the population, and consider a single player's optimization problem: [EQUATION] where [EQUATION] [MATH] is a Brownian motion, and [MATH] ranges over the set of progressively measurable processes satisfying [MATH]."", '1603.06312-3-10-1': 'The associated dynamic programming equation is [EQUATION] with terminal condition [MATH].', '1603.06312-3-10-2': 'Using the first-order condition, we obtain that the candidate optimizer is [MATH], and the Hamilton-Jacobi-Bellman (HJB) equation can be written as [EQUATION].', '1603.06312-3-10-3': 'The above equation can be linearized using the Cole-Hopf transformation [MATH], giving [EQUATION].', '1603.06312-3-10-4': 'Together with the boundary condition [MATH], we can easily write down the solution: [EQUATION] where [MATH] is a standard normal random variable.', '1603.06312-3-10-5': 'Let us further write [MATH] as an integral: [EQUATION]', '1603.06312-3-10-6': 'Using the dominated convergence theorem, we can differentiate under the integral sign and get [EQUATION]', '1603.06312-3-10-7': 'Similarly, we obtain [EQUATION]', '1603.06312-3-10-8': 'Using [REF]-[REF], together with the boundedness and monotonicity of [MATH], we easily get the following estimates.', '1603.06312-3-10-9': 'Note that all bounds are independent of [MATH].', '1603.06312-3-11-0': 'The functions [MATH] and [MATH] satisfy [EQUATION] where [MATH].', '1603.06312-3-12-0': 'Since [MATH] is bounded, the drift coefficient [MATH] is Lipschitz continuous in [MATH].', '1603.06312-3-12-1': 'It follows that the optimally controlled state process, denoted by [MATH], has a strong solution on [MATH].', '1603.06312-3-12-2': 'Observe that [EQUATION].', '1603.06312-3-12-3': 'So the optimal cumulative effort is bounded by some constant independent of [MATH].', '1603.06312-3-12-4': 'It also implies that [MATH] has a well-defined limit as [MATH].', '1603.06312-3-12-5': 'Standard verification theorem yields that the solution to the HJB equation is the value function of the problem [REF]-[REF], and that [MATH] is the optimal Markovian feedback control.', '1603.06312-3-12-6': 'Finally, using the dominated convergence theorem again, we can show that for [MATH], [EQUATION].', '1603.06312-3-12-7': 'The same limits also hold for [MATH] since [MATH] is bounded away from zero.', '1603.06312-3-12-8': 'In other words, the optimal effort level is small when the progress is very large in absolute value.', '1603.06312-3-12-9': 'This agrees with many real life observations that when a player has a very big lead, it is easy for her to show slackness; and when one is too far behind, she often gives up on the game instead of trying to catch up.', '1603.06312-3-13-0': '## Existence of a Nash equilibrium', '1603.06312-3-14-0': 'For each fixed [MATH], solving the stochastic control problem [REF]-[REF] yields a value function [MATH] and a best response [MATH].', '1603.06312-3-14-1': 'Suppose the game is started at time zero, with zero initial progress, the optimally controlled state process [MATH] of the generic player satisfies the SDE [EQUATION]', '1603.06312-3-14-2': 'Finding a Nash equilibrium for the limiting game is equivalent to finding a fixed point of the mapping [MATH], where [MATH] denotes the law of its argument.', '1603.06312-3-14-3': 'We shall sometimes refer to such a fixed point as an equilibrium measure.', '1603.06312-3-15-0': 'The mapping [MATH] has a fixed point.', '1603.06312-3-16-0': ""Similar to [CITATION], we will use Schauder's fixed point theorem."", '1603.06312-3-16-1': 'Observe that for any [MATH], we have [EQUATION].', '1603.06312-3-16-2': 'This implies the set of [MATH] is tight in [MATH], hence relatively compact for the topology of weak convergence by Prokhorov theorem.', '1603.06312-3-16-3': 'Recall that [MATH].', '1603.06312-3-16-4': 'Equip [MATH] with the topology induced by the 1-Wasserstein metric: [EQUATION]', '1603.06312-3-16-5': 'Here [MATH] denotes the space of Lipschitz continuous functions on [MATH] whose Lipschitz constant is bounded by one.', '1603.06312-3-16-6': 'It is known that [MATH] is a complete separable metric space (see e.g. [CITATION]).', '1603.06312-3-16-7': 'We shall work with a subset of [MATH] defined by [EQUATION].', '1603.06312-3-16-8': 'It is easy to check that [MATH] is non-empty, convex and closed (for the topology induced by the [MATH] metric).', '1603.06312-3-16-9': 'Moreover, one can show using [CITATION] that any weakly convergent sequence [MATH] is also [MATH]-convergent.', '1603.06312-3-16-10': 'Therefore, [MATH] is also relatively compact for the topology induced by the [MATH] metric.', '1603.06312-3-16-11': 'So we have found a non-empty, convex and compact set [MATH] such that [MATH] maps [MATH] into itself.', '1603.06312-3-16-12': 'It remains to show [MATH] is continuous on [MATH].', '1603.06312-3-16-13': 'In the rest of the proof, the constant [MATH] may change from line to line.', '1603.06312-3-17-0': 'Let [MATH] such that [MATH] as [MATH].', '1603.06312-3-17-1': 'We wish to show [MATH].', '1603.06312-3-17-2': 'Note that [EQUATION].', '1603.06312-3-17-3': 'From Lemma [REF], we know that [MATH].', '1603.06312-3-17-4': 'Since [MATH], thanks to the dominated convergence theorem, it suffices to show for [MATH], [EQUATION].', '1603.06312-3-18-0': 'By Lemma [REF] and the mean value theorem, we have that [EQUATION]', '1603.06312-3-18-1': 'So to show [MATH], it suffices to show that for each fixed [MATH], [EQUATION] and [EQUATION]', '1603.06312-3-18-2': 'We first show [REF].', '1603.06312-3-18-3': 'Using the estimates in Lemma [REF], we get [EQUATION]', '1603.06312-3-18-4': 'Since all integrands are bounded, to show the expectations converge to zero, it suffices to check that the integrands converge to zero a.s. Fix [MATH], we know from [REF] that [EQUATION]', '1603.06312-3-18-5': 'Since [MATH], [MATH] also converges to [MATH] weakly, and the cumulative distribution function [MATH] converges to [MATH] at every point [MATH] at which [MATH] is continuous.', '1603.06312-3-18-6': 'It follows from the continuity of [MATH] that [MATH] converges to [MATH] at every point [MATH] at which [MATH] is continuous.', '1603.06312-3-18-7': 'Since [MATH] has at most countably many points of discontinuity, the random variable inside the expectation converges to zero a.s.', '1603.06312-3-18-8': 'The dominated convergence theorem then allows us to interchange the limit and the expectation, giving that [EQUATION].', '1603.06312-3-18-9': 'Similarly, from [REF] we obtain [EQUATION]', '1603.06312-3-18-10': 'Again, using that [MATH] has countably many points of discontinuity, one can show that [EQUATION].', '1603.06312-3-18-11': 'Putting everything together, we have proved [REF].', '1603.06312-3-19-0': ""Next, we show [REF] by Gronwall's inequality."", '1603.06312-3-19-1': 'Let [MATH] be given.', '1603.06312-3-19-2': 'For any [MATH], [EQUATION]', '1603.06312-3-19-3': 'By [REF] and the bounded convergence theorem, we obtain [EQUATION].', '1603.06312-3-19-4': 'So for [MATH] large enough, we have [EQUATION]', '1603.06312-3-19-5': ""By Gronwall's inequality, [EQUATION]."", '1603.06312-3-19-6': 'This completes the proof of [REF], and thus the continuity of [MATH].', '1603.06312-3-19-7': ""By Schauder's fixed point theorem, there exists a fixed point of [MATH] in the set [MATH]."", '1603.06312-3-20-0': '## Uniqueness of Nash equilibrium.', '1603.06312-3-21-0': 'Let [MATH] be a class of measures in which uniqueness will be established.', '1603.06312-3-21-1': 'We first state a monotonicity assumption which is in the spirit of [CITATION].', '1603.06312-3-22-0': 'For any [MATH], we have [EQUATION].', '1603.06312-3-23-0': 'Take [MATH] to be the set of all measures in [MATH] that are absolutely continuous with respect to the Lebesgue measure, then Assumption [REF] is satisfied if the reward function [MATH] is Lipschitz continuous and [EQUATION] is differentiable and has non-negative partial derivatives [MATH].', '1603.06312-3-23-1': 'This includes any continuously differentiable function [MATH] which satisfies (i) [MATH] is convex, and (ii) [MATH] is non-decreasing.', '1603.06312-3-23-2': 'To see why [MATH] is sufficient to verify Assumption [REF], first note that for any [MATH], [MATH] and [MATH] are absolutely continuous.', '1603.06312-3-23-3': 'Using integration by parts for absolutely continuous functions, we have [EQUATION]', '1603.06312-3-23-4': 'Re-arranging terms and using that [MATH], we get [EQUATION]', '1603.06312-3-23-5': 'If one measures the rank of [MATH] with respect to a given distribution [MATH] using the ""regular"" cumulative distribution function [MATH], then for the case [MATH], Assumption [REF] is satisfied with [MATH] (see [CITATION]).', '1603.06312-3-24-0': 'Under Assumption [REF], [MATH] has at most one fixed point in [MATH].', '1603.06312-3-25-0': 'Suppose [MATH] and [MATH] are two fixed points of [MATH] in [MATH].', '1603.06312-3-25-1': 'To simplify notation, write [MATH] and [MATH].', '1603.06312-3-25-2': 'Let [MATH] and [MATH] be the optimally controlled state processes (starting at zero) in response to [MATH] and [MATH], respectively.', '1603.06312-3-25-3': 'Let [MATH].', '1603.06312-3-25-4': ""Using Ito's lemma and the PDE satisfied by [MATH] and [MATH], it is easy to show that [EQUATION] and [EQUATION]"", '1603.06312-3-25-5': 'Write [MATH], we obtain by subtracting [REF] from [REF] that [EQUATION].', '1603.06312-3-25-6': 'Letting [MATH] and using the continuity of [MATH] and [MATH] at the terminal time, we get [EQUATION]', '1603.06312-3-25-7': 'Now, exchange the role of [MATH] and [MATH].', '1603.06312-3-25-8': 'We also have [EQUATION]', '1603.06312-3-25-9': 'Adding [REF] and [REF], and using that [MATH], we get [EQUATION] where the last inequality follows from Assumption [REF].', '1603.06312-3-25-10': 'This implies [EQUATION]', '1603.06312-3-25-11': 'By the uniqueness of the solution of the SDE [REF], we must have [MATH] a.s. and [MATH].', '1603.06312-3-26-0': '## Approximate Nash equilibrium of the [MATH]-player game', '1603.06312-3-27-0': 'The MFG solution allows us to construct, using decentralized strategies, an approximate Nash equilibrium of the [MATH]-player game when [MATH] is large.', '1603.06312-3-27-1': 'In the MFG literature, this is typically done using results from the propagation of chaos.', '1603.06312-3-27-2': 'Here we have a simpler problem since the mean-field interaction does not enter the dynamics of the state process.', '1603.06312-3-27-3': 'And it is this special structure that allows us to handle rank-based terminal payoff which fails to be Lipschitz continuous in general.', '1603.06312-3-28-0': 'A progressively measurable vector [MATH] is called an [MATH]-Nash equilibrium of the [MATH]-player game if', '1603.06312-3-29-0': 'We now state an additional Holder condition on [MATH] which allows us to get the convergence rate.', '1603.06312-3-29-1': 'It holds, for example, when [MATH] where [MATH] and [MATH].', '1603.06312-3-30-0': 'There exist constants [MATH] and [MATH] such that [MATH] for any [MATH] and [MATH].', '1603.06312-3-31-0': 'Let Assumption [REF] hold.', '1603.06312-3-31-1': 'For any fixed point [MATH] of [MATH], [EQUATION] form an [MATH]-Nash equilibrium of the [MATH]-player game as [MATH].', '1603.06312-3-32-0': 'Let [MATH] be a fixed point of [MATH], and let [MATH] be defined as in the theorem statement.', '1603.06312-3-32-1': 'To keep the notation simple, we omit the superscript of any state process if it is controlled by the optimal Markovian feedback strategy [MATH].', '1603.06312-3-32-2': 'Let [EQUATION] be the value of the limiting game where [MATH] satisfies [REF], and [EQUATION] be the net gain of player [MATH] in an [MATH]-player game, if everybody use the candidate approximate Nash equilibrium [MATH].', '1603.06312-3-32-3': 'Here [MATH].', '1603.06312-3-32-4': 'Since our state processes do not depend on the empirical measure (the interaction is only through the terminal payoff), each [MATH] is simply an independent, identical copy of [MATH].', '1603.06312-3-32-5': 'Hence [EQUATION].', '1603.06312-3-32-6': 'Let us first show that [MATH] and [MATH] are close.', '1603.06312-3-32-7': 'We have [EQUATION].', '1603.06312-3-32-8': 'It follows from the [MATH]-Holder continuity of [MATH] that [EQUATION] where for [MATH], [MATH] denotes the empirical cumulative distribution function of [MATH] i.i.d. random variables with cumulative distribution function [MATH].', '1603.06312-3-32-9': 'By Dvoretzky-Kiefer-Wolfowitz inequality, we have [EQUATION].', '1603.06312-3-32-10': 'It follows that [EQUATION]', '1603.06312-3-32-11': 'Next, consider the system where player [MATH] makes a unilateral deviation from the candidate approximate Nash equilibrium [MATH]; say, she chooses an admissible control [MATH].', '1603.06312-3-32-12': 'Denote her controlled state process by [MATH], and the state processes of all other players by [MATH] as before for [MATH].', '1603.06312-3-32-13': 'Let [MATH] be the corresponding empirical measure of the terminal states, and [EQUATION] be the corresponding net gain for player [MATH].', '1603.06312-3-32-14': ""We have [EQUATION] where the inequality follows from the optimality of [MATH] for the [MATH]-th player's problem."", '1603.06312-3-32-15': ""Similar to how we estimate [MATH], we have [EQUATION] where we used Jensen's inequality in the fourth step."", '1603.06312-3-32-16': 'Combining the two estimates, we obtain [EQUATION].', '1603.06312-3-32-17': 'This shows [MATH] is an [MATH]-approximate Nash equilibrium.', '1603.06312-3-33-0': 'Without Assumption [REF], one can still use the continuity and boundedness of [MATH] to get convergence; that is, the MFG solution still provides an approximate Nash equilibrium of the [MATH]-player game.', '1603.06312-3-33-1': 'However, the convergence rate is no longer valid.', '1603.06312-3-34-0': '# Mean field approximation when there is common noise', '1603.06312-3-35-0': 'In this section, we assume [MATH] and [MATH] is independent of [MATH]; the latter means the reward is purely rank-dependent.', '1603.06312-3-35-1': 'Unlike the case with only idiosyncratic noises, since the common noise does not average out as [MATH], the limiting environment becomes a random measure instead of a deterministic one.', '1603.06312-3-35-2': 'So the MFG problem now reads:', '1603.06312-3-36-0': 'For [MATH], denote by [MATH] the probably measure obtained by shifting [MATH] to the left by [MATH].', '1603.06312-3-36-1': 'Observe that when [MATH] is independent of [MATH], we have [MATH].', '1603.06312-3-36-2': 'So we are precisely in the framework of translation invariant MFGs.', '1603.06312-3-36-3': 'In fact, purely rank-based functions should be another important example of translation invariant functions besides the convolution and local interaction given in [CITATION].', '1603.06312-3-36-4': 'In the general case without translation invariance, results have only been obtained in the weak formulation, see [CITATION].', '1603.06312-3-37-0': 'In the remaining discussion, let us refer to the problems [REF]-[REF] and [REF]-[REF] together with their respective fixed point problems as MFG[MATH] and MFG[MATH], respectively.', '1603.06312-3-37-1': 'A direct application of [CITATION] yields the following existence result.', '1603.06312-3-38-0': 'Let [MATH] be a (deterministic) equilibrium measure of MFG[MATH].', '1603.06312-3-38-1': 'Then [EQUATION] is a (random) equilibrium measure of MFG[MATH].', '1603.06312-3-38-2': 'Moreover, the optimal control associated with [MATH] for MFG[MATH] is also an optimal open loop control associated with [MATH] for MFG[MATH].', '1603.06312-3-39-0': 'The intuition is that the whole population is affected in parallel by the common noise.', '1603.06312-3-39-1': 'Thus, the effect of common noise is essentially cancelled out in the optimization problem due to translation invariance.', '1603.06312-3-39-2': 'Such a random equilibrium measure is clearly [MATH]-measurable, hence is a strong MFG solution in the language of [CITATION].', '1603.06312-3-40-0': 'In Proposition [REF], the equilibrium control for both MFG[MATH] and MFG[MATH] is [MATH] where [MATH] and [MATH] is the solution to [REF] controlled by [MATH].', '1603.06312-3-40-1': 'Such a control is a feedback control for MFG[MATH], but only an open loop control for MFG[MATH].', '1603.06312-3-40-2': 'Here it is reasonable to use open loop controls for MFG[MATH] because for an [MATH]-player game, the individual can observe all state processes.', '1603.06312-3-40-3': 'When [MATH] is large, the individual noises average out.', '1603.06312-3-40-4': 'Thus, observing the entire system should give each player some information about the common noise.', '1603.06312-3-40-5': 'Passing to the MFG limit, the individual should be allowed more information than that generated by her own state process.', '1603.06312-3-41-0': 'Let Assumption [REF] hold.', '1603.06312-3-41-1': 'Let [MATH] be an equilibrium measure of MFG[MATH], and let [MATH] where [MATH] is the solution to [REF] with [MATH] replaced by [MATH] and [MATH] replaced by [MATH].', '1603.06312-3-41-2': 'Then [MATH] form an [MATH]-Nash equilibrium of the [MATH]-player game with common noise.', '1603.06312-3-42-0': 'Let [MATH] and [MATH] be defined as in Proposition [REF] and Remark [REF].', '1603.06312-3-42-1': 'Also let [MATH] and [MATH].', '1603.06312-3-42-2': 'By Proposition [REF], we have [EQUATION].', '1603.06312-3-42-3': 'Also let [EQUATION] be the net gain of player [MATH] in an [MATH]-player game, if everybody use the candidate approximate Nash equilibrium [MATH].', '1603.06312-3-42-4': 'Here [MATH] denotes the empirical measure of the terminal state of the system.', '1603.06312-3-42-5': 'Translation invariance and the definition of [MATH] imply [EQUATION].', '1603.06312-3-42-6': 'Similarly, since [EQUATION] we also have [EQUATION].', '1603.06312-3-42-7': 'Hence we can rewrite [MATH] and [MATH] as [EQUATION] where we also used that [MATH] has the same distribution as [MATH].', '1603.06312-3-42-8': 'From the proof of Theorem [REF], we know [EQUATION].', '1603.06312-3-43-0': 'Next, suppose player [MATH] makes a unilateral deviation to some admissible control [MATH].', '1603.06312-3-43-1': 'Denote her controlled state process with and without common noise by [MATH] and [MATH], respectively.', '1603.06312-3-43-2': 'We have [MATH].', '1603.06312-3-43-3': 'Let [MATH] and [EQUATION].', '1603.06312-3-43-4': 'We have by the definition of [MATH] and the Holder continuity of [MATH] that [EQUATION]', '1603.06312-3-43-5': 'By translation invariance, [MATH].', '1603.06312-3-43-6': 'By definition of [MATH], [MATH].', '1603.06312-3-43-7': 'Also note that for [MATH], [MATH] since [MATH] is an equilibrium measure for MFG[MATH].', '1603.06312-3-43-8': 'Therefore, by expressing everything in terms of [MATH], [MATH] and [MATH], we are again back in the framework without common noise.', '1603.06312-3-43-9': 'The proof of Theorem [REF] implies that [EQUATION] where [MATH] denotes the empirical cumulative distribution function of [MATH] i.i.d. random variables with cumulative distribution function [MATH].', '1603.06312-3-43-10': 'We conclude that [MATH] and that [MATH] is an [MATH]-Nash equilibrium of the [MATH]-player game with common noise.', '1603.06312-3-44-0': 'The arbitrary control [MATH] in the above proof may depend on the common noise.', '1603.06312-3-44-1': ""However, the additional information of the common noise gives each player very little advantage when everyone else use their respective [MATH]'s which are independent of the common noise.""}",,
1512.03064,"{'1512.03064-1-0-0': 'A recent observational study of haloes of nearby Milky Way-like galaxies shows that only half of the current sample exhibits strong negative metallicity ([Fe/H]) gradients.', '1512.03064-1-0-1': 'This is at odds with predictions from hydrodynamical simulations where such gradients are ubiquitous.', '1512.03064-1-0-2': 'In this Letter, we use high resolution cosmological hydrodynamical simulations to study the [Fe/H] distribution of galactic haloes.', '1512.03064-1-0-3': 'We find that kinematically selected stellar haloes, including both in-situ and accreted particles, have an oblate [Fe/H] distribution.', '1512.03064-1-0-4': 'Spherical [Fe/H] radial profiles show strong negative gradients within 100 kpc, in agreement with previous numerical results.', '1512.03064-1-0-5': 'However, the projected median [Fe/H] profiles along the galactic disc minor axis, typically obtained in observations, are significantly flatter.', '1512.03064-1-0-6': 'The median [Fe/H] values at a given radius are larger for the spherical profiles than for the minor axis profiles by as much as 0.4 dex within the inner 50 kpc.', '1512.03064-1-0-7': ""Similar results are obtained if only the accreted stellar component is considered indicating that the differences between spherical and minor axis profiles are not purely driven by 'kicked-out' disc star particles formed in situ."", '1512.03064-1-0-8': 'Our study highlights the importance of performing careful comparisons between models and observations of halo [Fe/H] distributions.', '1512.03064-1-1-0': '# Introduction', '1512.03064-1-2-0': 'The stellar halo metallicity distributions of massive disc galaxies contain important information about their formation histories.', '1512.03064-1-2-1': 'Thus, they offer a direct test of galaxy formation models.', '1512.03064-1-2-2': 'While models of stellar halos built entirely from accretion events predict mostly flat metallicity profiles , models which include the contribution of in-situ star formation predict that haloes should have strong negative metallicity gradients .', '1512.03064-1-3-0': ""Hydrodynamical simulations have successfully reproduced the metallicity profile observed in M31's halo ."", '1512.03064-1-3-1': 'However, there seems to be some disagreement with the halo profiles observed in other nearby disc galaxies .', '1512.03064-1-3-2': 'Only half of the current sample presents strong gradients whereas the remaining half shows nearly flat metallicity profiles.', '1512.03064-1-3-3': 'The reason behind this discrepancy could plausibly lie in the different ways that observed and modelled metallicity profiles are derived.', '1512.03064-1-3-4': 'Results from simulations usually present spherically averaged metallicity profiles .', '1512.03064-1-3-5': 'In the Milky Way (MW) and all other massive disc galaxies, measurements of stellar halo metallicity profiles are made only for particular lines of sight which are, most commonly, perpendicular to the galactic disc plane (see e.g., for the MW, for M31, for GHOSTS galaxies), although see [CITATION] for M31.', '1512.03064-1-4-0': 'In this Letter we investigate the differences between spherically averaged and line of sight metallicity profiles of stellar haloes using high resolution cosmological hydrodynamical simulations of the formation of MW-like galaxies.', '1512.03064-1-4-1': 'Thanks to their high mass resolution, these simulations allow us to investigate in detail stellar halo properties of individual simulated galaxies, rather than averaged properties from large lower resolution simulations.', '1512.03064-1-4-2': 'Our goal is to explore whether the reported tension between models and observations is real or due to inconsistencies in the way they are compared.', '1512.03064-1-5-0': '# Numerical Simulations', '1512.03064-1-6-0': 'We use four high resolution cosmological zoom-in simulations of MW-sized galaxies from the ""Auriga"" simulation suite, performed using the state-of-the-art cosmological magneto-hydrodynamical code AREPO .', '1512.03064-1-6-1': 'A detailed description of these simulations can be found in [CITATION].', '1512.03064-1-6-2': 'Here we briefly describe their main features.', '1512.03064-1-7-0': 'Candidate galaxies were first selected from a parent dark matter only cosmological simulation, carried out in a periodic cube of side 100[MATH]Mpc.', '1512.03064-1-7-1': 'A [MATH]CDM cosmology was adopted, with parameters [MATH], [MATH], [MATH], and Hubble constant [MATH] km s[MATH] Mpc[MATH], [MATH] .', '1512.03064-1-7-2': ""Haloes were selected to have masses comparable to that of the MW's and to be relatively isolated at [MATH]."", '1512.03064-1-7-3': ""By applying a multi-mass particle 'zoom-in' technique, each halo was re-simulated at a higher resolution."", '1512.03064-1-8-0': 'Gas was added to the initial conditions and its evolution was followed by solving the Euler equations on an unstructured Voronoi mesh.', '1512.03064-1-8-1': 'The typical mass of a dark matter particle is [MATH] M[MATH], and the baryonic mass resolution is [MATH] M[MATH].', '1512.03064-1-8-2': 'The physical gravitational softening length grows with the scale factor up to a maximum of 369 pc, after which it is kept constant.', '1512.03064-1-8-3': 'The softening length of gas cells is scaled by the mean radius of the cell, with maximum physical softening of 1.85 kpc.', '1512.03064-1-8-4': 'The simulations include a comprehensive model for galaxy formation physics (see G15) which includes the most important baryonic processes.', '1512.03064-1-8-5': 'The model is specifically developed for the AREPO code and was calibrated to reproduce several observables such as the stellar mass to halo mass function, the galaxy luminosity functions and the history of the cosmic star formation rate density.', '1512.03064-1-9-0': 'A summary of the properties of the galaxies analysed in this work is presented in Table [REF].', '1512.03064-1-9-1': 'These four galaxies are a subset of the simulation suite introduced in G15 and were chosen to illustrate our results.', '1512.03064-1-9-2': 'The full suite will be analysed in more detail in a follow-up work.', '1512.03064-1-10-0': '# Stellar halo definition', '1512.03064-1-11-0': 'The definition of a stellar halo is rather arbitrary and several different criteria have been used in the past to isolate halo from disc and bulge star particles in simulated disc galaxies.', '1512.03064-1-11-1': 'Following previous work, we first define the stellar halo purely based on a kinematic decomposition, regardless of an in-situ or accreted origin for the stars.', '1512.03064-1-11-2': 'Discs are aligned with the XY plane as in [CITATION].', '1512.03064-1-11-3': 'For each star particle, we compute the circularity parameter [MATH] .', '1512.03064-1-11-4': 'Here [MATH] is the angular momentum around the disc symmetry axis and [MATH] is the maximum specific angular momentum possible at the same specific binding energy, [MATH].', '1512.03064-1-11-5': 'Three different subsets of star particles, with [MATH] and 0.65, are selected as the spheroidal component.', '1512.03064-1-11-6': 'The contamination from particles with disc kinematics in these subsets decreases with decreasing [MATH].', '1512.03064-1-11-7': 'The first and weakest constraint (i.e., [MATH]) is equivalent to the one adopted by, e.g., [CITATION].', '1512.03064-1-11-8': 'The third and more restrictive constraint (i.e. [MATH]) has been used by, e.g., [CITATION].', '1512.03064-1-11-9': 'Following [CITATION], particles from the spheroidal component that lie within 5 kpc from the galactic centre are defined to be bulge.', '1512.03064-1-11-10': 'We also analyse a set of particles with no constraint on [MATH], which is readily comparable to observations.', '1512.03064-1-12-0': 'In addition, we analyse a second sample of star particles in which only accreted particles are selected.', '1512.03064-1-12-1': 'This allows us to compare with previous results based purely on the accreted component of simulated galaxies .', '1512.03064-1-12-2': 'One should bear in mind, however, that some of these models are based on dark matter only simulations .', '1512.03064-1-12-3': 'Thus, the dynamical evolution of the baryonic component in such simulations is simplified due to, e.g., the lack of a disc gravitational potential .', '1512.03064-1-12-4': 'In this work we consider a particle to be accreted if, at its formation time, it was bound to any subhalo other than the host.', '1512.03064-1-12-5': 'The stars that form out of gas that was stripped from infalling satellites and has not yet mixed with the surroundings are not included in this sample.', '1512.03064-1-13-0': 'We note that there is an overlap between the kinematically-selected and accreted sets of stellar particles.', '1512.03064-1-13-1': 'Since the accreted component of a galaxy dominates beyond 20-30 kpc , the star particles in the outer regions ([MATH] kpc) are mostly the same in both samples.', '1512.03064-1-13-2': 'In all cases only star particles that at [MATH] are gravitationally bound to the host galaxy are selected.', '1512.03064-1-14-0': '# Results', '1512.03064-1-15-0': 'In Figure [REF] we show the stellar [MATH]band surface brightness map of the four galaxies analysed in this work.', '1512.03064-1-15-1': 'Only star particles that at [MATH] are gravitationally bound to the main galaxy are used to create the maps.', '1512.03064-1-15-2': 'A visual inspection reveals differences between, e.g, disc size (see Table 1) and amount of substructure and stellar halo shapes.', '1512.03064-1-15-3': 'The diversity in morphological properties of these MW-like simulated galaxies reflects the stochasticity inherent to the process of galaxy formation .', '1512.03064-1-16-0': 'Figure [REF] shows median metallicity [Fe/H] profiles for the kinematically selected stellar halo stars.', '1512.03064-1-16-1': 'The profiles are shown between 15 and 100 kpc from the galactic centre.', '1512.03064-1-16-2': 'These are the regions generally targeted by observations of external stellar haloes, given the difficulty of isolating halo stars in the inner regions of a disc galaxy.', '1512.03064-1-16-3': 'The black lines show the overall median [Fe/H] profile, computed in spherical shells around the galactic centre.', '1512.03064-1-16-4': 'We refer to this as the spherical profile.', '1512.03064-1-16-5': 'The blue and red solid lines show median [Fe/H] profiles along the minor and major axes respectively.', '1512.03064-1-16-6': 'These are computed in [MATH] projected wedges, as illustrated in Figure [REF].', '1512.03064-1-16-7': 'To increase the number of particles and smooth out sudden variations due to the presence of substructure, we include on both axes particles located within the diametrically opposed wedge.', '1512.03064-1-16-8': 'As already stated, observed stellar halo [Fe/H] profiles are typically obtained along galaxy minor axes .', '1512.03064-1-16-9': 'The spherically averaged [Fe/H] profiles generally presented in theoretical works cannot currently be measured either for external galaxies or for the MW.', '1512.03064-1-17-0': 'The top panels of Fig. [REF] show the results obtained with [MATH].', '1512.03064-1-17-1': 'These expose very significant differences between the spherical (black solid lines) and minor axis (blue solid lines) [MATH] profiles.', '1512.03064-1-17-2': 'At least within the inner 50 kpc, the spherical profiles show larger median [MATH] values, which can differ by as much as 0.4 dex.', '1512.03064-1-17-3': 'More importantly, the radial dependence of the median [MATH] is also different.', '1512.03064-1-17-4': 'The dashed lines on each panel show linear fits to the profiles.', '1512.03064-1-17-5': 'In general, flatter [MATH] profiles are obtained along the minor axis compared to their spherical counterpart.', '1512.03064-1-17-6': 'The red solid lines show [Fe/H] profiles along the major axis.', '1512.03064-1-17-7': 'Clearly, in all cases, the spherical profiles are strongly dominated by the [Fe/H] profiles along the major axis, at least out to [MATH] kpc.', '1512.03064-1-17-8': 'This indicates a flattened metallicity distribution in these stellar haloes.', '1512.03064-1-17-9': 'The difference between the major and minor axis profiles is mostly, although not entirely (see below), due to the contribution of in-situ kicked out disc star particles.', '1512.03064-1-17-10': 'These are particles formed in the disc that have been scattered into low [MATH] orbits and thus classified as halo stars .', '1512.03064-1-18-0': 'The bottom panels of Fig. [REF] show the results obtained for [MATH] 0.8 (green lines), 0.7 (orange lines) and 0.65 (purple lines).', '1512.03064-1-18-1': 'This contrasts our results for increasingly conservative classifications of stellar halo particles.', '1512.03064-1-18-2': 'Strong differences between minor and spherical [Fe/H] profiles are found even for our most restrictive selection criteria ([MATH].', '1512.03064-1-18-3': 'In some cases, such as galaxy Au 2, the difference between the median [Fe/H] profiles within 50 kpc becomes slightly smaller but is still significant, [MATH] dex.', '1512.03064-1-18-4': 'Large differences between the steepness of the profiles are found regardless of [MATH].', '1512.03064-1-18-5': 'We also show, with blue dash-dot lines, the minor axis [Fe/H] profiles obtained when no [MATH] selection is made.', '1512.03064-1-18-6': 'In this case, we omit the spherical profiles as they purely reflect the metallicity distribution of the discs within the inner [MATH] kpc.', '1512.03064-1-18-7': 'Interestingly, the minor axis [Fe/H] profiles are indistinguishable, regardless of the selection criteria.', '1512.03064-1-18-8': 'This indicates that the mass fraction of in-situ kicked-out disc stars along the minor axis is negligible at distances larger than 15 kpc .', '1512.03064-1-19-0': 'Figure [REF] shows [Fe/H] profiles taking into account only accreted particles.', '1512.03064-1-19-1': 'The top panels show the results obtained when no constraint in [MATH] is applied.', '1512.03064-1-19-2': 'Three out of the four galaxies show large differences between the spherical and minor axis [Fe/H] profiles, both in their median values and radial behaviour.', '1512.03064-1-19-3': 'Differences between these two profiles can be as large 0.3 dex (Au 24).', '1512.03064-1-19-4': 'This indicates that the accreted component of these simulated galaxies presents also a flattened [MATH] distribution.', '1512.03064-1-19-5': 'We find that as massive metal rich satellites are accreted, the host galactic disc responds by tilting its orientation .', '1512.03064-1-19-6': 'As a result, in many cases the cores of massive satellites are disrupted on a plane that is well-aligned with the host disc angular momentum .', '1512.03064-1-19-7': 'Figure [REF] shows an example of such an accretion event.', '1512.03064-1-19-8': 'Prior to infall, at [MATH] Gyr, the satellite reaches a total peak mass of [MATH].', '1512.03064-1-19-9': 'By [MATH] Gyr, the angle between the angular momentum vector of its inner bound core and that of the disc is [MATH].', '1512.03064-1-19-10': 'Note that, based on our kinematic selection criteria, not all the accreted particles would belong to the stellar halo.', '1512.03064-1-19-11': 'For [MATH], the accreted stellar mass fraction that belongs to the disc component varies between 5% (in Au 15) and 35% (in Au 2).', '1512.03064-1-20-0': 'The bottom panels of Fig. [REF] show accreted [Fe/H] profiles using the three different circularities previously defined.', '1512.03064-1-20-1': 'The differences between the spherical and minor axis [Fe/H] profiles disappear for halo Au 2 (left-most panel) in the bottom panels, indicating a strong orbital circularization of the inner metal rich cores of massive accreted satellites (see Fig. [REF]).', '1512.03064-1-20-2': 'Interestingly, for haloes Au 15 and Au 24 the difference between the spherical and minor axis [Fe/H] profiles remains almost the same for all values of [MATH].', '1512.03064-1-20-3': 'These examples show that the accreted component also contributes to the differences seen (both in median values and radial gradient) between the spherical and minor axis [Fe/H] profiles.', '1512.03064-1-21-0': '# Summary and Conclusions', '1512.03064-1-22-0': 'While previous studies based on hydrodynamical simulations indicate that negative metallicity gradients are a ubiquitous feature of simulated MW-like stellar haloes , half of the current observed sample of nearby disc galaxies suggests flat metallicity profiles .', '1512.03064-1-22-1': 'Motivated by this apparent discrepancy, we use four high resolution cosmological hydrodynamical simulations of MW-like galaxies to characterise the metallicity profiles of their stellar haloes.', '1512.03064-1-22-2': 'Our goal is to study whether this discrepancy could be due to the different ways in which simulations and data are compared.', '1512.03064-1-22-3': 'In contrast to numerical studies where spherically averaged [Fe/H] profiles are shown, observations are obtained along a particular direction, which ideally is perpendicular to the disc plane so as to minimise disc contamination.', '1512.03064-1-23-0': 'Following previous simulation studies, we define stellar haloes purely based on kinematic criteria.', '1512.03064-1-23-1': 'We find that spherical [Fe/H] profiles show large negative gradients, in agreement with previous work.', '1512.03064-1-23-2': 'However, significant differences are obtained when the profiles are computed in projection along the minor axis of the galactic disc.', '1512.03064-1-23-3': 'Not only are the median [Fe/H] values larger in the spherical profiles (up to [MATH] dex), at least within the inner 50 kpc, but also the gradients are steeper in general than along the minor axis.', '1512.03064-1-23-4': ""We find that the spherical profiles are dominated by the halo [Fe/H] distribution along the disc's major axis."", '1512.03064-1-23-5': 'We also show that the minor axis [Fe/H] profiles are indistinguishable for the different circularity criteria we have adopted, even when no kinematic selection is imposed.', '1512.03064-1-23-6': 'This indicates that a straightforward comparison between observations and models of stellar haloes is possible along the minor axis.', '1512.03064-1-23-7': 'Furthermore, of the four examples we have analysed, one has a flat [Fe/H] profile, one has a rather steep trend and the other are intermediate.', '1512.03064-1-23-8': 'This diversity is reminiscent of the variety of profiles seen in the observational data.', '1512.03064-1-24-0': 'Interestingly, in most cases, very similar results are obtained when only the accreted component of the stellar halo is considered, indicating that our results are not entirely driven by in-situ kicked out disc star particles.', '1512.03064-1-24-1': 'Note that even in dark matter only simulations, where baryonic effects such as a thin disc potential are not taken into account, stellar haloes generally present a flattened density distribution .', '1512.03064-1-24-2': 'Such flattening is the result of the anisotropic distribution of satellite orbits.', '1512.03064-1-24-3': 'Thus, these models are also expected to show differences between spherical and line-of-sight [Fe/H] profiles.', '1512.03064-1-25-0': 'The existence of stellar halo [Fe/H] gradients (or the lack of them) in massive disc galaxies depends strongly on how the profiles are constructed.', '1512.03064-1-25-1': 'Such profiles depend on the adopted definition of the stellar halo, which often differs among observations and theoretical works.', '1512.03064-1-25-2': 'Since stellar halo [Fe/H] profiles are an important diagnostic of galaxy formation history, a careful and faithful comparison between observations and models is crucial in order to interpret observations correctly and constrain the models.'}","{'1512.03064-2-0-0': 'A recent observational study of haloes of nearby Milky Way-like galaxies shows that only half (four out of eight) of the current sample exhibits strong negative metallicity ([Fe/H]) gradients.', '1512.03064-2-0-1': 'This is at odds with predictions from hydrodynamical simulations where such gradients are ubiquitous.', '1512.03064-2-0-2': 'In this Letter, we use high resolution cosmological hydrodynamical simulations to study the [Fe/H] distribution of galactic haloes.', '1512.03064-2-0-3': 'We find that kinematically selected stellar haloes, including both in-situ and accreted particles, have an oblate [Fe/H] distribution.', '1512.03064-2-0-4': 'Spherical [Fe/H] radial profiles show strong negative gradients within 100 kpc, in agreement with previous numerical results.', '1512.03064-2-0-5': 'However, the projected median [Fe/H] profiles along the galactic disc minor axis, typically obtained in observations, are significantly flatter.', '1512.03064-2-0-6': 'The median [Fe/H] values at a given radius are larger for the spherical profiles than for the minor axis profiles by as much as 0.4 dex within the inner 50 kpc.', '1512.03064-2-0-7': 'Similar results are obtained if only the accreted stellar component is considered indicating that the differences between spherical and minor axis profiles are not purely driven by heated disc star particles formed in situ.', '1512.03064-2-0-8': 'Our study highlights the importance of performing careful comparisons between models and observations of halo [Fe/H] distributions.', '1512.03064-2-1-0': '# Introduction', '1512.03064-2-2-0': 'The stellar halo metallicity distributions of massive disc galaxies contain important information about their formation histories.', '1512.03064-2-2-1': 'Thus, they offer a direct test of galaxy formation models.', '1512.03064-2-2-2': 'While models of stellar halos built entirely from accretion events predict mostly flat metallicity profiles , models which include the contribution of in-situ star formation predict that haloes should have strong negative metallicity gradients .', '1512.03064-2-3-0': ""Hydrodynamical simulations have successfully reproduced the metallicity profile observed in M31's halo ."", '1512.03064-2-3-1': 'However, there seems to be some disagreement with the halo profiles observed in other nearby disc galaxies .', '1512.03064-2-3-2': 'Only half of the current sample of eight studied galaxies (including M31 and the MW) presents strong gradients whereas the remaining half shows nearly flat metallicity profiles.', '1512.03064-2-3-3': 'The reason behind this discrepancy could plausibly lie in the different ways that observed and modelled metallicity profiles are derived.', '1512.03064-2-3-4': 'Results from simulations usually present spherically averaged metallicity profiles .', '1512.03064-2-3-5': 'In the Milky Way (MW) and all other massive disc galaxies, measurements of stellar halo metallicity profiles are made only for particular lines of sight which are, most commonly, perpendicular to the galactic disc plane (see e.g., for the MW, for M31, for GHOSTS galaxies), although see [CITATION] for M31.', '1512.03064-2-4-0': 'In this Letter we investigate the differences between spherically averaged and line of sight metallicity profiles of stellar haloes using high resolution cosmological hydrodynamical simulations of the formation of MW-like galaxies.', '1512.03064-2-4-1': 'Thanks to their high mass resolution, these simulations allow us to investigate in detail stellar halo properties of individual simulated galaxies, rather than averaged properties from large lower resolution simulations.', '1512.03064-2-4-2': 'Our goal is to explore whether the reported tension between models and observations is real or due to inconsistencies in the way they are compared.', '1512.03064-2-5-0': '# Numerical Simulations', '1512.03064-2-6-0': 'We use four high resolution cosmological zoom-in simulations of MW-sized galaxies from the ""Auriga"" simulation suite, performed using the state-of-the-art cosmological magneto-hydrodynamical code AREPO .', '1512.03064-2-6-1': 'A detailed description of these simulations can be found in [CITATION].', '1512.03064-2-6-2': 'Here we briefly describe their main features.', '1512.03064-2-7-0': 'Candidate galaxies were first selected from a parent dark matter only cosmological simulation, carried out in a periodic cube of side 100[MATH]Mpc.', '1512.03064-2-7-1': 'A [MATH]CDM cosmology was adopted, with parameters [MATH], [MATH], [MATH], and Hubble constant [MATH] km s[MATH] Mpc[MATH], [MATH] .', '1512.03064-2-7-2': ""Haloes were selected to have masses comparable to that of the MW's and to be relatively isolated at [MATH]."", '1512.03064-2-7-3': ""By applying a multi-mass particle 'zoom-in' technique, each halo was re-simulated at a higher resolution."", '1512.03064-2-8-0': 'Gas was added to the initial conditions and its evolution was followed by solving the Euler equations on an unstructured Voronoi mesh.', '1512.03064-2-8-1': 'The typical mass of a dark matter particle is [MATH] M[MATH], and the baryonic mass resolution is [MATH] M[MATH].', '1512.03064-2-8-2': 'The physical gravitational softening length grows with the scale factor up to a maximum of 369 pc, after which it is kept constant.', '1512.03064-2-8-3': 'The softening length of gas cells is scaled by the mean radius of the cell, with maximum physical softening of 1.85 kpc.', '1512.03064-2-8-4': 'The simulations include a comprehensive model for galaxy formation physics (see G16) which includes the most important baryonic processes.', '1512.03064-2-8-5': 'The model is specifically developed for the AREPO code and was calibrated to reproduce several observables such as the stellar mass to halo mass function, the galaxy luminosity functions and the history of the cosmic star formation rate density.', '1512.03064-2-9-0': 'A summary of the properties of the galaxies analysed in this work is presented in Table [REF].', '1512.03064-2-9-1': 'These four galaxies are a subset of the simulation suite introduced in G16 and were chosen to clearly illustrate our results.', '1512.03064-2-9-2': 'The full suite shows consistent results with those presented here and will be analysed in more detail in a follow-up work.', '1512.03064-2-10-0': '# Stellar halo definition', '1512.03064-2-11-0': 'The definition of a stellar halo is rather arbitrary and several different criteria have been used in the past to isolate halo from disc and bulge star particles in simulated disc galaxies.', '1512.03064-2-11-1': 'Following previous work, we first define the stellar halo purely based on a kinematic decomposition, regardless of an in-situ or accreted origin for the stars.', '1512.03064-2-11-2': 'Discs are aligned with the XY plane as in [CITATION].', '1512.03064-2-11-3': 'For each star particle, we compute the circularity parameter [MATH] .', '1512.03064-2-11-4': 'Here [MATH] is the angular momentum around the disc symmetry axis and [MATH] is the maximum specific angular momentum possible at the same specific binding energy, [MATH].', '1512.03064-2-11-5': 'Three different subsets of star particles, with [MATH] and 0.65, are selected as the spheroidal component.', '1512.03064-2-11-6': 'The contamination from particles with disc kinematics in these subsets decreases with decreasing [MATH].', '1512.03064-2-11-7': 'The first and weakest constraint (i.e., [MATH]) is equivalent to the one adopted by, e.g., [CITATION].', '1512.03064-2-11-8': 'The third and more restrictive constraint (i.e. [MATH]) has been used by, e.g., [CITATION].', '1512.03064-2-11-9': 'Following [CITATION], particles from the spheroidal component that lie within 5 kpc from the galactic centre are defined to be bulge.', '1512.03064-2-11-10': 'Here we do not attempt to select halo star particles as accurately as possible.', '1512.03064-2-11-11': 'Instead, our goal is to replicate previously (and commonly) used selection criteria.', '1512.03064-2-11-12': 'We also analyse a set of particles with no constraint on [MATH], which is readily comparable to observations.', '1512.03064-2-12-0': 'In addition, we analyse a second sample of star particles containing only accreted particles.', '1512.03064-2-12-1': 'This allows us to compare with previous results based purely on the accreted component of simulated galaxies .', '1512.03064-2-12-2': 'One should bear in mind, however, that some of these models are based on dark matter only simulations .', '1512.03064-2-12-3': 'Thus, the dynamical evolution of the baryonic component in such simulations is simplified due to, e.g., the lack of a disc gravitational potential .', '1512.03064-2-12-4': 'In this work we consider a particle to be accreted if, at its formation time, it was bound to any subhalo other than the host.', '1512.03064-2-12-5': 'The stars that form out of gas that was stripped from infalling satellites and has not yet mixed with the surroundings are not included in this sample.', '1512.03064-2-13-0': 'We note that there is an overlap between the kinematically-selected and accreted sets of stellar particles.', '1512.03064-2-13-1': 'Since the accreted component of a galaxy dominates beyond 20-30 kpc , the star particles in the outer regions ([MATH] kpc) are mostly the same in both samples.', '1512.03064-2-13-2': 'In all cases only star particles that at [MATH] are gravitationally bound to the host galaxy are selected.', '1512.03064-2-14-0': '# Results', '1512.03064-2-15-0': 'In Figure [REF] we show the stellar [MATH]band surface brightness map of the four galaxies analysed in this work.', '1512.03064-2-15-1': 'Only star particles that at [MATH] are gravitationally bound to the main galaxy are used to create the maps.', '1512.03064-2-15-2': 'A visual inspection reveals differences between, e.g, disc size (see Table 1) and amount of substructure and stellar halo shapes.', '1512.03064-2-15-3': 'The diversity in morphological properties of these MW-like simulated galaxies reflects the stochasticity inherent to the process of galaxy formation .', '1512.03064-2-16-0': 'Figure [REF] shows median metallicity [Fe/H] profiles for the kinematically selected stellar halo stars.', '1512.03064-2-16-1': 'The profiles are shown between 15 and 100 kpc from the galactic centre.', '1512.03064-2-16-2': 'These are the regions generally targeted by observations of external stellar haloes, given the difficulty of isolating halo stars in the inner regions of a disc galaxy.', '1512.03064-2-16-3': 'The black lines show the overall median [Fe/H] profile, computed in spherical shells around the galactic centre.', '1512.03064-2-16-4': 'We refer to this as the spherical profile.', '1512.03064-2-16-5': 'The blue and red solid lines show median [Fe/H] profiles along the minor and major axes respectively.', '1512.03064-2-16-6': 'These are computed in [MATH] projected wedges, as illustrated in Figure [REF].', '1512.03064-2-16-7': 'To increase the number of particles and smooth out sudden variations due to the presence of substructure, we include on both axes particles located within the diametrically opposed wedge.', '1512.03064-2-16-8': 'As already stated, observed stellar halo [Fe/H] profiles are typically obtained along galaxy minor axes .', '1512.03064-2-16-9': 'The spherically averaged [Fe/H] profiles generally presented in theoretical works cannot currently be measured either for external galaxies or for the MW.', '1512.03064-2-17-0': 'The top panels of Fig. [REF] show the results obtained with [MATH].', '1512.03064-2-17-1': 'These expose very significant differences between the spherical (black solid lines) and minor axis (blue solid lines) [MATH] profiles.', '1512.03064-2-17-2': 'At least within the inner 50 kpc, the spherical profiles show larger median [MATH] values, which can differ by as much as 0.4 dex.', '1512.03064-2-17-3': 'More importantly, the radial dependence of the median [MATH] is also different.', '1512.03064-2-17-4': 'The dashed lines on each panel show linear fits to the profiles.', '1512.03064-2-17-5': 'In general, flatter [MATH] profiles are obtained along the minor axis compared to their spherical counterpart.', '1512.03064-2-17-6': 'The red solid lines show [Fe/H] profiles along the major axis.', '1512.03064-2-17-7': 'Clearly, in all cases, the spherical profiles are strongly dominated by the [Fe/H] profiles along the major axis, at least out to [MATH] kpc.', '1512.03064-2-17-8': 'This indicates a flattened metallicity distribution in these stellar haloes.', '1512.03064-2-17-9': 'The difference between the major and minor axis profiles is mostly, although not entirely (see below), due to the contribution of in-situ heated disc star particles.', '1512.03064-2-17-10': 'These are particles formed in the disc that have been scattered into low [MATH] orbits and thus classified as halo stars .', '1512.03064-2-18-0': 'The bottom panels of Fig. [REF] show the results obtained for [MATH] 0.8 (green lines), 0.7 (orange lines) and 0.65 (purple lines).', '1512.03064-2-18-1': 'This contrasts our results for increasingly conservative classifications of stellar halo particles.', '1512.03064-2-18-2': 'Strong differences between minor and spherical [Fe/H] profiles are found even for our most restrictive selection criteria ([MATH].', '1512.03064-2-18-3': 'In some cases, such as galaxy Au 2, the difference between the median [Fe/H] profiles within 50 kpc becomes slightly smaller but is still significant, [MATH] dex.', '1512.03064-2-18-4': 'Large differences between the steepness of the profiles are found regardless of [MATH].', '1512.03064-2-18-5': 'We also show, with blue dash-dot lines, the minor axis [Fe/H] profiles obtained when no [MATH] selection is made.', '1512.03064-2-18-6': 'In this case, we omit the spherical profiles as they purely reflect the metallicity distribution of the discs within the inner [MATH] kpc.', '1512.03064-2-18-7': 'Interestingly, the minor axis [Fe/H] profiles are indistinguishable, regardless of the selection criteria.', '1512.03064-2-18-8': 'This indicates that the mass fraction of in-situ heated disc stars along the minor axis is negligible at distances larger than 15 kpc .', '1512.03064-2-19-0': 'Figure [REF] shows [Fe/H] profiles taking into account only accreted particles.', '1512.03064-2-19-1': 'The top panels show the results obtained when no constraint in [MATH] is applied.', '1512.03064-2-19-2': 'Three out of the four galaxies show large differences between the spherical and minor axis [Fe/H] profiles, both in their median values and radial behaviour.', '1512.03064-2-19-3': 'Differences between these two profiles can be as large 0.3 dex (Au 24).', '1512.03064-2-19-4': 'This indicates that the accreted component of these simulated galaxies presents also a flattened [MATH] distribution.', '1512.03064-2-19-5': 'We find that as massive metal rich satellites are accreted, the host galactic disc responds by tilting its orientation .', '1512.03064-2-19-6': 'As a result, in many cases the cores of massive satellites are disrupted on a plane that is well-aligned with the host disc angular momentum .', '1512.03064-2-19-7': 'Figure [REF] shows one example of such accretion events.', '1512.03064-2-19-8': 'Prior to infall, at [MATH] Gyr, this satellite reaches a total peak mass of [MATH].', '1512.03064-2-19-9': 'By [MATH] Gyr, the angle between the angular momentum vector of its inner bound core and that of the disc is [MATH].', '1512.03064-2-20-0': 'Note that, based on our kinematic selection criteria, not all the accreted particles would belong to the stellar halo.', '1512.03064-2-20-1': 'For [MATH], the accreted stellar mass fraction that belongs to the disc component varies between 5% (in Au 15) and 35% (in Au 2).', '1512.03064-2-20-2': 'The bottom panels of Fig. [REF] show accreted [Fe/H] profiles using the three different circularities previously defined.', '1512.03064-2-20-3': 'The differences between the spherical and minor axis [Fe/H] profiles disappear for halo Au 2 (left-most panel) in the bottom panels, indicating a strong orbital circularization of the inner metal rich cores of massive accreted satellites (see Fig. [REF]).', '1512.03064-2-20-4': 'Interestingly, for haloes Au 15 and Au 24 the difference between the spherical and minor axis [Fe/H] profiles remains almost the same for all values of [MATH].', '1512.03064-2-20-5': 'These examples show that the accreted component also contributes to the differences seen (both in median values and radial gradient) between the spherical and minor axis [Fe/H] profiles.', '1512.03064-2-21-0': '# Summary and Conclusions', '1512.03064-2-22-0': 'While previous studies based on hydrodynamical simulations indicate that negative metallicity gradients are a ubiquitous feature of simulated MW-like stellar haloes , half of the current observed sample of eight nearby disc galaxies suggests flat metallicity profiles .', '1512.03064-2-22-1': 'Motivated by this apparent discrepancy, we use four high resolution cosmological hydrodynamical simulations of MW-like galaxies to characterise the metallicity profiles of their stellar haloes.', '1512.03064-2-22-2': 'Our goal is to study whether this discrepancy could be due to the different ways in which simulations and data are compared.', '1512.03064-2-22-3': 'In contrast to numerical studies where spherically averaged [Fe/H] profiles are shown, observations are obtained along a particular direction, ideally perpendicular to the disc plane so as to minimise disc contamination.', '1512.03064-2-23-0': 'Following previous simulation studies, we define stellar haloes purely based on kinematic criteria.', '1512.03064-2-23-1': 'We find that spherical [Fe/H] profiles show large negative gradients, in agreement with previous work.', '1512.03064-2-23-2': 'However, significant differences are obtained when the profiles are computed in projection along the minor axis of the galactic disc.', '1512.03064-2-23-3': 'Not only are the median [Fe/H] values larger in the spherical profiles (up to [MATH] dex), at least within the inner 50 kpc, but also the gradients are steeper in general than along the minor axis.', '1512.03064-2-23-4': 'Color profiles obtained from mock RGB stars generated as in [CITATION] (not shown here) yield equivalent results.', '1512.03064-2-23-5': ""We find that the spherical profiles are dominated by the halo [Fe/H] distribution along the disc's major axis."", '1512.03064-2-23-6': 'We also show that the minor axis [Fe/H] profiles are indistinguishable for the different circularity criteria we have adopted, even when no kinematic selection is imposed.', '1512.03064-2-23-7': 'This indicates that a straightforward comparison between observations and models of stellar haloes is possible along the minor axis.', '1512.03064-2-23-8': 'Furthermore, of the four examples we have analysed, one has a flat [Fe/H] profile, one has a rather steep trend and the other are intermediate.', '1512.03064-2-23-9': 'This diversity is reminiscent of the variety of profiles seen in the observational data.', '1512.03064-2-24-0': 'Interestingly, in most cases, very similar results are obtained when only the accreted component of the stellar halo is considered, indicating that our results are not entirely driven by in-situ heated disc star particles.', '1512.03064-2-24-1': 'Note that even in dark matter only simulations, where baryonic effects such as a thin disc potential are not taken into account, stellar haloes generally present a flattened density distribution as a result of an anisotropic distribution of satellite orbits .', '1512.03064-2-24-2': 'Thus, those models are also expected to show differences between spherical and line-of-sight [Fe/H] profiles.', '1512.03064-2-25-0': 'The existence of stellar halo [Fe/H] gradients (or the lack of them) in massive disc galaxies depends strongly on how the profiles are constructed.', '1512.03064-2-25-1': 'Such profiles depend on the adopted definition of the stellar halo, which often differs among observations and theoretical works.', '1512.03064-2-25-2': 'Since stellar halo [Fe/H] profiles are an important diagnostic of galaxy formation history, a careful and faithful comparison between observations and models is crucial in order to interpret observations correctly and constrain the models.', '1512.03064-2-25-3': 'This will be presented in a follow-up work.'}","[['1512.03064-1-15-0', '1512.03064-2-15-0'], ['1512.03064-1-15-1', '1512.03064-2-15-1'], ['1512.03064-1-15-2', '1512.03064-2-15-2'], ['1512.03064-1-15-3', '1512.03064-2-15-3'], ['1512.03064-1-19-0', '1512.03064-2-19-0'], ['1512.03064-1-19-1', '1512.03064-2-19-1'], ['1512.03064-1-19-2', '1512.03064-2-19-2'], ['1512.03064-1-19-3', '1512.03064-2-19-3'], ['1512.03064-1-19-4', '1512.03064-2-19-4'], ['1512.03064-1-19-5', '1512.03064-2-19-5'], 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['1512.03064-1-12-0', '1512.03064-2-12-0'], ['1512.03064-1-8-4', '1512.03064-2-8-4'], ['1512.03064-1-22-0', '1512.03064-2-22-0'], ['1512.03064-1-22-3', '1512.03064-2-22-3'], ['1512.03064-1-24-0', '1512.03064-2-24-0'], ['1512.03064-1-24-1', '1512.03064-2-24-1'], ['1512.03064-1-24-3', '1512.03064-2-24-2'], ['1512.03064-1-17-9', '1512.03064-2-17-9']]",[],"[['1512.03064-1-19-7', '1512.03064-2-19-7'], ['1512.03064-1-3-2', '1512.03064-2-3-2'], ['1512.03064-1-9-2', '1512.03064-2-9-2']]",[],[],"{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1512.03064,,,
1912.03228,"{'1912.03228-1-0-0': 'Let [MATH] be one of the ind-groups [MATH], [MATH], [MATH], and [MATH] be an arbitrary set of [MATH] splitting parabolic subgroups of [MATH].', '1912.03228-1-0-1': 'We determine all such sets with the property that [MATH] acts with finitely many orbits on the ind-variety [MATH] where [MATH].', '1912.03228-1-0-2': 'In the case of a finite-dimensional classical linear algebraic group [MATH], the analogous problem has been solved in a sequence of papers of Littelmann, Magyar-Weyman-Zelevinsky and Matsuki.', '1912.03228-1-0-3': 'An essential difference from the finite-dimensional case is that already for [MATH], the condition that [MATH] acts on [MATH] with finitely many orbits is a rather restrictive condition on the pair [MATH].', '1912.03228-1-0-4': 'We describe this condition explicitly.', '1912.03228-1-0-5': 'Using this result, we tackle the most interesting case where [MATH], and present the answer in the form of a table.', '1912.03228-1-0-6': 'For [MATH], there always are infinitely many [MATH]-orbits on [MATH].', '1912.03228-1-1-0': '[2010]14L30; 14M15; 22E65; 22F30', '1912.03228-1-2-0': '# Introduction', '1912.03228-1-3-0': 'The following is a fundamental question in the theory of group actions: given a linear reductive algebraic group [MATH], on which direct products [MATH] of compact [MATH]-homogeneous spaces does [MATH] act with finitely many orbits?', '1912.03228-1-3-1': 'The problem is non-trivial only for [MATH], since it is a classical fact that [MATH] always acts with finitely many orbits on [MATH] (parabolic Schubert decomposition of a partial flag variety).', '1912.03228-1-3-2': 'It has turned out that the problem is most interesting for [MATH], as for [MATH] the group [MATH] always acts with infinitely many orbits.', '1912.03228-1-4-0': 'In the special case where one of the factors is a full flag variety, e.g. [MATH], the above problem is equivalent to classifying [MATH]-orbits on [MATH]; this special case is solved in [CITATION].', '1912.03228-1-4-1': 'In this situation, the theory of spherical varieties is an effective tool.', '1912.03228-1-4-2': 'In particular, the existence of a dense orbit is sufficient for ensuring that there are finitely many orbits.', '1912.03228-1-4-3': 'Therefore, the problem is related to studying the complexity of a direct product of two HV-varieties, i.e. closures of [MATH]-orbits of highest weight vectors in irreducible [MATH]-modules; this problem is considered in [CITATION].', '1912.03228-1-5-0': 'If no factor [MATH] is a full flag variety, the problem is considered in the classical cases in [CITATION] (types A and C, through the theory of quiver representations) and in [CITATION] (types B and D).', '1912.03228-1-5-1': 'As far as we know, the general question is still open in the exceptional cases.', '1912.03228-1-6-0': 'In the present paper we address the above problem in a natural infinite-dimensional setting.', '1912.03228-1-6-1': 'We let [MATH] be one of the classical (or finitary) ind-groups [MATH], [MATH], [MATH] and ask the same question, where each [MATH] is now a locally compact [MATH]-homogeneous ind-space.', '1912.03228-1-6-2': 'The latter are known as ind-varieties of generalized flags and have been studied in particular in [CITATION] and [CITATION]; see also [CITATION] and the references therein.', '1912.03228-1-7-0': 'For these ind-varieties, our question becomes interesting already for [MATH].', '1912.03228-1-7-1': 'Indeed, for which direct products [MATH] of ind-varieties of generalized flags, does [MATH] act with finitely many orbits on [MATH]?', '1912.03228-1-7-2': 'We prove that this is a quite restrictive property of the ind-variety [MATH].', '1912.03228-1-7-3': 'More precisely, we show that [MATH] acts with finitely many orbits on [MATH] only if the stabilizers [MATH] and [MATH] of two respective (arbitrary) points on [MATH] and [MATH] have each only finitely many invariant subspaces in the natural representation [MATH] of [MATH].', '1912.03228-1-7-4': 'In addition, it is required that the invariant subspaces of one of the groups, say [MATH], are only of finite dimension or finite codimension.', '1912.03228-1-7-5': 'The precise result is Theorem [REF], where we introduce adequate terminology: we call the parabolic ind-subgroup [MATH] large, and the parabolic ind-subgroup [MATH] semilarge.', '1912.03228-1-8-0': 'Having settled the case [MATH] in this way, we saw ourselves strongly motivated to solve the problem for any [MATH].', '1912.03228-1-8-1': 'The case [MATH] is settled by a general statement, Lemma [REF], claiming roughly that in the direct limit case the number of orbits can only increase.', '1912.03228-1-8-2': 'Hence for [MATH] there are infinitely many orbits on [MATH].', '1912.03228-1-8-3': 'The case [MATH] is the most intriguing.', '1912.03228-1-8-4': 'Here we prove that [MATH] has finitely many [MATH]-orbits, if and only if the same is true for all products [MATH], [MATH] and [MATH], and in addition [MATH] can be exhausted by triple flag varieties with finitely many orbits over the corresponding finite-dimensional groups.', '1912.03228-1-8-5': 'Those triple flag varieties have been classified by Magyar-Weymann-Zelevinsky for [MATH] and [MATH] [CITATION], and by Matsuki for [MATH] and [MATH] [CITATION].', '1912.03228-1-8-6': 'In this way, we settle the problem completely for the classical ind-groups [MATH], [MATH], [MATH].'}","{'1912.03228-2-0-0': 'Let [MATH] be one of the ind-groups [MATH], [MATH], [MATH], and [MATH] be an arbitrary set of [MATH] splitting parabolic subgroups of [MATH].', '1912.03228-2-0-1': 'We determine all such sets with the property that [MATH] acts with finitely many orbits on the ind-variety [MATH] where [MATH].', '1912.03228-2-0-2': 'In the case of a finite-dimensional classical linear algebraic group [MATH], the analogous problem has been solved in a sequence of papers of Littelmann, Magyar-Weyman-Zelevinsky and Matsuki.', '1912.03228-2-0-3': 'An essential difference from the finite-dimensional case is that already for [MATH], the condition that [MATH] acts on [MATH] with finitely many orbits is a rather restrictive condition on the pair [MATH].', '1912.03228-2-0-4': 'We describe this condition explicitly.', '1912.03228-2-0-5': 'Using this result, we tackle the most interesting case where [MATH], and present the answer in the form of a table.', '1912.03228-2-0-6': 'For [MATH], there always are infinitely many [MATH]-orbits on [MATH].', '1912.03228-2-1-0': '[2010]14L30; 14M15; 22E65; 22F30', '1912.03228-2-2-0': '# Introduction', '1912.03228-2-3-0': 'The following is a fundamental question in the theory of group actions: given a linear reductive algebraic group [MATH], on which direct products [MATH] of compact [MATH]-homogeneous spaces does [MATH] act with finitely many orbits?', '1912.03228-2-3-1': 'The problem is non-trivial only for [MATH], since it is a classical fact that [MATH] always acts with finitely many orbits on [MATH] (parabolic Schubert decomposition of a partial flag variety).', '1912.03228-2-3-2': 'It has turned out that the problem is most interesting for [MATH], as for [MATH] the group [MATH] always acts with infinitely many orbits.', '1912.03228-2-4-0': 'In the special case where one of the factors is a full flag variety, e.g. [MATH], the above problem is equivalent to finding whether there are finitely many [MATH]-orbits on [MATH]; this special case is solved in [CITATION] and [CITATION].', '1912.03228-2-4-1': 'In this situation, the theory of spherical varieties is an effective tool.', '1912.03228-2-4-2': 'In particular, the existence of a dense [MATH]-orbit is sufficient for ensuring that there are finitely many [MATH]-orbits.', '1912.03228-2-4-3': 'The problem is also related to studying the complexity of a direct product of two HV-varieties, i.e. closures of [MATH]-orbits of highest weight vectors in irreducible [MATH]-modules; this problem is considered in [CITATION].', '1912.03228-2-5-0': 'If no factor [MATH] is a full flag variety, the problem is considered in the classical cases in [CITATION] (types A and C, through the theory of quiver representations) and in [CITATION] (types B and D).', '1912.03228-2-5-1': 'For exceptional groups, the general question has been considered in [CITATION].', '1912.03228-2-6-0': 'We also mention the references [CITATION] and [CITATION], where the authors study the double flag varieties of the form [MATH] with a finite number of [MATH]-orbits for a symmetric subgroup [MATH] of [MATH].', '1912.03228-2-6-1': 'The problem of finitely many [MATH]-orbits on [MATH] is recovered if [MATH] is taken to be the diagonal embedding of [MATH] into [MATH].', '1912.03228-2-7-0': 'In the present paper we address the above general problem in a natural infinite-dimensional setting.', '1912.03228-2-7-1': 'We let [MATH] be one of the classical (or finitary) ind-groups [MATH], [MATH], [MATH] and ask the same question, where each [MATH] is now a locally compact [MATH]-homogeneous ind-space.', '1912.03228-2-7-2': 'The latter are known as ind-varieties of generalized flags and have been studied in particular in [CITATION] and [CITATION]; see also [CITATION] and the references therein.', '1912.03228-2-8-0': 'For these ind-varieties, our question becomes interesting already for [MATH].', '1912.03228-2-8-1': 'Indeed, for which direct products [MATH] of ind-varieties of generalized flags, does [MATH] act with finitely many orbits on [MATH]?', '1912.03228-2-8-2': 'We prove that this is a quite restrictive property of the ind-variety [MATH].', '1912.03228-2-8-3': 'More precisely, we show that [MATH] acts with finitely many orbits on [MATH] only if the stabilizers [MATH] and [MATH] of two respective (arbitrary) points on [MATH] and [MATH] have each only finitely many invariant subspaces in the natural representation [MATH] of [MATH].', '1912.03228-2-8-4': 'In addition, it is required that the invariant subspaces of one of the groups, say [MATH], are only of finite dimension or finite codimension.', '1912.03228-2-8-5': 'The precise result is Theorem [REF], where we introduce adequate terminology: we call the parabolic ind-subgroup [MATH] large, and the parabolic ind-subgroup [MATH] semilarge.', '1912.03228-2-9-0': 'Having settled the case [MATH] in this way, we saw ourselves strongly motivated to solve the problem for any [MATH].', '1912.03228-2-9-1': 'The case [MATH] is settled by a general statement, Lemma [REF], claiming roughly that in the direct limit case the number of orbits can only increase.', '1912.03228-2-9-2': 'Hence for [MATH] there are infinitely many orbits on [MATH].', '1912.03228-2-9-3': 'The case [MATH] is the most intriguing.', '1912.03228-2-9-4': 'Here we prove that [MATH] has finitely many [MATH]-orbits, if and only if the same is true for all products [MATH], [MATH] and [MATH], and in addition [MATH] can be exhausted by triple flag varieties with finitely many orbits over the corresponding finite-dimensional groups.', '1912.03228-2-9-5': 'Those triple flag varieties have been classified by Magyar-Weymann-Zelevinsky for [MATH] and [MATH] [CITATION], and by Matsuki for [MATH] and [MATH] [CITATION].', '1912.03228-2-9-6': 'In this way, we settle the problem completely for the classical ind-groups [MATH], [MATH], [MATH].'}","[['1912.03228-1-4-1', '1912.03228-2-4-1'], ['1912.03228-1-3-0', '1912.03228-2-3-0'], ['1912.03228-1-3-1', '1912.03228-2-3-1'], ['1912.03228-1-3-2', '1912.03228-2-3-2'], ['1912.03228-1-5-0', '1912.03228-2-5-0'], ['1912.03228-1-8-0', '1912.03228-2-9-0'], ['1912.03228-1-8-1', '1912.03228-2-9-1'], ['1912.03228-1-8-2', '1912.03228-2-9-2'], ['1912.03228-1-8-3', '1912.03228-2-9-3'], ['1912.03228-1-8-4', '1912.03228-2-9-4'], ['1912.03228-1-8-5', '1912.03228-2-9-5'], ['1912.03228-1-8-6', '1912.03228-2-9-6'], ['1912.03228-1-0-0', '1912.03228-2-0-0'], ['1912.03228-1-0-1', '1912.03228-2-0-1'], ['1912.03228-1-0-2', '1912.03228-2-0-2'], ['1912.03228-1-0-3', '1912.03228-2-0-3'], ['1912.03228-1-0-4', '1912.03228-2-0-4'], ['1912.03228-1-0-5', '1912.03228-2-0-5'], ['1912.03228-1-0-6', '1912.03228-2-0-6'], ['1912.03228-1-6-1', '1912.03228-2-7-1'], ['1912.03228-1-6-2', '1912.03228-2-7-2'], ['1912.03228-1-7-0', '1912.03228-2-8-0'], ['1912.03228-1-7-1', '1912.03228-2-8-1'], ['1912.03228-1-7-2', '1912.03228-2-8-2'], ['1912.03228-1-7-3', '1912.03228-2-8-3'], ['1912.03228-1-7-4', '1912.03228-2-8-4'], ['1912.03228-1-7-5', '1912.03228-2-8-5'], ['1912.03228-1-4-0', '1912.03228-2-4-0'], ['1912.03228-1-4-2', '1912.03228-2-4-2'], ['1912.03228-1-4-3', '1912.03228-2-4-3'], ['1912.03228-1-6-0', '1912.03228-2-7-0']]","[['1912.03228-1-4-1', '1912.03228-2-4-1'], ['1912.03228-1-3-0', '1912.03228-2-3-0'], ['1912.03228-1-3-1', '1912.03228-2-3-1'], ['1912.03228-1-3-2', '1912.03228-2-3-2'], ['1912.03228-1-5-0', '1912.03228-2-5-0'], ['1912.03228-1-8-0', '1912.03228-2-9-0'], ['1912.03228-1-8-1', '1912.03228-2-9-1'], ['1912.03228-1-8-2', '1912.03228-2-9-2'], ['1912.03228-1-8-3', '1912.03228-2-9-3'], ['1912.03228-1-8-4', '1912.03228-2-9-4'], ['1912.03228-1-8-5', '1912.03228-2-9-5'], ['1912.03228-1-8-6', '1912.03228-2-9-6'], ['1912.03228-1-0-0', '1912.03228-2-0-0'], ['1912.03228-1-0-1', '1912.03228-2-0-1'], ['1912.03228-1-0-2', '1912.03228-2-0-2'], ['1912.03228-1-0-3', '1912.03228-2-0-3'], ['1912.03228-1-0-4', '1912.03228-2-0-4'], ['1912.03228-1-0-5', '1912.03228-2-0-5'], ['1912.03228-1-0-6', '1912.03228-2-0-6'], ['1912.03228-1-6-1', '1912.03228-2-7-1'], ['1912.03228-1-6-2', '1912.03228-2-7-2'], ['1912.03228-1-7-0', '1912.03228-2-8-0'], ['1912.03228-1-7-1', '1912.03228-2-8-1'], ['1912.03228-1-7-2', '1912.03228-2-8-2'], ['1912.03228-1-7-3', '1912.03228-2-8-3'], ['1912.03228-1-7-4', '1912.03228-2-8-4'], ['1912.03228-1-7-5', '1912.03228-2-8-5']]","[['1912.03228-1-4-0', '1912.03228-2-4-0'], ['1912.03228-1-4-2', '1912.03228-2-4-2'], ['1912.03228-1-4-3', '1912.03228-2-4-3'], ['1912.03228-1-6-0', '1912.03228-2-7-0']]",[],[],[],"['1912.03228-1-1-0', '1912.03228-2-1-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1912.03228,,,
1803.03077,"{'1803.03077-1-0-0': 'We construct a vector gauge invariant transverse field configuration [MATH], consisting of the well-known superfield [MATH] and of a Stueckelberg-like chiral superfield [MATH].', '1803.03077-1-0-1': 'The renormalizability of the Super Yang Mills action in the Landau gauge is analyzed in the presence of a gauge invariant mass term [MATH], with [MATH] a power series in [MATH].', '1803.03077-1-0-2': 'Unlike the original Stueckelberg action, the resulting action turns out to be renormalizable to all orders.', '1803.03077-1-1-0': '# Introduction', '1803.03077-1-2-0': 'In this work we study the renormalizability properties of a [MATH] non-abelian gauge theory defined by a multiplet containing a massive vectorial excitation.', '1803.03077-1-2-1': 'The model we study is the supersymmetric version of a Stueckelberg-like action, in the sense that the massive gauge field is constructed by means of a compensating scalar field, thus preserving gauge invariance.', '1803.03077-1-3-0': 'The history of Stueckelberg-like models is very well reviewed in [CITATION].', '1803.03077-1-3-1': 'Traditionally, most of the investigations have studied such models as potential alternative to the Higgs mechanism of mass generation, but as discussed in [CITATION] there seems to be an unavoidable clash between renormalizability and unitarity in non-abelian Stueckelberg-like models.', '1803.03077-1-3-2': 'The original Stueckelberg model is abelian and has been rigorously proved [CITATION] to be renormalizable and unitary, but its non-abelian version is known to be perturbatively non-renormalizable [CITATION].', '1803.03077-1-3-3': 'Physically, the problem is due to the high energy behavior of the longitudinal vector degree of freedom.', '1803.03077-1-3-4': 'In the abelian case it is perfectly compensated by the dynamics of the Stueckelberg field but in non-abelian theories this seems to be not so, resulting in incurable divergent interacting amplitudes or unbounded cross sections.', '1803.03077-1-4-0': 'Nevertheless there have been recent interests in the study of massive vector models without the Higgs.', '1803.03077-1-4-1': 'The main motivation comes here from the continuous efforts to understand the low energy behavior of strongly interacting gauge theories, such as QCD.', '1803.03077-1-4-2': 'Confinement is a very important phenomenon in this context, but the physical mechanism behind it is still an open problem.', '1803.03077-1-4-3': 'A way to obtain information about this phenomenon is through lattice investigations which have revealed that the gluon propagator shows a massive behavior in the deep infrared non-perturbative region, while also displaying positivity violations which precludes a proper particle propagation interpretation [CITATION].', '1803.03077-1-4-4': 'Therefore, in a confining theory, the issue of the physical unitarity is a quite complex and difficult topic.', '1803.03077-1-4-5': 'Of course, physical unitarity must hold in terms of the physical excitations of the spectrum which are bound states of quarks and gluons like, for instance, mesons, barions, glueballs, etc.', '1803.03077-1-4-6': 'Though, the positivity violation of the two-point gluon correlation functions is taken as a strong evidence of confinement, signalling that gluons are not excitations of the physical spectrum of the theory.', '1803.03077-1-4-7': 'Nevertheless, renormalizability should be expected to hold since one wants to recover the good UV behavior of QCD.', '1803.03077-1-4-8': 'This trend of investigations led to many works that proposed modifications of the Yang-Mills theory to accommodate the lattice results [CITATION].', '1803.03077-1-4-9': 'Recent developments along these lines involve the introduction of modified Stueckelberg-like models [CITATION] constructed as a generalization of a class of confining effective theories known as Gribov-Zwanziger scenarios [CITATION], see [CITATION] for a review.', '1803.03077-1-4-10': 'Unlike the standard Stueckelberg action, these modified models enjoy the pleasant property of being renormalizable to all orders, see [CITATION] for a detailed account on the construction of these modified models and on their differences with the standard Stueckelberg theory.', '1803.03077-1-4-11': 'Let us also also mention here that, recently, a BRST invariant reformulation of the Gribov-Zwanziger theory has been achieved [CITATION], allowing its extension from the Landau gauge to an arbitrary covariant gauge.', '1803.03077-1-5-0': 'In this work we will carry out a supersymmetric generalization of the Stueckelberg-like model proposed in [CITATION].', '1803.03077-1-5-1': 'We prove that the present supersymmetric generalization is renormalizable, a task that will be done by means of a set of suitable Ward identities.', '1803.03077-1-5-2': 'Supersymmetric generalizations of Stueckelberg-like models was studied since very early [CITATION] but mostly concentrated on the better behaved abelian models (see [CITATION], for instance, for a proposal of an abelian Stueckelberg sector in MSSM), with some constructions of non-abelian theories with tensor multiplets [CITATION] and also with composite gauge fields [CITATION].', '1803.03077-1-6-0': 'The work is organized as follows.', '1803.03077-1-6-1': 'In Section II we construct the [MATH] Supersymmetric massive classical action.', '1803.03077-1-6-2': 'In Section III we discuss the gauge fixing and the ensuing BRST symmetry.', '1803.03077-1-6-3': 'Sections IV and V are devoted to the derivation of a set of suitable Ward identities and to the characterization of the most general invariant local counterterm following the setup of the algebraic renormalization.', '1803.03077-1-6-4': 'In Section VI we provide a detailed analysis of the counterterm by showing that it can be reabsorbed into the starting classical action through a redefinition of the fields and parameters, thus establishing the all orders renormalizability of the model.', '1803.03077-1-6-5': 'Section VII contains our conclusion.', '1803.03077-1-6-6': 'The final Appendices collect the conventions and a few additional technical details.', '1803.03077-1-7-0': '# PURE [MATH] SUSY STUECKELBERG-LIKE YANG-MILLS THEORY', '1803.03077-1-8-0': 'In order to define the [MATH] Supersymmetric Stueckelberg-like Yang-Mills theory, we start with a real abelian gauge superfield, [EQUATION] and with a massless chiral superfield that acts as a Stueckelberg field [EQUATION]', '1803.03077-1-8-1': 'It is then possible to construct a gauge-invariant superfield [EQUATION] which is invariant under the abelian gauge transformations', '1803.03077-1-9-0': '[EQUATION]', '1803.03077-1-9-1': 'We now need a generalization of the definition of [MATH] to the non-abelian case.', '1803.03077-1-9-2': 'We start with the gauge-invariant superfield [REF] with every component now in the adjoint representation of the gauge group G, [MATH], [MATH] where the [MATH] are the generators in the adjoint.', '1803.03077-1-9-3': 'Now, the fundamental object is [MATH] instead of [MATH].', '1803.03077-1-9-4': 'The non-abelian generalization of [REF] is [EQUATION] where [MATH], [MATH] and [MATH] is a usual gauge superfield.', '1803.03077-1-9-5': 'The gauge transformations then have to be [EQUATION] such that [MATH] is gauge invariant.', '1803.03077-1-9-6': 'For infinitesimal transformations, this explicitly yields [EQUATION] with [MATH].', '1803.03077-1-9-7': 'To first order (abelian gauge limit) in [MATH] this reproduces [REF].', '1803.03077-1-10-0': 'Now, using all of the above definitions we can construct a gauge invariant [MATH] Supersymmetric Stueckelberg-like Yang-Mills model [EQUATION] with [EQUATION] and [EQUATION] where [MATH] are a set of infinite arbitrary dimensionless parameters.', '1803.03077-1-10-1': 'As one can figure out, the fact that the generalized mass term [MATH] is an infinite power series [MATH] follows from the dimensionless character of [MATH] itself.', '1803.03077-1-10-2': 'Though, from the pertubative point of view, only the first quadratic terms of the series [REF], i.e. [MATH] will enter the expression of the superfield propagator.', '1803.03077-1-10-3': 'The remaining terms represent an infinite set of interaction vertices, a feature which is typical of the non-abelian Stueckelberg-like theories.', '1803.03077-1-11-0': 'Notice that the first term of the action, the pure supersymmetric Yang-Mills term [MATH], is invariant under [MATH].', '1803.03077-1-12-0': '# Supersymmetric gauge invariant Stueckelberg-like Yang-Mills action in the Landau gauge', '1803.03077-1-13-0': 'The supersymmetric extension of the Landau gauge is [1] [EQUATION]', '1803.03077-1-13-1': 'We thus need to add the following terms to the action [EQUATION] where we introduced the auxiliary superfield [MATH], with the following field equations [EQUATION]', '1803.03077-1-13-2': 'Following the standard BRST procedure, the gauge fixing condition can be implemented in a BRST invariant way by defining the auxiliary field as the BRST variation of the anti-ghost field [MATH]"" namely [MATH], so that we can add the following BRST invariant term to the action in order to fix the gauge.', '1803.03077-1-13-3': '[EQUATION]', '1803.03077-1-13-4': 'Looking at the gauge fixing [REF], it is important to realize that for any local quantum field theory involving dimensionless fields, one has the freedom of performing arbitrary re-parametrization of these fields.', '1803.03077-1-13-5': 'Examples of this are the two-dimensional non-linear sigma model [CITATION],[CITATION] and quantum field theories with a Stueckelberg field entering the gauge fixing term [CITATION].', '1803.03077-1-13-6': 'In the case of the gauge fixing [REF], this means that we have the freedom of replacing [MATH] by an arbitrary dimensionless function of [MATH]', '1803.03077-1-14-0': '[EQUATION] where [MATH] are free dimensionless coefficients.', '1803.03077-1-14-1': 'This freedom, inherent to the dimensionless nature of [MATH], is evident at the quantum level because of the fact that this field renormalizes in a non-linear way [CITATION].', '1803.03077-1-14-2': 'Therefore, [REF] is expressing precisely the freedom one has in the choice of a re-parametrization for [MATH].', '1803.03077-1-15-0': 'In our case, this means that instead of equation [REF] we could have just as well started with a term', '1803.03077-1-16-0': '[EQUATION] and this would not have affected the correlation functions of the gauge invariant quantities.', '1803.03077-1-16-1': 'The coefficients [MATH] are gauge parameters, not affecting the correlation functions of the gauge invariant quantities.', '1803.03077-1-16-2': 'The freedom that we have in the gauge fixing term will become apparent when performing the renormalization analysis.', '1803.03077-1-16-3': 'In fact, in section [REF], we will use a generalized gauge-fixing term', '1803.03077-1-17-0': '[EQUATION] and by employing the corresponding Ward identities, we can handle the ambiguity that is inherent to the gauge fixing.', '1803.03077-1-17-1': 'The counterterm will then correspond to a renormalization of the gauge parameters [MATH], as will become clear in section [REF].', '1803.03077-1-18-0': 'One of the striking features ensuring the renormalizability of the non-supersymmetric modified Stueckelberg-like models introduced in [CITATION] was the implementation of a transversality constraint on the analogue of the gauge invariant field [MATH].', '1803.03077-1-18-1': 'This transversality constraint gives rise to a deep difference between the modified models constructed in [CITATION] and the conventional non-renormalizable Stueckelberg model.', '1803.03077-1-18-2': 'It is precisely the implementation of this transversality constraint which ensures the UV renormalizability of the modified model.', '1803.03077-1-18-3': 'We remind here the reader to reference [CITATION] for a detailed account on the differences between the conventional and the modified Stueckelberg action.', '1803.03077-1-18-4': 'We then pursue here the same route outlined in [CITATION] and impose the transversality constraint also in the supersymmetric case.', '1803.03077-1-18-5': 'More precisely, this amounts to require that the superfield [MATH] obeys the constraint', '1803.03077-1-19-0': '[EQUATION] which, at the level of the action, can be implemented by introducing the following terms', '1803.03077-1-20-0': '[EQUATION] with [MATH].', '1803.03077-1-20-1': 'The field [MATH] is a Lagrange multiplier implementing the transversality constraint [REF], while the fields [MATH] are a set of ghost fields needed to compensate the Jacobian which arises from the functional integral over [MATH] and [MATH] in order to get a unity, see [CITATION] for the non-supersymmetric case.', '1803.03077-1-21-0': 'Thus, adopting the Landau gauge, as well as the transversality condition, the total action becomes', '1803.03077-1-22-0': '[EQUATION]', '1803.03077-1-23-0': 'This action enjoys the exact BRST nilpotent symmetry [EQUATION]', '1803.03077-1-24-0': 'and [EQUATION]', '1803.03077-1-25-0': '# Renormalizability analysis', '1803.03077-1-26-0': 'In order to analyze the renormalizability of the action [REF], we start by establishing the set of Ward identities that will be employed for the study of the quantum corrections.', '1803.03077-1-26-1': 'Following the algebraic renormalization procedure [CITATION], we have first to add some external sources coupling to non-linear BRST transformations of the fields and of the composite operators entering the classical action.', '1803.03077-1-26-2': 'Therefore, we need to introduce a set of external BRST invariant sources [MATH] coupled to the non-linear BRST variations of [MATH] as well as sources [MATH] coupled to the BRST invariant composite operators [MATH],', '1803.03077-1-27-0': '[EQUATION]', '1803.03077-1-28-0': 'We shall thus start with the BRST invariant complete action [MATH] defined by', '1803.03077-1-29-0': '[EQUATION]', '1803.03077-1-30-0': 'All quantum numbers, dimensions and [MATH]-weights of all fields and sources are displayed in tables [REF] and [REF].', '1803.03077-1-31-0': '## Ward identities and algebraic characterization of the invariant counterterm', '1803.03077-1-32-0': 'The complete action [MATH] obeys a large set of Ward identities, being:', '1803.03077-1-33-0': '# The algebraic characterization of the invariant counterterm and renormalizability', '1803.03077-1-34-0': 'In order to characterize the most general invariant counterterm which can be freely added to all order in perturbation theory, we follow the setup of the algebraic renormalization [CITATION] and perturb the classical action [REF], by adding an integrated local quantity in the fields and sources, [MATH] , that has [MATH]-weight 0, ghost number (0,0), is hermitian and has dimension 3 in case of a chiral superfield, or 2 in case of a vector superfield.', '1803.03077-1-34-1': 'We demand thus that the perturbed action, [MATH], where [MATH] is an expansion parameter, fulfills, to the first order in [MATH], the same Ward identities obeyed by the classical action [MATH], i.e. equations [REF] to [REF].', '1803.03077-1-34-2': 'This amounts to impose the following constraints on [MATH]:', '1803.03077-1-35-0': '[EQUATION] where [MATH] is the so-called nilpotent linearized Slavnov-Taylor operator [CITATION], defined as [EQUATION] with [MATH].', '1803.03077-1-35-1': 'From equation [REF] one learns that [MATH] belongs to the cohomology [CITATION] of the linearized Slavnov- Taylor operator [MATH] in the space of the integrated local quantities in the fields and sources with ghost number (0,0), [MATH]-weight 0 and dimension 3 in case of a chiral superfield, or 2 in case of a vector superfield.', '1803.03077-1-35-2': 'Therefore, we can set [EQUATION] where [MATH] denotes a zero-dimensional integrated quantity in the fields and sources with ghost number (-1,0) and [MATH]-weight 0.', '1803.03077-1-35-3': 'The term [MATH] in equation [REF] corresponds to the trivial solution, i.e. to the exact part of the cohomology of [MATH].', '1803.03077-1-35-4': 'On the other hand, the quantity [MATH] identifies the non-trivial solution, i.e. the cohomology of [MATH] , meaning that [MATH], for some local integrated Q.', '1803.03077-1-36-0': 'In its most general form, [MATH] is given by [EQUATION] with [EQUATION] and [MATH] arbitrary coefficients.', '1803.03077-1-36-1': 'Then, after implementing the constraints [REF] and [REF] we find [EQUATION]', '1803.03077-1-36-2': 'Let us now discuss the trivial part of the counterterm, [MATH].', '1803.03077-1-36-3': 'The term [MATH], taking into account the quantum numbers of the fields and sources, can be parametrized in its most general form as: [EQUATION] where, [EQUATION]', '1803.03077-1-36-4': 'Imposing the constraint [REF] [EQUATION] and observing from eq. [REF] that [EQUATION] we can use the relation [EQUATION] to impose [EQUATION]', '1803.03077-1-36-5': 'From eq. [REF] we find the relations [EQUATION] so that [EQUATION]', '1803.03077-1-37-0': 'We can further reduce the number of parameters in [MATH] by noticing that if we set [CITATION] [EQUATION] in [REF], the resulting action is [EQUATION] which is nothing but the Super Yang-Mills gauge-fixed action in the Landau gauge (see appendix B), with the addition of the following terms [EQUATION]', '1803.03077-1-37-1': 'However, upon integration over [MATH], these terms give rise to a unity.', '1803.03077-1-37-2': 'Thus, in the limit [REF], the counterterm [MATH] has to reduce to the standard counterterm of [MATH] Super Yang-Mills, which we can be found in appendix B. Implementing this condition, we find the following restrictions on the coefficients in equation [REF] [EQUATION] and for equation [REF] [EQUATION] so that [EQUATION] and [EQUATION]', '1803.03077-1-37-3': 'For the purpose of the analysis of the renormalization factors at the end of this section, we will rewrite the counterterm [MATH] of the action in its parametrized form, namely as contact terms written in terms of the starting classical action [MATH], being given by the following expression [EQUATION]', '1803.03077-1-38-0': '# Analysis of the counterterm and renormalization factors', '1803.03077-1-39-0': 'Having determined the most general form of the local invariant counterterm, eq. [REF], we observe, however, that the terms on the last line,', '1803.03077-1-40-0': '[EQUATION] cannot be rewritten in an exact parametric form in terms of the starting action [MATH].', '1803.03077-1-40-1': 'This feature is due to dependence of the gauge fixing on the dimensionless field [MATH].', '1803.03077-1-40-2': 'As a consequence, the renormalization of the gauge fixing itself is determined up to an ambiguity of the type of eq.[REF].', '1803.03077-1-40-3': 'As was mentioned before, this term can be handled by starting with the generalized gauge fixing of eq.[REF].', '1803.03077-1-40-4': 'This means that we could have equally started with a term in the action like [EQUATION] with [MATH] given by eq.[REF].', '1803.03077-1-40-5': 'Since [MATH] is now a composite field, we need to introduce it into the starting action through a suitable external source.', '1803.03077-1-40-6': 'In order to maintain BRST invariance, we make use of a BRST doublet of external sources [MATH], of dimension 2, R-weight 0 and ghost number [MATH] [EQUATION] and introduce the term [EQUATION] so that the full action is now given by [EQUATION]', '1803.03077-1-41-0': 'The action [MATH] obeys the following Ward identities:', '1803.03077-1-42-0': 'Since [MATH] and [MATH] are a pair of BRST doublet, they do not appear in the non-trivial part of the counterterm [CITATION], so this will remain as in equation [REF].', '1803.03077-1-42-1': 'On the other hand, the [MATH]-term becomes', '1803.03077-1-43-0': '[EQUATION]', '1803.03077-1-43-1': 'In a similar fashion as in the analysis of eqs. [REF]-[REF], we can use the relation [MATH], to find', '1803.03077-1-44-0': '[EQUATION] from which we obtain the relations [EQUATION] so that [REF] becomes [EQUATION]', '1803.03077-1-44-1': 'We now set [EQUATION] where [MATH] is the first, [MATH]-independent term of the Taylor expansion of [MATH] in powers of [MATH], and [MATH] denotes the remaining [MATH]-dependent terms.', '1803.03077-1-44-2': 'We find that [MATH] is connected to the renormalization of the fields within the gauge-fixing, while [MATH] renormalizes the gauge parameters in eq. [REF].', '1803.03077-1-44-3': 'Employing the generalized gauge-fixing [REF], we can now write the full counterterm in a complete parametric form, namely [EQUATION] where the dots ... in the last line denote the infinite set of terms of the kind [EQUATION]', '1803.03077-1-44-4': 'The usefulness of rewriting the counterterm [REF] in the parametric form becomes clear by casting it into the form [EQUATION] with [EQUATION]', '1803.03077-1-44-5': 'Now, in order to determine the renormalization factors we can use that [EQUATION] with [EQUATION] where [MATH] is a short-hand notation for all renormalized quantities: fields, parameters and external sources.', '1803.03077-1-44-6': 'We thus find the the following renormalization factors [EQUATION] as well as a multiplicative renormalization of the infinite set of gauge parameters [MATH] and [MATH] of equations [REF] and [REF], being [EQUATION] and [EQUATION]', '1803.03077-1-44-7': 'This shows that the inclusion of the generalized field [MATH] in the gauge fixing leads to the standard renormalization of the fields, parameters and sources.', '1803.03077-1-44-8': 'The renormalization of [MATH] itself is encoded in the renormalization of the infinite set of gauge parameters [MATH], as in eq. [REF].', '1803.03077-1-45-0': 'Note that both [MATH] and [MATH], as well as their sources [MATH] and [MATH], are renormalized in a non-linear way through a power series in [MATH] and [MATH], respectively.', '1803.03077-1-45-1': 'This is expected, due to the fact that both superfields are dimensionless.', '1803.03077-1-45-2': 'However, one has to note that the dimensionless superfield [MATH] contains a massive supermultiplet [MATH].', '1803.03077-1-45-3': 'Despite the fact that [MATH] itself renormalizes in a non-linear way due to its dimensionless nature, the component fields [MATH] do renormalize in fact in a standard multiplicative way through a constant (i.e. field independent) renormalization factors, a feature which can be checked out by employing the the Wess-Zumino gauge.', '1803.03077-1-46-0': '# Conclusion', '1803.03077-1-47-0': 'In this work we took a first step towards the understanding of Stueckelberg-like models in supersymmetric non-abelian gauge theories.', '1803.03077-1-47-1': 'The gauge invariant transverse field configuration [MATH] has been investigated in supersymmetric Yang-Mills theory with the Landau gauge.', '1803.03077-1-47-2': 'An auxiliary chiral superfield [MATH] was introduced that compensates the gauge variation of the vector superfield [MATH], thus preserving gauge invariance of the composite field [MATH].', '1803.03077-1-47-3': 'This gauge invariant composite field allows the construction of a local BRST-invariant massive model, summarized by the action [REF].', '1803.03077-1-47-4': 'Both [MATH] and [MATH] are dimensionless, which leads to ambiguities in defining both the mass term and the gauge fixing term.', '1803.03077-1-47-5': 'However, working with a generalized gauge fixing term, we find that the model turns out to be renormalizable to all orders of perturbation theory, as was discussed in sections [REF] and [REF].', '1803.03077-1-48-0': 'As a possible future application of the present result, let us mention that the possibility of having constructed a manifestly BRST invariant supersymmetric renormalizable version of the modified Stueckelberg models introduced in [CITATION] can open the possibility to investigate the important issue of the non-perturbative phenomenon of the Gribov copies directly in superspace, by generalizing to [MATH] the Gribov-Zwanziger setup.', '1803.03077-1-48-1': 'This would enable us to study aspects of the non-perturbative region of [MATH] confining supersymmetric theories, see also [CITATION] for a preliminary attempt in this direction.', '1803.03077-1-49-0': '# Notation', '1803.03077-1-50-0': '[EQUATION]', '1803.03077-1-51-0': '# [MATH] Supersymmetric Yang-Mills', '1803.03077-1-52-0': 'For the benefit of the reader, we provide in this appendix a short overview of the well known renormalizability of pure [MATH] standard massless Super Yang-Mills in the Landau gauge.', '1803.03077-1-52-1': 'Let us start by giving the complete BRST invariant action, namely [EQUATION]', '1803.03077-1-52-2': 'The full action [MATH] obeys the Ward identities [REF],[REF],[REF],[REF] as well as the Slavnov-Taylor identity: [EQUATION]', '1803.03077-1-52-3': 'As usual, the counterterm [MATH] can be written as [EQUATION] with [EQUATION] and [EQUATION]', '1803.03077-1-52-4': 'When constructing the counterterm, we will run into to same parametrizing problem as in section [REF].', '1803.03077-1-52-5': 'Repeating the procedure of section [REF], we introduce a doublet [MATH] and the full action is now given by [EQUATION]', '1803.03077-1-52-6': 'The action obeys the Ward identities [REF],[REF],[REF] as well as the Slavnov-Taylor identity: [EQUATION]', '1803.03077-1-52-7': 'Making use of the relation [MATH], we find [EQUATION]', '1803.03077-1-52-8': 'Defining [EQUATION] we find the counterterm to be [EQUATION]', '1803.03077-1-52-9': 'Following the analysis at the end of section [REF], for the renormalization factors we obtain [EQUATION]', '1803.03077-1-52-10': 'Thereby concluding the proof of the renormalizability of [MATH] pure massless Super Yang-Mills action.'}","{'1803.03077-2-0-0': 'We construct a vector gauge invariant transverse field configuration [MATH], consisting of the well-known superfield [MATH] and of a Stueckelberg-like chiral superfield [MATH].', '1803.03077-2-0-1': 'The renormalizability of the Super Yang Mills action in the Landau gauge is analyzed in the presence of a gauge invariant mass term [MATH], with [MATH] a power series in [MATH].', '1803.03077-2-0-2': 'Unlike the original Stueckelberg action, the resulting action turns out to be renormalizable to all orders.', '1803.03077-2-1-0': '# Introduction', '1803.03077-2-2-0': 'In this work we study the renormalizability properties of a [MATH] non-abelian gauge theory defined by a multiplet containing a massive vectorial excitation.', '1803.03077-2-2-1': 'The model we study is the supersymmetric version of a Stueckelberg-like action, in the sense that the massive gauge field is constructed by means of a compensating scalar field, thus preserving gauge invariance.', '1803.03077-2-3-0': 'The history of Stueckelberg-like models is very well reviewed in [CITATION].', '1803.03077-2-3-1': 'Traditionally, most of the investigations have studied such models as potential alternative to the Higgs mechanism of mass generation, but as discussed in [CITATION] there seems to be an unavoidable clash between renormalizability and unitarity in non-abelian Stueckelberg-like models.', '1803.03077-2-3-2': 'The original Stueckelberg model is abelian and has been rigorously proved [CITATION] to be renormalizable and unitary, but its non-abelian version is known to be perturbatively non-renormalizable [CITATION].', '1803.03077-2-3-3': 'Physically, the problem is due to the high energy behavior of the longitudinal vector degree of freedom.', '1803.03077-2-3-4': 'In the abelian case it is perfectly compensated by the dynamics of the Stueckelberg field but in non-abelian theories this seems to be not so, resulting in incurable divergent interacting amplitudes or unbounded cross sections.', '1803.03077-2-4-0': 'Nevertheless there have been recent interests in the study of massive vector models without the Higgs.', '1803.03077-2-4-1': 'The main motivation comes here from the continuous efforts to understand the low energy behavior of strongly interacting gauge theories, such as QCD.', '1803.03077-2-4-2': 'Confinement is a very important phenomenon in this context, but the physical mechanism behind it is still an open problem.', '1803.03077-2-4-3': 'A way to obtain information about this phenomenon is through lattice investigations which have revealed that the gluon propagator shows a massive behavior in the deep infrared non-perturbative region, while also displaying positivity violations which precludes a proper particle propagation interpretation [CITATION].', '1803.03077-2-4-4': 'Therefore, in a confining theory, the issue of the physical unitarity is a quite complex and difficult topic.', '1803.03077-2-4-5': 'Of course, physical unitarity must hold in terms of the physical excitations of the spectrum which are bound states of quarks and gluons like, for instance, mesons, barions, glueballs, etc.', '1803.03077-2-4-6': 'Though, the positivity violation of the two-point gluon correlation functions is taken as a strong evidence of confinement, signalling that gluons are not excitations of the physical spectrum of the theory.', '1803.03077-2-4-7': 'Nevertheless, renormalizability should be expected to hold since one wants to recover the good UV behavior of QCD.', '1803.03077-2-4-8': 'This trend of investigations led to many works that proposed modifications of the Yang-Mills theory to accommodate the lattice results [CITATION].', '1803.03077-2-4-9': 'Recent developments along these lines involve the introduction of modified Stueckelberg-like models [CITATION] constructed as a generalization of a class of confining effective theories known as Gribov-Zwanziger scenarios [CITATION], see [CITATION] for a review.', '1803.03077-2-4-10': 'Unlike the standard Stueckelberg action, these modified models enjoy the pleasant property of being renormalizable to all orders, see [CITATION] for a detailed account on the construction of these modified models and on their differences with the standard Stueckelberg theory.', '1803.03077-2-4-11': 'Let us also also mention here that, recently, a BRST invariant reformulation of the Gribov-Zwanziger theory has been achieved [CITATION], allowing its extension from the Landau gauge to an arbitrary covariant gauge.', '1803.03077-2-5-0': 'In this work we will carry out a supersymmetric generalization of the Stueckelberg-like model proposed in [CITATION].', '1803.03077-2-5-1': 'We prove that the present supersymmetric generalization is renormalizable, a task that will be done by means of a set of suitable Ward identities.', '1803.03077-2-5-2': 'Supersymmetric generalizations of Stueckelberg-like models was studied since very early [CITATION] but mostly concentrated on the better behaved abelian models (see [CITATION], for instance, for a proposal of an abelian Stueckelberg sector in MSSM), with some constructions of non-abelian theories with tensor multiplets [CITATION] and also with composite gauge fields [CITATION].', '1803.03077-2-6-0': 'The work is organized as follows.', '1803.03077-2-6-1': 'In Section II we construct the [MATH] Supersymmetric massive classical action.', '1803.03077-2-6-2': 'In Section III we discuss the gauge fixing and the ensuing BRST symmetry.', '1803.03077-2-6-3': 'Sections IV and V are devoted to the derivation of a set of suitable Ward identities and to the characterization of the most general invariant local counterterm following the setup of the algebraic renormalization.', '1803.03077-2-6-4': 'In Section VI we provide a detailed analysis of the counterterm by showing that it can be reabsorbed into the starting classical action through a redefinition of the fields and parameters, thus establishing the all orders renormalizability of the model.', '1803.03077-2-6-5': 'Section VII contains our conclusion.', '1803.03077-2-6-6': 'The final Appendices collect the conventions and a few additional technical details.', '1803.03077-2-7-0': '# PURE [MATH] SUSY STUECKELBERG-LIKE YANG-MILLS THEORY', '1803.03077-2-8-0': 'In order to define the [MATH] Supersymmetric Stueckelberg-like Yang-Mills theory, we start with a real abelian gauge superfield, [EQUATION] and with a massless chiral superfield that acts as a Stueckelberg field [EQUATION]', '1803.03077-2-8-1': 'It is then possible to construct a gauge-invariant superfield [EQUATION] which is invariant under the abelian gauge transformations', '1803.03077-2-9-0': '[EQUATION]', '1803.03077-2-9-1': 'We now need a generalization of the definition of [MATH] to the non-abelian case.', '1803.03077-2-9-2': 'We start with the gauge-invariant superfield [REF] with every component now in the adjoint representation of the gauge group G, [MATH], [MATH] where the [MATH] are the generators in the adjoint.', '1803.03077-2-9-3': 'Now, the fundamental object is [MATH] instead of [MATH].', '1803.03077-2-9-4': 'The non-abelian generalization of [REF] is [EQUATION] where [MATH], [MATH] and [MATH] is a usual gauge superfield.', '1803.03077-2-9-5': 'The gauge transformations then have to be [EQUATION] such that [MATH] is gauge invariant.', '1803.03077-2-9-6': 'For infinitesimal transformations, this explicitly yields [EQUATION] with [MATH].', '1803.03077-2-9-7': 'To first order (abelian gauge limit) in [MATH] this reproduces [REF].', '1803.03077-2-10-0': 'Now, using all of the above definitions we can construct a gauge invariant [MATH] Supersymmetric Stueckelberg-like Yang-Mills model [EQUATION] with [EQUATION] and [EQUATION] where [MATH] are a set of infinite arbitrary dimensionless parameters.', '1803.03077-2-10-1': 'As one can figure out, the fact that the generalized mass term [MATH] is an infinite power series [MATH] follows from the dimensionless character of [MATH] itself.', '1803.03077-2-10-2': 'Though, from the pertubative point of view, only the first quadratic terms of the series [REF], i.e. [MATH] will enter the expression of the superfield propagator.', '1803.03077-2-10-3': 'The remaining terms represent an infinite set of interaction vertices, a feature which is typical of the non-abelian Stueckelberg-like theories.', '1803.03077-2-11-0': 'Notice that the first term of the action, the pure supersymmetric Yang-Mills term [MATH], is invariant under [MATH].', '1803.03077-2-12-0': '# Supersymmetric gauge invariant Stueckelberg-like Yang-Mills action in the Landau gauge', '1803.03077-2-13-0': 'The supersymmetric extension of the Landau gauge is [1] [EQUATION]', '1803.03077-2-13-1': 'We thus need to add the following terms to the action [EQUATION] where we introduced the auxiliary superfield [MATH], with the following field equations [EQUATION]', '1803.03077-2-13-2': 'Following the standard BRST procedure, the gauge fixing condition can be implemented in a BRST invariant way by defining the auxiliary field as the BRST variation of the anti-ghost field [MATH]"" namely [MATH], so that we can add the following BRST invariant term to the action in order to fix the gauge.', '1803.03077-2-13-3': '[EQUATION]', '1803.03077-2-13-4': 'Looking at the gauge fixing [REF], it is important to realize that for any local quantum field theory involving dimensionless fields, one has the freedom of performing arbitrary re-parametrization of these fields.', '1803.03077-2-13-5': 'Examples of this are the two-dimensional non-linear sigma model [CITATION],[CITATION] and quantum field theories with a Stueckelberg field entering the gauge fixing term [CITATION].', '1803.03077-2-13-6': 'In the case of the gauge fixing [REF], this means that we have the freedom of replacing [MATH] by an arbitrary dimensionless function of [MATH]', '1803.03077-2-14-0': '[EQUATION] where [MATH] are free dimensionless coefficients.', '1803.03077-2-14-1': 'This freedom, inherent to the dimensionless nature of [MATH], is evident at the quantum level because of the fact that this field renormalizes in a non-linear way [CITATION].', '1803.03077-2-14-2': 'Therefore, [REF] is expressing precisely the freedom one has in the choice of a re-parametrization for [MATH].', '1803.03077-2-15-0': 'In our case, this means that instead of equation [REF] we could have just as well started with a term', '1803.03077-2-16-0': '[EQUATION] and this would not have affected the correlation functions of the gauge invariant quantities.', '1803.03077-2-16-1': 'The coefficients [MATH] are gauge parameters, not affecting the correlation functions of the gauge invariant quantities.', '1803.03077-2-16-2': 'The freedom that we have in the gauge fixing term will become apparent when performing the renormalization analysis.', '1803.03077-2-16-3': 'In fact, in section [REF], we will use a generalized gauge-fixing term', '1803.03077-2-17-0': '[EQUATION] and by employing the corresponding Ward identities, we can handle the ambiguity that is inherent to the gauge fixing.', '1803.03077-2-17-1': 'The counterterm will then correspond to a renormalization of the gauge parameters [MATH], as will become clear in section [REF].', '1803.03077-2-18-0': 'One of the striking features ensuring the renormalizability of the non-supersymmetric modified Stueckelberg-like models introduced in [CITATION] was the implementation of a transversality constraint on the analogue of the gauge invariant field [MATH].', '1803.03077-2-18-1': 'This transversality constraint gives rise to a deep difference between the modified models constructed in [CITATION] and the conventional non-renormalizable Stueckelberg model.', '1803.03077-2-18-2': 'It is precisely the implementation of this transversality constraint which ensures the UV renormalizability of the modified model.', '1803.03077-2-18-3': 'We remind here the reader to reference [CITATION] for a detailed account on the differences between the conventional and the modified Stueckelberg action.', '1803.03077-2-18-4': 'We then pursue here the same route outlined in [CITATION] and impose the transversality constraint also in the supersymmetric case.', '1803.03077-2-18-5': 'More precisely, this amounts to require that the superfield [MATH] obeys the constraint', '1803.03077-2-19-0': '[EQUATION] which, at the level of the action, can be implemented by introducing the following terms', '1803.03077-2-20-0': '[EQUATION] with [MATH].', '1803.03077-2-20-1': 'The field [MATH] is a Lagrange multiplier implementing the transversality constraint [REF], while the fields [MATH] are a set of ghost fields needed to compensate the Jacobian which arises from the functional integral over [MATH] and [MATH] in order to get a unity, see [CITATION] for the non-supersymmetric case.', '1803.03077-2-21-0': 'Thus, adopting the Landau gauge, as well as the transversality condition, the total action becomes', '1803.03077-2-22-0': '[EQUATION]', '1803.03077-2-23-0': 'This action enjoys the exact BRST nilpotent symmetry [EQUATION]', '1803.03077-2-24-0': 'and [EQUATION]', '1803.03077-2-25-0': '# Renormalizability analysis', '1803.03077-2-26-0': 'In order to analyze the renormalizability of the action [REF], we start by establishing the set of Ward identities that will be employed for the study of the quantum corrections.', '1803.03077-2-26-1': 'Following the algebraic renormalization procedure [CITATION], we have first to add some external sources coupling to non-linear BRST transformations of the fields and of the composite operators entering the classical action.', '1803.03077-2-26-2': 'Therefore, we need to introduce a set of external BRST invariant sources [MATH] coupled to the non-linear BRST variations of [MATH] as well as sources [MATH] coupled to the BRST invariant composite operators [MATH],', '1803.03077-2-27-0': '[EQUATION]', '1803.03077-2-28-0': 'We shall thus start with the BRST invariant complete action [MATH] defined by', '1803.03077-2-29-0': '[EQUATION]', '1803.03077-2-30-0': 'All quantum numbers, dimensions and [MATH]-weights of all fields and sources are displayed in tables [REF] and [REF].', '1803.03077-2-31-0': '## Ward identities and algebraic characterization of the invariant counterterm', '1803.03077-2-32-0': 'The complete action [MATH] obeys a large set of Ward identities, being:', '1803.03077-2-33-0': '# The algebraic characterization of the invariant counterterm and renormalizability', '1803.03077-2-34-0': 'In order to characterize the most general invariant counterterm which can be freely added to all order in perturbation theory, we follow the setup of the algebraic renormalization [CITATION] and perturb the classical action [REF], by adding an integrated local quantity in the fields and sources, [MATH] , that has [MATH]-weight 0, ghost number (0,0), is hermitian and has dimension 3 in case of a chiral superfield, or 2 in case of a vector superfield.', '1803.03077-2-34-1': 'We demand thus that the perturbed action, [MATH], where [MATH] is an expansion parameter, fulfills, to the first order in [MATH], the same Ward identities obeyed by the classical action [MATH], i.e. equations [REF] to [REF].', '1803.03077-2-34-2': 'This amounts to impose the following constraints on [MATH]:', '1803.03077-2-35-0': '[EQUATION] where [MATH] is the so-called nilpotent linearized Slavnov-Taylor operator [CITATION], defined as [EQUATION] with [MATH].', '1803.03077-2-35-1': 'From equation [REF] one learns that [MATH] belongs to the cohomology [CITATION] of the linearized Slavnov- Taylor operator [MATH] in the space of the integrated local quantities in the fields and sources with ghost number (0,0), [MATH]-weight 0 and dimension 3 in case of a chiral superfield, or 2 in case of a vector superfield.', '1803.03077-2-35-2': 'Therefore, we can set [EQUATION] where [MATH] denotes a zero-dimensional integrated quantity in the fields and sources with ghost number (-1,0) and [MATH]-weight 0.', '1803.03077-2-35-3': 'The term [MATH] in equation [REF] corresponds to the trivial solution, i.e. to the exact part of the cohomology of [MATH].', '1803.03077-2-35-4': 'On the other hand, the quantity [MATH] identifies the non-trivial solution, i.e. the cohomology of [MATH] , meaning that [MATH], for some local integrated Q.', '1803.03077-2-36-0': 'In its most general form, [MATH] is given by [EQUATION] with [EQUATION] and [MATH] arbitrary coefficients.', '1803.03077-2-36-1': 'Then, after implementing the constraints [REF] and [REF] we find [EQUATION]', '1803.03077-2-36-2': 'Let us now discuss the trivial part of the counterterm, [MATH].', '1803.03077-2-36-3': 'The term [MATH], taking into account the quantum numbers of the fields and sources, can be parametrized in its most general form as: [EQUATION] where, [EQUATION]', '1803.03077-2-36-4': 'Imposing the constraint [REF] [EQUATION] and observing from eq. [REF] that [EQUATION] we can use the relation [EQUATION] to impose [EQUATION]', '1803.03077-2-36-5': 'From eq. [REF] we find the relations [EQUATION] so that [EQUATION]', '1803.03077-2-37-0': 'We can further reduce the number of parameters in [MATH] by noticing that if we set [CITATION] [EQUATION] in [REF], the resulting action is [EQUATION] which is nothing but the Super Yang-Mills gauge-fixed action in the Landau gauge (see appendix B), with the addition of the following terms [EQUATION]', '1803.03077-2-37-1': 'However, upon integration over [MATH], these terms give rise to a unity.', '1803.03077-2-37-2': 'Thus, in the limit [REF], the counterterm [MATH] has to reduce to the standard counterterm of [MATH] Super Yang-Mills, which we can be found in appendix B. Implementing this condition, we find the following restrictions on the coefficients in equation [REF] [EQUATION] and for equation [REF] [EQUATION] so that [EQUATION] and [EQUATION]', '1803.03077-2-37-3': 'For the purpose of the analysis of the renormalization factors at the end of this section, we will rewrite the counterterm [MATH] of the action in its parametrized form, namely as contact terms written in terms of the starting classical action [MATH], being given by the following expression [EQUATION]', '1803.03077-2-38-0': '# Analysis of the counterterm and renormalization factors', '1803.03077-2-39-0': 'Having determined the most general form of the local invariant counterterm, eq. [REF], we observe, however, that the terms on the last line,', '1803.03077-2-40-0': '[EQUATION] cannot be rewritten in an exact parametric form in terms of the starting action [MATH].', '1803.03077-2-40-1': 'This feature is due to dependence of the gauge fixing on the dimensionless field [MATH].', '1803.03077-2-40-2': 'As a consequence, the renormalization of the gauge fixing itself is determined up to an ambiguity of the type of eq.[REF].', '1803.03077-2-40-3': 'As was mentioned before, this term can be handled by starting with the generalized gauge fixing of eq.[REF].', '1803.03077-2-40-4': 'This means that we could have equally started with a term in the action like [EQUATION] with [MATH] given by eq.[REF].', '1803.03077-2-40-5': 'Since [MATH] is now a composite field, we need to introduce it into the starting action through a suitable external source.', '1803.03077-2-40-6': 'In order to maintain BRST invariance, we make use of a BRST doublet of external sources [MATH], of dimension 2, R-weight 0 and ghost number [MATH] [EQUATION] and introduce the term [EQUATION] so that the full action is now given by [EQUATION]', '1803.03077-2-41-0': 'The action [MATH] obeys the following Ward identities:', '1803.03077-2-42-0': 'Since [MATH] and [MATH] are a pair of BRST doublet, they do not appear in the non-trivial part of the counterterm [CITATION], so this will remain as in equation [REF].', '1803.03077-2-42-1': 'On the other hand, the [MATH]-term becomes', '1803.03077-2-43-0': '[EQUATION]', '1803.03077-2-43-1': 'In a similar fashion as in the analysis of eqs. [REF]-[REF], we can use the relation [MATH], to find', '1803.03077-2-44-0': '[EQUATION] from which we obtain the relations [EQUATION] so that [REF] becomes [EQUATION]', '1803.03077-2-44-1': 'We now set [EQUATION] where [MATH] is the first, [MATH]-independent term of the Taylor expansion of [MATH] in powers of [MATH], and [MATH] denotes the remaining [MATH]-dependent terms.', '1803.03077-2-44-2': 'We find that [MATH] is connected to the renormalization of the fields within the gauge-fixing, while [MATH] renormalizes the gauge parameters in eq. [REF].', '1803.03077-2-44-3': 'Employing the generalized gauge-fixing [REF], we can now write the full counterterm in a complete parametric form, namely [EQUATION] where the dots ... in the last line denote the infinite set of terms of the kind [EQUATION]', '1803.03077-2-44-4': 'The usefulness of rewriting the counterterm [REF] in the parametric form becomes clear by casting it into the form [EQUATION] with [EQUATION]', '1803.03077-2-44-5': 'Now, in order to determine the renormalization factors we can use that [EQUATION] with [EQUATION] where [MATH] is a short-hand notation for all renormalized quantities: fields, parameters and external sources.', '1803.03077-2-44-6': 'We thus find the the following renormalization factors [EQUATION] as well as a multiplicative renormalization of the infinite set of gauge parameters [MATH] and [MATH] of equations [REF] and [REF], being [EQUATION] and [EQUATION]', '1803.03077-2-44-7': 'This shows that the inclusion of the generalized field [MATH] in the gauge fixing leads to the standard renormalization of the fields, parameters and sources.', '1803.03077-2-44-8': 'The renormalization of [MATH] itself is encoded in the renormalization of the infinite set of gauge parameters [MATH], as in eq. [REF].', '1803.03077-2-45-0': 'Note that both [MATH] and [MATH], as well as their sources [MATH] and [MATH], are renormalized in a non-linear way through a power series in [MATH] and [MATH], respectively.', '1803.03077-2-45-1': 'This is expected, due to the fact that both superfields are dimensionless.', '1803.03077-2-45-2': 'However, one has to note that the dimensionless superfield [MATH] contains a massive supermultiplet [MATH].', '1803.03077-2-45-3': 'Despite the fact that [MATH] itself renormalizes in a non-linear way due to its dimensionless nature, the component fields [MATH] do renormalize in fact in a standard multiplicative way through a constant (i.e. field independent) renormalization factors, a feature which can be checked out by employing the the Wess-Zumino gauge.', '1803.03077-2-46-0': '# Conclusion', '1803.03077-2-47-0': 'In this work we took a first step towards the understanding of Stueckelberg-like models in supersymmetric non-abelian gauge theories.', '1803.03077-2-47-1': 'The gauge invariant transverse field configuration [MATH] has been investigated in supersymmetric Yang-Mills theory with the Landau gauge.', '1803.03077-2-47-2': 'An auxiliary chiral superfield [MATH] was introduced that compensates the gauge variation of the vector superfield [MATH], thus preserving gauge invariance of the composite field [MATH].', '1803.03077-2-47-3': 'This gauge invariant composite field allows the construction of a local BRST-invariant massive model, summarized by the action [REF].', '1803.03077-2-47-4': 'Both [MATH] and [MATH] are dimensionless, which leads to ambiguities in defining both the mass term and the gauge fixing term.', '1803.03077-2-47-5': 'However, working with a generalized gauge fixing term, we find that the model turns out to be renormalizable to all orders of perturbation theory, as was discussed in sections [REF] and [REF].', '1803.03077-2-48-0': 'As a possible future application of the present result, let us mention that the possibility of having constructed a manifestly BRST invariant supersymmetric renormalizable version of the modified Stueckelberg models introduced in [CITATION] can open the possibility to investigate the important issue of the non-perturbative phenomenon of the Gribov copies directly in superspace, by generalizing to [MATH] the Gribov-Zwanziger setup.', '1803.03077-2-48-1': 'This would enable us to study aspects of the non-perturbative region of [MATH] confining supersymmetric theories, see also [CITATION] for a preliminary attempt in this direction.'}","[['1803.03077-1-5-0', '1803.03077-2-5-0'], ['1803.03077-1-5-1', '1803.03077-2-5-1'], ['1803.03077-1-5-2', '1803.03077-2-5-2'], ['1803.03077-1-43-1', '1803.03077-2-43-1'], ['1803.03077-1-8-0', '1803.03077-2-8-0'], ['1803.03077-1-8-1', '1803.03077-2-8-1'], ['1803.03077-1-20-1', '1803.03077-2-20-1'], ['1803.03077-1-10-0', '1803.03077-2-10-0'], ['1803.03077-1-10-1', '1803.03077-2-10-1'], ['1803.03077-1-10-2', '1803.03077-2-10-2'], 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'1803.03077-2-4-0'], ['1803.03077-1-4-1', '1803.03077-2-4-1'], ['1803.03077-1-4-2', '1803.03077-2-4-2'], ['1803.03077-1-4-3', '1803.03077-2-4-3'], ['1803.03077-1-4-4', '1803.03077-2-4-4'], ['1803.03077-1-4-5', '1803.03077-2-4-5'], ['1803.03077-1-4-6', '1803.03077-2-4-6'], ['1803.03077-1-4-7', '1803.03077-2-4-7'], ['1803.03077-1-4-8', '1803.03077-2-4-8'], ['1803.03077-1-4-9', '1803.03077-2-4-9'], ['1803.03077-1-4-10', '1803.03077-2-4-10'], ['1803.03077-1-4-11', '1803.03077-2-4-11'], ['1803.03077-1-40-0', '1803.03077-2-40-0'], ['1803.03077-1-40-1', '1803.03077-2-40-1'], ['1803.03077-1-40-2', '1803.03077-2-40-2'], ['1803.03077-1-40-3', '1803.03077-2-40-3'], ['1803.03077-1-40-4', '1803.03077-2-40-4'], ['1803.03077-1-40-5', '1803.03077-2-40-5'], ['1803.03077-1-40-6', '1803.03077-2-40-6'], ['1803.03077-1-35-0', '1803.03077-2-35-0'], ['1803.03077-1-35-1', '1803.03077-2-35-1'], ['1803.03077-1-35-2', '1803.03077-2-35-2'], ['1803.03077-1-35-3', '1803.03077-2-35-3'], ['1803.03077-1-35-4', '1803.03077-2-35-4'], ['1803.03077-1-13-0', '1803.03077-2-13-0'], ['1803.03077-1-13-1', '1803.03077-2-13-1'], ['1803.03077-1-13-2', '1803.03077-2-13-2'], ['1803.03077-1-13-4', '1803.03077-2-13-4'], ['1803.03077-1-13-5', '1803.03077-2-13-5'], ['1803.03077-1-13-6', '1803.03077-2-13-6'], ['1803.03077-1-6-0', '1803.03077-2-6-0'], ['1803.03077-1-6-1', '1803.03077-2-6-1'], ['1803.03077-1-6-2', '1803.03077-2-6-2'], ['1803.03077-1-6-3', '1803.03077-2-6-3'], ['1803.03077-1-6-4', '1803.03077-2-6-4'], ['1803.03077-1-6-5', '1803.03077-2-6-5'], ['1803.03077-1-6-6', '1803.03077-2-6-6'], ['1803.03077-1-37-0', '1803.03077-2-37-0'], ['1803.03077-1-37-1', '1803.03077-2-37-1'], ['1803.03077-1-37-2', '1803.03077-2-37-2'], ['1803.03077-1-37-3', '1803.03077-2-37-3'], ['1803.03077-1-34-0', '1803.03077-2-34-0'], ['1803.03077-1-34-1', '1803.03077-2-34-1'], ['1803.03077-1-9-1', '1803.03077-2-9-1'], ['1803.03077-1-9-2', '1803.03077-2-9-2'], ['1803.03077-1-9-3', '1803.03077-2-9-3'], ['1803.03077-1-9-4', '1803.03077-2-9-4'], ['1803.03077-1-9-5', '1803.03077-2-9-5'], ['1803.03077-1-9-6', '1803.03077-2-9-6'], ['1803.03077-1-9-7', '1803.03077-2-9-7'], ['1803.03077-1-11-0', '1803.03077-2-11-0'], ['1803.03077-1-17-0', '1803.03077-2-17-0'], ['1803.03077-1-17-1', '1803.03077-2-17-1'], ['1803.03077-1-0-0', '1803.03077-2-0-0'], ['1803.03077-1-0-1', '1803.03077-2-0-1'], ['1803.03077-1-0-2', '1803.03077-2-0-2'], ['1803.03077-1-42-0', '1803.03077-2-42-0'], ['1803.03077-1-42-1', '1803.03077-2-42-1'], ['1803.03077-1-3-0', '1803.03077-2-3-0'], ['1803.03077-1-3-1', '1803.03077-2-3-1'], ['1803.03077-1-3-2', '1803.03077-2-3-2'], ['1803.03077-1-3-3', '1803.03077-2-3-3'], ['1803.03077-1-3-4', '1803.03077-2-3-4'], ['1803.03077-1-19-0', '1803.03077-2-19-0'], ['1803.03077-1-36-0', '1803.03077-2-36-0'], ['1803.03077-1-36-1', '1803.03077-2-36-1'], ['1803.03077-1-36-2', '1803.03077-2-36-2'], ['1803.03077-1-36-3', '1803.03077-2-36-3'], ['1803.03077-1-36-4', '1803.03077-2-36-4'], ['1803.03077-1-36-5', 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'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1803.03077,"{'1803.03077-3-0-0': 'We construct a vector gauge invariant transverse field configuration [MATH], consisting of the well-known superfield [MATH] and of a Stueckelberg-like chiral superfield [MATH].', '1803.03077-3-0-1': 'The renormalizability of the Super Yang Mills action in the Landau gauge is analyzed in the presence of a gauge invariant mass term [MATH], with [MATH] a power series in [MATH].', '1803.03077-3-0-2': 'Unlike the original Stueckelberg action, the resulting action turns out to be renormalizable to all orders.', '1803.03077-3-1-0': '# Introduction', '1803.03077-3-2-0': 'In this work we study the renormalizability properties of a [MATH] non-abelian gauge theory defined by a multiplet containing a massive vectorial excitation.', '1803.03077-3-2-1': 'The model we study is the supersymmetric version of a Stueckelberg-like action, in the sense that the massive gauge field is constructed by means of a compensating scalar field, thus preserving gauge invariance.', '1803.03077-3-3-0': 'The history of Stueckelberg-like models is very well reviewed in [CITATION].', '1803.03077-3-3-1': 'Traditionally, most of the investigations have studied such models as potential alternative to the Higgs mechanism of mass generation, but as discussed in [CITATION] there seems to be an unavoidable clash between renormalizability and unitarity in non-abelian Stueckelberg-like models.', '1803.03077-3-3-2': 'The original Stueckelberg model is abelian and has been rigorously proved [CITATION] to be renormalizable and unitary, but its non-abelian version is known to be perturbatively non-renormalizable [CITATION].', '1803.03077-3-3-3': 'Physically, the problem is due to the high energy behavior of the longitudinal vector degree of freedom.', '1803.03077-3-3-4': 'In the abelian case it is perfectly compensated by the dynamics of the Stueckelberg field but in non-abelian theories this seems to be not so, resulting in incurable divergent interacting amplitudes or unbounded cross sections.', '1803.03077-3-4-0': 'Nevertheless there have been recent interests in the study of massive vector models without the Higgs.', '1803.03077-3-4-1': 'The main motivation comes here from the continuous efforts to understand the low energy behavior of strongly interacting gauge theories, such as QCD.', '1803.03077-3-4-2': 'Confinement is a very important phenomenon in this context, but the physical mechanism behind it is still an open problem.', '1803.03077-3-4-3': 'A way to obtain information about this phenomenon is through lattice investigations which have revealed that the gluon propagator shows a massive behavior in the deep infrared non-perturbative region, while also displaying positivity violations which precludes a proper particle propagation interpretation [CITATION].', '1803.03077-3-4-4': 'Therefore, in a confining theory, the issue of the physical unitarity is a quite complex and difficult topic.', '1803.03077-3-4-5': 'Of course, physical unitarity must hold in terms of the physical excitations of the spectrum which are bound states of quarks and gluons like, for instance, mesons, barions, glueballs, etc.', '1803.03077-3-4-6': 'Though, the positivity violation of the two-point gluon correlation functions is taken as a strong evidence of confinement, signalling that gluons are not excitations of the physical spectrum of the theory.', '1803.03077-3-4-7': 'Nevertheless, renormalizability should be expected to hold since one wants to recover the good UV behavior of QCD.', '1803.03077-3-4-8': 'This trend of investigations led to many works that proposed modifications of the Yang-Mills theory to accommodate the lattice results [CITATION].', '1803.03077-3-4-9': 'Recent developments along these lines involve the introduction of modified Stueckelberg-like models [CITATION] constructed as a generalization of a class of confining effective theories known as Gribov-Zwanziger scenarios [CITATION], see [CITATION] for a review.', '1803.03077-3-4-10': 'Unlike the standard Stueckelberg action, these modified models enjoy the pleasant property of being renormalizable to all orders, see [CITATION] for a detailed account on the construction of these modified models and on their differences with the standard Stueckelberg theory.', '1803.03077-3-4-11': 'Let us also also mention here that, recently, a BRST invariant reformulation of the Gribov-Zwanziger theory has been achieved [CITATION], allowing its extension from the Landau gauge to an arbitrary covariant gauge.', '1803.03077-3-5-0': 'The present model is intended only to construct a renormalizable theory which generalizes the non-supersymmetric construction given in [CITATION].', '1803.03077-3-5-1': 'Issues like the perturbative unitarity of the models so obtained are not explicitly addressed.', '1803.03077-3-5-2': 'As far as we know, the non-supersymmetric model is not perturbative unitary.', '1803.03077-3-5-3': 'Though, it can be successfully employed as an effective renormalizable model in order to investigate the non-perturbative infrared region of confining Yang-Mills theories.', '1803.03077-3-5-4': 'So far, the prediction of the non-supersymmetric model are in good agreement with the actual lattice data on the correlation functions of the theory, like the two-point gluon propagator.', '1803.03077-3-6-0': 'Our aim here is to construct a sypersimmetric generalization of this model for a future investigation of the confinement aspects of pure STM, which is known to be a confining theory.', '1803.03077-3-6-1': 'This is the main purpose of the present model.', '1803.03077-3-7-0': 'In a confining YM theory, the issue of the unitarity has to be faced through the study of suitable colorless bound-state, a topic which is still too far from the goal of the present work, whose aim is that of obtaining a renormalizable massive SPYM theory whihc generalizes the model of [CITATION].', '1803.03077-3-8-0': 'In this work we will carry out a supersymmetric generalization of the Stueckelberg-like model proposed in [CITATION].', '1803.03077-3-8-1': 'We prove that the present supersymmetric generalization is renormalizable, a task that will be done by means of a set of suitable Ward identities.', '1803.03077-3-8-2': 'Supersymmetric generalizations of Stueckelberg-like models was studied since very early [CITATION] but mostly concentrated on the better behaved abelian models (see [CITATION], for instance, for a proposal of an abelian Stueckelberg sector in MSSM), with some constructions of non-abelian theories with tensor multiplets [CITATION] and also with composite gauge fields [CITATION].', '1803.03077-3-9-0': 'The work is organized as follows.', '1803.03077-3-9-1': 'In Section II we construct the [MATH] Supersymmetric massive classical action.', '1803.03077-3-9-2': 'In Section III we discuss the gauge fixing and the ensuing BRST symmetry.', '1803.03077-3-9-3': 'Sections IV and V are devoted to the derivation of a set of suitable Ward identities and to the characterization of the most general invariant local counterterm following the setup of the algebraic renormalization.', '1803.03077-3-9-4': 'In Section VI we provide a detailed analysis of the counterterm by showing that it can be reabsorbed into the starting classical action through a redefinition of the fields and parameters, thus establishing the all orders renormalizability of the model.', '1803.03077-3-9-5': 'Section VII contains our conclusion.', '1803.03077-3-9-6': 'The final Appendices collect the conventions and a few additional technical details.', '1803.03077-3-10-0': '# PURE [MATH] SUSY STUECKELBERG-LIKE YANG-MILLS THEORY', '1803.03077-3-11-0': 'In order to define the [MATH] Supersymmetric Stueckelberg-like Yang-Mills theory, we start with a real abelian gauge superfield, [EQUATION] and with a massless chiral superfield that acts as a Stueckelberg field [EQUATION]', '1803.03077-3-11-1': 'It is then possible to construct a gauge-invariant superfield [EQUATION] which is invariant under the abelian gauge transformations', '1803.03077-3-12-0': '[EQUATION]', '1803.03077-3-12-1': 'We now need a generalization of the definition of [MATH] to the non-abelian case.', '1803.03077-3-12-2': 'We start with the gauge-invariant superfield [REF] with every component now in the adjoint representation of the gauge group G, [MATH], [MATH] where the [MATH] are the generators in the adjoint.', '1803.03077-3-12-3': 'Now, the fundamental object is [MATH] instead of [MATH].', '1803.03077-3-12-4': 'The non-abelian generalization of [REF] is [EQUATION] where [MATH], [MATH] and [MATH] is a usual gauge superfield.', '1803.03077-3-12-5': 'The gauge transformations then have to be [EQUATION] such that [MATH] is gauge invariant.', '1803.03077-3-12-6': 'For infinitesimal transformations, this explicitly yields [EQUATION] with [MATH].', '1803.03077-3-12-7': 'To first order (abelian gauge limit) in [MATH] this reproduces [REF].', '1803.03077-3-13-0': 'Now, using all of the above definitions we can construct a gauge invariant [MATH] Supersymmetric Stueckelberg-like Yang-Mills model [EQUATION] with [EQUATION] and [EQUATION] where [MATH] are a set of infinite arbitrary dimensionless parameters.', '1803.03077-3-13-1': 'As one can figure out, the fact that the generalized mass term [MATH] is an infinite power series [MATH] follows from the dimensionless character of [MATH] itself.', '1803.03077-3-13-2': 'Though, from the pertubative point of view, only the first quadratic terms of the series [REF], i.e. [MATH] will enter the expression of the superfield propagator.', '1803.03077-3-13-3': 'The remaining terms represent an infinite set of interaction vertices, a feature which is typical of the non-abelian Stueckelberg-like theories.', '1803.03077-3-14-0': 'Notice that the first term of the action, the pure supersymmetric Yang-Mills term [MATH], is invariant under [MATH].', '1803.03077-3-14-1': 'For more details about the [MATH] supersymmetry and conventions, see [CITATION].', '1803.03077-3-15-0': '# Supersymmetric gauge invariant Stueckelberg-like Yang-Mills action in the Landau gauge', '1803.03077-3-16-0': 'The supersymmetric extension of the Landau gauge is [1] [EQUATION]', '1803.03077-3-16-1': 'We thus need to add the following terms to the action [EQUATION] where we introduced the auxiliary chiral superfield [MATH], with the following field equations [EQUATION]', '1803.03077-3-16-2': 'Following the standard BRST procedure, the gauge fixing condition can be implemented in a BRST invariant way by defining the auxiliary field as the BRST variation of the anti-ghost field [MATH]"" namely [MATH], so that we can add the following BRST invariant term to the action in order to fix the gauge.', '1803.03077-3-16-3': '[EQUATION]', '1803.03077-3-16-4': 'Looking at the gauge fixing [REF], it is important to realize that for any local quantum field theory involving dimensionless fields, one has the freedom of performing arbitrary re-parametrization of these fields.', '1803.03077-3-16-5': 'Examples of this are the two-dimensional non-linear sigma model [CITATION],[CITATION] and quantum field theories with a Stueckelberg field entering the gauge fixing term [CITATION].', '1803.03077-3-16-6': 'In the case of the gauge fixing [REF], this means that we have the freedom of replacing [MATH] by an arbitrary dimensionless function of [MATH]', '1803.03077-3-17-0': '[EQUATION] where [MATH] are free dimensionless coefficients.', '1803.03077-3-17-1': 'This freedom, inherent to the dimensionless nature of [MATH], is evident at the quantum level because of the fact that this field renormalizes in a non-linear way [CITATION].', '1803.03077-3-17-2': 'Therefore, [REF] is expressing precisely the freedom one has in the choice of a re-parametrization for [MATH].', '1803.03077-3-18-0': 'In our case, this means that instead of equation [REF] we could have just as well started with a term', '1803.03077-3-19-0': '[EQUATION] and this would not have affected the correlation functions of the gauge invariant quantities.', '1803.03077-3-19-1': 'The coefficients [MATH] are gauge parameters, not affecting the correlation functions of the gauge invariant quantities.', '1803.03077-3-19-2': 'The freedom that we have in the gauge fixing term will become apparent when performing the renormalization analysis.', '1803.03077-3-19-3': 'In fact, in section [REF], we will use a generalized gauge-fixing term', '1803.03077-3-20-0': '[EQUATION] and by employing the corresponding Ward identities, we can handle the ambiguity that is inherent to the gauge fixing.', '1803.03077-3-20-1': 'The counterterm will then correspond to a renormalization of the gauge parameters [MATH], as will become clear in section [REF].', '1803.03077-3-21-0': 'One of the striking features ensuring the renormalizability of the non-supersymmetric modified Stueckelberg-like models introduced in [CITATION] was the implementation of a transversality constraint on the analogue of the gauge invariant field [MATH].', '1803.03077-3-21-1': 'This transversality constraint gives rise to a deep difference between the modified models constructed in [CITATION] and the conventional non-renormalizable Stueckelberg model.', '1803.03077-3-21-2': 'It is precisely the implementation of this transversality constraint which ensures the UV renormalizability of the modified model.', '1803.03077-3-21-3': 'We remind here the reader to reference [CITATION] for a detailed account on the differences between the conventional and the modified Stueckelberg action.', '1803.03077-3-21-4': 'We then pursue here the same route outlined in [CITATION] and impose the transversality constraint also in the supersymmetric case.', '1803.03077-3-21-5': 'More precisely, this amounts to require that the superfield [MATH] obeys the constraint', '1803.03077-3-22-0': '[EQUATION] which, at the level of the action, can be implemented by introducing the following terms', '1803.03077-3-23-0': '[EQUATION] with [MATH].', '1803.03077-3-23-1': 'The field [MATH] is a Lagrange multiplier implementing the transversality constraint [REF], while the fields [MATH] are a set of ghost fields needed to compensate the Jacobian which arises from the functional integral over [MATH] and [MATH] in order to get a unity, see [CITATION] for the non-supersymmetric case.', '1803.03077-3-24-0': 'Thus, adopting the Landau gauge, as well as the transversality condition, the total action becomes', '1803.03077-3-25-0': '[EQUATION]', '1803.03077-3-26-0': 'This action enjoys the exact BRST nilpotent symmetry [EQUATION]', '1803.03077-3-27-0': 'and [EQUATION]', '1803.03077-3-28-0': '# Renormalizability analysis', '1803.03077-3-29-0': 'In order to analyze the renormalizability of the action [REF], we start by establishing the set of Ward identities that will be employed for the study of the quantum corrections.', '1803.03077-3-29-1': 'Following the algebraic renormalization procedure [CITATION], we have first to add some external sources coupling to non-linear BRST transformations of the fields and of the composite operators entering the classical action.', '1803.03077-3-29-2': 'Therefore, we need to introduce a set of external BRST invariant sources [MATH] coupled to the non-linear BRST variations of [MATH] as well as sources [MATH] coupled to the BRST invariant composite operators [MATH],', '1803.03077-3-30-0': '[EQUATION]', '1803.03077-3-31-0': 'We shall thus start with the BRST invariant complete action [MATH] defined by', '1803.03077-3-32-0': '[EQUATION]', '1803.03077-3-33-0': 'All quantum numbers, dimensions and [MATH]-weights of all fields and sources are displayed in tables [REF] and [REF].', '1803.03077-3-34-0': '## Ward identities and algebraic characterization of the invariant counterterm', '1803.03077-3-35-0': 'The complete action [MATH] obeys a large set of Ward identities, being:', '1803.03077-3-36-0': 'such that [MATH], [MATH], [MATH], [MATH] are defined by appendix [REF].', '1803.03077-3-36-1': 'Thus, we can see that the covariant action [MATH] satisfy the Ward identities and Lorentz invariance.', '1803.03077-3-37-0': '# The algebraic characterization of the invariant counterterm and renormalizability', '1803.03077-3-38-0': 'In order to characterize the most general invariant counterterm which can be freely added to all order in perturbation theory, we follow the setup of the algebraic renormalization [CITATION] and perturb the classical action [REF], by adding an integrated local quantity in the fields and sources, [MATH] , that has [MATH]-weight 0, ghost number (0,0), is hermitian and has dimension 3 in case of a chiral superfield, or 2 in case of a vector superfield.', '1803.03077-3-38-1': 'We demand thus that the perturbed action, [MATH], where [MATH] is an expansion parameter, fulfills, to the first order in [MATH], the same Ward identities obeyed by the classical action [MATH], i.e. equations [REF] to [REF].', '1803.03077-3-38-2': 'This amounts to impose the following constraints on [MATH]:', '1803.03077-3-39-0': '[EQUATION] where [MATH] is the so-called nilpotent linearized Slavnov-Taylor operator [CITATION], defined as [EQUATION] with [MATH].', '1803.03077-3-39-1': 'From equation [REF] one learns that [MATH] belongs to the cohomology [CITATION] of the linearized Slavnov- Taylor operator [MATH] in the space of the integrated local quantities in the fields and sources with ghost number (0,0), [MATH]-weight 0 and dimension 3 in case of a chiral superfield, or 2 in case of a vector superfield.', '1803.03077-3-39-2': 'Therefore, we can set [EQUATION] where [MATH] denotes a zero-dimensional integrated quantity in the fields and sources with ghost number (-1,0) and [MATH]-weight 0.', '1803.03077-3-39-3': 'The term [MATH] in equation [REF] corresponds to the trivial solution, i.e. to the exact part of the cohomology of [MATH].', '1803.03077-3-39-4': 'On the other hand, the quantity [MATH] identifies the non-trivial solution, i.e. the cohomology of [MATH] , meaning that [MATH], for any local integrated Q.', '1803.03077-3-40-0': 'In its most general form, [MATH] is given by [EQUATION] with [EQUATION] and [MATH] arbitrary coefficients.', '1803.03077-3-40-1': 'Then, after implementing the constraints [REF] and [REF] we find [EQUATION]', '1803.03077-3-40-2': 'Let us now discuss the trivial part of the counterterm, [MATH].', '1803.03077-3-40-3': 'The term [MATH], taking into account the quantum numbers of the fields and sources, can be parametrized in its most general form as: [EQUATION] where, [EQUATION]', '1803.03077-3-40-4': 'Imposing the constraint [REF] [EQUATION] and observing from eq. [REF] that [EQUATION] we can use the relation [EQUATION] to impose [EQUATION]', '1803.03077-3-40-5': 'From eq. [REF] we find the relations [EQUATION] so that [EQUATION]', '1803.03077-3-41-0': 'We can further reduce the number of parameters in [MATH] by noticing that if we set [CITATION] [EQUATION] in [REF], the resulting action is [EQUATION] which is nothing but the Super Yang-Mills gauge-fixed action in the Landau gauge (see appendix [REF]), with the addition of the following terms [EQUATION]', '1803.03077-3-41-1': 'However, upon integration over [MATH], these terms give rise to a unity.', '1803.03077-3-41-2': 'Thus, in the limit [REF] the starting action takes the following form: [EQUATION]', '1803.03077-3-41-3': 'Let us consider now the correlation functions of the Yang-Mills superfield [MATH], namely [EQUATION] where [MATH] stands for integration over all fields.', '1803.03077-3-41-4': 'Though, since [MATH], one can directly perform in [REF] the integration over [MATH], i.e. of the fields appearing in [MATH].', '1803.03077-3-42-0': 'This integration is easily seen to give a unity.', '1803.03077-3-42-1': 'It is in fact nothing but a Super Faddeev-Popov term (see [CITATION]) which, due to [MATH], gives a unity.', '1803.03077-3-43-0': 'Therefore, in the limit, [MATH], it follows that [EQUATION] meaning that the correlator [MATH] reduces to that of standard SPYM.', '1803.03077-3-43-1': 'As consequence, the dimensionless, and thus [MATH]-independent, coefficients appearing in the counterterm [MATH] are subject to the following additional conditions', '1803.03077-3-44-0': '[EQUATION] and for equation [REF] [EQUATION] so that [EQUATION] and [EQUATION]', '1803.03077-3-44-1': 'For the purpose of the analysis of the renormalization factors at the end of this section, we will rewrite the counterterm [MATH] of the action in its parametrized form, namely as contact terms written in terms of the starting classical action [MATH], being given by the following expression [EQUATION]', '1803.03077-3-45-0': '# Analysis of the counterterm and renormalization factors', '1803.03077-3-46-0': 'Having determined the most general form of the local invariant counterterm, eq. [REF], we observe, however, that the terms on the last line,', '1803.03077-3-47-0': '[EQUATION] cannot be rewritten in an exact parametric form in terms of the starting action [MATH].', '1803.03077-3-47-1': 'This feature is due to dependence of the gauge fixing on the dimensionless field [MATH].', '1803.03077-3-47-2': 'As a consequence, the renormalization of the gauge fixing itself is determined up to an ambiguity of the type of eq.[REF].', '1803.03077-3-47-3': 'As was mentioned before, this term can be handled by starting with the generalized gauge fixing of eq.[REF].', '1803.03077-3-47-4': 'This means that we could have equally started with a term in the action like [EQUATION] with [MATH] given by eq.[REF].', '1803.03077-3-47-5': 'Since [MATH] is now a composite field, we need to introduce it into the starting action through a suitable external source.', '1803.03077-3-47-6': 'In order to maintain BRST invariance, we make use of a BRST doublet of external sources [MATH], of dimension 2, R-weight 0 and ghost number [MATH] [EQUATION] and introduce the term [EQUATION] so that the full action is now given by [EQUATION]', '1803.03077-3-48-0': 'The action [MATH] obeys the following Ward identities:', '1803.03077-3-49-0': 'Since [MATH] and [MATH] are a pair of BRST doublet, they do not appear in the non-trivial part of the counterterm [CITATION], so this will remain as in equation [REF].', '1803.03077-3-49-1': 'On the other hand, the [MATH]-term becomes', '1803.03077-3-50-0': '[EQUATION]', '1803.03077-3-50-1': 'In a similar fashion as in the analysis of eqs. [REF]-[REF], we can use the relation [MATH], to find', '1803.03077-3-51-0': '[EQUATION] from which we obtain the relations [EQUATION] so that [REF] becomes [EQUATION]', '1803.03077-3-51-1': 'We now set [EQUATION] where [MATH] is the first, [MATH]-independent term of the Taylor expansion of [MATH] in powers of [MATH], and [MATH] denotes the remaining [MATH]-dependent terms.', '1803.03077-3-51-2': 'We find that [MATH] is connected to the renormalization of the fields within the gauge-fixing, while [MATH] renormalizes the gauge parameters in eq. [REF].', '1803.03077-3-51-3': 'Employing the generalized gauge-fixing [REF], we can now write the full counterterm in a complete parametric form, namely [EQUATION] where the dots ... in the last line denote the infinite set of terms of the kind [EQUATION]', '1803.03077-3-51-4': 'The usefulness of rewriting the counterterm [REF] in the parametric form becomes clear by casting it into the form [EQUATION] with [EQUATION]', '1803.03077-3-51-5': 'Now, in order to determine the renormalization factors we can use that [EQUATION] with [EQUATION] where [MATH] is a short-hand notation for all renormalized quantities: fields, parameters and external sources.', '1803.03077-3-51-6': 'We thus find the the following renormalization factors [EQUATION] as well as a multiplicative renormalization of the infinite set of gauge parameters [MATH] and [MATH] of equations [REF] and [REF], being [EQUATION] and [EQUATION]', '1803.03077-3-51-7': 'This shows that the inclusion of the generalized field [MATH] in the gauge fixing leads to the standard renormalization of the fields, parameters and sources.', '1803.03077-3-51-8': 'The renormalization of [MATH] itself is encoded in the renormalization of the infinite set of gauge parameters [MATH], as in eq. [REF].', '1803.03077-3-52-0': 'Note that both [MATH] and [MATH], as well as their sources [MATH] and [MATH], are renormalized in a non-linear way through a power series in [MATH] and [MATH], respectively.', '1803.03077-3-52-1': 'This is expected, due to the fact that both superfields are dimensionless.', '1803.03077-3-52-2': 'However, one has to note that the dimensionless superfield [MATH] contains a massive supermultiplet [MATH].', '1803.03077-3-52-3': 'Despite the fact that [MATH] itself renormalizes in a non-linear way due to its dimensionless nature, the component fields [MATH] do renormalize in fact in a standard multiplicative way through a constant (i.e. field independent) renormalization factors, a feature which can be checked out by employing the the Wess-Zumino gauge.', '1803.03077-3-53-0': '# Conclusion', '1803.03077-3-54-0': 'In this work we took a first step towards the understanding of Stueckelberg-like models in supersymmetric non-abelian gauge theories.', '1803.03077-3-54-1': 'The gauge invariant transverse field configuration [MATH] has been investigated in supersymmetric Yang-Mills theory with the Landau gauge.', '1803.03077-3-54-2': 'An auxiliary chiral superfield [MATH] was introduced that compensates the gauge variation of the vector superfield [MATH], thus preserving gauge invariance of the composite field [MATH].', '1803.03077-3-54-3': 'This gauge invariant composite field allows the construction of a local BRST-invariant massive model, summarized by the action [REF].', '1803.03077-3-54-4': 'Both [MATH] and [MATH] are dimensionless, which leads to ambiguities in defining both the mass term and the gauge fixing term.', '1803.03077-3-54-5': 'However, working with a generalized gauge fixing term, we find that the model turns out to be renormalizable to all orders of perturbation theory, as was discussed in sections [REF] and [REF].', '1803.03077-3-55-0': 'As a possible future application of the present result, let us mention that the possibility of having constructed a manifestly BRST invariant supersymmetric renormalizable version of the modified Stueckelberg models introduced in [CITATION] can open the possibility to investigate the important issue of the non-perturbative phenomenon of the Gribov copies directly in superspace, by generalizing to [MATH] the Gribov-Zwanziger setup.', '1803.03077-3-55-1': 'This would enable us to study aspects of the non-perturbative region of [MATH] confining supersymmetric theories, see also [CITATION] for a preliminary attempt in this direction.'}",,
1305.3884,"{'1305.3884-1-0-0': 'Radio pulsars are neutron stars that emit radiation modulated and powered by the rotation of their magnetic field, and which consequently decelerate[CITATION].', '1305.3884-1-0-1': 'The very fast millisecond spin periods measured in old radio pulsars[CITATION] are thought to be the outcome of an earlier X-ray bright phase, during which the neutron star accretes matter and angular momentum from a low mass companion star in a binary system[CITATION].', '1305.3884-1-0-2': 'This evolutionary scenario has been supported by the detection of X-ray millisecond pulsations from several accreting neutron stars in the past fifteen years[CITATION], as well as by the indirect evidence for the presence of a disk in the past around a millisecond radio pulsar now powered by rotation[CITATION].', '1305.3884-1-0-3': 'However, a transition between a rotation-powered and an accretion-powered state was never observed.', '1305.3884-1-0-4': 'Here we present the detection of millisecond X-ray pulsations from an accreting neutron star which showed multiple accretion event in the past few years, and was already known as a radio millisecond pulsar.', '1305.3884-1-0-5': 'The coherent signal that modulates the X-ray radiation is related to mass accretion, as demonstrated by its magnitude, variability and energy distribution, and by the detection of the signal during a thermonuclear explosion that occurred onto the neutron star surface.', '1305.3884-1-0-6': 'This source proves that in the context of the evolutionary link between millisecond pulsars fuelled either by rotation or by mass accretion, these states alternate over a time scale of a few years in a cyclic fashion.', '1305.3884-1-1-0': 'The extremely short spin periods shown by the [MATH] known rotation-powered radio millisecond pulsars of the Galaxy is explained in terms of a previous evolutionary phase, in which the neutron star increases its rotation rate through the accretion of the matter transferred by a low mass ([MATH]-[MATH]) companion star through a disc[CITATION].', '1305.3884-1-1-1': 'During such an accretion phase the system is X-ray bright, and the presence of plasma close to the neutron star quenches the radio pulses.', '1305.3884-1-1-2': 'When the mass transfer from the companion star decreases, the system cannot keep the viscous disc permanently in a high accretion state[CITATION], and enhanced mass accretion with a conspicuous X-ray emission ([MATH] erg s[MATH]) proceeds only during a few month-long periods called outbursts.', '1305.3884-1-1-3': 'Outside outbursts these sources spend long periods of time ([MATH] years) in an X-ray quiescent state ([MATH] erg s[MATH]).', '1305.3884-1-1-4': 'It was proposed that when the pressure of the transferred matter decreases, the pulsar wind may reactivate a rotation-powered, possibly pulsed activity[CITATION].', '1305.3884-1-1-5': 'Rotation and accretion-powered pulsar states would then alternate on the time scales set of the observed cycles between X-ray outbursts and quiescence.', '1305.3884-1-1-6': 'On the other hand, when the mass transfer from the companion star ceases altogether, the cycle between X-ray outbursts and quiescence ends, and the system then shines permanently as a recycled radio millisecond pulsar.', '1305.3884-1-2-0': 'The detection of millisecond X-ray pulsations from accreting neutron stars orbiting a low mass star[CITATION] demonstrated that the accretion of mass in an X-ray binary is able to spin-up a neutron star to a millisecond spin period.', '1305.3884-1-2-1': 'A modulation of the X-ray emission is produced when the accreting material lost by the companion star is pulled-up from the accretion disk by the neutron star magnetic field, and channelled toward the magnetic poles before being accreted onto the surface.', '1305.3884-1-2-2': 'A dozen of accreting millisecond X-ray pulsars has been discovered by now, transients alternating between X-ray outbursts and quiescence.', '1305.3884-1-2-3': 'The amount of optical light reprocessed by the companion star[CITATION], as well as the spin and orbital evolution of these neutron stars[CITATION], indirectly suggest the re-ignition of the radio pulsar during the X-ray quiescence, but deep searches for radio pulsations were unfruitful, so far[CITATION].', '1305.3884-1-3-0': 'The X-ray transient IGR J18245-2452 was first detected by INTEGRAL on 2013, March 28.', '1305.3884-1-3-1': 'It is located in the globular cluster M28, at a distance[CITATION] of 5.5 kpc.', '1305.3884-1-3-2': 'The X-ray luminosity of a [MATH] erg s[MATH] (0.3–-10 keV), and the detection by the X-ray Telescope on-board Swift of a burst originated by a thermonuclear explosion at the surface of the compact object[CITATION], firmly classified this source as an accreting neutron star in an X-ray binary.', '1305.3884-1-3-3': 'Seven days after the discovery of IGR J18245-2452, we detected a coherent modulation at a period of [MATH] ms during an XMM-Newton Target of Opportunity observation (see Fig. [REF] for the light curve).', '1305.3884-1-3-4': 'The signal has an almost sinusoidal profile with a peak-to-peak amplitude of [MATH] per cent (see the inset of Fig. [REF]), translating into a detection with a significance of more than [MATH].', '1305.3884-1-3-5': 'We observed delays of the pulse arrival times produced by the circular orbit of the neutron star around a companion star of a mass larger than [MATH], with an orbital period of [MATH] hours.', '1305.3884-1-3-6': 'The spin and orbital parameters of the source were further improved by making use of a second XMM-Newton observation (see Fig. [REF]), yielding the ephemerides given in Table [REF].', '1305.3884-1-4-0': 'By looking at the list of the radio rotation powered pulsars detected[CITATION] in M28, we found that PSR J1824-2452I has identical ephemerides to those of IGR J18245-2452.', '1305.3884-1-4-1': 'Given the accuracy of the measured parameters we concluded that they are the same source.', '1305.3884-1-4-2': 'However, the X-ray pulsations we have observed from IGR J18245-2452 are unambiguously powered by mass accretion.', '1305.3884-1-4-3': 'In particular, the pulse amplitude varies in strong correlation with the X-ray flux, implying that pulsations come from a source emitting [MATH] erg s[MATH] in X-rays; this value is larger by more than two orders of magnitudes than the luminosity shown by the X-ray counterparts of rotation-powered radio millisecond pulsars[CITATION], while it nicely agrees with the X-ray output of accreting millisecond pulsars.', '1305.3884-1-4-4': 'The spectrum of the X-ray emission of the source is typical of accreting millisecond pulsars, and the broad emission line observed at an energy compatible with the iron K-[MATH] transition ([MATH]-–[MATH] keV) is most easily interpreted in terms of reflection of hard X-rays by the truncated accretion disk[CITATION].', '1305.3884-1-4-5': 'Furthermore, the pulse is also detected by Swift-XRT during the decay of a thermonuclear burst, following a runaway nuclear burning of light nuclei accreted onto the neutron star surface.', '1305.3884-1-4-6': 'Such bursts are unambiguous indicators that mass accretion is taking place, and the periodic oscillations observed in some of them trace the spin of the accreting neutron star[CITATION].', '1305.3884-1-5-0': 'We derived an accurate position for IGR J18245-2452 using a Chandra image taken on 2013, April 29, while the source was still in outburst (see Fig. [REF]).', '1305.3884-1-5-1': 'Its position is compatible with a variable radio source that we have detected with the Australia Compact Telescope Array on 2013, April 5 at frequencies above 5.5 GHz, with spectral properties typical of an accreting millisecond pulsar in outburst[CITATION].', '1305.3884-1-5-2': 'The radio millisecond pulsar was not detected by observations performed simultaneously to the 2013 Chandra observation and at lower radio frequencies, by the 64-m Parkes radio telescope, and a few days later by the Westerbork Syntheis Radio Telescope, compatible with the neutron star being in an accretion phase.', '1305.3884-1-5-3': 'Furthermore, analysis of archival Chandra observations indicated that IGR J18245-2452 underwent other accretion events in the past ten years.', '1305.3884-1-6-0': 'On the other hand, the pulsed radio emission detected[CITATION] from PSR J1824-2452I in 2006 with the Green Bank Telescope rules out the possibility that this source was accreting at that time, as the presence of accreting material quenches a pulsed radio signal.', '1305.3884-1-6-1': 'The radio observations in 2006 indicated that PSR J1824-2452I was surrounded by conspicuous material, causing the irregular disappearing of the radio signal from this object and leading to a poorly determined source position[CITATION].', '1305.3884-1-6-2': 'During 2006, the X-ray luminosity of IGR J18245-2452 observed by Chandra was [MATH] erg s[MATH], a level typical of X-ray counterparts of rotation powered, radio millisecond pulsars[CITATION] and of accreting millisecond pulsars during X-ray quiescence[CITATION].', '1305.3884-1-7-0': 'This system provides the conclusive evidence of the evolutionary link between neutron stars in low mass X-ray binaries and millisecond radio pulsars, and establish the existence of an intermediate evolutionary phase during which rotation and accretion powered states alternate over a time scale of a few years.', '1305.3884-1-7-1': 'The discovery of accreting millisecond pulsars[CITATION] proved that accretion can spin-up a neutron star to a period of few millisecond.', '1305.3884-1-7-2': 'The past presence of a disk (though, not in an accreting phase) around a millisecond pulsar now powered by rotation[CITATION], gave an indirect indication of a transition from an accretion to a rotation powered state, in the expected evolutionary direction.', '1305.3884-1-7-3': 'IGR J18245-2452 is the first system to show at different times unambiguous tracers of rotation and accretion powered activity, and to show transitions in both directions.', '1305.3884-1-7-4': 'These swings take place on a time scale much shorter than the Gyr-long evolution of these binary systems[CITATION].', '1305.3884-1-7-5': 'The companion star of IGR J18245-2452 has a mass compatible with a main sequence star.', '1305.3884-1-7-6': 'It is indeed most likely that this intermediate evolutionary phase is observed in systems that have not reached the endpoint of the accretion phase yet[CITATION].', '1305.3884-1-8-0': 'The X-ray luminosity of IGR J18245-2452 during quiescence ([MATH] erg s[MATH]) implies that rate of mass in-fall was not larger than [MATH] M[MATH] yr[MATH], during such state.', '1305.3884-1-8-1': 'The presence of radio millisecond pulsation during the X-ray quiescent state of IGR J18245-2452 testifies that the pressure exerted by the mass transferred by the companion star could not prevent the pulsar magnetosphere from pushing the plasma beyond the light cylinder of the pulsar (located at a distance of [MATH]200 km).', '1305.3884-1-8-2': 'A pulsar magnetic field of the order of [MATH]-[MATH] G is able to satisfy this condition, and to explain the quiescent X-ray luminosity in terms of the pulsar rotational power (for a typical conversion efficiency of about 1-10 per cent).', '1305.3884-1-8-3': 'In this radio-active stage, the mass that the companion star keeps transferring towards the neutron star is ejected by the pressure of the pulsar wind[CITATION], determining the observed, irregular disappearance of the radio signal.', '1305.3884-1-8-4': 'A slight increase of the mass transfer rate may subsequently push the magnetosphere back within the light cylinder[CITATION].', '1305.3884-1-8-5': 'After a disk had a sufficient time to build up, another X-ray outburst takes place, as in the case of IGR J18245-2452 during the observations reported here.', '1305.3884-1-8-6': 'The discovery of IGR J18245-2452, alternating between rotation and accretion-powered emission, represents the most stringent probe of the recycling scenario, and the existence of an unstable intermediate evolutionary phase in the evolution of low mass X-ray binaries.', '1305.3884-1-9-0': 'This paper is based on ToO observations made by XMM-Newton, Chandra X-ray Observatory, Swift, INTEGRAL, ATCA, Westerbork Synthesis Radio Telescope and Parkes antenna.', '1305.3884-1-9-1': 'We thank the respective directors and operation teams for the support.', '1305.3884-1-9-2': 'Work done in the framework of the grants AYA2012-39303, SGR2009-811, and iLINK2011-0303.', '1305.3884-1-9-3': 'We also acknowledge support from the following funding agencies: CEA/Irfu, IN2P3/CNRS and CNES (France); INAF (Italy).', '1305.3884-1-9-4': 'AR acknowledges Sardinia Regional Government for financial support (P.O.R. Sardegna F.S.E. Operational Programme of the Autonomous Region of Sardinia, ESF 2007-13, Axis IV Human resources objective I.3, line of activity I.3.1.', '1305.3884-1-9-5': 'D. F. T was additionally supported by a Friedrich Wilhelm Bessel Award of the Alexander von Humboldt Foundation.', '1305.3884-1-9-6': 'L. P thanks the Societe Academique de Geneve and the Swiss Society for Astrophysics and Astronomy for travel grants supporting the on-going collaboration between the ISDC and the ATCA team.', '1305.3884-1-9-7': 'Finally, we acknowledge the use of data supplied by the UK Swift Science Data Centre at the University of Leicester.', '1305.3884-1-9-8': '[Author Contributions] A. P., C. F. and E. B. collected and analysed XMM-Newton data.', '1305.3884-1-9-9': 'A. P. and C. F. detected the pulsar in XMM-Newton data and derived its orbital solution.', '1305.3884-1-9-10': 'A. P. discovered the equivalence of its parameters with a radio pulsar, the thermonuclear burst and the burst oscillations.', '1305.3884-1-9-11': 'A. P. and N. R. wrote the paper.', '1305.3884-1-9-12': 'N. R. analysed Chandra data, detecting the X-ray quiescent counterpart of the source and past accretion events.', '1305.3884-1-9-13': 'L. P., M. H. W., M. D. F. and G. F. W. analysed ATCA data.', '1305.3884-1-9-14': 'E. B., S. C. and P. R. analysed Swift data.', '1305.3884-1-9-15': 'E. B. and C. F. analysed INTEGRAL data.', '1305.3884-1-9-16': 'J. W. T. H. collected and analysed a Westerbork Synthesis Radio Telescope pointing.', '1305.3884-1-9-17': 'M. B. and J. M. S. performed and analysed a 64-m Parkes radio telescope observation.', '1305.3884-1-9-18': 'A. R. provided valuable software tools.', '1305.3884-1-9-19': 'All the authors contributed to the interpretation.', '1305.3884-1-9-20': '[Competing Interests] The authors declare that they have no competing financial interests.', '1305.3884-1-9-21': '[Correspondence] Correspondence and requests for materials should be addressed to A. Papitto (email: papitto@ice.csic.es).'}","{'1305.3884-2-0-0': 'It is thought that neutron stars in low-mass binary systems can accrete matter and angular momentum from the companion star and be spun-up to millisecond rotational periods[CITATION].', '1305.3884-2-0-1': 'During the accretion stage, the system is called a low-mass X-ray binary and bright X-ray emission is observed.', '1305.3884-2-0-2': ""When the rate of mass transfer decreases in the later evolutionary stages, these binaries host instead a radio millisecond pulsar[CITATION], whose emission is powered by the neutron star's rotating magnetic field[CITATION]."", '1305.3884-2-0-3': 'This scenario is supported by the detection of X-ray millisecond pulsations from several accreting neutron stars[CITATION] and the evidence for a past accretion disc in a rotation-powered millisecond pulsar[CITATION].', '1305.3884-2-0-4': 'It has been proposed that a rotation-powered pulsar may temporarily switch on[CITATION] during periods of low mass inflow[CITATION] in some such systems.', '1305.3884-2-0-5': 'However, only indirect evidence for this transition had been observed[CITATION].', '1305.3884-2-0-6': 'Here we report the detection of accretion-powered, millisecond X-ray pulsations from a neutron star previously seen as a rotation-powered radio pulsar.', '1305.3884-2-0-7': 'Within a few days following a month-long X-ray outburst, radio pulses were again detected.', '1305.3884-2-0-8': 'This not only demonstrates the evolutionary link between accretion and rotation-powered millisecond pulsars, but also that some systems can swing between the two states on very short timescales.', '1305.3884-2-1-0': 'The X-ray transient IGR J18245-2452 was first detected by INTEGRAL on 28 March 2013, and is located in the globular cluster M28.', '1305.3884-2-1-1': 'The X-ray luminosity of a [MATH] erg s[MATH] (0.3–-10 keV), and the detection by the X-ray Telescope (XRT) on-board Swift of a burst originated by a thermonuclear explosion at the surface of the compact object[CITATION], firmly classified this source as an accreting neutron star with a low-mass companion.', '1305.3884-2-1-2': 'An observation performed by XMM-Newton on 4 April 2013 revealed a coherent modulation of its X-ray emission at a period of [MATH] ms (see Fig. [REF] and [REF]).', '1305.3884-2-1-3': 'We observed delays of the pulse arrival times produced by the orbit of the neutron star around a companion star of a mass [MATH], with an orbital period of [MATH] hours (see Fig. [REF]).', '1305.3884-2-1-4': 'The spin and orbital parameters of the source were further improved by making use of a second XMM-Newton observation, as well as two observations performed by Swift/XRT (see Table [REF]).', '1305.3884-2-2-0': 'Cross-referencing with the known rotation-powered radio pulsars in M28, we found that PSR J1824-2452I has ephemerides[CITATION] identical to those of IGR J18245-2452 (see Table [REF]).', '1305.3884-2-2-1': 'However, the X-ray pulsations we have observed from IGR J18245-2452 are not powered by the rotation of the magnetic field as for the radio emission of PSR J1824-2452I.', '1305.3884-2-2-2': 'The pulse amplitude was observed to vary in strong correlation with the X-ray flux, implying that pulsations came from a source emitting [MATH] erg s[MATH] in X-rays; this value is larger by more than two orders of magnitudes than the luminosity shown by the X-ray counterparts of rotation-powered radio millisecond pulsars[CITATION], while it nicely agrees with the X-ray output of accretion-powered millisecond pulsars[CITATION].', '1305.3884-2-2-3': 'The X-ray spectrum of IGR J18245-2452 was also typical of this class, and the broad emission line observed at an energy compatible with the iron K-[MATH] transition ([MATH]-–[MATH] keV) is most easily interpreted in terms of reflection of hard X-rays by a truncated accretion disk[CITATION].', '1305.3884-2-2-4': 'Furthermore, pulsations were detected by Swift-XRT during the decay of a thermonuclear burst, following a runaway nuclear burning of light nuclei accreted onto the neutron star surface.', '1305.3884-2-2-5': 'Such bursts are unambiguous indicators that mass accretion is taking place[CITATION], and the oscillations observed in some of them trace the spin period of the accreting neutron star[CITATION].', '1305.3884-2-3-0': 'We derived a precise position for IGR J18245-2452 using a Chandra image taken on 29 April 2013, while the source was fading in X-rays.', '1305.3884-2-3-1': 'Analysis of archival Chandra observations from 2008 indicate that IGR J18245-2452 already showed variations of its X-ray luminosity by an order of magnitude, as shown in Fig. [REF], suggesting it underwent other episodes of mass accretion in the past few years.', '1305.3884-2-3-2': 'This 2008 enhancement of the X-ray emission followed the nearest previous detection of the radio pulsar, on 13 June 2008, by less than two months, indicating a very rapid transition from rotation to accretion-powered activity (see Table 3 in Supplementary Information for a summary of past observations of the source in the X-ray and radio band).', '1305.3884-2-3-3': 'The Chandra position of IGR J18245-2452 is compatible with a variable unpulsed radio source that we have detected with the Australia Compact Telescope Array on 2013, April 5, with spectral properties typical of an accreting millisecond pulsar in outburst[CITATION].', '1305.3884-2-4-0': 'A combination of serendipitous and target-of-opportunity observations with the Green Bank Telescope (GBT), Parkes radio telescope, and Westerbork Synthesis Radio Telescope (WSRT) partially map the reactivation of IGR J18245-2452 as the radio pulsar PSR J1824-2452I (see Table 3 in Supplementary Information).', '1305.3884-2-4-1': 'No pulsed radio emission was seen in any of the three 2013 April observations, compatible with the neutron star being in an accretion phase and inactive as a radio pulsar.', '1305.3884-2-4-2': 'We caution however, that a non-detection of radio pulsations from PSR J1824-2452I can also be due to eclipsing and that the lack of observable radio pulsations does not necessarily prove the abscence of an active radio pulsar mechanism[CITATION].', '1305.3884-2-4-3': 'Radio pulsations were detected in 5 of the 13 observations conducted with GBT, Parkes, and WSRT in 2013 May.', '1305.3884-2-4-4': 'These observations demonstrate that the radio pulsar mechanism was active no more than a few weeks after the peak of the X-ray outburst.', '1305.3884-2-5-0': 'In the last decade, IGR J18245-2452 has thus shown unambiguous tracers of both rotation and accretion powered activity, providing conclusive evidence for the evolutionary link between neutron stars in low mass X-ray binaries and millisecond radio pulsars.', '1305.3884-2-5-1': 'The source swung between rotation and accretion powered states on few-day to few-month time scales; this establishes the existence of an evolutionary phase during which a source can alternate between these two states over a time scale much shorter than the Gyr-long evolution of these binary systems, as they are spun-up by mass accretion to millisecond spin periods[CITATION].', '1305.3884-2-5-2': 'It is probable that a rotation powered pulsar switches on also during the X-ray quiescent states of other accreting millisecond pulsars[CITATION], even if radio pulsations were not detected[CITATION], so far.', '1305.3884-2-6-0': 'The short time-scales observed for the transitions between accretion and rotation powered states of IGR J18245-2452 are comparable with those typical of X-ray luminosity variations.', '1305.3884-2-6-1': ""Like other X-ray transients, IGR J18245-2452 is X-ray bright ([MATH] erg s[MATH]) only during a few month-long periods called 'outbursts', while outside these episodes it spends years in an X-ray quiescent state ([MATH] erg s[MATH])."", '1305.3884-2-6-2': 'These variations are caused by swings of the mass in-flow rate onto the neutron star[CITATION], and our findings strongly suggest that this quantity mainly regulates the transitions between accretion and rotation powered activity, compatible with earlier suggestions[CITATION].', '1305.3884-2-6-3': 'The X-ray luminosity of IGR J18245-2452 during quiescence ([MATH] erg s[MATH]) implies that rate of mass accretion was not larger than [MATH] M[MATH] yr[MATH], during such a state.', '1305.3884-2-6-4': 'The presence of radio millisecond pulsations indicates that the pulsar magnetosphere kept the plasma beyond the light cylinder radius (located at a distance of [MATH]200 km), despite the pressure exerted by the mass inflowing from the companion star.', '1305.3884-2-6-5': 'A pulsar magnetic field of the order of [MATH]-[MATH] G is able to satisfy this condition, and to explain the quiescent X-ray luminosity in terms of the pulsar rotational power (for a typical conversion efficiency of about 1[MATH]).', '1305.3884-2-6-6': 'The irregular disappearance of the radio pulses of PSR J1824-2452I during the rotation powered stage suggests that, during that phase, most of the matter that the companion transfers towards the neutron star is ejected by the pressure of the pulsar wind[CITATION].', '1305.3884-2-6-7': 'A slight increase of the mass transfer rate may subsequently push the magnetosphere back inside the light cylinder[CITATION].', '1305.3884-2-6-8': 'After a disk had sufficient time to build up, an X-ray outburst is expected to take place, as in the case of IGR J18245-2452 during the observations reported here.', '1305.3884-2-6-9': 'As the mass accretion rate decreases during the decay of the X-ray outburst, the pressure of the magnetosphere is able to, at least partially, sweep away the residual matter from the surroundings of the neutron star, and a rotation-powered pulsed radio emission can reactivate.', '1305.3884-2-6-10': 'Our observations prove that such transitions can take place in both directions, on a time scale shorter than expected, perhaps only a few days.', '1305.3884-2-7-0': 'The discovery of IGR J18245-2452, swinging between rotation and accretion-powered emission, represents the most stringent probe of the recycling scenario[CITATION], and the existence of an unstable intermediate phase in the evolution of low mass X-ray binaries, offering the unprecedented opportunity to study in detail the transitions between these two states.', '1305.3884-2-8-0': 'This letter is based on ToO observations made by XMM-Newton, Chandra, INTEGRAL, Swift, ATCA, WSRT, GBT, and PKS.', '1305.3884-2-8-1': 'We thank the respective directors and operation teams for their support.', '1305.3884-2-8-2': 'Work done in the framework of the grants AYA2012-39303, SGR2009-811, and iLINK2011-0303, and with the support of CEA/Irfu, IN2P3/CNRS and CNES (France), INAF (Italy), NWO (The Netherlands), and NSERC (Canada).', '1305.3884-2-8-3': 'A. Pa. is supported by a Juan de la Cierva Research Fellowship.', '1305.3884-2-8-4': 'A. R. acknowledges Sardinia Regional Government for financial support (P.O.R. Sardegna ESF 2007-13).', '1305.3884-2-8-5': 'D. F. T was additionally supported by a Friedrich Wilhelm Bessel Award of the Alexander von Humboldt Foundation.', '1305.3884-2-8-6': 'L. P. thanks the Societe Academique de Geneve and the Swiss Society for Astrophysics and Astronomy.', '1305.3884-2-8-7': 'Finally, we acknowledge the use of data supplied by the UK Swift Science Data Centre at the University of Leicester.', '1305.3884-2-8-8': 'A. Pa. thanks S. Giannetti, D. Lai, R. V. E. Lovelace, M. M. Romanova for stimulating discussions, and Wolf Sol.', '1305.3884-2-8-9': 'Dig.', '1305.3884-2-8-10': 'for operational support.', '1305.3884-2-8-11': '[Author Contributions] A. Pa., C. F. and E. B. collected and analysed XMM-Newton data.', '1305.3884-2-8-12': 'A. Pa. and C. F. detected the pulsar in XMM-Newton data and derived its orbital solution.', '1305.3884-2-8-13': 'A. Pa. discovered the equivalence of its parameters with a radio pulsar, the thermonuclear burst and the burst oscillations.', '1305.3884-2-8-14': 'N. R. analysed Chandra data, detecting the X-ray quiescent counterpart of the source and past accretion events.', '1305.3884-2-8-15': 'L. P., M. H. W., M. D. F. and G. F. W. analysed ATCA data.', '1305.3884-2-8-16': 'E. B., S. C., P. R., A. Pa. and A. R. analysed Swift data.', '1305.3884-2-8-17': 'E. B. and C. F. analysed INTEGRAL data.', '1305.3884-2-8-18': 'J. W. T. H. analysed WSRT data.', '1305.3884-2-8-19': 'M. B. and J. M. S. analysed PKS data.', '1305.3884-2-8-20': 'J. W. T. H., S. M. R., A. Po.', '1305.3884-2-8-21': 'and I. H. S. and P. C. C. F. analysed GBT data.', '1305.3884-2-8-22': 'A. R. provided valuable software tools.', '1305.3884-2-8-23': 'A. Pa., N. R., and J. W. T. H. wrote the manuscript, with significant contribution by all the authors in interpreting the results and editing of the manuscript.', '1305.3884-2-8-24': '[Competing Interests] The authors declare that they have no competing financial interests.', '1305.3884-2-8-25': '[Correspondence] Correspondence and requests for materials should be addressed to A. Papitto (email: papitto@ice.csic.es).', '1305.3884-2-9-0': 'Supplementary Information', '1305.3884-2-10-0': '# INTEGRAL detection of IGR J18245-2452', '1305.3884-2-11-0': 'IGR J18245-2452 was first detected[CITATION] by the hard X-ray imager IBIS[CITATION] using the ISGRI[CITATION] detector on-board INTEGRAL[CITATION], on 28 March 2013, during observations of the Galactic Center.', '1305.3884-2-11-1': 'We analysed the corresponding data (pointings from 83 to 107 in satellite revolution 1276) by using the Off-line Science Analysis software provided by the Integral Science Data Centre.', '1305.3884-2-11-2': 'The source was detected in the ISGRI mosaicked image at a significance level of 21[MATH] in the 20-40 keV energy band and 15[MATH] in the 40-80 keV energy band (see Figure [REF]).', '1305.3884-2-11-3': 'The best source position was obtained at RA=276.14[MATH], Dec=−24.88[MATH] (J2000), with an associated uncertainty of 1.4[MATH] at 90[MATH] confidence level, well within the globular cluster M28.', '1305.3884-2-11-4': 'During the first detection of the source, its flux estimated from the ISGRI data was [MATH] erg cm[MATH] s[MATH] in the 20-100 keV energy band, during an effective exposure time of 32 ks, corresponding to a luminosity of [MATH] erg s[MATH], at a distance[CITATION] of 5.5 kpc.', '1305.3884-2-11-5': 'The source was outside the field of view of the JEM-X instrument[CITATION], sensitive in the 3-30 keV energy band, during the entire observation.', '1305.3884-2-12-0': '# XMM-Newton observations of IGR J18245-2452', '1305.3884-2-13-0': 'Following the detection of IGR J18245-2452 by INTEGRAL, we obtained two target of opportunity observations with the X-ray Multi-Mirror Mission[CITATION] (XMM-Newton), starting on 3 April 2013, at 23:49 (Obs.', '1305.3884-2-13-1': 'ID 0701981401, OBS 1, hereafter) and 13 April at 06:25 (Obs.', '1305.3884-2-13-2': 'ID 0701981501, OBS 2), and lasting 26.7 and 67.2 ks, respectively (all the epochs are given in the Coordinated Universal Time).', '1305.3884-2-13-3': 'The European Photon Imaging Camera (EPIC) pn[CITATION] was operated in fast timing mode, the two EPIC MOS cameras[CITATION] in small window imaging mode, and the Refection Grating Spectrometers[CITATION] (RGS) in standard spectroscopy mode.', '1305.3884-2-13-4': 'A thick blocking filter was used to shield the EPIC cameras from contamination of optical light.', '1305.3884-2-13-5': 'All the data were reduced by using the latest version of the XMM-Newton Science Analysis Software (SAS ver.', '1305.3884-2-13-6': '13).', '1305.3884-2-13-7': 'The average count rates observed by the EPIC pn, EPIC MOS1, EPIC MOS2, RGS1 and RGS2 (in their first order of dispersion) during the first (second) observations were 68.1 (71.0), 11.8 (11.7), 11.6 (11.5), 0.68 (0.86) and 0.69 (0.88) s[MATH], respectively.', '1305.3884-2-13-8': 'Since the source is detected also at energy larger than 10 keV, we built a 0.5-10 keV light curve from a background region falling in one of the outer chips of the EPIC MOS cameras which was not pointing the source, in order to ensure that soft proton flares of Solar origin did not contaminate the observations.', '1305.3884-2-14-0': '## Timing Analysis.', '1305.3884-2-15-0': 'The timing observing mode of the EPIC pn camera has a timing resolution of 29.52 [MATH]s.', '1305.3884-2-15-1': 'This resolution is achieved by losing spatial information along one of the axes of the CCD.', '1305.3884-2-15-2': 'To extract the source emission, we considered X-ray photons falling within 86.1[MATH] from the source position measured along one of the pointing axes (corresponding to a full width of 21 EPIC pn pixels), and containing 95[MATH] of the energy at every observed wavelength.', '1305.3884-2-15-3': 'The background emission was extracted from a region of width 12[MATH] (corresponding to 3 EPIC pn pixels).', '1305.3884-2-16-0': 'Although X-ray pulsations from several accreting millisecond pulsars have already been observed by XMM-Newton[CITATION], this is the first time that pulsations from an accreting millisecond pulsar have been discovered by this observatory.', '1305.3884-2-16-1': 'In order to perform a timing analysis of the signal, the times of arrival of X-ray photons were converted to the Solar System barycentre, by using the position determined by Chandra (see Sec. [REF] below) and Solar System ephemerides JPL DE405.', '1305.3884-2-16-2': 'We derived a zeroth order orbital solution by measuring the spin period in 2-ks long intervals through an epoch folding technique[CITATION], and modelling the orbital modulation affecting the values obtained.', '1305.3884-2-16-3': 'To assess the significance of any detection we considered that in the absence of any periodic signal, the variance of a folded profile follows a [MATH] distribution with [MATH] degrees of freedom, where [MATH] is the number of phase bins used to sample the signal period[CITATION] ([MATH] in this case).', '1305.3884-2-16-4': 'We then used the preliminary determination of the pulsar orbital parameters to convert the photon arrival times to the line of nodes of the binary system.', '1305.3884-2-16-5': 'This procedure was iterated until the spin period was observed to be constant throughout the observations.', '1305.3884-2-16-6': 'To further refine the parameters, we then folded data around the current best estimate of the spin period, considering 12 phase bins and describing the pulse profile with two harmonic components.', '1305.3884-2-16-7': 'The variation of the phase of the first harmonic over time was modelled in terms of the difference between the orbital and spin parameters used to correct the photon arrival times, and the actual ones[CITATION].', '1305.3884-2-16-8': 'The procedure was iterated until no significant corrections to the parameters were found within the uncertainties[CITATION].', '1305.3884-2-16-9': 'We checked that the phase difference between the first and second harmonic component was compatible with a constant.', '1305.3884-2-17-0': 'The amplitude of the harmonic component at the fundamental frequencies varied between 18[MATH] and the non detection, in strong positive correlation with the count rate observed in the 0.5-10 keV (see Figure [REF]).', '1305.3884-2-17-1': ""The Spearman's rank correlation coefficient evaluated from the observed count rate and amplitude is [MATH] for 45 points, implying a probability of less than [MATH] that the two variables are not correlated."", '1305.3884-2-18-0': '## Spectral Analysis.', '1305.3884-2-19-0': 'We built an X-ray spectrum of the emission observed by the EPIC pn by excluding the two brightest columns of CCD pixels (RAWX=37-38), to avoid the effect of photon pile up during the time intervals in which the count rate was the highest.', '1305.3884-2-19-1': 'Spectral bins were grouped to over-sample the effective instrument resolution by a factor not larger than three.', '1305.3884-2-19-2': 'Response matrices were built following the guidelines provided by the XMM-Newton calibration technical notes.', '1305.3884-2-19-3': 'The two observations were modelled simultaneously as they had a compatible average spectral shape.', '1305.3884-2-19-4': ""We modelled the 0.6-11 keV observed spectra with an accretion disk spectral component[CITATION] (diskbb in the terminology used by the spectral fitting routine used, Heasarc's XSPEC v.12.8.0), and with the emission produced by a thermal distributions of electrons which Compton up-scatter soft seed X-ray photons, with a black-body spectral shape[CITATION] (nthcomp)."", '1305.3884-2-19-5': 'A similar spectral decomposition well described the X-ray spectra shown by accreting millisecond pulsars[CITATION], observed by the EPIC pn camera[CITATION].', '1305.3884-2-19-6': 'The temperature of the electron cloud was fixed at [MATH] keV, outside the energy band observed by the EPIC pn camera, following the values usually observed at higher energies from accreting millisecond pulsars[CITATION] ([MATH] keV).', '1305.3884-2-19-7': 'Absorption of the interstellar medium was modelled according to the Tuebingen-Boulder model (Wilms, J. et al. 2011, in preparation, http:[MATH]pulsar.sternwarte.uni-erlangen.de[MATH]wilms[MATH]research[MATH]tbabs[MATH]).', '1305.3884-2-19-8': 'The value of the absorption column was fixed to the value measured by fitting the average spectra observed by the RGS 1 and 2 with an absorbed power law ([MATH] cm[MATH]); solar abundances were considered.', '1305.3884-2-19-9': 'A broad Gaussian emission feature centred at an energy compatible with iron K-[MATH] emission (6.4-6.97 keV), and a narrow emission feature most probably due to calibration residuals around the edge of Au (Guainazzi, M. et al. 2012, available from http:[MATH]xmm2.esac.esa.int[MATH]docs[MATH]documents[MATH]CAL-TN-0083.pdf), decreased significantly the variance of model residuals.', '1305.3884-2-19-10': 'The best-fit parameters obtained modelling the spectra observed during the two XMM-Newton are given in Table [REF], while the observed spectrum and residuals are plotted in Figure [REF].', '1305.3884-2-20-0': '## The position.', '1305.3884-2-21-0': 'To determine the source position we accumulated images from the EPIC MOS1 and MOS 2 in the energy bands 0.5-1 keV, 1-2 keV, 2-4.5 keV, and 4.5-12 keV, during the time intervals in which the source count rate was [MATH] s[MATH] (0.5-10 keV).', '1305.3884-2-21-1': 'This selection prevented pile-up and resulted in a total effective exposure time of 5.7 and 13.8 ks for the two instruments in OBS 1 and OBS 2, respectively.', '1305.3884-2-21-2': 'We created an exposure map using the attitude information to mask out the regions of the detector where not enough exposure was available, and performed a first localisation of the sources in the raw images using the sliding box method[CITATION].', '1305.3884-2-21-3': 'A background map was then produced from the previous images by removing the identified sources (taking into account the local instrument point spread function), and used together with previous products to perform a second optimised source localisation.', '1305.3884-2-21-4': 'The data of the two MOS cameras in the two observations and different energy bands provided a total of 16 independent estimates of the source position.', '1305.3884-2-21-5': 'By averaging such estimates we obtained a position of RA=[MATH] Dec=[MATH].', '1305.3884-2-21-6': 'We added in quadrature the [MATH] uncertainty we derived with the described analysis (1.5[MATH]) to the systematic error reported in the XMM catalogue[CITATION] (1.0[MATH]), to obtain a total uncertainty of 1.8[MATH] confidence level).', '1305.3884-2-21-7': 'The error circle at a [MATH] confidence level is plotted as a white circle in Fig. 3 of the main body of the Letter.', '1305.3884-2-22-0': '# Swift observations of IGR J18245-2452', '1305.3884-2-23-0': 'After the initial arc-second localisation with the X-ray Telescope[CITATION] (XRT) on-board Swift[CITATION], the source triggered the Burst Alert Telescope[CITATION] (BAT) on 30 April 2013 at 02:22:21 UT and subsequently at 15:10:37 UT and 15:17:33.61 UT.', '1305.3884-2-23-1': 'Several observations with XRT were obtained starting from 30 April, including a intensive one between the BAT triggers (Obs.', '1305.3884-2-23-2': 'Id 32787) and a long term one (Obs.', '1305.3884-2-23-3': 'Id 32785).', '1305.3884-2-23-4': 'Analysis of the XRT observation performed on 30 April yielded a determination[CITATION] of the position of the source RA=[MATH] Dec=[MATH], with an uncertainty of 3.5[MATH] at a 90[MATH] confidence level, compatible with that determined by ref. 59, and those measured by EPIC MOS (see Sec. [REF] above), Chandra (see Sec. [REF] below), and ATCA (see Sec. [REF]).', '1305.3884-2-23-5': 'The 0.5-10 keV flux corrected for absorption attained a maximum level of [MATH] erg cm[MATH] s[MATH] during an observation starting on 30 March 2013 (Obs.ID 00552369000), with a spectral distribution well described by a power law with index [MATH].', '1305.3884-2-23-6': 'The last detection was obtained during a 0.8-ks observation performed on 1 May 2005 (Obs.', '1305.3884-2-23-7': 'ID 00032785021), with an observed flux of [MATH] erg cm[MATH] s[MATH].', '1305.3884-2-23-8': 'This value of the flux was estimated assuming a [MATH], power-law shaped spectrum, as derived summing all Swift observations taken in the week starting from 21 April 2013.', '1305.3884-2-23-9': 'We note that a flux contamination of [MATH] erg cm[MATH] s[MATH] is expected to be produced by other unresolved source in M28, among which PSR B1821-24 gives the largest contribution[CITATION].', '1305.3884-2-23-10': 'Details of a sample of Swift/XRT observations are given in Supplementary Table [REF].', '1305.3884-2-24-0': 'During an observation that started on 7 April at 20:32 (Obs.', '1305.3884-2-24-1': 'ID 00032785005), a bursting event was observed[CITATION] by the XRT when it was observing in windowed timing mode with a temporal resolution of 1.78 ms. The burst profile has the typical shape observed from thermonuclear explosions taking place at the surface of the neutron star[CITATION], with a fast rise of [MATH] s and an exponential decay on a time scale of [MATH] s (see red dashed line in Figure [REF]).', '1305.3884-2-24-2': 'We performed a time resolved analysis of the emission observed by XRT during the burst.', '1305.3884-2-24-3': 'We fit all spectra with a black body emission absorbed by the interstellar medium, fixing the value of the absorption column to that indicated by the XMM-Newton analysis (see Tab. [REF]).', '1305.3884-2-24-4': 'We used the emission observed by XRT during a 100-s interval before the burst as a background to the burst emission.', '1305.3884-2-24-5': 'The values of temperature we observed are plotted as blue dots in Figure [REF].', '1305.3884-2-24-6': 'The temperature decay observed during the burst tail agrees with the cooling expected after the thermonuclear burning[CITATION], confirming the nature of the observed event.', '1305.3884-2-25-0': 'We used the Chandra pulsar position (see Sec. [REF], below) and orbital parameters determined from the XMM-Newton observations (see Sec. [REF] and Table 1 in the main body of the Letter) to report the photons arrival times to the Solar system Barycentre and to the pulsar line of nodes.', '1305.3884-2-25-1': 'We divided the first 160 s since the burst onset into four intervals, finding in the second one a significant signal at the pulsar period, with an amplitude of [MATH] (see the inset of Figure [REF]).', '1305.3884-2-25-2': 'After taking into account the number of trials made, the detection is significant at a [MATH] level.', '1305.3884-2-25-3': 'A signal with an amplitude of [MATH] is detected during the 740 s observed by XRT before the burst onset.', '1305.3884-2-25-4': 'The increase of the signal amplitude indicates that the signal observed during the burst is related to this event (i.e., it is a burst oscillation[CITATION]) and is not coming from a background source present within the field of view of XRT.', '1305.3884-2-25-5': 'Similar results were reported by ref. 65 and 66.', '1305.3884-2-26-0': 'We also searched for oscillations during the other observations performed by Swift, and found a signal significant above a 3[MATH] confidence level during the observation starting on 30 March 2013 (Obs.', '1305.3884-2-26-1': 'ID 00552369000), with an amplitude of [MATH].', '1305.3884-2-27-0': '# Chandra observations of IGR J18245-2452', '1305.3884-2-28-0': ""The Chandra X-ray observatory observed IGR J18245-2452 on 29 April 2013, starting from 00:24:14 for 53ks, via Director's Discretionary Time."", '1305.3884-2-28-1': 'The observations were performed using the High Resolution Camera[CITATION] (HRC-S) in timing mode, which has a [MATH] field of view with a 16-[MATH]s timing resolution, but no spectral information.', '1305.3884-2-28-2': 'Data were reduced with the CIAO 4.5 software.', '1305.3884-2-28-3': 'We first checked the data for the presence of solar flares and extracted a new observation-specific bad-pixel file.', '1305.3884-2-28-4': 'We then ran a degap correction, and cleaned the image for the hot pixels.', '1305.3884-2-29-0': 'The new Chandra image of the M28 globular cluster was compared to the several observations of the field performed by Chandra during the past decade (see Table [REF] for a complete list, and Fig. 3 of the main body of the Letter for a comparison).', '1305.3884-2-29-1': 'We identified one source (source 23 from ref. 60) to be an order of magnitude brighter than in many previous Chandra observations.', '1305.3884-2-29-2': 'We used the CIAO tools wavdetect and celldetect to infer a good position for the transient source, which was RA=[MATH], Dec=[MATH], with 0.6[MATH] uncertainty (at 1-[MATH] confidence level, inferred from a 0.3[MATH] and a 0.5[MATH] statistical and pointing accuracy, respectively).', '1305.3884-2-29-3': 'The position of this transient source is consistent within a 3-[MATH] confidence level with the XMM-Newton (see Sec. [REF]), Swift (see Sec. [REF]) and ATCA position (see Sec. [REF] below) of IGR J18245-2452 (see Fig. 3 of the main body of the Letter).', '1305.3884-2-29-4': 'We therefore identify it as the accreting X-ray pulsar, IGR J18245-2452.', '1305.3884-2-30-0': 'We have extracted the source events from a 2[MATH] region around the position of the source (and background spectra far from the globular cluster).', '1305.3884-2-30-1': 'In this Chandra observation the source had a count rate of 0.0251(5) counts per second (see Tab. [REF]), which assuming the spectral shape that IGR J18245-2452 had in the closest Swift XRT observation (a power-law with [MATH]; derived summing all Swift observations taken in the week starting from 21 April 2013), leads to a 0.5-10keV observed flux [MATH]ergcm[MATH]s[MATH].', '1305.3884-2-31-0': 'We searched for a coherent signal at the spin period of the source in the time series corrected for the spacecraft and orbital motion, and converted to the Solar System Barycentre.', '1305.3884-2-31-1': 'No signal was detected at a confidence level of 3[MATH], with an upper limit of 17[MATH] on the pulse amplitude, derived following the prescription given by Vaughan et al. [65].', '1305.3884-2-31-2': 'Pulsations at an amplitude lower than such a value were observed during most of the XMM-Newton observations (see Figure [REF]), when the source was brighter by three orders of magnitude.', '1305.3884-2-31-3': 'The non-detection of pulsations does not rule out that the X-ray emission of IGR J18245-2452 was pulsed during the Chandra observation, at a level similar to that previously seen.', '1305.3884-2-32-0': 'We have re-analysed all archival observations performed by Chandra in the past decade to identify and follow the flux evolution of the source.', '1305.3884-2-32-1': 'Timing and spectral data were always extracted from a 2[MATH] region around the position of the source.', '1305.3884-2-32-2': 'A previous X-ray outburst of IGR J18245-2452 is detected during the August 2008 observations performed with the Advanced CCD Imaging Spectrometer[CITATION] (ACIS; see Fig. 3 of the main body of the Letter).', '1305.3884-2-32-3': 'The light curve of this previous outburst shows a strong variability on many timescales (see Figure [REF]).', '1305.3884-2-32-4': 'The most noticeable event is a total quenching of the X-ray emission for a timescale of [MATH]10hrs, compatible with a complete orbital period.', '1305.3884-2-32-5': 'We did not detect a significant energy dependency of the light curve shape.', '1305.3884-2-33-0': 'Spectra were extracted from all the archival observations performed with ACIS-S (all taken in VFAINT mode), which have a good spectral and imaging resolution, but a 3.2s timing resolution, insufficient to search for millisecond pulsations.', '1305.3884-2-33-1': 'We found a good fit ([MATH] with 378 degrees of freedom) when modelling all ACIS-S spectra together with an absorbed power-law model (leaving the photoelectric absorption parameter, [MATH], tied to be the same for all spectra).', '1305.3884-2-33-2': 'We found a flux variability of more than one order of magnitude between the 2002 and the 2008 observations of IGR J18245-2452 (see Figure [REF] and Tab. [REF]).', '1305.3884-2-33-3': 'In particular, we found [MATH]cm[MATH] (with abundances and photoelectric cross-sections from ref. 69 and 70, respectively), a stable photon index [MATH], and a 0.5-10keV flux varying from 2.4 to 64[MATH]ergcm[MATH]s[MATH] (all errors are at 90[MATH] confidence level).', '1305.3884-2-34-0': '# ATCA observations of IGR J18245-2452', '1305.3884-2-35-0': 'On 2013 April 5 we observed IGR J18245-2452 with the Australia Telescope Compact Array (ATCA), at 9 and 5.5 GHz simultaneously.', '1305.3884-2-35-1': 'The data were analysed with the Miriad package distributed by ATNF[CITATION].', '1305.3884-2-35-2': 'Only one source was detected in the inner core of the cluster, at a significance level of nearly 20[MATH] in both frequencies, allowing for an accurate determination of its position at RA=[MATH] Dec=[MATH], with a 90% confidence error of 0.5[MATH].', '1305.3884-2-35-3': 'This position is consistent with that determined by Chandra (see Sec. [REF] and Fig. 3 of the main body of the Letter).', '1305.3884-2-35-4': 'The mean flux density of this source was 0.75[MATH]0.04 (0.62[MATH]0.03) mJy at 9 (5.5) GHz respectively, yielding a mean spectral index of [MATH].', '1305.3884-2-35-5': 'During the first 90 minutes of the observation, the source was strongly variable, reaching up to 2.5 times the mean flux density.', '1305.3884-2-36-0': 'The spectral properties of the ATCA source are similar to those observed from other accreting millisecond pulsars[CITATION], and interpreted in terms of emission originating from shocks within material ejected by the X-ray pulsar.', '1305.3884-2-36-1': 'Besides PSR J1824-2452I=IGR J18245-2452, which was powered by accretion at that moment, none of the other rotation-powered pulsars in the cluster were detected during the ATCA observation.', '1305.3884-2-37-0': '# Parkes radio telescope observations of PSR J1824-2452I', '1305.3884-2-38-0': 'IGR J18245-2452 was observed three times with the 64-m Parkes radio telescope in 2013 April/May (see Supplementary Table [REF]).', '1305.3884-2-38-1': 'We used the position determined by Chandra (see Sec. [REF] above).', '1305.3884-2-38-2': 'The observations were carried out using simultaneously the Berkeley-Parkes Swinburne Recorder (BPSR) and the Parkes Digital Filterbank (PDFB4) in search mode, for the first epoch, and BPSR and the analogue filterban (AFB), for the subsequent ones.', '1305.3884-2-38-3': 'The backends were operating at central frequencies of 1382 MHz (BPSR) and 1369 MHz (PDFB4 and AFB) , over bandwidths of 400 and 256 MHz respectively, subdivided in 1024 frequency channels (512 for AFB).', '1305.3884-2-38-4': 'The total usable bandwidth for BPSR, after removal of a known interference from the Thuraya3 satellite, is 350 MHz.', '1305.3884-2-39-0': 'The resulting time series, 2-bit sampled every 64 [MATH]s for BPSR and PDFB4, and 1-bit sampled every 80 [MATH]s for AFB, were folded with the dspsr package[CITATION] using the X-ray ephemeris presented in this work and using a value of the dispersion measure[CITATION] of 119 pc cm[MATH].', '1305.3884-2-39-1': 'PSR J1824-2452I was detected in two of three Parkes observations (see Supplementary Table [REF]).', '1305.3884-2-39-2': 'In the case of the non-detection, the flux density upper limit derived using the radiometer equation modified for pulsars[CITATION] is [MATH] mJy for a signal with a signal-to-noise ratio SNR=8 and a pulse duty cycle of 15 per cent.', '1305.3884-2-39-3': 'We caution that Parkes non-detection, as well as other radio non-detections presented below, cannot be taken as strong evidence that the radio pulsar was not active.', '1305.3884-2-39-4': 'PSR J1824-2452I is well known to eclipse in the radio[CITATION], particularly around superior conjunction, and this is likely the cause of at least some of the radio non-detections presented in Supplementary Table [REF].', '1305.3884-2-39-5': 'Though there is no published flux density for PSR J1824-2452I, given its comparable brightness to other pulsars in M28, it is likely that if the source was emitting as a radio pulsar during these observations that it would be at the threshold of detectability during this 64-m Parkes radio telescope observation - especially if the signal was further perturbed by intra-binary material, as in similar systems.', '1305.3884-2-39-6': 'Therefore, we consider these observations only moderately constraining as to whether the source was emitting as a radio pulsar at this epoch.', '1305.3884-2-40-0': '# Westerbork Synthesis radio telescope observations of PSR J1824-2452I', '1305.3884-2-41-0': 'We observed IGR J18245-2452 during four sessions in 2013 May using the Westerbork Synthesis radio telescope (WSRT).', '1305.3884-2-41-1': 'The pointing position was [MATH], [MATH].', '1305.3884-2-41-2': 'We used WSRT in the tied-array mode (gain, G = 1.2K/Jy), combining 13 of the individual 25-m dishes in phase and recording with the PuMaII pulsar data recorder[CITATION].', '1305.3884-2-41-3': 'We recorded a 160-MHz bandwidth and coherently de-dispersed and folded the data offline using the dspsr package[CITATION] and the X-ray-derived ephemeris presented here.', '1305.3884-2-41-4': 'PSR J1824-2452I was just barely detected in one of four WSRT observations (see Supplementary Table [REF]).', '1305.3884-2-41-5': 'Using the radiometer equation modified for pulsar signals[CITATION], we can place a flux density limit of [MATH] mJy on radio emission from the pulsar in the case of the non-detections.', '1305.3884-2-41-6': ""Note that the flux of PSR J1824-2452I is very close to WSRT's detection threshold and, as mentioned above, the radio pulsar may simply have been eclipsed in the case of non-detections."", '1305.3884-2-42-0': '# Green Bank Telescope observations of PSR J1824-2452I', '1305.3884-2-43-0': 'We observed IGR J18245-2452 during seven sessions in 2013 May using the Green Bank Telescope (GBT).', '1305.3884-2-43-1': 'The Chandra position presented here was used for pointing, and data were acquired using the GUPPI data recorder[CITATION] in a 800-MHz band centered at 2.0GHz.', '1305.3884-2-43-2': 'PSR J1824-2452I was detected in two of seven observations (see Supplementary Table [REF]).', '1305.3884-2-43-3': 'Flux density limits in the case of non-detections are [MATH]Jy, but come with the same caveats about eclipsing as described above.', '1305.3884-2-43-4': 'Supplementary Table [REF] also lists a number of archival detections of PSR J1824-2452I with the GBT since its discovery[CITATION] as a radio pulsar in 2006.', '1305.3884-2-43-5': 'These observations have been acquired as part of a regular timing program of the radio pulsars in M28 (Ransom et al.) and together with archival X-ray observations they conclusively demonstrate that the system has switched between rotation-powered radio pulsar and accretion-powered X-ray pulsar (and back) during 2006-2013.', '1305.3884-2-44-0': 'lclccc [Abridged Radio/X-ray History of PSR J1824-2452I/igr]Abridged Radio/X-ray History of PSR J1824-2452I/IGR J18245-2452: Flux densities of radio observations were measured at 2.0 GHz (GBT), 1.4 GHz (WSRT/Parkes), and 5.5/9 GHz (ATCA).', '1305.3884-2-44-1': 'Radio non detections are marked by a dash in the flux column, and are only moderately constraining as to whether the source was emitting as a radio pulsar at this epoch (see text for details).', '1305.3884-2-44-2': 'GBT non-detections prior to 2012-10-07 are not listed.', '1305.3884-2-44-3': 'Observed X-ray fluxes are evaluated in the 0.5-10 keV band (20-100 keV for INTEGRAL), and given in mCrab ([MATH]Crab = [MATH] erg cm[MATH] s[MATH] in the 0.5-10 keV band).', '1305.3884-2-45-0': '1cUT Date & 1cMJD Start & 1cTelescope & 1cType& 1cFlux& 1cComments', '1305.3884-2-46-0': '6c - continued from previous page', '1305.3884-2-47-0': '1cUT Date & 1cMJD Start & 1cTelescope & 1cType& 1cFlux& 1cComments', '1305.3884-2-48-0': '6rContinued on next page', '1305.3884-2-49-0': '2002-07-04 & 52,459.752 & Chandra/ACIS-S & X-rays & [MATH] mCrab & X-ray Quiescence', '1305.3884-2-50-0': '2002-08-04 & 52,490.990 & Chandra/ACIS-S & X-rays & [MATH] mCrab& X-ray Quiescence', '1305.3884-2-51-0': '2002-09-09 & 52,526.705 & Chandra/ACIS-S & X-rays & [MATH] mCrab& X-ray Quiescence', '1305.3884-2-52-0': '2002-11-08 & 52,586.237 & Chandra/HRC-S & X-rays & [MATH] mCrab & X-ray Quiescence', '1305.3884-2-53-0': '2006-01/02 & 53,738 - 53,781 & GBT & Radio & & Discovery of PSR J1824-2452I (ref. 69)', '1305.3884-2-54-0': '2006-05-27 & 53,882.520 & Chandra/HRC-S & X-rays & [MATH] mCrab & X-ray Quiescence', '1305.3884-2-55-0': '2006/2007 & & 3lVarious GBT Radio Observations', '1305.3884-2-56-0': '2007-12-30 & 54,464.746 & GBT & Radio & [MATH]Jy & Radio Pulsations', '1305.3884-2-57-0': '2008-04-17 & 54,573.354 & GBT & Radio & [MATH]Jy & Radio Pulsations', '1305.3884-2-58-0': '2008-06-13 & 54,631.274 & GBT & Radio & [MATH]Jy & Radio Pulsations', '1305.3884-2-59-0': '2008-08-07 & 54,685.865 & Chandra & X-rays & [MATH] mCrab & X-ray Enhanced', '1305.3884-2-60-0': '2008-08-10 & 54,688.993 & Chandra & X-rays & [MATH] mCrab & X-ray Enhanced', '1305.3884-2-61-0': '2009-05-06 & 54,957.418 & GBT & Radio & [MATH]Jy & Radio Pulsations', '1305.3884-2-62-0': '2009/2012 & & 4lVarious GBT Radio Observations', '1305.3884-2-63-0': '2012-10-07 & 56,207.967 & GBT & Radio & - & -', '1305.3884-2-64-0': '2013-01-06 & 56,298.702 & GBT & Radio & - & -', '1305.3884-2-65-0': '2013-03-28 & 56,379.122 & INTEGRAL/ISGRI& X-rays & [MATH] mCrab & Discovery of IGR J18245-2452', '1305.3884-2-66-0': '2013-03-30 & 56,381.632 & Swift/XRT & X-rays & [MATH] mCrab & X-ray Outburst / Pulsations', '1305.3884-2-67-0': '2013-04-04 & 56,386.018 & XMM-Newton/EPIC pn & X-rays & [MATH] mCrab & X-ray Outburst / Pulsations', '1305.3884-2-68-0': '2013-04-05 & 56,387.720 & ATCA & Radio & [MATH] mJy (5.5 GHz) & Non-pulsed', '1305.3884-2-69-0': '& & & & [MATH] mJy (9 GHz) & Non-pulsed', '1305.3884-2-70-0': '2013-04-08 & 56,390.481 & GBT & Radio & - & -', '1305.3884-2-71-0': '2013-04-13 & 56,395.294 & XMM-Newton/EPIC pn & X-rays & [MATH] mCrab & X-ray Outburst / Pulsations', '1305.3884-2-72-0': '2013-04-15 & 56,397.469 & GBT & Radio & - & -', '1305.3884-2-73-0': '2013-04-29 & 56,411.010 & Chandra/HRC-S& X-rays[MATH] & [MATH] mCrab & X-ray Enhanced', '1305.3884-2-74-0': '2013-04-29 & 56,411.560 & Parkes & Radio & - & -', '1305.3884-2-75-0': '2013-05-01 & 56,413.557 & Swift/XRT & X-rays[MATH] & [MATH] mCrab & Latest X-ray detection', '1305.3884-2-76-0': '2013-05-02 & 56,414.164 & WSRT & Radio & [MATH]Jy & Radio Pulsations', '1305.3884-2-77-0': '2013-05-04 & 56,416.162 & WSRT & Radio & - & -', '1305.3884-2-78-0': '2013-05-06 & 56,418.170 & WSRT & Radio & - & -', '1305.3884-2-79-0': '2013-05-06 & 56,418.296 & GBT & Radio & - & -', '1305.3884-2-80-0': '2013-05-06 & 56,418.831 & Swift/XRT & X-rays[MATH] & [MATH] mCrab & X-ray non detection', '1305.3884-2-81-0': '2013-05-07 & 56,419.114 & WSRT & Radio & - & -', '1305.3884-2-82-0': '2013-05-09 & 56,421.442 & GBT & Radio & - & -', '1305.3884-2-83-0': '2013-05-10 & 56,422.544 & Parkes & Radio & [MATH]Jy & Radio Pulsations', '1305.3884-2-84-0': '2013-05-11 & 56,423.405 & GBT & Radio & [MATH]Jy & Radio Pulsations', '1305.3884-2-85-0': '2013-05-13 & 56,425.432 & GBT & Radio & [MATH]Jy & Radio Pulsations', '1305.3884-2-86-0': '2013-05-13 & 56,425.684 & Parkes & Radio & [MATH]Jy & Radio Pulsations', '1305.3884-2-87-0': '2013-05-18 & 56,430.277 & GBT & Radio & - & -', '1305.3884-2-88-0': '2013-05-24 & 56,436.437 & GBT & Radio & - & -', '1305.3884-2-89-0': '2013-05-31 & 56,443.183 & GBT & Radio & - & -', '1305.3884-2-90-0': '[MATH] Flux estimated assuming a [MATH] power-law spectral shape.', '1305.3884-2-90-1': '[MATH] Other unresolved sources[CITATION] in M28 are expected to give a contribution of [MATH] mCrab to the quoted flux value/upper limit.', '1305.3884-2-90-2': 'Upper limits are evaluated at 3-[MATH] confidence level.'}","[['1305.3884-1-0-2', '1305.3884-2-0-3'], ['1305.3884-1-0-2', '1305.3884-2-0-6'], ['1305.3884-1-0-4', '1305.3884-2-0-6'], ['1305.3884-1-3-0', '1305.3884-2-1-0'], ['1305.3884-1-3-2', '1305.3884-2-1-1'], ['1305.3884-1-3-5', '1305.3884-2-1-3'], ['1305.3884-1-3-6', '1305.3884-2-1-4'], ['1305.3884-1-4-0', '1305.3884-2-2-0'], ['1305.3884-1-4-2', '1305.3884-2-2-1'], ['1305.3884-1-4-3', '1305.3884-2-2-2'], ['1305.3884-1-4-4', '1305.3884-2-2-3'], ['1305.3884-1-4-5', '1305.3884-2-2-4'], ['1305.3884-1-4-6', '1305.3884-2-2-5'], ['1305.3884-1-5-0', '1305.3884-2-3-0'], ['1305.3884-1-5-1', '1305.3884-2-3-3'], ['1305.3884-1-5-3', '1305.3884-2-3-1'], ['1305.3884-1-7-0', '1305.3884-2-5-0'], ['1305.3884-1-8-0', '1305.3884-2-6-3'], ['1305.3884-1-8-1', '1305.3884-2-6-4'], ['1305.3884-1-8-2', '1305.3884-2-6-5'], ['1305.3884-1-8-4', '1305.3884-2-6-7'], ['1305.3884-1-8-5', '1305.3884-2-6-8'], ['1305.3884-1-8-6', '1305.3884-2-7-0'], ['1305.3884-1-9-0', '1305.3884-2-8-0'], ['1305.3884-1-9-1', '1305.3884-2-8-1'], ['1305.3884-1-9-5', '1305.3884-2-8-5'], ['1305.3884-1-9-6', '1305.3884-2-8-6'], ['1305.3884-1-9-7', '1305.3884-2-8-7'], ['1305.3884-1-9-8', '1305.3884-2-8-11'], ['1305.3884-1-9-9', '1305.3884-2-8-12'], ['1305.3884-1-9-10', '1305.3884-2-8-13'], ['1305.3884-1-9-12', '1305.3884-2-8-14'], ['1305.3884-1-9-15', '1305.3884-2-8-17'], ['1305.3884-1-9-18', '1305.3884-2-8-22'], ['1305.3884-1-9-20', '1305.3884-2-8-24'], ['1305.3884-1-9-21', '1305.3884-2-8-25']]",[],[],[],"[['1305.3884-1-0-2', '1305.3884-2-0-3'], ['1305.3884-1-0-2', '1305.3884-2-0-6'], ['1305.3884-1-0-4', '1305.3884-2-0-6']]","[['1305.3884-1-3-0', '1305.3884-2-1-0'], ['1305.3884-1-3-2', '1305.3884-2-1-1'], ['1305.3884-1-3-5', '1305.3884-2-1-3'], ['1305.3884-1-3-6', '1305.3884-2-1-4'], ['1305.3884-1-4-0', '1305.3884-2-2-0'], ['1305.3884-1-4-2', '1305.3884-2-2-1'], ['1305.3884-1-4-3', '1305.3884-2-2-2'], ['1305.3884-1-4-4', '1305.3884-2-2-3'], ['1305.3884-1-4-5', '1305.3884-2-2-4'], ['1305.3884-1-4-6', '1305.3884-2-2-5'], ['1305.3884-1-5-0', '1305.3884-2-3-0'], ['1305.3884-1-5-1', '1305.3884-2-3-3'], ['1305.3884-1-5-3', '1305.3884-2-3-1'], ['1305.3884-1-7-0', '1305.3884-2-5-0'], ['1305.3884-1-8-0', '1305.3884-2-6-3'], ['1305.3884-1-8-1', '1305.3884-2-6-4'], ['1305.3884-1-8-2', '1305.3884-2-6-5'], ['1305.3884-1-8-4', '1305.3884-2-6-7'], ['1305.3884-1-8-5', '1305.3884-2-6-8'], ['1305.3884-1-8-6', '1305.3884-2-7-0'], ['1305.3884-1-9-0', '1305.3884-2-8-0'], ['1305.3884-1-9-1', '1305.3884-2-8-1'], ['1305.3884-1-9-5', '1305.3884-2-8-5'], ['1305.3884-1-9-6', '1305.3884-2-8-6'], ['1305.3884-1-9-7', '1305.3884-2-8-7'], ['1305.3884-1-9-8', '1305.3884-2-8-11'], ['1305.3884-1-9-9', '1305.3884-2-8-12'], ['1305.3884-1-9-10', '1305.3884-2-8-13'], ['1305.3884-1-9-12', '1305.3884-2-8-14'], ['1305.3884-1-9-15', '1305.3884-2-8-17'], ['1305.3884-1-9-18', '1305.3884-2-8-22'], ['1305.3884-1-9-20', '1305.3884-2-8-24'], ['1305.3884-1-9-21', '1305.3884-2-8-25']]","['1305.3884-1-9-2', '1305.3884-1-9-13', '1305.3884-2-8-9', '1305.3884-2-8-10', '1305.3884-2-8-15', '1305.3884-2-8-16', '1305.3884-2-8-20', '1305.3884-2-9-0', '1305.3884-2-13-1', '1305.3884-2-13-6', '1305.3884-2-13-7', '1305.3884-2-23-3', '1305.3884-2-23-7', '1305.3884-2-26-1', '1305.3884-2-46-0', '1305.3884-2-48-0', '1305.3884-2-49-0', '1305.3884-2-50-0', '1305.3884-2-51-0', '1305.3884-2-52-0', '1305.3884-2-53-0', '1305.3884-2-54-0', '1305.3884-2-55-0', '1305.3884-2-56-0', '1305.3884-2-57-0', '1305.3884-2-58-0', '1305.3884-2-59-0', '1305.3884-2-60-0', '1305.3884-2-61-0', '1305.3884-2-62-0', '1305.3884-2-63-0', '1305.3884-2-64-0', '1305.3884-2-65-0', '1305.3884-2-66-0', '1305.3884-2-67-0', '1305.3884-2-68-0', '1305.3884-2-69-0', '1305.3884-2-70-0', '1305.3884-2-71-0', '1305.3884-2-72-0', '1305.3884-2-73-0', '1305.3884-2-74-0', '1305.3884-2-75-0', '1305.3884-2-76-0', '1305.3884-2-77-0', '1305.3884-2-78-0', '1305.3884-2-79-0', '1305.3884-2-80-0', '1305.3884-2-81-0', '1305.3884-2-82-0', '1305.3884-2-83-0', '1305.3884-2-84-0', '1305.3884-2-85-0', '1305.3884-2-86-0', '1305.3884-2-87-0', '1305.3884-2-88-0', '1305.3884-2-89-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1305.3884,,,
0809.1816,"{'0809.1816-1-0-0': 'Considering the Casimir effect due to phononic excitations of a weakly interacting dilute BEC, we derive a re-normalized expression for the zero temperature Casimir energy [MATH] of a BEC confined to a parallel plate geometry with periodic boundary conditions.', '0809.1816-1-0-1': 'Our expression is formally equivalent to the free energy of a bosonic field at finite temperature, with a nontrivial density of modes that we compute analytically.', '0809.1816-1-0-2': 'As a function of the interaction strength, [MATH] smoothly describes the transition from the weakly interacting Bogoliubov regime to the non-interacting ideal BEC.', '0809.1816-1-0-3': 'For the weakly interacting case, [MATH] reduces to leading order to the Casimir energy due to zero-point fluctuations of massless phonon modes.', '0809.1816-1-0-4': 'In the limit of an ideal Bose gas, our result correctly describes the Casimir energy going to zero.', '0809.1816-1-1-0': '# Introduction', '0809.1816-1-2-0': 'The Casimir effect is a consequence of the distorted vacuum fluctuation spectrum of quantized fields in bounded domains or spaces with non-trivial topologies [CITATION].', '0809.1816-1-2-1': ""In Casimir's original calculation, the system under consideration is the electromagnetic vacuum [CITATION]."", '0809.1816-1-2-2': 'Imposing Dirichlet boundary conditions along one spatial direction by confining the system between two (idealized) parallel plates causes a change in the (infinite) vacuum energy-density.', '0809.1816-1-2-3': 'The variation of the vacuum energy-density with respect to the plate separation is called Casimir pressure, and, after renormalization, yields a finite expression for an attractive interaction energy per unit area between the plates.', '0809.1816-1-2-4': 'The electromagnetic Casimir force, caused by quantum fluctuations of the electromagnetic vacuum, is varying as [MATH] with the plate separation [MATH] and the speed of light [MATH].', '0809.1816-1-2-5': 'It has been measured in a number of experiments using various experimental settings [CITATION].', '0809.1816-1-2-6': 'The comparison between quantum vacuum experiments like these and the predictions of different theoretical models provides possibility to test fundamental physics (like higher dimensions or additional interactions), in much the same way as accelerator experiments in high-energy physics do at the other end of the energy scale [CITATION].', '0809.1816-1-3-0': ""We consider here, instead of the electromagnetic vacuum, a weakly interacting Bose-Einstein-Condensate (BEC) at zero temperature, and expect, in a similar manner, the quantum fluctuations on top of the ground state of the BEC to give rise to observable Casimir forces: Within the Bogoliubov approximation, the excited states of a BEC can be treated as quasi-particles characterized by the dispersion relation [EQUATION] that behaves linear for small momenta, with the 'sound velocity' [MATH] being inverse to the healing length [MATH]."", '0809.1816-1-3-1': 'The wave-number [MATH] also characterizes the transition between the linear (phonon) and the quadratic (free-particle) regimes [CITATION].', '0809.1816-1-3-2': 'For small momenta, the quasi-particles (phonons) propagate through the quasi-particle vacuum in the same way as a massless scalar field does in the electromagnetic vacuum, except for the propagation velocity being different.', '0809.1816-1-3-3': 'Hence, the zero temperature quantum fluctuations in a spatially confined BEC can be expected to result in an observable Casimir force.', '0809.1816-1-4-0': 'Casimir forces in BECs have indeed been found by several authors: For the parallel plate geometry, an asymptotic expansion of the Casimir force has been calculated in [CITATION], the small expansion parameter being the ratio between healing length and plate separation.', '0809.1816-1-4-1': 'In the leading order, it reproduces exactly the [MATH] behavior as in the electromagnetic vacuum.', '0809.1816-1-4-2': 'The next order corrections scale with the ratio [MATH].', '0809.1816-1-4-3': 'Replacing the perfectly reflecting plates by impurities embedded in a one-dimensional quantum liquid, Casimir forces between these impurities have been calculated in [CITATION] as a function of the impurity-liquid coupling.', '0809.1816-1-4-4': 'In the limit of an infinitely strong coupling, the result for the (one-dimensional) parallel plate scenario was recovered; for a weak coupling, however, the interaction between the impurities vanishes exponentially with the impurity separation on a scale set by the healing length.', '0809.1816-1-4-5': 'For the ideal (i.e.non-interacting) Bose gas, it was found that there is no Casimir force at all between impurities of arbitrary interaction strength, including the idealized parallel plate scenario [CITATION].', '0809.1816-1-4-6': 'This is consistent with the quite general method of [CITATION], which is mapping (polynomial) dispersion relations to Casimir forces in the parallel plate geometry: this method shows that media with quadratic dispersion relations, and hence the ideal Bose gas, do not give rise to any zero-temperature Casimir forces.', '0809.1816-1-4-7': 'A non-vanishing Casimir force in the ideal Bose gas can arise due to thermal fluctuations, as calculated in Ref. [CITATION].', '0809.1816-1-4-8': 'All these forces are small but finite observable quantities, which-if experimentally confirmed-would provide direct evidence of the quantum fluctuations in weakly interacting BECs.', '0809.1816-1-5-0': 'The system under consideration in the present article is a homogeneous, weakly-interacting dilute BEC at zero temperature, confined to a parallel plate geometry with periodic boundary conditions in one of the three spatial dimensions.', '0809.1816-1-5-1': 'The condition of diluteness can be formulated as [MATH], where [MATH] is the [MATH]-wave scattering length and [MATH] is the density, [MATH] being the total particle number and [MATH] the volume of the gas.', '0809.1816-1-5-2': 'The quantity [MATH] is usually called the gas parameter.', '0809.1816-1-5-3': 'The perturbative calculation of typical properties of such a BEC, like the ground state energy or the depletion of the condensate, is then essentially an expansion in the gas parameter [CITATION].', '0809.1816-1-6-0': 'For this system, we will give a re-normalized expression for the Casimir energy-density per unit area.', '0809.1816-1-6-1': ""Our expression has the form of an integral over a 'density of modes' [MATH] times the Bose distribution function: [EQUATION] where [MATH] has a simple analytic form (see eqn:rho_an), and correctly describes the vanishing of the Casimir force in the limit of the interaction strength going to zero."", '0809.1816-1-6-2': 'The possibility to express the zero temperature Casimir energy in the above form, resembling the density of states of a bosonic system at finite temperature, is connected to a topological analogy between our parallel plate scenario and finite temperature field theory: In the parallel plate geometry, one spatial coordinate of the field is subject to periodic boundary conditions, while in finite temperature field theory, the imaginary time coordinate is subject to a similar periodicity condition.', '0809.1816-1-6-3': 'This analogy has been pointed out some time ago in [CITATION]; it does not carry over, however, to Casimir calculations for non-linear dispersion relations.', '0809.1816-1-7-0': 'The article is organized as follows: In sec:free, we briefly recall some well-known perturbative expressions for the free energy, chemical potential and ground state energy of a dilute, weakly interacting BEC at zero temperature.', '0809.1816-1-7-1': 'Working from the formula for the free energy, we derive in sec:casimir the expression for the Casimir energy-density.', '0809.1816-1-7-2': 'In the limit [MATH], the function [MATH] reproduces the power-series expression for the Casimir energy derived in [CITATION], as is shown in subsec:rho_expansion.', '0809.1816-1-7-3': 'In subsec:toms, we relate the above mentioned analogy to the finite temperature case in some detail, as this topic does not seem to get overly much attention in the recent literature.', '0809.1816-1-7-4': 'Finally, subsec:limit deals with the behavior of the function [MATH] in the limit of the effective coupling constant going to zero.', '0809.1816-1-7-5': 'Here, our expression [MATH] correctly describes the vanishing of the Casimir force.', '0809.1816-1-7-6': 'We comment on the failure of the large-distance expansion to describe the non-interacting Bose gas.', '0809.1816-1-8-0': '# Free energy in a weakly interacting, dilute BEC at [MATH]', '0809.1816-1-9-0': 'In the one-loop approximation, the (unrenormalized) free energy density of a weakly interacting BEC at [MATH] (in three spatial dimensions) is given by [EQUATION] with the mean-field and one-loop contributions [CITATION]: [EQUATION]', '0809.1816-1-9-1': 'Here and in the following, we work in units where [MATH].', '0809.1816-1-9-2': 'In the above expression, [MATH] is the chemical potential (which, in subsec:rho_expansion, will be connected to the speed of sound in the medium), and [MATH] is the effective coupling constant that can, for low energy collisions in a dilute medium, be identified with the [MATH]-wave scattering length [MATH]: [MATH].', '0809.1816-1-9-3': 'The term [MATH] represents the one-loop counter-term needed to render [MATH] finite.', '0809.1816-1-9-4': 'If the integral in eqn:F1b is regularized with a momentum cut-off in the ultraviolet, the linear, cubic and quintic UV-divergences in [MATH] can be absorbed by re-normalizing [MATH], [MATH] and the vacuum energy respectively.', '0809.1816-1-9-5': 'If dimensional regularization is used in eqn:F1b, no counter-term is needed at this level of perturbation theory.', '0809.1816-1-10-0': 'After renormalization, the chemical potential [MATH] can be obtained from eqn:F1 by inverting [EQUATION] which yields [EQUATION]', '0809.1816-1-10-1': 'Reinserting [MATH] into eqn:F1 will then reproduce the classical result for the leading quantum corrections to the ground state energy density [MATH] of a hard-sphere Bose gas, which was first derived by Lee, Huang and Yang in [CITATION]: [EQUATION]', '0809.1816-1-11-0': '# The Casimir contribution as an integral over a density-function', '0809.1816-1-12-0': 'Now, in order to describe a BEC between a pair of parallel plates separated by a finite distance [MATH] (with periodic boundary conditions), we have to quantize the momentum component perpendicular to the plates: [EQUATION]', '0809.1816-1-12-1': 'Correspondingly, the momentum integration perpendicular to the plates is replaced by a discrete sum: [EQUATION]', '0809.1816-1-12-2': 'The area of the plates is taken to be [MATH] with [MATH], so the system now inhabits the volume [MATH].', '0809.1816-1-12-3': 'The mean field contribution [MATH] (see eqn:F1a) depends only trivially on the new boundary conditions, with the volume [MATH] entering through [MATH] in eqn:mu_def.', '0809.1816-1-13-0': 'The Casimir energy [MATH] of the BEC is related to the free energy at the one-loop level.', '0809.1816-1-13-1': 'We are interested in its change per unit area that is due to the introduction of the boundary conditions: [EQUATION] where the first term gives the (one-loop contribution to) the free energy in a homogeneous system.', '0809.1816-1-13-2': 'After applying the substitutions eqn:quant1 and ([REF]) to eqn:F1b, we are left with the following expression for [MATH], now describing the leading quantum corrections to the free energy of a BEC confined between parallel plates: [EQUATION] where we have used the abbreviations [EQUATION] with [MATH] and [MATH].', '0809.1816-1-13-3': 'The summation over [MATH] can be converted into two integrals by using the Abel-Plana formula in the form (see [CITATION]) [EQUATION]', '0809.1816-1-13-4': 'Application of eqn:ap splits eqn:F1c into the two terms written in eqn:F1d, as the [MATH]-term cancels out.', '0809.1816-1-13-5': 'The first term, [MATH], is divergent, but of the same form as the one-loop free energy eqn:F1b itself.', '0809.1816-1-13-6': 'Hence, it can be re-normalized as described above.', '0809.1816-1-14-0': ""The term [MATH] in eqn:F1d, an energy per area, describes the effects induced by restricting the periodicity in one spatial dimension to be much smaller than the remaining ones, i.e.,confining the system to a volume [MATH] with one 'short' side [MATH]."", '0809.1816-1-14-1': '[MATH] is convergent for any finite [MATH], strictly negative, and goes to zero for [MATH]: [EQUATION]', '0809.1816-1-14-2': 'The integration domain is sketched in fig:wedge-domain (the shaded regions).', '0809.1816-1-15-0': 'This domain as well as the form of the integrand in eqn:cas1 are due to the branch-points of the integrand in eqn:F1c: In order to stay clear of the branch-cuts, we have evaluated the last term in the Abel-Plana formula eqn:ap along an integration contour slightly to the right of the imaginary axis.', '0809.1816-1-15-1': 'Inserting the function [MATH] given by eqn:F1c, [EQUATION] the numerator in the last term of eqn:ap evaluates to [EQUATION] being zero everywhere else along the contour of integration (see [CITATION]).', '0809.1816-1-15-2': 'By changing the order of integration in eqn:cas1, the [MATH]-integral can be performed, and we find the expression ([REF]) for the Casimir energy: [EQUATION]', '0809.1816-1-15-3': ""In the form of eqn:DOS1, [MATH] is expressed as an integral over a 'density of states' (DOS) for a bosonic system, as setting the integration variable [MATH] will reproduce the Boltzmann factor in the denominator."", '0809.1816-1-16-0': ""The 'mode density' [MATH] is obtained by integrating separately over the lower triangular region in fig:wedge-domain (dark grey) and the 'hyperbolic tail' (light grey)."", '0809.1816-1-16-1': 'In terms of the dimensionless variable [MATH], we have [EQUATION]', '0809.1816-1-16-2': 'Above [MATH], the function [MATH] changes into a constant independent of [MATH] (see the solid line in fig:rho_plot).', '0809.1816-1-16-3': 'The Casimir energy [MATH] as a function of the normalized distance [MATH] is shown in fig:E_plot, obtained by numerically integrating eqn:DOS1.', '0809.1816-1-17-0': 'To provide a cross-check for the above results, we will in the following section show that the function [MATH] reproduces the asymptotic expansion for the Casimir energy given in [CITATION], which is valid for small values of the parameter [MATH].', '0809.1816-1-17-1': 'In subsec:limit we then show that the formula for [MATH] in eqn:rho_an also yields correct results in the opposite limit of the ideal BEC, i.e.[MATH].', '0809.1816-1-18-0': '## Expansion at large distance and moderate interactions', '0809.1816-1-19-0': 'In this section, we will assume the plate separation to be much greater than the healing length of the BEC, i.e.,the ratio [MATH] can be treated as a small parameter.', '0809.1816-1-19-1': 'Note that this limit cannot describe a strongly interacting Bose gas in the proper sense, since this would lead to the break-down of the one-loop approximation at the basis of our approach.', '0809.1816-1-20-0': 'As can be seen in fig:rho_plot, the kink in the function [MATH] happens at the large value [MATH] in our limit.', '0809.1816-1-20-1': 'In evaluating the Casimir energy with eqn:DOS1, large values of [MATH], and hence the behavior of [MATH] after the kink, get exponentially suppressed by the denominator.', '0809.1816-1-20-2': 'Thus we can approximate the function [MATH] by expanding its small-[MATH] part as a power-series around [MATH]: [EQUATION]', '0809.1816-1-20-3': 'The integrals in the expression ([REF]) for the Casimir energy can be performed explicitly.', '0809.1816-1-20-4': 'Upon setting [MATH] (the speed of sound in the medium), we recover the result derived in [CITATION]: [EQUATION]', '0809.1816-1-20-5': 'The leading term [MATH] is, as noted in [CITATION], equal to the Casimir energy-density of a massless scalar field with propagation velocity [MATH] confined between two parallel plates with periodic boundary conditions [CITATION].', '0809.1816-1-20-6': 'Its presence is a manifestation of the Goldstone-theorem, the long wavelength part of the Bogoliubov spectrum representing the gap-less Goldstone modes [CITATION].', '0809.1816-1-20-7': 'The next-to-leading term in eqn:cas_ex is in [CITATION] referred to as the Bogoliubov correction due to the non-linearity of the dispersion.', '0809.1816-1-20-8': 'Indeed, if we express the Bogoliubov dispersion relation eqn:bogo1 as a power series around [MATH] and put the first few terms into the expression for the Casimir force derived in [CITATION], the leading term in the same manner reproduces the result for the scalar field (in one dimension), followed by terms that are smaller in magnitude and of opposite sign.', '0809.1816-1-20-9': 'As can be seen in fig:E_plot, the contribution of these corrections is getting smaller as the ratio between plate separation and healing length increases, leaving only the dominant scalar term [MATH] depicted by the horizontal dashed line in fig:E_plot.', '0809.1816-1-21-0': 'When finally calculating the Casimir pressure from eqn:cas_ex, one has to consider that, as the number of particles is held constant, the derivative of the speed of sound with respect to [MATH] is not zero, as discussed in [CITATION].', '0809.1816-1-22-0': '## A formal analogy to finite temperature systems', '0809.1816-1-23-0': 'As already mentioned, our expression eqn:DOS1 for [MATH] at [MATH] formally resembles the DOS for a bosonic system at finite temperature.', '0809.1816-1-23-1': 'This can be understood by recalling that a finite temperature system can be described in imaginary time, combined with periodic boundary conditions with period [MATH].', '0809.1816-1-23-2': 'The same topology is realized in the parallel plate system at [MATH], when one spatial dimension is subject to periodic boundary conditions with period [MATH] ([CITATION], see also [CITATION]).', '0809.1816-1-24-0': 'Let us briefly re-phrase the argument of [CITATION] within our notation.', '0809.1816-1-24-1': 'The canonical partition function [MATH] for a system at temperature [MATH] in [MATH] spatial dimensions is usually expressed as the path integral [EQUATION] where [MATH] is the Lagrangian (a scalar functional of the field [MATH]), and the field is constrained in such a way that [MATH].', '0809.1816-1-24-2': 'If we take [MATH], the path-integral in eqn:toms_Z can, after Fourier expansion of the field, be evaluated to yield [CITATION] [EQUATION]', '0809.1816-1-24-3': 'Here, [MATH] with [MATH] are the Matsubara-frequencies due to the periodicity condition in eqn:toms_Z, and [MATH] with the continuous parameter [MATH] is the dispersion relation of the massless scalar field.', '0809.1816-1-24-4': 'Note the similarity to eqn:quant1, where we had periodic boundary conditions not in imaginary time but in one spatial dimension.', '0809.1816-1-25-0': 'The sum over [MATH] in eqn:toms_logz_a is usually evaluated by multiplying with a factor [MATH], which has poles of residue 1 at [MATH], and then integrating over a contour in the complex [MATH]-plane which includes all the poles [CITATION].', '0809.1816-1-25-1': 'This technique is actually the same that is used in [CITATION] to prove the Abel-Plana formula eqn:ap.', '0809.1816-1-25-2': 'The well known result is [EQUATION] where we have already subtracted the zero-point fluctuations.', '0809.1816-1-25-3': 'After integrating by parts and employing the integral-identity from the footnote on page eqn:integral_formula, we obtain ([MATH] is the volume of the unit sphere in [MATH] dimensions) [EQUATION]', '0809.1816-1-25-4': 'Note that, upon setting [MATH], the above expression has the same form as [MATH] in eqn:DOS1, with [MATH] being proportional to [MATH].', '0809.1816-1-25-5': 'Now, let the system inhabit a volume [MATH].', '0809.1816-1-25-6': 'With [MATH] we get for the free energy per unit area [EQUATION]', '0809.1816-1-25-7': 'Tentatively exchanging [MATH] with [MATH] in eqn:toms_logZ_3d will reproduce the Casimir energy density for a massless scalar field, i.e.the first term in eqn:cas_ex [CITATION]: [EQUATION]', '0809.1816-1-25-8': 'So, we have seen that the zero temperature Casimir energy of a massless scalar field confined between two parallel plates can be obtained by a simple exchange of variables, once we know the thermal contribution to the free energy density of that field.', '0809.1816-1-25-9': 'Unfortunately, this simple mapping does not carry over to fields characterized by nonlinear dispersion relations: The temperature dependent part of the free energy for a BEC is (in the Bogoliubov approximation) still given by eqn:toms_logz_bose with the dispersion now being [MATH] [CITATION].', '0809.1816-1-25-10': 'But a simple interchange of [MATH] with [MATH] in eqn:toms_logz_bose will no longer yield [MATH], because the discretized Matsubara frequencies [MATH] (that become a discretized momentum) always enter in eqn:toms_logz_a in the same way as a spatial momentum component enters in a linear dispersion relation.', '0809.1816-1-25-11': 'It thus seems very difficult to mimic the fully nonlinear behavior of the dispersion relation.', '0809.1816-1-26-0': '## The non-interacting limit', '0809.1816-1-27-0': 'The transition from the weakly interacting BEC (here described at the one-loop level only) to the ideal BEC should be accomplished by letting the effective coupling [MATH] (see sec:free) go to zero, corresponding to [MATH].', '0809.1816-1-27-1': 'The Casimir energy is expected to vanish in this limit, as shown in [CITATION].', '0809.1816-1-28-0': 'The series expansion eqn:rho_ex1 was constructed for [MATH], which is a physically reasonable assumption for finite L and a weak but finite interaction.', '0809.1816-1-28-1': 'But with the effective coupling strength [MATH], the chemical potential [MATH] in eqn:mu_def will vanish, too, and the healing length [MATH] will diverge.', '0809.1816-1-28-2': 'As [MATH] is kept finite, the non-interacting BEC is hence described by the limit [MATH], which is the opposite to the case considered in subsec:rho_expansion.', '0809.1816-1-28-3': 'The asymptotic form of eqn:rho_ex1 for the mode density [MATH]-as well as the Casimir energy in eqn:cas_ex-diverges in the limit of zero interaction strength.', '0809.1816-1-28-4': 'Our calculation of the exact mode density [MATH] suggests that this divergence is due to a branch point in the complex [MATH]-plane that moves towards [MATH] and makes the power series expansion behind eqn:cas_ex break down.', '0809.1816-1-28-5': 'We show here that the exact mode density eqn:rho_an leads to a Casimir energy that smoothly vanishes with the interaction strength (see fig:E_plot).', '0809.1816-1-29-0': 'To examine the asymptotic behavior of [MATH] for [MATH], we again start from eqn:DOS1 and eqn:rho_an, separately treating the behavior of [MATH] to the left and the right of the kink at [MATH]: [EQUATION]', '0809.1816-1-29-1': 'As the upper limit of the integration in [MATH] is going to zero, we can replace [MATH] by the first term in the expansion of eqn:rho_ex1, and Taylor-expand the denominator about [MATH].', '0809.1816-1-29-2': 'After integrating over this expansion, [MATH] will yield terms of [MATH].', '0809.1816-1-29-3': 'Integrating [MATH], from the lower border of the integral we get a contribution [EQUATION]', '0809.1816-1-29-4': 'Altogether, we find for the behavior of [MATH] in the non-interacting limit [EQUATION]', '0809.1816-1-29-5': 'Note that the leading order for [MATH] at fixed [MATH] goes like [MATH].', '0809.1816-1-29-6': 'Conversely, at fixed [MATH], a logarithmic divergence remains for [MATH].', '0809.1816-1-29-7': 'The logarithmic term changes sign for [MATH] and, for large values of [MATH], the above expression diverges, just as the expansion eqn:cas_ex does for small values of [MATH] (see fig:E_plot).', '0809.1816-1-29-8': 'Hence, eqn:E_nonint and eqn:cas_ex provide two asymptotic expansions to [MATH] for opposite limits, while the exact formula is given by eqn:DOS1 integrated over eqn:rho_an.', '0809.1816-1-30-0': '# Summary', '0809.1816-1-31-0': ""Starting from the free energy in a weakly-interacting dilute BEC, we derived a re-normalized expression for the 'phononic' Casimir energy density of the BEC confined at zero temperature to a parallel plate geometry with periodic boundary conditions."", '0809.1816-1-31-1': 'Our formula for the Casimir energy, eqn:DOS1, has the form of an integral over a density function [MATH] times the Bose distribution.', '0809.1816-1-31-2': 'The function [MATH] is given by a rather simple analytic expression in eqn:rho_an.', '0809.1816-1-31-3': 'In subsec:rho_expansion, we provided a cross-check for our result by showing that a series expansion of [MATH] in the parameter [MATH] reproduces the asymptotic series for the Casimir energy density derived in [CITATION].', '0809.1816-1-31-4': 'There, the convergent expression was obtained by using the Euler-MacLaurin formula to extract the long wavelength behavior out of the UV-divergent sum over all Bogoliubov modes satisfying the boundary conditions.', '0809.1816-1-31-5': 'This expression fails to reproduce the non-interacting limit.', '0809.1816-1-32-0': 'As pointed out in [CITATION], the Casimir energy density of the weakly interacting BEC is, due to the linear dispersion of its low lying excitations, in the leading order determined by a term analogous to the Casimir energy of a massless scalar field propagating with velocity [MATH] in the electromagnetic vacuum.', '0809.1816-1-32-1': 'Our result displays this behavior in the regime of the weakly interacting BEC where the plate separation is much larger than the healing length, as can be seen in fig:E_plot.', '0809.1816-1-32-2': 'In addition, for [MATH] with [MATH] kept finite (the non-interacting limit), our result eqn:rho_an correctly describes the Casimir energy going to zero and displays the Casimir energy as a smoothly varying function of the interaction strength in the intermediate range.', '0809.1816-1-32-3': 'The subtleties of the asymptotic expansions illustrate the rich physical content behind the nonlinear dispersion relation of the Bogoliubov-vacuum.', '0809.1816-1-32-4': 'This research was supported by Deutsche Forschungsgemeinschaft (DFG), grant He 2849/3.', '0809.1816-1-33-0': '# References'}","{'0809.1816-2-0-0': 'Considering the Casimir effect due to phononic excitations of a weakly interacting dilute BEC, we derive a re-normalized expression for the zero temperature Casimir energy [MATH] of a BEC confined to a parallel plate geometry with periodic boundary conditions.', '0809.1816-2-0-1': 'Our expression is formally equivalent to the free energy of a bosonic field at finite temperature, with a nontrivial density of modes that we compute analytically.', '0809.1816-2-0-2': 'As a function of the interaction strength, [MATH] smoothly describes the transition from the weakly interacting Bogoliubov regime to the non-interacting ideal BEC.', '0809.1816-2-0-3': 'For the weakly interacting case, [MATH] reduces to leading order to the Casimir energy due to zero-point fluctuations of massless phonon modes.', '0809.1816-2-0-4': 'In the limit of an ideal Bose gas, our result correctly describes the Casimir energy going to zero.', '0809.1816-2-1-0': '# Introduction', '0809.1816-2-2-0': 'The Casimir effect is a consequence of the distorted vacuum fluctuation spectrum of quantized fields in bounded domains or spaces with non-trivial topologies [CITATION].', '0809.1816-2-2-1': ""In Casimir's original calculation, the system under consideration is the electromagnetic vacuum [CITATION]."", '0809.1816-2-2-2': 'Imposing Dirichlet boundary conditions along one spatial direction by confining the system between two (idealized) parallel plates causes a change in the (infinite) vacuum energy-density.', '0809.1816-2-2-3': 'The variation of the vacuum energy-density with respect to the plate separation is called Casimir pressure, and, after renormalization, yields a finite expression for an attractive interaction energy per unit area between the plates.', '0809.1816-2-2-4': 'The electromagnetic Casimir force, caused by quantum fluctuations of the electromagnetic vacuum, is varying as [MATH] with the plate separation [MATH] and the speed of light [MATH].', '0809.1816-2-2-5': 'It has been measured in a number of experiments using various experimental settings [CITATION].', '0809.1816-2-2-6': 'The comparison between quantum vacuum experiments like these and the predictions of different theoretical models provides possibility to test fundamental physics (like higher dimensions or additional interactions), in much the same way as accelerator experiments in high-energy physics do at the other end of the energy scale [CITATION].', '0809.1816-2-3-0': ""We consider here, instead of the electromagnetic vacuum, a weakly interacting Bose-Einstein-Condensate (BEC) at zero temperature, and expect, in a similar manner, the quantum fluctuations on top of the ground state of the BEC to give rise to observable Casimir forces: Within the Bogoliubov approximation, the excited states of a BEC can be treated as quasi-particles characterized by the dispersion relation [EQUATION] that behaves linear for small momenta, with the 'sound velocity' [MATH] being inverse to the healing length [MATH]."", '0809.1816-2-3-1': 'In eqn:bogo1, the wave-number [MATH] characterizes the transition between the linear (phonon) and the quadratic (free-particle) regimes [CITATION].', '0809.1816-2-3-2': '[MATH] is also related to the [MATH]-wave scattering length [MATH] of the atoms and to the BEC density [MATH] via [MATH].', '0809.1816-2-3-3': 'For small momenta, the quasi-particles (phonons) propagate in the same way as the massless electromagnetic field, except for the propagation velocity being different.', '0809.1816-2-3-4': 'Hence, the zero temperature quantum fluctuations in a spatially confined BEC can be expected to result in an observable Casimir force.', '0809.1816-2-4-0': 'Different scenarios for Casimir forces in BECs have been analyzed by previous work: for the parallel plate geometry, an asymptotic expansion of the Casimir force has been calculated in [CITATION], the small expansion parameter being the ratio between healing length and plate separation.', '0809.1816-2-4-1': 'In the leading order, it reproduces exactly the same [MATH] behavior as in the electromagnetic vacuum.', '0809.1816-2-4-2': 'The next order corrections scale with the ratio [MATH].', '0809.1816-2-4-3': 'Replacing the perfectly reflecting plates by impurities embedded in a quantum liquid, Casimir forces between these impurities have been calculated in [CITATION] as a function of the impurity-liquid coupling.', '0809.1816-2-4-4': 'If the impurities are realized by atoms which, in a certain internal state, interact with the atoms of the quantum liquid through [MATH]-wave scattering, the Casimir interaction should be detectable as a shift of spectral lines that depends on the distance between the impurities [CITATION].', '0809.1816-2-4-5': 'In the limit of an infinitely strong impurity-liquid coupling, the result for the (one-dimensional) parallel plate scenario was recovered [CITATION]; for a weak coupling, however, the interaction between the impurities vanishes exponentially with the impurity separation on a scale set by the healing length.', '0809.1816-2-4-6': 'For the ideal (i.e.non-interacting) Bose gas, it was found that there is no Casimir force at all between impurities of arbitrary interaction strength, including the idealized parallel plate scenario [CITATION].', '0809.1816-2-4-7': 'This is consistent with the quite general method of [CITATION], which is mapping (polynomial) dispersion relations to Casimir forces in the parallel plate geometry: this method shows that media with quadratic dispersion relations, and hence the ideal Bose gas, do not give rise to any zero-temperature Casimir forces.', '0809.1816-2-4-8': 'A non-vanishing Casimir force in the ideal Bose gas can arise due to thermal fluctuations, as calculated in [CITATION].', '0809.1816-2-4-9': 'All these forces are small but finite observable quantities, which-if experimentally confirmed-would provide direct evidence of the quantum fluctuations in weakly interacting BECs.', '0809.1816-2-5-0': 'The system under consideration in the present article is a homogeneous, weakly-interacting dilute BEC at zero temperature, confined to a parallel plate geometry with periodic boundary conditions in one of the three spatial dimensions.', '0809.1816-2-5-1': 'The condition of diluteness can be formulated as [MATH], where [MATH] is the [MATH]-wave scattering length and [MATH] is the density, [MATH] being the total particle number and [MATH] the volume of the gas.', '0809.1816-2-5-2': 'The quantity [MATH] is usually called the gas parameter.', '0809.1816-2-5-3': 'The perturbative calculation of typical properties of such a BEC, like the ground state energy or the depletion of the condensate, is then essentially an expansion in the gas parameter [CITATION].', '0809.1816-2-6-0': 'For this system, we will give a re-normalized expression for the Casimir energy-density per unit area.', '0809.1816-2-6-1': ""Our expression has the form of an integral over a 'density of modes' [MATH] times the Bose distribution function: [EQUATION] where [MATH] has a simple analytic form (see eqn:rho_an), and correctly describes the vanishing of the Casimir force in the limit of the interaction strength going to zero."", '0809.1816-2-6-2': 'The possibility to express the zero temperature Casimir energy in the above form, resembling the density of states of a bosonic system at finite temperature, is connected to a topological analogy between our parallel plate scenario and finite temperature field theory: In the parallel plate geometry, one spatial coordinate of the field is subject to periodic boundary conditions, while in finite temperature field theory, the imaginary time coordinate is subject to a similar periodicity condition.', '0809.1816-2-6-3': 'This analogy has been pointed out some time ago in [CITATION]; it does not carry over, however, to Casimir calculations for non-linear dispersion relations.', '0809.1816-2-6-4': 'The periodic boundary conditions for the BEC have mainly been chosen because they make the relation to the finite temperature case particularly evident.', '0809.1816-2-6-5': 'In experiments, periodic boundary conditions can be realized in toroidal traps, but they also appear in optical lattices.', '0809.1816-2-6-6': 'If the boundaries are taken as real physical plates, the perfect mirror scenario (i.e.Dirichlet boundary conditions as discussed in [CITATION]) is closer to an experimentally realizable situation.', '0809.1816-2-6-7': 'When Dirichlet boundary conditions are imposed on the fluctuations on top of the BEC ground state, the expansion of [MATH] for moderate interaction shows in the leading term again the same behaviour as a massless scalar field propagating at the speed of sound [MATH], but with a different numerical prefactor.', '0809.1816-2-7-0': 'The article is organized as follows: In sec:free, we briefly recall some well-known perturbative expressions for the free energy, chemical potential and ground state energy of a dilute, weakly interacting BEC at zero temperature.', '0809.1816-2-7-1': 'Working from the formula for the free energy, we derive in sec:casimir the expression for the Casimir energy.', '0809.1816-2-7-2': 'In the limit [MATH], the function [MATH] reproduces the power-series expression for the Casimir energy derived in [CITATION], as is shown in subsec:rho_expansion.', '0809.1816-2-7-3': 'In subsec:toms, we relate the above mentioned analogy to the finite temperature case in some detail, as this topic does not seem to get overly much attention in the recent literature.', '0809.1816-2-7-4': 'Finally, subsec:limit deals with the behavior of the function [MATH] in the limit of the effective coupling constant between the particles going to zero.', '0809.1816-2-7-5': 'Here, our expression [MATH] correctly describes the vanishing of the Casimir force.', '0809.1816-2-7-6': 'We comment on the failure of the large-distance expansion to describe the non-interacting Bose gas.', '0809.1816-2-7-7': 'The case of Dirichlet boundary conditions is briefly discussed in the Appendix.', '0809.1816-2-8-0': '# Free energy in a weakly interacting, dilute BEC at [MATH]', '0809.1816-2-9-0': 'In the one-loop approximation, the (unrenormalized) free energy density of a weakly interacting BEC at [MATH] (in three spatial dimensions) is given by [EQUATION] with the mean-field and one-loop contributions [CITATION]: [EQUATION]', '0809.1816-2-9-1': 'Here and in the following, we work in units where [MATH].', '0809.1816-2-9-2': 'In the above expression, [MATH] is the chemical potential (which, in subsec:rho_expansion, will be connected to the speed of sound in the medium), and [MATH] is the effective coupling constant that can, for low energy collisions in a dilute medium, be identified with the [MATH]-wave scattering length [MATH]: [MATH].', '0809.1816-2-9-3': 'The term [MATH] represents the one-loop counter-term needed to render [MATH] finite.', '0809.1816-2-9-4': 'If the integral in eqn:F1b is regularized with a momentum cut-off in the ultraviolet, the linear, cubic and quintic UV-divergences in [MATH] can be absorbed by re-normalizing [MATH], [MATH] and the vacuum energy respectively.', '0809.1816-2-9-5': 'If dimensional regularization is used in eqn:F1b, no counter-term is needed at this level of perturbation theory.', '0809.1816-2-10-0': 'After renormalization, the chemical potential [MATH] can be obtained from eqn:F1 by inverting [EQUATION] which yields [EQUATION]', '0809.1816-2-10-1': 'Reinserting [MATH] into eqn:F1 will then reproduce the classical result for the leading quantum corrections to the ground state energy density [MATH] of a hard-sphere Bose gas, which was first derived by Lee, Huang and Yang in [CITATION]: [EQUATION]', '0809.1816-2-11-0': '# The Casimir contribution as an integral over a density-function', '0809.1816-2-12-0': 'Now, in order to describe a BEC between a pair of parallel plates separated by a finite distance [MATH] (with periodic boundary conditions), we have to quantize the momentum component perpendicular to the plates: [EQUATION]', '0809.1816-2-12-1': 'Correspondingly, the momentum integration perpendicular to the plates is replaced by a discrete sum: [EQUATION]', '0809.1816-2-12-2': 'The area of the plates is taken to be [MATH] with [MATH], so the system now inhabits the volume [MATH].', '0809.1816-2-12-3': 'The mean field contribution [MATH] (see eqn:F1a) depends only trivially on the new boundary conditions, with the volume [MATH] entering through [MATH] in eqn:mu_def.', '0809.1816-2-13-0': 'The Casimir energy [MATH] of the BEC is related to the free energy at the one-loop level.', '0809.1816-2-13-1': 'We are interested in its change per unit area that is due to the introduction of the boundary conditions: [EQUATION] where the first term gives the (one-loop contribution to) the free energy in a homogeneous system.', '0809.1816-2-13-2': 'After applying the substitutions eqn:quant1 and eqn:quant2 to eqn:F1b, we are left with the following expression for [MATH], now describing the leading quantum corrections to the free energy of a BEC confined between parallel plates: [EQUATION] where we have used the abbreviations [EQUATION] with [MATH] and [MATH].', '0809.1816-2-13-3': 'The summation over [MATH] can be converted into two integrals by using the Abel-Plana formula in the form (see [CITATION]) [EQUATION]', '0809.1816-2-13-4': 'Application of eqn:ap splits eqn:F1c into the two terms written in eqn:F1d, as the [MATH]-term cancels out.', '0809.1816-2-13-5': 'The first term, [MATH], is divergent, but of the same form as the one-loop free energy eqn:F1b itself.', '0809.1816-2-13-6': 'Hence, it can be re-normalized as described above.', '0809.1816-2-14-0': ""The term [MATH] in eqn:F1d, an energy per area, describes the effects induced by restricting the periodicity in one spatial dimension to be much smaller than the remaining ones, i.e.,confining the system to a volume [MATH] with one 'short' side [MATH]."", '0809.1816-2-14-1': '[MATH] is convergent for any finite [MATH], strictly negative, and goes to zero for [MATH]: [EQUATION]', '0809.1816-2-14-2': 'The integration domain is sketched in fig:wedge-domain (the shaded regions).', '0809.1816-2-15-0': 'This domain as well as the form of the integrand in eqn:cas1 are due to the branch-points of the integrand in eqn:F1c: In order to stay clear of the branch-cuts, we have evaluated the last term in the Abel-Plana formula eqn:ap along an integration contour slightly to the right of the imaginary axis.', '0809.1816-2-15-1': 'Inserting the function [MATH] given by eqn:F1c, [EQUATION] the numerator in the last term of eqn:ap evaluates to [EQUATION] being zero everywhere else along the contour of integration (see [CITATION]).', '0809.1816-2-15-2': 'By changing the order of integration in eqn:cas1, the [MATH]-integral can be performed, and we find the expression eq:Casimir-as-Bose-intgral for the Casimir energy: [EQUATION]', '0809.1816-2-15-3': ""In the form of eqn:DOS1, [MATH] is expressed as an integral over a 'density of states' (DOS) for a bosonic system, as setting the integration variable [MATH] will reproduce the Boltzmann factor in the denominator."", '0809.1816-2-16-0': ""The 'mode density' [MATH] is obtained by integrating separately over the lower triangular region in fig:wedge-domain (dark gray) and the 'hyperbolic tail' (light gray)."", '0809.1816-2-16-1': 'In terms of the dimensionless variable [MATH], we have [EQUATION]', '0809.1816-2-16-2': 'Above [MATH], the function [MATH] changes into a constant independent of [MATH] (see the solid line in fig:rho_plot).', '0809.1816-2-16-3': 'The Casimir energy [MATH] as a function of the normalized distance [MATH] is shown in fig:E_plot, obtained by numerically integrating eqn:DOS1.', '0809.1816-2-17-0': 'To provide a cross-check for the above results, we will in the following section show that the function [MATH] reproduces the asymptotic expansion for the Casimir energy given in [CITATION], which is valid for small values of the parameter [MATH].', '0809.1816-2-17-1': 'In subsec:limit we then show that the formula for [MATH] in eqn:rho_an also yields correct results in the opposite limit of the ideal BEC, i.e.[MATH].', '0809.1816-2-18-0': '## Expansion at large distance and moderate interactions', '0809.1816-2-19-0': 'In this section, we will assume the plate separation to be much greater than the healing length of the BEC, i.e.,the ratio [MATH] can be treated as a small parameter.', '0809.1816-2-19-1': 'Note that this limit cannot describe a strongly interacting Bose gas in the proper sense, since this would lead to the break-down of the one-loop approximation at the basis of our approach.', '0809.1816-2-20-0': 'As can be seen in fig:rho_plot, the kink in the function [MATH] happens at the large value [MATH] in our limit.', '0809.1816-2-20-1': 'In evaluating the Casimir energy with eqn:DOS1, large values of [MATH], and hence the behavior of [MATH] after the kink, get exponentially suppressed by the denominator.', '0809.1816-2-20-2': 'Thus we can approximate the function [MATH] by expanding its small-[MATH] part as a power-series around [MATH]: [EQUATION]', '0809.1816-2-20-3': 'The integrals in the expression eqn:DOS1 for the Casimir energy can be performed explicitly.', '0809.1816-2-20-4': 'Upon setting [MATH] (the speed of sound in the medium), we recover the result derived in [CITATION]: [EQUATION]', '0809.1816-2-20-5': 'The leading term [MATH] is, as noted in [CITATION], equal to the Casimir energy (per unit area) of a massless scalar field with propagation velocity [MATH] confined between two parallel plates with periodic boundary conditions [CITATION].', '0809.1816-2-20-6': 'Its presence is a manifestation of the Goldstone-theorem, the long wavelength part of the Bogoliubov spectrum representing the gap-less Goldstone modes [CITATION].', '0809.1816-2-20-7': 'The next-to-leading term in eqn:cas_ex is in [CITATION] referred to as the Bogoliubov correction due to the non-linearity of the dispersion.', '0809.1816-2-20-8': 'Indeed, if we express the Bogoliubov dispersion relation eqn:bogo1 as a power series around [MATH] and put the first few terms into the expression for the Casimir force derived in [CITATION], the leading term in the same manner reproduces the result for the scalar field (in one dimension), followed by terms that are smaller in magnitude and of opposite sign.', '0809.1816-2-20-9': 'As can be seen in fig:E_plot, the contribution of these corrections is getting smaller as the ratio between plate separation and healing length increases, leaving only the dominant scalar term [MATH] depicted by the horizontal dashed line in fig:E_plot.', '0809.1816-2-21-0': 'When finally calculating the Casimir pressure from eqn:cas_ex, one has to consider that, as the number of particles is held constant, the derivative of the speed of sound with respect to [MATH] is not zero, as discussed in [CITATION].', '0809.1816-2-22-0': '## A formal analogy to finite temperature systems', '0809.1816-2-23-0': 'As already mentioned, our expression eqn:DOS1 for [MATH] at [MATH] formally resembles the DOS for a bosonic system at finite temperature.', '0809.1816-2-23-1': 'This can be understood by recalling that a finite temperature system can be described in imaginary time, combined with periodic boundary conditions with period [MATH].', '0809.1816-2-23-2': 'The same topology is realized in the parallel plate system at [MATH], when one spatial dimension is subject to periodic boundary conditions with period [MATH] ([CITATION], see also [CITATION]).', '0809.1816-2-24-0': 'Let us briefly re-phrase the argument of [CITATION] within our notation.', '0809.1816-2-24-1': 'The canonical partition function [MATH] for a system at temperature [MATH] in [MATH] spatial dimensions is usually expressed as the path integral [EQUATION] where [MATH] is the Lagrangian (a scalar functional of the field [MATH]), and the field is constrained in such a way that [MATH].', '0809.1816-2-24-2': 'If we take [MATH], the path-integral in eqn:toms_Z can, after Fourier expansion of the field, be evaluated to yield [CITATION] [EQUATION]', '0809.1816-2-24-3': 'Here, [MATH] with [MATH] are the Matsubara-frequencies due to the periodicity condition in eqn:toms_Z, and [MATH] with the continuous parameter [MATH] is the dispersion relation of the massless scalar field.', '0809.1816-2-24-4': 'Note the similarity to eqn:quant1, where we had periodic boundary conditions not in imaginary time but in one spatial dimension.', '0809.1816-2-25-0': 'The sum over [MATH] in eqn:toms_logz_a is usually evaluated by multiplying with a factor [MATH], which has poles of residue 1 at [MATH], and then integrating over a contour in the complex [MATH]-plane which includes all the poles [CITATION].', '0809.1816-2-25-1': 'This technique is actually the same that is used in [CITATION] to prove the Abel-Plana formula eqn:ap.', '0809.1816-2-25-2': 'The well known result is [EQUATION] where we have already subtracted the zero-point fluctuations.', '0809.1816-2-25-3': 'After integrating by parts and employing the integral-identity from the footnote on page eqn:integral_formula, we obtain ([MATH] is the volume of the unit sphere in [MATH] dimensions) [EQUATION]', '0809.1816-2-25-4': 'Note that, upon setting [MATH], the above expression has the same form as [MATH] in eqn:DOS1, with [MATH] being proportional to [MATH].', '0809.1816-2-25-5': 'Now, let the system inhabit a volume [MATH].', '0809.1816-2-25-6': 'With [MATH] we get for the free energy per unit area [EQUATION]', '0809.1816-2-25-7': 'Tentatively exchanging [MATH] with [MATH] in eqn:toms_logZ_3d will reproduce the Casimir energy for a massless scalar field, i.e.the first term in eqn:cas_ex [CITATION]: [EQUATION]', '0809.1816-2-25-8': 'So, we have seen that the zero temperature Casimir energy of a massless scalar field confined between two parallel plates can be obtained by a simple exchange of variables, once we know the thermal contribution to the free energy density of that field.', '0809.1816-2-25-9': 'Unfortunately, this simple mapping does not carry over to fields characterized by nonlinear dispersion relations: The temperature dependent part of the free energy for a BEC is (in the Bogoliubov approximation) still given by eqn:toms_logz_bose with the dispersion now being [MATH] [CITATION].', '0809.1816-2-25-10': 'But a simple interchange of [MATH] with [MATH] in eqn:toms_logz_bose will no longer yield [MATH], because the discretized Matsubara frequencies [MATH] (that become a discretized momentum) always enter in eqn:toms_logz_a in the same way as a spatial momentum component enters in a linear dispersion relation.', '0809.1816-2-25-11': 'It thus seems very difficult to mimic the fully nonlinear behavior of the dispersion relation.', '0809.1816-2-26-0': '## The non-interacting limit', '0809.1816-2-27-0': 'The transition from the weakly interacting BEC (here described at the one-loop level only) to the ideal BEC should be accomplished by letting the effective coupling [MATH] (see sec:free) go to zero, corresponding to [MATH].', '0809.1816-2-27-1': 'The Casimir energy is expected to vanish in this limit, as shown in [CITATION].', '0809.1816-2-28-0': 'The series expansion eqn:rho_ex1 was constructed for [MATH], which is a physically reasonable assumption for finite L and a weak but finite interaction.', '0809.1816-2-28-1': 'But with the effective coupling strength [MATH], the chemical potential [MATH] in eqn:mu_def will vanish, too, and the healing length [MATH] will diverge.', '0809.1816-2-28-2': 'As [MATH] is kept finite, the non-interacting BEC is hence described by the limit [MATH], which is the opposite to the case considered in subsec:rho_expansion.', '0809.1816-2-28-3': 'The asymptotic form of eqn:rho_ex1 for the mode density [MATH]-as well as the Casimir energy in eqn:cas_ex-diverges in the limit of zero interaction strength.', '0809.1816-2-28-4': 'Our calculation of the exact mode density [MATH] suggests that this divergence is due to a branch point in the complex [MATH]-plane that moves towards [MATH] and makes the power series expansion behind eqn:cas_ex break down.', '0809.1816-2-28-5': 'We show here that the exact mode density eqn:rho_an leads to a Casimir energy that smoothly vanishes with the interaction strength (see fig:E_plot).', '0809.1816-2-29-0': 'To examine the asymptotic behavior of [MATH] for [MATH], we again start from eqn:DOS1 and eqn:rho_an, separately treating the behavior of [MATH] to the left and the right of the kink at [MATH]: [EQUATION]', '0809.1816-2-29-1': 'As the upper limit of the integration in [MATH] is going to zero, we can replace [MATH] by the first term in the expansion of eqn:rho_ex1, and Taylor-expand the denominator about [MATH].', '0809.1816-2-29-2': 'After integrating over this expansion, [MATH] will yield terms of [MATH].', '0809.1816-2-29-3': 'Integrating [MATH], from the lower border of the integral we get a contribution [EQUATION]', '0809.1816-2-29-4': 'Altogether, we find for the behavior of [MATH] in the non-interacting limit [EQUATION]', '0809.1816-2-29-5': 'Note that the leading order for [MATH] at fixed [MATH] goes like [MATH].', '0809.1816-2-29-6': 'Conversely, at fixed [MATH], a logarithmic divergence remains for [MATH].', '0809.1816-2-29-7': 'The logarithmic term changes sign for [MATH] and, for large values of [MATH], the above expression diverges, just as the expansion eqn:cas_ex does for small values of [MATH] (see fig:E_plot).', '0809.1816-2-29-8': 'Hence, eqn:E_nonint and eqn:cas_ex provide two asymptotic expansions to [MATH] for opposite limits, while the exact formula is given by eqn:DOS1 integrated over eqn:rho_an.', '0809.1816-2-30-0': '# Summary', '0809.1816-2-31-0': ""Starting from the free energy in a weakly-interacting dilute BEC, we derived a re-normalized expression for the 'phononic' Casimir energy of the BEC confined at zero temperature to a parallel plate geometry with periodic boundary conditions."", '0809.1816-2-31-1': 'Our formula for the Casimir energy (per unit plate area), eqn:DOS1, has the form of an integral over a density function [MATH] times the Bose distribution.', '0809.1816-2-31-2': 'The function [MATH] is given by a rather simple analytic expression in eqn:rho_an.', '0809.1816-2-31-3': 'In subsec:rho_expansion, we provided a cross-check for our result by showing that a series expansion of [MATH] in the parameter [MATH] reproduces the asymptotic series for the Casimir energy derived in [CITATION].', '0809.1816-2-31-4': 'There, the Euler-MacLaurin formula was used to extract the long wavelength behavior out of the UV-divergent sum over all Bogoliubov modes satisfying the boundary conditions.', '0809.1816-2-31-5': 'This approach fails to reproduce the non-interacting limit.', '0809.1816-2-32-0': 'As pointed out in [CITATION], the Casimir energy of the weakly interacting BEC is, due to the linear dispersion of its low lying excitations, in the leading order determined by a term analogous to the Casimir energy of a massless scalar field propagating with vacuum velocity [MATH].', '0809.1816-2-32-1': 'Our result displays this behavior in the regime of the weakly interacting BEC where the plate separation is much larger than the healing length, as can be seen in fig:E_plot.', '0809.1816-2-32-2': 'In addition, for [MATH] with [MATH] kept finite (the non-interacting limit), our result eqn:rho_an correctly describes the Casimir energy going to zero and displays the Casimir energy as a smoothly varying function of the interaction strength in the intermediate range.', '0809.1816-2-32-3': 'The subtleties of the asymptotic expansions illustrate the rich physical content behind the nonlinear dispersion relation of the Bogoliubov vacuum.', '0809.1816-2-32-4': 'This research was supported by Deutsche Forschungsgemeinschaft (DFG), grant He 2849/3.', '0809.1816-2-33-0': '# Appendix: Dirichlet boundary-conditions', '0809.1816-2-34-0': 'Here we discuss how our results for the Casimir energy get modified with Dirichlet boundary conditions instead of periodic ones: taking the wavenumbers [MATH] in eqn:quant1 as [MATH] with [MATH] running from [MATH] to infinity, application of the Abel-Plana formula generates an additional divergent surface-term [MATH] in eqn:F1d: [EQUATION] (The same term with opposite sign will appear for von Neumann boundary conditions, similar to [CITATION] where the ideal gas at finite temperature is considered.)', '0809.1816-2-34-1': 'As [MATH] does not depend on [MATH], it does not affect the Casimir energy.', '0809.1816-2-34-2': 'Our main formula for [MATH], eqn:DOS1, is modified by an overall factor [MATH], while in the mode density [MATH] (see eqn:rho_an) the distance argument picks up an additional factor of two: [EQUATION]', '0809.1816-2-34-3': 'The asymptotic expansion for large plate separation and moderate interactions, as derived in subsec:rho_expansion, now yields (compare with eqn:cas_ex): [EQUATION]', '0809.1816-2-34-4': 'Again, the leading term is equal to the known result for the Casimir energy of a massless scalar field with propagation velocity [MATH] confined between two parallel ideal mirrors (see [CITATION]).', '0809.1816-2-34-5': 'For the non-interacting limit, as treated in subsec:limit, we get the asymptotic expression (compare with eqn:E_nonint) [EQUATION]', '0809.1816-2-34-6': 'This goes to zero as for periodic boundary conditions, with slightly different numerical factors.', '0809.1816-2-35-0': 'The divergent surface energy eqn:surf does not fit into the renormalization scheme for the bulk part of the free energy [MATH] as described after eqn:F1b.', '0809.1816-2-35-1': 'For a full treatment of the free energy of a BEC between perfect mirrors, it is not sufficient to change only the wavenumbers [MATH] in the Bogoliubov excitations, because both the ground-state wave function and the excitations have to be zero on the boundaries.', '0809.1816-2-35-2': 'A detailed analysis will be reported elsewhere.', '0809.1816-2-36-0': '# References'}","[['0809.1816-1-16-1', '0809.1816-2-16-1'], ['0809.1816-1-16-2', '0809.1816-2-16-2'], ['0809.1816-1-16-3', '0809.1816-2-16-3'], ['0809.1816-1-0-0', '0809.1816-2-0-0'], ['0809.1816-1-0-1', '0809.1816-2-0-1'], ['0809.1816-1-0-2', '0809.1816-2-0-2'], ['0809.1816-1-0-3', '0809.1816-2-0-3'], ['0809.1816-1-0-4', '0809.1816-2-0-4'], ['0809.1816-1-14-0', '0809.1816-2-14-0'], ['0809.1816-1-14-1', '0809.1816-2-14-1'], ['0809.1816-1-14-2', '0809.1816-2-14-2'], ['0809.1816-1-24-0', '0809.1816-2-24-0'], ['0809.1816-1-24-1', '0809.1816-2-24-1'], ['0809.1816-1-24-2', '0809.1816-2-24-2'], ['0809.1816-1-24-3', '0809.1816-2-24-3'], ['0809.1816-1-24-4', '0809.1816-2-24-4'], ['0809.1816-1-21-0', '0809.1816-2-21-0'], ['0809.1816-1-23-0', '0809.1816-2-23-0'], ['0809.1816-1-23-1', '0809.1816-2-23-1'], ['0809.1816-1-23-2', '0809.1816-2-23-2'], ['0809.1816-1-7-0', '0809.1816-2-7-0'], ['0809.1816-1-7-2', '0809.1816-2-7-2'], ['0809.1816-1-7-3', '0809.1816-2-7-3'], ['0809.1816-1-7-5', '0809.1816-2-7-5'], ['0809.1816-1-7-6', '0809.1816-2-7-6'], ['0809.1816-1-4-2', '0809.1816-2-4-2'], ['0809.1816-1-4-5', '0809.1816-2-4-6'], ['0809.1816-1-4-6', '0809.1816-2-4-7'], ['0809.1816-1-4-8', '0809.1816-2-4-9'], ['0809.1816-1-13-0', '0809.1816-2-13-0'], ['0809.1816-1-13-1', '0809.1816-2-13-1'], ['0809.1816-1-13-3', '0809.1816-2-13-3'], ['0809.1816-1-13-4', '0809.1816-2-13-4'], ['0809.1816-1-13-5', '0809.1816-2-13-5'], ['0809.1816-1-13-6', '0809.1816-2-13-6'], ['0809.1816-1-28-0', '0809.1816-2-28-0'], ['0809.1816-1-28-1', '0809.1816-2-28-1'], ['0809.1816-1-28-2', '0809.1816-2-28-2'], ['0809.1816-1-28-3', '0809.1816-2-28-3'], ['0809.1816-1-28-4', '0809.1816-2-28-4'], ['0809.1816-1-28-5', '0809.1816-2-28-5'], ['0809.1816-1-5-0', '0809.1816-2-5-0'], ['0809.1816-1-5-1', '0809.1816-2-5-1'], ['0809.1816-1-5-2', '0809.1816-2-5-2'], ['0809.1816-1-5-3', '0809.1816-2-5-3'], ['0809.1816-1-10-0', '0809.1816-2-10-0'], ['0809.1816-1-10-1', '0809.1816-2-10-1'], ['0809.1816-1-31-2', '0809.1816-2-31-2'], ['0809.1816-1-15-0', '0809.1816-2-15-0'], ['0809.1816-1-15-1', '0809.1816-2-15-1'], ['0809.1816-1-15-3', '0809.1816-2-15-3'], ['0809.1816-1-12-0', '0809.1816-2-12-0'], ['0809.1816-1-12-1', '0809.1816-2-12-1'], ['0809.1816-1-12-2', '0809.1816-2-12-2'], ['0809.1816-1-12-3', '0809.1816-2-12-3'], ['0809.1816-1-17-0', '0809.1816-2-17-0'], ['0809.1816-1-17-1', '0809.1816-2-17-1'], ['0809.1816-1-25-0', '0809.1816-2-25-0'], ['0809.1816-1-25-1', '0809.1816-2-25-1'], ['0809.1816-1-25-2', '0809.1816-2-25-2'], ['0809.1816-1-25-3', '0809.1816-2-25-3'], ['0809.1816-1-25-4', '0809.1816-2-25-4'], ['0809.1816-1-25-5', '0809.1816-2-25-5'], ['0809.1816-1-25-6', '0809.1816-2-25-6'], ['0809.1816-1-25-8', '0809.1816-2-25-8'], ['0809.1816-1-25-9', '0809.1816-2-25-9'], ['0809.1816-1-25-10', '0809.1816-2-25-10'], ['0809.1816-1-25-11', '0809.1816-2-25-11'], ['0809.1816-1-19-0', '0809.1816-2-19-0'], ['0809.1816-1-19-1', '0809.1816-2-19-1'], ['0809.1816-1-3-0', '0809.1816-2-3-0'], ['0809.1816-1-3-3', '0809.1816-2-3-4'], ['0809.1816-1-29-0', '0809.1816-2-29-0'], ['0809.1816-1-29-1', '0809.1816-2-29-1'], ['0809.1816-1-29-2', '0809.1816-2-29-2'], ['0809.1816-1-29-3', '0809.1816-2-29-3'], ['0809.1816-1-29-4', '0809.1816-2-29-4'], ['0809.1816-1-29-5', '0809.1816-2-29-5'], ['0809.1816-1-29-6', '0809.1816-2-29-6'], ['0809.1816-1-29-7', '0809.1816-2-29-7'], ['0809.1816-1-29-8', '0809.1816-2-29-8'], ['0809.1816-1-27-0', '0809.1816-2-27-0'], ['0809.1816-1-27-1', '0809.1816-2-27-1'], ['0809.1816-1-20-0', '0809.1816-2-20-0'], ['0809.1816-1-20-1', '0809.1816-2-20-1'], ['0809.1816-1-20-2', '0809.1816-2-20-2'], ['0809.1816-1-20-4', '0809.1816-2-20-4'], ['0809.1816-1-20-6', '0809.1816-2-20-6'], ['0809.1816-1-20-7', '0809.1816-2-20-7'], ['0809.1816-1-20-8', '0809.1816-2-20-8'], ['0809.1816-1-20-9', '0809.1816-2-20-9'], ['0809.1816-1-32-1', '0809.1816-2-32-1'], ['0809.1816-1-32-2', '0809.1816-2-32-2'], ['0809.1816-1-32-4', '0809.1816-2-32-4'], ['0809.1816-1-2-0', '0809.1816-2-2-0'], ['0809.1816-1-2-1', '0809.1816-2-2-1'], ['0809.1816-1-2-2', '0809.1816-2-2-2'], ['0809.1816-1-2-3', '0809.1816-2-2-3'], ['0809.1816-1-2-4', '0809.1816-2-2-4'], ['0809.1816-1-2-5', '0809.1816-2-2-5'], ['0809.1816-1-2-6', '0809.1816-2-2-6'], ['0809.1816-1-6-0', '0809.1816-2-6-0'], ['0809.1816-1-6-1', '0809.1816-2-6-1'], ['0809.1816-1-6-2', '0809.1816-2-6-2'], ['0809.1816-1-6-3', '0809.1816-2-6-3'], ['0809.1816-1-9-0', 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'0809.1816-2-4-3'], ['0809.1816-1-4-4', '0809.1816-2-4-5'], ['0809.1816-1-4-7', '0809.1816-2-4-8'], ['0809.1816-1-13-2', '0809.1816-2-13-2'], ['0809.1816-1-31-0', '0809.1816-2-31-0'], ['0809.1816-1-31-1', '0809.1816-2-31-1'], ['0809.1816-1-31-3', '0809.1816-2-31-3'], ['0809.1816-1-31-4', '0809.1816-2-31-4'], ['0809.1816-1-31-5', '0809.1816-2-31-5'], ['0809.1816-1-15-2', '0809.1816-2-15-2'], ['0809.1816-1-25-7', '0809.1816-2-25-7'], ['0809.1816-1-3-1', '0809.1816-2-3-1'], ['0809.1816-1-3-2', '0809.1816-2-3-3'], ['0809.1816-1-20-3', '0809.1816-2-20-3'], ['0809.1816-1-20-5', '0809.1816-2-20-5'], ['0809.1816-1-32-0', '0809.1816-2-32-0'], ['0809.1816-1-32-3', '0809.1816-2-32-3']]",[],[],[],[],"{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/0809.1816,,,
1403.5466,"{'1403.5466-1-0-0': 'We present new results from our search for z[MATH] galaxies from deep spectroscopic observations of candidate z-dropouts in the CANDELS fields.', '1403.5466-1-0-1': 'Despite the extremely low flux limits achieved by our sensitive observations, only 2 galaxies have robust redshift identifications, one from its Ly[MATH] emission line at z=6.65, the other from its Lyman-break, i.e. the continuum discontinuity at the Ly[MATH] wavelength consistent with a redshift 6.42, but with no emission line.', '1403.5466-1-0-2': 'In addition, for [MATH] galaxies we present deep limits in the Ly[MATH] EW derived from the non detections in ultra-deep observations.', '1403.5466-1-1-0': 'Using this new data as well as previous samples, we assemble a total of 68 candidate z[MATH] galaxies with deep spectroscopic observations, of which 12 have a line detection.', '1403.5466-1-1-1': 'With this much enlarged sample we can place solid constraints on the declining fraction of Ly[MATH] emission in z[MATH]7 Lyman break galaxies compared to z[MATH], both for bright and faint galaxies.', '1403.5466-1-1-2': 'Applying a simple analytical model, we show that the present data favor a patchy reionization process rather than a smooth one.', '1403.5466-1-2-0': '# Introduction', '1403.5466-1-3-0': 'The use of Ly[MATH] transmission by the intergalactic medium (IGM) as a probe of its ionization state during the reionization epoch has been proposed already many years ago (Miralda-Escude Rees 1998; Santos et al. 2004).', '1403.5466-1-3-1': 'Strong Ly[MATH] emission, powered by star formation is present is many distant galaxies: being a resonant line, it is sensitive to even small quantities of neutral hydrogen in the IGM, and it is easily suppressed .', '1403.5466-1-3-2': 'We thus expect the observed properties of Ly[MATH] emitting galaxies to change at higher redshifts, when the IGM becomes more neutral.', '1403.5466-1-3-3': 'A very common approach for studying the reionization history of the Universe using Ly[MATH] emitting galaxies is to determine the evolution of the Ly[MATH] luminosity function and the clustering properties of narrow band selected Ly[MATH] emitters (LAEs e.g. Ota et al. 2008, Ouchi et a. 2010; Kashikawa et al. 2011, Clements et al. 2012).', '1403.5466-1-3-4': 'A recent complementary approach and the one used in this paper, is instead to measure the redshift evolution of the Ly[MATH] fraction in Lyman break galaxies (LBGs), i.e. the percentage of LBGs that have an appreciable Ly[MATH] emission line (e.g. Stark et al. 2010).', '1403.5466-1-3-5': 'Indeed this fraction is supposed to increase, as we move to higher redshift since galaxies are increasingly young (hence with stronger intrinsic Ly[MATH]) and almost dust free (Finkelstein et al. 2012) which facilitates the escape of Ly[MATH] photons.', '1403.5466-1-3-6': 'On the other hand this fraction is expected to fall-off as we approach the time when the intergalactic medium becomes significantly neutral and the galaxies’ Ly[MATH] emission is progressively attenuated.', '1403.5466-1-3-7': 'Compared to other probes of reionization such as the evolution of the LAE luminosity function this approach can overcome concerns about intrinsic density evolution of the underlying population (Stark et al. 2010).', '1403.5466-1-4-0': 'Intriguingly, early measurements with this technique suggest a strong drop in the Ly[MATH] fraction near z[MATH]7, more significant for relatively fainter galaxies.', '1403.5466-1-4-1': 'In particular in a series of recent works, a lack of Ly[MATH] emission was found at [MATH] compared to [MATH] by several independent teams: in our previous observations (Pentericci et al. 2011; Vanzella et al. 2011; Fontana et al. 2010; P11, V11 and F10 from here on) we found 4 Ly[MATH] emitting galaxies (plus another object with a tentative line detection) out of a sample of 20 robust candidates.', '1403.5466-1-4-2': 'Similar or lower fractions were found by [CITATION],[CITATION], Bradac et al. (2012), with the only notable exception of the work by Ono et al. (2012) who found a fraction of [MATH] emitters that was similar to the z[MATH] value for bright galaxies.', '1403.5466-1-4-3': 'In our favored interpretation, the lack of line emission is due to a substantial increase in the neutral hydrogen content of the Universe in the time between [MATH] and [MATH].', '1403.5466-1-4-4': ""Comparing our data to the predictions of the semi-analytical models by Dijkstra et al. (2011) we concluded that to explain the observations a substantial change of the neutral hydrogen fraction of the order of [MATH] in a time [MATH] was required, assuming that the galaxies' physical properties remain constant during this time."", '1403.5466-1-4-5': 'Recent observations pushing to [MATH] are consistent with this interpretation (Treu et al. 2013; Schmidt et al. 2014).', '1403.5466-1-5-0': 'However other factors could also play a role in the Ly[MATH] quenching: in particular we cannot rule out the possibility that a change in some of the intrinsic galaxy properties (the Lyman continuum escape fraction, wind properties and dust content) could at least partially contribute to the lack of Ly[MATH] emission.', '1403.5466-1-5-1': 'Indeed the interpretation of the results as only due to the change in the neutral hydrogen fraction was questioned by several successive works (e.g. Jensen et al. 2013, Forero-Romeiro et al. 2012, Bolton Haehnelt 2013, Taylor Lidz 2014, Dijkstra et al. 2014).', '1403.5466-1-5-2': 'In particular Bolton Hahnelt (2013) suggested that the opacity of the intervening IGM red-ward of rest-frame Ly[MATH] can rise rapidly in average regions of the Universe simply because of the increasing incidence of absorption systems, which are optically thick to Lyman continuum photons.', '1403.5466-1-5-3': 'They claimed that the data do not require a large change in the IGM neutral fraction from z[MATH]6 to [MATH]7.', '1403.5466-1-5-4': 'However such a rapid evolution of the photo-ionizing background could be very difficult to achieve without requiring either a late reionization, or an emissivity at [MATH] which is too high to be consistent with observations of the Ly[MATH] forest (e.g. Sobacchi Mesinger 2014).', '1403.5466-1-5-5': 'Preliminary estimates suggest that the neutral fraction constraint relaxes only mildly when taking into account the absorption systems (Mesinger et al. in preparation).', '1403.5466-1-6-0': 'We also mention the very recent work by Taylor Lidz (2014), pointing out that sample variance is notnegligible for existing surveys: considering the large spatial fluctuations of the medium owing to an inhomogeneous reionization, the required neutral fraction at z[MATH]7 can somehow be reduced to less extreme values.', '1403.5466-1-6-1': 'Indeed the observational results are presently based on small data-sets, with considerable field to field variations (Pentericci et al. 2011), and mostly focusing on the brightest candidates ([MATH]).', '1403.5466-1-7-0': 'The complex topology of reionization is also a highly debated matter.', '1403.5466-1-7-1': 'Depending on the nature of the main sources of reionization, it is expected that the characteristic scale of the reionization process might change substantially (Iliev et al. 2006; Furlanetto et al. 2006).', '1403.5466-1-7-2': 'Accurate theoretical predictions for the morphology and sizes of H II regions depend on the abundance and clustering of the ionizing sources themselves, in addition to the underlying inhomogeneous density field and clumpiness of the gas in the IGM (McQuinn et al. 2007, Sobacchi Mesinger 2014).', '1403.5466-1-7-3': 'Observations of Ly[MATH] emitting galaxies and their clustering have the potential to reveal the signature of patchy reionization, although early results have been inconclusive (Kashikawa et al. 2011, Ouchi et al. 2010)', '1403.5466-1-8-0': 'In this paper we present new observations of z[MATH] candidates, significantly increasing the statistics of previous works (expecially in the faint regime, thanks to the inclusion of lensed candidates), which will allow us to assess the emergence of Ly[MATH] emission at high redshift with greater accuracy and address some of the above issues.', '1403.5466-1-8-1': 'In Section 2 we present the new observations and the previous data available; in Section 3 we describe the simulations used to accurately evaluate the sensitivity of our spectroscopic observations.', '1403.5466-1-8-2': 'In Section 4 we first evaluate the new limits on the Ly[MATH] fractions at high redshift, and we derive the neutral hydrogen fraction that is needed to explain the observed decrease; then applying a simple phenomenological model we derive new constraints on the topology of reionization.', '1403.5466-1-8-3': 'In Section 5 we summarize our findings.', '1403.5466-1-9-0': 'All magnitudes are in the AB system, and we adopt [MATH] km/s/Mpc, [MATH] and [MATH].', '1403.5466-1-10-0': '# Observations', '1403.5466-1-11-0': 'In this section we summarize the new observations presented in this work as well as previous data that we will use in this paper.', '1403.5466-1-12-0': '## UDS field', '1403.5466-1-13-0': 'We selected candidate z[MATH]7 galaxies in the UDS field from CANDELS multi-wavelength observations (Galametz et al. 2013).', '1403.5466-1-13-1': 'Objects were detected in the J band and then the color selection criteria presented by Grazian et al. (2012).', '1403.5466-1-13-2': 'were applied Observations were taken in service mode with the FORS2 spectrograph on the ESO Very Large Telescope.', '1403.5466-1-13-3': 'We used the 600Z holographic grating, that provides the highest sensitivity in the range [MATH] with a spectral resolution [MATH] and a sampling of 1.6 per pixel for a 1[MATH] slit.', '1403.5466-1-13-4': 'Out of the entire sample of z-dropout candidates (which consists of 50 galaxies), we placed a total of 12 galaxies in the slits (the selection was just driven by the geometry of the mask).', '1403.5466-1-13-5': 'The rest of the mask was filled with i-dropouts (Vanzella et al. in preparation) and other targets such as massive high redshift galaxies and AGN candidates.', '1403.5466-1-13-6': 'The sources have been observed through slitlets 1[MATH] wide by 10-12[MATH] long.', '1403.5466-1-13-7': 'The observation strategy was identical to the one adopted in P11 and previous papers: series of spectra were taken at two different positions, offset by [MATH] (16 pixels) in the direction perpendicular to the dispersion.', '1403.5466-1-13-8': 'The total net integration time was 15.5 hours for each object.', '1403.5466-1-13-9': 'Data were reduced using our dedicated pipeline which was described in detail in F10 and V11.', '1403.5466-1-14-0': 'For one of the candidates with a relatively bright continuum magnitude (J=25.98), UDS29249, we detect a faint continuum emission in the red part of the spectrum beyond [MATH]9100 as shown in Figure 2.', '1403.5466-1-14-1': 'The total integrated S/N of the flux is [MATH]; while the detection is relatively secure in the wavelength range 9160-9240 , the exact position of the break (that is ascribed to the IGM) is difficult to locate both because of the faintness of the emission and also because of the residuals of the bright sky emission lines in the region immediately below 9100.', '1403.5466-1-14-2': 'We conservatively estimate a redshift that ranges between 6.31 and 6.53 (in the table we report [MATH]).', '1403.5466-1-14-3': 'Note that this is perfectly consistent with the photometric redshift distribution obtained from the CANDELS photometry, also shown in Figure 2.', '1403.5466-1-15-0': 'All other candidates are undetected (meaning that no feature is detected).', '1403.5466-1-15-1': 'In Table 1 we report the candidates, RA and Dec, their J band magnitudes and the limiting EW.', '1403.5466-1-15-2': 'For the undetected objects we assume a redshift of 6.9, which is the median redshift of the selection function (see Grazian et al. 2012).', '1403.5466-1-16-0': '## ESO Archive', '1403.5466-1-17-0': 'We searched the ESO archive for observations of high redshift objects; in particular we retrieved the data from the observations carried out within the program ESO 088.A-1013 (PI Bunker).', '1403.5466-1-17-1': 'This program used the same observational setup used above, with a total net integration time of 27 hours.', '1403.5466-1-17-2': 'It observed a mixture of z and i-dropouts.', '1403.5466-1-17-3': 'Recently the results were presented by [CITATION] and the authors report all non detections for the candidate z[MATH] galaxies.', '1403.5466-1-18-0': 'Caruana et al. have observed candidate high redshift galaxies selected in previous works (Bouwens et al. 2011, McLure et al. 2010 and Wilkins et al. 2011) which used each a different selection criteria, from different color-color cuts to photometric redshifts.', '1403.5466-1-18-1': 'Since we want to work on a sample that is as homogeneously selected as possible, we have selected our own list of z-dropouts again using the color criteria presented by Grazian et al. (2012).', '1403.5466-1-18-2': 'We then cross correlated our list with the targets in [CITATION].', '1403.5466-1-18-3': 'We found 9 matching objects, of which 5 are in common with the sample already observed in F10.', '1403.5466-1-18-4': 'We then retrieved the raw (public) data from the ESO archive and then processed through our own pipeline (V11) as all other data in this work.', '1403.5466-1-18-5': 'Here we present the results for the 4 new targets that are not in common with F10 (see Table 1).', '1403.5466-1-18-6': 'Further results in particular the extremely deep combined spectra of the objects in common between the Caruana et al. program and F10 (52 hours) will be presented elsewhere (Vanzella et al. in preparation).', '1403.5466-1-19-0': 'The results are again presented in Table 1.', '1403.5466-1-19-1': 'We detect a significant emission line in one of the 4 new objects, galaxy 34271 in the GOODS-South field corresponding to galaxy ERSz-2352941047 in [CITATION].', '1403.5466-1-19-2': 'The line is detected at 9301 and shows the typical asymmetry of Ly[MATH] which would place the object at redshift [MATH].', '1403.5466-1-19-3': 'In Figure 1 we show the 1-dimensional and 2-dimensional spectra of the galaxy.', '1403.5466-1-19-4': 'The EW of the line is 43 , calculated from its measured line flux and the Y band magnitude from the GOODS CANDELS catalog.', '1403.5466-1-20-0': 'As we mentioned above, [CITATION] report no emission line for this galaxy, but a median EW limit of 28.5 (without specifying if it is a limit at 3[MATH] or 5[MATH]).', '1403.5466-1-20-1': 'We ascribe the difference to the fact that for most of the data reduction steps the authors used the standard ESO pipeline, while we use our own pipeline that has been tailored specifically to the detection of faint high redshift emission lines.', '1403.5466-1-20-2': 'The other 3 targets are undetected and we can set stringent limits of their Ly[MATH] EW limit ranging from [MATH] to [MATH] depending on the continuum magnitude and assuming that they are placed at redshift 6.9.', '1403.5466-1-20-3': 'Obviously the actual EW limit depends sensibly on the exact redshift of the objects (see F10 Figure 1).', '1403.5466-1-21-0': '## Data from previous literature', '1403.5466-1-22-0': 'Besides the new data, we include all previously published spectroscopic data on z-dropouts, in order to assemble the largest possible sample of candidate z[MATH] galaxies with deep spectroscopic observations.', '1403.5466-1-23-0': 'In particular we consider: (1) the 20 z-dropouts selected in the GOODS-South, NTTDF and BDF4 fields (Castellano et al. 2010a, 2010b) whose observations were carried out by our groups in P11, and previously presented by V11 and F10.', '1403.5466-1-23-1': 'Of these, 4 show a convincing Ly[MATH] emission line, while the tentative detection of a fifth candidate originally showed in F10 was not confirmed by the combination of our own data with the deeper observations of [CITATION] (Vanzella et al. in preparation).', '1403.5466-1-23-2': '(2) The 11 bright z-dropout observed by Ono et al. (2012) in the SDF field of which 3 have bright Ly[MATH] in emission.', '1403.5466-1-23-3': 'These candidates were detected in deep Y band observations and selected using color criteria that are very similar to ours.', '1403.5466-1-23-4': '(3) A subset of the objects presented by Schenker et al. (2012).', '1403.5466-1-23-5': 'In particular we select those galaxies whose colors are consistent with the z-dropout selection criteria used in this paper (note that Schenker et al. also observed Y-band dropouts whose photometric redshifts are [MATH] 7).', '1403.5466-1-23-6': 'In total we consider 11 objects of which 2 are detected with Ly[MATH].', '1403.5466-1-24-0': 'Overall considering new and previous data, we assemble a sample of 68 z-dropouts that have been spectroscopically observed with either VLT, Keck or Subaru down to very faint flux limits.', '1403.5466-1-24-1': 'Note that 46 out of 68 have been observed with exactly the same set-up (with FORS2@VLT using grism 600z).', '1403.5466-1-25-0': '## Bullet cluster', '1403.5466-1-26-0': 'Bradac et al. (2012) observed the lensed z-dropouts detected behind the Bullet cluster and selected by Hall et al. 2011.', '1403.5466-1-26-1': 'The observations were carried out with FORS2@VLT using the same observational set-up as in our various programs (V11, P11), and above (UDS and GOODS-S) with a total net integration time of 16.5 hours.', '1403.5466-1-26-2': 'Data were reduced using our own pipeline (V11).', '1403.5466-1-27-0': 'The confirmation of one galaxy showing Ly[MATH] emission consistent with a redshift of 6.74 was presented by Bradac et al. (2012).', '1403.5466-1-27-1': 'Here we also consider the observations and limits, in terms of Ly[MATH] line detection of the rest of the sample.', '1403.5466-1-27-2': 'In Table 1 we report the resulting limits on the Ly[MATH] EW for each galaxy, assuming again a median redshift of 6.9.', '1403.5466-1-27-3': 'However note that in this case the median expected redshift of the sample is [MATH]: the redshift probability distribution function of these candidates is much larger and extends well beyond z=8, differing considerably from the other samples presented here (for reference see Figure 5 in Hall et al. 2012).', '1403.5466-1-27-4': 'This is due to the fact that the candidates were selected by applying criteria based on a [MATH] color (due to the nature of the HST data available).', '1403.5466-1-27-5': 'In the following section we will consider, whenever necessary, the appropriate selection function of this sample.', '1403.5466-1-28-0': '# Simulations', '1403.5466-1-29-0': 'To determine the EW limit achieved by our observations for each of our targets, and reported in Table 1, we performed detailed 2D simulations that we briefly describe here.', '1403.5466-1-29-1': 'We take real individual MXU raw frames, corresponding to slits where no targets were detected and insert an emission line of a given flux, at a given wavelength and at a given spatial position corresponding to the middle of the slit.', '1403.5466-1-29-2': 'The emission line is modeled as a Gaussian that is then truncated to half to simulate the typically asymmetric emission lines that are routinely observed at lower redshift, and then further convolved with the seeing (with values varying from 0.6 to 1.2 [MATH] as in the real observations).', '1403.5466-1-29-3': 'The individual frames are then processed as normally done during the reduction procedure: after standard flat-fielding, we remove the sky emission lines by subtracting the sky background between two consecutive exposures, exploiting the fact that the target spectrum is offset due to dithering (ABBA technique).', '1403.5466-1-29-4': 'Then spectra have been wavelength calibrated (using lamp exposures) and finally, they are flux-calibrated using the observations of spectrophotometric standards and combined.', '1403.5466-1-30-0': 'The resulting 2D frame is then scanned with a window of [MATH] pixel to see if there is a detection, and in this case the signal to noise ratio of the line is registered.', '1403.5466-1-30-1': 'The simulations are repeated after shifting the Ly[MATH] emission line along the dispersion axis in steps of [MATH], and in the end we cover the redshift range from z=5.68 to z=7.30.', '1403.5466-1-30-2': 'At each redshift step, the scan is repeated and the S/N of the line is registered again.', '1403.5466-1-31-0': 'The emission lines are then varied in terms of total flux (from 0.24 to [MATH]), and full width half maximum (varying from 230 to 520 [MATH]).', '1403.5466-1-31-1': 'These values are in the range of the real observed Ly[MATH] lines.', '1403.5466-1-31-2': 'The entire procedure is repeated for each combination of line flux and FWHM.', '1403.5466-1-31-3': 'The simulations were performed for more than one slit in order to cover the entire CCD (top and bottom chips) of the FORS2 MXU frame and the results are always the same to within 5%.', '1403.5466-1-31-4': 'Although our candidates were placed at the center of the slits in most cases, we also checked for possible differences by placing the initial emission line at various positions along individual slits, i.e. we shifted the Ly[MATH] up and down by few pixels.', '1403.5466-1-31-5': 'Again no significant difference is found.', '1403.5466-1-31-6': 'In Figure 3 we show one of the results of these tests, with the three colored curves (red, green and black) representing the resulting S/N of a line with flux 1.6 [MATH], positioned in three different slits: for each slit the result is the average of 3 different positions along the spatial axis (0,+5 and -5 pixel).', '1403.5466-1-31-7': 'It is evident that the differences in the resulting S/N between slits are only marginal.', '1403.5466-1-31-8': 'In the same Figure we also show the skyline emission for reference.', '1403.5466-1-32-0': '# The declining fraction of Ly[MATH] emitters: new limits and discussion', '1403.5466-1-33-0': '## The fraction of Ly[MATH] emitters at z[MATH]', '1403.5466-1-34-0': 'With our new sample we can evaluate with greater accuracy the fraction of Ly[MATH] emission in LBGs at redshift 7, the decline between [MATH] and [MATH] and its implication.', '1403.5466-1-34-1': 'In Table 2 we report the fraction of galaxies having and EW [MATH] and [MATH] separately for the two absolute magnitude bins that were adopted by previous works (Stark et al. 2010, P11, Ono et al. 2012).', '1403.5466-1-34-2': 'In the bright bin (galaxies with magnitudes [MATH]) there are 39 galaxies of which 7 are detected in Ly[MATH]: 5 of these have [MATH] 2 have [MATH] and none has [MATH].', '1403.5466-1-34-3': 'In the faint bin (galaxies with [MATH]) there are 25 objects, of which 5 have a Ly[MATH] emission with [MATH] and 2 with [MATH].', '1403.5466-1-34-4': 'Note that 3 of the targets in the Bradac et al. sample are intrinsically fainter than [MATH] thus are excluded from this bin.', '1403.5466-1-34-5': 'In the Table we report the fractions taking into account the fact that the limit in the EW detectable for the galaxies is not always below 25 ; for example for some of the objects in Ono et al. (2012) the limits achieved are above this value.', '1403.5466-1-34-6': 'In calculating the fractions we also consider that for some galaxies the redshift probability distribution extends well beyond z[MATH], which is approximately the limit out to which we can detect the Ly[MATH] emission in our current observations.', '1403.5466-1-34-7': 'In particular as already stated above, the sample observed by Bradac et al. (2012) was selected in such a way that the probability of galaxies being at [MATH] is quite high, [MATH]% (see Figure 5 in Hall et al. 2012).', '1403.5466-1-34-8': 'This is due to the broad J-band filter ([MATH]) that was available for the selection.', '1403.5466-1-34-9': 'Therefore we weighted each sample by evaluating the total probability of galaxies being outside the redshift range that is observable by the spectroscopic setup.', '1403.5466-1-34-10': 'In practice for most of the samples this probability is negligible (see Figure 6 in Ouchi et al. 2010 for the Ono et al. sample, Figure 7 in Castellano et al 2010a for the NTT,GOODS-South and BDF samples), while it is non negligible for the UDS sample (which has a tail to z[MATH]8 , see Figure 1 in Grazian at al. 2012) and quite high for the Bradac et al. sample.', '1403.5466-1-35-0': 'We also report the fractions after assuming that 20% of the undetected objects are lower redshift interlopers: this value (20%) is the upper limit for possible interlopers found in a large sample of z[MATH]6 galaxies in our previous work (P11), and we assume that there is no significant change between the two epochs.', '1403.5466-1-35-1': 'Note that none of our galaxies has a detected Ly[MATH] emission with EW larger than 75 : to calculate the upper limit for the fraction, we assume the statistics for small numbers of events by Gehrels (1986).', '1403.5466-1-36-0': 'Comparing the above results to those at [MATH] presented by Stark et al. (2010), it is clear that there is a very significant deficit of Ly[MATH] emission at z[MATH] compared to earlier epochs.', '1403.5466-1-36-1': 'In the following sections we will try to interpret this deficit, first within the context of large-scale seminumeric simulations of reionization that includes the reionization field as well as galactic properties (Dijkstra et al. 2011).', '1403.5466-1-36-2': 'We will then apply to our data a simple phenomenological model developed by Treu et al. (2012) that usese the evolution of the distribution of Ly[MATH] equivalent widths to make some simple predictions about the complex topology of reionization.', '1403.5466-1-37-0': '## The neutral hydrogen fraction', '1403.5466-1-38-0': 'In P11 we interpreted the drop in the Ly[MATH] fraction in LBGs as due most probably to the sudden increase of neutral hydrogen in the universe between z[MATH]6 and z[MATH] (see also Schenker et al. 2012).', '1403.5466-1-38-1': 'We then compared the results to the predictions of Dijkstra et al. (2011), to determine what fraction of neutral hydrogen would be needed to explain the drop, provided that all other physical parameters (e.g. dust content, the escape fraction of Lyman continuum photons etc) would not change between z=6 and z=7.', '1403.5466-1-38-2': 'We obtained a rather high neutral hydrogen fraction by volume, [MATH].', '1403.5466-1-39-0': 'We now make use of improved models to compare our new results.', '1403.5466-1-39-1': 'As in Dijsktra et al. (2011), reionization morphologies were generated using the public code, DexM (Mesinger Furlanetto 2007).', '1403.5466-1-39-2': 'The box size is 200 Mpc, and the ionization field is computed on a [MATH] grid.', '1403.5466-1-39-3': 'Reionization morphologies at a given [MATH] are generated by varying the ionization efficiency of halos, down to a minimum halo mass of [MATH], roughly corresponding to the average minimum mass of halos at z=7 which retain enough gas to form stars efficiently (Sobacchi Mesinger 2013).', '1403.5466-1-39-4': 'Compared to P11 the model now includes also more massive halos, with stellar masses up to [MATH].', '1403.5466-1-39-5': 'This is because the previous model was tailored to analyze the nature of fainter dropout galaxies (Dijkstra et at.', '1403.5466-1-39-6': '2011) compared to those presented in this work.', '1403.5466-1-39-7': 'The results however change only minimally with the new choice of halo mass, as expected given that the halo bias (and the associated opacity distribution for a given [MATH]) does not evolve much over this mass range (e.g. Mesinger Furlanetto 2008, McQuinn et al. 2008).', '1403.5466-1-39-8': 'In Figure 4 we present the comparison of the outcome of the new model with the present fractions.', '1403.5466-1-39-9': 'It is clear that only a very high neutral hydrogen fraction ([MATH]) can reproduce the lack of Ly[MATH] emission at [MATH] compared to earlier epochs.', '1403.5466-1-39-10': 'This high value seems at odds with other observational results: for instance, Raskutti et al. (2012) study the IGM temperature in quasar near-zones and find that reionization must have been completed by [MATH] at high confidence, while several teams (Hu et al. (2010), Kashikawa et al. (2011) and Ouchi et al. 2010) study the Ly[MATH] line shapes of LAEs at z = 6.5 finding no evidence of damping wings.', '1403.5466-1-40-0': 'As recently pointed out by Taylor Lidz (2014), before reionization completes, the simulated Ly[MATH] fraction might have large spatial fluctuations depending on the degree of homogeneity/inhomogeneity of the reionization process.', '1403.5466-1-40-1': 'Since existing measurements of the Ly[MATH] fraction span relatively small regions on the sky, and sample these regions only sparsely (typically only a few dropouts are observed per field), they might by chance probe mostly galaxies with above than average Ly[MATH] attenuation and therefore point to higher neutral hydrogen fractions compared to the average values.', '1403.5466-1-40-2': 'It is therefore important to include the effect of cosmic variance for different sight-lines within our survey.', '1403.5466-1-40-3': 'In their work Taylor Lidz found that the sample variance is non negligible for existing surveys, and it does somewhat mitigate the required neutral fraction at z[MATH] 7.', '1403.5466-1-41-0': 'Compared to previous studies and to the surveys analyzed by Taylor Lidz, this work presents more independent fields of view: even considering as single pointings those in adjacent areas (e.g. the GOODS-South/ERS areas and the 2 SDF pointings in Ono et al. 2012), we are now sampling 8 independent lines of sight, with areas varying between [MATH] in each field.', '1403.5466-1-41-1': 'We have verified what is the uncertainty in our results that might be derived from the cosmic variance.', '1403.5466-1-41-2': 'We made a computation of the field (cosmic) variance in the opacity ([MATH] ) from reionization.', '1403.5466-1-41-3': 'The computation just uses the opacity at a single wavelength, roughly 200 [MATH] red-ward of the line center, where most of the intrinsic emission is expected to lie.', '1403.5466-1-41-4': 'With 8 pointings, the average pointing to pointing standard deviation [MATH] is of the order of 6%.', '1403.5466-1-41-5': 'We also varied the number of candidate galaxies probing the reionization per each pointing.', '1403.5466-1-41-6': 'Indeed the discreteness in sampling the reionization morphology becomes an issue especially for the largest neutral fractions since in this case the fewer number of would-be Ly[MATH] emitters can miss the relatively rare regions in a given pointing which have a high transmission.', '1403.5466-1-41-7': 'However even considering a small sampling (5 emitters per pointing) [MATH] is still around 10%.', '1403.5466-1-42-0': 'Therefore we are quite confident that for our sample, with a large number of independent field and a reasonable number of candidates observed per pointing, the field to field fluctuations are not very large and would not effect sensibly the results of Figure 4.', '1403.5466-1-43-0': '## A patchy reionization process?', '1403.5466-1-44-0': 'Applying the simple phenomenological models developed by Treu et al. (2012) to describe the evolution of the distribution of Ly[MATH] equivalent widths we can now use the Ly[MATH] detections and non-detections to make some inferences about the complex topology of reionization.', '1403.5466-1-44-1': 'This model starts from the intrinsic rest-frame distribution in terms of the one measured at [MATH] by Stark et al. (2011).', '1403.5466-1-44-2': 'It then considers two extreme cases which should bracket the range of possible scenarios for the reionization morphology: in the first (""patchy"") model, no Ly[MATH] is received from a fraction [MATH] of the sources, while the rest is unaffected.', '1403.5466-1-44-3': 'In the second (""smooth"") model, the Ly[MATH] emission is attenuated in every galaxy in the same way, by a factor [MATH].', '1403.5466-1-44-4': 'These two models can be thought respectively as simple idealizations of smooth and patchy reionization: although very simple and somewhat unphysical (especially the smooth one) these two models should bracket the expected behavior of the IGM near the epoch of reionization (see Treu et al. 2012, Treu et al. 2013 for a more detailed explanation).', '1403.5466-1-45-0': ""For each object in our sample, Bayes's rule gives the posterior probability of [MATH] and [MATH] (which we collectively indicate as [MATH]) and redshift given the observed spectrum and the continuum magnitude m."", '1403.5466-1-45-1': 'The likelihood is as usual the probability of obtaining the data for any given value of the parameter.', '1403.5466-1-45-2': 'The model adopts a uniform prior p([MATH]) between zero and unity, while the prior for the redshift [MATH] is obtained from the redshift probability distribution (as described in Section 3 for each of the different parent samples).', '1403.5466-1-45-3': 'We use the implementation of the method that takes as input the line equivalent widths or equivalent width limits, in order to incorporate all the available information, even when a noise spectrum is not available (Treu et al. 2012).', '1403.5466-1-46-0': 'One of the output of the model is the normalization constant [MATH], known as the Bayesian evidence and quantifies how well each of the two models matches the data.', '1403.5466-1-46-1': 'The evidence ratio is a powerful way to perform model selection e.g., comparing the patchy and smooth models and eventually discriminate between the two.', '1403.5466-1-47-0': 'Treu et al. 2012 applied their model to the sample presented by Ono et al. (2012) which included also data from P11 and V11.', '1403.5466-1-47-1': 'The data clearly preferred an attenuation factor [MATH] (0.65-0.68), independent of the model considered, but the evidence ratio indicated no significant preference for any of the two models.', '1403.5466-1-48-0': 'We have repeated the exercise for our new enlarged sample, which is almost double compared to the previous one and most importantly contains a larger fraction of very faint galaxies (thanks e.g. to the inclusion of the lensed galaxies).', '1403.5466-1-48-1': 'With the new sample we obtain [MATH] and [MATH] respectively for the patchy and smooth model, as shown in Figure [REF]: this means that both models require a considerable quenching of the Ly[MATH] compared to z[MATH], as expected from the many non-detections.', '1403.5466-1-48-2': 'Note that these results assume that the level of contamination in the samples, is the same at [MATH] and at [MATH].', '1403.5466-1-48-3': 'We can interpret the [MATH] and [MATH] as the average excess optical depth of Ly[MATH] with respect to z [MATH]6, i.e., [MATH], although a conversion from this to a neutral hydrogen fraction requires detailed and uncertain modeling (e.g. Santos 2004).', '1403.5466-1-48-4': 'A key result is that with the new sample the evidence ratio between the two models is quite high, [MATH] which means that the patchy model is highly favored (26 times more) by the data over the smooth one.', '1403.5466-1-48-5': 'This is also suggested by the likelihood ratio test.', '1403.5466-1-48-6': '[MATH] strongly favors the patchy model (in other terms the ratio corresponds to a difference in [MATH] between the two models of [MATH]).', '1403.5466-1-48-7': 'As expected the power to discriminate between the two models is given by the inclusion of fainter galaxies, as well as the fact that for many galaxies we have very deep EW limits.', '1403.5466-1-48-8': 'As a result of the inference, the model also allows us to calculate the fraction of emitters using all the available information.', '1403.5466-1-48-9': 'For objects brighter than [MATH] the model predicts respectively [MATH] for galaxies with [MATH] and [MATH] for galaxies with [MATH] .', '1403.5466-1-48-10': 'For fainter galaxies, the predictions are [MATH] and [MATH] respectively.', '1403.5466-1-48-11': 'These values are very close to the numbers reported in Table 2 (considering obviously the fractions derived assuming no interlopers in the sample).', '1403.5466-1-49-0': 'While our observational results indicate clearly that the distribution of neutral hydrogen in these phases of reionization was highly inhomogeneous, as expected by most theoretical predictions (e.g. Iliev et al. 2006), to fully constrain the morphology of reionization we will have to wait for the direct observations of the 21 cm emission from neutral hydrogen in the high redshift universe, which is one of the prime tasks of the upcoming LOFAR surveys observations (e.g. Jensen et al. 2013).', '1403.5466-1-50-0': '# Summary and concluding remarks', '1403.5466-1-51-0': 'In this paper we have presented new results from our search for z[MATH] galaxies from deep spectroscopic observations of candidate z-dropouts in the CANDELS fields.', '1403.5466-1-51-1': 'Even though our sensitive VLT observations reached extremely low flux limits, only two galaxies have new robust redshift identifications, one from the Ly[MATH] emission line at z=6.65 and the other from its Lyman-alpha break, i.e. the continuum discontinuity at the Ly[MATH] wavelength consistent with a redshift [MATH]6.42.', '1403.5466-1-51-2': 'In this second object no emission line is observed.', '1403.5466-1-51-3': 'In addition for [MATH] galaxies we present new deep limits on the Ly[MATH] EW derived from the non detections in ultra-deep observations (from 15 to 27 hours) obtained with FORS2 spectrograph on the VLT.', '1403.5466-1-51-4': 'Using this new data as well as previously published samples, we have assembled a total of 68 candidate z[MATH] galaxies with deep spectroscopic observations, of which 12 have a redshift identification from the Ly[MATH] emission line.', '1403.5466-1-51-5': 'With this much enlarged sample we have placed solid constraints on the fraction of Ly[MATH] emission in z[MATH]7 Lyman break galaxies both for bright and faint galaxies, confirming the large decline in the presence of Ly[MATH] emission from z[MATH] to [MATH].', '1403.5466-1-51-6': 'If this decline is only due to the evolution of the IGM, and assuming that all other galaxies properties remain unchanged in this redshift interval, a very large fraction ([MATH]) of neutral hydrogen is needed to explain the observations.', '1403.5466-1-51-7': 'Finally applying the simple phenomenological model developed by Treu et al. (2012), we show that the present data favor a patchy reionization process rather than a smooth one, as expected from most simulations.', '1403.5466-1-52-0': 'Obviously we cannot rule out that an evolution of other properties, namely [MATH] and dust, come into play and contribute to the Ly[MATH] quenching.', '1403.5466-1-52-1': 'Indeed, in a recent paper (Dijkstra et al. 2014) we discuss the possibility that the decline in strong Ly[MATH] emission from [MATH] galaxies is due, in part, also to an increase of the Lyman continuum escape fraction in star forming galaxies.', '1403.5466-1-52-2': 'In particular assuming that the escape fraction evolves with redshift as [MATH] (as in Kuhlen Faucher-Giguere 2012), and taking [MATH] and [MATH] such that we have [MATH] at z=6 and [MATH] at z=7, the observed decline in Ly[MATH] emission could be reproduced with a more modest evolution in the global neutral fraction, of the order of [MATH].', '1403.5466-1-52-3': 'This work is clearly rather speculative, since [MATH] is a very elusive quantity to measure and we only have tentative indications on its value from upper limits (e.g. Nestor et al. 2013, Vanzella et al. 2012, Boutsia et al. 2011) and on its evolution (e.g. Cowie et al. 2009, Siana et al. 2010).', '1403.5466-1-52-4': 'However it shows that an evolving escape fraction of ionizing photons should be considered as part of the explanation for evolution in the Ly[MATH] emission of high redshift galaxies in addition to the evolution of the IGM (see Dijkstra et al. 2014 for more details).', '1403.5466-1-52-5': 'On a similar line, another recent work (Hutter et al. 2014) attempts to jointly constrain three fundamental parameters that determine the visibility of the Ly[MATH] line in high redshift galaxies, namely [MATH], [MATH] and and the ISM dust distribution This work shows that if dust is clumped in the interstellar medium of high-redshift galaxies, we can not differentiate between a Universe which is either completely ionized or half neutral, or has an [MATH] ranging between [MATH] %.', '1403.5466-1-53-0': 'It is clear that to fully characterize and understand the reionization epoch, and to clarify the relation between the disappearing Ly[MATH] emission line and cosmic reionization we still have to make a substantial effort, both on the observational and on the modelling side.', '1403.5466-1-53-1': 'Even if the current samples of candidate galaxies at [MATH] are quite large, and despite all the observational efforts by several teams, the number of spectroscopically confirmed objects remains very small.', '1403.5466-1-53-2': 'To overcome this limitation and substantially increase the statistics we are currently carrying out an ESO Large Program with FORS2@VLT (PI Pentericci) that in the end should allow us to increase considerably the number of confirmed high redshift galaxies by observing [MATH] candidates.', '1403.5466-1-53-3': 'In particular, since the targets will be selected from the CANDELS field with extremely deep near-IR observations, we will include also galaxies as faint as [MATH].', '1403.5466-1-53-4': ""The new deep spectroscopic observations will allow us to assess the continuous evolution of the Ly[MATH] emission over the range [MATH], and from a comparison to state-of-the art models, we will be able to determine if the Ly[MATH] was mainly quenched by the neutral IGM, or if any evolution of the galaxies' physical properties also played a significant role."", '1403.5466-1-54-0': 'Part of this work has been funded through INAF Grants (PRIN 2010 and 2012)'}","{'1403.5466-2-0-0': 'We present new results from our search for [MATH] galaxies from deep spectroscopic observations of candidate z-dropouts in the CANDELS fields.', '1403.5466-2-0-1': 'Despite the extremely low flux limits achieved by our sensitive observations, only 2 galaxies have robust redshift identifications, one from its Ly[MATH] emission line at z=6.65, the other from its Lyman-break, i.e. the continuum discontinuity at the Ly[MATH] wavelength consistent with a redshift 6.42, but with no emission line.', '1403.5466-2-0-2': 'In addition, for [MATH] galaxies we present deep limits in the Ly[MATH] EW derived from the non detections in ultra-deep observations.', '1403.5466-2-1-0': 'Using this new data as well as previous samples, we assemble a total of 68 candidate z[MATH] galaxies with deep spectroscopic observations, of which 12 have a line detection.', '1403.5466-2-1-1': 'With this much enlarged sample we can place solid constraints on the declining fraction of Ly[MATH] emission in z[MATH]7 Lyman break galaxies compared to z[MATH], both for bright and faint galaxies.', '1403.5466-2-1-2': 'Applying a simple analytical model, we show that the present data favor a patchy reionization process rather than a smooth one.', '1403.5466-2-2-0': '# Introduction', '1403.5466-2-3-0': 'The use of Ly[MATH] transmission by the intergalactic medium (IGM) as a probe of its ionization state during the reionization epoch has been proposed already many years ago (Miralda-Escude Rees 1998; Santos et al. 2004).', '1403.5466-2-3-1': 'Strong Ly[MATH] emission, powered by star formation is present is many distant galaxies: being a resonant line, it is sensitive to even small quantities of neutral hydrogen in the IGM, and it is easily suppressed .', '1403.5466-2-3-2': 'We thus expect the observed properties of Ly[MATH] emitting galaxies to change at higher redshifts, when the IGM becomes more neutral.', '1403.5466-2-3-3': 'A very common approach for studying the reionization history of the Universe using Ly[MATH] emitting galaxies is to determine the evolution of the Ly[MATH] luminosity function and the clustering properties of narrow band selected Ly[MATH] emitters (LAEs e.g. Ota et al. 2008, Ouchi et a. 2010; Kashikawa et al. 2011, Clements et al. 2012, Faisst et al. 2014).', '1403.5466-2-3-4': 'A recent complementary approach and the one used in this paper, is instead to measure the redshift evolution of the Ly[MATH] fraction in Lyman break galaxies (LBGs), i.e. the percentage of LBGs that have an appreciable Ly[MATH] emission line (e.g. Stark et al. 2010).', '1403.5466-2-3-5': 'Indeed this fraction is supposed to increase, as we move to higher redshift since galaxies are increasingly young (hence with stronger intrinsic Ly[MATH]) and almost dust free (Finkelstein et al. 2012) which facilitates the escape of Ly[MATH] photons.', '1403.5466-2-3-6': 'On the other hand this fraction is expected to fall-off as we approach the time when the intergalactic medium becomes significantly neutral and the galaxies’ Ly[MATH] emission is progressively attenuated.', '1403.5466-2-3-7': 'Compared to other probes of reionization such as the evolution of the LAE luminosity function this approach can overcome concerns about intrinsic density evolution of the underlying population (Stark et al. 2010).', '1403.5466-2-4-0': 'Intriguingly, early measurements with this technique suggest a strong drop in the Ly[MATH] fraction near z[MATH]7, more significant for relatively fainter galaxies.', '1403.5466-2-4-1': 'In particular in a series of recent works, a lack of Ly[MATH] emission was found at [MATH] compared to [MATH] by several independent teams: in our previous observations (Pentericci et al. 2011; Vanzella et al. 2011; Fontana et al. 2010; P11, V11 and F10 from here on) we found 4 Ly[MATH] emitting galaxies (plus another object with a tentative line detection) out of a sample of 20 robust candidates.', '1403.5466-2-4-2': 'Similar or lower fractions were found by [CITATION],[CITATION], Bradac et al. (2012), Ono et al. (2012) although considerable field-to-field variations are present due to the small number of candidates observed in each sample (see for example Figure 8 in Ono et al. 2012).', '1403.5466-2-4-3': 'In our favored interpretation, the lack of line emission is due to a substantial increase in the neutral hydrogen content of the Universe in the time between [MATH] and [MATH].', '1403.5466-2-4-4': ""Comparing our data to the predictions of the semi-analytical models by Dijkstra et al. (2011) we concluded that to explain the observations a substantial change of the neutral hydrogen fraction of the order of [MATH] in a time [MATH] was required, assuming that the galaxies' physical properties remain constant during this time."", '1403.5466-2-4-5': 'Recent observations pushing to [MATH] are consistent with this interpretation (Treu et al. 2013; Schmidt et al. 2014).', '1403.5466-2-5-0': 'However other factors could also play a role in the Ly[MATH] quenching: in particular we cannot rule out the possibility that a change in some of the intrinsic galaxy properties (the Lyman continuum escape fraction, wind properties and dust content) could at least partially contribute to the lack of Ly[MATH] emission.', '1403.5466-2-5-1': 'Indeed the interpretation of the results as only due to the change in the neutral hydrogen fraction was questioned by several successive works (e.g. Jensen et al. 2013, Forero-Romeiro et al. 2012, Bolton Haehnelt 2013, Taylor Lidz 2014, Dijkstra et al. 2014).', '1403.5466-2-5-2': 'In particular Bolton Haehnelt (2013) suggested that the opacity of the intervening IGM red-ward of rest-frame Ly[MATH] can rise rapidly in average regions of the Universe simply because of the increasing incidence of absorption systems, which are optically thick to Lyman continuum photons.', '1403.5466-2-5-3': 'They claimed that the data do not require a large change in the IGM neutral fraction from z[MATH]6 to [MATH]7.', '1403.5466-2-5-4': 'However such a rapid evolution of the photo-ionizing background could be very difficult to achieve without requiring either a late reionization, or an emissivity at [MATH] which is too high to be consistent with observations of the Ly[MATH] forest (e.g. Sobacchi Mesinger 2014).', '1403.5466-2-5-5': 'Preliminary estimates suggest that the neutral fraction constraint relaxes only mildly when taking into account the absorption systems (Mesinger et al. in preparation).', '1403.5466-2-6-0': 'We also mention the very recent work by Taylor Lidz (2014), pointing out that sample variance is not negligible for existing surveys: considering the large spatial fluctuations of the medium owing to an inhomogeneous reionization, the required neutral fraction at z[MATH]7 can somehow be reduced to less extreme values.', '1403.5466-2-6-1': 'Indeed the observational results are presently based on small data-sets, with considerable field to field variations (Pentericci et al. 2011), and mostly focusing on the brightest candidates ([MATH]).', '1403.5466-2-7-0': 'The complex topology of reionization is also a highly debated matter.', '1403.5466-2-7-1': 'Depending on the nature of the main sources of reionization, it is expected that the characteristic scale of the reionization process might change substantially (Iliev et al. 2006; Furlanetto et al. 2006).', '1403.5466-2-7-2': 'Accurate theoretical predictions for the morphology and sizes of H II regions depend on the abundance and clustering of the ionizing sources themselves, in addition to the underlying inhomogeneous density field and clumpiness of the gas in the IGM (McQuinn et al. 2007, Sobacchi Mesinger 2014).', '1403.5466-2-7-3': 'Observations of Ly[MATH] emitting galaxies and their clustering have the potential to reveal the signature of patchy reionization, although early results have been inconclusive (Kashikawa et al. 2011, Ouchi et al. 2010)', '1403.5466-2-8-0': 'In this paper we present new observations of z[MATH] candidates, significantly increasing the statistics of previous works (especially in the faint regime, thanks to the inclusion of lensed candidates), which will allow us to assess the emergence of Ly[MATH] emission at high redshift with greater accuracy and address some of the above issues.', '1403.5466-2-8-1': 'In Section 2 we present the new observations and the previous data available; in Section 3 we describe the simulations used to accurately evaluate the sensitivity of our spectroscopic observations.', '1403.5466-2-8-2': 'In Section 4 we first evaluate the new limits on the Ly[MATH] fractions at high redshift, and we derive the neutral hydrogen fraction that is needed to explain the observed decrease; then applying a simple phenomenological model we derive new constraints on the topology of reionization.', '1403.5466-2-8-3': 'In Section 5 we summarize our findings.', '1403.5466-2-9-0': 'All magnitudes are in the AB system, and we adopt [MATH] km/s/Mpc, [MATH] and [MATH].', '1403.5466-2-10-0': '# Observations', '1403.5466-2-11-0': 'In this section we summarize the new observations presented in this work as well as previous data that we will use in this paper.', '1403.5466-2-12-0': '## UDS field', '1403.5466-2-13-0': 'We selected candidate z[MATH]7 galaxies in the UDS field from CANDELS multi-wavelength observations (Galametz et al. 2013).', '1403.5466-2-13-1': 'Objects were detected in the J band and then the color selection criteria presented by Grazian et al. (2012).', '1403.5466-2-13-2': 'were applied Observations were taken in service mode with the FORS2 spectrograph on the ESO Very Large Telescope.', '1403.5466-2-13-3': 'We used the 600Z holographic grating, that provides the highest sensitivity in the range [MATH] with a spectral resolution [MATH] and a sampling of 1.6 per pixel for a 1[MATH] slit.', '1403.5466-2-13-4': 'Out of the entire sample of z-dropout candidates (which consists of 50 galaxies), we placed a total of 12 galaxies in the slits (the selection was just driven by the geometry of the mask).', '1403.5466-2-13-5': 'The rest of the mask was filled with i-dropouts (Vanzella et al. in preparation) and other targets such as massive high redshift galaxies and AGN candidates.', '1403.5466-2-13-6': 'The sources have been observed through slitlets 1[MATH] wide by 10-12[MATH] long.', '1403.5466-2-13-7': 'The observation strategy was identical to the one adopted in P11 and previous papers: series of spectra were taken at two different positions, offset by [MATH] (16 pixels) in the direction perpendicular to the dispersion.', '1403.5466-2-13-8': 'The total net integration time was 15.5 hours for each object.', '1403.5466-2-13-9': 'Data were reduced using our dedicated pipeline which was described in detail in F10 and V11.', '1403.5466-2-13-10': 'Here we only mention that our pipeline performs the sky-subtraction as typically done for the near-IR, subtracting the sky background between two consecutive exposures, exploiting the fact that the target spectrum is offset due to dithering in the classic ABBA pattern.', '1403.5466-2-13-11': 'Our algorithm implements a “A-B” sky subtraction joined with a zero (e.g., median) or first order fit of the sky along columns that regularized possible local differences in the sky counts among the partial frames before they are combined.', '1403.5466-2-13-12': 'We find that this procedure ensures the best final results when searching for faint emission lines, especially in the reddest part of the spectra where many strong skylines are present.', '1403.5466-2-13-13': 'The two dimensional sky-subtracted partial frames are also combined (in the pixel domain) to produce the weighted RMS map, associated to the final reduced spectrum.', '1403.5466-2-13-14': 'This allows us to calculate the two dimensional signal to noise spectra, useful to access the reliability of the spectral features Finally we also take extra care in the alignment of the different frames before the combination.', '1403.5466-2-14-0': 'For one of the candidates with a relatively bright continuum magnitude (J=25.98), UDS29249, we detect a faint continuum emission in the red part of the spectrum beyond [MATH]9100 as shown in Figure 1.', '1403.5466-2-14-1': 'The total integrated S/N of the flux is [MATH]; while the detection is relatively secure in the wavelength range 9160-9240 , the exact position of the break (that is ascribed to the IGM) is difficult to locate both because of the faintness of the emission and also because of the residuals of the bright sky emission lines in the region immediately below 9100.', '1403.5466-2-14-2': 'We conservatively estimate a redshift that ranges between 6.31 and 6.53 (in the table we report [MATH]).', '1403.5466-2-14-3': 'Note that this is perfectly consistent with the photometric redshift distribution obtained from the CANDELS photometry, also shown in Figure 1.', '1403.5466-2-14-4': 'Recently Wilkins et al. (2014) discussed the possibility that a newly identified Y-dwarf population, as well as the late T-dwarfs stars might contaminate the photometric selection and spectroscopic follow-up of faint and distant galaxies (see also Bowler et al. 2014).', '1403.5466-2-14-5': 'Our target appears very compact but still resolved in the HST J-band (as the majority of the [MATH] candidates); in addition its colors are not consistent with those of Y and T dwarf.', '1403.5466-2-14-6': 'If we place the target in the [MATH] vs [MATH] plot, as in Figure 3 of Wilkins et al., the object is almost coincident with the high-z star-forming galaxy track, and very distant from the position of both the L- and T-dwarf spectral standards as well as the tracks of the model Y-dwarfs.', '1403.5466-2-14-7': 'Thus this gives us extra confidence that this is a true high redshift galaxy without Ly[MATH] emission in its spectrum.', '1403.5466-2-15-0': 'All other candidates are undetected (meaning that no feature is detected).', '1403.5466-2-15-1': 'In Table 1 we report the candidates, RA and Dec, their J band magnitudes and the limiting EW.', '1403.5466-2-15-2': 'For the undetected objects we assume a redshift of 6.9, which is the median redshift of the selection function (see Grazian et al. 2012).', '1403.5466-2-16-0': '## ESO Archive', '1403.5466-2-17-0': 'We searched the ESO archive for observations of high redshift objects; in particular we retrieved the data from the observations carried out within the program ESO 088.A-1013 (PI Bunker).', '1403.5466-2-17-1': 'This program used the same observational setup used above, with a total net integration time of 27 hours.', '1403.5466-2-17-2': 'It observed a mixture of z and i-dropouts.', '1403.5466-2-17-3': 'Recently the results were presented by [CITATION] and the authors report all non detections for the candidate z[MATH] galaxies.', '1403.5466-2-18-0': 'Caruana et al. have observed candidate high redshift galaxies selected in previous works (Bouwens et al. 2011, McLure et al. 2010 and Wilkins et al. 2011) which used each a different selection criteria, from different color-color cuts to photometric redshifts.', '1403.5466-2-18-1': 'Since we want to work on a sample that is as homogeneously selected as possible, we have selected our own list of z-dropouts again using the color criteria presented by Grazian et al. (2012).', '1403.5466-2-18-2': 'We then cross correlated our list with the targets in [CITATION].', '1403.5466-2-18-3': 'We found 9 matching objects, of which 5 are in common with the sample already observed in F10.', '1403.5466-2-18-4': 'We then retrieved the raw (public) data from the ESO archive and then processed through our own pipeline (V11) as all other data in this work.', '1403.5466-2-18-5': 'Here we present the results for the 4 new targets that are not in common with F10 (see Table 1).', '1403.5466-2-18-6': 'Further results in particular the extremely deep combined spectra of the objects in common between the Caruana et al. program and F10 (52 hours) will be presented elsewhere (Vanzella et al. in preparation).', '1403.5466-2-19-0': 'The results are again presented in Table 1.', '1403.5466-2-19-1': 'We detect a significant emission line in one of the 4 new objects, galaxy 34271 in the GOODS-South field corresponding to galaxy ERSz-2225141173 in [CITATION].', '1403.5466-2-19-2': 'The line is detected at 9301 and shows the typical asymmetry of Ly[MATH] which would place the object at redshift [MATH].', '1403.5466-2-19-3': 'In Figure 2 we show the 1-dimensional and 2-dimensional spectra of the galaxy.', '1403.5466-2-19-4': 'The EW of the line is 43 , calculated from its measured line flux and the Y band magnitude from the GOODS CANDELS catalog.', '1403.5466-2-20-0': 'As we mentioned above, [CITATION] report no emission line for this galaxy, but a median EW limit of 28.5.', '1403.5466-2-20-1': 'We ascribe the difference to the fact that for most of the data reduction steps the authors used the standard ESO pipeline, while we use our own pipeline that has been tailored specifically to the detection of faint high redshift emission lines (see the description above).', '1403.5466-2-20-2': 'The other 3 targets are undetected and we can set stringent limits of their Ly[MATH] EW limit ranging from [MATH] to [MATH] depending on the continuum magnitude and assuming that they are placed at redshift 6.9.', '1403.5466-2-20-3': 'Obviously the actual EW limit depends sensibly on the exact redshift of the objects (see F10 Figure 1).', '1403.5466-2-21-0': '## Data from previous literature', '1403.5466-2-22-0': 'Besides the new data, we include all previously published spectroscopic data on z-dropouts, in order to assemble the largest possible sample of candidate z[MATH] galaxies with deep spectroscopic observations.', '1403.5466-2-23-0': 'In particular we consider: (1) the 20 z-dropouts selected in the GOODS-South, NTTDF and BDF4 fields (Castellano et al. 2010a, 2010b) whose observations were carried out by our groups in P11, and previously presented by V11 and F10.', '1403.5466-2-23-1': 'Of these, 4 show a convincing Ly[MATH] emission line, while the tentative detection of a fifth candidate originally shown in F10 was not confirmed by the combination of our own data with the deeper observations of [CITATION] (Vanzella et al. in preparation).', '1403.5466-2-23-2': '(2) The 11 bright z-dropout observed by Ono et al. (2012) in the SDF field of which 3 have bright Ly[MATH] in emission.', '1403.5466-2-23-3': 'These candidates were detected in deep Y band observations and selected using color criteria that are very similar to ours.', '1403.5466-2-23-4': '(3) A subset of the objects presented by Schenker et al. (2012).', '1403.5466-2-23-5': 'In particular we select those galaxies whose colors are consistent with the z-dropout selection criteria used in this paper (note that Schenker et al. also observed Y-band dropouts whose photometric redshifts are [MATH] 7).', '1403.5466-2-23-6': 'In total we consider 11 objects of which 2 are detected with Ly[MATH].', '1403.5466-2-24-0': 'Overall considering new and previous data, we assemble a sample of 68 z-dropouts that have been spectroscopically observed with either VLT, Keck or Subaru down to very faint flux limits.', '1403.5466-2-24-1': 'Note that 46 out of 68 have been observed with exactly the same set-up (with FORS2@VLT using grism 600z).', '1403.5466-2-25-0': '## Bullet cluster', '1403.5466-2-26-0': 'Bradac et al. (2012) observed the lensed z-dropouts detected behind the Bullet cluster and selected by Hall et al. 2011.', '1403.5466-2-26-1': 'The observations were carried out with FORS2@VLT using the same observational set-up as in our various programs (V11, P11), and above (UDS and GOODS-S) with a total net integration time of 16.5 hours.', '1403.5466-2-26-2': 'Data were reduced using our own pipeline (V11).', '1403.5466-2-27-0': 'The confirmation of one galaxy showing Ly[MATH] emission consistent with a redshift of 6.74 was presented by Bradac et al. (2012).', '1403.5466-2-27-1': 'Here we also consider the observations and limits, in terms of Ly[MATH] line detection of the rest of the sample.', '1403.5466-2-27-2': 'In Table 1 we report the resulting limits on the Ly[MATH] EW for each galaxy, assuming again a median redshift of 6.9.', '1403.5466-2-27-3': 'However note that in this case the median expected redshift of the sample is [MATH]: the redshift probability distribution function of these candidates is much larger and extends well beyond z=8, differing considerably from the other samples presented here (for reference see Figure 5 in Hall et al. 2012).', '1403.5466-2-27-4': 'This is due to the fact that the candidates were selected by applying criteria based on a [MATH] color (due to the nature of the HST data available).', '1403.5466-2-27-5': 'In the following section we will consider, whenever necessary, the appropriate selection function of this sample.', '1403.5466-2-28-0': '# Simulations', '1403.5466-2-29-0': 'To determine the EW limit achieved by our observations for each of our targets, and reported in Table 1, we performed detailed 2D simulations that we briefly describe here.', '1403.5466-2-29-1': 'We take real individual MXU raw frames, corresponding to slits where no targets were detected and insert an emission line of a given flux, at a given wavelength and at a given spatial position corresponding to the middle of the slit.', '1403.5466-2-29-2': 'The emission line is modeled as a Gaussian that is then truncated to half to simulate the typically asymmetric emission lines that are routinely observed at lower redshift, and then further convolved with the seeing (with values varying from 0.6to 1.2as in the real observations).', '1403.5466-2-29-3': 'The individual frames are then processed as normally done during the reduction procedure: after standard flat-fielding, we remove the sky emission lines by subtracting the sky background between two consecutive exposures, exploiting the fact that the target spectrum is offset due to dithering (ABBA technique).', '1403.5466-2-29-4': 'Then spectra have been wavelength calibrated (using lamp exposures) and finally, they are flux-calibrated using the observations of spectrophotometric standards and combined.', '1403.5466-2-30-0': 'The resulting 2D frame is then scanned with a window of [MATH] pixel to see if there is a detection, and in this case the signal to noise ratio of the line is registered.', '1403.5466-2-30-1': 'The simulations are repeated after shifting the Ly[MATH] emission line along the dispersion axis in steps of [MATH], and in the end we cover the redshift range from z=5.68 to z=7.30.', '1403.5466-2-30-2': 'At each redshift step, the scan is repeated and the S/N of the line is registered again.', '1403.5466-2-31-0': 'The emission lines are then varied in terms of total flux (from 0.24 to [MATH]), and full width half maximum (varying from 230 to 520 [MATH]).', '1403.5466-2-31-1': 'These values are in the range of the real observed Ly[MATH] lines.', '1403.5466-2-31-2': 'The entire procedure is repeated for each combination of line flux and FWHM.', '1403.5466-2-31-3': 'The simulations were performed for more than one slit in order to cover the entire CCD (top and bottom chips) of the FORS2 MXU frame and the results are always the same to within 5%.', '1403.5466-2-31-4': 'Although our candidates were placed at the center of the slits in most cases, we also checked for possible differences by placing the initial emission line at various positions along individual slits, i.e. we shifted the Ly[MATH] up and down by a few pixels.', '1403.5466-2-31-5': 'Again no significant difference is found.', '1403.5466-2-31-6': 'In Figure 3 we show one of the results of these tests, with the three colored curves (red, green and black) representing the resulting S/N of a line with flux 1.6 [MATH], positioned in three different slits: for each slit the result is the average of 3 different positions along the spatial axis (0,+5 and -5 pixel).', '1403.5466-2-31-7': 'It is evident that the differences in the resulting S/N between slits are only marginal.', '1403.5466-2-31-8': 'In the same Figure we also show the skyline emission for reference.', '1403.5466-2-32-0': '# The declining fraction of Ly[MATH] emitters: new limits and discussion', '1403.5466-2-33-0': '## The fraction of Ly[MATH] emitters at z[MATH]', '1403.5466-2-34-0': 'With our new sample we can evaluate with greater accuracy the fraction of Ly[MATH] emission in LBGs at redshift 7, the decline between [MATH] and [MATH] and its implication.', '1403.5466-2-34-1': 'In Table 2 we report the fraction of galaxies having and EW [MATH] and [MATH] separately for the two absolute magnitude bins that were adopted by previous works (Stark et al. 2010, P11, Ono et al. 2012).', '1403.5466-2-34-2': 'In the bright bin (galaxies with magnitudes [MATH]) there are 39 galaxies of which 7 are detected in Ly[MATH]: 5 of these have [MATH] 2 have [MATH] and none has [MATH].', '1403.5466-2-34-3': 'In the faint bin (galaxies with [MATH]) there are 25 objects, of which 5 have a Ly[MATH] emission with [MATH] and 2 with [MATH].', '1403.5466-2-34-4': 'Note that 3 of the targets in the Bradac et al. sample are intrinsically fainter than [MATH] thus are excluded from this bin.', '1403.5466-2-34-5': 'In the Table we report the fractions taking into account the fact that the limit in the EW detectable for the galaxies is not always below 25 ; for example for some of the objects in Ono et al. (2012) the limits achieved are above this value.', '1403.5466-2-34-6': 'In calculating the fractions we also consider that for some galaxies the redshift probability distribution extends well beyond z[MATH], which is approximately the limit out to which we can detect the Ly[MATH] emission in our current observations.', '1403.5466-2-34-7': 'In particular as already stated above, the sample observed by Bradac et al. (2012) was selected in such a way that the probability of galaxies being at [MATH] is quite high, [MATH]% (see Figure 5 in Hall et al. 2012).', '1403.5466-2-34-8': 'This is due to the broad J-band filter ([MATH]) that was available for the selection.', '1403.5466-2-34-9': 'Therefore we weighted each sample by evaluating the total probability of galaxies being outside the redshift range that is observable by the spectroscopic setup.', '1403.5466-2-34-10': 'In practice for most of the samples this probability is negligible (see Figure 6 in Ouchi et al. 2010 for the Ono et al. sample, Figure 7 in Castellano et al. 2010a for the NTT,GOODS-South and BDF samples), while it is non negligible for the UDS sample (which has a tail to z[MATH]8 , see Figure 1 in Grazian at al. 2012) and quite high for the Bradac et al. sample.', '1403.5466-2-35-0': 'We also report the fractions after assuming that 20% of the undetected objects are lower redshift interlopers: this value (20%) is the upper limit for possible interlopers found in a large sample of z[MATH]6 galaxies in our previous work (P11), and we assume that there is no significant change between the two epochs.', '1403.5466-2-35-1': 'Note that none of our galaxies has a detected Ly[MATH] emission with EW larger than 75 : to calculate the upper limit for the fraction, we assume the statistics for small numbers of events by Gehrels (1986).', '1403.5466-2-36-0': 'Comparing the above results to those at [MATH] presented by Stark et al. (2010), it is clear that there is a very significant deficit of Ly[MATH] emission at z[MATH] compared to earlier epochs.', '1403.5466-2-36-1': 'We note here that very recently Schenker et al. (2014) introduced a new method to analyze the decrease of Ly[MATH] emission in LBGs, based on using the measured slopes of the rest-frame ultraviolet continua of galaxies, rather than their absolute [MATH] magnitudes as we do here.', '1403.5466-2-36-2': 'According to their conclusions, the observed difference between the z[MATH] and z[MATH] EW distribution is even slightly larger than with the traditional way of computing fractions in bins of [MATH].', '1403.5466-2-36-3': 'This is mainly because blue galaxies at [MATH] exhibit stronger Ly[MATH] emission and candidates at [MATH] tend to be bluer than at lower redshift, hence they are expected to exhibit Ly[MATH] even more often.', '1403.5466-2-37-0': 'In the following sections we will try to interpret this deficit, first within the context of large-scale semi-numeric simulations of reionization that includes the reionization field as well as galactic properties (Dijkstra et al. 2011).', '1403.5466-2-37-1': 'We will then apply to our data a simple phenomenological model developed by Treu et al. (2012) that uses the evolution of the distribution of Ly[MATH] equivalent widths to make some simple predictions about the complex topology of reionization.', '1403.5466-2-38-0': '## The neutral hydrogen fraction', '1403.5466-2-39-0': 'In P11 we interpreted the drop in the Ly[MATH] fraction in LBGs as due most probably to the sudden increase of neutral hydrogen in the universe between z[MATH]6 and z[MATH] (see also Schenker et al. 2012).', '1403.5466-2-39-1': 'We then compared the results to the predictions of Dijkstra et al. (2011), to determine what fraction of neutral hydrogen would be needed to explain the drop, provided that all other physical parameters (e.g. dust content, the escape fraction of Lyman continuum photons etc) would not change between z=6 and z=7.', '1403.5466-2-39-2': 'We obtained a rather high neutral hydrogen fraction by volume, [MATH].', '1403.5466-2-40-0': 'We now make use of improved models to compare our new results.', '1403.5466-2-40-1': 'As in Dijsktra et al. (2011), reionization morphologies were generated using the public code, DexM (Mesinger Furlanetto 2007).', '1403.5466-2-40-2': 'The box size is 200 Mpc, and the ionization field is computed on a [MATH] grid.', '1403.5466-2-40-3': 'Reionization morphologies at a given [MATH] are generated by varying the ionization efficiency of halos, down to a minimum halo mass of [MATH], roughly corresponding to the average minimum mass of halos at z=7 which retain enough gas to form stars efficiently (Sobacchi Mesinger 2013).', '1403.5466-2-40-4': 'Compared to P11 the model now includes also more massive halos, with stellar masses up to [MATH].', '1403.5466-2-40-5': 'This is because the previous model was tailored to analyze the nature of fainter dropout galaxies (Dijkstra et at.', '1403.5466-2-40-6': '2011) compared to those presented in this work.', '1403.5466-2-40-7': 'The results however change only minimally with the new choice of halo mass, as expected given that the halo bias (and the associated opacity distribution for a given [MATH]) does not evolve much over this mass range (e.g. Mesinger Furlanetto 2008, McQuinn et al. 2008).', '1403.5466-2-40-8': 'In Figure 4 we present the comparison of the outcome of the new model with the present fractions for the faint sample.', '1403.5466-2-40-9': 'The red circles (and limit) show the fractions assuming that all our non detected target are at z[MATH] (the same assumption that is made in this model at z[MATH]) while the blue circles and limit assume 20% interlopers.', '1403.5466-2-40-10': 'It is clear that only a very high neutral hydrogen fraction ([MATH]) can best reproduce the lack of Ly[MATH] emission at [MATH] compared to earlier epochs, even if there are still considerable uncertainties i.e. large error bars due to the small size of the samples.', '1403.5466-2-40-11': 'This high value seems at odds with other observational results: for instance, Raskutti et al. (2012) study the IGM temperature in quasar near-zones and find that reionization must have been completed by [MATH] at high confidence, while several teams (Hu et al. (2010), Kashikawa et al. (2011) and Ouchi et al. 2010) study the Ly[MATH] line shapes of LAEs at z = 6.5 finding no evidence of damping wings.', '1403.5466-2-41-0': 'As recently pointed out by Taylor Lidz (2014), before reionization completes, the simulated Ly[MATH] fraction might have large spatial fluctuations depending on the degree of homogeneity/inhomogeneity of the reionization process.', '1403.5466-2-41-1': 'Since existing measurements of the Ly[MATH] fraction span relatively small regions on the sky, and sample these regions only sparsely (typically only a few dropouts are observed per field), they might by chance probe mostly galaxies with above than average Ly[MATH] attenuation and therefore point to higher neutral hydrogen fractions compared to the average values.', '1403.5466-2-41-2': 'It is therefore important to include the effect of cosmic variance for different sight-lines within our survey.', '1403.5466-2-41-3': 'In their work Taylor Lidz found that the sample variance is non negligible for existing surveys, and it does somewhat mitigate the required neutral fraction at z[MATH] 7.', '1403.5466-2-42-0': 'Compared to previous studies and to the surveys analyzed by Taylor Lidz, this work presents more independent fields of view: even considering as single pointing those in adjacent areas (e.g. the GOODS-South/ERS areas and the 2 SDF pointings in Ono et al. 2012), we are now sampling 8 independent lines of sight, with areas varying between [MATH] in each field.', '1403.5466-2-42-1': 'We have verified what is the uncertainty in our results that might be derived from the cosmic variance.', '1403.5466-2-43-0': 'We have tried to quantify what is the variance in the opacity ([MATH]) in the simulations, due to the limited number of fields analyzed.', '1403.5466-2-43-1': 'In the simulations we take 8 random regions with areas corresponding to our observed fields and in each field we sample a reasonable number of lines-of-sight (corresponding to the average number of candidates spectroscopically observed).', '1403.5466-2-43-2': 'We then compute the pointing-to-pointing (cosmic) standard deviation [MATH], which for 8 field is of the order of 6%.', '1403.5466-2-43-3': 'The variance is so small because the total volume sampled is quite large.', '1403.5466-2-43-4': 'We also varied the number of candidate galaxies probing the reionization per each pointing: indeed the discreteness in sampling the reionization morphology becomes an issue especially for large neutral fractions since in this case the fewer number of would-be Ly[MATH] emitters can miss the relatively rare regions in a given pointing that have a high transmission.', '1403.5466-2-43-5': 'However even considering a small sampling (only 5 emitters per pointing) [MATH] is still around 10%.', '1403.5466-2-43-6': 'Note that this computation just uses the opacity at a single wavelength, roughly 200 [MATH] red-ward of the line center, where most of the intrinsic emission is expected to lie.', '1403.5466-2-43-7': 'This is not really the variance in the Ly[MATH] fraction, since the latter requires some more detailed modeling, but it gives a crude idea of what is the expected cosmic variance from reionization.', '1403.5466-2-43-8': 'Therefore we are quite confident that for our sample which has a large number of independent field and a reasonable number of candidates observed per pointing, the field to field fluctuations are not very large and would not effect sensibly the results of Figure 4.', '1403.5466-2-44-0': '## A patchy reionization process?', '1403.5466-2-45-0': 'Applying the simple phenomenological models developed by Treu et al. (2012) to describe the evolution of the distribution of Ly[MATH] equivalent widths we can now use the Ly[MATH] detections and non-detections to make some inferences about the complex topology of reionization.', '1403.5466-2-45-1': 'This model starts from the intrinsic rest-frame distribution in terms of the one measured at [MATH] by Stark et al. (2011).', '1403.5466-2-45-2': 'It then considers two extreme cases which should bracket the range of possible scenarios for the reionization morphology: in the first (""patchy"") model, no Ly[MATH] is received from a fraction [MATH] of the sources, while the rest is unaffected.', '1403.5466-2-45-3': 'In the second (""smooth"") model, the Ly[MATH] emission is attenuated in every galaxy in the same way, by a factor [MATH].', '1403.5466-2-45-4': 'These two models can be thought respectively as simple idealizations of smooth and patchy reionization: although very simple and somewhat unphysical (especially the smooth one) these two models should bracket the expected behavior of the IGM near the epoch of reionization (see Treu et al. 2012, Treu et al. 2013 for a more detailed explanation).', '1403.5466-2-46-0': ""For each object in our sample, Bayes's rule gives the posterior probability of [MATH] and [MATH] (which we collectively indicate as [MATH]) and redshift given the observed spectrum and the continuum magnitude."", '1403.5466-2-46-1': 'The likelihood is as usual the probability of obtaining the data for any given value of the parameter.', '1403.5466-2-46-2': 'The model adopts a uniform prior p([MATH]) between zero and unity, while the prior for the redshift [MATH] is obtained from the redshift probability distribution (as described in Section 3 for each of the different parent samples).', '1403.5466-2-46-3': 'We use the implementation of the method that takes as input the line equivalent widths or equivalent width limits, in order to incorporate all the available information, even when a noise spectrum is not available (Treu et al. 2012).', '1403.5466-2-47-0': 'One of the output of the model is the normalization constant [MATH], known as the Bayesian evidence and quantifies how well each of the two models matches the data.', '1403.5466-2-47-1': 'The evidence ratio is a powerful way to perform model selection e.g., comparing the patchy and smooth models and eventually discriminate between the two.', '1403.5466-2-48-0': 'Treu et al. 2012 applied their model to the sample presented by Ono et al. (2012) which included also data from P11 and V11.', '1403.5466-2-48-1': 'The data clearly preferred an attenuation factor [MATH] (0.65-0.68), independent of the model considered, but the evidence ratio indicated no significant preference for any of the two models.', '1403.5466-2-49-0': 'We have repeated the exercise for our new enlarged sample, which is almost double compared to the previous one and most importantly contains a larger fraction of very faint galaxies (thanks e.g. to the inclusion of the lensed galaxies of the Bradac et al. sample, and several other faint targets from UDS and the archival data) .', '1403.5466-2-49-1': 'With the new sample we obtain [MATH] and [MATH] respectively for the patchy and smooth model, as shown in Figure [REF]: this means that both models require a considerable quenching of the Ly[MATH] compared to z[MATH], as expected from the many non-detections.', '1403.5466-2-49-2': 'Note that these results assume that the level of contamination in the samples, is the same at [MATH] and at [MATH].', '1403.5466-2-49-3': 'We can interpret the [MATH] and [MATH] as the average excess optical depth of Ly[MATH] with respect to z [MATH]6, i.e., [MATH], although a conversion from this to a neutral hydrogen fraction requires detailed and uncertain modeling (e.g. Santos 2004).', '1403.5466-2-49-4': 'A key result is that with the new sample the evidence ratio between the two models is quite high, [MATH] which means that the patchy model is highly favored ([MATH] times) by the data over the smooth one.', '1403.5466-2-49-5': 'This is also suggested by the likelihood ratio test: [MATH] strongly favors the patchy model (in other terms the ratio corresponds to a difference in [MATH] between the two models of [MATH]).', '1403.5466-2-49-6': 'As expected the power to discriminate between the two models is given by the inclusion of fainter galaxies, as well as the fact that for many galaxies we have very deep EW limits.', '1403.5466-2-49-7': 'As a result of the inference, the model also allows us to calculate the fraction of emitters using all the available information.', '1403.5466-2-49-8': 'For objects brighter than [MATH] the model predicts respectively [MATH] for galaxies with [MATH] and [MATH] for galaxies with [MATH] .', '1403.5466-2-49-9': 'For fainter galaxies, the predictions are [MATH] and [MATH] respectively.', '1403.5466-2-49-10': 'These values are very close to the numbers reported in Table 2 (considering obviously the fractions derived assuming no interlopers in the sample).', '1403.5466-2-49-11': 'In Figure [REF] we show the predicted distribution of rest-frame equivalent width for the best patchy (blue) and smooth (red) models, for the bright and faint sub-sample separately.', '1403.5466-2-49-12': 'The black histograms are based on the detected Ly[MATH] emitters in each sample.', '1403.5466-2-49-13': 'In particular the blue model makes predictions that are closer to the real data for the faint sub-sample, since it predicts better the high EW tail and it does not show a deficit of detections at intermediate EW ([MATH] ) While our observational results indicate clearly that the distribution of neutral hydrogen in these phases of reionization was highly inhomogeneous, as expected by most theoretical predictions (e.g. Iliev et al. 2006), to fully constrain the morphology of reionization we will have to wait for the direct observations of the 21 cm emission from neutral hydrogen in the high redshift universe, which is one of the prime tasks of the upcoming LOFAR surveys observations (e.g. Jensen et al. 2013).', '1403.5466-2-50-0': '# Summary and concluding remarks', '1403.5466-2-51-0': 'In this paper we have presented new results from our search for z[MATH] galaxies from deep spectroscopic observations of candidate z-dropouts in the CANDELS fields.', '1403.5466-2-51-1': 'Even though our sensitive VLT observations reached extremely low flux limits, only two galaxies have new robust redshift identifications, one from the Ly[MATH] emission line at z=6.65 and the other from its Lyman-alpha break, i.e. the continuum discontinuity at the Ly[MATH] wavelength consistent with a redshift [MATH]6.42.', '1403.5466-2-51-2': 'In this second object no emission line is observed.', '1403.5466-2-51-3': 'In addition for [MATH] galaxies we present new deep limits on the Ly[MATH] EW derived from the non detections in ultra-deep observations (from 15 to 27 hours) obtained with FORS2 spectrograph on the VLT.', '1403.5466-2-51-4': 'Using this new data as well as previously published samples, we have assembled a total of 68 candidate z[MATH] galaxies with deep spectroscopic observations, of which 12 have a redshift identification from the Ly[MATH] emission line.', '1403.5466-2-51-5': 'With this much enlarged sample we have placed solid constraints on the fraction of Ly[MATH] emission in z[MATH]7 Lyman break galaxies both for bright and faint galaxies, confirming the large decline in the presence of Ly[MATH] emission from z[MATH] to [MATH].', '1403.5466-2-51-6': 'If this decline is only due to the evolution of the IGM, and assuming that all other galaxies properties remain unchanged in this redshift interval, a very large fraction ([MATH]) of neutral hydrogen is needed to explain the observations.', '1403.5466-2-51-7': 'Finally applying the simple phenomenological model developed by Treu et al. (2012), we show that the present data favor a patchy reionization process rather than a smooth one, as expected from most simulations (e.g. Friedrich et al. 2011, Choudhury et al. 2009, Iliev et al. 2006 to name a few: also see Trac Gnedin 2009 for a review on simulations of reionization).', '1403.5466-2-52-0': 'Obviously we cannot rule out that an evolution of other properties, namely [MATH] and dust, come into play and contribute to the Ly[MATH] quenching Indeed, in a recent paper (Dijkstra et al. 2014) we discuss the possibility that the decline in strong Ly[MATH] emission from [MATH] galaxies is due, in part, also to an increase of the Lyman continuum escape fraction in star forming galaxies.', '1403.5466-2-52-1': 'In particular assuming that the escape fraction evolves with redshift as [MATH] (as in Kuhlen Faucher-Giguere 2012), and taking [MATH] and [MATH] such that we have [MATH] at z=6 and [MATH] at z=7, the observed decline in Ly[MATH] emission could be reproduced with a more modest evolution in the global neutral fraction, of the order of [MATH].', '1403.5466-2-52-2': 'This work is clearly rather speculative, since [MATH] is a very elusive quantity to measure and we only have tentative indications on its value from upper limits (e.g. Nestor et al. 2013, Vanzella et al. 2012, Boutsia et al. 2011) and on its evolution (e.g. Cowie et al. 2009, Siana et al. 2010).', '1403.5466-2-52-3': 'However it shows that an evolving escape fraction of ionizing photons should be considered as part of the explanation for evolution in the Ly[MATH] emission of high redshift galaxies in addition to the evolution of the IGM (see Dijkstra et al. 2014 for more details).', '1403.5466-2-53-0': 'It is clear that to fully characterize and understand the reionization epoch, and to clarify the relation between the disappearing Ly[MATH] emission line and cosmic reionization we still have to make a substantial effort, both on the observational and on the modeling side.', '1403.5466-2-53-1': 'Even if the current samples of candidate galaxies at [MATH] are quite large, and despite all the observational efforts by several teams, the number of spectroscopically confirmed objects remains very small.', '1403.5466-2-53-2': 'To overcome this limitation and substantially increase the statistics we are currently carrying out an ESO Large Program with FORS2@VLT (PI Pentericci) that in the end should allow us to increase considerably the number of confirmed high redshift galaxies by observing [MATH] candidates.', '1403.5466-2-53-3': 'In particular, since the targets will be selected from the CANDELS field with extremely deep near-IR observations, we will include also galaxies as faint as [MATH].', '1403.5466-2-53-4': ""The new deep spectroscopic observations will allow us to assess the continuous evolution of the Ly[MATH] emission over the range [MATH], and from a comparison to state-of-the art models, we will be able to determine if the Ly[MATH] was mainly quenched by the neutral IGM, or if any evolution of the galaxies' physical properties also played a significant role."", '1403.5466-2-54-0': 'Part of this work has been funded through INAF Grants (PRIN-INAF 2010 and 2012).', '1403.5466-2-54-1': 'RJM acknowledges the support of the European Research Council via the award of a Consolidator Grant JSD acknowledges the support of the European Research Council via the award of an Advanced Grant, the support of the Royal Society via a Wolfson Research Merit Award, and the contribution of the EC FP7 SPACE project ASTRODEEP (Ref.No: 312725).'}","[['1403.5466-1-7-0', '1403.5466-2-7-0'], ['1403.5466-1-7-1', '1403.5466-2-7-1'], ['1403.5466-1-7-2', '1403.5466-2-7-2'], ['1403.5466-1-7-3', '1403.5466-2-7-3'], ['1403.5466-1-5-0', '1403.5466-2-5-0'], ['1403.5466-1-5-1', '1403.5466-2-5-1'], ['1403.5466-1-5-3', '1403.5466-2-5-3'], ['1403.5466-1-5-4', '1403.5466-2-5-4'], ['1403.5466-1-5-5', '1403.5466-2-5-5'], ['1403.5466-1-24-0', '1403.5466-2-24-0'], 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'1403.5466-2-43-2']]",[],[],"{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1403.5466,,,
1112.1185,"{'1112.1185-1-0-0': 'The aim of this book article is to study infinite games and to prove formally properties in this framework.', '1112.1185-1-0-1': 'In particular, we show that the behavior which leads to speculative crashes or escalation is fully rational.', '1112.1185-1-0-2': 'Indeed it proceeds logically from the statement that resources are infinite.', '1112.1185-1-0-3': 'The reasoning is based on the concept of coinduction conceived by computer scientists to model infinite computations and used by economic agents unknowingly.', '1112.1185-1-1-0': '[Introduction]Introduction [Introduction]Introduction', '1112.1185-1-2-0': 'The aim of thisbook article is to study infinite games and to prove formally some properties in this framework.', '1112.1185-1-2-1': 'As a consequence, we show that the behavior (the madness) of people which leads to speculative crashes or escalation can be proved fully rational.', '1112.1185-1-2-2': 'Indeed it proceeds from the statement that resources are infinite.', '1112.1185-1-2-3': 'The reasoning is based on the concept of coinduction conceived by computer scientists to model infinite computations and used by economic agents unknowingly.', '1112.1185-1-2-4': 'When used consciously, this concept is not as simple as induction and we could paraphrase Newton : ""Modeling the madness of people is more difficult than modeling the motion of planets"".', '1112.1185-1-3-0': 'In this chapter section we present the three words of the title, namely rationality, escalation and infiniteness.', '1112.1185-1-4-0': '[Rationality and escalation]Rationality and escalation [Rationality and escalation]Rationality and escalation', '1112.1185-1-5-0': 'We consider the ability of agents to reason and to conduct their action according to a line of reasoning.', '1112.1185-1-5-1': 'We call this rationality.', '1112.1185-1-5-2': 'This could have been called wisdom as this attributed to King Solomon.', '1112.1185-1-5-3': 'It is not clear that agents act always rationally.', '1112.1185-1-5-4': 'If an agent acts always following a strict reasoning one says that he (she) is rational.', '1112.1185-1-5-5': 'To specify strictly this ability, one associates the agent with a mechanical reasoning device, more specifically a Turing machine or a similar decision mechanism based on abstract computations.', '1112.1185-1-5-6': 'One admits that in making a decision the agent chooses the option which is the better, that is no other will give better payoff, one says that this option is an equilibrium in the sense of game theory.', '1112.1185-1-5-7': 'A well-known game theory situation where rationality of agents is questionable is the so-called escalation.', '1112.1185-1-5-8': 'This is a situation where there is a sequence of decisions wich can be infinite.', '1112.1185-1-5-9': 'If many agents act one after the others in an infinite sequence of decisions and if this sequence leads to situations which are worst and worst for the agents, one speaks of escalation.', '1112.1185-1-5-10': 'One notices the emergence of a property of complex systems, namely the behavior of the system is not the conjunction of this of all the constituents.', '1112.1185-1-5-11': 'Here the individual wisdom becomes a global madness.', '1112.1185-1-6-0': '[Infiniteness]Infiniteness [Infiniteness]Infiniteness infiniteness', '1112.1185-1-7-0': 'It is notorious that there is a wall between finiteness and infiniteness, a fact known to model theorists like [CITATION] [CITATION] and to specialists of functions of real variable.', '1112.1185-1-7-1': '[CITATION] gave an example of the fact that a finite sum of functions differentiable everywhere is differentiable everywhere whereas an infinite sum is differentiable nowhere.', '1112.1185-1-7-2': 'This confusion between finite and infinite is at the origin of the conclusion of the irrationality of the escalation founded on the belief that a property of a infinite mathematical object can be extrapolated from a similar property of finite approximations.', '1112.1185-1-7-3': 'As [CITATION] recalls, ""Most of the classical theorems of logic [for infinite structures] fail for finite structures"" (see [CITATION] for a full development of the finite model theory).', '1112.1185-1-7-4': 'The reciprocal holds obviously: ""Most of the results which hold for finite structures, fail for infinite structures"".', '1112.1185-1-7-5': 'This has been beautifully evidenced in mathematics, when [CITATION] has exhibited his function: [EQUATION]', '1112.1185-1-7-6': 'Every finite sum is differentiable and the limit, i.e., the infinite sum, is not.', '1112.1185-1-7-7': 'In another domain, [CITATION] have proved that the sequence of prime numbers contains arbitrarily long arithmetic progressions.', '1112.1185-1-7-8': 'By extrapolation, there would exist an infinite arithmetic progression of prime numbers, which is trivially not true.', '1112.1185-1-7-9': 'To give another picture, infinite games are to finite games what fractal curves are to smooth curves .', '1112.1185-1-7-10': 'In game theory the error done by the ninetieth century mathematicians Weierstrass would lead to the same issue.', '1112.1185-1-7-11': 'With what we are concerned, a result which holds on finite games does not hold necessarily on infinite games and vice-versa.', '1112.1185-1-7-12': 'More specifically equilibria on finite games are not preserved at the limit on infinite games whereas new types of equilibria emerge on the infinite game not present in the approximation (see the [MATH] game in Section [REF]) and Section [REF].', '1112.1185-1-7-13': 'In particular, we cannot conclude that, whereas the only rational attitude in finite dollar auction would be to stop immediately, it is irrational to escalate in the case of an infinite auction.', '1112.1185-1-7-14': 'We have to keep in mind that in the case of escalation, the game is infinite, therefore reasoning made for finite objects are inappropriate and tools specifically conceived for infinite objects should be adopted.', '1112.1185-1-7-15': ""Like Weierstrass' discovery led to the development of function series, logicians have devised methods for correct deductions on infinite structures."", '1112.1185-1-7-16': 'The right framework for reasoning logically on infinite mathematical objects is called coinductioncoinduction.', '1112.1185-1-8-0': 'The inadequate reasoning on infinite games is as follows: people study finite approximation of infinite games as infinite games truncated at a finite location.', '1112.1185-1-8-1': 'If they obtain the same result on all the approximations, they extrapolate the result to the infinite game as if the limit would have the same property.', '1112.1185-1-8-2': 'But this says nothing since the infiniteness is not the limit of finiteness.', '1112.1185-1-8-3': 'Instead of reexamining their reasoning or considering carefully the hypotheses their reasoning is based upon (is the set of resource infinite?)', '1112.1185-1-8-4': 'they conclude that humans are irrational.', '1112.1185-1-8-5': 'If there is an escalation, then the game is infinite, then the reasoning must be specific to infinite games, that is based on coinduction.', '1112.1185-1-8-6': 'This is only on this basis that one can conclude that humans are rational or irrational.', '1112.1185-1-8-7': 'In no case, a property on the infinite game generated by escalation cane be extrapolated from the same property on finite games.', '1112.1185-1-8-8': 'escalation', '1112.1185-1-9-0': 'In this book article we address these issues.', '1112.1185-1-9-1': 'The games we consider may have arbitrary long histories as well as infinite histories.', '1112.1185-1-9-2': 'history In our games there are two choices at each node, this will not loose generality, since we can simulate finitely branching games in this framework.', '1112.1185-1-9-3': ""By Konig's lemma, finitely branching, specifically binary, infinite games have at least an infinite history."", '1112.1185-1-9-4': 'We are taking the problem of defining formally infinite games, infinite strategy profiles, and infinite histories extremely seriously.', '1112.1185-1-9-5': 'By ""seriously"" we mean that we prepare the land for precise, correct and rigorous reasoning.', '1112.1185-1-9-6': 'For instance, an important issue which is not considered in the literature is how the utilities associated with an infinite history are computed.', '1112.1185-1-9-7': 'To be formal and rigorous, we expect some kinds of recursive definitions, more precisely co-recursive definitions, but then comes the questions of what the payoff associated with an infinite strategy profile is and whether such a payoff exists (see Section [REF]).', '1112.1185-1-10-0': '[Games]Games [Games]Games', '1112.1185-1-11-0': 'Finite extensive games are represented by finite trees and are analyzed through induction.', '1112.1185-1-11-1': 'For instance, in finite extensive games, a concept like subgame perfect equilibrium is defined inductively and receives appropriately the name of backward induction.', '1112.1185-1-11-2': 'Similarly convertibility (an agent changes choices in his strategy) has also an inductive definition and this concept is a key for this of Nash equilibrium.', '1112.1185-1-11-3': 'But induction, which has been designed for finitely based objects, no more works on infinite games, i.e., games underlying infinite trees.', '1112.1185-1-11-4': 'Logicians have proposed a tool, which they call coinduction, to reason on infinite objects.', '1112.1185-1-11-5': 'In short, since objects are infinite and their construction cannot be analyzed, coinduction ""observes"" them, that is looks at how they ""react"" to operations (see Section [REF] for more explanation).', '1112.1185-1-11-6': 'In this book, article, we formalize with coinduction, the concept of infinite game, of infinite strategy profile, of equilibrium in infinite games, of utility (payoff), and of subgame.', '1112.1185-1-11-7': 'We verify on the proof assistant Coq that everything works smoothly and yields interesting consequences.', '1112.1185-1-11-8': 'Thanks to coinduction, examples of apparently paradoxical human behavior are explained logically, demonstrating a rational behavior.', '1112.1185-1-12-0': 'Finite extensive games have been introduced by [CITATION].', '1112.1185-1-12-1': 'But many interesting extensive games are infinite and therefore the theory of infinite extensive games play an important role in game theory, with examples like the dollar auction game Shubik , the generalized centipede game or infinipede or the [MATH] game.', '1112.1185-1-12-2': 'From a formal point of view, the concepts associated with infinite extensive games are not appropriately treated in papers and books.', '1112.1185-1-12-3': 'In particular, there is no clear notion of Nash equilibrium in infinite extensive game and the gap between finiteness and infiniteness is not correctly understood.', '1112.1185-1-12-4': 'For instance in one of the textbooks on game theory, one finds the following definition of games with finite horizon:', '1112.1185-1-13-0': 'If the length of the longest derivation is [...] finite, we say that the game has a finite horizon.', '1112.1185-1-13-1': 'Even a game with a finite horizon may have infinitely many terminal histories, because some player has infinitely many actions after some history.', '1112.1185-1-14-0': 'Notice that in an infinite game with infinite branching it is not always the case that a longest derivation exists.', '1112.1185-1-14-1': 'If a game has only finite histories, but has infinitely many such finite histories of increasing length, there is no longest history.', '1112.1185-1-14-2': 'Before giving a formal definition later in the book, article, let us say intuitively why this definition is inconsistent.', '1112.1185-1-14-3': 'Roughly speaking, a history is a path in the ordered tree which underlies the game.', '1112.1185-1-14-4': 'A counterexample is precisely when the tree is infinitely branching i.e., when ""some player has infinitely many actions"".', '1112.1185-1-15-0': 'Escalation takes place in specific sequential games in which players continue although their payoff decreases on the whole.', '1112.1185-1-15-1': 'The dollar auction game dollar auction has been presented by [CITATION] Shubik as the paradigm of escalation.', '1112.1185-1-15-2': 'He noted that, even though their cost (the opposite of the payoff) basically increases, players may keep bidding.', '1112.1185-1-15-3': ""This attitude was considered as inadequate and when talking about escalation, [CITATION] says this is a paradox, [CITATION] and [CITATION] consider the bidders as irrational, [CITATION] speaks of illogic conflict of escalation and [CITATION] calls it Macbeth effect after Shakespeare's play."", '1112.1185-1-15-4': 'Rebutting these authors, we prove in this book, article, using a reasoning conceived for infinite structures that escalation is logic and that agents are rational, therefore this is not a paradox and we are led to assert that Macbeth is in some way rational.', '1112.1185-1-16-0': 'This escalation phenomenon occurs in infinite sequential games and only there.', '1112.1185-1-16-1': 'escalation To quote [CITATION]:', '1112.1185-1-17-0': 'We could add an upper limit to the amount that anyone is allowed to bid.', '1112.1185-1-17-1': 'However the analysis is confined to the (possibly infinite) game without a specific termination point, as no particularly interesting general phenomena appear if an upper bound is introduced.', '1112.1185-1-18-0': 'Therefore it must be studied in infinite games with adequate tools, i.e., in a framework designed for mathematical infinite objects.', '1112.1185-1-18-1': 'Like [CITATION] we will limit ourselves to two players only.', '1112.1185-1-18-2': 'In auctions, this consists in the two players bidding forever.', '1112.1185-1-18-3': 'This statement of rationality is based on the largely accepted assumption that a player is rational if he adopts a strategy which corresponds to a subgame perfect equilibrium.', '1112.1185-1-18-4': 'To characterize this equilibrium most of the above cited authors consider a finite restriction of the game for which they compute the subgame perfect equilibrium by backward induction.', '1112.1185-1-18-5': 'Then they extrapolate the result obtained on the amputated games to the infinite game.', '1112.1185-1-18-6': 'To justify their practice, they add a new hypothesis on the amount of money the bidders are ready to pay, called the limited bankroll.', '1112.1185-1-18-7': 'By enforcing the finiteness of the game, they exclude clearly escalation.', '1112.1185-1-18-8': 'In the amputated game dollar auction, they conclude that there is a unique subgame perfect equilibrium.', '1112.1185-1-18-9': 'This consists in both agents giving up immediately, not starting the auction and adopting the same choice at each step.', '1112.1185-1-18-10': 'In our formalization in infinite games, we show that extending that case up to infinity is not a subgame perfect equilibrium and we found two subgame perfect equilibria, namely the cases when one agent continues at each step and the other leaves at each step.', '1112.1185-1-18-11': 'Those equilibria which correspond to rational attitudes account for the phenomenon of escalation.', '1112.1185-1-18-12': 'Actually this discrepancy between equilibrium in amputated games extrapolated to infinite extensions and infinite games occurs in a much simpler game than the dollar auction namely the [MATH] game [MATH] game which will be studied in this book.', '1112.1185-1-18-13': 'article.', '1112.1185-1-19-0': '[Coinduction]Coinduction [Coinduction]Coinduction coinduction', '1112.1185-1-20-0': 'Like induction, coinduction is based on a fixpoint, but whereas induction is based on the least fixpoint, coinduction is based on the greatest fixpoint, for an ordering we are not going to describe here as it would go beyond the scope of this book.', '1112.1185-1-20-1': 'article.', '1112.1185-1-20-2': 'Attached to induction is the concept of inductive definition, which characterizes objects like finite lists, finite trees, finite games, finite strategy profiles, etc.', '1112.1185-1-20-3': 'Similarly attached to coinduction is the concept of coinductive definition which characterizes streams (infinite lists), infinite trees, infinite games, infinite strategy profiles etc.', '1112.1185-1-20-4': 'An inductive definition yields the least set that satisfies the definition and a coinductive definition yields the greatest set that satisfies the definition.', '1112.1185-1-20-5': 'Associated with these definitions we have inference principles.', '1112.1185-1-20-6': 'For induction there is the famous induction principle used in backward induction.', '1112.1185-1-20-7': 'On coinductively defined sets of objects there is a principle like induction principle which uses the fact that the set satisfies the definition (proofs by case or by pattern) and that it is the largest set with this property.', '1112.1185-1-20-8': 'Since coinductive definitions allow us building infinite objects, one can imagine constructing a specific category of objects with ""loops"", like the infinite word [MATH] (i.e., [MATH]) infinite word which is made by repeating the sequence [MATH] infinitely many times.', '1112.1185-1-20-9': 'Other examples with trees are given in Section [REF], with infinite games and strategy profiles in Chapter Section [REF].', '1112.1185-1-20-10': 'Such an object is a fixpoint, this means that it contains an object like itself.', '1112.1185-1-20-11': 'For instance [MATH] contains itself.', '1112.1185-1-20-12': 'We say that such an object is defined as a cofixpoint.', '1112.1185-1-20-13': 'To prove a property [MATH] on a cofixpoint [MATH], one assumes [MATH] holds on [MATH] (the [MATH] in [MATH]), considered as a sub-object of [MATH].', '1112.1185-1-20-14': 'If one can prove [MATH] on the whole object (on [MATH]), then one has proved that [MATH] holds on [MATH].', '1112.1185-1-20-15': 'This is called the coinduction principle a concept which comes from [CITATION], [CITATION], and [CITATION] and was introduced in the framework we are considering by [CITATION].', '1112.1185-1-20-16': '[CITATION] gives a good survey with a complete historical account.', '1112.1185-1-20-17': 'To be sure to not be entangled, it is advisable to use a proof assistant which implements coinduction, to build and to check the proof.', '1112.1185-1-20-18': 'Indeed reasoning with coinduction is sometimes so counter-intuitive that the use of a proof assistant is not only advisable but compulsory.', '1112.1185-1-20-19': 'For instance, we were, at first, convinced that in the dollar auction the strategy profile consisting in both agents stopping at every step was a Nash equilibrium, like in the finite case, and only failing in proving it mechanically convinced us of the contrary and we were able to prove the opposite, namely that the strategy profile ""stopping at every step"" is not a Nash equilibrium.', '1112.1185-1-20-20': 'In the examples of Chapter Section [REF], we have checked every statement using Coq and in what follows a sentence like ""we have proved that ..."" means that we have succeeded in building a formal proof in Coq.', '1112.1185-1-21-0': '[Backward coinduction and invariants]Backward coinduction as a method for proving invariants', '1112.1185-1-22-0': '### Backward coinduction as a method for proving invariants', '1112.1185-1-23-0': 'In infinite strategy profiles, the coinduction principles can be seen as follows: a property which holds on a strategy profile of an infinite extensive game is an invariant, i.e., a property which is always true, along the histories and to prove that this is an invariant one proceeds back to the past.', '1112.1185-1-23-1': 'Therefore the name backward coinduction is appropriate, since it proceeds backward the histories, from future to past.', '1112.1185-1-24-0': '[Backward induction vs backward coinduction]Backward induction vs backward coinduction', '1112.1185-1-25-0': '### Backward induction vs backward coinduction', '1112.1185-1-26-0': 'One may wonder the difference between the classical method, we call backward induction and the new method we call backward coinduction.', '1112.1185-1-26-1': 'The main difference is that backward induction starts the reasoning from the leaves, works only on finite games and does not work on infinite games (or on finite strategy profiles), because it requires a well-foundedness to work properly, whereas backward coinduction works on infinite games (or on infinite strategy profiles).', '1112.1185-1-26-2': 'Coinduction is unavoidable on infinite games, since the methods that consists in ""cutting the tail"" and extrapolating the result from finite games or finite strategies profile to infinite games or infinite strategy profiles cannot solve the problem or even approximate it.', '1112.1185-1-26-3': 'It is indeed the same erroneous reasoning as this of the predecessors of Weierstrass who concluded that since: Weierstrass [EQUATION] is differentiable everywhere then [EQUATION] is differentiable everywhere whereas [MATH] is differentiable nowhere.', '1112.1185-1-27-0': ""Much earlier, during the IV[MATH] century BC, the improper use of inductive reasoning allowed Parmenides and Zeno to negate motion and lead to Zeno's paradox of Achilles and the tortoise."", '1112.1185-1-27-1': 'This paradox was reported by Aristotle as follows: Achilles and the tortoise', '1112.1185-1-28-0': '""In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.""', '1112.1185-1-29-0': 'Aristotle, Physics VI:9, 239b15', '1112.1185-1-30-0': ""In Zeno's framework, Zeno's reasoning is correct, because by induction, one can prove that Achilles will never overtake the tortoise."", '1112.1185-1-30-1': 'Indeed this applies to the infinite sequence of races described by Aristotle.', '1112.1185-1-30-2': 'In each race of the sequence, the pursuer starts from where the pursued started previously the race and the pursuer ends where the pursued started in the current race.', '1112.1185-1-30-3': 'By induction one can prove, but only for a sequence of races, the truth of the statement ""Achilles will never overtake the tortoise"".', '1112.1185-1-30-4': 'In each race ""Achilles does not overtake the tortoise"".', '1112.1185-1-30-5': 'For the infinite race for which coinduction would be needed, the result ""Achilles overtakes the tortoise"" holds.', '1112.1185-1-30-6': 'By the way, experience tells us that Achilles would overtake the tortoise in a real race and Zeno has long been refuted by the real world.', '1112.1185-1-31-0': '[Von Neumann and coinduction]Von Neumann and coinduction', '1112.1185-1-32-0': '### Von Neumann and coinduction', '1112.1185-1-33-0': 'As one knows, von Neumann is the creator of game theory, whereas extensive games and equilibrium in non cooperative games are due to [CITATION] and [CITATION].', '1112.1185-1-33-1': 'In the spirit of their creators all those games are finite and backward induction is the basic principle for computing subgame perfect equilibria .', '1112.1185-1-33-2': 'This is not surprising since [CITATION] is also at the origin of the role of well-foundedness in set theory despite he left a door open for a not well-founded membership relation.', '1112.1185-1-33-3': 'As explained by [CITATION], research on anti-foundation initiated by [CITATION] are at the origin of coinduction and were not well known until the work of [CITATION].', '1112.1185-1-34-0': '[Proof assistants vs automated theorem provers]Proof assistants vs automated theorem provers', '1112.1185-1-35-0': '### Proof assistants vs automated theorem provers', '1112.1185-1-36-0': 'Coq is a proof assistant built by [CITATION], see [CITATION] for a good introduction and notice that they call it ""interactive theorem provers"", which is a strict synonymous.', '1112.1185-1-36-1': 'Despite both deal with theorems and their proofs and are mechanized using a computer, proof assistants are not automated theorem provers.', '1112.1185-1-36-2': 'In particular, they are much more expressive than automated theorem provers and this is the reason why they are interactive.', '1112.1185-1-36-3': 'For instance, there is no automated theorem prover implementing coinduction.', '1112.1185-1-36-4': 'Proof assistants are automated only for elementary steps and interactive for the rest.', '1112.1185-1-36-5': 'A specificity of a proof assistant is that it builds a mathematical object called a (formal) proof which can be checked independently, copied, stored and exchanged.', '1112.1185-1-36-6': 'Following [CITATION] and [CITATION], we can consider that they are the tools of the mathematicians of the XXI[MATH] century.', '1112.1185-1-36-7': 'Therefore using a proof assistant is a highly mathematical modern activity.', '1112.1185-1-37-0': ""The mathematical development presented here corresponds to a Coq script which can be found on the following url's:"", '1112.1185-1-38-0': 'http://perso.ens-lyon.fr/pierre.lescanne/COQ/Book/', '1112.1185-1-39-0': 'http://perso.ens-lyon.fr/pierre.lescanne/COQ/Book/SCRIPTS/', '1112.1185-1-40-0': '[Induction vs coinduction]Induction vs coinduction', '1112.1185-1-41-0': '### Induction vs coinduction', '1112.1185-1-42-0': 'To formalize structured finite objects, like finite games, one uses induction, i.e.,', '1112.1185-1-43-0': 'and a definition of the way to build new objects', '1112.1185-1-44-0': 'In the case of infinite objects like infinite games, one characterizes infinite objects not by their construction, but by their behavior.', '1112.1185-1-44-1': 'This characterization by ""observation"" is called coinduction.', '1112.1185-1-44-2': 'Coinduction is associated with the greatest fixpoint.', '1112.1185-1-44-3': 'The proof assistant Coq offers a framework for coinductive definitions and reasonings which are keys of our formalization.', '1112.1185-1-45-0': '[Acknowledgments]Acknowledgments'}","{'1112.1185-2-0-0': 'The aim of this book article is to study infinite games and to prove formally properties in this framework.', '1112.1185-2-0-1': 'In particular, we show that the behavior which leads to speculative crashes or escalation is fully rational.', '1112.1185-2-0-2': 'Indeed it proceeds logically from the statement that resources are infinite.', '1112.1185-2-0-3': 'The reasoning is based on the concept of coinduction conceived by computer scientists to model infinite computations and used by economic agents unknowingly.', '1112.1185-2-1-0': '[Introduction]Introduction [Introduction]Introduction', '1112.1185-2-2-0': 'The aim of thisbook article is to study infinite games and to prove formally some properties in this framework.', '1112.1185-2-2-1': 'As a consequence, we show that the behavior (the madness) of people which leads to speculative crashes or escalation can be proved fully rational.', '1112.1185-2-2-2': 'Indeed it proceeds from the statement that resources are infinite.', '1112.1185-2-2-3': 'The reasoning is based on the concept of coinduction conceived by computer scientists to model infinite computations and used by economic agents unknowingly.', '1112.1185-2-2-4': 'When used consciously, this concept is not as simple as induction and we could paraphrase Newton : ""Modeling the madness of people is more difficult than modeling the motion of planets"".', '1112.1185-2-3-0': 'In this chapter section we present the three words of the title, namely rationality, escalation and infiniteness.', '1112.1185-2-4-0': '[Rationality and escalation]Rationality and escalation [Rationality and escalation]Rationality and escalation', '1112.1185-2-5-0': 'We consider the ability of agents to reason and to conduct their action according to a line of reasoning.', '1112.1185-2-5-1': 'We call this rationality.', '1112.1185-2-5-2': 'This could have been called wisdom as this attributed to King Solomon.', '1112.1185-2-5-3': 'It is not clear that agents act always rationally.', '1112.1185-2-5-4': 'If an agent acts always following a strict reasoning one says that he (she) is rational.', '1112.1185-2-5-5': 'To specify strictly this ability, one associates the agent with a mechanical reasoning device, more specifically a Turing machine or a similar decision mechanism based on abstract computations.', '1112.1185-2-5-6': 'One admits that in making a decision the agent chooses the option which is the better, that is no other will give better payoff, one says that this option is an equilibrium in the sense of game theory.', '1112.1185-2-5-7': 'A well-known game theory situation where rationality of agents is questionable is the so-called escalation.', '1112.1185-2-5-8': 'This is a situation where there is a sequence of decisions wich can be infinite.', '1112.1185-2-5-9': 'If many agents act one after the others in an infinite sequence of decisions and if this sequence leads to situations which are worst and worst for the agents, one speaks of escalation.', '1112.1185-2-5-10': 'One notices the emergence of a property of complex systems, namely the behavior of the system is not the conjunction of this of all the constituents.', '1112.1185-2-5-11': 'Here the individual wisdom becomes a global madness.', '1112.1185-2-6-0': '[Infiniteness]Infiniteness [Infiniteness]Infiniteness infiniteness', '1112.1185-2-7-0': 'It is notorious that there is a wall between finiteness and infiniteness, a fact known to model theorists like [CITATION] [CITATION] and to specialists of functions of real variable.', '1112.1185-2-7-1': '[CITATION] gave an example of the fact that a finite sum of functions differentiable everywhere is differentiable everywhere whereas an infinite sum is differentiable nowhere.', '1112.1185-2-7-2': 'This confusion between finite and infinite is at the origin of the conclusion of the irrationality of the escalation founded on the belief that a property of a infinite mathematical object can be extrapolated from a similar property of finite approximations.', '1112.1185-2-7-3': 'As [CITATION] recalls, ""Most of the classical theorems of logic [for infinite structures] fail for finite structures"" (see [CITATION] for a full development of the finite model theory).', '1112.1185-2-7-4': 'The reciprocal holds obviously: ""Most of the results which hold for finite structures, fail for infinite structures"".', '1112.1185-2-7-5': 'This has been beautifully evidenced in mathematics, when [CITATION] has exhibited his function: [EQUATION]', '1112.1185-2-7-6': 'Every finite sum is differentiable and the limit, i.e., the infinite sum, is not.', '1112.1185-2-7-7': 'In another domain, [CITATION] have proved that the sequence of prime numbers contains arbitrarily long arithmetic progressions.', '1112.1185-2-7-8': 'By extrapolation, there would exist an infinite arithmetic progression of prime numbers, which is trivially not true.', '1112.1185-2-7-9': 'To give another picture, infinite games are to finite games what fractal curves are to smooth curves .', '1112.1185-2-7-10': 'In game theory the error done by the ninetieth century mathematicians Weierstrass would lead to the same issue.', '1112.1185-2-7-11': 'With what we are concerned, a result which holds on finite games does not hold necessarily on infinite games and vice-versa.', '1112.1185-2-7-12': 'More specifically equilibria on finite games are not preserved at the limit on infinite games whereas new types of equilibria emerge on the infinite game not present in the approximation (see the [MATH] game in Section [REF]) and Section [REF].', '1112.1185-2-7-13': 'In particular, we cannot conclude that, whereas the only rational attitude in finite dollar auction would be to stop immediately, it is irrational to escalate in the case of an infinite auction.', '1112.1185-2-7-14': 'We have to keep in mind that in the case of escalation, the game is infinite, therefore reasoning made for finite objects are inappropriate and tools specifically conceived for infinite objects should be adopted.', '1112.1185-2-7-15': ""Like Weierstrass' discovery led to the development of function series, logicians have devised methods for correct deductions on infinite structures."", '1112.1185-2-7-16': 'The right framework for reasoning logically on infinite mathematical objects is called coinductioncoinduction.', '1112.1185-2-8-0': 'The inadequate reasoning on infinite games is as follows: people study finite approximation of infinite games as infinite games truncated at a finite location.', '1112.1185-2-8-1': 'If they obtain the same result on all the approximations, they extrapolate the result to the infinite game as if the limit would have the same property.', '1112.1185-2-8-2': 'But this says nothing since the infiniteness is not the limit of finiteness.', '1112.1185-2-8-3': 'Instead of reexamining their reasoning or considering carefully the hypotheses their reasoning is based upon (is the set of resource infinite?)', '1112.1185-2-8-4': 'they conclude that humans are irrational.', '1112.1185-2-8-5': 'If there is an escalation, then the game is infinite, then the reasoning must be specific to infinite games, that is based on coinduction.', '1112.1185-2-8-6': 'This is only on this basis that one can conclude that humans are rational or irrational.', '1112.1185-2-8-7': 'In no case, a property on the infinite game generated by escalation cane be extrapolated from the same property on finite games.', '1112.1185-2-8-8': 'escalation', '1112.1185-2-9-0': 'In this book article we address these issues.', '1112.1185-2-9-1': 'The games we consider may have arbitrary long histories as well as infinite histories.', '1112.1185-2-9-2': 'history In our games there are two choices at each node, this will not loose generality, since we can simulate finitely branching games in this framework.', '1112.1185-2-9-3': ""By Konig's lemma, finitely branching, specifically binary, infinite games have at least an infinite history."", '1112.1185-2-9-4': 'We are taking the problem of defining formally infinite games, infinite strategy profiles, and infinite histories extremely seriously.', '1112.1185-2-9-5': 'By ""seriously"" we mean that we prepare the land for precise, correct and rigorous reasoning.', '1112.1185-2-9-6': 'For instance, an important issue which is not considered in the literature is how the utilities associated with an infinite history are computed.', '1112.1185-2-9-7': 'To be formal and rigorous, we expect some kinds of recursive definitions, more precisely co-recursive definitions, but then comes the questions of what the payoff associated with an infinite strategy profile is and whether such a payoff exists (see Section [REF]).', '1112.1185-2-10-0': '[Games]Games [Games]Games', '1112.1185-2-11-0': 'Finite extensive games are represented by finite trees and are analyzed through induction.', '1112.1185-2-11-1': 'For instance, in finite extensive games, a concept like subgame perfect equilibrium is defined inductively and receives appropriately the name of backward induction.', '1112.1185-2-11-2': 'Similarly convertibility (an agent changes choices in his strategy) has also an inductive definition and this concept is a key for this of Nash equilibrium.', '1112.1185-2-11-3': 'But induction, which has been designed for finitely based objects, no more works on infinite games, i.e., games underlying infinite trees.', '1112.1185-2-11-4': 'Logicians have proposed a tool, which they call coinduction, to reason on infinite objects.', '1112.1185-2-11-5': 'In short, since objects are infinite and their construction cannot be analyzed, coinduction ""observes"" them, that is looks at how they ""react"" to operations (see Section [REF] for more explanation).', '1112.1185-2-11-6': 'In this book, article, we formalize with coinduction, the concept of infinite game, of infinite strategy profile, of equilibrium in infinite games, of utility (payoff), and of subgame.', '1112.1185-2-11-7': 'We verify on the proof assistant Coq that everything works smoothly and yields interesting consequences.', '1112.1185-2-11-8': 'Thanks to coinduction, examples of apparently paradoxical human behavior are explained logically, demonstrating a rational behavior.', '1112.1185-2-12-0': 'Finite extensive games have been introduced by [CITATION].', '1112.1185-2-12-1': 'But many interesting extensive games are infinite and therefore the theory of infinite extensive games play an important role in game theory, with examples like the dollar auction game Shubik , the generalized centipede game or infinipede or the [MATH] game.', '1112.1185-2-12-2': 'From a formal point of view, the concepts associated with infinite extensive games are not appropriately treated in papers and books.', '1112.1185-2-12-3': 'In particular, there is no clear notion of Nash equilibrium in infinite extensive game and the gap between finiteness and infiniteness is not correctly understood.', '1112.1185-2-12-4': 'For instance in one of the textbooks on game theory, one finds the following definition of games with finite horizon:', '1112.1185-2-13-0': 'If the length of the longest derivation is [...] finite, we say that the game has a finite horizon.', '1112.1185-2-13-1': 'Even a game with a finite horizon may have infinitely many terminal histories, because some player has infinitely many actions after some history.', '1112.1185-2-14-0': 'Notice that in an infinite game with infinite branching it is not always the case that a longest derivation exists.', '1112.1185-2-14-1': 'If a game has only finite histories, but has infinitely many such finite histories of increasing length, there is no longest history.', '1112.1185-2-14-2': 'Before giving a formal definition later in the book, article, let us say intuitively why this definition is inconsistent.', '1112.1185-2-14-3': 'Roughly speaking, a history is a path in the ordered tree which underlies the game.', '1112.1185-2-14-4': 'A counterexample is precisely when the tree is infinitely branching i.e., when ""some player has infinitely many actions"".', '1112.1185-2-15-0': 'Escalation takes place in specific sequential games in which players continue although their payoff decreases on the whole.', '1112.1185-2-15-1': 'The dollar auction game dollar auction has been presented by [CITATION] Shubik as the paradigm of escalation.', '1112.1185-2-15-2': 'He noted that, even though their cost (the opposite of the payoff) basically increases, players may keep bidding.', '1112.1185-2-15-3': ""This attitude was considered as inadequate and when talking about escalation, [CITATION] says this is a paradox, [CITATION] and [CITATION] consider the bidders as irrational, [CITATION] speaks of illogic conflict of escalation and [CITATION] calls it Macbeth effect after Shakespeare's play."", '1112.1185-2-15-4': 'Rebutting these authors, we prove in this book, article, using a reasoning conceived for infinite structures that escalation is logic and that agents are rational, therefore this is not a paradox and we are led to assert that Macbeth is in some way rational.', '1112.1185-2-16-0': 'This escalation phenomenon occurs in infinite sequential games and only there.', '1112.1185-2-16-1': 'escalation To quote [CITATION]:', '1112.1185-2-17-0': 'We could add an upper limit to the amount that anyone is allowed to bid.', '1112.1185-2-17-1': 'However the analysis is confined to the (possibly infinite) game without a specific termination point, as no particularly interesting general phenomena appear if an upper bound is introduced.', '1112.1185-2-18-0': 'Therefore it must be studied in infinite games with adequate tools, i.e., in a framework designed for mathematical infinite objects.', '1112.1185-2-18-1': 'Like [CITATION] we will limit ourselves to two players only.', '1112.1185-2-18-2': 'In auctions, this consists in the two players bidding forever.', '1112.1185-2-18-3': 'This statement of rationality is based on the largely accepted assumption that a player is rational if he adopts a strategy which corresponds to a subgame perfect equilibrium.', '1112.1185-2-18-4': 'To characterize this equilibrium most of the above cited authors consider a finite restriction of the game for which they compute the subgame perfect equilibrium by backward induction.', '1112.1185-2-18-5': 'Then they extrapolate the result obtained on the amputated games to the infinite game.', '1112.1185-2-18-6': 'To justify their practice, they add a new hypothesis on the amount of money the bidders are ready to pay, called the limited bankroll.', '1112.1185-2-18-7': 'By enforcing the finiteness of the game, they exclude clearly escalation.', '1112.1185-2-18-8': 'In the amputated game dollar auction, they conclude that there is a unique subgame perfect equilibrium.', '1112.1185-2-18-9': 'This consists in both agents giving up immediately, not starting the auction and adopting the same choice at each step.', '1112.1185-2-18-10': 'In our formalization in infinite games, we show that extending that case up to infinity is not a subgame perfect equilibrium and we found two subgame perfect equilibria, namely the cases when one agent continues at each step and the other leaves at each step.', '1112.1185-2-18-11': 'Those equilibria which correspond to rational attitudes account for the phenomenon of escalation.', '1112.1185-2-18-12': 'Actually this discrepancy between equilibrium in amputated games extrapolated to infinite extensions and infinite games occurs in a much simpler game than the dollar auction namely the [MATH] game [MATH] game which will be studied in this book.', '1112.1185-2-18-13': 'article.', '1112.1185-2-19-0': '[Coinduction]Coinduction [Coinduction]Coinduction coinduction', '1112.1185-2-20-0': 'Like induction, coinduction is based on a fixpoint, but whereas induction is based on the least fixpoint, coinduction is based on the greatest fixpoint, for an ordering we are not going to describe here as it would go beyond the scope of this book.', '1112.1185-2-20-1': 'article.', '1112.1185-2-20-2': 'Attached to induction is the concept of inductive definition, which characterizes objects like finite lists, finite trees, finite games, finite strategy profiles, etc.', '1112.1185-2-20-3': 'Similarly attached to coinduction is the concept of coinductive definition which characterizes streams (infinite lists), infinite trees, infinite games, infinite strategy profiles etc.', '1112.1185-2-20-4': 'An inductive definition yields the least set that satisfies the definition and a coinductive definition yields the greatest set that satisfies the definition.', '1112.1185-2-20-5': 'Associated with these definitions we have inference principles.', '1112.1185-2-20-6': 'For induction there is the famous induction principle used in backward induction.', '1112.1185-2-20-7': 'On coinductively defined sets of objects there is a principle like induction principle which uses the fact that the set satisfies the definition (proofs by case or by pattern) and that it is the largest set with this property.', '1112.1185-2-20-8': 'Since coinductive definitions allow us building infinite objects, one can imagine constructing a specific category of objects with ""loops"", like the infinite word [MATH] (i.e., [MATH]) infinite word which is made by repeating the sequence [MATH] infinitely many times.', '1112.1185-2-20-9': 'Other examples with trees are given in Section [REF], with infinite games and strategy profiles in Chapter Section [REF].', '1112.1185-2-20-10': 'Such an object is a fixpoint, this means that it contains an object like itself.', '1112.1185-2-20-11': 'For instance [MATH] contains itself.', '1112.1185-2-20-12': 'We say that such an object is defined as a cofixpoint.', '1112.1185-2-20-13': 'To prove a property [MATH] on a cofixpoint [MATH], one assumes [MATH] holds on [MATH] (the [MATH] in [MATH]), considered as a sub-object of [MATH].', '1112.1185-2-20-14': 'If one can prove [MATH] on the whole object (on [MATH]), then one has proved that [MATH] holds on [MATH].', '1112.1185-2-20-15': 'This is called the coinduction principle a concept which comes from [CITATION], [CITATION], and [CITATION] and was introduced in the framework we are considering by [CITATION].', '1112.1185-2-20-16': '[CITATION] gives a good survey with a complete historical account.', '1112.1185-2-20-17': 'To be sure to not be entangled, it is advisable to use a proof assistant which implements coinduction, to build and to check the proof.', '1112.1185-2-20-18': 'Indeed reasoning with coinduction is sometimes so counter-intuitive that the use of a proof assistant is not only advisable but compulsory.', '1112.1185-2-20-19': 'For instance, we were, at first, convinced that in the dollar auction the strategy profile consisting in both agents stopping at every step was a Nash equilibrium, like in the finite case, and only failing in proving it mechanically convinced us of the contrary and we were able to prove the opposite, namely that the strategy profile ""stopping at every step"" is not a Nash equilibrium.', '1112.1185-2-20-20': 'In the examples of Chapter Section [REF], we have checked every statement using Coq and in what follows a sentence like ""we have proved that ..."" means that we have succeeded in building a formal proof in Coq.', '1112.1185-2-21-0': '[Backward coinduction and invariants]Backward coinduction as a method for proving invariants', '1112.1185-2-22-0': '### Backward coinduction as a method for proving invariants', '1112.1185-2-23-0': 'In infinite strategy profiles, the coinduction principles can be seen as follows: a property which holds on a strategy profile of an infinite extensive game is an invariant, i.e., a property which is always true, along the histories and to prove that this is an invariant one proceeds back to the past.', '1112.1185-2-23-1': 'Therefore the name backward coinduction is appropriate, since it proceeds backward the histories, from future to past.', '1112.1185-2-24-0': '[Backward induction vs backward coinduction]Backward induction vs backward coinduction', '1112.1185-2-25-0': '### Backward induction vs backward coinduction', '1112.1185-2-26-0': 'One may wonder the difference between the classical method, we call backward induction and the new method we call backward coinduction.', '1112.1185-2-26-1': 'The main difference is that backward induction starts the reasoning from the leaves, works only on finite games and does not work on infinite games (or on finite strategy profiles), because it requires a well-foundedness to work properly, whereas backward coinduction works on infinite games (or on infinite strategy profiles).', '1112.1185-2-26-2': 'Coinduction is unavoidable on infinite games, since the methods that consists in ""cutting the tail"" and extrapolating the result from finite games or finite strategies profile to infinite games or infinite strategy profiles cannot solve the problem or even approximate it.', '1112.1185-2-26-3': 'It is indeed the same erroneous reasoning as this of the predecessors of Weierstrass who concluded that since: Weierstrass [EQUATION] is differentiable everywhere then [EQUATION] is differentiable everywhere whereas [MATH] is differentiable nowhere.', '1112.1185-2-27-0': ""Much earlier, during the IV[MATH] century BC, the improper use of inductive reasoning allowed Parmenides and Zeno to negate motion and lead to Zeno's paradox of Achilles and the tortoise."", '1112.1185-2-27-1': 'This paradox was reported by Aristotle as follows: Achilles and the tortoise', '1112.1185-2-28-0': '""In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.""', '1112.1185-2-29-0': 'Aristotle, Physics VI:9, 239b15', '1112.1185-2-30-0': ""In Zeno's framework, Zeno's reasoning is correct, because by induction, one can prove that Achilles will never overtake the tortoise."", '1112.1185-2-30-1': 'Indeed this applies to the infinite sequence of races described by Aristotle.', '1112.1185-2-30-2': 'In each race of the sequence, the pursuer starts from where the pursued started previously the race and the pursuer ends where the pursued started in the current race.', '1112.1185-2-30-3': 'By induction one can prove, but only for a sequence of races, the truth of the statement ""Achilles will never overtake the tortoise"".', '1112.1185-2-30-4': 'In each race ""Achilles does not overtake the tortoise"".', '1112.1185-2-30-5': 'For the infinite race for which coinduction would be needed, the result ""Achilles overtakes the tortoise"" holds.', '1112.1185-2-30-6': 'By the way, experience tells us that Achilles would overtake the tortoise in a real race and Zeno has long been refuted by the real world.', '1112.1185-2-31-0': '[Von Neumann and coinduction]Von Neumann and coinduction', '1112.1185-2-32-0': '### Von Neumann and coinduction', '1112.1185-2-33-0': 'As one knows, von Neumann is the creator of game theory, whereas extensive games and equilibrium in non cooperative games are due to [CITATION] and [CITATION].', '1112.1185-2-33-1': 'In the spirit of their creators all those games are finite and backward induction is the basic principle for computing subgame perfect equilibria .', '1112.1185-2-33-2': 'This is not surprising since [CITATION] is also at the origin of the role of well-foundedness in set theory despite he left a door open for a not well-founded membership relation.', '1112.1185-2-33-3': 'As explained by [CITATION], research on anti-foundation initiated by [CITATION] are at the origin of coinduction and were not well known until the work of [CITATION].', '1112.1185-2-34-0': '[Proof assistants vs automated theorem provers]Proof assistants vs automated theorem provers', '1112.1185-2-35-0': '### Proof assistants vs automated theorem provers', '1112.1185-2-36-0': 'Coq is a proof assistant built by [CITATION], see [CITATION] for a good introduction and notice that they call it ""interactive theorem provers"", which is a strict synonymous.', '1112.1185-2-36-1': 'Despite both deal with theorems and their proofs and are mechanized using a computer, proof assistants are not automated theorem provers.', '1112.1185-2-36-2': 'In particular, they are much more expressive than automated theorem provers and this is the reason why they are interactive.', '1112.1185-2-36-3': 'For instance, there is no automated theorem prover implementing coinduction.', '1112.1185-2-36-4': 'Proof assistants are automated only for elementary steps and interactive for the rest.', '1112.1185-2-36-5': 'A specificity of a proof assistant is that it builds a mathematical object called a (formal) proof which can be checked independently, copied, stored and exchanged.', '1112.1185-2-36-6': 'Following [CITATION] and [CITATION], we can consider that they are the tools of the mathematicians of the XXI[MATH] century.', '1112.1185-2-36-7': 'Therefore using a proof assistant is a highly mathematical modern activity.', '1112.1185-2-37-0': ""The mathematical development presented here corresponds to a Coq script which can be found on the following url's:"", '1112.1185-2-38-0': 'http://perso.ens-lyon.fr/pierre.lescanne/COQ/Book/', '1112.1185-2-39-0': 'http://perso.ens-lyon.fr/pierre.lescanne/COQ/Book/SCRIPTS/', '1112.1185-2-40-0': '[Induction vs coinduction]Induction vs coinduction', '1112.1185-2-41-0': '### Induction vs coinduction', '1112.1185-2-42-0': 'To formalize structured finite objects, like finite games, one uses induction, i.e.,', '1112.1185-2-43-0': 'and a definition of the way to build new objects', '1112.1185-2-44-0': 'In the case of infinite objects like infinite games, one characterizes infinite objects not by their construction, but by their behavior.', '1112.1185-2-44-1': 'This characterization by ""observation"" is called coinduction.', '1112.1185-2-44-2': 'Coinduction is associated with the greatest fixpoint.', '1112.1185-2-44-3': 'The proof assistant Coq offers a framework for coinductive definitions and reasonings which are keys of our formalization.', '1112.1185-2-45-0': '[Acknowledgments]Acknowledgments'}","[['1112.1185-1-9-0', '1112.1185-2-9-0'], ['1112.1185-1-9-1', '1112.1185-2-9-1'], ['1112.1185-1-9-2', '1112.1185-2-9-2'], ['1112.1185-1-9-3', '1112.1185-2-9-3'], ['1112.1185-1-9-4', '1112.1185-2-9-4'], ['1112.1185-1-9-5', '1112.1185-2-9-5'], ['1112.1185-1-9-6', '1112.1185-2-9-6'], ['1112.1185-1-9-7', '1112.1185-2-9-7'], ['1112.1185-1-33-0', '1112.1185-2-33-0'], ['1112.1185-1-33-1', '1112.1185-2-33-1'], ['1112.1185-1-33-2', '1112.1185-2-33-2'], ['1112.1185-1-33-3', '1112.1185-2-33-3'], ['1112.1185-1-15-0', '1112.1185-2-15-0'], ['1112.1185-1-15-1', 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'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1112.1185,"{'1112.1185-3-0-0': 'The aim of this book article is to study infinite games and to prove formally properties in this framework.', '1112.1185-3-0-1': 'In particular, we show that the behavior which leads to speculative crashes or escalation is rational.', '1112.1185-3-0-2': 'Indeed it proceeds logically from the statement that resources are infinite.', '1112.1185-3-0-3': 'The reasoning is based on the concept of coinduction conceived by computer scientists to model infinite computations and used by rational agents unknowingly.', '1112.1185-3-1-0': '[Introduction]Introduction [Introduction]Introduction', '1112.1185-3-2-0': 'The aim of thisbook article is to study infinite games and to prove formally some properties in this framework.', '1112.1185-3-2-1': 'As a consequence, we show that the behavior (the madness) of people which leads to speculative crashes or escalation can be proved rational.', '1112.1185-3-2-2': 'Indeed it proceeds from the statement that resources are infinite.', '1112.1185-3-2-3': 'The reasoning is based on the concept of coinduction conceived by computer scientists to model infinite computations and used by rational agents unknowingly.', '1112.1185-3-2-4': 'When used consciously, this concept is not as simple as induction and we could paraphrase Newton : ""Modeling the madness of people is more difficult than modeling the motion of planets"".', '1112.1185-3-3-0': 'In this chapter section we present the three words of the title, namely rationality, escalation and infiniteness.', '1112.1185-3-4-0': '[Rationality and escalation]Rationality and escalation [Rationality and escalation]Rationality and escalation', '1112.1185-3-5-0': 'We consider the ability of agents to reason and to conduct their action according to a line of reasoning.', '1112.1185-3-5-1': 'We call this rationality.', '1112.1185-3-5-2': 'This could have been called wisdom as this attributed to King Solomon.', '1112.1185-3-5-3': 'It is not clear that agents act always rationally.', '1112.1185-3-5-4': 'If an agent acts always following a strict reasoning one says that he (she) is rational.', '1112.1185-3-5-5': 'To specify strictly this ability, one associates the agent with a mechanical reasoning device, more specifically a Turing machine or a similar decision mechanism based on abstract computations.', '1112.1185-3-5-6': 'One admits that in making a decision the agent chooses the option which is the better, that is no other will give better payoff, one says that this option is an equilibrium in the sense of game theory.', '1112.1185-3-5-7': 'A well-known game theory situation where rationality of agents is questionable is the so-called escalation.', '1112.1185-3-5-8': 'This is a situation where there is a sequence of decisions wich can be infinite.', '1112.1185-3-5-9': 'If many agents act one after the others in an infinite sequence of decisions and if this sequence leads to situations which are worst and worst for the agents, one speaks of escalation.', '1112.1185-3-5-10': 'One notices the emergence of a property of complex systems, namely the behavior of the system is not the conjunction of this of all the constituents.', '1112.1185-3-5-11': 'Here the individual wisdom becomes a global madness.', '1112.1185-3-6-0': '[Infiniteness]Infiniteness [Infiniteness]Infiniteness infiniteness', '1112.1185-3-7-0': 'It is notorious that there is a wall between finiteness and infiniteness, a fact known to model theorists like [CITATION] [CITATION] and to specialists of functions of real variable.', '1112.1185-3-7-1': '[CITATION] gave an example of the fact that a finite sum of functions differentiable everywhere is differentiable everywhere whereas an infinite sum is differentiable nowhere.', '1112.1185-3-7-2': 'This confusion between finite and infinite is at the origin of the conclusion of the irrationality of the escalation founded on the belief that a property of a infinite mathematical object can be extrapolated from a similar property of finite approximations.', '1112.1185-3-7-3': 'As [CITATION] recalls, ""Most of the classical theorems of logic [for infinite structures] fail for finite structures"" (see [CITATION] for a full development of the finite model theory).', '1112.1185-3-7-4': 'The reciprocal holds obviously: ""Most of the results which hold for finite structures, fail for infinite structures"".', '1112.1185-3-7-5': 'This has been beautifully evidenced in mathematics, when [CITATION] has exhibited his function: [EQUATION]', '1112.1185-3-7-6': 'Every finite sum is differentiable and the limit, i.e., the infinite sum, is not.', '1112.1185-3-7-7': 'In another domain, [CITATION] have proved that the sequence of prime numbers contains arbitrarily long arithmetic progressions.', '1112.1185-3-7-8': 'By extrapolation, there would exist an infinite arithmetic progression of prime numbers, which is trivially not true.', '1112.1185-3-7-9': 'To give another picture, infinite games are to finite games what fractal curves are to smooth curves and .', '1112.1185-3-7-10': 'In game theory the error done by the ninetieth century mathematicians Weierstrass would lead to the same issue.', '1112.1185-3-7-11': 'With what we are concerned, a result which holds on finite games does not hold necessarily on infinite games and vice-versa.', '1112.1185-3-7-12': 'More specifically equilibria on finite games are not preserved at the limit on infinite games whereas new types of equilibria emerge on the infinite game not present in the approximation (see the [MATH] game in Section [REF]) and Section [REF].', '1112.1185-3-7-13': 'In particular, we cannot conclude that, whereas the only rational attitude in finite dollar auction would be to stop immediately, it is irrational to escalate in the case of an infinite auction.', '1112.1185-3-7-14': 'We have to keep in mind that in the case of escalation, the game is infinite, therefore reasoning made for finite objects are inappropriate and tools specifically conceived for infinite objects should be adopted.', '1112.1185-3-7-15': ""Like Weierstrass' discovery led to the development of function series, logicians have devised methods for correct deductions on infinite structures."", '1112.1185-3-7-16': 'The right framework for reasoning logically on infinite mathematical objects is called coinductioncoinduction.', '1112.1185-3-8-0': 'The inadequate reasoning on infinite games is as follows: people study finite approximation of infinite games as infinite games truncated at a finite location.', '1112.1185-3-8-1': 'If they obtain the same result on all the approximations, they extrapolate the result to the infinite game as if the limit would have the same property.', '1112.1185-3-8-2': 'But this says nothing since the infiniteness is not the limit of finiteness.', '1112.1185-3-8-3': 'Instead of reexamining their reasoning or considering carefully the hypotheses their reasoning is based upon, namely wondering whether the set of resource is infinite, they conclude that humans are irrational.', '1112.1185-3-8-4': 'If there is an escalation, then the game is infinite, then the reasoning must be specific to infinite games, that is based on coinduction.', '1112.1185-3-8-5': 'This is only on this basis that one can conclude that humans are rational or irrational.', '1112.1185-3-8-6': 'In no case, a property on the infinite game generated by escalation cane be extrapolated from the same property on finite games.', '1112.1185-3-8-7': 'escalation', '1112.1185-3-9-0': 'In this book article we address these issues.', '1112.1185-3-9-1': 'The games we consider may have arbitrary long histories as well as infinite histories.', '1112.1185-3-9-2': 'history In our games there are two choices at each node, this will not loose generality, since we can simulate finitely branching games in this framework.', '1112.1185-3-9-3': ""By Konig's lemma, finitely branching, specifically binary, infinite games have at least an infinite history."", '1112.1185-3-9-4': 'We are taking the problem of defining formally infinite games, infinite strategy profiles, and infinite histories extremely seriously.', '1112.1185-3-9-5': 'By ""seriously"" we mean that we prepare the land for precise, correct and rigorous reasoning.', '1112.1185-3-9-6': 'For instance, an important issue which is not considered in the literature is how the utilities associated with an infinite history are computed.', '1112.1185-3-9-7': 'To be formal and rigorous, we expect some kinds of recursive definitions, more precisely co-recursive definitions, but then comes the questions of what the payoff associated with an infinite strategy profile is and whether such a payoff exists (see Section [REF]).', '1112.1185-3-10-0': '[Games]Games [Games]Games', '1112.1185-3-11-0': 'Finite extensive games are represented by finite trees and are analyzed through induction.', '1112.1185-3-11-1': 'For instance, in finite extensive games, a concept like subgame perfect equilibrium is defined inductively and receives appropriately the name of backward induction.', '1112.1185-3-11-2': 'Similarly convertibility (an agent changes choices in his strategy) has also an inductive definition and this concept is a key for this of Nash equilibrium.', '1112.1185-3-11-3': 'But induction, which has been designed for finitely based objects, no more works on infinite games, i.e., games underlying infinite trees.', '1112.1185-3-11-4': 'Logicians have proposed a tool, which they call coinduction, to reason on infinite objects.', '1112.1185-3-11-5': 'In short, since objects are infinite and their construction cannot be analyzed, coinduction ""observes"" them, that is looks at how they ""react"" to operations (see Section [REF] for more explanation).', '1112.1185-3-11-6': 'In this book, article, we formalize with coinduction, the concept of infinite game, of infinite strategy profile, of equilibrium in infinite games, of utility (payoff), and of subgame.', '1112.1185-3-11-7': 'We verify on the proof assistant Coq that everything works smoothly and yields interesting consequences.', '1112.1185-3-11-8': 'Thanks to coinduction, examples of apparently paradoxical human behavior are explained logically, demonstrating a rational behavior.', '1112.1185-3-12-0': 'Finite extensive games have been introduced by [CITATION].', '1112.1185-3-12-1': 'But many interesting extensive games are infinite and therefore the theory of infinite extensive games play an important role in game theory, with examples like the dollar auction game Shubik , the generalized centipede game or infinipede or the [MATH] game.', '1112.1185-3-12-2': 'From a formal point of view, the concepts associated with infinite extensive games are not appropriately treated in papers and books.', '1112.1185-3-12-3': 'In particular, there is no clear notion of Nash equilibrium in infinite extensive game and the gap between finiteness and infiniteness is not correctly understood.', '1112.1185-3-12-4': 'For instance in one of the textbooks on game theory, one finds the following definition of games with finite horizon:', '1112.1185-3-13-0': 'If the length of the longest derivation is [...] finite, we say that the game has a finite horizon.', '1112.1185-3-13-1': 'Even a game with a finite horizon may have infinitely many terminal histories, because some player has infinitely many actions after some history.', '1112.1185-3-14-0': 'Notice that in an infinite game with infinite branching it is not always the case that a longest derivation exists.', '1112.1185-3-14-1': 'If a game has only finite histories, but has infinitely many such finite histories of increasing length, there is no longest history.', '1112.1185-3-14-2': 'Before giving a formal definition later in the book, article, let us say intuitively why this definition is inconsistent.', '1112.1185-3-14-3': 'Roughly speaking, a history is a path in the ordered tree which underlies the game.', '1112.1185-3-14-4': 'A counterexample is precisely when the tree is infinitely branching i.e., when ""some player has infinitely many actions"".', '1112.1185-3-15-0': 'Escalation takes place in specific sequential games in which players continue although their payoff decreases on the whole.', '1112.1185-3-15-1': 'The dollar auction game dollar auction has been presented by [CITATION] Shubik as the paradigm of escalation.', '1112.1185-3-15-2': 'He noted that, even though their cost (the opposite of the payoff) basically increases, players may keep bidding.', '1112.1185-3-15-3': ""This attitude was considered as inadequate and when talking about escalation, [CITATION] says this is a paradox, [CITATION] and [CITATION] consider the bidders as irrational, [CITATION] speaks of illogic conflict of escalation and [CITATION] calls it Macbeth effect after Shakespeare's play."", '1112.1185-3-15-4': 'Rebutting these authors, we prove in this book, article, using a reasoning conceived for infinite structures that escalation is logic and that agents are rational, therefore this is not a paradox and we are led to assert that Macbeth is in some way rational.', '1112.1185-3-16-0': 'This escalation phenomenon occurs in infinite sequential games and only there.', '1112.1185-3-16-1': 'escalation To quote [CITATION]:', '1112.1185-3-17-0': 'We could add an upper limit to the amount that anyone is allowed to bid.', '1112.1185-3-17-1': 'However the analysis is confined to the (possibly infinite) game without a specific termination point, as no particularly interesting general phenomena appear if an upper bound is introduced.', '1112.1185-3-18-0': 'Therefore it must be studied in infinite games with adequate tools, i.e., in a framework designed for mathematical infinite objects.', '1112.1185-3-18-1': 'Like [CITATION] we will limit ourselves to two players only.', '1112.1185-3-18-2': 'In auctions, this consists in the two players bidding forever.', '1112.1185-3-18-3': 'This statement of rationality is based on the largely accepted assumption that a player is rational if he adopts a strategy which corresponds to a subgame perfect equilibrium.', '1112.1185-3-18-4': 'To characterize this equilibrium most of the above cited authors consider a finite restriction of the game for which they compute the subgame perfect equilibrium by backward induction.', '1112.1185-3-18-5': 'Then they extrapolate the result obtained on the amputated games to the infinite game.', '1112.1185-3-18-6': 'To justify their practice, they add a new hypothesis on the amount of money the bidders are ready to pay, called the limited bankroll.', '1112.1185-3-18-7': 'By enforcing the finiteness of the game, they exclude clearly escalation.', '1112.1185-3-18-8': 'In the amputated game dollar auction, they conclude that there is a unique subgame perfect equilibrium.', '1112.1185-3-18-9': 'This consists in both agents giving up immediately, not starting the auction and adopting the same choice at each step.', '1112.1185-3-18-10': 'In our formalization in infinite games, we show that extending that case up to infinity is not a subgame perfect equilibrium and we found two subgame perfect equilibria, namely the cases when one agent continues at each step and the other leaves at each step.', '1112.1185-3-18-11': 'Those equilibria which correspond to rational attitudes account for the phenomenon of escalation.', '1112.1185-3-18-12': 'Actually this discrepancy between equilibrium in amputated games extrapolated to infinite extensions and infinite games occurs in a much simpler game than the dollar auction namely the [MATH] game [MATH] game which will be studied in this book.', '1112.1185-3-18-13': 'article.', '1112.1185-3-19-0': '[Coinduction]Coinduction [Coinduction]Coinduction coinduction', '1112.1185-3-20-0': 'Like induction, coinduction is based on a fixpoint, but whereas induction is based on the least fixpoint, coinduction is based on the greatest fixpoint, for an ordering that we are not going to describe here, since it would go beyond the scope of this book.', '1112.1185-3-20-1': 'article.', '1112.1185-3-20-2': 'Attached to induction is the concept of inductive definition, which characterizes objects like finite lists, finite trees, finite games, finite strategy profiles, etc.', '1112.1185-3-20-3': 'Similarly attached to coinduction is the concept of coinductive definition which characterizes streams (infinite lists), infinite trees, infinite games, infinite strategy profiles etc.', '1112.1185-3-20-4': 'An inductive definition yields the least set that satisfies the definition and a coinductive definition yields the greatest set that satisfies the definition.', '1112.1185-3-20-5': 'Associated with these definitions we have inference principles.', '1112.1185-3-20-6': 'For induction there is the famous induction principle used in backward induction.', '1112.1185-3-20-7': 'On coinductively defined sets of objects there is a principle like induction principle which uses the fact that the set satisfies the definition (proofs by case or by pattern) and that it is the largest set with this property.', '1112.1185-3-20-8': 'Since coinductive definitions allow us building infinite objects, one can imagine constructing a specific category of objects with ""loops"", like the infinite word [MATH] (i.e., [MATH]) infinite word which is made by repeating the sequence [MATH] infinitely many times.', '1112.1185-3-20-9': 'Other examples with trees are given in Section [REF], with infinite games and strategy profiles in Chapter Section [REF].', '1112.1185-3-20-10': 'Such an object is a fixpoint, this means that it contains an object like itself.', '1112.1185-3-20-11': 'For instance [MATH] contains itself.', '1112.1185-3-20-12': 'We say that such an object is defined as a cofixpoint.', '1112.1185-3-20-13': 'To prove a property [MATH] on a cofixpoint [MATH], one assumes [MATH] holds on [MATH] (the [MATH] in [MATH]), considered as a sub-object of [MATH].', '1112.1185-3-20-14': 'If one can prove [MATH] on the whole object (on [MATH]), then one has proved that [MATH] holds on [MATH].', '1112.1185-3-20-15': 'This is called the coinduction principle a concept which comes from [CITATION], [CITATION], and [CITATION] and was introduced in the framework we are considering by [CITATION].', '1112.1185-3-20-16': '[CITATION] gives a good survey with a complete historical account.', '1112.1185-3-20-17': 'To be sure to not be entangled, it is advisable to use a proof assistant which implements coinduction, to build and to check the proof.', '1112.1185-3-20-18': 'Indeed reasoning with coinduction is sometimes so counter-intuitive that the use of a proof assistant is not only advisable but compulsory.', '1112.1185-3-20-19': 'For instance, we were, at first, convinced that in the dollar auction the strategy profile consisting in both agents stopping at every step was a Nash equilibrium, like in the finite case, and only failing in proving it mechanically convinced us of the contrary and we were able to prove the opposite, namely that the strategy profile ""stopping at every step"" is not a Nash equilibrium.', '1112.1185-3-20-20': 'In the examples of Chapter Section [REF], we have checked every statement using Coq and in what follows a sentence like ""we have proved that ..."" means that we have succeeded in building a formal proof in Coq.', '1112.1185-3-21-0': '[Backward coinduction and invariants]Backward coinduction as a method for proving invariants', '1112.1185-3-22-0': '### Backward coinduction as a method for proving invariants', '1112.1185-3-23-0': 'In infinite strategy profiles, the coinduction principles can be seen as follows: a property which holds on a strategy profile of an infinite extensive game is an invariant, i.e., a property which is always true, along the histories and to prove that this is an invariant one proceeds back to the past.', '1112.1185-3-23-1': 'Therefore the name backward coinduction is appropriate, since it proceeds backward the histories, from future to past.', '1112.1185-3-24-0': '[Backward induction vs backward coinduction]Backward induction vs backward coinduction', '1112.1185-3-25-0': '### Backward induction vs backward coinduction', '1112.1185-3-26-0': 'One may wonder the difference between the classical method, we call backward induction and the new method we call backward coinduction.', '1112.1185-3-26-1': 'The main difference is that backward induction starts the reasoning from the leaves, works only on finite games and does not work on infinite games (or on finite strategy profiles), because it requires a well-foundedness to work properly, whereas backward coinduction works on infinite games (or on infinite strategy profiles).', '1112.1185-3-26-2': 'Coinduction is unavoidable on infinite games, since the methods that consists in ""cutting the tail"" and extrapolating the result from finite games or finite strategies profile to infinite games or infinite strategy profiles cannot solve the problem or even approximate it.', '1112.1185-3-26-3': 'It is indeed the same erroneous reasoning as this of the predecessors of Weierstrass who concluded that since: Weierstrass [EQUATION] is differentiable everywhere then [EQUATION] is differentiable everywhere whereas [MATH] is differentiable nowhere.', '1112.1185-3-27-0': ""Much earlier, during the IV[MATH] century BC, the improper use of inductive reasoning allowed Parmenides and Zeno to negate motion and lead to Zeno's paradox of Achilles and the tortoise."", '1112.1185-3-27-1': 'This paradox was reported by Aristotle as follows: Achilles and the tortoise', '1112.1185-3-28-0': '""In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.""', '1112.1185-3-29-0': 'Aristotle, Physics VI:9, 239b15', '1112.1185-3-30-0': ""In Zeno's framework, Zeno's reasoning is correct, because by induction, one can prove that Achilles will never overtake the tortoise."", '1112.1185-3-30-1': 'Indeed this applies to the infinite sequence of races described by Aristotle.', '1112.1185-3-30-2': 'In each race of the sequence, the pursuer starts from where the pursued started previously the race and the pursuer ends where the pursued started in the current race.', '1112.1185-3-30-3': 'By induction one can prove, but only for a sequence of races, the truth of the statement ""Achilles will never overtake the tortoise"".', '1112.1185-3-30-4': 'In each race ""Achilles does not overtake the tortoise"".', '1112.1185-3-30-5': 'For the infinite race for which coinduction would be needed, the result ""Achilles overtakes the tortoise"" holds.', '1112.1185-3-30-6': 'By the way, experience tells us that Achilles would overtake the tortoise in a real race and Zeno has long been refuted by the real world.', '1112.1185-3-31-0': '[Von Neumann and coinduction]Von Neumann and coinduction', '1112.1185-3-32-0': '### Von Neumann and coinduction', '1112.1185-3-33-0': 'As one knows, von Neumann is the creator of game theory, whereas extensive games and equilibrium in non cooperative games are due to [CITATION] and [CITATION].', '1112.1185-3-33-1': 'In the spirit of their creators all those games are finite and backward induction is the basic principle for computing subgame perfect equilibria .', '1112.1185-3-33-2': 'This is not surprising since [CITATION] is also at the origin of the role of well-foundedness in set theory despite he left a door open for a not well-founded membership relation.', '1112.1185-3-33-3': 'As explained by [CITATION], research on anti-foundation initiated by [CITATION] are at the origin of coinduction and were not well known until the work of [CITATION].', '1112.1185-3-34-0': '[Proof assistants vs automated theorem provers]Proof assistants vs automated theorem provers', '1112.1185-3-35-0': '### Proof assistants vs automated theorem provers', '1112.1185-3-36-0': 'Coq is a proof assistant built by [CITATION], see [CITATION] for a good introduction and notice that they call it ""interactive theorem provers"", which is a strict synonymous.', '1112.1185-3-36-1': 'Despite both deal with theorems and their proofs and are mechanized using a computer, proof assistants are not automated theorem provers.', '1112.1185-3-36-2': 'In particular, they are much more expressive than automated theorem provers and this is the reason why they are interactive.', '1112.1185-3-36-3': 'For instance, there is no automated theorem prover implementing coinduction.', '1112.1185-3-36-4': 'Proof assistants are automated only for elementary steps and interactive for the rest.', '1112.1185-3-36-5': 'A specificity of a proof assistant is that it builds a mathematical object called a (formal) proof which can be checked independently, copied, stored and exchanged.', '1112.1185-3-36-6': 'Following [CITATION] and [CITATION], we can consider that they are the tools of the mathematicians of the XXI[MATH] century.', '1112.1185-3-36-7': 'Therefore using a proof assistant is a highly mathematical modern activity.', '1112.1185-3-37-0': ""The mathematical development presented here corresponds to a Coq script which can be found on the following url's:"", '1112.1185-3-38-0': 'http://perso.ens-lyon.fr/pierre.lescanne/COQ/Book/', '1112.1185-3-39-0': 'http://perso.ens-lyon.fr/pierre.lescanne/COQ/Book/SCRIPTS/', '1112.1185-3-40-0': '[Induction vs coinduction]Induction vs coinduction', '1112.1185-3-41-0': '### Induction vs coinduction', '1112.1185-3-42-0': 'To formalize structured finite objects, like finite games, one uses induction, i.e.,', '1112.1185-3-43-0': 'and a definition of the way to build new objects', '1112.1185-3-44-0': 'In the case of infinite objects like infinite games, one characterizes infinite objects not by their construction, but by their behavior.', '1112.1185-3-44-1': 'This characterization by ""observation"" is called coinduction.', '1112.1185-3-44-2': 'Coinduction is associated with the greatest fixpoint.', '1112.1185-3-44-3': 'The proof assistant Coq offers a framework for coinductive definitions and reasonings which are keys of our formalization.', '1112.1185-3-45-0': '[Acknowledgments]Acknowledgments'}",,
1612.06149,"{'1612.06149-1-0-0': 'Maximum-a-posteriori estimation has become the main Bayesian estimation methodology in many areas of modern data science such as mathematical imaging and machine learning, where high dimensionality is addressed by using models that are log-concave and where the posterior mode can be computed very efficiently by using convex optimisation algorithms.', '1612.06149-1-0-1': 'However, despite its success and rapid adoption, maximum-a-posteriori estimation is not theoretically well understood yet.', '1612.06149-1-0-2': 'This paper presents a new decision-theoretic derivation of maximum-a-posteriori estimation in Bayesian models that are log-concave.', '1612.06149-1-0-3': 'Our analysis is based on differential geometry and proceeds as follows.', '1612.06149-1-0-4': 'First, we exploit the log-concavity of the model to induce a Riemannian geometry on the parameter space, and use differential geometry to identify the natural or canonical loss function to perform Bayesian point estimation in that space.', '1612.06149-1-0-5': 'We then show that for log-concave models the canonical loss is the Bregman divergence associated with the negative log posterior density, and that the maximum-a-posteriori estimator is the Bayesian estimator that minimises the expected loss.', '1612.06149-1-0-6': 'We also show that the posterior mean or minimum mean square error estimator is the Bayesian estimator that minimises the dual canonical loss, and establish general performance guarantees for both maximum-a-posteriori and minimum mean square error estimation.', '1612.06149-1-0-7': 'These results provide a new understanding of these estimation methodologies under log-concavity, and reveal new insights about their good empirical performance and about the roles that log-concavity plays in high dimensional inference problems.', '1612.06149-1-1-0': '# Introduction', '1612.06149-1-2-0': 'We consider the Bayesian estimation of an unknown quantify of interest [MATH] from an observation [MATH] [CITATION].', '1612.06149-1-2-1': 'We focus on Bayesian models whose posterior distribution is log-concave, i.e., [EQUATION] for some proper convex function [MATH].', '1612.06149-1-2-2': 'Such models are ubiquitous in modern data science, particularly for problems where [MATH] is high dimensional (e.g., [MATH]).', '1612.06149-1-2-3': 'For example, statistical imaging and machine learning methods rely strongly on log-concave models of the form [MATH] for some linear operators [MATH] and [MATH], convex regulariser [MATH], and convex set constraint [MATH], and where typically the dimension [MATH] [CITATION].', '1612.06149-1-3-0': 'Because drawing conclusions directly from [MATH] is difficult, Bayesian methods generally deliver summaries of [MATH], namely Bayes point estimators, which summarises the information in [MATH] optimally in the following decision-theoretic sense [CITATION]:', '1612.06149-1-4-0': 'Let [MATH] be a loss function that quantifies the difference between two points in [MATH].', '1612.06149-1-4-1': 'A Bayes estimator associated with [MATH] is any estimator that minimises the posterior expected loss, i.e., [EQUATION].', '1612.06149-1-5-0': 'The loss function [MATH] is specified by the analyst and usually verifies the following general conditions:', '1612.06149-1-6-0': 'Observe that [MATH] is not necessarily symmetric, i.e., [MATH] because the arguments of [MATH] have clearly different roles in the decision problem.', '1612.06149-1-7-0': 'Ideally [MATH] should be chosen carefully based on specific aspects of the problem and application considered.', '1612.06149-1-7-1': 'This is particularly important for instance in imaging problems that are ill-posed or ill-conditioned, where this choice can significantly impact estimation results.', '1612.06149-1-7-2': 'However, specifying a bespoke loss function for high dimensional problems is not easy, and as a result most methods reported in the literature use default losses and estimators.', '1612.06149-1-8-0': 'In particular, Bayesian methods in engineering fields such as imaging have traditionally used the minimum mean square estimator, which is given by the posterior mean [MATH].', '1612.06149-1-8-1': 'This estimator is widely regarded as a gold standard, in part because of its good empirical performance and favourable theoretical properties, and also perhaps in part because of cultural heritage.', '1612.06149-1-8-2': 'From Bayesian decision theory, minimum mean square estimation is optimal with respect to the entire class of quadratic loss functions of the form [MATH] with [MATH] positive definite [CITATION].', '1612.06149-1-8-3': 'This class provides a second order approximation to any strongly convex loss function, and hence [MATH] is also a proxy for many other Bayesian estimators.', '1612.06149-1-8-4': 'Also, the quadratic loss is directly related to the Euclidean squared distance, giving [MATH] a clear geometric interpretation.', '1612.06149-1-8-5': 'In addition, it has been established in [CITATION] that [MATH] is also optimal w.r.t. the second argument of any Bregman divergence (i.e., any loss function of the form [MATH] for a convex function [MATH]), a more general class of loss functions that includes quadratic losses.', '1612.06149-1-9-0': 'Unfortunately, calculating [MATH] in high dimensional models can be very difficult because it requires solving integrals that are often too computationally expensive for the applications considered.', '1612.06149-1-9-1': 'This has stimulated much research on the topic, from fast Monte Carlo simulation methods to efficient approximations with deterministic algorithms [CITATION].', '1612.06149-1-9-2': 'But with ever increasingly large problems and datasets, many applied fields have progressively focused on alternatives to minimum mean square error estimation.', '1612.06149-1-10-0': 'In particular, modern imaging and machine learning methods rely strongly on maximum-a-posteriori estimation [EQUATION] whose calculation is a convex problem that can often be solved very efficiently, even in very high dimensions (e.g., [MATH]), by using convex optimisation techniques [CITATION].', '1612.06149-1-10-1': 'Interestingly, modern non-statistical imaging and machine learning methods also predominately solve problems by convex optimisation, and their solutions are often equivalent to performing maximum-a-posteriori estimation for some implicit Bayesian model.', '1612.06149-1-11-0': 'There is abundant evidence that maximum-a-posteriori estimation delivers accurate results for a wide range of log-concave models and applications.', '1612.06149-1-11-1': 'However, from a theoretical viewpoint maximum-a-posteriori estimation is not well understood.', '1612.06149-1-11-2': 'Currently the predominant view is that maximum-a-posteriori estimation is not formal Bayes estimation in the decision-theoretic sense postulated by Definition [REF] because it does not minimise a known expected loss.', '1612.06149-1-11-3': 'The prevailing interpretation is that maximum-a-posteriori estimation is in fact an approximation arising from the loss [MATH] when [MATH].', '1612.06149-1-11-4': 'However, this derivation does not lead to a proper Bayesian estimator because [MATH] becomes degenerate as [MATH] vanishes.', '1612.06149-1-11-5': 'Perhaps more importantly, the resulting loss is very difficult to motivate for inference problems in [MATH] [CITATION].', '1612.06149-1-12-0': 'Furthermore, most other theoretical results for maximum-a-posteriori estimation only hold for very specific models, or have been derived by adopting analyses that are extrinsic to the Bayesian decision theory framework (e.g. by analysing maximum-a-posteriori estimation as constrained or regularised least-squares regression [CITATION]).', '1612.06149-1-12-1': 'For example, when [MATH] is symmetric we have [MATH], and thus maximum-a-posteriori estimation inherits the favourable properties of minimum square error estimation.', '1612.06149-1-12-2': 'This result has been partially extended to some denoising models of the form [MATH] in [CITATION], where it is shown that maximum-a-posteriori estimation coincides with minimum square error estimation with a different model [MATH].', '1612.06149-1-12-3': 'It follows that for these models maximum-a-posteriori estimation is decision-theoretic Bayesian estimation w.r.t. the weighted loss [MATH].', '1612.06149-1-12-4': 'This is a post-hoc loss, but the result is interesting because it highlights that a single estimator may have a plurality of origins.', '1612.06149-1-12-5': 'Lastly, Burger Lucka [CITATION] recently established that maximum-a-posteriori estimation is decision-theoretic Bayesian estimation for all linear Gaussian models of the form [MATH], where [MATH] is a known linear operator, [MATH] a known noise covariance, and [MATH] is convex and Lipchitz continuous.', '1612.06149-1-12-6': 'More precisely, that paper shows that for these models maximum-a-posteriori estimation is optimal w.r.t. the loss [MATH], where [MATH] is the Bregman divergence associated with [MATH].', '1612.06149-1-12-7': 'Observe that in this case the loss function defining the estimator is determined by the model considered.', '1612.06149-1-13-0': 'In this paper we revisit the choice of the loss function for Bayesian point estimation in the context of log-concave models.', '1612.06149-1-13-1': 'A main novelty is that, instead of specifying the loss directly, we seek to derive it automatically from the geometry of the model.', '1612.06149-1-13-2': 'Precisely, we use differential geometry to derive the loss from the geometry of the parameter space.', '1612.06149-1-13-3': 'We show that under some regularity assumptions, the log-concavity of the posterior distribution induces a specific Riemannian geometry on the parameter space, and that taking into account this space geometry naturally leads to a natural or canonical loss function to perform Bayesian estimation.', '1612.06149-1-13-4': ""Following on from this, we then establish that the canonical loss for the space is given by the Bregman divergence associated with the model's negative log density, and that the Bayesian estimator associated with this loss is the maximum-a-posteriori estimator."", '1612.06149-1-13-5': 'We also show that the minimum mean square error estimator is the Bayesian estimator associated with the dual canonical loss.', '1612.06149-1-14-0': 'The remainder of the paper is organised as follows: section [REF] introduces the elements of differential geometry that are essential to our analysis.', '1612.06149-1-14-1': 'In section [REF] we present our main theoretical result: a decision-theoretic and differential-geometric derivation of maximum-a-posteriori and minimum mean square error estimation, as well as general bounds on the estimation error involved.', '1612.06149-1-14-2': 'Proofs are reported in the appendix.', '1612.06149-1-15-0': '# Riemannian geometry and the canonical divergence', '1612.06149-1-16-0': 'In this section we recall some elements of differential geometry that are necessary for our analysis.', '1612.06149-1-16-1': 'For a detailed introduction to this topic we refer the reader to [CITATION].', '1612.06149-1-17-0': 'An [MATH]-dimensional Riemannian manifold [MATH], with metric [MATH] and global coordinate system [MATH], is a vector space that behaves locally as an Euclidean space.', '1612.06149-1-17-1': 'Precisely, for any point [MATH] we have a tangent space [MATH] with inner product [MATH] and norm [MATH].', '1612.06149-1-17-2': 'This geometry is local and may vary smoothly from [MATH] to neighbouring tangent spaces.', '1612.06149-1-17-3': ""These variations are encoded in the manifold's affine connection [MATH], with coefficients given by [MATH]."", '1612.06149-1-18-0': 'Moreover, similarly to Euclidean spaces, the manifold [MATH] supports divergence functions.', '1612.06149-1-19-0': 'A function [MATH] is a divergence function on [MATH] if the following conditions hold for any [MATH]:', '1612.06149-1-20-0': 'The class of divergence functions coincides with that of loss functions for Bayesian point estimation considered in section [REF] with mild additional regularity conditions, hence divergence functions are suitable candidates to define Bayesian estimators.', '1612.06149-1-20-1': 'Divergence functions also provide a link to the differential geometry of the space, which allows relating geometry and Bayesian estimation, or more generally geometry and Bayesian decision theory.', '1612.06149-1-20-2': 'This relationship has been used previously to analyse decision problems from a Riemannian geometric viewpoint, leading to the so-called decision geometry framework [CITATION].', '1612.06149-1-20-3': 'Here we adopt an opposite perspective: we start by defining a Riemannian manifold [MATH] and then use the relationship to identify the divergence functions that arise naturally in that space.', '1612.06149-1-20-4': 'In particular, we focus on the so-called canonical divergence on [MATH], which is a generalisation of the Euclidean squared distance to this kind of manifold [CITATION].', '1612.06149-1-20-5': '[Canonical divergence [CITATION]] For any two points [MATH], the [MATH]-canonical divergence is given by [EQUATION] where [MATH] is the [MATH]-geodesic connecting [MATH] and [MATH].', '1612.06149-1-21-0': 'It is easy to check that [REF] reduces to the Euclidean squared distance [MATH] when [MATH] is the Euclidean space with inner product [MATH].', '1612.06149-1-21-1': 'If [MATH] is not Euclidean then [MATH] is consistent with the local Euclidean geometry of the space.', '1612.06149-1-21-2': 'That is, for any point [MATH] in the neighbourhood of [MATH] we have [MATH], where [MATH] is the Euclidean norm of the tangent space [MATH] (a higher order approximation of [MATH] is also possible by using the affine connection [MATH]).', '1612.06149-1-21-3': 'And if we use the decision geometry framework [CITATION] to derive the Riemannian geometry induced by [MATH] on [MATH] we obtain [EQUATION] (here [MATH] and [MATH] denote differentiation w.r.t. the first and second components of [MATH] respective), which indicates that [MATH] is specified by [MATH] and that it induces the same space geometry that originated it in the first place.', '1612.06149-1-22-0': 'Finally, to gain a geometric intuition for [MATH] it is useful to compare it to the length of the affine geodesic between [MATH] and [MATH].', '1612.06149-1-22-1': 'Precisely, by noting that the squared length of a curve [MATH] is given by [MATH], we observe that [MATH] is essentially the squared length of the [MATH]-geodesic [MATH] weighted linearly along the path from [MATH] to [MATH].', '1612.06149-1-22-2': 'This weighting in [REF] guarantees that [MATH] is convex in [MATH] (a necessary condition to define a divergence function).', '1612.06149-1-22-3': 'It also introduces an asymmetry, i.e., generally [MATH].', '1612.06149-1-22-4': 'We will see that this asymmetry has deep implications for Bayesian estimation.', '1612.06149-1-23-0': '# A differential-geometric derivation of maximum-a-posteriori and minimum mean square error estimation', '1612.06149-1-24-0': '## Canonical Bayesian estimators: from differential geometry to decision theory', '1612.06149-1-25-0': 'In this section we use differential geometry to relate [MATH] to the loss functions used for Bayesian point estimation.', '1612.06149-1-25-1': 'Precisely, we exploit the log-concavity of [MATH] to induce a Riemannian geometry on the parameter space.', '1612.06149-1-25-2': 'This in turn defines a canonical loss for that space and two Bayesian estimators w.r.t. to this loss: a primal estimator related to [MATH] and a dual estimator related to the dual divergence [MATH].', '1612.06149-1-25-3': 'We consider first the case where [MATH] is smooth and strongly log-concave, and analyse the effect of relaxing these assumptions in section [REF].', '1612.06149-1-26-0': '[Canonical Bayesian estimators] Suppose that [MATH] is strongly convex, continuous, and [MATH] on [MATH].', '1612.06149-1-26-1': 'Let [MATH] denote the Riemannian manifold induced by [MATH], with metric coefficients given by [MATH].', '1612.06149-1-26-2': 'Then, the canonical divergence on [MATH] is the Bregman divergence associated [MATH], i.e., [EQUATION].', '1612.06149-1-26-3': 'In addition, the Bayesian estimator associated with [MATH] is unique and is given by the maximum-a-posteriori estimator, [EQUATION]', '1612.06149-1-26-4': 'The Bayesian estimation associated with the dual canonical divergence [MATH] is also unique and given by the minimum mean squared error estimator [EQUATION]', '1612.06149-1-26-5': 'The proof is postponed to section [REF].', '1612.06149-1-27-0': 'Theorem [REF] provides several valuable new insights into maximum-a-posteriori and minimum mean square error estimation for the convex estimation problems considered.', '1612.06149-1-27-1': 'First, maximum-a-posteriori estimation stems from Bayesian decision theory, and hence it stands on the same theoretical footing as the core Bayesian methodologies such as minimum mean square error estimation.', '1612.06149-1-27-2': 'Second, the conventional definition of the maximum-a-posteriori estimator as the maximiser [MATH] is mainly algorithmic for these problems, useful to highlight that these estimators take the form of a convex optimisation problem that can be solved efficiently by convex optimisation (which is an important computational advantage over most other Bayesian point estimators).', '1612.06149-1-27-3': 'Third, Theorem [REF] also reveals a surprising form of duality between maximum-a-posteriori and minimum mean square error estimation, which are intimately related to each other by the (asymmetry of the) canonical divergence function that [MATH] induces on the parameter space.', '1612.06149-1-27-4': 'This duality also manifests itself in other ways.', '1612.06149-1-27-5': 'For example it is easy to show that [MATH] is the Bayesian estimator associated with [MATH], where [MATH] is the convex dual or convex conjugate of [MATH] (see section [REF] for details).', '1612.06149-1-27-6': 'Similarly, noting that [MATH] we see that [MATH] plays the role of [MATH] on the manifold [MATH].', '1612.06149-1-27-7': 'The case of Gaussian models is particular because [MATH] is Euclidean, which is a self-dual space, and consequently [MATH], hence the primal and dual canonical estimators coincide.', '1612.06149-1-27-8': 'Finally, Theorem [REF] also highlights that under log-concavity and smoothness the posterior mode is a global property of [MATH].', '1612.06149-1-28-0': '## Error bounds for maximum-a-posteriori and minimum mean square error estimation', '1612.06149-1-29-0': 'We now establish performance guarantees for maximum-a-posteriori and minimum mean square error estimation when [MATH] is log-concave.', '1612.06149-1-29-1': 'Precisely, we establish universal estimation error bounds w.r.t. the dual error function [MATH].', '1612.06149-1-29-2': 'Here we do not assume that [MATH] is smooth; if [MATH] we replace [MATH] with the generalised divergence [MATH] where [MATH] belongs to the subdifferential set of [MATH] at [MATH] [add ref].', '1612.06149-1-29-3': 'We first present the following universal bounds on the expected estimation error: [mean error bound] Suppose that [MATH] is convex on [MATH].', '1612.06149-1-29-4': 'Then, [EQUATION].', '1612.06149-1-29-5': 'In addition, if [MATH] then [EQUATION]', '1612.06149-1-29-6': 'Proof.', '1612.06149-1-29-7': 'The proof is postponed to section [REF].', '1612.06149-1-30-0': 'Theorem [REF] establishes that [MATH] minimises the expected dual canonical loss [MATH], and Proposition [REF] complements this result by providing an explicit and general upper bound on the loss incurred by using this Bayesian estimator.', '1612.06149-1-30-1': 'Proposition [REF] also states that this bound also applies to [MATH], and that the expected loss per coordinate (e.g., per pixel in imaging problems) cannot exceed [MATH].', '1612.06149-1-30-2': 'This form of stability of high dimensional stability of maximum-a-posteriori and minimum mean square error estimation provides a theoretical argument for their good empirical performance in imaging and other large convex optimisation problems.', '1612.06149-1-31-0': 'Moreover, we also have the following universal large error bound for maximum-a-posteriori estimation: [Large error bound] Suppose that [MATH] is convex on [MATH].', '1612.06149-1-31-1': 'Then, for any [MATH] [EQUATION]', '1612.06149-1-31-2': 'Proof.', '1612.06149-1-31-3': 'The proof is postponed to section [REF].', '1612.06149-1-32-0': 'Proposition [REF] essentially indicates that in high dimensional settings the true value of [MATH] is almost certainly within the set [MATH], because the probability of a larger error vanishes exponentially fast as [MATH] increases.', '1612.06149-1-32-1': 'Again, this theoretical result supports the vast empirical evidence that maximum-a-posteriori estimation delivers accurate results in large-scale convex problems.', '1612.06149-1-32-2': 'It also follows from Proposition [REF] that in such problems [MATH] and [MATH] are close to each other (i.e., that [MATH] with high probability).', '1612.06149-1-33-0': 'Propositions [REF] and [REF], together with Theorem [REF], show that the predominant view of maximum-a-posteriori estimators as hastily inferences that are mainly motivated by computational efficiency is fundamentally incorrect.', '1612.06149-1-33-1': 'On the contrary, maximum-a-posteriori estimation offers a principled inference methodology that, similarly to minimum mean square error inference, is rooted in Bayesian decision theory.', '1612.06149-1-33-2': 'Moreover, the bounds on the expected estimation error and large error probability provide a new theoretical insight into its good empirical performance in large scale settings, such as imaging and machine learning problems.', '1612.06149-1-33-3': 'And of course, with the fundamental computational advantage of being a equivalent to convex optimisation problem that can be solved efficiently.', '1612.06149-1-34-0': '## Relaxation of regularity conditions', '1612.06149-1-35-0': 'To conclude, we examine the effect of relaxing the regularity assumptions of Theorem [REF].', '1612.06149-1-35-1': 'We consider three main cases: lack of smoothness, lack of strong convexity, and lack of continuity.', '1612.06149-1-36-0': 'Several models used in imaging and machine learning are not smooth because they involve priors based on the [MATH] norm, the nuclear norm, and the total-variation pseudo-norm [add reference].', '1612.06149-1-36-1': 'he results of Theorem [REF] hold for these models with the following minor modifications.', '1612.06149-1-36-2': 'First, these models [MATH] almost everywhere; that is, the set of non-differentiable points has dimension [MATH], and consequently it has no probability mass and can be omitted in the computation of expectations.', '1612.06149-1-36-3': 'Second, these non-differentiable points do not have Euclidean tangent spaces, and instead of a global manifold we need to consider the collection local manifolds associated with the regions of [MATH] where [MATH] is [MATH].', '1612.06149-1-36-4': 'Each one of these regions has a local canonical divergence given by the Bregman divergence [MATH].', '1612.06149-1-36-5': 'Therefore, in this case we need to posit [MATH] as the global loss function for any [MATH] (technically the global loss is the generalised Bregman divergence [MATH], where [MATH] belongs to the subdifferential set of [MATH] at [MATH] [CITATION], however the expectation [MATH] is taken over the points where [MATH] is [MATH] and hence [MATH]).', '1612.06149-1-36-6': 'We then consider the primal and dual Bayesian estimators related to this loss and obtain that [MATH] and [MATH] (note that [MATH] is optimised over [MATH], and consequently [MATH] is potentially at a non-differentiable point).', '1612.06149-1-36-7': ""Finally, despite not being a global canonical divergence, [MATH] is still consistent with the space's Riemannian geometry which is local."", '1612.06149-1-36-8': 'And the key guarantees of Propositions [REF] and [REF] also hold.', '1612.06149-1-37-0': 'Moreover, in the case of models that are strictly convex but not strongly convex only the second and third results of Theorem [REF] remain true.', '1612.06149-1-37-1': 'It is easy to check that the Bayesian estimator associated with[MATH] is [MATH], and that [MATH], similarly to strongly convex models.', '1612.06149-1-37-2': 'Therefore, the decision-theoretic derivation of [MATH] remains valid, and also [MATH] and [MATH] are dual to each other.', '1612.06149-1-37-3': 'The performance guarantees of Propositions [REF] and [REF] also hold because [MATH] is convex.', '1612.06149-1-37-4': 'However, without strong convexity, [MATH] becomes semi-positive definite and [MATH] becomes a singular manifold.', '1612.06149-1-37-5': 'Consequently, the validity of the interpretation of [MATH] as a canonical divergence as defined above is not clear.', '1612.06149-1-37-6': 'The generalisation of canonical divergences and of Theorem [REF] to singular manifolds is currently under investigation.', '1612.06149-1-38-0': 'Finally, in cases where [MATH] is constrained to a convex region [MATH]only the first and third results of Theorem [REF] hold.', '1612.06149-1-38-1': 'Proceeding similarly to the proof of Theorem [REF] it is easy to show that [MATH] is the canonical divergence of the manifold [MATH], and that the Bayesian estimator related to the dual divergence is [MATH].', '1612.06149-1-38-2': 'However, the Bayesian estimator that minimises the canonical divergence is now a shifted maximum-a-posteriori estimator [EQUATION] where generally [MATH] (see Section [REF] for details).', '1612.06149-1-38-3': 'Therefore, for these models [MATH] is generally not a proper Bayesian estimator in the decision-theoretic sense.', '1612.06149-1-38-4': 'Nevertheless, the guarantees of Propositions [REF] and [REF] still hold for [MATH], providing some theoretical motivation for using this estimator (recall that [MATH] is close to [MATH] in the sense of Proposition [REF]).'}","{'1612.06149-2-0-0': 'Maximum-a-posteriori estimation has become the main Bayesian estimation methodology in many areas of modern data science such as mathematical imaging and machine learning, where high dimensionality is addressed by using models that are log-concave and where the posterior mode can be computed very efficiently by using convex optimisation algorithms.', '1612.06149-2-0-1': 'However, despite its success and rapid adoption, maximum-a-posteriori estimation is not theoretically well understood yet.', '1612.06149-2-0-2': 'This paper presents a new decision-theoretic derivation of maximum-a-posteriori estimation in Bayesian models that are log-concave.', '1612.06149-2-0-3': 'Our analysis is based on differential geometry and proceeds as follows.', '1612.06149-2-0-4': 'First, we exploit the log-concavity of the model to induce a Riemannian geometry on the parameter space, and use differential geometry to identify the natural or canonical loss function to perform Bayesian point estimation in that space.', '1612.06149-2-0-5': 'We then show that for log-concave models the canonical loss is the Bregman divergence associated with the negative log posterior density, and that the maximum-a-posteriori estimator is the Bayesian estimator that minimises the expected loss.', '1612.06149-2-0-6': 'We also show that the posterior mean or minimum mean square error estimator is the Bayesian estimator that minimises the dual canonical loss, and establish general performance guarantees for both maximum-a-posteriori and minimum mean square error estimation.', '1612.06149-2-0-7': 'These results provide a new understanding of these estimation methodologies under log-concavity, and reveal new insights about their good empirical performance and about the roles that log-concavity plays in high dimensional inference problems.', '1612.06149-2-1-0': '# Introduction', '1612.06149-2-2-0': 'We consider the Bayesian estimation of an unknown quantify of interest [MATH] from an observation [MATH] [CITATION].', '1612.06149-2-2-1': 'We focus on Bayesian models whose posterior distribution is log-concave, i.e., [EQUATION] for some proper convex function [MATH].', '1612.06149-2-2-2': 'Such models are ubiquitous in modern data science, particularly for problems where [MATH] is high dimensional (e.g., [MATH]).', '1612.06149-2-2-3': 'For example, statistical imaging and machine learning methods rely strongly on log-concave models of the form [MATH] for some linear operators [MATH] and [MATH], convex regulariser [MATH], and convex set constraint [MATH], and where typically the dimension [MATH] [CITATION].', '1612.06149-2-3-0': 'Because drawing conclusions directly from [MATH] is difficult, Bayesian methods generally deliver summaries of [MATH], namely Bayes point estimators, which summarises the information in [MATH] optimally in the following decision-theoretic sense [CITATION]:', '1612.06149-2-4-0': 'Let [MATH] be a loss function that quantifies the difference between two points in [MATH].', '1612.06149-2-4-1': 'A Bayes estimator associated with [MATH] is any estimator that minimises the posterior expected loss, i.e., [EQUATION].', '1612.06149-2-5-0': 'The loss function [MATH] is specified by the analyst and usually verifies the following general conditions:', '1612.06149-2-6-0': 'Observe that [MATH] is not necessarily symmetric, i.e., [MATH] because the arguments of [MATH] have clearly different roles in the decision problem.', '1612.06149-2-7-0': 'Ideally [MATH] should be chosen carefully based on specific aspects of the problem and application considered.', '1612.06149-2-7-1': 'This is particularly important for instance in imaging problems that are ill-posed or ill-conditioned, where this choice can significantly impact estimation results.', '1612.06149-2-7-2': 'However, specifying a bespoke loss function for high dimensional problems is not easy, and as a result most methods reported in the literature use default losses and estimators.', '1612.06149-2-8-0': 'In particular, Bayesian methods in engineering fields such as imaging have traditionally used the minimum mean square estimator, which is given by the posterior mean [MATH].', '1612.06149-2-8-1': 'This estimator is widely regarded as a gold standard, in part because of its good empirical performance and favourable theoretical properties, and also perhaps in part because of cultural heritage.', '1612.06149-2-8-2': 'From Bayesian decision theory, minimum mean square estimation is optimal with respect to the entire class of quadratic loss functions of the form [MATH] with [MATH] positive definite [CITATION].', '1612.06149-2-8-3': 'This class provides a second order approximation to any strongly convex loss function, and hence [MATH] is also a proxy for many other Bayesian estimators.', '1612.06149-2-8-4': 'Also, the quadratic loss is directly related to the Euclidean squared distance, giving [MATH] a clear geometric interpretation.', '1612.06149-2-8-5': 'In addition, it has been established in [CITATION] that [MATH] is also optimal w.r.t. the second argument of any Bregman divergence (i.e., any loss function of the form [MATH] for a convex function [MATH]), a more general class of loss functions that includes quadratic losses.', '1612.06149-2-9-0': 'Unfortunately, calculating [MATH] in high dimensional models can be very difficult because it requires solving integrals that are often too computationally expensive for the applications considered.', '1612.06149-2-9-1': 'This has stimulated much research on the topic, from fast Monte Carlo simulation methods to efficient approximations with deterministic algorithms [CITATION].', '1612.06149-2-9-2': 'But with ever increasingly large problems and datasets, many applied fields have progressively focused on alternatives to minimum mean square error estimation.', '1612.06149-2-10-0': 'In particular, modern imaging and machine learning methods rely strongly on maximum-a-posteriori estimation [EQUATION] whose calculation is a convex problem that can often be solved very efficiently, even in very high dimensions (e.g., [MATH]), by using convex optimisation techniques [CITATION].', '1612.06149-2-10-1': 'Interestingly, modern non-statistical imaging and machine learning methods also predominately solve problems by convex optimisation, and their solutions are often equivalent to performing maximum-a-posteriori estimation for some implicit Bayesian model.', '1612.06149-2-11-0': 'There is abundant evidence that maximum-a-posteriori estimation delivers accurate results for a wide range of log-concave models and applications.', '1612.06149-2-11-1': 'However, from a theoretical viewpoint maximum-a-posteriori estimation is not well understood.', '1612.06149-2-11-2': 'Currently the predominant view is that maximum-a-posteriori estimation is not formal Bayes estimation in the decision-theoretic sense postulated by Definition [REF] because it does not minimise a known expected loss.', '1612.06149-2-11-3': 'The prevailing interpretation is that maximum-a-posteriori estimation is in fact an approximation arising from the loss [MATH] when [MATH].', '1612.06149-2-11-4': 'However, this derivation does not lead to a proper Bayesian estimator because [MATH] becomes degenerate as [MATH] vanishes.', '1612.06149-2-11-5': 'Perhaps more importantly, the resulting loss is very difficult to motivate for inference problems in [MATH] [CITATION].', '1612.06149-2-12-0': 'Furthermore, most other theoretical results for maximum-a-posteriori estimation only hold for very specific models, or have been derived by adopting analyses that are extrinsic to the Bayesian decision theory framework (e.g. by analysing maximum-a-posteriori estimation as constrained or regularised least-squares regression [CITATION]).', '1612.06149-2-12-1': 'For example, when [MATH] is symmetric we have [MATH], and thus maximum-a-posteriori estimation inherits the favourable properties of minimum square error estimation.', '1612.06149-2-12-2': 'This result has been partially extended to some denoising models of the form [MATH] in [CITATION], where it is shown that maximum-a-posteriori estimation coincides with minimum square error estimation with a different model [MATH].', '1612.06149-2-12-3': 'It follows that for these models maximum-a-posteriori estimation is decision-theoretic Bayesian estimation w.r.t. the weighted loss [MATH].', '1612.06149-2-12-4': 'This is a post-hoc loss, but the result is interesting because it highlights that a single estimator may have a plurality of origins.', '1612.06149-2-12-5': 'Lastly, Burger Lucka [CITATION] recently established that maximum-a-posteriori estimation is decision-theoretic Bayesian estimation for all linear Gaussian models of the form [MATH], where [MATH] is a known linear operator, [MATH] a known noise covariance, and [MATH] is convex and Lipchitz continuous.', '1612.06149-2-12-6': 'More precisely, that paper shows that for these models maximum-a-posteriori estimation is optimal w.r.t. the loss [MATH], where [MATH] is the Bregman divergence associated with [MATH].', '1612.06149-2-12-7': 'Observe that in this case the loss function defining the estimator is determined by the model considered.', '1612.06149-2-13-0': 'In this paper we revisit the choice of the loss function for Bayesian point estimation in the context of log-concave models.', '1612.06149-2-13-1': 'A main novelty is that, instead of specifying the loss directly, we seek to derive it automatically from the geometry of the model.', '1612.06149-2-13-2': 'Precisely, we use differential geometry to derive the loss from the geometry of the parameter space.', '1612.06149-2-13-3': 'We show that under some regularity assumptions, the log-concavity of the posterior distribution induces a specific Riemannian geometry on the parameter space, and that taking into account this space geometry naturally leads to a natural or canonical loss function to perform Bayesian estimation.', '1612.06149-2-13-4': ""Following on from this, we then establish that the canonical loss for the space is given by the Bregman divergence associated with the model's negative log density, and that the Bayesian estimator associated with this loss is the maximum-a-posteriori estimator."", '1612.06149-2-13-5': 'We also show that the minimum mean square error estimator is the Bayesian estimator associated with the dual canonical loss.', '1612.06149-2-14-0': 'The remainder of the paper is organised as follows: section [REF] introduces the elements of differential geometry that are essential to our analysis.', '1612.06149-2-14-1': 'In section [REF] we present our main theoretical result: a decision-theoretic and differential-geometric derivation of maximum-a-posteriori and minimum mean square error estimation, as well as general bounds on the estimation error involved.', '1612.06149-2-14-2': 'Proofs are reported in the appendix.', '1612.06149-2-15-0': '# Riemannian geometry and the canonical divergence', '1612.06149-2-16-0': 'In this section we recall some elements of differential geometry that are necessary for our analysis.', '1612.06149-2-16-1': 'For a detailed introduction to this topic we refer the reader to [CITATION].', '1612.06149-2-17-0': 'An [MATH]-dimensional Riemannian manifold [MATH], with metric [MATH] and global coordinate system [MATH], is a vector space that behaves locally as an Euclidean space.', '1612.06149-2-17-1': 'Precisely, for any point [MATH] we have a tangent space [MATH] with inner product [MATH] and norm [MATH].', '1612.06149-2-17-2': 'This geometry is local and may vary smoothly from [MATH] to neighbouring tangent spaces.', '1612.06149-2-17-3': ""These variations are encoded in the manifold's affine connection [MATH], with coefficients given by [MATH]."", '1612.06149-2-18-0': 'Moreover, similarly to Euclidean spaces, the manifold [MATH] supports divergence functions.', '1612.06149-2-19-0': 'A function [MATH] is a divergence function on [MATH] if the following conditions hold for any [MATH]:', '1612.06149-2-20-0': 'The class of divergence functions coincides with that of loss functions for Bayesian point estimation considered in section [REF] with mild additional regularity conditions, hence divergence functions are suitable candidates to define Bayesian estimators.', '1612.06149-2-20-1': 'Divergence functions also provide a link to the differential geometry of the space, which allows relating geometry and Bayesian estimation, or more generally geometry and Bayesian decision theory.', '1612.06149-2-20-2': 'This relationship has been used previously to analyse decision problems from a Riemannian geometric viewpoint, leading to the so-called decision geometry framework [CITATION].', '1612.06149-2-20-3': 'Here we adopt an opposite perspective: we start by defining a Riemannian manifold [MATH] and then use the relationship to identify the divergence functions that arise naturally in that space.', '1612.06149-2-20-4': 'In particular, we focus on the so-called canonical divergence on [MATH], which is a generalisation of the Euclidean squared distance to this kind of manifold [CITATION].', '1612.06149-2-20-5': '[Canonical divergence [CITATION]] For any two points [MATH], the [MATH]-canonical divergence is given by [EQUATION] where [MATH] is the [MATH]-geodesic connecting [MATH] and [MATH].', '1612.06149-2-21-0': 'It is easy to check that [REF] reduces to the Euclidean squared distance [MATH] when [MATH] is the Euclidean space with inner product [MATH].', '1612.06149-2-21-1': 'If [MATH] is not Euclidean then [MATH] is consistent with the local Euclidean geometry of the space.', '1612.06149-2-21-2': 'That is, for any point [MATH] in the neighbourhood of [MATH] we have [MATH], where [MATH] is the Euclidean norm of the tangent space [MATH] (a higher order approximation of [MATH] is also possible by using the affine connection [MATH]).', '1612.06149-2-21-3': 'And if we use the decision geometry framework [CITATION] to derive the Riemannian geometry induced by [MATH] on [MATH] we obtain [EQUATION] (here [MATH] and [MATH] denote differentiation w.r.t. the first and second components of [MATH] respective), which indicates that [MATH] is specified by [MATH] and that it induces the same space geometry that originated it in the first place.', '1612.06149-2-22-0': 'Finally, to gain a geometric intuition for [MATH] it is useful to compare it to the length of the affine geodesic between [MATH] and [MATH].', '1612.06149-2-22-1': 'Precisely, by noting that the squared length of a curve [MATH] is given by [MATH], we observe that [MATH] is essentially the squared length of the [MATH]-geodesic [MATH] weighted linearly along the path from [MATH] to [MATH].', '1612.06149-2-22-2': 'This weighting in [REF] guarantees that [MATH] is convex in [MATH] (a necessary condition to define a divergence function).', '1612.06149-2-22-3': 'It also introduces an asymmetry, i.e., generally [MATH].', '1612.06149-2-22-4': 'We will see that this asymmetry has deep implications for Bayesian estimation.', '1612.06149-2-23-0': '# A differential-geometric derivation of maximum-a-posteriori and minimum mean square error estimation', '1612.06149-2-24-0': '## Canonical Bayesian estimators: from differential geometry to decision theory', '1612.06149-2-25-0': 'In this section we use differential geometry to relate [MATH] to the loss functions used for Bayesian point estimation.', '1612.06149-2-25-1': 'Precisely, we exploit the log-concavity of [MATH] to induce a Riemannian geometry on the parameter space.', '1612.06149-2-25-2': 'This in turn defines a canonical loss for that space and two Bayesian estimators w.r.t. to this loss: a primal estimator related to [MATH] and a dual estimator related to the dual divergence [MATH].', '1612.06149-2-25-3': 'We consider first the case where [MATH] is smooth and strongly log-concave, and analyse the effect of relaxing these assumptions in section [REF].', '1612.06149-2-26-0': '[Canonical Bayesian estimators] Suppose that [MATH] is strongly convex, continuous, and [MATH] on [MATH].', '1612.06149-2-26-1': 'Let [MATH] denote the Riemannian manifold induced by [MATH], with metric coefficients given by [MATH].', '1612.06149-2-26-2': 'Then, the canonical divergence on [MATH] is the Bregman divergence associated [MATH], i.e., [EQUATION].', '1612.06149-2-26-3': 'In addition, the Bayesian estimator associated with [MATH] is unique and is given by the maximum-a-posteriori estimator, [EQUATION]', '1612.06149-2-26-4': 'The Bayesian estimation associated with the dual canonical divergence [MATH] is also unique and given by the minimum mean squared error estimator [EQUATION]', '1612.06149-2-26-5': 'The proof is reported in the appendix.', '1612.06149-2-27-0': 'Theorem [REF] provides several valuable new insights into maximum-a-posteriori and minimum mean square error estimation for the convex estimation problems considered.', '1612.06149-2-27-1': 'First, maximum-a-posteriori estimation stems from Bayesian decision theory, and hence it stands on the same theoretical footing as the core Bayesian methodologies such as minimum mean square error estimation.', '1612.06149-2-27-2': 'Second, the conventional definition of the maximum-a-posteriori estimator as the maximiser [MATH] is mainly algorithmic for these problems, useful to highlight that these estimators take the form of a convex optimisation problem that can be solved efficiently by convex optimisation (which is an important computational advantage over most other Bayesian point estimators).', '1612.06149-2-27-3': 'Third, Theorem [REF] also reveals a surprising form of duality between maximum-a-posteriori and minimum mean square error estimation, which are intimately related to each other by the (asymmetry of the) canonical divergence function that [MATH] induces on the parameter space.', '1612.06149-2-27-4': 'This duality also manifests itself in other ways.', '1612.06149-2-27-5': 'For example it is easy to show that [MATH] is the Bayesian estimator associated with [MATH], where [MATH] is the convex dual or convex conjugate of [MATH] (see section [REF] for details).', '1612.06149-2-27-6': 'Similarly, noting that [MATH] we see that [MATH] plays the role of [MATH] on the manifold [MATH].', '1612.06149-2-27-7': 'The case of Gaussian models is particular because [MATH] is Euclidean, which is a self-dual space, and consequently [MATH], hence the primal and dual canonical estimators coincide.', '1612.06149-2-27-8': 'Finally, Theorem [REF] also highlights that under log-concavity and smoothness the posterior mode is a global property of [MATH].', '1612.06149-2-28-0': '## Error bounds for maximum-a-posteriori and minimum mean square error estimation', '1612.06149-2-29-0': 'We now establish performance guarantees for maximum-a-posteriori and minimum mean square error estimation when [MATH] is log-concave.', '1612.06149-2-29-1': 'Precisely, we establish universal estimation error bounds w.r.t. the dual error function [MATH].', '1612.06149-2-29-2': 'Here we do not assume that [MATH] is smooth; if [MATH] we replace [MATH] with the generalised divergence [MATH] where [MATH] belongs to the subdifferential set of [MATH] at [MATH] [add ref].', '1612.06149-2-29-3': 'We first present the following universal bounds on the expected estimation error: [mean error bound] Suppose that [MATH] is convex on [MATH].', '1612.06149-2-29-4': 'Then, [EQUATION].', '1612.06149-2-29-5': 'In addition, if [MATH] then [EQUATION]', '1612.06149-2-29-6': 'Proof.', '1612.06149-2-29-7': 'The proof is reported in the appendix.', '1612.06149-2-30-0': 'Theorem [REF] establishes that [MATH] minimises the expected dual canonical loss [MATH], and Proposition [REF] complements this result by providing an explicit and general upper bound on the loss incurred by using this Bayesian estimator.', '1612.06149-2-30-1': 'Proposition [REF] also states that this bound also applies to [MATH], and that the expected loss per coordinate (e.g., per pixel in imaging problems) cannot exceed [MATH].', '1612.06149-2-30-2': 'This form of stability of high dimensional stability of maximum-a-posteriori and minimum mean square error estimation provides a theoretical argument for their good empirical performance in imaging and other large convex optimisation problems.', '1612.06149-2-31-0': 'Moreover, we also have the following universal large error bound for maximum-a-posteriori estimation: [Large error bound] Suppose that [MATH] is convex on [MATH].', '1612.06149-2-31-1': 'Then, for any [MATH] [EQUATION]', '1612.06149-2-31-2': 'Proof.', '1612.06149-2-31-3': 'The proof is reported in the appendix.', '1612.06149-2-32-0': 'Proposition [REF] essentially indicates that in high dimensional settings the true value of [MATH] is almost certainly within the set [MATH], because the probability of a larger error vanishes exponentially fast as [MATH] increases.', '1612.06149-2-32-1': 'Again, this theoretical result supports the vast empirical evidence that maximum-a-posteriori estimation delivers accurate results in large-scale convex problems.', '1612.06149-2-32-2': 'It also follows from Proposition [REF] that in such problems [MATH] and [MATH] are close to each other (i.e., that [MATH] with high probability).', '1612.06149-2-33-0': 'Propositions [REF] and [REF], together with Theorem [REF], show that the predominant view of maximum-a-posteriori estimators as hastily inferences that are mainly motivated by computational efficiency is fundamentally incorrect.', '1612.06149-2-33-1': 'On the contrary, maximum-a-posteriori estimation offers a principled inference methodology that, similarly to minimum mean square error inference, is rooted in Bayesian decision theory.', '1612.06149-2-33-2': 'Moreover, the bounds on the expected estimation error and large error probability provide a new theoretical insight into its good empirical performance in large scale settings, such as imaging and machine learning problems.', '1612.06149-2-33-3': 'And of course, with the fundamental computational advantage of being a equivalent to convex optimisation problem that can be solved efficiently.', '1612.06149-2-34-0': '## Relaxation of regularity conditions', '1612.06149-2-35-0': 'To conclude, we examine the effect of relaxing the regularity assumptions of Theorem [REF].', '1612.06149-2-35-1': 'We consider three main cases: lack of smoothness, lack of strong convexity, and lack of continuity.', '1612.06149-2-36-0': 'Several models used in imaging and machine learning are not smooth because they involve priors based on the [MATH] norm, the nuclear norm, and the total-variation pseudo-norm [add reference].', '1612.06149-2-36-1': 'he results of Theorem [REF] hold for these models with the following minor modifications.', '1612.06149-2-36-2': 'First, these models [MATH] almost everywhere; that is, the set of non-differentiable points has dimension [MATH], and consequently it has no probability mass and can be omitted in the computation of expectations.', '1612.06149-2-36-3': 'Second, these non-differentiable points do not have Euclidean tangent spaces, and instead of a global manifold we need to consider the collection local manifolds associated with the regions of [MATH] where [MATH] is [MATH].', '1612.06149-2-36-4': 'Each one of these regions has a local canonical divergence given by the Bregman divergence [MATH].', '1612.06149-2-36-5': 'Therefore, in this case we need to posit [MATH] as the global loss function for any [MATH] (technically the global loss is the generalised Bregman divergence [MATH], where [MATH] belongs to the subdifferential set of [MATH] at [MATH] [CITATION], however the expectation [MATH] is taken over the points where [MATH] is [MATH] and hence [MATH]).', '1612.06149-2-36-6': 'We then consider the primal and dual Bayesian estimators related to this loss and obtain that [MATH] and [MATH] (note that [MATH] is optimised over [MATH], and consequently [MATH] is potentially at a non-differentiable point).', '1612.06149-2-36-7': ""Finally, despite not being a global canonical divergence, [MATH] is still consistent with the space's Riemannian geometry which is local."", '1612.06149-2-36-8': 'And the key guarantees of Propositions [REF] and [REF] also hold.', '1612.06149-2-37-0': 'Moreover, in the case of models that are strictly convex but not strongly convex only the second and third results of Theorem [REF] remain true.', '1612.06149-2-37-1': 'It is easy to check that the Bayesian estimator associated with[MATH] is [MATH], and that [MATH], similarly to strongly convex models.', '1612.06149-2-37-2': 'Therefore, the decision-theoretic derivation of [MATH] remains valid, and also [MATH] and [MATH] are dual to each other.', '1612.06149-2-37-3': 'The performance guarantees of Propositions [REF] and [REF] also hold because [MATH] is convex.', '1612.06149-2-37-4': 'However, without strong convexity, [MATH] becomes semi-positive definite and [MATH] becomes a singular manifold.', '1612.06149-2-37-5': 'Consequently, the validity of the interpretation of [MATH] as a canonical divergence as defined above is not clear.', '1612.06149-2-37-6': 'The generalisation of canonical divergences and of Theorem [REF] to singular manifolds is currently under investigation.', '1612.06149-2-38-0': 'Finally, in cases where [MATH] is constrained to a convex region [MATH]only the first and third results of Theorem [REF] hold.', '1612.06149-2-38-1': 'Proceeding similarly to the proof of Theorem [REF] it is easy to show that [MATH] is the canonical divergence of the manifold [MATH], and that the Bayesian estimator related to the dual divergence is [MATH].', '1612.06149-2-38-2': 'However, the Bayesian estimator that minimises the canonical divergence is now a shifted maximum-a-posteriori estimator [EQUATION] where generally [MATH] (see Section [REF] for details).', '1612.06149-2-38-3': 'Therefore, for these models [MATH] is generally not a proper Bayesian estimator in the decision-theoretic sense.', '1612.06149-2-38-4': 'Nevertheless, the guarantees of Propositions [REF] and [REF] still hold for [MATH], providing some theoretical motivation for using this estimator (recall that [MATH] is close to [MATH] in the sense of Proposition [REF]).'}","[['1612.06149-1-20-0', '1612.06149-2-20-0'], ['1612.06149-1-20-1', '1612.06149-2-20-1'], ['1612.06149-1-20-2', '1612.06149-2-20-2'], ['1612.06149-1-20-3', '1612.06149-2-20-3'], ['1612.06149-1-20-4', '1612.06149-2-20-4'], ['1612.06149-1-20-5', '1612.06149-2-20-5'], ['1612.06149-1-6-0', '1612.06149-2-6-0'], 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'1612.06149-4-30-7']]","['1612.06149-1-3-0', '1612.06149-1-5-0', '1612.06149-1-19-0', '1612.06149-1-29-4', '1612.06149-1-29-6', '1612.06149-1-31-2', '1612.06149-2-3-0', '1612.06149-2-5-0', '1612.06149-2-19-0', '1612.06149-2-29-4', '1612.06149-2-29-6', '1612.06149-2-31-2', '1612.06149-3-3-0', '1612.06149-3-5-0', '1612.06149-3-19-0', '1612.06149-3-21-0', '1612.06149-3-31-4', '1612.06149-3-31-6', '1612.06149-3-33-2', '1612.06149-4-4-1', '1612.06149-4-5-3', '1612.06149-4-19-0', '1612.06149-4-20-0', '1612.06149-4-22-0', '1612.06149-4-37-1', '1612.06149-4-38-1', '1612.06149-4-38-2', '1612.06149-4-44-1', '1612.06149-4-45-1', '1612.06149-4-45-2']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1612.06149,"{'1612.06149-3-0-0': 'Maximum-a-posteriori (MAP) estimation has become the main Bayesian estimation methodology in many areas of modern data science such as mathematical imaging and machine learning, where high dimensionality is addressed by using models that are log-concave and whose posterior mode can be computed very efficiently by using convex optimisation algorithms.', '1612.06149-3-0-1': 'However, despite its success and rapid adoption, MAP estimation is not theoretically well understood yet, and the prevalent view is that it is generally not proper Bayesian estimation in a decision-theoretic sense.', '1612.06149-3-0-2': 'This paper presents a new decision-theoretic derivation of MAP estimation in Bayesian models that are log-concave.', '1612.06149-3-0-3': 'Our analysis is based on differential geometry and proceeds as follows.', '1612.06149-3-0-4': 'First, we exploit the log-concavity of the model to induce a Riemannian geometry on the parameter space.', '1612.06149-3-0-5': 'We then use differential geometry to identify the natural or canonical loss function to perform Bayesian point estimation in that Riemannian manifold.', '1612.06149-3-0-6': 'For log-concave models this canonical loss is the Bregman divergence of the negative log posterior density, a similarity measure rooted in convex analysis that in addition to the relative position of points also takes into account the geometry of the space, and which generalises the Euclidean squared distance to non-Euclidean settings.', '1612.06149-3-0-7': 'We then show that the MAP estimator is the Bayesian estimator that minimises the expected canonical loss, and that the posterior mean or minimum mean squared error (MMSE) estimator is the Bayesian estimator that minimises the dual canonical loss.', '1612.06149-3-0-8': 'Finally, we establish universal performance and stability guarantees for MAP and MMSE estimation in high dimensional log-concave models.', '1612.06149-3-0-9': 'These theoretical results provide a new understanding of MAP and MMSE estimation under log-concavity, and reveal new insights about their good empirical performance and about the roles that log-concavity plays in high dimensional inference problems.', '1612.06149-3-1-0': '# Introduction', '1612.06149-3-2-0': 'We consider the Bayesian estimation of an unknown quantify of interest [MATH] from an observation [MATH] [CITATION].', '1612.06149-3-2-1': 'We focus on Bayesian models whose posterior distribution is log-concave, i.e., [EQUATION] for some proper convex function [MATH].', '1612.06149-3-2-2': 'Such models are ubiquitous in modern data science, particularly for problems where [MATH] is high dimensional (e.g., [MATH]).', '1612.06149-3-2-3': 'For example, statistical imaging and machine learning methods rely strongly on log-concave models of the form [MATH] for some linear operators [MATH] and [MATH], convex regulariser [MATH], and convex set constraint [MATH], and are typically of dimension [MATH] [CITATION].', '1612.06149-3-3-0': 'Because drawing conclusions directly from [MATH] is difficult, Bayesian methods generally deliver summaries of [MATH], namely Bayes point estimators, which summarises the information in [MATH] optimally in the following decision-theoretic sense [CITATION]:', '1612.06149-3-4-0': 'Let [MATH] be a loss function that quantifies the difference between two points in [MATH].', '1612.06149-3-4-1': 'A Bayes estimator associated with [MATH] is any estimator that minimises the posterior expected loss, i.e., [EQUATION].', '1612.06149-3-5-0': 'The loss function [MATH] is specified by the analyst and usually verifies the following general conditions:', '1612.06149-3-6-0': 'Observe that [MATH] is not necessarily symmetric, i.e., [MATH].', '1612.06149-3-6-1': 'We do not enforce symmetry because the arguments of [MATH] have clearly different roles in the decision problem.', '1612.06149-3-7-0': 'Ideally [MATH] should be chosen carefully based on specific aspects of the problem and application considered.', '1612.06149-3-7-1': 'This is particularly important for instance in imaging problems that are severely ill-posed or ill-conditioned, where this choice can significantly impact estimation results.', '1612.06149-3-7-2': 'However, specifying a bespoke loss function for high dimensional problems is not easy, and as a result most methods reported in the literature use default losses and estimators.', '1612.06149-3-8-0': 'In particular, Bayesian methods in engineering fields such as imaging have traditionally used the minimum mean squared error (MMSE) estimator, which is given by the posterior mean [MATH].', '1612.06149-3-8-1': 'This estimator is widely regarded as a gold standard in these fields, in part because of its good empirical performance and favourable theoretical properties, and also perhaps in part because of cultural heritage.', '1612.06149-3-8-2': 'From Bayesian decision theory, MMSE estimation is optimal with respect to the entire class of quadratic loss functions of the form [MATH] with [MATH] positive definite [CITATION].', '1612.06149-3-8-3': 'This class provides a second order approximation to any strongly convex loss function, and hence [MATH] is also a proxy for other Bayesian estimators.', '1612.06149-3-8-4': 'Also, the quadratic loss is directly related to the Euclidean squared distance, giving [MATH] a clear geometric interpretation.', '1612.06149-3-8-5': 'In addition, it has been established in [CITATION] that [MATH] is also optimal w.r.t. the second argument of any Bregman divergence (i.e., any loss function of the form [MATH] for a convex function [MATH]), a more general class of loss functions that includes quadratic losses and that is related to non-Euclidean geometries [CITATION].', '1612.06149-3-9-0': 'Unfortunately, calculating [MATH] in high dimensional models can be very difficult because it requires solving integrals that are often too computationally expensive for the applications considered.', '1612.06149-3-9-1': 'This has stimulated much research on the topic, from fast Monte Carlo simulation methods to efficient approximations with deterministic algorithms [CITATION].', '1612.06149-3-9-2': 'But with ever increasingly large problems and datasets, many applied fields have progressively focused on alternatives to MMSE estimation.', '1612.06149-3-10-0': 'In particular, modern imaging and machine learning methods rely strongly on maximum-a-posteriori (MAP) estimation [EQUATION] whose calculation is a convex problem that can often be solved very efficiently, even in very high dimensions (e.g., [MATH]), by using convex optimisation techniques [CITATION].', '1612.06149-3-10-1': 'Interestingly, modern non-statistical imaging and machine learning methods also predominately solve problems by convex optimisation, and their solutions are often equivalent to performing MAP estimation for some implicit Bayesian model.', '1612.06149-3-11-0': 'There is abundant evidence that MAP estimation delivers accurate results for a wide range of log-concave models and applications.', '1612.06149-3-11-1': 'However, from a theoretical viewpoint MAP estimation is not well understood.', '1612.06149-3-11-2': 'Currently the predominant view is that MAP estimation is not formal Bayesian estimation in the decision-theoretic sense postulated by Definition [REF] because it does not minimise a known expected loss.', '1612.06149-3-11-3': 'The prevailing interpretation is that MAP estimation is in fact an approximation arising from the degenerate loss [MATH] with [MATH] [CITATION] (this derivation holds for all log-concave models, but is not generally true [CITATION]).', '1612.06149-3-11-4': 'However, this asymptotic derivation does not lead to a proper Bayesian estimator.', '1612.06149-3-11-5': 'More importantly, the resulting loss is very difficult to motivate for inference problems in [MATH], and does not explain the good empirical performance reported in the literature.', '1612.06149-3-12-0': 'Furthermore, most other theoretical results for MAP estimation only hold for very specific models, or have been derived by adopting analyses that are extrinsic to the Bayesian decision theory framework (e.g. by analysing MAP estimation as constrained or regularised least-squares regression [CITATION]).', '1612.06149-3-12-1': 'As a trivial example, when [MATH] is symmetric we have [MATH], and thus MAP estimation inherits the favourable properties of MMSE estimation.', '1612.06149-3-12-2': 'This result has been partially extended to some denoising models of the form [MATH] in [CITATION], where it is shown that MAP estimation coincides with MMSE estimation with a different model [MATH].', '1612.06149-3-12-3': 'It follows that for these models MAP estimation is decision-theoretic Bayesian estimation w.r.t. the weighted loss [MATH].', '1612.06149-3-12-4': 'This is a post-hoc loss, but the result is interesting because it highlights that a single estimator may have a plurality of origins.', '1612.06149-3-12-5': 'Lastly, Burger Lucka [CITATION] recently established that MAP estimation is decision-theoretic Bayesian estimation for all linear Gaussian models of the form [MATH], where [MATH] is a known linear operator, [MATH] a known noise covariance, and [MATH] is convex and Lipchitz continuous.', '1612.06149-3-12-6': 'More precisely, that paper shows that for these models MAP estimation is optimal w.r.t. the loss [MATH], where [MATH] is the Bregman divergence associated with [MATH].', '1612.06149-3-12-7': 'It may appear that this loss is rather artificial and difficult to analyse and motivate, however the new results presented in section [REF] show that it is a specific instance of a more general loss that stems directly from the consideration of the model geometry.', '1612.06149-3-13-0': 'In order to understand MAP estimation, in this paper we first revisit the choice of the loss function for Bayesian point estimation in the context of log-concave models.', '1612.06149-3-13-1': 'A main novelty is that, instead of specifying the loss directly, we use differential geometry to derive it automatically from the geometry of the model.', '1612.06149-3-13-2': 'Precisely, we show that under some regularity assumptions, the log-concavity of [MATH] induces a specific Riemannian differential geometry on the parameter space, and that taking into account this space geometry naturally leads to an intrinsic or canonical loss function to perform Bayesian point estimation.', '1612.06149-3-13-3': 'Following on from this, we establish that the canonical loss for the parameter space is the Bregman divergence associated with [MATH], and that the Bayesian estimator w.r.t. this loss is the MAP estimator.', '1612.06149-3-13-4': 'We then show that the MMSE estimator is the Bayesian estimator associated with the dual canonical loss, and propose universal estimation performance guarantees for MAP and MMSE estimation in log-concave models.', '1612.06149-3-14-0': 'The remainder of the paper is organised as follows: section [REF] introduces the elements of differential geometry that are essential to our analysis.', '1612.06149-3-14-1': 'In section [REF] we present our main theoretical result: a decision-theoretic and differential-geometric derivation of MAP and MMSE estimation, as well as universal bounds on the estimation error involved.', '1612.06149-3-14-2': 'Conclusions are finally reported in section [REF].', '1612.06149-3-14-3': 'Proofs are presented in the appendix.', '1612.06149-3-15-0': '# Riemannian geometry and the canonical divergence', '1612.06149-3-16-0': 'In this section we recall some elements of differential geometry that are necessary for our analysis.', '1612.06149-3-16-1': 'For a detailed introduction to this topic we refer the reader to [CITATION].', '1612.06149-3-17-0': 'An [MATH]-dimensional Riemannian manifold [MATH], with metric [MATH] and global coordinate system [MATH], is a vector space that behaves locally as an Euclidean space.', '1612.06149-3-17-1': 'Precisely, for any point [MATH] we have a tangent space [MATH] with inner product [MATH] and norm [MATH].', '1612.06149-3-17-2': 'This geometry is local and may vary smoothly from [MATH] to neighbouring tangent spaces.', '1612.06149-3-17-3': ""These variations are encoded in the manifold's affine connection [MATH], with coefficients given by [MATH]."", '1612.06149-3-18-0': 'Moreover, similarly to Euclidean spaces, the manifold [MATH] supports divergence functions.', '1612.06149-3-19-0': 'A function [MATH] is a divergence function on [MATH] if the following conditions hold for any [MATH]:', '1612.06149-3-20-0': 'The class of divergence functions coincides with that of loss functions for Bayesian point estimation considered in section [REF] with mild additional regularity conditions.', '1612.06149-3-20-1': 'Hence divergence functions are sensible candidates to define Bayesian estimators.', '1612.06149-3-20-2': 'Divergence functions also provide a link to the differential geometry of the space, which allows relating space geometry and Bayesian decision theory.', '1612.06149-3-20-3': 'This relationship has been used previously to analyse Bayesian decision problems from a Riemannian geometric viewpoint, leading to the so-called decision geometry framework [CITATION].', '1612.06149-3-20-4': 'Here we adopt an opposite perspective: we start by considering a Riemannian manifold [MATH] and then use the relationship to identify the divergence functions that arise naturally in that space.', '1612.06149-3-20-5': 'In particular, we focus on the so-called canonical divergence on [MATH], which is a generalisation of the Euclidean squared distance to this kind of manifold [CITATION].', '1612.06149-3-21-0': '[Canonical divergence [CITATION]] For any two points [MATH], the [MATH]-canonical divergence is given by [EQUATION] where [MATH] is the [MATH]-geodesic connecting [MATH] and [MATH].', '1612.06149-3-22-0': 'To gain a geometric intuition for [MATH] it is useful to compare it to the length of the [MATH]-geodesic between [MATH] and [MATH].', '1612.06149-3-22-1': 'Precisely, by noting that the squared length of a curve [MATH] on the manifold [MATH] is given by [MATH], we observe that [MATH] is essentially the squared length of the [MATH]-geodesic [MATH] weighted linearly along the path from [MATH] to [MATH].', '1612.06149-3-22-2': 'This weighting in [REF] guarantees that [MATH] is convex in [MATH], a necessary condition to define a divergence function (the weighting also leads to other important properties such as linearity w.r.t. [MATH], see section [REF]).', '1612.06149-3-22-3': 'The linear weighting also introduces an asymmetry, i.e., generally [MATH], which will have deep implications for Bayesian estimation.', '1612.06149-3-23-0': 'Finally, it is easy to check that [REF] reduces to the Euclidean squared distance [MATH] when [MATH] is the Euclidean space with inner product [MATH].', '1612.06149-3-23-1': 'More generally, [MATH] is always consistent with the local Euclidean geometry of the manifold [MATH].', '1612.06149-3-23-2': 'That is, for any point [MATH] in the neighbourhood of [MATH] we have [MATH], where [MATH] is the Euclidean norm of the tangent space [MATH] (a higher order approximation of [MATH] is also possible by using the affine connection [MATH] [CITATION]).', '1612.06149-3-23-3': 'And if we use the decision geometry framework [CITATION] to derive the Riemannian geometry induced by [MATH] on [MATH] we obtain [EQUATION] (here [MATH] and [MATH] denote differentiation w.r.t. the first and second components of [MATH] respective), which indicates that [MATH] is fully specified by [MATH] and that it induces the same space geometry that originated it in the first place.', '1612.06149-3-24-0': '# A geometric derivation of MAP and MMSE estimation', '1612.06149-3-25-0': '## From differential geometry to Bayesian decision theory', '1612.06149-3-26-0': 'In this section we use differential geometry to relate [MATH] to the loss functions used for Bayesian point estimation of [MATH].', '1612.06149-3-26-1': 'Precisely, we exploit the log-concavity of [MATH] to induce a Riemannian geometry on the parameter space.', '1612.06149-3-26-2': 'This in turn defines a canonical loss for that space and two Bayesian estimators w.r.t. to this loss: a primal estimator related to [MATH] and a dual estimator related to the dual divergence [MATH].', '1612.06149-3-26-3': 'We first consider the case where [MATH] is smooth and strongly log-concave, and later analyse the effect of relaxing these assumptions.', '1612.06149-3-27-0': '[Canonical Bayesian estimators] Suppose that [MATH] is strongly convex, continuous, and [MATH] on [MATH].', '1612.06149-3-27-1': 'Let [MATH] denote the Riemannian manifold induced by [MATH], with metric coefficients [MATH].', '1612.06149-3-27-2': 'Then, the canonical divergence on [MATH] is the Bregman divergence associated with [MATH], i.e., [EQUATION].', '1612.06149-3-27-3': 'In addition, the Bayesian estimator associated with [MATH] is unique and is given by the maximum-a-posteriori estimator, [EQUATION]', '1612.06149-3-27-4': 'The Bayesian estimator associated with the dual canonical divergence [MATH] is also unique and is given by the minimum mean squared error estimator [EQUATION]', '1612.06149-3-27-5': 'The proof is reported in the appendix.', '1612.06149-3-28-0': 'The way in which the Bregman divergence [MATH] measures the similarity between [MATH] and [MATH] is directly related to the log-concavity of [MATH].', '1612.06149-3-28-1': 'Precisely, because [MATH] is strongly convex, then [MATH] for any [MATH].', '1612.06149-3-28-2': 'The divergence [MATH] essentially quantifies this gap, which as mentioned previously is directly related to the length of the affine geodesic from [MATH] to [MATH] (and hence not only to the relative position of [MATH] and [MATH] but also to the space geometry).', '1612.06149-3-28-3': 'Moreover, [MATH] is linear w.r.t. [MATH].', '1612.06149-3-28-4': 'For example, if [MATH] for two convex functions [MATH] and [MATH] and [MATH], then [MATH]; it follows that for the specific case of Gaussian observation models [MATH] is equivalent to the loss identified in [CITATION].', '1612.06149-3-29-0': 'Finally, Theorem [REF] provides several valuable new insights into MAP and MMSE estimation in log-concave models.', '1612.06149-3-29-1': 'First, MAP estimation stems from Bayesian decision theory, and hence it stands on the same theoretical footing as the core Bayesian methodologies such as MMSE estimation.', '1612.06149-3-29-2': 'Second, the conventional definition of the MAP estimator as the maximiser [MATH] is mainly algorithmic for these models, useful to highlight that these estimators take the form of a convex optimisation problem that can be solved efficiently by convex optimisation (which is an important computational advantage over other Bayesian point estimators).', '1612.06149-3-29-3': 'Third, Theorem [REF] also reveals a surprising form of duality between MAP and MMSE estimation, which are intimately related to each other by the (asymmetry of the) canonical divergence function that [MATH] induces on the parameter space.', '1612.06149-3-29-4': 'This duality also manifests itself in other ways.', '1612.06149-3-29-5': 'For example it is easy to show that [MATH] is the Bayesian estimator associated with [MATH], where [MATH] is the convex dual or convex conjugate of [MATH] (see the appendix for details).', '1612.06149-3-29-6': 'Similarly, noting that [MATH] we see that [MATH] plays the role of [MATH] on the manifold [MATH].', '1612.06149-3-29-7': 'The case of Gaussian models is particular because [MATH] is Euclidean, which is a self-dual space; consequently [MATH] and the primal and dual canonical estimators coincide.', '1612.06149-3-29-8': 'Finally, Theorem [REF] also shows that under log-concavity and smoothness the posterior mode is a global property of [MATH], which is otherwise not an intuitive property.', '1612.06149-3-30-0': '## Error bounds for MAP and MMSE estimation', '1612.06149-3-31-0': 'We now establish performance guarantees for MAP and MMSE estimation when [MATH] is log-concave.', '1612.06149-3-31-1': 'Precisely, we establish universal estimation error bounds w.r.t. the dual error function [MATH].', '1612.06149-3-31-2': 'Here we do not assume that [MATH] is smooth; if [MATH] we replace [MATH] with the generalised divergence [MATH] where [MATH] belongs to the subdifferential set of [MATH] at [MATH] [CITATION].', '1612.06149-3-31-3': 'We first present the following universal bounds on the expected estimation error: [Expected error bound] Suppose that [MATH] is convex on [MATH].', '1612.06149-3-31-4': 'Then, [EQUATION].', '1612.06149-3-31-5': 'In addition, if [MATH] then [EQUATION]', '1612.06149-3-31-6': 'Proof.', '1612.06149-3-31-7': 'The proof is reported in the appendix.', '1612.06149-3-32-0': 'Theorem [REF] establishes that [MATH] minimises the expected dual canonical loss [MATH], and Proposition [REF] complements this result by providing an explicit and general upper bound on the loss incurred by using this Bayesian estimator.', '1612.06149-3-32-1': 'Proposition [REF] also states that this bound also applies to [MATH], and that the expected loss per coordinate (e.g., per pixel in imaging problems) cannot exceed [MATH].', '1612.06149-3-32-2': 'Observe that this is also a high dimensional stability result for MAP and MMSE estimation, which provides a theoretical argument for their good empirical performance in imaging, machine learning, and other large scale problems.', '1612.06149-3-33-0': 'Moreover, we also have the following universal large error bound for MAP estimation: [Large error bound] Suppose that [MATH] is convex on [MATH].', '1612.06149-3-33-1': 'Then, for any [MATH] [EQUATION]', '1612.06149-3-33-2': 'Proof.', '1612.06149-3-33-3': 'The proof is reported in the appendix.', '1612.06149-3-34-0': 'Proposition [REF] essentially indicates that in high dimensional settings the true value of [MATH] is almost certainly within the set [MATH], because the probability of a larger error vanishes exponentially fast as [MATH] increases.', '1612.06149-3-34-1': 'Again, this theoretical result supports the vast empirical evidence that MAP estimation delivers accurate results in large-scale convex problems.', '1612.06149-3-34-2': 'It also follows from Proposition [REF] that in such problems [MATH] and [MATH] are close to each other (i.e., that [MATH] with high probability).', '1612.06149-3-35-0': '## Relaxation of regularity conditions', '1612.06149-3-36-0': 'We now examine the effect of relaxing the regularity assumptions of Theorem [REF].', '1612.06149-3-36-1': 'We consider three main cases: lack of smoothness, lack of strong convexity, and lack of continuity.', '1612.06149-3-37-0': '### Non-smooth models', '1612.06149-3-38-0': 'Several models used in imaging and machine learning are not smooth because they involve priors with non-differentiable points, such as priors based on the [MATH] norm, the nuclear norm, and the total-variation pseudo-norm [CITATION].', '1612.06149-3-38-1': 'The results of Theorem [REF] hold for these models with the following minor modifications.', '1612.06149-3-39-0': 'First, observe that these non-smooth models are [MATH] almost everywhere; that is, the set of non-differentiable points has dimension [MATH], and consequently it has no probability mass and can be omitted in the computation of expectations.', '1612.06149-3-39-1': 'Second, because the non-differentiable points do not have Euclidean tangent spaces, instead of a global manifold we consider the collection local manifolds associated with the regions of [MATH] where [MATH] is [MATH].', '1612.06149-3-39-2': 'Each one of these regions has a local canonical divergence given by the Bregman divergence [MATH].', '1612.06149-3-39-3': 'Therefore, for these models we posit [MATH] as the global loss function for any [MATH] [technically the global loss is the generalised Bregman divergence [MATH], where [MATH] belongs to the subdifferential set of [MATH] at [MATH] [CITATION], however the expectation [MATH] is taken over the points where [MATH] is [MATH] and hence [MATH]].', '1612.06149-3-39-4': 'We then consider the primal and dual Bayesian estimators related to this global loss and obtain that [MATH] and [MATH], similarly to Theorem [REF].', '1612.06149-3-39-5': 'Observe that in the computation of [MATH] and [MATH] the argument [MATH] is optimised over [MATH] including non-differentiable points.', '1612.06149-3-40-0': ""Finally, we note that despite not being a global canonical divergence, [MATH] is still consistent with the space's Riemannian geometry which is local."", '1612.06149-3-40-1': 'In addition, the key high dimensional performance guarantees of Propositions [REF] and [REF] also hold because [MATH] is convex.', '1612.06149-3-41-0': '### Strictly log-concave models', '1612.06149-3-42-0': 'For models that are strictly log-concave but not strongly log-concave only the second and third results of Theorem [REF] remain true.', '1612.06149-3-42-1': 'It is easy to check that the Bayesian estimator associated with [MATH] is [MATH], and that [MATH], similarly to strongly log-concave models.', '1612.06149-3-42-2': 'Therefore, the decision-theoretic derivation of [MATH] remains valid, and [MATH] and [MATH] remain dual to each other.', '1612.06149-3-42-3': 'The high dimensional performance guarantees of Propositions [REF] and [REF] also hold because [MATH] is convex.', '1612.06149-3-42-4': 'However, without strong convexity, [MATH] becomes semi-positive definite and [MATH] becomes a singular manifold.', '1612.06149-3-42-5': 'Currently, the validity of the interpretation of [MATH] as a canonical divergence in singular manifolds is not clear.', '1612.06149-3-42-6': 'The generalisation of canonical divergences and of Theorem [REF] to singular manifolds is currently under investigation.', '1612.06149-3-43-0': '### Models involving constraints on the parameter space', '1612.06149-3-44-0': 'Finally, in cases where [MATH] is constrained to a convex set [MATH] only the first and the third results of Theorem [REF] hold.', '1612.06149-3-44-1': 'Proceeding similarly to the proof of Theorem [REF], it is easy to show that [MATH] is the canonical divergence of the manifold [MATH], and that the Bayesian estimator related to the dual divergence is [MATH].', '1612.06149-3-44-2': 'However, the Bayesian estimator that minimises the canonical divergence is now a shifted MAP estimator [EQUATION] where generally [MATH] (see the appendix for details).', '1612.06149-3-44-3': 'Therefore, for these models [MATH] is potentially not a proper Bayesian estimator in the decision-theoretic sense.', '1612.06149-3-44-4': 'Nevertheless, the high dimensional guarantees of Propositions [REF] and [REF] still hold for [MATH], providing partial theoretical justification for using this estimator (also observe that [MATH] is in the neighbourhood of [MATH] in the sense of Proposition [REF]).', '1612.06149-3-45-0': '# Conclusion', '1612.06149-3-46-0': 'MAP estimation is one of the the most successful Bayesian estimation methodologies in modern data science, with a track record of accurate results across a wide range of challenging applications involving very high dimensionality.', '1612.06149-3-46-1': 'Our aim here has been to contribute to the theoretical understanding of this widely used methodology, particularly by placing it in the Bayesian decision theory framework that underpins the core Bayesian inference methodologies.', '1612.06149-3-47-0': 'In order to analyse MAP estimators we have adopted an entirely new approach: we allowed the model to self-specify the loss function, or equivalently the Bayesian estimator, that is used to summarise the information that the model represents.', '1612.06149-3-47-1': 'This was achieved by using the connections between model log-concavity, Riemannian geometry, and divergence functions.', '1612.06149-3-47-2': 'We first established that if [MATH] is strongly log-concave, continuous, and [MATH] on [MATH], then [MATH] induces a dually-flat Riemannian structure on the parameter space, where the canonical divergence is the Bregman divergence associated with [MATH], and where the MAP estimator is the unique Bayesian estimator w.r.t. to this loss function.', '1612.06149-3-47-3': 'We also established that the MMSE estimator is the Bayesian estimator w.r.t. the dual canonical loss, and that both estimators enjoy favourable stability properties in high dimensions.', '1612.06149-3-47-4': 'We then examined the effect of relaxing these assumptions to models that are not smooth, strictly but not strongly convex, or that involve constraints on the parameter space.', '1612.06149-3-48-0': 'The theoretical results presented in this work provide several valuable new insights into MAP and MMSE estimation.', '1612.06149-3-48-1': 'In particular, both estimators stem from Bayesian decision theory and from the consideration of the geometry of the parameter space, and exhibit an interesting form duality.', '1612.06149-3-48-2': 'Also, the bounds on the expected estimation error and the large error probability for MAP estimators support the remarkable empirical performance observed in large scale settings, such as imaging and machine learning problems.', '1612.06149-3-48-3': 'The results also show that the predominant view of MAP estimators as hastily heuristic or approximate inferences, motivated only by computational efficiency, is fundamentally incorrect (though the fact that MAP estimators are available as solutions to convex minimisation problems is a fundamental practical advantage).', '1612.06149-3-48-4': 'We hope that these new theoretical results will promote a wider adoption of this powerful Bayesian point estimation methodology across all domains of statistical data science.'}","{'1612.06149-4-0-0': 'Maximum-a-posteriori (MAP) estimation is the main Bayesian estimation methodology in imaging sciences, where high dimensionality is often addressed by using Bayesian models that are log-concave and whose posterior mode can be computed efficiently by convex optimisation.', '1612.06149-4-0-1': 'However, despite its success and wide adoption, MAP estimation is not theoretically well understood yet.', '1612.06149-4-0-2': 'In particular, the prevalent view in the community is that MAP estimation is not proper Bayesian estimation in the sense of Bayesian decision theory because it does not minimise a meaningful expected loss function (unlike the minimum mean squared error (MMSE) estimator that minimises the mean squared loss).', '1612.06149-4-0-3': 'This paper addresses this theoretical gap by presenting a general decision-theoretic derivation of MAP estimation in Bayesian models that are log-concave.', '1612.06149-4-0-4': 'A main novelty is that our analysis is based on differential geometry, and proceeds as follows.', '1612.06149-4-0-5': 'First, we use the underlying convex geometry of the Bayesian model to induce a Riemannian geometry on the parameter space.', '1612.06149-4-0-6': 'We then use differential geometry to identify the so-called natural or canonical loss function to perform Bayesian point estimation in that Riemannian manifold.', '1612.06149-4-0-7': 'For log-concave models, this canonical loss coincides with the Bregman divergence associated with the negative log posterior density.', '1612.06149-4-0-8': 'Following on from this, we show that the MAP estimator is the only Bayesian estimator that minimises the expected canonical loss, and that the posterior mean or MMSE estimator minimises the dual canonical loss.', '1612.06149-4-0-9': 'We then study the question of MAP and MSSE estimation performance in high dimensions.', '1612.06149-4-0-10': 'Precisely, we establish a universal bound on the expected canonical error as a function of image dimension, providing new insights the good empirical performance observed in convex problems.', '1612.06149-4-0-11': 'Together, these results provide a new understanding of MAP and MMSE estimation in log-concave settings, and of the multiple beneficial roles that convex geometry plays in imaging problems.', '1612.06149-4-0-12': 'Finally, we illustrate this new theory by analysing the regularisation-by-denoising Bayesian models, a class of state-of-the-art imaging models where priors are defined implicitly through image denoising algorithms, and an image denoising model with a wavelet shrinkage prior.', '1612.06149-4-1-0': '# Introduction', '1612.06149-4-2-0': 'We consider the estimation of an unknown image [MATH] from some data [MATH], related to [MATH] by a statistical model with likelihood [MATH].', '1612.06149-4-2-1': 'Adopting a Bayesian statistical approach, we postulate a prior distribution [MATH] modelling the prior knowledge available, and base our inferences on the posterior distribution [CITATION] [EQUATION] which models our knowledge about [MATH] after observing [MATH].', '1612.06149-4-2-2': 'In this paper we focus on the case where [MATH] belongs to the class of log-concave distribution, i.e., [EQUATION] for some proper convex function [MATH] [CITATION], and where we note that for notation convenience we do not write explicitly the dependence of [MATH] on [MATH].', '1612.06149-4-3-0': 'Log-concave models [REF] are ubiquitous in modern imaging sciences.', '1612.06149-4-3-1': 'For instance, many imaging methods to solve linear imaging inverse problems involving additive Gaussian noise use models of the form [MATH] for some linear operators [MATH] and [MATH], convex regulariser [MATH], and convex set [MATH] (see [CITATION] for examples related to image deblurring, inpainting, compressive sensing, super-resolution, and tomographic reconstruction, with total-variation and wavelet priors).', '1612.06149-4-3-2': 'Similar log-concave Bayesian models can be considered for problems involving other observation noise models, such as Poisson noise [CITATION], and for phase retrieval problems [CITATION].', '1612.06149-4-3-3': 'Log-concave models [REF] are also used extensively in other areas of data science such as machine learning [CITATION].', '1612.06149-4-4-0': 'Because drawing conclusions directly from [MATH] is difficult, Bayesian imaging methods deliver summaries of [MATH] - namely Bayes point estimators - that summarise [MATH] optimally in a single value [MATH].', '1612.06149-4-4-1': 'This estimated value is optimal the following decision-theoretic sense [CITATION]:', '1612.06149-4-5-0': 'Let [MATH] be a loss function that quantifies the difference between two points in [MATH].', '1612.06149-4-5-1': 'A Bayes estimator associated with [MATH] is any estimator that minimises the posterior expected loss, i.e., [EQUATION]', '1612.06149-4-5-2': 'Recall that the posterior expectation [MATH].', '1612.06149-4-5-3': 'Sensible loss functions [MATH] usually verify the following three desiderata [CITATION]:', '1612.06149-4-6-0': 'Estimator uniqueness is important because it implies admissibility (i.e., Bayesian estimator [MATH] is not dominated by any other estimator) [CITATION].', '1612.06149-4-6-1': 'Observe that [MATH] is not necessarily symmetric; that is, [MATH].', '1612.06149-4-6-2': 'We do not enforce symmetry because the arguments of [MATH] have clearly different roles in the decision problem.', '1612.06149-4-7-0': 'In a purely theoretical Bayesian exercise, [MATH] should be chosen based on specific aspects of the problem and application considered.', '1612.06149-4-7-1': 'This is particularly important in imaging problems that are ill-posed or ill-conditioned, as the choice of [MATH] can significantly impact estimation results.', '1612.06149-4-7-2': 'However, specifying a bespoke loss function for high dimensional problems is not easy.', '1612.06149-4-7-3': 'Consequently, most methods in the imaging literature use default losses and estimators.', '1612.06149-4-8-0': 'In particular, Bayesian imaging methods have traditionally used the minimum mean squared error (MMSE) estimator, given by the posterior mean [MATH].', '1612.06149-4-8-1': 'This estimator is widely regarded as a gold standard in the field, in part because of its good empirical performance and favourable theoretical properties, and also perhaps in part because of cultural heritage.', '1612.06149-4-8-2': 'From Bayesian decision theory, MMSE estimation is optimal with respect to the entire class of Euclidean or Mahalanobis squared distances, given by quadratic loss functions of the form [MATH] with [MATH] (i.e., the set of [MATH] positive definite matrices), which includes the popular mean square loss [MATH] when [MATH] [CITATION].', '1612.06149-4-8-3': 'This gives [MATH] a straightforward geometric interpretation.', '1612.06149-4-8-4': 'Moreover, MSE estimation is optimal w.r.t. a more general class of functions [CITATION] that provides a second order approximation to any strongly convex loss function; hence [MATH] can act as a universal proxy for other Bayesian estimators in this sense.', '1612.06149-4-9-0': 'Unfortunately, calculating [MATH] in high dimensional models can be very difficult because it requires solving integrals that are often too computationally expensive.', '1612.06149-4-9-1': 'This has stimulated much research on the topic, from fast Monte Carlo simulation methods to efficient approximations with deterministic algorithms [CITATION].', '1612.06149-4-9-2': 'But with ever increasingly large problems and datasets, many imaging methods have focused on alternatives to MMSE estimation.', '1612.06149-4-10-0': 'In particular, modern methods rely strongly on maximum-a-posteriori (MAP) estimation [EQUATION] whose calculation is a convex problem that can often be solved very efficiently, even in very high dimensions (e.g., [MATH]), by using proximal convex optimisation techniques [CITATION].', '1612.06149-4-10-1': 'Modern non-statistical imaging methods also predominately solve problems by convex optimisation, and their solutions are often equivalent to performing MAP estimation for some implicit Bayesian model.', '1612.06149-4-10-2': 'The precise sense in which these solutions are equivalent to MAP estimators is an interesting discussion topic that is beyond the scope of this paper.', '1612.06149-4-11-0': 'Following a decade of intensive activity, there is now abundant evidence that MAP estimation delivers accurate results for a wide range of imaging problems.', '1612.06149-4-11-1': 'However, from a theoretical viewpoint MAP estimation is not well understood.', '1612.06149-4-11-2': 'Currently the predominant view is that MAP estimation is not formal Bayesian estimation in the decision-theoretic sense postulated by Definition [REF] because it does not generally minimise a known expected loss.', '1612.06149-4-11-3': 'The prevailing interpretation is that MAP estimation is in fact an approximation arising from the degenerate loss [MATH] with [MATH] [CITATION] (this derivation holds for all log-concave models, but is not generally true [CITATION]).', '1612.06149-4-11-4': 'However, this asymptotic derivation does not lead to a proper Bayesian estimator.', '1612.06149-4-11-5': 'More importantly, the resulting loss is very difficult to motivate for inference problems in continuous domains such as [MATH], and does not help explain the good empirical performance reported in the literature.', '1612.06149-4-12-0': 'Furthermore, most other theoretical results for MAP estimation only hold for very specific models, or have been derived by adopting analyses that are extrinsic to the Bayesian decision theory framework (e.g. by analysing MAP estimation as constrained or regularised least-squares regression, see for example [CITATION]).', '1612.06149-4-12-1': 'As a trivial example of results that only hold for specific models, when [MATH] is symmetric we have [MATH], and thus MAP estimation inherits the favourable properties of MMSE estimation.', '1612.06149-4-12-2': 'This result has been partially extended to some denoising models of the form [MATH] in [CITATION], where it is shown that MAP estimation coincides with MMSE estimation with a different model [MATH] involving a different prior distribution.', '1612.06149-4-12-3': 'It follows that for these models MAP estimation is decision-theoretic Bayesian estimation w.r.t. the weighted loss [MATH].', '1612.06149-4-12-4': 'This is of course a post-hoc loss, but the result is interesting in that it highlights that a single estimator may have a plurality of origins.', '1612.06149-4-12-5': 'More importantly, it raises the question if MAP estimation is merely a computational proxy for MMSE estimation, which unlike [MATH] has an appealing theoretical underpinning.', '1612.06149-4-12-6': 'This question was recently answered in Burger Lucka [CITATION]: MAP estimation is proper decision-theoretic Bayesian estimation for all models of the form [MATH], with known linear operator [MATH] and noise covariance [MATH], and where [MATH] is convex and Lipchitz continuous.', '1612.06149-4-12-7': 'More precisely, that paper shows that for this class of models MAP estimation is optimal w.r.t. the loss [MATH], where [MATH] is the [MATH]-Bregman divergence [CITATION].', '1612.06149-4-12-8': 'The paper also shows that [MATH] outperforms [MATH] w.r.t., the expected Bregman error [MATH], an error measure that grasps some distinctive features of [MATH] (e.g., sparsity, regularity, smoothness, etc.).', '1612.06149-4-12-9': 'It may appear that the loss identified in [CITATION] is rather artificial and difficult to analyse and motivate; however, the new results presented in Section [REF] show that it is a specific instance of a more general loss that stems directly from the consideration of the geometry of the Bayesian model.', '1612.06149-4-13-0': 'It is also worth mentioning that several recent works have studied MAP estimation in infinite-dimensional settings, which is important for our understanding of how the technique behaves in increasingly large problems.', '1612.06149-4-13-1': 'An important advance in this area is the connection of the topological description of the MAP estimate to a variational problem, developed in [CITATION] for non-linear inverse problems in a Gaussian framework, and subsequently extended to non-Gaussian settings in [CITATION].', '1612.06149-4-13-2': 'Agapiou et al. made another key contributions in this area by studying infinite-dimensional MAP estimation with Besov priors, which are very relevant to imaging sciences because they promote sparsity and preservation of edges[CITATION].', '1612.06149-4-13-3': 'We also mention the recent work [CITATION] that further improves our understanding of modes in infinite dimensions.', '1612.06149-4-14-0': 'In order to better understand MAP estimation, in this paper we first revisit the choice of the loss function for Bayesian point estimation in the context of models that are log-concave, where MAP is a convex problem (we limit our analysis to finite-dimensional problems).', '1612.06149-4-14-1': 'A main novelty of our analysis is that, instead of specifying the loss directly, we use differential geometry to derive the loss from the geometry of the model.', '1612.06149-4-14-2': 'Precisely, we show that under some regularity assumptions, the log-concavity of [MATH] induces a specific Riemannian differential geometry on the parameter space, and that taking into account this space geometry naturally leads to an intrinsic or canonical loss function to perform Bayesian point estimation in that space.', '1612.06149-4-14-3': 'Following on from this, we establish that the canonical loss for the parameter space is the Bregman divergence associated with [MATH], and that the Bayesian estimator w.r.t. this loss is the MAP estimator.', '1612.06149-4-14-4': 'We then show that the MMSE estimator is the Bayesian estimator associated with the dual canonical loss, and propose universal estimation performance guarantees for MAP and MMSE estimation in log-concave models.', '1612.06149-4-14-5': 'We conclude by illustrating our theory with an application to linear inverse problems with sparsity-promoting wavelet priors, and an analysis of the regularisation-by-denoising models proposed recently in [CITATION].', '1612.06149-4-15-0': 'The remainder of the paper is organised as follows: Section [REF] introduces the elements of differential geometry that are essential to our analysis.', '1612.06149-4-15-1': 'In Section [REF] we present our main theoretical results: a decision-theoretic and differential-geometric derivation of MAP and MMSE estimation, with universal bounds on the estimation error involved.', '1612.06149-4-15-2': 'Section [REF] discusses the impact of relaxing the regularity assumptions adopted in Section [REF].', '1612.06149-4-15-3': 'Conclusions are finally reported in section [REF].', '1612.06149-4-15-4': 'Proofs are presented in the appendix.', '1612.06149-4-16-0': '# Riemannian geometry and the canonical divergence function', '1612.06149-4-17-0': 'In this section we recall some elements of differential geometry that are necessary for our analysis.', '1612.06149-4-17-1': 'For a detailed introduction to this topic we refer the reader to [CITATION].', '1612.06149-4-18-0': 'An [MATH]-dimensional Riemannian manifold [MATH], with metric [MATH] and global coordinate system [MATH], is a vector space that behaves locally as an Euclidean space.', '1612.06149-4-18-1': 'More precisely, at any point [MATH], we have a tangent space [MATH] with inner product [MATH] and norm [MATH].', '1612.06149-4-18-2': 'This tangent space describes how the manifold [MATH] behaves locally at [MATH].', '1612.06149-4-18-3': 'The geometry is local and may vary smoothly from [MATH] to neighbouring tangent spaces (i.e., the inner product and norm used are local properties and vary spatially).', '1612.06149-4-18-4': 'The variations are encoded in the affine connection [MATH], with coefficients given by [MATH] describing the spatial evolution of the metric [MATH].', '1612.06149-4-19-0': 'A crucial property of [MATH] is that, similarly to Euclidean spaces, manifolds supports divergence functions:', '1612.06149-4-20-0': 'A function [MATH] is a divergence function on [MATH] if the following conditions hold for any [MATH]:', '1612.06149-4-21-0': 'Observe that the class of divergence functions is equivalent to that of loss functions for Bayesian point estimation specified in Section [REF], with some mild additional regularity conditions.', '1612.06149-4-21-1': 'This suggests that divergence functions are sensible losses to define estimators.', '1612.06149-4-21-2': 'Moreover, divergence functions also provide a link to the differential geometry of the space, which allows relating space geometry and Bayesian decision theory.', '1612.06149-4-21-3': 'This relationship has been used previously to analyse Bayesian decision problems from a Riemannian geometric viewpoint, leading to the so-called decision geometry framework [CITATION].', '1612.06149-4-21-4': 'Here we adopt an opposite perspective: we start by considering a Riemannian manifold [MATH] and then use the relationship to identify the divergence functions that arise naturally in that space.', '1612.06149-4-21-5': 'In particular, we focus on the so-called canonical divergence on [MATH], which generalises the Euclidean squared distance to this kind of manifold [CITATION].', '1612.06149-4-22-0': '[Canonical divergence [CITATION]] For any two points [MATH], the [MATH]-canonical divergence is given by [EQUATION] where [MATH] is the [MATH]-geodesic from [MATH] to [MATH] and [MATH].', '1612.06149-4-23-0': 'The reason why [MATH] is the [MATH]-canonical divergence is that it fully specifies the geometry of [MATH]; i.e., there is a one-to-one relationship between [MATH] and the metric [MATH].', '1612.06149-4-24-0': 'Observe that [MATH] has connections to the length of the [MATH]-geodesic between [MATH] and [MATH].', '1612.06149-4-24-1': 'Precisely, by noting that the squared length of a curve [MATH] on the manifold [MATH] is given by [MATH], we observe that [MATH] is essentially the squared length of the [MATH]-geodesic [MATH] weighted linearly along the path from [MATH] to [MATH].', '1612.06149-4-24-2': 'This weighting in [REF] is important because it guarantees that [MATH] is convex in [MATH], a necessary condition to define a divergence function (the weighting also leads to other important properties such as linearity w.r.t. [MATH], see section [REF]).', '1612.06149-4-24-3': 'Note that the weighting also introduces an asymmetry, i.e., generally [MATH], which will have deep implications for Bayesian estimation.', '1612.06149-4-25-0': 'Finally, it is easy to check that [REF] reduces to the Euclidean squared distance [MATH] when [MATH] is the Euclidean space with product [MATH].', '1612.06149-4-25-1': 'More generally, [MATH] is always consistent with the local Euclidean geometry of the manifold [MATH].', '1612.06149-4-25-2': 'That is, for any point [MATH] in the neighbourhood of [MATH] we have [MATH], where [MATH] is the Euclidean norm of the tangent space [MATH] (a higher order approximation of [MATH] is also possible by using the connection [MATH] [CITATION]).', '1612.06149-4-25-3': 'And because [MATH] is the canonical divergence, if we use the decision geometry framework [CITATION] to derive the Riemannian geometry induced by [MATH] on [MATH] we obtain [EQUATION] (here [MATH] and [MATH] denote differentiation w.r.t. the first and second components of [MATH] respectively), indicating that [MATH] fully specifies the geometry of [MATH], and vice-versa.', '1612.06149-4-26-0': '# A differential-geometric derivation of MAP and MMSE estimation', '1612.06149-4-27-0': '## Canonical Bayesian estimation: from differential geometry to decision theory', '1612.06149-4-28-0': 'In this section we use differential geometry to relate the geometry of [MATH] to the loss functions used for Bayesian estimation of [MATH].', '1612.06149-4-28-1': 'Precisely, we exploit the log-concavity of [MATH] to induce a Riemannian geometry on the solutions space.', '1612.06149-4-28-2': 'This in turn defines a canonical loss for that space and two Bayesian estimators: a primal estimator related to [MATH] and a dual estimator related to the dual divergence [MATH].', '1612.06149-4-28-3': 'We focus on the case where [MATH] is smooth and strongly log-concave, and later analyse the effect of relaxing these assumptions.', '1612.06149-4-29-0': '[Canonical Bayesian estimators] Suppose that [MATH] is strongly convex, continuous, and [MATH] on [MATH].', '1612.06149-4-29-1': 'Let [MATH] denote the Riemannian manifold induced by [MATH], with metric coefficients [MATH].', '1612.06149-4-29-2': 'Then, the canonical divergence on [MATH] is the [MATH]-Bregman divergence, i.e., [EQUATION].', '1612.06149-4-29-3': 'In addition, the Bayesian estimator associated with [MATH] is unique and is given by the maximum-a-posteriori estimator, [EQUATION]', '1612.06149-4-29-4': 'The Bayesian estimator associated with the dual canonical divergence [MATH] is also unique and is given by the minimum mean squared error estimator [EQUATION]', '1612.06149-4-29-5': 'The proof is reported in the appendix.', '1612.06149-4-30-0': 'Theorem [REF] provides several interesting new insights into MAP and MMSE estimation in log-concave models.', '1612.06149-4-30-1': 'First and foremost, MAP estimation stems from Bayesian decision theory, and hence it stands on the same theoretical footing as the core Bayesian methodologies such as MMSE estimation (albeit w.r.t. a different class of loss functions).', '1612.06149-4-30-2': 'The MAP loss, [MATH], is a generalisation of the Euclidean squared distance that arises naturally from the consideration of the geometry of [MATH].', '1612.06149-4-30-3': 'Consequently, the conventional definition of the MAP estimator as the maximiser [MATH] is mainly algorithmic for these models, useful to highlight that these estimators take the form of a convex optimisation problem.', '1612.06149-4-30-4': '(Of course, this is a key computational advantage over other Bayesian point estimators because it allows computing [MATH] by using using modern proximal convex optimisation algorithms that scale very efficiently to high-dimensions - see e.g., [CITATION] for details).', '1612.06149-4-30-5': 'Moreover, Theorem [REF] also reveals a surprising form of duality between MAP and MMSE estimation, with the two estimators intimately related to each other by the (asymmetry of the) canonical divergence function that [MATH] induced on the solutions space.', '1612.06149-4-30-6': 'Note that Gaussian models are particular because [MATH] is Euclidean in that case, which is a self-dual space; consequently [MATH] and the primal and dual canonical estimators coincide as a result.', '1612.06149-4-30-7': 'Finally, Theorem [REF] also shows that, under log-concavity and smoothness assumptions, the posterior mode is a global property of [MATH], similarly to the posterior mean.', '1612.06149-4-31-0': 'The way in which the Bregman divergence [MATH] measures the similarity between [MATH] and [MATH] is directly related to the geometry of [MATH].', '1612.06149-4-31-1': 'Precisely, because [MATH] is strongly convex, then [MATH] for any [MATH].', '1612.06149-4-31-2': 'The divergence [MATH] essentially quantifies this gap, which as mentioned previously, is directly related to the length of the affine geodesic from [MATH] to [MATH] (and hence not only to the relative position of [MATH] and [MATH] but also to the space geometry induced by [MATH]).', '1612.06149-4-31-3': 'Moreover, this geometry can depend on the value of [MATH], however for the important class of models of the [MATH] the geometry is completely specified by [MATH] and [MATH] independently of [MATH].', '1612.06149-4-31-4': 'Furthermore, observe that because [MATH] is linear w.r.t. [MATH], then for any decomposition [MATH] based on two convex functions [MATH] and [MATH] and [MATH], we obtain [MATH].', '1612.06149-4-31-5': 'It follows that for the specific case of Gaussian linear observation models, the canonical divergence [MATH] is equivalent to the specific loss identified in [CITATION].', '1612.06149-4-32-0': 'We also mention at this point that for Gaussian denoising models; i.e., [MATH], the estimator [MATH] results from the computation of the proximal operator [MATH] [CITATION].', '1612.06149-4-32-1': 'This is equivalent to a gradient step on the Moreau-Yoshida regularisation of [MATH]; i.e., [MATH], with [MATH].', '1612.06149-4-32-2': 'In like manner, [MATH] can be expressed as the gradient step [MATH], where [MATH] is a different smooth approximation of [MATH] (please see [CITATION] for details).', '1612.06149-4-33-0': 'Also note that a different Bregman divergence, namely the KL divergence [MATH], is often used in Bayesian point estimation to define an estimator that is independent of the parametrisation of the likelihood [CITATION].', '1612.06149-4-33-1': 'This estimator is particularly relevant when the object of interest is [MATH], as opposed to the value of [MATH] itself, for example in prediction problems.', '1612.06149-4-33-2': 'This estimator is not often used in imaging sciences.', '1612.06149-4-34-0': 'Finally, we notice that because [MATH] is derived from [MATH] may depend on the value of [MATH], which is controversial in some lines of Bayesian thinking because it implies that the decision problem underpinning the estimator is defined a-posteriori.', '1612.06149-4-34-1': 'This can happen for example in problems involving non-Gaussian observation models.', '1612.06149-4-34-2': 'Our view on this matter is that although decision problems are generally defined a-priori, the case of Bayesian estimators is particular because the decision involved is precisely how to summarise [MATH], and this decision can be considered a-posteriori if this allows delivering an estimator with favourable accuracy or computational properties.', '1612.06149-4-34-3': 'Of course, loss functions that do not depend on the model considered also have advantages, namely the mean squared error loss that also leads to an estimator with good properties (albeit often very expensive to compute).', '1612.06149-4-34-4': 'In any case, it is fundamental that one understands how the estimator that one uses summarises [MATH], and the aim of this work is to improve our understanding of the widely used MAP estimator.', '1612.06149-4-35-0': '## Error bounds for MAP and MMSE estimation', '1612.06149-4-36-0': 'Theorem [REF] establishes that under certain conditions [MATH] is a proper Bayesian estimator.', '1612.06149-4-36-1': 'Following on from this, it is natural to study the accuracy of [MATH] as a Bayesian estimator.', '1612.06149-4-36-2': 'The Bayesian approach to this question is to infer the accuracy of [MATH] according to the posterior distribution [MATH].', '1612.06149-4-36-3': 'For [MATH] this generally corresponds to computing the expected MSE loss, related to the posterior covariance.', '1612.06149-4-36-4': 'This type of analysis can be useful, for example, to identify high dimensional stability conditions (i.e., conditions under which the error grows linearly with [MATH]).', '1612.06149-4-37-0': 'Here we perform this type of analysis for [MATH] w.r.t. the canonical loss.', '1612.06149-4-37-1': 'Precisely, we establish universal estimation error bounds w.r.t. the dual error function [MATH] for MAP and MMSE estimation, for which we have the following result:', '1612.06149-4-38-0': '[Expected error bound] Suppose that [MATH] is convex on [MATH] and [MATH].', '1612.06149-4-38-1': 'Then, [EQUATION]', '1612.06149-4-38-2': 'Proof.', '1612.06149-4-38-3': 'The proof is reported in the appendix.', '1612.06149-4-39-0': 'We read Proposition [REF] as a high dimensional stability result for MAP and MMSE estimation, stating that the expected estimation error, as measured by the dual loss [MATH], grows at most linearly with the number of image pixels.', '1612.06149-4-39-1': 'Therefore, even if the likelihood [MATH] is poorly identifiable because [MATH], or because the linear operator [MATH] is very rank deficient, or because [MATH] is corrupted by Poison noise, in smooth log-concave settings the expected error cannot grow polynomially or with a linear constant greater than [MATH].', '1612.06149-4-40-0': 'To formally study the expected error as [MATH] increases we consider a generic log-concave stochastic process [MATH], where for each [MATH], the random vector [MATH] has marginal distribution [MATH] for some convex function [MATH].', '1612.06149-4-40-1': 'We also assume that the entropy rate of [MATH] is finite; i.e., [MATH] [CITATION].', '1612.06149-4-40-2': 'This limit captures the asymptotic information gain per pixel and characterises global statistical features of the image, particularly correlations at any rage.', '1612.06149-4-40-3': 'In log-concave settings, this condition holds for example when [MATH] ; it also holds when [MATH] is strongly stationary [CITATION].', '1612.06149-4-40-4': 'By analysing Proposition [REF] in this setting we see that [EQUATION].', '1612.06149-4-40-5': 'Then, because the entropic rate of [MATH] is finite [MATH] [CITATION], and hence the dimension-normalised expected errors verify [EQUATION].', '1612.06149-4-40-6': 'We emphasise again this form of dimension stability is not generally available in estimation problems, and is a direct consequence of the log-concavity of the model and its relationship with the MAP and MMSE estimators.', '1612.06149-4-40-7': 'Finally, observe that the above error bounds are tight; e.g., the trivial i.i.d. process [MATH], for [MATH], [MATH], attains the bound.', '1612.06149-4-40-8': 'Lastly, we conjecture that this bound can be improved for specific subclasses of log-concave models by using entropy rate results from the probability literature; future work will investigate this.', '1612.06149-4-41-0': '## Connections to other works', '1612.06149-4-42-0': 'We conclude this section by discussing some connections between this paper and other theoretical works related to MAP estimation.', '1612.06149-4-42-1': 'As previously explained, Theorem [REF] directly builds on [CITATION], which considered the class of log-concave models [MATH] with Gaussian likelihood [MATH] and prior [MATH], and establishes that in this case [MATH] is the Bayesian estimator for the Bregman loss [MATH].', '1612.06149-4-42-2': 'Theorem [REF] generalises this result to a larger class of posterior distributions and provides motivation for this unconventional loss function by showing that it stems directly from the consideration of the geometry of the parameter space.', '1612.06149-4-42-3': 'Proposition [REF] provides further motivation for this loss by establishing explicit bounds on the expected estimation error.', '1612.06149-4-43-0': 'It is worth mentioning that the generalisation of [CITATION] to other log-concave models was developed simultaneously and independently in Burger et al. [CITATION] (see [CITATION] for MAP estimation, and [CITATION] for MSSE estimation, which is also closely related to [CITATION]).', '1612.06149-4-43-1': 'Moreover, that work also analyses the expected estimation error involved in MAP and MMSE estimation but w.r.t. other divergence functions.', '1612.06149-4-43-2': 'More precisely, Burger et al. [CITATION] uses the Bregman divergence [MATH] related to the regulariser or negative log-prior, whereas we use the canonical Bregman divergence [MATH] related to the negative log-posterior.', '1612.06149-4-43-3': 'As mentioned previously, [MATH] grasps important features of [MATH] (e.g;, sparsity, regularity, smoothness), and is always independent of the observed data [MATH], whereas [MATH] is independent of [MATH] only in special cases (e.g., Gaussian linear observation models).', '1612.06149-4-44-0': 'Moreover, Burger et al. also show that [MATH] and conclude that [MATH] outperforms [MATH] when the estimation error is measured in this way, which is independent of [MATH].', '1612.06149-4-44-1': 'To analyse how these expected errors behave as dimensionality increases we combine this result with Proposition [REF] and obtain the following bound:', '1612.06149-4-45-0': 'Suppose that [MATH] is convex on [MATH] and [MATH].', '1612.06149-4-45-1': 'Then, [EQUATION]', '1612.06149-4-45-2': 'Proof.', '1612.06149-4-45-3': 'The proof follows directly from combining [CITATION] with Proposition [REF] and the fact that [MATH], for any splitting [MATH] where [MATH] and [MATH] are convex functions.', '1612.06149-4-45-4': 'This result can also be derived from the integration by parts argument in [CITATION].', '1612.06149-4-46-0': 'Again, we read Corollary [REF] as a high dimensional stability result for MAP and MMSE estimation, stating that the expected estimation error, measured in this case by the dual loss [MATH], grows at most linearly with the number of image pixels.', '1612.06149-4-46-1': 'Polynomial growth or faster linear growth is not possible within the class of smooth log-concave models, even if the problem is very ill-conditioned.', '1612.06149-4-46-2': 'At the same time, this linear rate cannot be further improved, as any i.i.d. process [MATH] will have an error that grows linearly with [MATH].', '1612.06149-4-47-0': '# Illustrative examples', '1612.06149-4-48-0': 'As a way of illustrating our theory, we now analyse the geometry of a simple image denoising model in the wavelet domain, and of the regularisation-by-denoising (RED) Bayesian models recently proposed in [CITATION].', '1612.06149-4-49-0': '## Wavelet image denoising model', '1612.06149-4-50-0': 'In this example we analyse the behaviour of MAP estimation in linear inverse problems with sparsity-promoting or shrinkage priors.', '1612.06149-4-50-1': 'Without loss of generality, we first consider a simple additive noise observation model [MATH], with noise [MATH] with variance [MATH], that allows a detailed analysis.', '1612.06149-4-50-2': 'To recover [MATH] we put a shrinkage prior on a wavelet representation [MATH] of [MATH], where [MATH] is some orthogonal wavelet transform.', '1612.06149-4-50-3': 'More precisely, we use the smoothed Laplace prior [EQUATION] where [MATH] and [MATH] are respectively scale and shape regularisation parameters; this prior is also known as the pseudo-Huber, Hardy, or Charbonnier prior [CITATION].', '1612.06149-4-50-4': 'The likelihood is [MATH], and hence the posterior for the wavelet coefficients is [EQUATION]', '1612.06149-4-50-5': 'To check that Theorem [REF] and Proposition [REF] apply, we note that [MATH] belongs to [MATH] and has a diagonal Hessian matrix with elements given by [EQUATION].', '1612.06149-4-50-6': 'Noticing that the elements [MATH] take values in [MATH] for all [MATH], we conclude that [MATH] is strongly convex.', '1612.06149-4-50-7': 'Notice that, similarly to the previous example, the geometry of the manifold [MATH] does not depend on the value of the observation [MATH], and hence the canonical divergences are independent of [MATH] too.', '1612.06149-4-51-0': 'Because of the action of the shrinkage prior [REF], the Bayesian model [MATH] will promote solutions that have some large wavelet coefficients and some coefficients close to zero.', '1612.06149-4-51-1': 'This behaviour is controlled by the regularisation parameter [MATH], and also by the choice of the Bayesian estimator used to summarise [MATH].', '1612.06149-4-51-2': 'In particular, MAP estimation may significantly accentuate shrinkage.', '1612.06149-4-51-3': 'This can be theoretically analysed in different ways, and in particular by using Theorem [REF].', '1612.06149-4-51-4': 'Accordingly, [MATH] minimises the expected canonical divergence on [MATH], given by the [MATH]-Bregman divergence [EQUATION]', '1612.06149-4-52-0': 'Because [MATH] we have that [MATH] is fully separable, i.e., [MATH] with [EQUATION].', '1612.06149-4-52-1': 'Because [MATH] is a divergence it promotes values of [MATH] that are close to [MATH].', '1612.06149-4-52-2': 'To develop an intuition for [MATH] it is useful to analyse its behaviour when [MATH] is small and when it is large relative to [MATH].', '1612.06149-4-52-3': 'Observe that [MATH] has a quadratic term related to the likelihood, and a non-quadratic term related to the shrinkage prior.', '1612.06149-4-52-4': 'When [MATH] the non-quadratic term vanishes and hence [EQUATION].', '1612.06149-4-52-5': 'As a result, if the observed data is such that the posterior distribution for [MATH] has most of its mass in large values of [MATH], the MAP estimate for [MATH] will essentially coincide with the MMSE estimate given by the posterior mean of [MATH].', '1612.06149-4-52-6': 'In this case there is no additional shrinkage from the estimator.', '1612.06149-4-52-7': 'Conversely, when [MATH] the estimator will significantly boost shrinkage.', '1612.06149-4-52-8': 'More precisely, when [MATH], the non-quadratic term behaves as[EQUATION] for [MATH], and for [MATH] as [EQUATION]', '1612.06149-4-52-9': 'In these two cases [MATH] strongly promotes [MATH] values that are close to zero, either explicitly via the shrinkage term [MATH], or by amplifying the convexity constant of the quadratic loss from [MATH] to [MATH].', '1612.06149-4-52-10': 'As a result, if the posterior distribution for [MATH] has mass in small values of [MATH], then the MAP estimate will intensify the shrinkage effect of the prior.', '1612.06149-4-52-11': 'This additional shrinkage is not observed with other loss functions, e.g., MMSE, and is consistent with the empirical observation that MAP estimation performs well with shrinkage priors.', '1612.06149-4-53-0': 'For illustration, Figure [REF] shows an experiment with the Flinstones image of size [MATH] pixels.', '1612.06149-4-53-1': 'Figure [REF](a) shows a corrupted observation [MATH] with noise [MATH] with [MATH], which has a signal-to-noise ratio of [MATH]dB.', '1612.06149-4-53-2': ""The restored imaged obtained by MAP estimation is displayed in Figure [REF](b), this estimate has a signal-to-noise ratio of [MATH]dB (we used a Haar wavelet decomposition with four levels and [MATH] and [MATH] for all scales except the coarse scale, for which we used a Jeffreys' prior [MATH] to avoid excessively biasing the estimates)."", '1612.06149-4-53-3': 'For comparison, we also report [MATH], which in this experiment performs poorly (signal-to-noise ratio of [MATH]dB).', '1612.06149-4-53-4': 'The MAP estimator obtained with a conventional Laplace or [MATH] prior (i.e., with [MATH]) has a worse signal-to-noise ratio ([MATH]dB, not displayed)', '1612.06149-4-54-0': 'Because [MATH] is fully separable, i.e., [MATH] - and thus [MATH] - the action of these estimator can be clearly visualised by plotting the estimation function that performs the denoising of the wavelet coefficients: for MAP estimation this is given by [MATH], where [MATH] is the [MATH]th wavelet coefficient of [MATH]; and for MMSE estimation it is the marginal posterior mean [MATH].', '1612.06149-4-54-1': 'These functions, displayed in Figure [REF](d), clearly show the importance of the choice of the loss used to summarise [MATH].', '1612.06149-4-55-0': 'We emphasise at this point that this experiment has been selected to highlight the additional shrinkage obtained by using MAP estimation instead of MMSE estimation.', '1612.06149-4-55-1': 'However, there are other models where, because of the likelihood and the choice of the wavelet representation and the parameters used, shrinkage arises mainly from the prior.', '1612.06149-4-55-2': 'In that case MAP and MMSE estimation perform equally well.', '1612.06149-4-55-3': 'To illustrate this point, Figure [REF] below shows the reconstruction results obtained in [CITATION] with MAP and MMSE estimation for a radio-interferometric imaging problem with a very similar model of the form [MATH], where the difference is that the likelihood term involves a linear operator [MATH] modelling the radio-telescope system (see [CITATION] for more details about the model and the algorithms used to compute the estimates).', '1612.06149-4-55-4': 'Observe that in this case both MAP and MMSE estimation deliver excellent and remarkably similar results.', '1612.06149-4-55-5': 'A similar empirical observation is reported in [CITATION] for a sparse tomography experiment using the Besov wavelet model of [CITATION], which is closely related to the model considered here.', '1612.06149-4-56-0': 'To conclude, shrinkage priors promote solutions that are sparse or approximately sparse through two mechanisms: directly through the definition of the bayesian model [MATH], and indirectly through the loss function used to summarise [MATH].', '1612.06149-4-56-1': 'In the case of MAP estimation, this loss function is a Bregman divergence that can significantly amplify shrinkage.', '1612.06149-4-56-2': 'In some cases this may lead to better estimation performance.', '1612.06149-4-56-3': 'Therefore, when designing Bayesian procedures for imaging problems, it is important to carefully consider both the model and the Bayesian estimator used to summarise the information provided by the model.', '1612.06149-4-57-0': '## Regularisation-by-denoising (RED) Bayesian models', '1612.06149-4-58-0': 'As a second illustrative example we analyse the geometry of the RED Bayesian models [CITATION].', '1612.06149-4-58-1': 'In this class of models the prior [MATH] is defined implicitly through an image denoising algorithm.', '1612.06149-4-58-2': 'Precisely, starting from some image denoising filter [MATH], we posit the prior [EQUATION] which promotes values of [MATH] that are approximately invariant to filtering by [MATH] (i.e., for which [MATH]), the rationale being that these are values that [MATH] considers to be realistic images.', '1612.06149-4-58-3': 'Note that this approach has close connections to plug-and-play priors, that are also defined through denoising algorithms [CITATION].', '1612.06149-4-59-0': 'The RED framework [CITATION] assumes that [MATH] verifies three regularity conditions that are necessary to make inference with [REF] analytically and computationally tractable.', '1612.06149-4-59-1': 'First, [MATH] is smooth, at least [MATH].', '1612.06149-4-59-2': 'Second, the Jacobian matrix [MATH] is symmetric and has all its eigenvalues in [MATH] for all [MATH].', '1612.06149-4-59-3': 'Third, [MATH] is locally homogenous, i.e., [MATH]; this property implies that [MATH].', '1612.06149-4-59-4': 'Under these assumptions, it is possible to express [REF] in the following pseudo-quadratic form [EQUATION] where [MATH] plays the role of an image-adapted graph Laplacian operator, highlighting the connection between [MATH] and the model geometry [CITATION].', '1612.06149-4-60-0': 'Notice that these regularity assumptions imply the regulariser or negative log-prior [MATH] is [MATH] and convex, which is important for example for the efficient computation of [MATH] by optimisation.', '1612.06149-4-60-1': 'To show that [MATH] we first use the fact that [MATH], where we have used the symmetry [MATH] and the identity [MATH] related to the local homogeneity of [MATH].', '1612.06149-4-60-2': 'Therefore, the Hessian matrix of [MATH] has elements given by [MATH], which are also continuously differentiable because [MATH], and hence [MATH].', '1612.06149-4-60-3': 'Moreover, the Hessian matrix of [MATH], given by [MATH], is negative-semidefinite because [MATH] has all its eigenvalues in [MATH], and consequently [MATH] is convex.', '1612.06149-4-60-4': 'As a result, if the negative log-likelihood [MATH] is [MATH] and convex w.r.t. [MATH], then [MATH] is also [MATH] and log-concave, and Theorem [REF] and Proposition [REF] apply.', '1612.06149-4-61-0': 'As illustrative example, consider linear inverse problems of the form [MATH], where [MATH] is a known linear operator, and [MATH] with noise variance [MATH].', '1612.06149-4-61-1': 'The resulting RED Bayesian model has posterior density [EQUATION]', '1612.06149-4-61-2': 'This distribution is strongly log-concave and [MATH], and hence Theorem [REF] and Proposition [REF] apply.', '1612.06149-4-61-3': 'More precisely, we have [MATH], which induces the metric [EQUATION]', '1612.06149-4-61-4': 'Observe that [REF] combines an Euclidean geometry term [MATH] from the Gaussian likelihood, and a non-Euclidean term from the Laplacian [MATH].', '1612.06149-4-61-5': 'Again, note that for this class of models the geometry of the manifold [MATH] does not depend on the value of the observation [MATH].', '1612.06149-4-62-0': 'Moreover, from Theorem [REF], the estimator [MATH] is the Bayesian estimator associated with the canonical divergence on [MATH], given by the [MATH]-Bregman divergence [EQUATION] where the Mahalanobis (Euclidean) distance [EQUATION] is a measure of prediction MSE related to the Gaussian likelihood, and [EQUATION] is related to the Laplacian [MATH], which encodes the geometry of the prior (to compute [MATH] we used the local homogeneity property [MATH] of the denoiser, see [CITATION] for details).', '1612.06149-4-63-0': 'Finally, observe that [MATH] is very similar to the Euclidean loss [MATH] in that it measures the difference between the squared norms of [MATH] and [MATH] and the projection of [MATH] on [MATH], with the only difference being that for [MATH] these norms and projections are measured on the tangent spaces [MATH] and [MATH] of the manifold [MATH], with inner products specified by [MATH].', '1612.06149-4-64-0': '# Relaxation of regularity conditions', '1612.06149-4-65-0': 'We now examine the effect of relaxing the regularity assumptions of Theorem [REF].', '1612.06149-4-65-1': 'We consider three main cases: lack of smoothness, lack of strong convexity, and lack of continuity.', '1612.06149-4-66-0': '## Non-smooth models', '1612.06149-4-67-0': 'The results of Theorem [REF] hold for non-smooth models with the following modifications.', '1612.06149-4-67-1': 'First, assume that [MATH] is almost everywhere [MATH] on [MATH]; i.e., the set of points of [MATH] where [MATH] is not smooth has dimension [MATH] and hence zero Lebesgue measure.', '1612.06149-4-67-2': ""To check that [MATH] is [MATH] almost everywhere it is necessary to analyse the regularity of the second order derivatives [MATH] (e.g., if the second derivatives are Lipchitz continuous then [MATH] is almost everywhere [MATH] by Rademacher's theorem [CITATION])."", '1612.06149-4-67-3': 'Because in models that are almost everywhere smooth the set of non-differentiable points has no probability mass, this set can be simply omitted in the computation of expectations.', '1612.06149-4-67-4': 'Second, because the non-differentiable points do not have Euclidean tangent spaces, instead of a global manifold we need to consider the collection local manifolds associated with the regions of [MATH] where [MATH] is [MATH].', '1612.06149-4-67-5': 'Each one of these regions has a local canonical divergence given by the Bregman divergence [MATH].', '1612.06149-4-67-6': 'Therefore, for these models we need to posit [MATH] as the global loss function for any [MATH] [technically the global loss is the generalised Bregman divergence [MATH], where [MATH] belongs to the subdifferential set of [MATH] at [MATH] [CITATION], however the expectation [MATH] is taken over the points where [MATH] is [MATH] and hence [MATH]].', '1612.06149-4-67-7': 'By calculating the primal and dual Bayesian estimators related to this global loss we obtain that [MATH] and [MATH], similarly to Theorem [REF].', '1612.06149-4-67-8': 'Observe that these modifications do not affect the fact that [MATH] and [MATH] can correspond to non-differentiable points.', '1612.06149-4-67-9': ""Also note that despite not being a global canonical divergence, [MATH] is still consistent with the space's Riemannian geometry which is local."", '1612.06149-4-68-0': 'Many imaging models involve non-smooth norms such as the [MATH] and the nuclear norm, or the total-variation pseudo-norm, that are almost everywhere [MATH] but not [MATH].', '1612.06149-4-68-1': 'More generally, all Lipchitz continuous functions are almost everywhere [MATH].', '1612.06149-4-68-2': 'In this case, only the second and third parts of Theorem [REF] hold.', '1612.06149-4-68-3': 'That is, we posit [MATH] as the loss function for any [MATH] and any [MATH], excluding non-differentiable points, and obtain that [MATH] and [MATH] by removing non-differentiable points from the calculation of the expectations.', '1612.06149-4-68-4': 'To recover the differential geometric derivation of [MATH] it is necessary to use a smooth approximation of the model, i.e., the smoothed L1 norm [MATH] for some arbitrarily small [MATH].', '1612.06149-4-69-0': 'Finally, also note that the bound [MATH] in Proposition [REF] is straightforwardly extended to non-smooth models by using the generalised dual Bregman divergence [MATH] with subgradient [MATH].', '1612.06149-4-69-1': 'Conversely, the other bound [MATH] is lost (see the appendix for details).', '1612.06149-4-70-0': '## Strictly log-concave models', '1612.06149-4-71-0': 'For models that are strictly but not strongly log-concave only the second and third results of Theorem [REF] remain true.', '1612.06149-4-71-1': 'It is easy to check that the Bayesian estimators w.r.t. [MATH] are still [MATH] and [MATH], similarly to strongly log-concave models (see in the appendix that strong log-concavity is not required to prove the second and third parts of Theorem [REF]).', '1612.06149-4-71-2': 'Thus, the decision-theoretic derivation of [MATH] remains valid, and [MATH] and [MATH] remain dual to each other.', '1612.06149-4-71-3': 'The high dimensional performance guarantees of Proposition [REF] also hold because [MATH] is convex.', '1612.06149-4-71-4': 'However, without strong convexity, [MATH] becomes semi-positive definite and [MATH] becomes a singular manifold.', '1612.06149-4-71-5': 'Currently, the validity of the interpretation of [MATH] as a canonical divergence in singular manifolds is not clear.', '1612.06149-4-71-6': 'The generalisation of canonical divergences and of Theorem [REF] to singular manifolds is currently under investigation.', '1612.06149-4-71-7': 'In any case, without strong convexity [MATH] is no longer a divergence in the sense of Definition [REF] as [MATH] does not imply [MATH], which is an important desired property for loss functions.', '1612.06149-4-72-0': '## Models involving constraints', '1612.06149-4-73-0': 'Finally, in cases where [MATH] is constrained to a convex set [MATH] only the first and the third results of Theorem [REF] hold.', '1612.06149-4-73-1': 'Proceeding similarly to the proof of Theorem [REF], it is easy to show that [MATH] is the canonical divergence of the manifold [MATH], and that the Bayesian estimator related to the dual divergence is [MATH].', '1612.06149-4-73-2': 'However, the Bayesian estimator that minimises the canonical divergence is now a shifted or tilted MAP estimator [EQUATION] where generally [MATH] (see the appendix for details).', '1612.06149-4-73-3': 'It is not clear at this point under what conditions [MATH].', '1612.06149-4-73-4': 'Nevertheless, the high dimensional guarantees of Proposition [REF] still hold for [MATH], providing some theoretical justification for using this estimator.', '1612.06149-4-74-0': '## Models with heavy-tails', '1612.06149-4-75-0': 'We conclude this section by discussing the difficulties of extending our results to models that are heavy-tailed and hence not log-concave, such as imaging models involving heavy-tailed priors related to compressible distributions [CITATION].', '1612.06149-4-75-1': 'Unfortunately, extending our results to heavy-tailed settings is extremely challenging for several reasons.', '1612.06149-4-75-2': 'First, the Hessian matrix of [MATH] does not define a Riemannian metric because there are regions of the space where it has negative eigenvalues.', '1612.06149-4-75-3': 'Also, directly postulating [MATH] as loss function is not appropriate either because [MATH] can take negative values.', '1612.06149-4-75-4': 'The analysis is further complicated by the fact that [MATH] may have an infinite number of maximisers in disconnected areas of the parameter space.', '1612.06149-4-75-5': 'As mentioned previously, the derivation of MAP estimation as an approximation arising from the degenerate loss [MATH] with [MATH] also fails in this case [CITATION].', '1612.06149-4-75-6': 'Interestingly, MMSE estimation may also struggle here given that models in this class may not have a posterior mean [CITATION].', '1612.06149-4-76-0': '# Conclusion', '1612.06149-4-77-0': 'MAP estimation is one of the the most successful Bayesian estimation methodologies in imaging science, with a track record of accurate results across a wide range of challenging imaging problems.', '1612.06149-4-77-1': 'Our aim here has been to contribute to the theoretical understanding of this widely used methodology, particularly by placing it in the Bayesian decision theory framework that underpins the core Bayesian inference methodologies.', '1612.06149-4-78-0': 'In order to analyse MAP estimators we have adopted an entirely new approach: we allowed the model to specify the loss function, or equivalently the Bayesian estimator, that is used to summarise the information that the model represents.', '1612.06149-4-78-1': 'This was achieved by using the connections between model log-concavity, Riemannian geometry, and divergence functions.', '1612.06149-4-78-2': 'We first established that if [MATH] is strongly log-concave, continuous, and [MATH] on [MATH], then [MATH] induces a dually-flat Riemannian structure on the parameter space, where the canonical divergence is the Bregman divergence associated with [MATH], and where the MAP estimator is the unique Bayesian estimator w.r.t. to this loss function.', '1612.06149-4-78-3': 'We also established that the MMSE estimator is the Bayesian estimator w.r.t. the dual canonical loss, and that both estimators enjoy favourable stability properties in high dimensions.', '1612.06149-4-78-4': 'We then examined the effect of relaxing these assumptions to models with weaker regularity conditions.', '1612.06149-4-79-0': 'The theoretical results presented in this work provide several valuable new insights into MAP and MMSE estimation.', '1612.06149-4-79-1': 'In particular, both estimators stem from Bayesian decision theory and from the consideration of the geometry of the parameter space, and exhibit an interesting form of duality.', '1612.06149-4-79-2': 'Also, the expected estimation error - as measured by the canonical loss - is stable in high dimensions; this is in agreement with the remarkable empirical performance observed imaging and other large scale settings.', '1612.06149-4-79-3': 'The fact that MAP estimators are available as solutions to convex problems is a fundamental practical advantage.', '1612.06149-4-79-4': 'However, our results also show that the predominant view of MAP estimators as hastily approximate inferences, motivated only by computational efficiency, is fundamentally incorrect.', '1612.06149-4-79-5': 'We hope that these results will provide some clarity to imaging scientists using MAP estimators, and that they stimulate further research into the theory of this powerful Bayesian methodology.'}",
1808.06499,"{'1808.06499-1-0-0': 'In this work we extend the SM by introducing only exotic scalars and leptons and show that within the reasonable error limits, both the observed anomalies in [MATH] and [MATH] can be explained along with satisfying all the constrains from nuetral meson oscillations, precision Z-pole data etc.', '1808.06499-1-0-1': 'In a trivial extension of our model with addition of three heavy right handed neutrinos, explaining the small masses of neutrinos and generation of Baryon excess via leptogenesis is possible as well.', '1808.06499-1-0-2': 'The disagreement between SM prediction and experimental data in muon (g-2) measurements can be reduced from 3.6[MATH] to around 2[MATH] in this model.', '1808.06499-1-0-3': 'Also our model has enough new particles for the scalar singlet DM to interact and generate enough annihilation to avoid over abundance problem (exotic portal), unlike the SM Higgs portal which has been ruled out.', '1808.06499-1-1-0': '# Introduction.', '1808.06499-1-2-0': 'Even though LHC reporting observation of no new particles beyond the standard-model (SM) Higgs in direct searches, some observables in flavor sector show tension with SM prediction up to about 4[MATH] in some cases.', '1808.06499-1-2-1': 'It may be an indication that new-physics (NP) scale is close to the SM scale and so a precision machine is better equipped than an energy frontier machine to probe the nature of NP that lies beyond present SM.', '1808.06499-1-2-2': 'Besides the well known short comings of SM such as it is unable to explain the existence of small but non-zero neutrino masses, dark-energy (DE), dark-matter (DM) and observed Baryon excess in the universe, some experiments in flavor sector have reported intriguing deviations in lepton universality observables such as [MATH] [CITATION][CITATION][CITATION][CITATION][CITATION][CITATION], [MATH] [CITATION] and muon (g-2) [CITATION].', '1808.06499-1-2-3': 'For the anomalies in [MATH] and related observables, in [CITATION] it has been determined that the combined global data, which is about 4[MATH] deviated from SM prediction [CITATION], is best fitted by a NP with [MATH] at 1[MATH].', '1808.06499-1-2-4': 'For the [MATH] observables the HFAG global average is given as [CITATION] [EQUATION] and [EQUATION] amounting to about 4.1[MATH] deviation from SM prediction and it has been shown that one of the best fit NP model would be one that add coherently to the SM effective four current in this observables [CITATION].', '1808.06499-1-2-5': 'Then there is the reported deviation from SM prediction in muon (g-2) with present global average of the deviation given as [EQUATION] which is about 3.6[MATH] away from SM prediction [CITATION].', '1808.06499-1-2-6': 'In this work we will propose a NP model that will be able to generate [MATH] via box-loop to explain the anomalies in [MATH] and related observables and a NP Wilson coefficient that add coherently with the SM effective four current, also via box-loop, to explain the [MATH] anomalies as well as explaining the anomaly in muon (g-2) within 2[MATH], smallness of neutrino masses and baryon-genesis.', '1808.06499-1-2-7': 'This work is organized as follows, in section [REF] the details of the NP model is given, in section [REF] the implications of the NP model to flavor physics observables along with constrains on NP parameters from flavor precision data.', '1808.06499-1-2-8': 'And in section [REF] we conclude.', '1808.06499-1-3-0': '# Model details.', '1808.06499-1-4-0': 'In some recent works it has been shown that NP models with exotic leptons and scalars contributing to [MATH] [CITATION], [MATH] [CITATION](at box-loop level) and muon (g-2) [CITATION] will be able to resolve the reported anomalies in those observables within the error limits.', '1808.06499-1-4-1': 'In this work we would like to propose an extension of SM which will be able to resolve the anomalies in both [MATH] and [MATH] as well as smallness of nutrino masses and Baryon-genesis.', '1808.06499-1-4-2': 'We add to SM two [MATH] doublets [MATH] and [MATH] along with [MATH] singlets [MATH] and [MATH], they are all leptons carrying same [MATH] charges as the SM lepton doublets and singlets respectively.', '1808.06499-1-4-3': 'We also add three SM singlet right handed neutrinos [MATH] to the SM lepton content to generate small neutrino masses at loop level.', '1808.06499-1-4-4': 'Where the subscripts e, [MATH] and [MATH] denotes the SM lepton number carried by the heavy right handed neutrinos.', '1808.06499-1-4-5': 'Since new leptons form a vector like pairs under the relevant SM gauge groups, the model is free of axial anomaly.', '1808.06499-1-4-6': 'The collider signatures of a locally gauged lepton number extension of SM with similar new particle content as our model is proposed in [CITATION][CITATION], but here we will keep the lepton number to be a global gauge as in the SM case.', '1808.06499-1-4-7': 'All the new leptons are assumed to be odd under the [MATH].', '1808.06499-1-4-8': 'Also to SM Higgs, we add two scalar leptoquarks one [MATH] singlet [MATH] and one [MATH] doublet [MATH] along with an inert-doublet [MATH] and a singlet S, with all the new scalars also being odd under the [MATH].', '1808.06499-1-4-9': 'One more real scalar singlet under the SM gauge group [MATH] is also added, which is even under the [MATH] and which develop a non-zero VEV to give masses to the new leptons.', '1808.06499-1-4-10': 'In Table [REF] and Table [REF] we have shown the charge assignments of the new leptons and new scalars respectively.', '1808.06499-1-5-0': '## Yukawa Interactions.', '1808.06499-1-6-0': 'The most general Yukawa interaction terms that are invariant under the full symmetries of the model can be written down as [EQUATION]', '1808.06499-1-6-1': 'After SM Higgs H and the new scalar [MATH] develops a non-zero VEV [MATH] and [MATH] respectively, we have the mass matrix of the new charged leptons given as [EQUATION] and matrix of new non-neutrino neutral leptons given as [EQUATION]', '1808.06499-1-6-2': 'In this work we take the limit [MATH] where [MATH] is mass in the order of SM charged lepton masses.', '1808.06499-1-6-3': 'Then as shown in [CITATION], both the matrices of new charged leptons and new nuetral leptons are diagonalized by same rotation matrix [EQUATION] with degenerate masses for the nuetral leptons at tree level and mass difference of order [MATH] for the new charged leptons, so in the relevant scenario where we take the new lepton masses well above the scale of SM lepton masses, we can take [MATH], i.e in the limit stipulated above we can take the neutral lepton and charged leptons having nearly degenerate masses, where subscript h and l denote heavy and light particle respectively.', '1808.06499-1-7-0': 'SM gauge interactions and collider productions and decay signatures of our model is same as those given in [CITATION].', '1808.06499-1-7-1': 'In the near degenerate masses for the new exotic neutral and charged leptons as well as near degenerate masses for the charged Higgs and heavier nuetral Higgs of inert-doublet cases, as will be assumed in this work, the contributions to S and T parameters are [MATH] and [MATH] which is well within the present experimental limit of [MATH] and [MATH].', '1808.06499-1-7-2': 'Also as shown in [CITATION], in our case where the SM Higgs coupling to the new exotic leptons are at same order as the SM lepton couplings to the SM Higgs, the contributions to the new exotic leptons and charged Higgs to [MATH] is within the experimental limit even for the light charged Higgs to have Yukawa couplings of [MATH].', '1808.06499-1-8-0': '# Constrains and implications in flavor physics.', '1808.06499-1-9-0': 'In SM if we multiply only the first two rows of the CKM matrix elements with -1, there is no observable that can detect this sign change.', '1808.06499-1-9-1': 'But here in our model, as will be shown in the following paragraphs, this change in relative sign between rows of CKM matrix elements have observable effect.', '1808.06499-1-9-2': 'Here we will fix the angles of CKM matrix elements as [MATH] and [MATH], i.e the signs of the first two rows of CKM matrix elements are changed relative to the third row compared to the usual convention where all the angles are fixed in the first quadrant [CITATION].', '1808.06499-1-9-3': 'The Yukawa couplings in the exotic lepton sector taken as [MATH] and [MATH] is favored by the reported anomalies.', '1808.06499-1-9-4': 'The Yukawa couplings in the down quark sector in mass diagonal states can be expressed as [EQUATION] where [MATH] are CKM matrix elements and [MATH].', '1808.06499-1-9-5': 'Since [MATH] and [MATH] are very precisely measured and there being no deviations observed in this modes, these data can be accommodated easily if we impose [EQUATION]', '1808.06499-1-9-6': 'In this work the Yukawa couplings are assumed to satisfy the above conditions.', '1808.06499-1-9-7': 'It can be shown that the constrains from [MATH] oscillation and [MATH] data along with the condition [MATH](where sign change of the first two rows of CKM elements are shown explicitely) can be satisfied for [MATH] and [MATH] with [MATH] and [MATH] [CITATION].', '1808.06499-1-10-0': 'Similarly [MATH] can be satisfied along with explaining the [MATH] data for [MATH] with [MATH] and [MATH].', '1808.06499-1-10-1': 'In the above calculations we have taken the values of the Wolfenstien parameters of CKM from PDG [CITATION].', '1808.06499-1-11-0': '## Neutral meson oscillation.', '1808.06499-1-12-0': 'Like in the SM, both [MATH] and [MATH] can contribute to the [MATH] oscillation at box loop level.', '1808.06499-1-12-1': 'Their contributions can be expressed as [CITATION][CITATION] [EQUATION] where [MATH] is the measured mass of the [MATH] with [MATH] and [MATH] are the decay constant and QCD scale correction factor respectively, their values are taken from [CITATION][CITATION].', '1808.06499-1-12-2': 'The [MATH] and [MATH] can be expressed as [EQUATION] where [MATH] being Inami-Lim functions, see [REF] for detail, and [MATH] with [MATH] denoting the mass of the leptoquark involved.', '1808.06499-1-13-0': 'With benchmark values of masses taken as [MATH] GeV and [MATH] GeV, we get [MATH] which is within the 1.1[MATH] of the error in the latest SM prediction given as [MATH].', '1808.06499-1-13-1': 'Due to [MATH] being complex, there is also an imaginary component of [MATH] which can contribute to CP violation observables in the [MATH] oscillation.', '1808.06499-1-13-2': 'It turn out that with values of the Yukawa couplings given in the section [REF], [MATH] and [MATH], so contribution due to [MATH] is negligible compared to [MATH].', '1808.06499-1-13-3': 'With the benchmark values for the parameters, we get [MATH] which gives [MATH] compared to [MATH], the NP contributions is an order of magnitude smaller than the present experimental limit.', '1808.06499-1-13-4': 'For the [MATH] oscillation, we have at 2[MATH] experimental bound as [MATH] compared to the NP contribution given as [MATH], the NP contribution is about an order of magnitude smaller than the present experimental bound at 2[MATH].', '1808.06499-1-14-0': '## Z pole constrains.', '1808.06499-1-15-0': 'For theoretical calculations of contribution from new fermions to the [MATH] decay into two fermions via higher order loops, we have used [CITATION] [EQUATION] where [EQUATION] and [EQUATION] with [MATH]; [MATH] and [MATH] depending on the final state and Yukawa coupling involved.', '1808.06499-1-15-1': 'At [MATH] GeV, [MATH] GeV, [MATH] GeV, [MATH] GeV and [MATH] GeV we have [MATH] due to Eqs([REF]) and [MATH], [MATH] compared to [MATH], the NP contributions are negligible.', '1808.06499-1-15-2': 'With [MATH] and [MATH] gives [MATH] which is an order of magnitude smaller than the experimental error where [MATH] [CITATION].', '1808.06499-1-15-3': 'We take [MATH], [MATH], [MATH], [MATH] 1 and [MATH].', '1808.06499-1-15-4': 'Then we get [MATH], [MATH] and [MATH], compared to the respective experimental errors, we have [MATH] which is an order of magnitude larger than the NP contribution and in all other cases the NP contributions are smaller than the respective experimental error estimates by two orders of magnitude or smaller and so negligible.', '1808.06499-1-15-5': 'Also there is a contribution to the muon (g-2) from the Yukawa coupling involving the two neutral components of the inert-doublet([MATH]) and give [MATH] [CITATION], which is within 2.1[MATH] of the experimental value where as SM shows a deviation of about 3.6[MATH] from the experimental value [CITATION].', '1808.06499-1-16-0': '## Implications to [MATH], [MATH], neutrino masses, Baryon-genesis and DM.', '1808.06499-1-17-0': 'As indicated in a recent model independent analysis of [MATH] data with new estimates of the form factors [CITATION], the vector type NP is the best fit to the data while tensor type NP is highly restricted and scalar type NP is almost ruled out.', '1808.06499-1-17-1': 'In our model the terms in the Eqs([REF]) involving [MATH] can not contribute to [MATH] but relevant terms involving [MATH] can contribute to the decay at box loop level given as [CITATION] [EQUATION] where [MATH] is the SM left handed vector four current operator and [MATH] is given as [EQUATION] where [MATH] is the Inami-Lim functions [CITATION][CITATION][CITATION] with [MATH], [MATH], [MATH] and [MATH].', '1808.06499-1-17-2': 'Using the CKM matrix elements from the PDG [CITATION] as in [CITATION] and the benchmark values of the masses of the new particles and Yukawa couplings as before we have [EQUATION] and [EQUATION] which is within 1[MATH] of the theoretical and experimental errors combined .', '1808.06499-1-18-0': 'As mentioned above the Yukawa terms involving [MATH] can not contribute substantially to [MATH] but Yukawa terms involving [MATH] in Eqs([REF]) can contribute to this decay mode via box loop.', '1808.06499-1-18-1': 'In our model due to presence of terms involving [MATH], the NP contributions to Wilson coefficients [MATH] and [MATH] via box loop can be expressed as [CITATION][CITATION] [EQUATION] where [MATH] and [MATH] with [MATH], [MATH] and [MATH] and as shown in section [REF] [MATH] is taken.', '1808.06499-1-18-2': 'Then with benchmark values of the new particle masses and Yukawa couplings in the flavor states implies [MATH] and gives [EQUATION] which is within 1.1[MATH] of the combine global best fit estimate of these NP Wilson coefficients to the data [CITATION].', '1808.06499-1-18-3': 'Besides the NP contributions to [MATH], see section [REF] for details, [MATH] can contribute to [MATH] which is measured to be consistent with the SM prediction of [MATH] [CITATION] compared to [MATH] with [MATH] and [MATH], the [MATH] is well within 1[MATH] of the experimental value, see [CITATION][CITATION] for detail calculations.', '1808.06499-1-18-4': 'The NP contribution to [MATH] which affect the [MATH] rate and the NP contribution is about 2 orders of magnitude smaller than the present experimental bound at 2[MATH] [CITATION], also check [CITATION] for detail calculations.', '1808.06499-1-18-5': 'The bound coming from [MATH] on NP is much weaker than that from [MATH] and so constrains from these modes are automatically satisfied [CITATION].', '1808.06499-1-19-0': 'Similar to the estimates in [CITATION], the NP contributions to [MATH], [MATH], [MATH] including to CP violations due to [MATH] being complex, [MATH], [MATH], [MATH] and [MATH] are all negligible compare to the respective experimental bounds [CITATION].', '1808.06499-1-19-1': 'The NP contributions to the anomalous magnetic moment of [MATH] is [MATH] which is many orders of magnitude smaller than the latest experimental bound of [MATH][CITATION].', '1808.06499-1-20-0': 'Now with the introduction of heavy righthanded neutrinos [MATH] there can be Yukawa terms such as [EQUATION] which can give Majorana mass term of [MATH] for the light neutrinos with [CITATION] [EQUATION] where [MATH] being the masses of the heavy Majorana neutrinos.', '1808.06499-1-20-1': 'Then with benchmark values of masses [MATH] GeV, [MATH] GeV and taking lightest of [MATH] GeV, we get [MATH] eV for [MATH].', '1808.06499-1-20-2': ""At this values of the parameters, generation of universe's Baryon access via Leptogenesis is also possible, see [CITATION][CITATION] for more details."", '1808.06499-1-20-3': 'Although [MATH] being one of the LSP, due to its large Yukawa couplings required from [MATH] and [MATH] data and non-observation of stable heavy charged particle in colliders etc. its contribution to the DM relic density would be small [CITATION][CITATION].', '1808.06499-1-20-4': 'Another trivial extension of our model is to include a new singlet scalar DM, although the Higgs portal of this DM is ruled out due to over abundance problem [CITATION][CITATION][CITATION][CITATION][CITATION], in our model there are many more new particles it can couple to so as to generate enough DM annihilation to avoid over abundance problem unlike SM Higgs only portal, see e.g [CITATION] for an exotic scalar portal extension of the scalar singlet DM which can be easily incorporated into our model.', '1808.06499-1-21-0': '# Conclusions.', '1808.06499-1-22-0': 'In this work we have proposed an extension of SM lepton content by introducing a right handed and a left handed pair of [MATH] doublet leptons ([MATH]) along with their respective charged right handed and left handed [MATH] singlet partners ([MATH]), plus we also added three heavy right handed neutrinos ([MATH]) to generate small neutrino masses at loop level.', '1808.06499-1-22-1': 'We extended the SM Higgs sector by introducing two [MATH] doublet leptoquarks ([MATH]) along with an inert-Higgs-doublet ([MATH]) and a complex singlet scalar S, plus a real singlet scalar ([MATH]) whose VEV gives the dominant masses to the new leptons.', '1808.06499-1-22-2': 'All the new particles are assumed to be odd under a [MATH] except [MATH] which is assumed to be even so that it can develop a non-zero VEV.', '1808.06499-1-22-3': 'With these new particles added to the SM, we have shown that all the observed anomalies in lepton universality observables in semi-leptonic B meson decays can be explained with satisfying constrains from nuetral meson oscillations, precision Z-pole data, etc. within reasonable error limits.', '1808.06499-1-22-4': ""In addition our model is also able to explain the small neutrino masses along with generations of universe's Baryon excess via leptogenesis."", '1808.06499-1-22-5': 'Also our model have enough new parameters to avoid over abundance problem in scalar singlet DM and so DM can be incorporated trivially in our model.'}","{'1808.06499-2-0-0': 'In this work we extend the SM by introducing only exotic scalars and leptons and show that within the reasonable error limits, both the observed anomalies in [MATH] and [MATH] can be explained along with satisfying all the constrains from nuetral meson oscillations, precision Z-pole data etc.', '1808.06499-2-0-1': 'In a trivial extension of our model with addition of three heavy right handed neutrinos, explaining the small masses of neutrinos and generation of Baryon excess via leptogenesis is possible as well.', '1808.06499-2-0-2': 'The disagreement between SM prediction and experimental data in muon (g-2) measurements can be reduced from 3.6[MATH] to around 2[MATH] in this model.', '1808.06499-2-0-3': 'Also our model has enough new particles for the scalar singlet DM to interact and generate enough annihilation to avoid over abundance problem (exotic portal), unlike the SM Higgs portal which has been ruled out.', '1808.06499-2-1-0': '# Introduction.', '1808.06499-2-2-0': 'Even though LHC reporting observation of no new particles beyond the standard-model (SM) Higgs in direct searches, some observables in flavor sector show tension with SM prediction up to about 4[MATH] in some cases.', '1808.06499-2-2-1': 'It may be an indication that new-physics (NP) scale is close to the SM scale and so a precision machine is better equipped than an energy frontier machine to probe the nature of NP that lies beyond present SM.', '1808.06499-2-2-2': 'Besides the well known short comings of SM such as it is unable to explain the existence of small but non-zero neutrino masses, dark-energy (DE), dark-matter (DM) and observed Baryon excess in the universe, some experiments in flavor sector have reported intriguing deviations in lepton universality observables such as [MATH] [CITATION][CITATION][CITATION][CITATION][CITATION][CITATION], [MATH] [CITATION] and muon (g-2) [CITATION].', '1808.06499-2-2-3': 'For the anomalies in [MATH] and related observables, in [CITATION] it has been determined that the combined global data, which is about 4[MATH] deviated from SM prediction [CITATION], is best fitted by a NP with [MATH] at 1[MATH].', '1808.06499-2-2-4': 'For the [MATH] observables the HFAG global average is given as [CITATION] [EQUATION] and [EQUATION] amounting to about 4.1[MATH] deviation from SM prediction and it has been shown that one of the best fit NP model would be one that add coherently to the SM effective four current in this observables [CITATION].', '1808.06499-2-2-5': 'Then there is the reported deviation from SM prediction in muon (g-2) with present global average of the deviation given as [EQUATION] which is about 3.6[MATH] away from SM prediction [CITATION].', '1808.06499-2-2-6': 'In this work we will propose a NP model that will be able to generate [MATH] via box-loop to explain the anomalies in [MATH] and related observables and a NP Wilson coefficient that add coherently with the SM effective four current, also via box-loop, to explain the [MATH] anomalies as well as explaining the anomaly in muon (g-2) within 2[MATH], smallness of neutrino masses and baryon-genesis.', '1808.06499-2-2-7': 'This work is organized as follows, in section [REF] the details of the NP model is given, in section [REF] the implications of the NP model to flavor physics observables along with constrains on NP parameters from flavor precision data.', '1808.06499-2-2-8': 'And in section [REF] we conclude.', '1808.06499-2-3-0': '# Model details.', '1808.06499-2-4-0': 'In some recent works it has been shown that NP models with exotic leptons and scalars contributing to [MATH] [CITATION], [MATH] [CITATION](at box-loop level) and muon (g-2) [CITATION] will be able to resolve the reported anomalies in those observables within the error limits.', '1808.06499-2-4-1': 'In this work we would like to propose an extension of SM which will be able to resolve the anomalies in both [MATH] and [MATH] as well as smallness of nutrino masses and Baryon-genesis.', '1808.06499-2-4-2': 'We add to SM two [MATH] doublets [MATH] and [MATH] along with [MATH] singlets [MATH] and [MATH], they are all leptons carrying same [MATH] charges as the SM lepton doublets and singlets respectively.', '1808.06499-2-4-3': 'We also add three SM singlet right handed neutrinos [MATH] to the SM lepton content to generate small neutrino masses at loop level.', '1808.06499-2-4-4': 'Where the subscripts e, [MATH] and [MATH] denotes the SM lepton number carried by the heavy right handed neutrinos.', '1808.06499-2-4-5': 'Since new leptons form a vector like pairs under the relevant SM gauge groups, the model is free of axial anomaly.', '1808.06499-2-4-6': 'The collider signatures of a locally gauged lepton number extension of SM with similar new particle content as our model is proposed in [CITATION][CITATION], but here we will keep the lepton number to be a global gauge as in the SM case.', '1808.06499-2-4-7': 'All the new leptons are assumed to be odd under the [MATH].', '1808.06499-2-4-8': 'Also to SM Higgs, we add two scalar leptoquarks one [MATH] singlet [MATH] and one [MATH] doublet [MATH] along with an inert-doublet [MATH] and a singlet S, with all the new scalars also being odd under the [MATH].', '1808.06499-2-4-9': 'One more real scalar singlet under the SM gauge group [MATH] is also added, which is even under the [MATH] and which develop a non-zero VEV to give masses to the new leptons.', '1808.06499-2-4-10': 'In Table [REF] and Table [REF] we have shown the charge assignments of the new leptons and new scalars respectively.', '1808.06499-2-5-0': '## Yukawa Interactions.', '1808.06499-2-6-0': 'The most general Yukawa interaction terms that are invariant under the full symmetries of the model can be written down as [EQUATION]', '1808.06499-2-6-1': 'After SM Higgs H and the new scalar [MATH] develops a non-zero VEV [MATH] and [MATH] respectively, we have the mass matrix of the new charged leptons given as [EQUATION] and matrix of new non-neutrino neutral leptons given as [EQUATION]', '1808.06499-2-6-2': 'In this work we take the limit [MATH] where [MATH] is mass in the order of SM charged lepton masses.', '1808.06499-2-6-3': 'Then as shown in [CITATION], both the matrices of new charged leptons and new nuetral leptons are diagonalized by same rotation matrix [EQUATION] with degenerate masses for the nuetral leptons at tree level and mass difference of order [MATH] for the new charged leptons, so in the relevant scenario where we take the new lepton masses well above the scale of SM lepton masses, we can take [MATH], i.e in the limit stipulated above we can take the neutral lepton and charged leptons having nearly degenerate masses, where subscript h and l denote heavy and light particle respectively.', '1808.06499-2-7-0': 'SM gauge interactions and collider productions and decay signatures of our model is same as those given in [CITATION].', '1808.06499-2-7-1': 'In the near degenerate masses for the new exotic neutral and charged leptons as well as near degenerate masses for the charged Higgs and heavier nuetral Higgs of inert-doublet cases, as will be assumed in this work, the contributions to S and T parameters are [MATH] and [MATH] which is well within the present experimental limit of [MATH] and [MATH].', '1808.06499-2-7-2': 'Also as shown in [CITATION], in our case where the SM Higgs coupling to the new exotic leptons are at same order as the SM lepton couplings to the SM Higgs, the contributions to the new exotic leptons and charged Higgs to [MATH] is within the experimental limit even for the light charged Higgs to have Yukawa couplings of [MATH].', '1808.06499-2-8-0': '# Constrains and implications in flavor physics.', '1808.06499-2-9-0': 'In SM if we multiply only the first two rows of the CKM matrix elements with -1, there is no observable that can detect this sign change.', '1808.06499-2-9-1': 'But here in our model, as will be shown in the following paragraphs, this change in relative sign between rows of CKM matrix elements have observable effect.', '1808.06499-2-9-2': 'Here we will fix the angles of CKM matrix elements as [MATH] and [MATH], i.e the signs of the first two rows of CKM matrix elements are changed relative to the third row compared to the usual convention where all the angles are fixed in the first quadrant [CITATION].', '1808.06499-2-9-3': 'The Yukawa couplings in the exotic lepton sector taken as [MATH] and [MATH] is favored by the reported anomalies.', '1808.06499-2-9-4': 'The Yukawa couplings in the down quark sector in mass diagonal states can be expressed as [EQUATION] where [MATH] are CKM matrix elements and [MATH].', '1808.06499-2-9-5': 'Since [MATH] and [MATH] are very precisely measured and there being no deviations observed in this modes, these data can be accommodated easily if we impose [EQUATION]', '1808.06499-2-9-6': 'In this work the Yukawa couplings are assumed to satisfy the above conditions.', '1808.06499-2-9-7': 'It can be shown that the constrains from [MATH] oscillation and [MATH] data along with the condition [MATH](where sign change of the first two rows of CKM elements are shown explicitely) can be satisfied for [MATH] and [MATH] with [MATH] and [MATH] [CITATION].', '1808.06499-2-10-0': 'Similarly [MATH] can be satisfied along with explaining the [MATH] data for [MATH] with [MATH] and [MATH].', '1808.06499-2-10-1': 'In the above calculations we have taken the values of the Wolfenstien parameters of CKM from PDG [CITATION].', '1808.06499-2-11-0': '## Neutral meson oscillation.', '1808.06499-2-12-0': 'Like in the SM, both [MATH] and [MATH] can contribute to the [MATH] oscillation at box loop level.', '1808.06499-2-12-1': 'Their contributions can be expressed as [CITATION][CITATION] [EQUATION] where [MATH] is the measured mass of the [MATH] with [MATH] and [MATH] are the decay constant and QCD scale correction factor respectively, their values are taken from [CITATION][CITATION].', '1808.06499-2-12-2': 'The [MATH] and [MATH] can be expressed as [EQUATION] where [MATH] being Inami-Lim functions, see [REF] for detail, and [MATH] with [MATH] denoting the mass of the leptoquark involved.', '1808.06499-2-13-0': 'With benchmark values of masses taken as [MATH] GeV and [MATH] GeV, we get [MATH] which is within the 1.1[MATH] of the error in the latest SM prediction given as [MATH].', '1808.06499-2-13-1': 'Due to [MATH] being complex, there is also an imaginary component of [MATH] which can contribute to CP violation observables in the [MATH] oscillation.', '1808.06499-2-13-2': 'It turn out that with values of the Yukawa couplings given in the section [REF], [MATH] and [MATH], so contribution due to [MATH] is negligible compared to [MATH].', '1808.06499-2-13-3': 'With the benchmark values for the parameters, we get [MATH] which gives [MATH] compared to [MATH], the NP contributions is an order of magnitude smaller than the present experimental limit.', '1808.06499-2-13-4': 'For the [MATH] oscillation, we have at 2[MATH] experimental bound as [MATH] compared to the NP contribution given as [MATH], the NP contribution is about an order of magnitude smaller than the present experimental bound at 2[MATH].', '1808.06499-2-14-0': '## Z pole constrains.', '1808.06499-2-15-0': 'For theoretical calculations of contribution from new fermions to the [MATH] decay into two fermions via higher order loops, we have used [CITATION] [EQUATION] where [EQUATION] and [EQUATION] with [MATH]; [MATH] and [MATH] depending on the final state and Yukawa coupling involved and index [MATH] refers to the exotic fermions in the loop.', '1808.06499-2-15-1': 'At [MATH] GeV, [MATH] GeV, [MATH] GeV, [MATH] GeV and [MATH] GeV we have [MATH] due to Eqs([REF]) and [MATH], [MATH] compared to [MATH], the NP contributions are negligible.', '1808.06499-2-15-2': 'With [MATH] and [MATH] gives [MATH] which is an order of magnitude smaller than the experimental error where [MATH] [CITATION].', '1808.06499-2-15-3': 'We take [MATH], [MATH], [MATH], [MATH] 1 and [MATH].', '1808.06499-2-15-4': 'Then we get [MATH], [MATH] and [MATH], compared to the respective experimental errors [CITATION], we have [MATH], [MATH] and [MATH] which are an order of magnitude larger than the respective NP contributions and in all other cases the NP contributions are smaller than the respective experimental error estimates by two orders of magnitude or smaller and so negligible.', '1808.06499-2-15-5': 'Also there is a contribution to the muon (g-2) from the Yukawa coupling involving the two neutral components of the inert-doublet([MATH]) and give [MATH] [CITATION], which is within 2.1[MATH] of the experimental value whereas SM shows a deviation of about 3.6[MATH] from the experimental value [CITATION].', '1808.06499-2-16-0': '## Implications to [MATH], [MATH], neutrino masses, Baryon-genesis and DM.', '1808.06499-2-17-0': 'As indicated in a recent model independent analysis of [MATH] data with new estimates of the form factors [CITATION], the vector type NP is the best fit to the data while tensor type NP is highly restricted and scalar type NP is almost ruled out.', '1808.06499-2-17-1': 'In our model the terms in the Eqs([REF]) involving [MATH] can not contribute to [MATH] but relevant terms involving [MATH] can contribute to the decay at box loop level given as [CITATION] [EQUATION] where [MATH] is the SM left handed vector four current operator and [MATH] is given as [EQUATION] where [MATH] is the Inami-Lim functions [CITATION][CITATION][CITATION] with [MATH], [MATH], [MATH] and [MATH].', '1808.06499-2-17-2': 'Using the CKM matrix elements from the PDG [CITATION] as in [CITATION] and the benchmark values of the masses of the new particles and Yukawa couplings as before we have [EQUATION] and [EQUATION] which is within 1[MATH] of the theoretical and experimental errors combined .', '1808.06499-2-18-0': 'As mentioned above the Yukawa terms involving [MATH] can not contribute substantially to [MATH] but Yukawa terms involving [MATH] in Eqs([REF]) can contribute to this decay mode via box loop.', '1808.06499-2-18-1': 'In our model due to presence of terms involving [MATH], the NP contributions to Wilson coefficients [MATH] and [MATH] via box loop can be expressed as [CITATION][CITATION] [EQUATION] where [MATH] and [MATH] with [MATH], [MATH] and [MATH] and as shown in section [REF] [MATH] is taken.', '1808.06499-2-18-2': 'Then with benchmark values of the new particle masses and Yukawa couplings in the flavor states implies [MATH] and gives [EQUATION] which is within 1.1[MATH] of the combine global best fit estimate of these NP Wilson coefficients to the data [CITATION].', '1808.06499-2-18-3': 'Besides the NP contributions to [MATH], see section [REF] for details, [MATH] can contribute to [MATH] which is measured to be consistent with the SM prediction of [MATH] [CITATION] compared to [MATH] with [MATH] and [MATH], the [MATH] is well within 1[MATH] of the experimental value, see [CITATION][CITATION] for detail calculations.', '1808.06499-2-18-4': 'The NP contribution to [MATH] which affect the [MATH] rate and the NP contribution is about 2 orders of magnitude smaller than the present experimental bound at 2[MATH] [CITATION], also check [CITATION] for detail calculations.', '1808.06499-2-18-5': 'The bound coming from [MATH] on NP is much weaker than that from [MATH] and so constrains from these modes are automatically satisfied [CITATION].', '1808.06499-2-19-0': 'Similar to the estimates in [CITATION], the NP contributions to [MATH], [MATH], [MATH] including to CP violations due to [MATH] being complex, [MATH], [MATH], [MATH] and [MATH] are all negligible compare to the respective experimental bounds [CITATION].', '1808.06499-2-19-1': 'The NP contributions to the anomalous magnetic moment of [MATH] is [MATH] which is many orders of magnitude smaller than the latest experimental bound of [MATH][CITATION].', '1808.06499-2-20-0': 'Now with the introduction of heavy righthanded neutrinos [MATH] there can be Yukawa terms such as [EQUATION] which can give Majorana mass term of [MATH] for the light neutrinos with [CITATION] [EQUATION] where [MATH] being the masses of the heavy Majorana neutrinos.', '1808.06499-2-20-1': 'Then with benchmark values of masses [MATH] GeV, [MATH] GeV and taking lightest of [MATH] GeV, we get [MATH] eV for [MATH].', '1808.06499-2-20-2': ""At this values of the parameters, generation of universe's Baryon access via Leptogenesis is also possible, see [CITATION][CITATION] for more details."", '1808.06499-2-20-3': 'Although [MATH] being one of the LSP, due to its large Yukawa couplings required from [MATH] and [MATH] data and non-observation of stable heavy charged particle in colliders etc. its contribution to the DM relic density would be small [CITATION][CITATION].', '1808.06499-2-20-4': 'Another trivial extension of our model is to include a new singlet scalar DM, although the Higgs portal of this DM is ruled out due to over abundance problem [CITATION][CITATION][CITATION][CITATION][CITATION], in our model there are many more new particles it can couple to so as to generate enough DM annihilation to avoid over abundance problem unlike SM Higgs only portal, see e.g [CITATION] for an exotic scalar portal extension of the scalar singlet DM which can be easily incorporated into our model.', '1808.06499-2-21-0': '# Conclusions.', '1808.06499-2-22-0': 'In this work we have proposed an extension of SM lepton content by introducing a right handed and a left handed pair of [MATH] doublet leptons ([MATH]) along with their respective charged right handed and left handed [MATH] singlet partners ([MATH]), plus we also added three heavy right handed neutrinos ([MATH]) to generate small neutrino masses at loop level.', '1808.06499-2-22-1': 'We extended the SM Higgs sector by introducing two [MATH] doublet leptoquarks ([MATH]) along with an inert-Higgs-doublet ([MATH]) and a complex singlet scalar S, plus a real singlet scalar ([MATH]) whose VEV gives the dominant masses to the new leptons.', '1808.06499-2-22-2': 'All the new particles are assumed to be odd under a [MATH] except [MATH] which is assumed to be even so that it can develop a non-zero VEV.', '1808.06499-2-22-3': 'With these new particles added to the SM, we have shown that all the observed anomalies in lepton universality observables in semi-leptonic B meson decays can be explained with satisfying constrains from nuetral meson oscillations, precision Z-pole data, etc. within reasonable error limits.', '1808.06499-2-22-4': ""In addition our model is also able to explain the small neutrino masses along with generations of universe's Baryon excess via leptogenesis."", '1808.06499-2-22-5': 'Also our model have enough new parameters to avoid over abundance problem in scalar singlet DM and so DM can be incorporated trivially in our model.'}","[['1808.06499-1-6-0', '1808.06499-2-6-0'], ['1808.06499-1-6-1', '1808.06499-2-6-1'], ['1808.06499-1-6-2', '1808.06499-2-6-2'], ['1808.06499-1-6-3', '1808.06499-2-6-3'], ['1808.06499-1-9-0', '1808.06499-2-9-0'], ['1808.06499-1-9-1', '1808.06499-2-9-1'], ['1808.06499-1-9-2', '1808.06499-2-9-2'], ['1808.06499-1-9-3', '1808.06499-2-9-3'], ['1808.06499-1-9-4', '1808.06499-2-9-4'], ['1808.06499-1-9-5', '1808.06499-2-9-5'], ['1808.06499-1-9-6', '1808.06499-2-9-6'], ['1808.06499-1-9-7', '1808.06499-2-9-7'], ['1808.06499-1-20-0', '1808.06499-2-20-0'], ['1808.06499-1-20-1', '1808.06499-2-20-1'], ['1808.06499-1-20-2', '1808.06499-2-20-2'], ['1808.06499-1-20-3', '1808.06499-2-20-3'], ['1808.06499-1-20-4', '1808.06499-2-20-4'], ['1808.06499-1-12-0', '1808.06499-2-12-0'], ['1808.06499-1-12-1', '1808.06499-2-12-1'], ['1808.06499-1-12-2', '1808.06499-2-12-2'], ['1808.06499-1-18-0', '1808.06499-2-18-0'], ['1808.06499-1-18-1', '1808.06499-2-18-1'], ['1808.06499-1-18-2', '1808.06499-2-18-2'], ['1808.06499-1-18-3', '1808.06499-2-18-3'], ['1808.06499-1-18-4', '1808.06499-2-18-4'], ['1808.06499-1-18-5', 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1211.1680,"{'1211.1680-1-0-0': 'S2LET: A code to perform fast wavelet analysis on the sphere', '1211.1680-1-1-0': 'We describe S2LET, a fast and robust implementation of the scale-discretised wavelet transform on the sphere.', '1211.1680-1-1-1': 'Wavelets are constructed through a tiling of the harmonic line and can be used to probe spatially localised, scale-depended features of signals on the sphere.', '1211.1680-1-1-2': 'The scale-discretised wavelet transform was developed previously and reduces to the needlet transform in the axisymmetric case.', '1211.1680-1-1-3': 'The reconstruction of a signal from its wavelets coefficients is made exact here through the use of a sampling theorem on the sphere.', '1211.1680-1-1-4': 'Moreover, a multiresolution algorithm is presented to capture all information of each wavelet scale in the minimal number of samples on the sphere.', '1211.1680-1-1-5': 'In addition S2LET supports the HEALPix pixelisation scheme, in which case the transform is not exact but nevertheless achieves good numerical accuracy.', '1211.1680-1-1-6': 'The core routines of S2LET are written in C and have interfaces in Matlab, IDL and Java.', '1211.1680-1-1-7': 'Real signals can be written to and read from FITS files and plotted as Mollweide projections.', '1211.1680-1-1-8': 'The S2LET code is made publicly available, is extensively documented, and ships with several examples in the four languages supported.', '1211.1680-1-1-9': 'At present the code is restricted to axisymmetric wavelets but will be extended to directional, steerable wavelets in a future release.', '1211.1680-1-2-0': '# Introduction', '1211.1680-1-3-0': 'Signals defined or measured on the sphere arise in numerous disciplines, where analysis techniques defined explicitly on the sphere are now in common use.', '1211.1680-1-3-1': 'In particular, wavelets on the sphere have been applied very successfully to problems in astrophysics and cosmology, where data-sets are increasingly large and need to be analysed at high resolution in order to confront accurate theoretical predictions (e.g. ).', '1211.1680-1-4-0': 'While wavelet theory is well established in Euclidean space (see e.g. [CITATION]), multiple wavelet frameworks have been developed on the sphere, only a fraction of which lead to exact transforms in both the continuous and discrete settings.', '1211.1680-1-4-1': 'In fact, discrete methodologies achieve exactness in practice but may not lead to a stable basis on the sphere .', '1211.1680-1-4-2': 'In the continuous setting several constructions are theoretically exact, and have been combined with sampling theorems on the sphere to enable exact reconstruction in the discrete setting also.', '1211.1680-1-4-3': 'In particular, scale-discretised wavelets lean on a tiling of the harmonic line to yield an exact wavelet transform in both the continuous and discrete settings.', '1211.1680-1-4-4': 'In the axisymmetric case, the scale-discretised wavelets reduce to needlets , which were developed independently using a similar tilling of the harmonic line.', '1211.1680-1-5-0': 'In this paper we describe the new publicly available S2LET code to perform the scale-discretised wavelet transform of complex signals on the sphere.', '1211.1680-1-5-1': 'At present S2LET is restricted to axisymmetric wavelets (i.e. azimuthally symmetric when centred on the poles) and includes generating functions for both axisymmetric scale-discretised wavelets and needlets .', '1211.1680-1-5-2': 'We intend to extend the code to directional, steerable wavelets and spin functions in a future release.', '1211.1680-1-5-3': 'The core routines of S2LET are written in C, exploit fast algorithms on the sphere, and have interfaces in Matlab, IDL and Java.', '1211.1680-1-5-4': 'We note that public codes are already available to compute scale-discretised (S2DW; ) and needlet (NeedATool; ) transforms.', '1211.1680-1-5-5': 'We envisage S2LET will supersede S2DW once it is extended to directional, steerable wavelets and spin functions.', '1211.1680-1-6-0': 'The remainder of this article is organised as follows.', '1211.1680-1-6-1': 'In section 2 we detail the construction of scale-discretised axisymmetric wavelets and the corresponding exact scale-discretised wavelet transform on the sphere.', '1211.1680-1-6-2': 'In section 3 we describe the S2LET code, including implementation details, computational complexity and numerical performance.', '1211.1680-1-6-3': 'We present a number of simple examples using S2LET in section 4, along with the code to execute them.', '1211.1680-1-6-4': 'We conclude in section 5.', '1211.1680-1-7-0': '# Wavelets on the sphere', '1211.1680-1-8-0': 'We review scale-discretised wavelets on the sphere, as considered previously by [CITATION].', '1211.1680-1-8-1': 'Directional, steerable wavelets were also considered by [CITATION], however we restrict our attention to axisymmetric wavelets here.', '1211.1680-1-8-2': 'The use of a sampling theorem on the sphere guarantees that spherical harmonic coefficients capture all the information content of band-limited signals, resulting in exact harmonic and wavelet transforms; hence, we recall harmonic analysis on the sphere concisely.', '1211.1680-1-8-3': 'One may alternatively adopt samplings of the sphere for which exact quadrature rules do not exist such as HEALPix , but which nevertheless exhibit other useful properties, leading to numerically accurate but not theoretically exact transforms.', '1211.1680-1-9-0': '## Harmonic analysis on the sphere', '1211.1680-1-10-0': 'The spherical harmonic decomposition of a square integrable signal [MATH] on the two-dimensional sphere [MATH] reads [EQUATION] where [MATH] are the spherical harmonic functions, which form the canonical orthogonal basis on [MATH].', '1211.1680-1-10-1': 'The spherical harmonic coefficients [MATH], with [MATH] such that [MATH] and [MATH] such that [MATH], form a dual representation of the signal [MATH] in the harmonic basis on the sphere.', '1211.1680-1-10-2': 'The angular position [MATH] is specified by colatitude [MATH] and longitude [MATH].', '1211.1680-1-10-3': 'The spherical harmonic coefficients are given by [EQUATION] with the surface element [MATH].', '1211.1680-1-10-4': 'The signal [MATH] is band-limited in the spherical harmonic basis with band-limit [MATH] if [MATH].', '1211.1680-1-10-5': 'For band-limited signals sampling theorems can be invoked so that both forward and inverse transforms can be reduced to finite summations that are theoretically exact.', '1211.1680-1-10-6': 'Sampling theorems effectively encode a quadrature rule for the exact evaluation of integrals on the sphere from a finite set of sampling nodes.', '1211.1680-1-10-7': 'Various sampling theorems exist in the literature (e.g. ).', '1211.1680-1-10-8': 'In this work we adopt the [CITATION] sampling theorem (hereafter MW), which is based on an equiangular sampling scheme and, for a given band-limit [MATH], requires the lowest number of samples on the sphere of all sampling theorems, namely [MATH] samples (for comparison [MATH] samples are required by [CITATION]).', '1211.1680-1-10-9': 'Fast algorithms to compute the corresponding spherical harmonic transform scale as [MATH] and are numerically stable to band-limits of at least [MATH] .', '1211.1680-1-10-10': 'Alternative sampling schemes that are not based on sampling theorems also exist, such as HEALPix or GLESP .', '1211.1680-1-10-11': 'However, these schemes do not lead to exact spherical harmonic transforms on the sphere.', '1211.1680-1-10-12': 'Nevertheless the resulting approximate transforms achieve good accuracy and benefit from other practical advantages, such as equal-area pixels.', '1211.1680-1-11-0': '## Scale-discretised wavelets on the sphere', '1211.1680-1-12-0': 'The scale-discretised wavelet transform aims to exact spatially localised, scale-dependent features in the signal of interest [MATH].', '1211.1680-1-12-1': 'The [MATH]-th wavelet coefficient [MATH] is defined as the convolution of [MATH] with the wavelet [MATH]: [EQUATION] where [MATH] denotes complex conjugation.', '1211.1680-1-12-2': 'Convolution on the sphere is defined by the inner product of [MATH] with the rotated wavelet [MATH].', '1211.1680-1-12-3': 'We restrict our attention to axisymmetric wavelets, i.e. wavelets that are azimuthally symmetric when centred on the poles.', '1211.1680-1-12-4': 'Consequently, the rotation operator [MATH] is parameterised by angular position [MATH] only and not also orientation.', '1211.1680-1-12-5': 'For the axisymmetric case the spherical harmonic decomposition of [MATH] is then simply given by a weighted product in harmonic space: [EQUATION] where [MATH], [MATH] and [MATH], and where [MATH] is the Kronecker delta symbol.', '1211.1680-1-13-0': 'The wavelet coefficients extract the detail information of the signal only; a scaling function and corresponding scaling coefficients must be introduced to represent the low-frequency (low-[MATH]), approximate information of the signal.', '1211.1680-1-13-1': 'The scaling coefficients [MATH] are defined by the convolution of [MATH] with the scaling function [MATH]: [EQUATION] or in harmonic space, [EQUATION] where [MATH] and [MATH].', '1211.1680-1-14-0': 'Provided the wavelets and scaling function satisfy an admissibility property (defined below), the function [MATH] may be reconstructed exactly from its wavelet and scaling coefficients by [EQUATION] or equivalently in harmonic space by [EQUATION]', '1211.1680-1-14-1': 'The parameters [MATH], [MATH] define the lowest and highest scales [MATH] of the wavelet decomposition and must be defined consistently to extract and reconstruct all the information content of [MATH].', '1211.1680-1-14-2': 'These parameters depend on the construction of the wavelets and scaling function and are defined explicitly in the next paragraphs.', '1211.1680-1-14-3': 'The admissibility condition under which a band-limited function [MATH] can be decomposed and reconstructed exactly is given by the following resolution of the identity: [EQUATION]', '1211.1680-1-14-4': 'We are now in position to define wavelets and a scaling function that satisfy the admissibility property.', '1211.1680-1-14-5': 'We use the smooth generating functions defined by [CITATION] in order to tile the harmonic line (the needlet generating functions defined by [CITATION] can also be used; in fact, the S2LET code supports both definitions).', '1211.1680-1-14-6': 'Consider the [MATH] Schwartz function with compact support on [MATH]: [EQUATION] for [MATH].', '1211.1680-1-14-7': 'We introduce the positive real parameter [MATH] to map [MATH] to [EQUATION] which has compact support in [MATH].', '1211.1680-1-14-8': 'We then define the smoothly decreasing function [MATH] by [EQUATION] which is unity for [MATH], zero for [MATH], and is smoothly decreasing from unity to zero for [MATH].', '1211.1680-1-14-9': 'We finally define the wavelet generating function by [EQUATION] and the scaling function generating function by [EQUATION]', '1211.1680-1-14-10': 'The wavelets and scaling function are constructed from their generating functions to satisfy the admissibility condition given by Eqn. ([REF]).', '1211.1680-1-14-11': 'A natural approach is to define [MATH] from the generating functions [MATH] to have support on [MATH], yielding [EQUATION]', '1211.1680-1-14-12': 'For these wavelets, Eqn. ([REF]) is satisfied for [MATH], where [MATH] is the lowest wavelet scale used in the decomposition.', '1211.1680-1-14-13': 'The scaling function [MATH] is constructed to extract the modes that cannot be probed by the wavelets (i.e. modes with [MATH]): [EQUATION]', '1211.1680-1-14-14': 'To satisfy exact reconstruction, [MATH] is set to ensure the wavelets reach the band-limit of the signal of interest, yielding [MATH].', '1211.1680-1-14-15': 'The choice of the lowest wavelet scale [MATH] is arbitrary, provided that [MATH].', '1211.1680-1-14-16': 'The wavelets and scaling function may then be reconstructed on the sphere through an inverse spherical harmonic transform.', '1211.1680-1-14-17': 'With this construction the wavelets and scaling function are well-localised both spatially on the sphere and also in harmonic space, as shown in Figure [REF].', '1211.1680-1-14-18': 'Consequently, the scale-discretised wavelet transform on the sphere can be used to extract spatially localised, scale-dependent features in the signal of interest.', '1211.1680-1-15-0': '# The S2LET code', '1211.1680-1-16-0': 'In this section we describe the S2LET code.', '1211.1680-1-16-1': 'We first introduce a multiresolution algorithm to capture each wavelet scale in the minimum number of samples on the sphere, which follows by taking advantage of the reduced band-limit of the wavelets for scales [MATH].', '1211.1680-1-16-2': 'This multiresolution algorithm was introduced by [CITATION] already and reduces the computation cost of the transform considerably.', '1211.1680-1-16-3': 'We then provide details of the implementation, the computational complexity and the numerical accuracy of the scale-discretised wavelet transform implemented in the S2LET code.', '1211.1680-1-16-4': 'We finally outline planed future extensions of the S2LET code.', '1211.1680-1-17-0': '## Multiresolution algorithm', '1211.1680-1-18-0': 'In harmonic space, the wavelet coefficients are simply given by the weighted product of the spherical harmonic coefficients of [MATH] and the wavelets, as expressed in Eqn. ([REF]).', '1211.1680-1-18-1': 'Although the wavelet coefficients can be analysed at the same resolution as the signal [MATH] (full-resolution), by construction they have different band-limits for different scales [MATH], as shown in Figure [REF].', '1211.1680-1-18-2': 'Reconstruction can thus be performed at a lower resolution without any loss of information.', '1211.1680-1-18-3': 'This approach yields a multiresolution algorithm where the wavelet coefficients are reconstructed with the minimal number of samples on the sphere: the [MATH]-th wavelet coefficients have band-limit [MATH] and are thus recovered on [MATH] samples on the sphere when the MW sampling theorem is adopted.', '1211.1680-1-18-4': 'This approach leads to significant improvements in terms of speed and memory use compared to the full-resolution case, as shown in the next section.', '1211.1680-1-18-5': 'Figure [REF] illustrates the use of the full-resolution and multiresolution transforms on a map of Earth topography data.', '1211.1680-1-19-0': '## Implementation and interfaces', '1211.1680-1-20-0': 'The core numerical routines of S2LET are implemented in C. By adopting a low level programming language such as C for the implementation of the core algorithms, computational efficiency is optimised.', '1211.1680-1-20-1': 'The C library includes the full-resolution and multiresolution wavelet transforms, with specific optimisations for real signals in order to take advantage of all symmetries of the spherical harmonic transform.', '1211.1680-1-20-2': 'The scale-discretised wavelet transform is computed in harmonic space through Eqn. ([REF]) and Eqn. ([REF]), for the input parameters [MATH].', '1211.1680-1-20-3': 'To reconstruct signals on the sphere, by default S2LET uses the exact spherical harmonic transform of the MW sampling theorem implemented in the SSHT code.', '1211.1680-1-20-4': 'In this case all transforms are exact and one can analyse and synthesise real and complex signals using the MW sampling at floating-point precision.', '1211.1680-1-20-5': 'S2LET has been extended to also support the HEALPix sampling scheme.', '1211.1680-1-20-6': 'Since the spherical harmonic transform implemented in HEALPix is not theoretically exact, the scale-discretised wavelet transform computed using the HEALPix sampling does not capture all the information of a band-limited signal and is thus not theoretically exact.', '1211.1680-1-20-7': 'Nevertheless, the transforms computed using HEALPix achieve good numerical accuracy in most situations.', '1211.1680-1-21-0': 'We provide interfaces for the C library in three languages: Matlab, IDL and Java.', '1211.1680-1-21-1': 'The Matlab and IDL codes also include routines to read/write signals on the sphere stored in either HEALPix FITS files or the FITS file format used to stored MW sampled signals.', '1211.1680-1-21-2': 'In addition, functionality to plot the Mollweide projection of real signals for both MW or HEALPix samplings is included.', '1211.1680-1-21-3': 'The Java interface includes an object-oriented representation of sampled maps, spherical harmonics and wavelet transforms.', '1211.1680-1-21-4': 'All routines and interfaces are well documented and illustrated with several examples for both the MW and HEALPix samplings.', '1211.1680-1-21-5': 'These examples cover multiple combinations of parameters and types of signals.', '1211.1680-1-21-6': 'S2LET requires SSHT, which implements fast and exact algorithms to perform the forward and inverse spherical harmonic transforms corresponding to the MW sampling theorem .', '1211.1680-1-21-7': 'SSHT in turn requires the FFTW package for the computation of fast Fourier transforms.', '1211.1680-1-21-8': 'The fast spherical harmonic transforms implemented in SSHT compute Wigner functions, and thus the spherical harmonic functions, through efficient recursion using either the method of [CITATION] or [CITATION].', '1211.1680-1-21-9': 'Here we present results using the recursion of [CITATION].', '1211.1680-1-21-10': 'The fast spherical harmonic transform algorithms implemented in SSHT scale as [MATH] .', '1211.1680-1-22-0': 'The complexity of the axisymmetric scale-discretised wavelet transform is dominated by spherical harmonic transforms since the wavelet transforms are computed efficiently in harmonic space, through Eqn. ([REF]) and Eqn. ([REF]) for the forward transform and through Eqn. ([REF]) for the inverse transform.', '1211.1680-1-22-1': 'Given a band-limit [MATH] and wavelet parameters [MATH], recall that the maximum scale is given by [MATH] and hence the wavelet transform (forward or inverse) involves [MATH] spherical harmonic transforms (one for the original signal, one for the scaling coefficients and [MATH] for the wavelet coefficients).', '1211.1680-1-22-2': 'If the scaling coefficients and all wavelet coefficients are reconstructed at full-resolution in real space, the axisymmetric scale-discretised wavelet transform scales as [MATH].', '1211.1680-1-22-3': 'However, in the previous section we established a multiresolution algorithm that takes advantage of the reduced band-limit of the wavelets for scales [MATH].', '1211.1680-1-22-4': 'With the multiresolution algorithm only the finest wavelet scales [MATH] are computed at maximal resolution corresponding to the band-limit of the signal.', '1211.1680-1-22-5': 'The complexity of the overall multiresolution wavelet transform is then dominated by these operations and effectively scales as [MATH].', '1211.1680-1-23-0': '## Numerical validation', '1211.1680-1-24-0': 'We evaluate the performance of S2LET in terms of accuracy and complexity using the MW sampling theorem, for which all transforms are theoretically exact.', '1211.1680-1-24-1': 'We show that S2LET achieves floating-point precision and scales as detailed in the previous section.', '1211.1680-1-25-0': 'We consider band-limits [MATH] with [MATH] and generate sets of spherical harmonic coefficients [MATH] following independent Gaussian distributions [MATH].', '1211.1680-1-25-1': 'We then perform the wavelet decomposition and reconstruct the harmonic coefficients, denoted by [MATH].', '1211.1680-1-25-2': 'We evaluate the accuracy of the transform using the error metric [MATH], which is theoretically zero for both transforms since all signals are band-limited by construction.', '1211.1680-1-25-3': 'The complexity is quantified by observing how the computation time [MATH] scales with band-limit, where the synthesis and analysis computation times are specified by [MATH] and [MATH] respectively.', '1211.1680-1-25-4': 'Since we evaluate the wavelet transform in real space, a preliminary step is required to reconstruct the signal [MATH] from the randomly generated [MATH].', '1211.1680-1-25-5': 'This step is not included in the computation time since its only purpose is to generate a valid band-limited test signal on the sphere.', '1211.1680-1-25-6': 'The analysis then denotes the decomposition of [MATH] into wavelet coefficients [MATH] and scaling coefficients [MATH] on the sphere.', '1211.1680-1-25-7': 'The synthesis refers to recovering the signal [MATH] from these coefficients.', '1211.1680-1-25-8': 'The final step, which is not included in the computation time either, is to decompose [MATH] into harmonic coefficients [MATH] in order to compare them with [MATH].', '1211.1680-1-25-9': 'The stability of both [MATH] and [MATH] is checked by averaging over hundreds of realisations of [MATH] for [MATH] with [MATH] and a few realisations with [MATH].', '1211.1680-1-25-10': 'The results proved to be very stable, i.e. the variances of the error and timing metrics are lower than 5%.', '1211.1680-1-25-11': 'Recall that for given band-limit [MATH] the number of samples on the sphere required by the exact quadrature is [MATH].', '1211.1680-1-25-12': 'All tests were run on an Intel 2.0GHz Core i7 processor with 8GB of RAM.', '1211.1680-1-26-0': 'The accuracy and timing performance of the scale-discretised wavelet transform implemented in S2LET are presented in Figure [REF].', '1211.1680-1-26-1': 'S2LET achieves very good numerical accuracy, with numerical errors comparable to floating-point precision.', '1211.1680-1-26-2': 'Moreover, the full-resolution and multiresolution algorithms are indistinguishable in terms of accuracy.', '1211.1680-1-26-3': 'However, the latter is four to five times faster than the former for the band-limits considered since only the wavelet coefficients for [MATH] are computed at full-resolution.', '1211.1680-1-26-4': 'As shown in Figure [REF], computation time scales as [MATH] for both algorithms, in agreement with theory.', '1211.1680-1-27-0': '## Future extensions', '1211.1680-1-28-0': 'We have restricted this work to axisymmetric scale-discretised wavelets so that the rotation operator is only parametrised by angular position on the sphere and not also orientation.', '1211.1680-1-28-1': 'The extension to directional wavelets has already been considered by [CITATION] but has not yet been implemented in the S2LET code.', '1211.1680-1-28-2': 'In future work we plan to relax this constraint and to consider directional, steerable wavelets on the sphere .', '1211.1680-1-28-3': 'We also plan to exploit recent ideas leading to fast (spin) spherical harmonic transforms to yield faster algorithms than those developed by [CITATION] to compute directional wavelet transforms on the sphere.', '1211.1680-1-28-4': 'Finally, we intend to add support to analyse spin signals on the sphere.', '1211.1680-1-28-5': 'In a future release, the code will also be parallelised, which will lead to further speed improvements, and a more complete framework in Java will be added, including IO and plotting routines.', '1211.1680-1-28-6': 'The S2LET code will thus be under active development with future releases forthcoming.', '1211.1680-1-28-7': 'In any case, we hope this first version of the S2LET code will prove useful for axisymmetric scale-discretised wavelet analysis on the sphere.', '1211.1680-1-28-8': 'Indeed, the code has already been used as an integral part of the new exact flaglet wavelet transform on the ball , the spherical space constructed by augmenting the sphere with the radial line.', '1211.1680-1-29-0': '# Examples', '1211.1680-1-30-0': 'The S2LET code is extensively documented and ships with several examples in the four languages supported.', '1211.1680-1-30-1': 'In this section we present a subset of short examples, along with the code to execute them in order to demonstrate the ease of using S2LET to perform wavelet transforms.', '1211.1680-1-31-0': '## Wavelet transform from the command line', '1211.1680-1-32-0': 'S2LET includes ready-to-use high-level programs to directly decompose a real signal into wavelet coefficients.', '1211.1680-1-32-1': 'The inputs are a FITS file containing the signal of interest and the parameters for the transform.', '1211.1680-1-32-2': 'The program writes the output coefficients in FITS files in the same directory as the input file and with a consistent naming scheme.', '1211.1680-1-32-3': 'These commands are available for both sampling schemes.', '1211.1680-1-32-4': 'For the MW sampling case illustrated in Example [REF], the wavelet transform is exact and the band limit corresponds to the resolution of the input map, which will be read automatically from the file.', '1211.1680-1-32-5': 'The transform may be performed in full-resolution or multiresolution by adjusting the multiresolution flag specified by the last parameter (respectively [MATH] and [MATH]), and the output wavelet coefficients are computed at full and minimal resolution accordingly.', '1211.1680-1-32-6': 'For the case of a HEALPix map, as illustrated in Example [REF], the transform is not exact and the band-limit must be supplied as the last parameter in the command.', '1211.1680-1-32-7': 'The output scaling and wavelet coefficients of a HEALPix map are reconstructed and stored in FITS files at the same resolution as the input map.', '1211.1680-1-32-8': 'For both MW and HEALPix samplings the output coefficients may be read and plotted using the Matlab or IDL routines.', '1211.1680-1-33-0': '## Wavelet transform in Matlab and IDL', '1211.1680-1-34-0': 'Examples [REF] and [REF] read real signals on the sphere from FITS files, calculate the wavelet coefficients and plot them using a Mollweide projection.', '1211.1680-1-34-1': 'The first case is a Matlab example where the input map is a simulation of the cosmic microwave background in the HEALPix sampling.', '1211.1680-1-34-2': 'The second case is a IDL example where the input map is a topography map of the Earth in MW sampling.', '1211.1680-1-34-3': 'S2LET ships with versions of these two examples in C, Matlab and IDL.', '1211.1680-1-35-0': '## Wavelet denoising in C', '1211.1680-1-36-0': 'Example [REF] illustrates the use of the wavelet transform to denoise a signal on the sphere.', '1211.1680-1-36-1': 'The input noisy map is a band-limited topography map of the Earth in MW sampling at resolution [MATH].', '1211.1680-1-36-2': 'It is read from a FITS file, decomposed into wavelet coefficients (for given parameters [MATH] and [MATH] which are then denoised by thresholding.', '1211.1680-1-36-3': 'The denoised signal is reconstructed from the denoised wavelet coefficients and written to a FITS file.', '1211.1680-1-37-0': 'In this example we consider a noisy signal [MATH], where the signal of interest [MATH] is contaminated with noise [MATH].', '1211.1680-1-37-1': 'We consider zero-mean white Gaussian noise on the sphere, where the variance of the harmonic coefficients of the noise is specified by [EQUATION]', '1211.1680-1-37-2': 'A simple way to evaluate the fidelity of the observed signal [MATH] is through the signal-to-noise ratio (SNR), which we define on the sphere by [EQUATION] where the signal energy is defined by [EQUATION]', '1211.1680-1-37-3': 'We seek a denoised version of [MATH], denoted by [MATH], with large [MATH] so that [MATH] isolates the informative signal [MATH].', '1211.1680-1-37-4': 'When taking the wavelet transform of the noisy signal [MATH], one expects the energy of the informative part to be concentrated in a small number of wavelet coefficients while the noise energy should be spread over various wavelet scales.', '1211.1680-1-37-5': 'Since the transform is linear, the wavelet coefficients of the [MATH]-th scale are simply given by the sum of the individual contributions: [EQUATION] where capital letters denote the wavelet coefficients, i.e. [MATH], [MATH] and [MATH].', '1211.1680-1-37-6': 'For the zero-mean white Gaussian noise defined by Eqn. [REF], the noise in wavelet space is also zero-mean and Gaussian, with variance [EQUATION]', '1211.1680-1-37-7': 'Denoising is performed by hard-thresholding the wavelet coefficients [MATH], where the threshold is taken as [MATH].', '1211.1680-1-37-8': 'The denoised wavelet coefficients [MATH] are thus given by [EQUATION]', '1211.1680-1-37-9': 'The denoised signal [MATH] is reconstructed from its wavelet coefficients [MATH] and the scaling coefficients of [MATH], which are not thresholded.', '1211.1680-1-37-10': 'The denoising procedure outlined above is particularly simple and more sophisticated denoising strategies can be developed; we adopt this simple denoising strategy merely to illustrate the use of the S2LET code.', '1211.1680-1-37-11': 'In this example we perform the wavelet transform with parameters [MATH] and [MATH].', '1211.1680-1-37-12': 'For a noisy signal [MATH] with [MATH]dB, the scale-discretised wavelet denoising recovers a denoised signal [MATH] with [MATH]dB.', '1211.1680-1-37-13': 'The initial, noisy and denoised maps are shown in Figure [REF].', '1211.1680-1-38-0': '# Summary', '1211.1680-1-39-0': 'In the era of precision astrophysics and cosmology, large and complex data-sets on the sphere must be analysed at high precision in order to confront accurate theoretical predictions.', '1211.1680-1-39-1': 'Scale-discretised wavelets are a powerful analysis technique where spatially localised, scale-dependent signal features of interest can be extracted and analysed.', '1211.1680-1-39-2': 'Combined with a sampling theorem, this framework leads to an exact multiresolution wavelet analysis, where signals on the sphere can be reconstructed from their scaling and wavelet coefficients exactly.', '1211.1680-1-40-0': 'We have described S2LET, a fast and robust implementation of the scale-discretised wavelet transform.', '1211.1680-1-40-1': 'Although the first public release of S2LET is restricted to axisymmetric wavelets, the generalisation to directional, steerable wavelets will be made available in a future release.', '1211.1680-1-40-2': 'The core numerical routines of S2LET are written in C and have interfaces in Matlab, IDL and Java.', '1211.1680-1-40-3': 'Both MW and HEALPix pixelisation schemes are supported.', '1211.1680-1-40-4': 'In this article we have presented a number of examples to illustrate the ease of use of S2LET for performing wavelet transform of real signals stored as FITS files and to plot scaling and wavelet coefficients on Mollweide projections of the sphere.', '1211.1680-1-40-5': 'We have also detailed a denoising example where denoising is performed through simple hard-thresholding in wavelet space.', '1211.1680-1-40-6': 'Although only a simple denoising strategy was performed to illustrate the use of the S2LET code, it nevertheless performed very well, highlighting the effectiveness of the scale-discretised wavelet transform on the sphere.', '1211.1680-1-41-0': 'BL is supported by the Perren Fund and the Impact Fund.', '1211.1680-1-41-1': 'JDM is supported by a Newton International Fellowship from the Royal Society and the British Academy.', '1211.1680-1-41-2': 'YW is supported by the Center for Biomedical Imaging (CIBM) of the Geneva and Lausanne Universities, EPFL and the Leenaards and Louis-Jeantet foundations.'}","{'1211.1680-2-0-0': 'S2LET: A code to perform fast wavelet analysis on the sphere', '1211.1680-2-1-0': 'We describe S2LET, a fast and robust implementation of the scale-discretised wavelet transform on the sphere.', '1211.1680-2-1-1': 'Wavelets are constructed through a tiling of the harmonic line and can be used to probe spatially localised, scale-dependent features of signals on the sphere.', '1211.1680-2-1-2': 'The reconstruction of a signal from its wavelets coefficients is made exact here through the use of a sampling theorem on the sphere.', '1211.1680-2-1-3': 'Moreover, a multiresolution algorithm is presented to capture all information of each wavelet scale in the minimal number of samples on the sphere.', '1211.1680-2-1-4': 'In addition S2LET supports the HEALPix pixelisation scheme, in which case the transform is not exact but nevertheless achieves good numerical accuracy.', '1211.1680-2-1-5': 'The core routines of S2LET are written in C and have interfaces in Matlab, IDL and Java.', '1211.1680-2-1-6': 'Real signals can be written to and read from FITS files and plotted as Mollweide projections.', '1211.1680-2-1-7': 'The S2LET code is made publicly available, is extensively documented, and ships with several examples in the four languages supported.', '1211.1680-2-1-8': 'At present the code is restricted to axisymmetric wavelets but will be extended to directional, steerable wavelets in a future release.', '1211.1680-2-2-0': '# Introduction', '1211.1680-2-3-0': 'Signals defined or measured on the sphere arise in numerous disciplines, where analysis techniques defined explicitly on the sphere are now in common use.', '1211.1680-2-3-1': 'In particular, wavelets on the sphere have been applied very successfully to problems in astrophysics and cosmology, where data-sets are increasingly large and need to be analysed at high resolution in order to confront accurate theoretical predictions (e.g. ).', '1211.1680-2-4-0': 'While wavelet theory is well established in Euclidean space (see e.g. ), multiple wavelet frameworks have been developed on the sphere, only a fraction of which lead to exact transforms in both the continuous and discrete settings.', '1211.1680-2-4-1': 'In fact, discrete methodologies achieve exactness in practice but may not lead to a stable basis on the sphere .', '1211.1680-2-4-2': 'In the continuous setting several constructions are theoretically exact, and have been combined with sampling theorems on the sphere to enable exact reconstruction in the discrete setting also.', '1211.1680-2-4-3': 'In particular, scale-discretised wavelets lean on a tiling of the harmonic line to yield an exact wavelet transform in both the continuous and discrete settings.', '1211.1680-2-4-4': 'In the axisymmetric case, the scale-discretised wavelets reduce to needlets , which were developed independently using an analogous tiling of the harmonic line.', '1211.1680-2-4-5': 'Similarly, the isotropic undecimated wavelet transform (UWT) developed by exploits B-splines of order 3 to cover the harmonic line with filters with greater overlap but nevertheless compact support.', '1211.1680-2-5-0': 'In this paper we describe the new publicly available S2LET code to perform the scale-discretised wavelet transform of complex signals on the sphere.', '1211.1680-2-5-1': 'At present S2LET is restricted to axisymmetric wavelets (i.e. azimuthally symmetric when centred on the poles) and includes generating functions for axisymmetric scale-discretised wavelets , needlets and B-spline wavelets .', '1211.1680-2-5-2': 'We intend to extend the code to directional, steerable wavelets and spin functions in a future release.', '1211.1680-2-5-3': 'The core routines of S2LET are written in C, exploit fast algorithms on the sphere, and have interfaces in Matlab, IDL and Java.', '1211.1680-2-6-0': 'We note that many very useful public codes are already available to compute wavelet transforms on the sphere, including isotropic undecimated wavelet, ridgelet and curvelet transforms , invertible filter banks , needlets (NeedATool; ) and scale-discretised wavelets (S2DW; ).', '1211.1680-2-6-1': 'S2LET aims primarily to provide a fast and flexible implementation of the scale-discretised transform with exact reconstruction on the sphere using the sampling theorem of [CITATION], although it has also been extended to support some of the features of these other codes.', '1211.1680-2-6-2': 'Furthermore, particular attention has been paid in the development of S2LET to prove a user-friendly code, supporting multiple programming languages, and which is extensively documented.', '1211.1680-2-7-0': 'The remainder of this article is organised as follows.', '1211.1680-2-7-1': 'In section 2 we detail the construction of scale-discretised axisymmetric wavelets and the corresponding exact scale-discretised wavelet transform on the sphere.', '1211.1680-2-7-2': 'In section 3 we describe the S2LET code, including implementation details, computational complexity and numerical performance.', '1211.1680-2-7-3': 'We present a number of simple examples using S2LET in section 4, along with the code to execute them.', '1211.1680-2-7-4': 'We conclude in section 5.', '1211.1680-2-8-0': '# Wavelets on the sphere', '1211.1680-2-9-0': 'We review the construction of scale-discretised wavelets on the sphere through tiling of the harmonic line .', '1211.1680-2-9-1': 'Directional, steerable wavelets were also considered by [CITATION], however we restrict our attention to axisymmetric wavelets here.', '1211.1680-2-9-2': 'Furthermore, the use of a sampling theorem on the sphere guarantees that spherical harmonic coefficients capture all the information content of band-limited signals, resulting in theoretically exact harmonic and wavelet transforms.', '1211.1680-2-9-3': 'One may alternatively adopt samplings of the sphere for which exact quadrature rules do not exist, such as HEALPix , but which nevertheless exhibit other useful properties, leading to numerically accurate but not theoretically exact transforms.', '1211.1680-2-10-0': '## Harmonic analysis on the sphere', '1211.1680-2-11-0': 'The spherical harmonic decomposition of a square integrable signal [MATH] on the two-dimensional sphere [MATH] reads [EQUATION] where [MATH] are the spherical harmonic functions, which form the canonical orthogonal basis on [MATH].', '1211.1680-2-11-1': 'The spherical harmonic coefficients [MATH], with [MATH] and [MATH] such that [MATH], form a dual representation of the signal [MATH] in the harmonic basis on the sphere.', '1211.1680-2-11-2': 'The angular position [MATH] is specified by colatitude [MATH] and longitude [MATH].', '1211.1680-2-11-3': 'The spherical harmonic coefficients are given by [EQUATION] with the surface element [MATH].', '1211.1680-2-11-4': 'We consider band-limited signals in the spherical harmonic basis, with band-limit [MATH] if [MATH].', '1211.1680-2-11-5': 'For band-limited signals sampling theorems can be invoked so that both forward and inverse transforms can be reduced to finite summations that are theoretically exact.', '1211.1680-2-11-6': 'Sampling theorems effectively encode a quadrature rule for the exact evaluation of integrals on the sphere from a finite set of sampling nodes.', '1211.1680-2-11-7': 'Various sampling theorems exist in the literature (e.g. ).', '1211.1680-2-11-8': 'In this work we adopt the [CITATION] sampling theorem (hereafter MW), which is based on an equiangular sampling scheme and, for a given band-limit [MATH], requires the lowest number of samples on the sphere of all sampling theorems, namely [MATH] samples (for comparison [MATH] samples are required by [CITATION]).', '1211.1680-2-11-9': 'Fast algorithms to compute the corresponding spherical harmonic transform scale as [MATH] and are numerically stable to band-limits of at least [MATH] .', '1211.1680-2-11-10': 'The GLESP pixelisation scheme also provides a sampling theorem based on the Gauss-Legendre quadrature, and could be used in place of the MW sampling theorem.', '1211.1680-2-11-11': 'However, GLESP uses more samples than Gauss-Legendre quadrature requires, which may lead to an overhead when considering large band-limits and numerous wavelet scales.', '1211.1680-2-11-12': 'We focus on the MW sampling scheme to obtain a theoretically exact transform.', '1211.1680-2-11-13': 'Alternative sampling schemes that are not based on sampling theorems also exist such as HEALPix , which is supported by S2LET, MRS and Needatool.', '1211.1680-2-11-14': 'HEALPix does not lead to exact transforms on the sphere but the resulting approximate transforms nevertheless achieve good accuracy and benefit from other practical advantages, such as equal-area pixels.', '1211.1680-2-12-0': '## Scale-discretised wavelets on the sphere', '1211.1680-2-13-0': 'The scale-discretised wavelet transform allows one to probe spatially localised, scale-dependent content in the signal of interest [MATH].', '1211.1680-2-13-1': 'The [MATH]-th wavelet scale [MATH] is defined as the convolution of [MATH] with the wavelet [MATH]: [EQUATION] where [MATH] denotes complex conjugation.', '1211.1680-2-13-2': 'Convolution on the sphere is defined by the inner product of [MATH] with the rotated wavelet [MATH].', '1211.1680-2-13-3': 'We restrict our attention to axisymmetric wavelets, i.e. wavelets that are azimuthally symmetric when centred on the poles.', '1211.1680-2-13-4': 'Consequently, the rotation operator [MATH] is parameterised by angular position [MATH] only and not also orientation.', '1211.1680-2-13-5': 'For the axisymmetric case the spherical harmonic decomposition of [MATH] is then simply given by a weighted product in harmonic space: [EQUATION] where [MATH], [MATH] and [MATH], and where [MATH] is the Kronecker delta symbol.', '1211.1680-2-14-0': 'The wavelet coefficients extract the detail information of the signal only; a scaling function and corresponding scaling coefficients must be introduced to represent the low-frequency (low-[MATH]), approximate information of the signal.', '1211.1680-2-14-1': 'The scaling coefficients [MATH] are defined by the convolution of [MATH] with the scaling function [MATH]: [EQUATION] or in harmonic space, [EQUATION] where [MATH] and [MATH].', '1211.1680-2-15-0': 'Provided the wavelets and scaling function satisfy an admissibility property (defined below), the function [MATH] may be reconstructed exactly from its wavelet and scaling coefficients by [EQUATION] or equivalently in harmonic space by [EQUATION]', '1211.1680-2-15-1': 'The parameters [MATH], [MATH] define the lowest and highest scales [MATH] of the wavelet decomposition and must be defined consistently to extract and reconstruct all the information content of [MATH].', '1211.1680-2-15-2': 'These parameters depend on the construction of the wavelets and scaling function and are defined explicitly in the next paragraphs.', '1211.1680-2-15-3': 'The admissibility condition under which a band-limited function [MATH] can be decomposed and reconstructed exactly is given by the following resolution of the identity: [EQUATION]', '1211.1680-2-15-4': 'We are now in a position to define wavelets and a scaling function that satisfy the admissibility property.', '1211.1680-2-15-5': 'In this paper, we use the smooth generating functions defined by [CITATION] in order to tile the harmonic line.', '1211.1680-2-15-6': 'Alternative definitions are also supported by S2LET, as presented at the end of this section.', '1211.1680-2-15-7': 'Consider the [MATH] Schwartz function with compact support on [MATH]: [EQUATION] for [MATH].', '1211.1680-2-15-8': 'We introduce the positive real parameter [MATH] to map [MATH] to [EQUATION] which has compact support in [MATH].', '1211.1680-2-15-9': 'We then define the smoothly decreasing function [MATH] by [EQUATION] which is unity for [MATH], zero for [MATH], and is smoothly decreasing from unity to zero for [MATH].', '1211.1680-2-15-10': 'We finally define the wavelet generating function by [EQUATION] and the scaling function generating function by [EQUATION]', '1211.1680-2-15-11': 'The wavelets and scaling function are constructed from their generating functions to satisfy the admissibility condition given by Eqn. ([REF]).', '1211.1680-2-15-12': 'A natural approach is to define [MATH] from the generating functions [MATH] to have support on [MATH], yielding [EQUATION]', '1211.1680-2-15-13': 'For these wavelets Eqn. ([REF]) is satisfied for [MATH], where [MATH] is the lowest wavelet scale used in the decomposition.', '1211.1680-2-15-14': 'The scaling function [MATH] is constructed to extract the modes that cannot be probed by the wavelets (i.e. modes with [MATH]): [EQUATION]', '1211.1680-2-15-15': 'To satisfy exact reconstruction, [MATH] is set to ensure the wavelets reach the band-limit of the signal of interest, yielding [MATH].', '1211.1680-2-15-16': 'The choice of the lowest wavelet scale [MATH] is arbitrary, provided that [MATH].', '1211.1680-2-15-17': 'The wavelets and scaling function may then be reconstructed on the sphere through an inverse spherical harmonic transform.', '1211.1680-2-15-18': 'The harmonic tiling and real space representation of these wavelets are shown in Figure [REF] and Figure [REF] respectively.', '1211.1680-2-16-0': 'In addition to the scale-discretised generating functions , S2LET also supports the needlet functions , which yield a similar tiling of the harmonic line, as shown in Figure [REF].', '1211.1680-2-16-1': 'The B-spline filters used to construct the isotropic undecimated wavelet transform are also supported, as also shown in Figure [REF].', '1211.1680-2-16-2': 'With these three constructions, the wavelets and scaling functions are well-localised both spatially on the sphere and also in harmonic space.', '1211.1680-2-16-3': 'Consequently, the associated wavelet transforms on the sphere can be used to extract spatially localised, scale-dependent features in signals of interest.', '1211.1680-2-17-0': '# The S2LET code', '1211.1680-2-18-0': 'In this section we describe the S2LET code.', '1211.1680-2-18-1': 'We first introduce a multiresolution algorithm to capture each wavelet scale in the minimum number of samples on the sphere, which follows by taking advantage of the reduced band-limit of the wavelets for scales [MATH].', '1211.1680-2-18-2': 'This multiresolution algorithm reduces the computation cost of the transform considerably.', '1211.1680-2-18-3': 'We then provide details of the implementation, the computational complexity and the numerical accuracy of the scale-discretised wavelet transform supported in S2LET.', '1211.1680-2-18-4': 'We finally outline planned future extensions of the code.', '1211.1680-2-19-0': '## Multiresolution algorithm', '1211.1680-2-20-0': 'In harmonic space, the wavelet coefficients are simply given by the weighted product of the spherical harmonic coefficients of [MATH] and the wavelets, as expressed in Eqn. ([REF]).', '1211.1680-2-20-1': 'Although the wavelet coefficients can be analysed at the same resolution as the signal [MATH] (i.e., at full resolution), by construction they have different band-limits for different scales [MATH], as shown in Figure [REF].', '1211.1680-2-20-2': 'The reconstruction can thus be performed at lower resolution, without any loss of information if a sampling theorem is used.', '1211.1680-2-20-3': 'This approach yields a multiresolution algorithm where the wavelet coefficients are reconstructed with the minimal number of samples on the sphere: the [MATH]-th wavelet coefficients have band-limit [MATH] when using the scale-discretised and needlet kernels, and [MATH] when using the B-splines.', '1211.1680-2-20-4': 'When the MW sampling theorem is used, the wavelets are recovered on [MATH] samples on the sphere.', '1211.1680-2-20-5': 'This approach leads to significant improvements in terms of speed and memory use compared to the full-resolution case, as shown in the next section.', '1211.1680-2-20-6': 'Figure [REF] illustrates the use of the full-resolution and multiresolution transforms on a map of Earth topography data with the scale-discretised filters and the MW scheme.', '1211.1680-2-20-7': 'When adopting the HEALPix sampling of the sphere, multiresolution can also be used.', '1211.1680-2-20-8': 'However HEALPix does not rely on a sampling theorem and therefore the resolution for the reconstruction of each wavelet scale must be chosen heuristically and adapted to the desired accuracy.', '1211.1680-2-20-9': 'For example, in the MRS code it is chosen such that [MATH].', '1211.1680-2-20-10': 'More detail on the accuracy of the wavelet transform with HEALPix are provided below.', '1211.1680-2-21-0': '## Implementation', '1211.1680-2-22-0': 'The core numerical routines of S2LET are implemented in C. By adopting a low level programming language such as C for the implementation of the core algorithms, computational efficiency is optimised.', '1211.1680-2-22-1': 'The C library includes the full-resolution and multiresolution wavelet transforms, with specific optimisations for real signals in order to take advantage of all symmetries of the spherical harmonic transform.', '1211.1680-2-22-2': 'The wavelet transform is computed in harmonic space through Eqn. ([REF]) and Eqn. ([REF]), for the input parameters [MATH].', '1211.1680-2-22-3': 'To reconstruct signals on the sphere, by default S2LET uses the exact spherical harmonic transform of the MW sampling theorem implemented in the SSHT code.', '1211.1680-2-22-4': 'In this case all transforms are theoretically exact and one can analyse and synthesise real and complex signals at floating-point precision.', '1211.1680-2-22-5': 'S2LET has been extended to also support the HEALPix sampling scheme, in which case the transform is not theoretically exact but nevertheless achieves good numerical accuracy.', '1211.1680-2-23-0': 'We provide interfaces for the C library in three languages: Matlab, IDL and Java.', '1211.1680-2-23-1': 'The Matlab and IDL codes also include routines to read/write signals on the sphere stored in either HEALPix FITS files or the FITS file format used to stored MW sampled signals.', '1211.1680-2-23-2': 'In addition, functionality to plot the Mollweide projection of real signals for both MW or HEALPix samplings is included.', '1211.1680-2-23-3': 'The Java interface includes an object-oriented representation of sampled maps, spherical harmonics and wavelet transforms.', '1211.1680-2-23-4': 'All routines and interfaces are well documented and illustrated with several examples for both the MW and HEALPix samplings.', '1211.1680-2-23-5': 'These examples cover multiple combinations of parameters and types of signals.', '1211.1680-2-23-6': 'S2LET requires SSHT, which implements fast and exact algorithms to perform the forward and inverse spherical harmonic transforms corresponding to the MW sampling theorem .', '1211.1680-2-23-7': 'SSHT in turn requires the FFTW package for the computation of fast Fourier transforms.', '1211.1680-2-23-8': 'The fast spherical harmonic transforms implemented in SSHT compute Wigner functions, and thus the spherical harmonic functions, through efficient recursion using either the method of [CITATION] or [CITATION].', '1211.1680-2-23-9': 'Here we present results using the recursion of [CITATION].', '1211.1680-2-23-10': 'The fast spherical harmonic transform algorithms implemented in SSHT scale as [MATH] .', '1211.1680-2-24-0': 'Although primarily intended to perform the scale-discretised wavelet transform of [CITATION], S2LET also supports the needlet and spline-based wavelet transforms developed by [CITATION] and [CITATION].', '1211.1680-2-24-1': 'As shown in Figure [REF], these generating functions yield the same number of wavelet scales (for the parameter choices described previously).', '1211.1680-2-24-2': 'However, with the scale-discretised and needlet generating functions the [MATH]-th wavelet scale has compact support in [MATH], whereas the support is much wider with the B-splines, i.e. [MATH] in the S2LET implementation.', '1211.1680-2-24-3': 'As a consequence, when using the multiresolution algorithm the wavelet coefficients must be captured on a greater number of pixels than with the scale-discretised or needlet kernels, while probing approximately the same scales, as shown in Figure [REF].', '1211.1680-2-25-0': 'The complexity of the axisymmetric wavelet transform is dominated by spherical harmonic transforms since the wavelet transforms are computed efficiently in harmonic space, through Eqn. ([REF]) and Eqn. ([REF]) for the forward transform and through Eqn. ([REF]) for the inverse transform.', '1211.1680-2-25-1': 'Given a band-limit [MATH] and wavelet parameters [MATH], recall that the maximum scale is given by [MATH] and hence the wavelet transform (forward or inverse) involves [MATH] spherical harmonic transforms (one for the original signal, one for the scaling coefficients and [MATH] for the wavelet coefficients).', '1211.1680-2-25-2': 'If the scaling coefficients and all wavelet coefficients are reconstructed at full-resolution in real space, the axisymmetric wavelet transform scales as [MATH].', '1211.1680-2-25-3': 'However, in the previous section we established a multiresolution algorithm that takes advantage of the reduced band-limit of the wavelets for scales [MATH].', '1211.1680-2-25-4': 'With the multiresolution algorithm with a sampling theorem, only the finest wavelet scales [MATH] are computed at maximal resolution corresponding to the band-limit of the signal.', '1211.1680-2-25-5': 'The complexity of the overall multiresolution wavelet transform is then dominated by these operations and effectively scales as [MATH].', '1211.1680-2-26-0': '## Numerical validation', '1211.1680-2-27-0': 'We first evaluate the performance of S2LET in terms of accuracy and complexity using the MW sampling theorem, for which all transforms are theoretically exact.', '1211.1680-2-27-1': 'We show that S2LET achieves floating-point precision and scales as detailed in the previous section.', '1211.1680-2-28-0': 'We consider band-limits [MATH] with [MATH] and generate sets of spherical harmonic coefficients [MATH] following independent Gaussian distributions [MATH].', '1211.1680-2-28-1': 'We then perform the wavelet decomposition and reconstruct the harmonic coefficients, denoted by [MATH].', '1211.1680-2-28-2': 'We evaluate the accuracy of the transform using the error metric [MATH], which is theoretically zero since all signals are band-limited by construction.', '1211.1680-2-28-3': 'The complexity is quantified by observing how the computation time [MATH] scales with band-limit, where the synthesis and analysis computation times are specified by [MATH] and [MATH] respectively.', '1211.1680-2-28-4': 'Since we evaluate the wavelet transform in real space, a preliminary step is required to reconstruct the signal [MATH] from the randomly generated [MATH].', '1211.1680-2-28-5': 'This step is not included in the computation time since its only purpose is to generate a valid band-limited test signal on the sphere.', '1211.1680-2-28-6': 'The analysis then denotes the decomposition of [MATH] into wavelet coefficients [MATH] and scaling coefficients [MATH] on the sphere.', '1211.1680-2-28-7': 'The synthesis refers to recovering the signal [MATH] from these coefficients.', '1211.1680-2-28-8': 'The final step, which is not included in the computation time either, is to decompose [MATH] into harmonic coefficients [MATH] in order to compare them with [MATH].', '1211.1680-2-28-9': 'The stability of both [MATH] and [MATH] is checked by averaging over hundreds of realisations of [MATH] for [MATH] with [MATH] and a few realisations with [MATH].', '1211.1680-2-28-10': 'The results proved to be very stable, i.e. the variances of the error and timing metrics are lower than 5%.', '1211.1680-2-28-11': 'Recall that for given band-limit [MATH] the number of samples on the sphere required by the exact quadrature is [MATH].', '1211.1680-2-28-12': 'All tests were run on an Intel 2.0GHz Core i7 processor with 8GB of RAM.', '1211.1680-2-28-13': 'On this machine, precision of floating point numbers is of the order of [MATH], and errors are expected to add up and accumulate when considering linear operations such as the spherical harmonic and wavelet transforms.', '1211.1680-2-28-14': 'The accuracy and timing performance of the scale-discretised wavelet transform implemented in S2LET are presented in Figure [REF].', '1211.1680-2-28-15': 'S2LET achieves very good numerical accuracy, with numerical errors comparable to accumulated floating-point errors only.', '1211.1680-2-28-16': 'Moreover, the full-resolution and multiresolution algorithms are indistinguishable in terms of accuracy.', '1211.1680-2-28-17': 'However, the latter is four to five times faster than the former for the band-limits considered since only the wavelet coefficients for [MATH] are computed at full-resolution.', '1211.1680-2-28-18': 'As shown in Figure [REF], computation time scales as [MATH] for both algorithms, in agreement with theory.', '1211.1680-2-29-0': 'S2LET can also be used with HEALPix, in which case the accuracy of the spherical harmonic transform is critical to the accuracy of the wavelet transform (since HEALpix does not rely on a sampling theorem it does not exhibit theoretically exact harmonic transforms, unlike SSHT or GLESP).', '1211.1680-2-29-1': 'The performances of the spherical harmonic transforms in HEALPix and GLESP have been widely studied in the past (see, e.g., ), and that of the MW sampling were presented in [CITATION].', '1211.1680-2-29-2': 'We do not compile the entirety of these results here, but we have reproduced the essential results on our machine; Table [REF] summarises the orders of accuracy of the HEALPix iterative spherical harmonic transform.', '1211.1680-2-29-3': 'Using the same setup as previously, we calculated the maximum error on the spherical harmonic coefficients when performing the transform back and forth, averaged over the values of [MATH], since the results were found to be sensitive only to the ratio [MATH].', '1211.1680-2-29-4': 'Even with several iterations, which multiplies the number of transforms and thus computation time, the spherical harmonic transform in HEALPix remains at least an order of magnitude less accurate than the MW and GLESP counterparts (which, being both theoretically exact, achieve comparable performances, see ).', '1211.1680-2-29-5': 'Since the wavelet transforms implemented in MRS and Needatool are also computed in harmonic space, their complexity and accuracy are dominated by that of the underlying spherical harmonic transforms.', '1211.1680-2-29-6': 'As a consequence, when adopting the HEALPix scheme, S2LET, MRS and Needatool achieve similar performances, resulting from the computation time and accumulated errors of [MATH] HEALPix spherical harmonic transforms.', '1211.1680-2-29-7': 'In the multiresolution case, the results depend on the resolution chosen to reconstruct each wavelet scale.', '1211.1680-2-30-0': '## Future extensions', '1211.1680-2-31-0': 'In future work we plan to extend S2LET to support directional, steerable wavelets on the sphere .', '1211.1680-2-31-1': 'We also plan to exploit recent ideas leading to fast (spin) spherical harmonic transforms to yield faster algorithms than those developed by [CITATION] and [CITATION] to compute directional wavelet transforms on the sphere.', '1211.1680-2-31-2': 'Finally, we intend to add support to analyse spin signals on the sphere .', '1211.1680-2-31-3': 'In a future release, the code will also be parallelised, which will lead to further speed improvements.', '1211.1680-2-31-4': 'The S2LET code will thus be under active development with future releases forthcoming.', '1211.1680-2-31-5': 'In any case, we hope this first version of the S2LET code will prove useful for axisymmetric scale-discretised wavelet analysis on the sphere.', '1211.1680-2-31-6': 'Indeed, the code has already been used as an integral part of the new exact flaglet wavelet transform on the ball , the spherical space constructed by augmenting the sphere with the radial line.', '1211.1680-2-32-0': '# Examples', '1211.1680-2-33-0': 'The S2LET code is extensively documented and ships with several examples in the four languages supported.', '1211.1680-2-33-1': 'In this section we present a subset of short examples, along with the code to execute them in order to demonstrate the ease of using S2LET to perform wavelet transforms.', '1211.1680-2-33-2': 'All examples were run with the scale-discretised wavelet generating functions.', '1211.1680-2-34-0': '## Wavelet transform from the command line', '1211.1680-2-35-0': 'S2LET includes ready-to-use high-level programs to directly decompose a real signal into wavelet coefficients.', '1211.1680-2-35-1': 'The inputs are a FITS file containing the signal of interest and the parameters for the transform.', '1211.1680-2-35-2': 'The program writes the output coefficients in FITS files in the same directory as the input file and with a consistent naming scheme.', '1211.1680-2-35-3': 'These commands are available for both HEALPix and MW sampling schemes.', '1211.1680-2-35-4': 'For the MW sampling case illustrated in Example [REF], the wavelet transform is theoretically exact and the band limit corresponds to the resolution of the input map, which will be read automatically from the file.', '1211.1680-2-35-5': 'The transform may be performed in full-resolution or multiresolution by adjusting the multiresolution flag specified by the last parameter (respectively [MATH] and [MATH]), and the output wavelet coefficients are computed at full and minimal resolution accordingly.', '1211.1680-2-35-6': 'For the case of a HEALPix map, as illustrated in Example [REF], the band-limit must be supplied as the last parameter in the command.', '1211.1680-2-35-7': 'The output scaling and wavelet coefficients of a HEALPix map are reconstructed and stored in FITS files at the same resolution as the input map.', '1211.1680-2-35-8': 'For both MW and HEALPix samplings the output coefficients may be read and plotted using the Matlab or IDL routines.', '1211.1680-2-36-0': '## Wavelet transform in Matlab and IDL', '1211.1680-2-37-0': 'Examples [REF] and [REF] read real signals on the sphere from FITS files, calculate the wavelet coefficients and plot them using a Mollweide projection.', '1211.1680-2-37-1': 'The first case is a Matlab example where the input map is a simulation of the cosmic microwave background in the HEALPix sampling.', '1211.1680-2-37-2': 'The second case is a IDL example where the input map is a topography map of the Earth in MW sampling.', '1211.1680-2-37-3': 'S2LET ships with versions of these two examples in C, Matlab and IDL.', '1211.1680-2-38-0': '## Wavelet denoising in C', '1211.1680-2-39-0': 'Example [REF] illustrates the use of the wavelet transform to denoise a signal on the sphere.', '1211.1680-2-39-1': 'The input noisy map is a band-limited topography map of the Earth in MW sampling at resolution [MATH].', '1211.1680-2-39-2': 'It is read from a FITS file, decomposed into wavelet coefficients (for given parameters [MATH] and [MATH] which are then denoised by thresholding.', '1211.1680-2-39-3': 'The denoised signal is reconstructed from the denoised wavelet coefficients and written to a FITS file.', '1211.1680-2-40-0': 'In this example we consider a noisy signal [MATH], where the signal of interest [MATH] is contaminated with noise [MATH].', '1211.1680-2-40-1': 'We consider zero-mean white Gaussian noise on the sphere, where the variance of the harmonic coefficients of the noise is specified by [EQUATION]', '1211.1680-2-40-2': 'A simple way to evaluate the fidelity of the observed signal [MATH] is through the signal-to-noise ratio (SNR), define on the sphere by [EQUATION] where the signal energy is defined by [EQUATION]', '1211.1680-2-40-3': 'We seek a denoised version of [MATH], denoted by [MATH], with large [MATH] so that [MATH] isolates the informative signal [MATH].', '1211.1680-2-40-4': 'When taking the wavelet transform of the noisy signal [MATH], one expects the energy of the informative part to be concentrated in a small number of wavelet coefficients, whereas the noise energy should be spread over various wavelet scales.', '1211.1680-2-40-5': 'In this particular toy example, the signal has significant power on large scales, as shown in Figure [REF], which are well described in the wavelet basis and less affected by the random white noise.', '1211.1680-2-40-6': 'Since the transform is linear, the wavelet coefficients of the [MATH]-th scale are simply given by the sum of the individual contributions: [EQUATION] where capital letters denote the wavelet coefficients, i.e. [MATH], [MATH] and [MATH].', '1211.1680-2-40-7': 'For the zero-mean white Gaussian noise defined by Eqn. [REF], the noise in wavelet space is also zero-mean and Gaussian, with variance [EQUATION]', '1211.1680-2-40-8': 'Denoising is performed by hard-thresholding the wavelet coefficients [MATH], where the threshold is taken as [MATH].', '1211.1680-2-40-9': 'The denoised wavelet coefficients [MATH] are thus given by [EQUATION]', '1211.1680-2-40-10': 'The denoised signal [MATH] is reconstructed from its wavelet coefficients [MATH] and the scaling coefficients of [MATH], which are not thresholded.', '1211.1680-2-40-11': 'The denoising procedure outlined above is particularly simple and more sophisticated denoising strategies can be developed; we adopt this simple denoising strategy merely to illustrate the use of the S2LET code.', '1211.1680-2-40-12': 'In this example we perform the wavelet transform with parameters [MATH] and [MATH].', '1211.1680-2-40-13': 'For a noisy signal [MATH] with [MATH]dB, the scale-discretised wavelet denoising recovers a denoised signal [MATH] with [MATH]dB.', '1211.1680-2-40-14': 'The initial, noisy and denoised maps are shown in Figure [REF].', '1211.1680-2-40-15': 'When switching to needlets and B-spline wavelets while keeping [MATH] and [MATH] unchanged, the denoised signals have [MATH]dB and [MATH]dB respectively.', '1211.1680-2-41-0': '# Summary', '1211.1680-2-42-0': 'In the era of precision astrophysics and cosmology, large and complex data-sets on the sphere must be analysed at high precision in order to confront accurate theoretical predictions.', '1211.1680-2-42-1': 'Scale-discretised wavelets are a powerful analysis technique where spatially localised, scale-dependent signal features of interest can be extracted and analysed.', '1211.1680-2-42-2': 'Combined with a sampling theorem, this framework leads to an exact multiresolution wavelet analysis, where signals on the sphere can be reconstructed from their scaling and wavelet coefficients exactly.', '1211.1680-2-43-0': 'We have described S2LET, a fast and robust implementation of the scale-discretised wavelet transform.', '1211.1680-2-43-1': 'Although the first public release of S2LET is restricted to axisymmetric wavelets, the generalisation to directional, steerable wavelets will be made available in a future release.', '1211.1680-2-43-2': 'The core numerical routines of S2LET are written in C and have interfaces in Matlab, IDL and Java.', '1211.1680-2-43-3': 'Both MW and HEALPix pixelisation schemes are supported.', '1211.1680-2-43-4': 'In this article we have presented a number of examples to illustrate the ease of use of S2LET for performing wavelet transform of real signals stored as FITS files and to plot scaling and wavelet coefficients on Mollweide projections of the sphere.', '1211.1680-2-43-5': 'We have also detailed a denoising example where denoising is performed through simple hard-thresholding in wavelet space.', '1211.1680-2-43-6': 'Although only a simple denoising strategy was performed to illustrate the use of the S2LET code, it nevertheless performed very well, highlighting the effectiveness of the scale-discretised wavelet transform on the sphere.', '1211.1680-2-44-0': 'BL is supported by the Perren Fund and the Impact Fund.', '1211.1680-2-44-1': 'JDM is supported in part by a Newton International Fellowship from the Royal Society and the British Academy.', '1211.1680-2-44-2': 'YW is supported by the Center for Biomedical Imaging (CIBM) of the Geneva and Lausanne Universities, EPFL and the Leenaards and Louis-Jeantet foundations.'}","[['1211.1680-1-20-0', '1211.1680-2-22-0'], ['1211.1680-1-20-1', '1211.1680-2-22-1'], ['1211.1680-1-20-3', '1211.1680-2-22-3'], ['1211.1680-1-5-0', '1211.1680-2-5-0'], ['1211.1680-1-5-2', '1211.1680-2-5-2'], ['1211.1680-1-5-3', '1211.1680-2-5-3'], ['1211.1680-1-28-6', '1211.1680-2-31-4'], ['1211.1680-1-28-7', '1211.1680-2-31-5'], ['1211.1680-1-28-8', '1211.1680-2-31-6'], ['1211.1680-1-16-0', '1211.1680-2-18-0'], ['1211.1680-1-16-1', '1211.1680-2-18-1'], ['1211.1680-1-36-0', 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quant-ph-0211159,"{'quant-ph-0211159-1-0-0': 'We investigate the transformation of entangled spin states of two massive particles under Lorentz transformations.', 'quant-ph-0211159-1-0-1': 'We observe the transfer of momentum entanglement into the spin state so that the spin entanglement could be preserved.', 'quant-ph-0211159-1-0-2': 'We also find a decoherence-free subspace of states that are exempt from the decoherence induced by Lorentz transformations.', 'quant-ph-0211159-1-0-3': 'Further we construct Lorentz invariant subspaces of states and operators.', 'quant-ph-0211159-1-0-4': 'An application of these subspaces is to the relativistic invariant protocol for quantum information processing, with which the non-relativistic quantum information theory could be invariantly applied to relativistic situations.', 'quant-ph-0211159-1-1-0': 'The thermodynamics of moving objects in the theory of relativity has been an intriguing problem for decades [CITATION].', 'quant-ph-0211159-1-1-1': 'It has been shown that probability distributions can depend on the inertial frames, and thus the entropy and information may also change if viewed in different inertial frames [CITATION].', 'quant-ph-0211159-1-1-2': 'Recently, the effect of Lorentz transformations on quantum states, quantum entanglement and quantum information has drawn particular interests [CITATION].', 'quant-ph-0211159-1-1-3': 'The investigations show that the spin entropy is not a relativistic scalar and has no invariant meanings [CITATION], and the entanglement between the spins may loss by transferring into momenta [CITATION].', 'quant-ph-0211159-1-1-4': 'These observations may have important consequence for the quantum information theory, since on the one hand entanglement of bipartite quantum states forms a vital resource for many quantum information processing protocols [CITATION], including quantum teleportation [CITATION], cryptography [CITATION] and computation [CITATION].', 'quant-ph-0211159-1-1-5': 'On the other hand, the relativistic quantum information theory may become a necessary theory in the near future, with possible applications to quantum clock synchronization [CITATION] and quantum-enhanced global positioning [CITATION].', 'quant-ph-0211159-1-2-0': 'The effect of Lorentz transformations on the spin states, as observed in Ref. [CITATION], could be regard as a decoherence effect.', 'quant-ph-0211159-1-2-1': 'In this paper, we show that, for spin states of two spin-1/2 mass particles, if the momenta of the two particles are entangled, the decoherence of spin state could be partially prevented, and the spin entanglement could be preserved.', 'quant-ph-0211159-1-2-2': 'This indicates that the momentum entanglement could transfer into the spins.', 'quant-ph-0211159-1-2-3': 'We also find a class of states whose spin entanglement remains invariant under Lorentz transformations.', 'quant-ph-0211159-1-2-4': 'These states in fact constitute a decoherence-free subspace under Lorentz transformations.', 'quant-ph-0211159-1-2-5': 'In addition, these states can be easily prepared in experiments.', 'quant-ph-0211159-1-2-6': 'Further we construct Lorentz invariant subspaces of states and operators, and find the application to propose a relativistic invariant protocol for quantum information processing.', 'quant-ph-0211159-1-2-7': 'With this protocol, the non-relativistic quantum information theory could be invariantly applied to relativistic situations.', 'quant-ph-0211159-1-3-0': 'The general wavefunction for two massive (mass [MATH]) spin-1/2 particles can be found in Ref. [CITATION].', 'quant-ph-0211159-1-3-1': 'In that paper, the authors set the momentum to be uniformly distributed in all directions in the space.', 'quant-ph-0211159-1-3-2': 'However in practice one needs to maintain the particles in a desired direction.', 'quant-ph-0211159-1-3-3': 'Considering this fact, we shall restrict ourselves to focus on the state that the two particles have a deterministic momentum direction.', 'quant-ph-0211159-1-3-4': 'Keeping this consideration in mind, we write the state, in the momentum representation, as the following form.', 'quant-ph-0211159-1-3-5': '[EQUATION] where [MATH] and [MATH] represent the momentum for the first and the second particle, and are along certain direction deterministically.', 'quant-ph-0211159-1-3-6': 'The spin part of this state is the Bell state [EQUATION] where [EQUATION]', 'quant-ph-0211159-1-3-7': 'Note that in this state, the spin and momentum are separable.', 'quant-ph-0211159-1-3-8': 'For the momentum distribution, we use a entangled Gaussian with width [MATH], as follows.', 'quant-ph-0211159-1-3-9': '[EQUATION] where [MATH] is the normalization.', 'quant-ph-0211159-1-3-10': 'In equation ([REF]), [MATH] could be viewed as a measure of the momentum entanglement.', 'quant-ph-0211159-1-3-11': 'when [MATH], it turns back to the separable case as the same as that in Ref. [CITATION].', 'quant-ph-0211159-1-3-12': 'However in the limit [MATH], we see that [EQUATION] where [MATH] is the normalization.', 'quant-ph-0211159-1-3-13': 'Equation ([REF]) in fact indicates a perfect correlation of the momenta, with width [MATH].', 'quant-ph-0211159-1-4-0': 'We now investigate how the state ([REF]) varies under Lorentz transformation.', 'quant-ph-0211159-1-4-1': 'Since any Lorentz transformation can be written as a rotation followed by a boost [CITATION], and tracing over the momentum after a rotation will not change the entanglement of spins [CITATION], we can look only at pure boosts and without loss of generality we may choose boosts in the [MATH]-direction.', 'quant-ph-0211159-1-4-2': 'Writing the momentum [MATH]-vector in polar coordinates as [EQUATION]', 'quant-ph-0211159-1-4-3': 'Let [MATH] be the boosts along the [MATH]-direction as defined in Ref. [CITATION], where [MATH] is the rapidity of the boost.', 'quant-ph-0211159-1-4-4': 'With equation ([REF]), we can obtain the Wigner rotation matrix [CITATION] [EQUATION] where [EQUATION] and [MATH] and [MATH].', 'quant-ph-0211159-1-4-5': 'And the similar is for the second particle with momentum [MATH].', 'quant-ph-0211159-1-4-6': 'The boosted state of the system is [EQUATION] and we have [EQUATION]', 'quant-ph-0211159-1-4-7': 'From equation ([REF]), we can see if [MATH] is along the same direction as the boost [MATH], i.e. [MATH], we obtain [MATH] and hence [MATH].', 'quant-ph-0211159-1-4-8': 'The boost does not affect the spin state.', 'quant-ph-0211159-1-4-9': 'Therefore we shall only consider the case that [MATH] and [MATH] are both perpendicular to the boost, i.e. [MATH] and [MATH] lies in the [MATH]-[MATH] plane.', 'quant-ph-0211159-1-4-10': 'For further convenience we assume [MATH] and [MATH] are parallel, and without loss of generality set to be along the [MATH]-axis, i.e. [MATH] and [MATH].', 'quant-ph-0211159-1-4-11': 'By integrating over the momenta, we obtain the reduced spin density matrix as [EQUATION] where d[MATH] and d[MATH] are the Lorentz-invariant momentum integration measures.', 'quant-ph-0211159-1-4-12': 'In our case the momenta are along the [MATH]-axis, so we have [EQUATION] and similar is d[MATH].', 'quant-ph-0211159-1-4-13': ""After obtaining the reduced density matrix, we can obtain the spin entanglement between the two particles by calculating the Wootters' concurrence [CITATION]."", 'quant-ph-0211159-1-4-14': 'We denote the concurrence by [MATH].', 'quant-ph-0211159-1-4-15': 'The change in concurrence depends only on [MATH], [MATH] and [MATH].', 'quant-ph-0211159-1-5-0': 'Fig. [REF] shows the dependence of concurrence on rapidity [MATH], for [MATH], [MATH] and [MATH].', 'quant-ph-0211159-1-5-1': 'It is interesting to see that as [MATH] increases, the concurrence increases.', 'quant-ph-0211159-1-5-2': 'However it is surprising that for the limit [MATH], i.e. the momentum of the two particles are perfectly correlated, the concurrence remains [MATH], independent of [MATH].', 'quant-ph-0211159-1-5-3': 'In fact, as long as [MATH], the concurrence [MATH] and the reduced density matrix [MATH] are both independent of [MATH] and [MATH].', 'quant-ph-0211159-1-5-4': 'The state is invariant under the Lorentz transformation.', 'quant-ph-0211159-1-5-5': 'The dependence of concurrence on momentum entanglement can be explained as follows.', 'quant-ph-0211159-1-5-6': 'Since under Lorentz transformation, the spin entanglement could transfer into momentum state, the momentum entanglement could potentially transfer into spin state at the same time.', 'quant-ph-0211159-1-5-7': 'The more entangled is the momentum state, the more momentum entanglement is transferred into spin state.', 'quant-ph-0211159-1-5-8': 'Therefore the spin entanglement is preserved by the initial entanglement between the momenta.', 'quant-ph-0211159-1-5-9': 'As the momenta is perfectly correlated (the momentum state not necessarily be maximally entangled), the entanglement transfer from spins to momenta and that from momenta to spins cancel each other, and the concurrence of spins remains unchanged under any boosts.', 'quant-ph-0211159-1-6-0': 'The above observation that the concurrence remains unchanged for [MATH] can be generalized, without the assumption about the direction ([MATH] and [MATH]) along which the particles move.', 'quant-ph-0211159-1-6-1': 'Directly from equation ([REF]), we see that if the momentum distribution can be written as the following form, [EQUATION] where [MATH] can be any distribution.', 'quant-ph-0211159-1-6-2': 'The boosted state then can be written as [EQUATION] where [MATH] due to the unitarity of [MATH].', 'quant-ph-0211159-1-6-3': 'Here we shall note that, in equation ([REF]) as well as in the remaining part of the paper, all the Dirac [MATH]-functions should be regard as limits of certain analytical functions, such as equation ([REF]) is the limit of equation ([REF]) with [MATH].', 'quant-ph-0211159-1-6-4': 'In addition, the [MATH]-function does not mean any singularity of the wavefunction, but means that some quantity is conservative in the system.', 'quant-ph-0211159-1-6-5': 'For example in equation ([REF]), [MATH] means that the total momentum is conservative.', 'quant-ph-0211159-1-6-6': 'The wavefunction could be commonly normalized, and thus represent real physical state, as long as [MATH] is physical and normalized, since [MATH]d[MATH]d[MATH]d[MATH].', 'quant-ph-0211159-1-7-0': 'Equation ([REF]) indicates that a state is invariant under Lorentz boosts, as long as it satisfies the following conditions: (i) [MATH] and [MATH] are equal (perfect correlation of the magnitude of the momenta); (ii) [MATH]; (iii) [MATH].', 'quant-ph-0211159-1-7-1': 'However the choice of [MATH]-axis is arbitrary, change of the choice of [MATH]-axis is equivalent to the change of the phase factors of the bases ([MATH] and [MATH]).', 'quant-ph-0211159-1-7-2': 'Therefore the states in equation ([REF]) with momentum distribution in equation ([REF]) constitute a decoherence subspace under Lorentz transformations.', 'quant-ph-0211159-1-7-3': 'The spin entanglement of such states is invariant under Lorentz transformations, thus could be useful in relativistic quantum information processing.', 'quant-ph-0211159-1-8-0': 'A special case of particular interest of equation ([REF]) is that the two particles are moving in opposite deterministic direction in the plane perpendicular to the boost.', 'quant-ph-0211159-1-8-1': 'In this situation, we have [EQUATION] with the direction of [MATH], [MATH] being deterministic and [MATH] being any distribution.', 'quant-ph-0211159-1-8-2': 'Note that the distribution [MATH] in equation ([REF]) is a very natural result if the state is prepared by a process where the total momentum is conservative and is initially zero.', 'quant-ph-0211159-1-8-3': 'Since [MATH] is arbitrary, we actually obtain a very practical and simple method to preserve maximal spin entanglement under Lorentz transformations.', 'quant-ph-0211159-1-8-4': 'The invariance of entanglement is naturally guaranteed by the momentum conservation law provided that the direction of [MATH], [MATH] are deterministic and the initial total momentum is zero.', 'quant-ph-0211159-1-8-5': 'Another advantage of the method is that it does not require any entanglement between the spin and momentum of the particles.', 'quant-ph-0211159-1-9-0': 'We now apply the above observation to propose a relativistic invariant protocol for quantum information processing.', 'quant-ph-0211159-1-9-1': 'We shall turn back to the situation that the two particles are moving together in the same deterministic direction.', 'quant-ph-0211159-1-9-2': 'Without loss of generality, we can set the boost [MATH] to be along the [MATH]-axis and set the motion of the two particles in the [MATH]-[MATH] plane, therefore we have [MATH] and [MATH].', 'quant-ph-0211159-1-9-3': 'Denote the rest inertial frame by [MATH] and the boosted inertial frame by [MATH].', 'quant-ph-0211159-1-9-4': 'If the momentum distribution in frame [MATH] takes the following form (an special instance of equation ([REF])).', 'quant-ph-0211159-1-9-5': '[EQUATION] with [MATH] be any arbitrary distribution.', 'quant-ph-0211159-1-9-6': 'It can be easily verify that, besides the state [MATH], the state [MATH]) is also invariant under any boost.', 'quant-ph-0211159-1-9-7': 'However the other two Bell states, i.e. [MATH] and [MATH] are not invariant [CITATION].', 'quant-ph-0211159-1-9-8': 'If we define [EQUATION]', 'quant-ph-0211159-1-9-9': 'It is obvious that [MATH] are orthonormal.', 'quant-ph-0211159-1-9-10': 'Therefore [MATH] can be used to represent the two bases of a qubit.', 'quant-ph-0211159-1-9-11': 'Under the Lorentz transformation, we have [EQUATION]', 'quant-ph-0211159-1-9-12': 'For any state [EQUATION] with [MATH] and [MATH], we have [EQUATION]', 'quant-ph-0211159-1-9-13': 'Therefore the states in equation ([REF]) constitute a Lorentz invariant subspace.', 'quant-ph-0211159-1-9-14': 'The multi-qubit states can be defined analogously.', 'quant-ph-0211159-1-9-15': 'Note that in a multi-qubit state, the momentum distributions of individual qubits are not necessary to be the same.', 'quant-ph-0211159-1-9-16': 'We can also construct Lorentz invariant subspace of operators that is consist of all the operators acting on a single qubit having the following form.', 'quant-ph-0211159-1-9-17': '[EQUATION]', 'quant-ph-0211159-1-9-18': 'The operators acting on multiple qubits can be defined analogously.', 'quant-ph-0211159-1-9-19': 'It is obvious that under Lorentz transformation, we have [EQUATION]', 'quant-ph-0211159-1-9-20': 'It is interesting to note that these operators reduce to common operators on the spins for the states in equation ([REF]), and thus could be invariantly and conveniently performed in any inertial frames.', 'quant-ph-0211159-1-9-21': 'Within these subspaces, the entropy, entanglement and measurement results all have invariant meanings in different inertial frames, despite that for a single particle and some other situations these quantities may have no invariant meanings [CITATION].', 'quant-ph-0211159-1-9-22': 'This invariance guarantees that the non-relativistic quantum information theory could be invariantly applied to relativistic situation.', 'quant-ph-0211159-1-9-23': 'As to the experimental aspect, the momentum distribution in equation ([REF]) could be realized by two simple steps: (i) Prepare the state in which the two particles are moving in opposite directions as given in equation ([REF]), in which only the momentum conservation is required; (ii) Deflect the two particles into the desired direction, without disturbing the spins.', 'quant-ph-0211159-1-10-0': 'A possible extension of the present work is that to the cases of massless particles, particularly to spin-1 massless particles, such as photons, since the current experiments for quantum information processing are mostly based on polarized photons.', 'quant-ph-0211159-1-10-1': 'Interesting and important results with possible applications may be found.', 'quant-ph-0211159-1-10-2': 'We believe that similar decoherence-free subspace for entangled two photons could be found, and the Lorentz invariant subspaces of states and operators could also be constructed analogously.', 'quant-ph-0211159-1-11-0': 'In conclusion, we have investigated the transformation of entangled spin states of two spin-1/2 particles under Lorentz transformations, with the presence of momentum entanglement between the two particles.', 'quant-ph-0211159-1-11-1': 'We show that under Lorentz transformations, the loss of spin entanglement could be compensated by the transfer of momentum entanglement into the spins, without the initial entanglement between spin and momentum.', 'quant-ph-0211159-1-11-2': 'The more entangled is the momentum state, the more entanglement is transferred into spin state.', 'quant-ph-0211159-1-11-3': 'We also find a class of states whose spin entanglement is preserved invariant under Lorentz transformations.', 'quant-ph-0211159-1-11-4': 'These states in fact constitute a decoherence-free subspace under Lorentz transformations.', 'quant-ph-0211159-1-11-5': 'A case of particular interest and importance is that when the process in which the state is prepared is momentum conservative, in which the invariance of the spin entanglement is simply guaranteed by momentum conservation.', 'quant-ph-0211159-1-11-6': 'This fact provides a natural and simple method for experiments to prepare bipartite states whose entanglement is invariant under Lorentz transformations.', 'quant-ph-0211159-1-11-7': 'We also construct Lorentz invariant subspaces of states and operators.', 'quant-ph-0211159-1-11-8': 'Within these subspaces, we propose a relativistic invariant protocol for quantum information processing, with which the non-relativistic quantum information theory could be invariantly applied to relativistic situations.', 'quant-ph-0211159-1-12-0': 'We thank Dr. Z.B. Chen and Dr. Y.S. Xia for helpful discussion.', 'quant-ph-0211159-1-12-1': 'This work was supported by the National Nature Science Foundation of China (Grants.', 'quant-ph-0211159-1-12-2': 'No. 10075041) and Chinese Academy of Science.'}","{'quant-ph-0211159-2-0-0': 'The entanglement between spins of a pair of particles may change because the spin and momentum become mixed when viewed by a moving observer [R.M. Gingrich and C. Adami, Phys.', 'quant-ph-0211159-2-0-1': 'Rev. Lett.', 'quant-ph-0211159-2-0-2': '89, 270402 (2002)].', 'quant-ph-0211159-2-0-3': 'In this paper, it is shown that, if the momenta are appropriately entangled, the entanglement between the spins of the Bell states can remain maximal when viewed by any moving observer.', 'quant-ph-0211159-2-0-4': 'Further, we suggest a relativistic-invariant protocol for quantum communication, with which the non-relativistic quantum information theory could be invariantly applied to relativistic situations.', 'quant-ph-0211159-2-1-0': '# Introduction', 'quant-ph-0211159-2-2-0': 'The relativistic thermodynamics has been an intriguing problem for decades [CITATION].', 'quant-ph-0211159-2-2-1': 'It has been shown that probability distributions can depend on the frames, and thus the entropy and information may change if viewed from different frames [CITATION].', 'quant-ph-0211159-2-2-2': 'Recently, the effect of Lorentz boosts on quantum states, quantum entanglement and quantum information has drawn particular interests [CITATION].', 'quant-ph-0211159-2-2-3': 'The relativistic quantum information theory may become necessary in the near future, with possible applications to quantum clock synchronization [CITATION] and quantum-enhanced global positioning [CITATION].', 'quant-ph-0211159-2-3-0': 'Entanglement of quantum systems forms a vital resource for many quantum information processing protocols [CITATION], including quantum teleportation [CITATION], cryptography [CITATION] and computation [CITATION].', 'quant-ph-0211159-2-3-1': 'However, it has shown that Lorentz boosts can affect the marginal entropy of a single quantum spin [CITATION], and a fully entangled spin-1/2 system may lose entanglement when observed by a Lorentz-boosted observer [CITATION].', 'quant-ph-0211159-2-3-2': 'Particularly, fully entangled spin states in the rest frame will most likely decohere due to the mixing with momentum if viewed from a moving frame, depending on the initial momentum wave function [CITATION].', 'quant-ph-0211159-2-3-3': 'The entanglement between two systems depends on the frame in which this entanglement is measured.', 'quant-ph-0211159-2-3-4': 'These effects may have important consequences for quantum communication, especially when the communicating parties are in relative movement.', 'quant-ph-0211159-2-4-0': 'In this paper, we show that for a pair of spin-1/2 massive particles, if the momenta are appropriately entangled, the entanglement between the spins can remain the same as in the rest frame when viewed from any Lorentz-transformed frame.', 'quant-ph-0211159-2-4-1': 'We also find a set of states, of which the marginal entropy, entanglement and measurement results of the spins are independent of the frames from which they are observed.', 'quant-ph-0211159-2-4-2': 'Further, we suggest a relativistic-invariant representation of the quantum bit (qubit), and suggest a relativistic-invariant protocol for quantum communication, with which the non-relativistic quantum information theory could be invariantly applied to relativistic situations.', 'quant-ph-0211159-2-4-3': 'Though, in this paper, we restrict ourselves to spin-1/2 cases, the generalization to larger spins could be done analogously.', 'quant-ph-0211159-2-4-4': 'Particularly, the generalization to spin-1 massless particles, such as photons [CITATION], may be of special interests since current experiments for quantum communications are mostly based on photons.', 'quant-ph-0211159-2-5-0': '# Entanglement between the spins, with the presence of momentum entanglement', 'quant-ph-0211159-2-6-0': 'We start by investigating the bipartite state that, in the momentum representation, has the following form viewed from the rest frame, [EQUATION] where [MATH] and [MATH] are the momenta for the first and second particles, respectively (For review of the definition of the momentum eigenstates for massive particles with spin and the transformation under Lorentz boosts, one may refer to Refs. [CITATION]).', 'quant-ph-0211159-2-6-1': 'The spin part of the state is the singlet Bell state [EQUATION] where [MATH], [MATH], with [EQUATION]', 'quant-ph-0211159-2-6-2': 'The momentum distribution [MATH] is normalized according to [EQUATION] where [MATH]) is the Lorentz-invariant momentum integration measures given by (We use natural units: [MATH].)', 'quant-ph-0211159-2-6-3': '[EQUATION]', 'quant-ph-0211159-2-6-4': 'Note that there is no entanglement between the spin and the momentum parts of [MATH].', 'quant-ph-0211159-2-6-5': 'The spins are maximally entangled, while the entanglement between momenta depends on [MATH].', 'quant-ph-0211159-2-6-6': 'In what follows, we use [MATH] to represent the momentum [MATH]-vector as in Eq. ([REF]) unless it is ambiguous.', 'quant-ph-0211159-2-7-0': 'To an observer in a frame Lorentz transformed by [MATH], the state [MATH] appears to be transformed by [MATH].', 'quant-ph-0211159-2-7-1': 'Therefore the state viewed by this observer appears to be [EQUATION] where [MATH] represents the unitary transformation induced by the Lorentz transformation.', 'quant-ph-0211159-2-7-2': 'Here, for compactness of notation, we define [MATH] as the spin-1/2 representation of the Wigner rotation [MATH] [CITATION].', 'quant-ph-0211159-2-7-3': 'Because [MATH] differs from [MATH] by only local unitary transformations, the entanglement will not change provided we do not trace out a part of the state.', 'quant-ph-0211159-2-7-4': 'However, in looking at the entanglement between the spins, tracing out over the momentum degrees of freedom is implied.', 'quant-ph-0211159-2-7-5': 'In [MATH] the spin and momentum may appear to be entangled, therefore the entanglement between the spins may change when viewed by the Lorentz-transformed observer.', 'quant-ph-0211159-2-7-6': ""By writing [MATH] as a density matrix and tracing over the momentum degrees of freedom, the entanglement between the spins (viewed by the Lorentz-transformed observer) could be obtained by calculating the Wootters' concurrence [CITATION] of the reduced density matrix for spins."", 'quant-ph-0211159-2-8-0': 'Any Lorentz transformation could be written as a rotation followed by a boost [CITATION], and tracing over the momentum after a rotation will not change the spin concurrence [CITATION], therefore we can look only at pure boosts.', 'quant-ph-0211159-2-8-1': 'Without loss of generality we may choose boosts in the [MATH]-direction and write the momentum 4-vector in polar coordinates as [EQUATION] with [MATH], [MATH] and [MATH].', 'quant-ph-0211159-2-8-2': 'Let [MATH] be the boost along the [MATH]-direction (as defined in Ref. [CITATION]), where [MATH] is the rapidity of the boost and let [MATH].', 'quant-ph-0211159-2-8-3': 'With Eq. ([REF]), we obtain [EQUATION] where [EQUATION] and [MATH].', 'quant-ph-0211159-2-8-4': 'The similar is for the second particle with momentum [MATH].', 'quant-ph-0211159-2-8-5': 'Substituting Eq. ([REF]) into Eq. ([REF]), we obtain the state viewed by the Lorentz-boosted observer as [EQUATION]', 'quant-ph-0211159-2-8-6': 'At the present stage, we use an entangled Gaussian with width [MATH] for the momentum distribution, as follows, [EQUATION] where [MATH] and [MATH] is the normalization.', 'quant-ph-0211159-2-8-7': 'In Eq. ([REF]), for a given [MATH], [MATH] could be reasonably regard as a measure of the entanglement between momenta.', 'quant-ph-0211159-2-8-8': 'When [MATH], the momentum part of the state is separable, i.e. the momentum entanglement is zero.', 'quant-ph-0211159-2-8-9': 'However at the limit [MATH], we have [EQUATION] where [MATH] is the normalization.', 'quant-ph-0211159-2-8-10': 'Eq. ([REF]) indicates a perfect correlation between the momenta.', 'quant-ph-0211159-2-8-11': 'Note that in Eq. ([REF]) the momenta are not necessarily maximally entangled.', 'quant-ph-0211159-2-9-0': 'By integrating over the momenta, we obtain the reduced density matrix, viewed by the Lorentz-boosted observer, as [EQUATION]', 'quant-ph-0211159-2-9-1': ""The entanglement between the spins viewed by the Lorentz-boosted observer is obtained by calculating the Wootters' concurrence [CITATION], denoted as [MATH]."", 'quant-ph-0211159-2-9-2': 'The change in the Lorentz-transformed concurrence [MATH] depends on [MATH], [MATH] and [MATH].', 'quant-ph-0211159-2-9-3': 'Fig. [REF] shows the concurrence as a function of rapidity [MATH], for different values of [MATH] and [MATH].', 'quant-ph-0211159-2-9-4': 'Similar to Ref. [CITATION], the decrease from the maximum value ([MATH] for Bell states) documents the boost-induced decoherence of the spin entanglement [CITATION].', 'quant-ph-0211159-2-9-5': 'However, it is interesting to see that for fixed [MATH] and [MATH], the concurrence decreases less for non-zero [MATH].', 'quant-ph-0211159-2-9-6': 'Further, it is surprising that at the limit [MATH], the concurrence does not decrease, no matter what [MATH] and [MATH] are.', 'quant-ph-0211159-2-9-7': 'Indeed, at the limit [MATH], not only the concurrence but also the reduced density matrix for spins are independent of [MATH] and [MATH].', 'quant-ph-0211159-2-10-0': 'One possible explanation for that the concurrence decreases less with the presence of momentum entanglement is as follows.', 'quant-ph-0211159-2-10-1': 'Boosting the state, we move some of the spin entanglement to the momentum [CITATION], however the momentum entanglement appears to be moved to spins simultaneously.', 'quant-ph-0211159-2-10-2': 'The transfer of momentum entanglement to spins hence compensates the decrease of spin entanglement, and the Lorentz-transformed concurrence decreases less.', 'quant-ph-0211159-2-10-3': 'When the momenta of the two particles are perfectly correlated, even though may be not maximally entangled, the transfer of entanglement from momenta to spins happens to fully compensate the decrease of spin entanglement, so the entanglement of the reduced spin state remains maximal when viewed by any Lorentz-boosted observer.', 'quant-ph-0211159-2-10-4': 'Particularly, for the singlet Bell states with momentum distribution given in Eq. ([REF]) (generally in Eq. ([REF]) in the following), the Lorentz boost does not affect the reduced spin state, only transforms [MATH] and [MATH].', 'quant-ph-0211159-2-10-5': 'The momentum and spin parts of such states always appear to be separate viewed from any Lorentz-boosted frame.', 'quant-ph-0211159-2-11-0': 'That the spin concurrence remains maximal at the limit [MATH] when viewed from any Lorentz-boosted frame can be generalized, without the assumption that the momentum distribution is an entangled Gaussian given in Eq. ([REF]).', 'quant-ph-0211159-2-11-1': 'Directly from Eq. ([REF]), we see that if the momentum distribution takes the following form, [EQUATION] where [MATH] can be any distribution as long as [MATH] is normalized according to Eq. ([REF]), the boosted state could be written as [EQUATION] where [MATH] due to the unitarity of [MATH].', 'quant-ph-0211159-2-11-2': 'For the singlet Bell state shown in Eq. ([REF]) with momentum distribution given in Eq. ([REF]), the reduced density matrix remains the same as in the rest frame when viewed by any Lorentz-boosted observer.', 'quant-ph-0211159-2-11-3': 'Thus the entanglement between the spins remains maximal if viewed from any Lorentz-transformed frame.', 'quant-ph-0211159-2-11-4': 'Indeed, the following four Bell states all have invariant reduced density matrices for spins viewed from any frame Lorentz boosted along the [MATH]-axis.', 'quant-ph-0211159-2-11-5': '[EQUATION] with [MATH] being arbitrary as long as [MATH] is normalized as in Eq. ([REF]).', 'quant-ph-0211159-2-11-6': 'Because Eq. ([REF]) is a simultaneous instance of the momentum distributions of the states in both Eq. ([REF]) and Eq. ([REF]), both [MATH] and [MATH] have invariant reduced density matrices for spins when viewed from any Lorentz-boosted frames.', 'quant-ph-0211159-2-11-7': 'This enables us to use these two states as the orthonormal bases, namely [MATH] and [MATH], of a qubit, as follows.', 'quant-ph-0211159-2-11-8': '[EQUATION]', 'quant-ph-0211159-2-11-9': 'The operators acting upon multiple qubits can be obtained analogously.', 'quant-ph-0211159-2-11-10': 'We refer the operators as in Eq. ([REF]) to be Lorentz-invariant in the sense that, if we look only at spins, the action of the operator on the state [MATH] with [MATH]) remains the same when viewed in any Lorentz-boosted frame.', 'quant-ph-0211159-2-11-11': 'Within the set of these Lorentz-invariant qubits and operators, the entropy, entanglement and measurement results all have invariant meanings, despite that for a single quantum spin and some other situations these quantities may have no invariant meanings in different frames [CITATION].', 'quant-ph-0211159-2-11-12': 'Therefore it is guaranteed that, using such states and operators, the non-relativistic quantum information theory could be invariantly applied to relativistic situation.', 'quant-ph-0211159-2-12-0': '# Conclusion', 'quant-ph-0211159-2-13-0': 'As observed in Ref. [CITATION], because Lorentz boosts entangle the spin and momentum degrees of freedom, entanglement between the spins may change if viewed from a moving frame.', 'quant-ph-0211159-2-13-1': 'Especially, maximally entangled spin states will most likely decohere due to the mixing with momentum degrees of freedom, depending on the initial momentum wave function [CITATION].', 'quant-ph-0211159-2-13-2': 'In this paper, we investigate the quantum entanglement between the spins of a pair of spin-1/2 massive particles in moving frames, for the case that the momenta of the particles are entangled.', 'quant-ph-0211159-2-13-3': 'We show that if the momenta of the pair are appropriately entangled, the entanglement between the spins of the Bell states remains maximal when viewed from any Lorentz-transformed frame.', 'quant-ph-0211159-2-13-4': 'Further, we suggest a relativistic-invariant protocol for quantum communication, with which the non-relativistic quantum information theory could be invariantly applied to relativistic situations.', 'quant-ph-0211159-2-13-5': 'Though the investigations are based on spin-1/2 particles, we believe the similar results for larger spins could be obtained analogously.', 'quant-ph-0211159-2-13-6': 'Especially, we hope our work would help to find a relativistic-invariant protocol for quantum information processing based on photons, i.e. the case of massless spin-1 particles.', 'quant-ph-0211159-2-14-0': 'We would like to thank M.J. Shi, Z.B. Chen, and Y.S. Xia for the fruitful discussions.', 'quant-ph-0211159-2-14-1': 'We also acknowledge the kind help on the subject and valuable suggestions from R.M. Gingrich, D.R. Terno, and A. Peres.', 'quant-ph-0211159-2-14-2': 'This work was supported by the Nature Science Foundation of China (Grant No. 10075041), the National Fundamental Research Program (Grant No. 2001CB309300), and the ASTAR Grant No. 012-104-0040.'}","[['quant-ph-0211159-1-1-1', 'quant-ph-0211159-2-2-1'], ['quant-ph-0211159-1-1-2', 'quant-ph-0211159-2-2-2'], ['quant-ph-0211159-1-1-5', 'quant-ph-0211159-2-2-3'], ['quant-ph-0211159-1-4-5', 'quant-ph-0211159-2-8-4'], ['quant-ph-0211159-1-3-6', 'quant-ph-0211159-2-6-1'], ['quant-ph-0211159-1-3-7', 'quant-ph-0211159-2-6-4'], ['quant-ph-0211159-1-5-0', 'quant-ph-0211159-2-9-3'], ['quant-ph-0211159-1-5-1', 'quant-ph-0211159-2-9-5'], ['quant-ph-0211159-1-5-3', 'quant-ph-0211159-2-9-7'], ['quant-ph-0211159-1-1-0', 'quant-ph-0211159-2-2-0'], ['quant-ph-0211159-1-2-1', 'quant-ph-0211159-2-4-0'], ['quant-ph-0211159-1-2-1', 'quant-ph-0211159-2-4-3'], ['quant-ph-0211159-1-2-3', 'quant-ph-0211159-2-4-1'], ['quant-ph-0211159-1-2-7', 'quant-ph-0211159-2-4-2'], ['quant-ph-0211159-1-12-1', 'quant-ph-0211159-2-14-2'], ['quant-ph-0211159-1-9-5', 'quant-ph-0211159-2-11-5'], ['quant-ph-0211159-1-9-14', 'quant-ph-0211159-2-11-9'], ['quant-ph-0211159-1-9-18', 'quant-ph-0211159-2-11-9'], ['quant-ph-0211159-1-9-21', 'quant-ph-0211159-2-11-11'], ['quant-ph-0211159-1-9-22', 'quant-ph-0211159-2-11-12'], ['quant-ph-0211159-1-4-1', 'quant-ph-0211159-2-8-0'], ['quant-ph-0211159-1-4-3', 'quant-ph-0211159-2-8-2'], ['quant-ph-0211159-1-4-4', 'quant-ph-0211159-2-8-3'], ['quant-ph-0211159-1-11-8', 'quant-ph-0211159-2-0-4'], ['quant-ph-0211159-1-0-4', 'quant-ph-0211159-2-0-4'], ['quant-ph-0211159-1-1-4', 'quant-ph-0211159-2-3-0'], ['quant-ph-0211159-1-11-8', 'quant-ph-0211159-2-13-4'], ['quant-ph-0211159-1-0-4', 'quant-ph-0211159-2-13-4']]",[],"[['quant-ph-0211159-1-1-1', 'quant-ph-0211159-2-2-1'], ['quant-ph-0211159-1-1-2', 'quant-ph-0211159-2-2-2'], ['quant-ph-0211159-1-1-5', 'quant-ph-0211159-2-2-3'], ['quant-ph-0211159-1-4-5', 'quant-ph-0211159-2-8-4']]",[],"[['quant-ph-0211159-1-3-6', 'quant-ph-0211159-2-6-1'], ['quant-ph-0211159-1-3-7', 'quant-ph-0211159-2-6-4'], ['quant-ph-0211159-1-5-0', 'quant-ph-0211159-2-9-3'], ['quant-ph-0211159-1-5-1', 'quant-ph-0211159-2-9-5'], ['quant-ph-0211159-1-5-3', 'quant-ph-0211159-2-9-7'], ['quant-ph-0211159-1-1-0', 'quant-ph-0211159-2-2-0'], ['quant-ph-0211159-1-2-1', 'quant-ph-0211159-2-4-0'], ['quant-ph-0211159-1-2-1', 'quant-ph-0211159-2-4-3'], ['quant-ph-0211159-1-2-3', 'quant-ph-0211159-2-4-1'], ['quant-ph-0211159-1-2-7', 'quant-ph-0211159-2-4-2'], ['quant-ph-0211159-1-12-1', 'quant-ph-0211159-2-14-2'], ['quant-ph-0211159-1-9-5', 'quant-ph-0211159-2-11-5'], ['quant-ph-0211159-1-9-14', 'quant-ph-0211159-2-11-9'], ['quant-ph-0211159-1-9-18', 'quant-ph-0211159-2-11-9'], ['quant-ph-0211159-1-9-21', 'quant-ph-0211159-2-11-11'], ['quant-ph-0211159-1-9-22', 'quant-ph-0211159-2-11-12'], ['quant-ph-0211159-1-4-1', 'quant-ph-0211159-2-8-0'], ['quant-ph-0211159-1-4-3', 'quant-ph-0211159-2-8-2'], ['quant-ph-0211159-1-4-4', 'quant-ph-0211159-2-8-3']]","[['quant-ph-0211159-1-11-8', 'quant-ph-0211159-2-0-4'], ['quant-ph-0211159-1-0-4', 'quant-ph-0211159-2-0-4'], ['quant-ph-0211159-1-1-4', 'quant-ph-0211159-2-3-0'], ['quant-ph-0211159-1-11-8', 'quant-ph-0211159-2-13-4'], ['quant-ph-0211159-1-0-4', 'quant-ph-0211159-2-13-4']]","['quant-ph-0211159-1-9-17', 'quant-ph-0211159-2-0-1', 'quant-ph-0211159-2-0-2', 'quant-ph-0211159-2-6-3', 'quant-ph-0211159-2-11-8']","{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}",https://arxiv.org/abs/quant-ph/0211159,,,
1012.3738,"{'1012.3738-1-0-0': 'For every [MATH]-group of order [MATH] with the derived subgroup of order [MATH], Rocco in [CITATION] has shown that the order of tensor square of [MATH] is at most [MATH].', '1012.3738-1-0-1': 'In the present paper not only we improve his bound for non-abelian [MATH]-groups but also we describe the structure of all non-abelian [MATH]-groups when the bound is attained for a special case.', '1012.3738-1-0-2': 'Moreover, our results give as well an upper bound for the order of [MATH].', '1012.3738-1-1-0': '# Introduction and Preliminaries', '1012.3738-1-2-0': 'The tensor square [MATH] of a group [MATH] is a group generated by the symbols [MATH] subject to the relations [EQUATION] for all [MATH] where [MATH].', '1012.3738-1-2-1': 'The non abelian tensor square is a special case of non abelian tensor product, which was introduced by R. Brown and J.-L. Loday in [CITATION].', '1012.3738-1-3-0': 'There exists a homomorphism of groups [MATH] sending [MATH] to [MATH].', '1012.3738-1-3-1': 'The kernel of [MATH] is denoted by [MATH]; its topological interest is in the formula [MATH] (see [CITATION]).', '1012.3738-1-4-0': 'According to the formula [MATH] computing the order of [MATH] has interests in topology in addition to its interpretation as a problem in the group theory.', '1012.3738-1-5-0': 'Rocco in [CITATION] and later Ellis in [CITATION] have shown that the order of tensor square of [MATH] is at most [MATH] for every [MATH]-group of order [MATH] with the derived subgroup of order [MATH].', '1012.3738-1-6-0': 'The purpose of this paper is a further investigation on the order of tensor square of non abelian [MATH]-groups.', '1012.3738-1-6-1': 'We focus on non abelian [MATH]-groups because in abelian case the non abelian tensor square coincides with the usual abelian tensor square of abelian groups.', '1012.3738-1-6-2': 'To be precise, for a non abelian [MATH]-group of order [MATH] and the derived subgroup of order [MATH], we prove that [MATH] and also we obtain the explicit structure of [MATH] when [MATH].', '1012.3738-1-6-3': ""It easily seen that the bound is less than of Rocco's bound, unless that [MATH] or [MATH], which causes two bounds to be equal."", '1012.3738-1-6-4': 'As a corollary by using the fact [MATH], we can see that [MATH].', '1012.3738-1-7-0': 'Thorough the paper, [MATH], [MATH] denote the dihedral and quaternion group of order 8, [MATH] and [MATH] denote the extra-special p-groups of order [MATH] of exponent [MATH] and [MATH], respectively.', '1012.3738-1-7-1': 'Also [MATH] and [MATH] denote the direct product of [MATH] copies of the cyclic group of order [MATH] and the subgroup generated by [MATH] for all [MATH] in [MATH], respectively.', '1012.3738-1-8-0': '# Main Results', '1012.3738-1-9-0': 'The aim of this section is finding an upper bound for the order tensor square of non abelian [MATH]-groups of order [MATH] in terms of the order of [MATH].', '1012.3738-1-9-1': 'Also in the case for which [MATH], the structure of groups is obtained when [MATH] reaches the upper bound.', '1012.3738-1-10-0': 'Recall that [CITATION] the order of tensor square of G is equal to [MATH], where [MATH] is the Schur multiplier of [MATH].', '1012.3738-1-11-0': 'Put [MATH].', '1012.3738-1-11-1': 'In analogy with the above proposition the following lemma is characterized the tensor square of extra-special [MATH]-groups of order [MATH].', '1012.3738-1-12-0': 'The tensor square of [MATH] and [MATH] are isomorphic to [MATH] and [MATH], respectively.', '1012.3738-1-13-0': 'It can be proved from [CITATION] that [MATH] is elementary abelian.', '1012.3738-1-13-1': 'Now, by invoking [CITATION], [MATH] and hence [MATH].', '1012.3738-1-13-2': 'On the other hand, [CITATION] implies that the Schur multiplier of [MATH] is of order [MATH], and so [MATH].', '1012.3738-1-14-0': 'In the case [MATH] in a similar fashion, we can prove that [MATH].', '1012.3738-1-15-0': '[CITATION] The tensor square of an extra-special [MATH]-group [MATH] of order [MATH] is elementary abelian of order [MATH], for [MATH].', '1012.3738-1-16-0': '[Proof (i)]The proof is an upstanding result of Proposition [REF] while [MATH].', '1012.3738-1-16-1': 'Let [MATH] where [MATH] and [MATH] for all [MATH].', '1012.3738-1-16-2': 'Then [EQUATION] as required.', '1012.3738-1-17-0': '[(ii)] Since [MATH] is a vector space on [MATH], let [MATH] be the complement of [MATH] in [MATH].', '1012.3738-1-17-1': 'Moreover [MATH] is extra-special and [MATH].', '1012.3738-1-17-2': 'There is an epimorphism [MATH], so [EQUATION].', '1012.3738-1-17-3': 'Let [MATH] and [MATH], we can suppose that [MATH] by using Proposition [REF].', '1012.3738-1-17-4': 'Now the following two cases can be considered.', '1012.3738-1-18-0': 'Case [MATH].', '1012.3738-1-18-1': 'First suppose that [MATH].', '1012.3738-1-19-0': 'Let [MATH] and [MATH].', '1012.3738-1-19-1': 'Applying Lemma [REF] and [CITATION], we have [EQUATION] as required.', '1012.3738-1-20-0': 'Case [MATH].', '1012.3738-1-20-1': 'Without loss of generality, we can suppose that [MATH].', '1012.3738-1-20-2': 'Now the result is obtained by using Proposition [REF] and the fact that [MATH].', '1012.3738-1-21-0': 'Let [MATH] be a non abelian [MATH]-group of order [MATH].', '1012.3738-1-21-1': 'If [MATH], then [EQUATION] and the equality holds if and only if [MATH] is isomorphic to [MATH], where [MATH] or [MATH] and [MATH] is an elementary abelian [MATH]-group.', '1012.3738-1-22-0': 'One can assume that [MATH] and [MATH] are elementary abelian and [MATH] by Proposition [REF].', '1012.3738-1-22-1': 'Let [MATH] be the complement of [MATH] in [MATH].', '1012.3738-1-22-2': 'Thus there exists an extra-special [MATH]-group [MATH] of order [MATH] such that [MATH].', '1012.3738-1-23-0': 'In the case [MATH], it is easily seen that [MATH].', '1012.3738-1-23-1': 'For [MATH], [EQUATION] where [MATH].', '1012.3738-1-24-0': 'Now Proposition [REF] and Lemma [REF] imply that [MATH] when [MATH] or [MATH] has exponent [MATH].', '1012.3738-1-24-1': '[EQUATION]', '1012.3738-1-24-2': 'We prove theorem by induction on [MATH].', '1012.3738-1-24-3': 'For [MATH] the result is obtained by Theorem [REF].', '1012.3738-1-25-0': 'Let [MATH] and [MATH] be a central subgroup of order [MATH] contained in [MATH].', '1012.3738-1-25-1': 'Induction hypothesis and Proposition [REF] yield [EQUATION]', '1012.3738-1-25-2': 'Let [MATH] be a non abelian [MATH]-group of order [MATH].', '1012.3738-1-25-3': 'If [MATH], then [EQUATION].', '1012.3738-1-25-4': 'In particular when [MATH], then [EQUATION] and the equality holds if and only if [MATH] is isomorphic to [MATH], in which [MATH] is extra-special of order [MATH] of exponent [MATH] or [MATH] and [MATH] is an elementary abelian [MATH]-group.', '1012.3738-1-26-0': 'If the order of tensor square of [MATH] is equal to [MATH], then [EQUATION]'}","{'1012.3738-2-0-0': 'For every [MATH]-group of order [MATH] with the derived subgroup of order [MATH], Rocco in [CITATION] has shown that the order of tensor square of [MATH] is at most [MATH].', '1012.3738-2-0-1': 'In the present paper not only we improve his bound for non-abelian [MATH]-groups but also we describe the structure of all non-abelian [MATH]-groups when the bound is attained for a special case.', '1012.3738-2-0-2': 'Moreover, our results give as well an upper bound for the order of [MATH].', '1012.3738-2-1-0': '# Introduction and Preliminaries', '1012.3738-2-2-0': 'The tensor square [MATH] of a group [MATH] is a group generated by the symbols [MATH] subject to the relations [EQUATION] for all [MATH] where [MATH].', '1012.3738-2-2-1': 'The non abelian tensor square is a special case of non abelian tensor product, which was introduced by R. Brown and J.-L. Loday in [CITATION].', '1012.3738-2-3-0': 'There exists a homomorphism of groups [MATH] sending [MATH] to [MATH].', '1012.3738-2-3-1': 'The kernel of [MATH] is denoted by [MATH]; its topological interest is in the formula [MATH] (see [CITATION]).', '1012.3738-2-4-0': 'According to the formula [MATH] computing the order of [MATH] has interests in topology in addition to its interpretation as a problem in the group theory.', '1012.3738-2-5-0': 'Rocco in [CITATION] and later Ellis in [CITATION] have shown that the order of tensor square of [MATH] is at most [MATH] for every [MATH]-group of order [MATH] with the derived subgroup of order [MATH].', '1012.3738-2-6-0': 'The purpose of this paper is a further investigation on the order of tensor square of non abelian [MATH]-groups.', '1012.3738-2-6-1': 'We focus on non abelian [MATH]-groups because in abelian case the non abelian tensor square coincides with the usual abelian tensor square of abelian groups.', '1012.3738-2-6-2': 'To be precise, for a non abelian [MATH]-group of order [MATH] and the derived subgroup of order [MATH], we prove that [MATH] and also we obtain the explicit structure of [MATH] when [MATH].', '1012.3738-2-6-3': ""It easily seen that the bound is less than of Rocco's bound, unless that [MATH] or [MATH], which causes two bounds to be equal."", '1012.3738-2-6-4': 'As a corollary by using the fact [MATH], we can see that [MATH].', '1012.3738-2-7-0': 'Thorough the paper, [MATH], [MATH] denote the dihedral and quaternion group of order 8, [MATH] and [MATH] denote the extra-special p-groups of order [MATH] of exponent [MATH] and [MATH], respectively.', '1012.3738-2-7-1': 'Also [MATH] and [MATH] denote the direct product of [MATH] copies of the cyclic group of order [MATH] and the subgroup generated by [MATH] for all [MATH] in [MATH], respectively.', '1012.3738-2-8-0': '# Main Results', '1012.3738-2-9-0': 'The aim of this section is finding an upper bound for the order tensor square of non abelian [MATH]-groups of order [MATH] in terms of the order of [MATH].', '1012.3738-2-9-1': 'Also in the case for which [MATH], the structure of groups is obtained when [MATH] reaches the upper bound.', '1012.3738-2-10-0': 'Recall that [CITATION] the order of tensor square of G is equal to [MATH], where [MATH] is the Schur multiplier of [MATH].', '1012.3738-2-11-0': 'Put [MATH].', '1012.3738-2-11-1': 'In analogy with the above proposition the following lemma is characterized the tensor square of extra-special [MATH]-groups of order [MATH].', '1012.3738-2-12-0': 'The tensor square of [MATH] and [MATH] are isomorphic to [MATH] and [MATH], respectively.', '1012.3738-2-13-0': 'It can be proved from [CITATION] that [MATH] is elementary abelian.', '1012.3738-2-13-1': 'Now, by invoking [CITATION], [MATH] and hence [MATH].', '1012.3738-2-13-2': 'On the other hand, [CITATION] implies that the Schur multiplier of [MATH] is of order [MATH], and so [MATH].', '1012.3738-2-14-0': 'In the case [MATH] in a similar fashion, we can prove that [MATH].', '1012.3738-2-15-0': '[CITATION] The tensor square of an extra-special [MATH]-group [MATH] of order [MATH] is elementary abelian of order [MATH], for [MATH].', '1012.3738-2-16-0': '[Proof (i)]The proof is an upstanding result of Proposition [REF] while [MATH].', '1012.3738-2-16-1': 'Let [MATH] where [MATH] and [MATH] for all [MATH].', '1012.3738-2-16-2': 'Then [EQUATION] as required.', '1012.3738-2-17-0': '[(ii)] Since [MATH] is a vector space on [MATH], let [MATH] be the complement of [MATH] in [MATH].', '1012.3738-2-17-1': 'Moreover [MATH] is extra-special and [MATH].', '1012.3738-2-17-2': 'There is an epimorphism [MATH], so [EQUATION].', '1012.3738-2-17-3': 'Let [MATH] and [MATH], we can suppose that [MATH] by using Proposition [REF].', '1012.3738-2-17-4': 'Now the following two cases can be considered.', '1012.3738-2-18-0': 'Case [MATH].', '1012.3738-2-18-1': 'First suppose that [MATH].', '1012.3738-2-19-0': 'Let [MATH] and [MATH].', '1012.3738-2-19-1': 'Applying Lemma [REF] and [CITATION], we have [EQUATION] as required.', '1012.3738-2-20-0': 'Case [MATH].', '1012.3738-2-20-1': 'Without loss of generality, we can suppose that [MATH].', '1012.3738-2-20-2': 'Now the result is obtained by using Proposition [REF] and the fact that [MATH].', '1012.3738-2-21-0': 'Let [MATH] be a non abelian [MATH]-group of order [MATH].', '1012.3738-2-21-1': 'If [MATH], then [EQUATION] and the equality holds if and only if [MATH] is isomorphic to [MATH], where [MATH] or [MATH] and [MATH] is an elementary abelian [MATH]-group.', '1012.3738-2-22-0': 'One can assume that [MATH] and [MATH] are elementary abelian and [MATH] by Proposition [REF].', '1012.3738-2-22-1': 'Let [MATH] be the complement of [MATH] in [MATH].', '1012.3738-2-22-2': 'Thus there exists an extra-special [MATH]-group [MATH] of order [MATH] such that [MATH].', '1012.3738-2-23-0': 'In the case [MATH], it is easily seen that [MATH].', '1012.3738-2-23-1': 'For [MATH], [EQUATION] where [MATH].', '1012.3738-2-24-0': 'Now Proposition [REF] and Lemma [REF] imply that [MATH] when [MATH] or [MATH] has exponent [MATH].', '1012.3738-2-24-1': '[EQUATION]', '1012.3738-2-24-2': 'We prove theorem by induction on [MATH].', '1012.3738-2-24-3': 'For [MATH] the result is obtained by Theorem [REF].', '1012.3738-2-25-0': 'Let [MATH] and [MATH] be a central subgroup of order [MATH] contained in [MATH].', '1012.3738-2-25-1': 'Induction hypothesis and Proposition [REF] yield [EQUATION]', '1012.3738-2-25-2': 'Let [MATH] be a non abelian [MATH]-group of order [MATH].', '1012.3738-2-25-3': 'If [MATH], then [EQUATION].', '1012.3738-2-25-4': 'In particular when [MATH], then [EQUATION] and the equality holds if and only if [MATH] is isomorphic to [MATH], in which [MATH] is extra-special of order [MATH] of exponent [MATH] or [MATH] and [MATH] is an elementary abelian [MATH]-group.', '1012.3738-2-26-0': 'If the order of tensor square of [MATH] is equal to [MATH], then [EQUATION]'}","[['1012.3738-1-4-0', '1012.3738-2-4-0'], ['1012.3738-1-22-0', '1012.3738-2-22-0'], ['1012.3738-1-22-1', '1012.3738-2-22-1'], ['1012.3738-1-22-2', '1012.3738-2-22-2'], ['1012.3738-1-13-0', '1012.3738-2-13-0'], ['1012.3738-1-13-1', '1012.3738-2-13-1'], ['1012.3738-1-13-2', '1012.3738-2-13-2'], ['1012.3738-1-24-2', '1012.3738-2-24-2'], ['1012.3738-1-24-3', '1012.3738-2-24-3'], ['1012.3738-1-0-0', '1012.3738-2-0-0'], ['1012.3738-1-0-1', '1012.3738-2-0-1'], ['1012.3738-1-0-2', '1012.3738-2-0-2'], ['1012.3738-1-20-1', '1012.3738-2-20-1'], ['1012.3738-1-20-2', '1012.3738-2-20-2'], ['1012.3738-1-7-0', '1012.3738-2-7-0'], ['1012.3738-1-7-1', '1012.3738-2-7-1'], ['1012.3738-1-9-0', '1012.3738-2-9-0'], ['1012.3738-1-9-1', '1012.3738-2-9-1'], ['1012.3738-1-17-0', '1012.3738-2-17-0'], ['1012.3738-1-17-1', '1012.3738-2-17-1'], ['1012.3738-1-17-2', '1012.3738-2-17-2'], ['1012.3738-1-17-3', '1012.3738-2-17-3'], ['1012.3738-1-17-4', '1012.3738-2-17-4'], ['1012.3738-1-11-1', 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'1012.3738-2-26-0']]",[],[],[],[],"['1012.3738-1-11-0', '1012.3738-1-18-0', '1012.3738-1-18-1', '1012.3738-1-19-0', '1012.3738-1-19-1', '1012.3738-1-20-0', '1012.3738-1-23-0', '1012.3738-1-23-1', '1012.3738-1-24-0', '1012.3738-1-24-1', '1012.3738-1-25-3', '1012.3738-2-11-0', '1012.3738-2-18-0', '1012.3738-2-18-1', '1012.3738-2-19-0', '1012.3738-2-19-1', '1012.3738-2-20-0', '1012.3738-2-23-0', '1012.3738-2-23-1', '1012.3738-2-24-0', '1012.3738-2-24-1', '1012.3738-2-25-3']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1012.3738,,,
physics-0701176,"{'physics-0701176-1-0-0': 'We consider two ways that one might convert a prediction of sea surface temperature (SST) into a prediction of landfalling hurricane numbers.', 'physics-0701176-1-0-1': 'First, one might regress historical numbers of landfalling hurricanes onto historical SSTs, and use the fitted regression relation to predict future landfalling hurricane numbers given predicted SSTs.', 'physics-0701176-1-0-2': 'We call this the direct approach.', 'physics-0701176-1-0-3': 'Second, one might regress basin hurricane numbers onto historical SSTs, estimate the proportion of basin hurricanes that make landfall, and use the fitted regression relation and estimated proportion to predict future landfalling hurricane numbers.', 'physics-0701176-1-0-4': 'We call this the indirect approach.', 'physics-0701176-1-0-5': 'Which of these two methods is likely to work better?', 'physics-0701176-1-0-6': 'We answer this question in the context of a simple abstract model.', 'physics-0701176-1-1-0': '# Introduction', 'physics-0701176-1-2-0': 'There is a great need to predict the distribution of the number of hurricanes that might make landfall in the US in the next few years.', 'physics-0701176-1-2-1': 'Such predictions are of use to all the entities that are affected by hurricanes, ranging from local and national governments to insurance and reinsurance companies.', 'physics-0701176-1-2-2': 'How, then, should we make such predictions?', 'physics-0701176-1-2-3': 'There is no obvious best method.', 'physics-0701176-1-2-4': 'For instance, one might consider making a prediction based on time-series analysis of the time-series of historical landfalling hurricane numbers; one might consider making a prediction of basin hurricane numbers using time-series analysis, and convert that prediction to a prediction of landfalling hurricane numbers; one might consider trying to predict SSTs first, and convert that prediction to a prediction of landfalling numbers; or one might try and use output from a numerical model of the climate system.', 'physics-0701176-1-2-5': 'All of these are valid approaches, and each has their own pros and cons.', 'physics-0701176-1-3-0': 'In this article, we consider the idea of first predicting SST and then predicting hurricane numbers given a prediction of SST.', 'physics-0701176-1-3-1': 'There are two obvious flavours of this.', 'physics-0701176-1-3-2': ""The first is what we will call the 'direct' (or 'one-step') method, in which one regresses historical numbers of landfalling hurricanes directly onto historical SSTs, and uses the fitted regression relation to convert a prediction of future SSTs into a prediction of future hurricane numbers."", 'physics-0701176-1-3-3': ""The second is what we will call the 'indirect' (or 'two-step') method, in which one regresses basin hurricane numbers onto historical SSTs, predicts basin numbers, and then predicts landfalling numbers from basin numbers."", 'physics-0701176-1-3-4': 'In the simplest version of the indirect method one might predict landfalling numbers as a constant proportion of the number of basin hurricanes, where this proportion is estimated using historical data.', 'physics-0701176-1-4-0': 'Consideration of the direct and indirect SST-based methods motivates the question: at a theoretical level, which of these two methods is likely to work best?', 'physics-0701176-1-4-1': 'This is a statistical question about the properties of regression and proportion models.', 'physics-0701176-1-4-2': 'We consider this abstract question in the context of a simple but realistic model, in which we use observed SSTs, the mean number of hurricanes in the basin is a linear function of SST, and each basin hurricane has a constant probability of making landfall.', 'physics-0701176-1-5-0': ""We don't think the answer as to which of the direct or indirect methods is better is a priori obvious."", 'physics-0701176-1-5-1': 'On the one hand, the direct method has fewer parameters to estimate, which might work in its favour.', 'physics-0701176-1-5-2': 'On the other hand, the indirect method allows us to use more data by incorporating the basin hurricane numbers into the analysis.', 'physics-0701176-1-6-0': '# Methods', 'physics-0701176-1-7-0': 'We compare the direct and indirect methods described above as follows.', 'physics-0701176-1-8-0': '## Generating artificial basin hurricane numbers', 'physics-0701176-1-9-0': 'First, we simulate 10,000 sets of artificial basin hurricane numbers for the period 1950-2005, giving a total of 10,000 x 56 = 560,000 years of simulated hurricane numbers.', 'physics-0701176-1-9-1': 'These numbers are created by sampling from poisson distributions with mean given by: [EQUATION] where [MATH] is the observed MDR SST for each year in the period 1950-2005.', 'physics-0701176-1-9-2': 'The values of [MATH] and [MATH] are derived from model 4 in table 7 in [CITATION], in which observed basin hurricane numbers were regressed onto observed SSTs using data for 1950-2005.', 'physics-0701176-1-9-3': 'They have values of 6.25 and 5, respectively.', 'physics-0701176-1-10-0': 'The basin hurricane numbers we create by this method should contain roughly the same long-term SST driven variability as the observed basin hurricane numbers, but different numbers of hurricanes in the individual years.', 'physics-0701176-1-10-1': ""We say 'roughly' the same, because (a) the linear model we are using to relate SST to hurricane numbers is undoubtedly not exactly correct, although given the analysis in [CITATION] is certainly seems to be reasonable, and (b) the parameters of the linear model are only estimated."", 'physics-0701176-1-11-0': '## Generating artificial landfalling hurricane numbers', 'physics-0701176-1-12-0': 'Given the 10,000 sets of simulated basin hurricane numbers described above, we then create 10,000 sets of simulated landfalling hurricane numbers by applying the rule that each basin hurricane has a probability of 0.254 of making landfall (this value is taken from observed data for 1950-2005).', 'physics-0701176-1-13-0': 'The landfalling hurricane numbers we create by this method should contain roughly the same long-term SST driven variability as the observed landfalling series, but different numbers of hurricane in the individual years.', 'physics-0701176-1-13-1': 'They should also contain roughly the right dependency structure between the number of hurricanes in the basin and the number at landfall (e.g. that years with more hurricanes in the basin will tend to have more hurricanes at landfall).', 'physics-0701176-1-14-0': '## Making predictions', 'physics-0701176-1-15-0': 'We now have 10,000 sets of 56 years of artificial data for basin and landfalling hurricanes.', 'physics-0701176-1-15-1': 'This data contains a realistic representation of the SST-driven variability of hurricane numbers, and of the dependency structure between the numbers of hurricanes in the basin and at landfall, but different actual numbers of hurricanes from the observations.', 'physics-0701176-1-15-2': 'We can consider this data as 10,000 realisations of what might have occurred over the last 56 years, had the SSTs been the same, but the evolution of the atmosphere different.', 'physics-0701176-1-15-3': 'This data is a test-bed that can help us understand aspects of the predictability of landfalling hurricanes given SST.', 'physics-0701176-1-16-0': 'The observed and simulated data is illustrated in figures [REF] to [REF].', 'physics-0701176-1-16-1': 'Figure [REF] shows the observed basin data (solid black line) and the observed landfall data (solid grey line).', 'physics-0701176-1-16-2': 'The dashed black line shows the variability in the observed basin data that is explained using SSTs.', 'physics-0701176-1-16-3': 'The dotted grey line shows the variability in the observed landfall data that is explained using SSTs using the direct method, and the dotted grey line shows the variability in the landfall data that is explained using SSTs using the indirect method.', 'physics-0701176-1-17-0': 'Figures [REF] to [REF] show 4 realisations of the simulated data.', 'physics-0701176-1-17-1': 'In each figure the dotted and dashed lines are the same as in figure [REF], and show the SST driven signal.', 'physics-0701176-1-17-2': 'The solid black line then shows the simulated basin hurricane numbers and the solid grey line shows the simulated landfalling hurricane numbers.', 'physics-0701176-1-18-0': 'We test predictions of landfalling hurricane numbers using the direct method as follows:', 'physics-0701176-1-19-0': 'We test the indirect method in almost exactly the same way, except that this time we also fit a model for predicting landfalling numbers from basin numbers.', 'physics-0701176-1-20-0': '## Comparing the predictions', 'physics-0701176-1-21-0': 'We compare these predictions in two ways:', 'physics-0701176-1-22-0': 'We also repeat the entire calculation a number of times as a rough way to evaluate the convergence of our results.', 'physics-0701176-1-23-0': '# Results', 'physics-0701176-1-24-0': 'The RMSE for the direct method is 1.61 hurricanes, while the RMSE for the indirect method is 1.58 hurricanes.', 'physics-0701176-1-24-1': 'This difference is small, but the sign of it does appear to be real: when we repeat the whole experiment a number of times, we always find that the indirect method beats the direct method.', 'physics-0701176-1-25-0': 'The indirect method beats the direct method 51.8% of the time.', 'physics-0701176-1-26-0': 'Given the design of the experiment, these results tell us how the two methods perform, on average over the whole range of SST values.', 'physics-0701176-1-26-1': ""Next year's SST, however, is likely to be warm relative to historical SSTs."", 'physics-0701176-1-26-2': 'We therefore consider the more specific question of how the methods are likely to perform for given warm SSTs.', 'physics-0701176-1-26-3': 'Based on [CITATION], we fit a linear trend to the historical SSTs, and extrapolate this trend out to 2011.', 'physics-0701176-1-26-4': 'This then gives SST values that are warmer than anything experienced in history (27.987[MATH]C to be precise).', 'physics-0701176-1-26-5': 'We then repeat the whole analysis for predictions for this warm SST only.', 'physics-0701176-1-26-6': 'The results are more or less as before: the indirect method still wins, only this time by a slightly larger margin.', 'physics-0701176-1-26-7': 'The ratio of RMSE scores (direct divided by indirect) increases from 1.02 to 1.04.', 'physics-0701176-1-27-0': '# Conclusions', 'physics-0701176-1-28-0': 'We have compared the likely performance of direct and indirect methods for predicting landfalling hurricane numbers from SST using simulated data.', 'physics-0701176-1-28-1': 'The direct method is based on building a linear regression model directly from SST to landfalling hurricane numbers.', 'physics-0701176-1-28-2': 'The indirect method is based on building a regression model from SST to basin numbers, and then predicting landfalling numbers from basin numbers using a constant proportion.', 'physics-0701176-1-28-3': 'We find that the indirect method is better than the direct method, but that the difference is very small.', 'physics-0701176-1-29-0': 'Which method should we then use in practice?', 'physics-0701176-1-29-1': 'If we had to chose one method, our results seem to imply that we should choose the indirect method, since it is slightly more accurate.', 'physics-0701176-1-29-2': 'The results are, however, very close, and given the possibility to use two methods, we would use both, as alterative points of view.'}","{'physics-0701176-2-0-0': 'We consider two ways that one might convert a prediction of sea surface temperature (SST) into a prediction of landfalling hurricane numbers.', 'physics-0701176-2-0-1': 'First, one might regress historical numbers of landfalling hurricanes onto historical SSTs, and use the fitted regression relation to predict future landfalling hurricane numbers given predicted SSTs.', 'physics-0701176-2-0-2': 'We call this the direct approach.', 'physics-0701176-2-0-3': 'Second, one might regress basin hurricane numbers onto historical SSTs, estimate the proportion of basin hurricanes that make landfall, and use the fitted regression relation and estimated proportion to predict future landfalling hurricane numbers.', 'physics-0701176-2-0-4': 'We call this the indirect approach.', 'physics-0701176-2-0-5': 'Which of these two methods is likely to work better?', 'physics-0701176-2-0-6': 'We answer this question for two simple models.', 'physics-0701176-2-0-7': 'The first model is reasonably realistic, but we have to resort to using simulations to answer the question in the context of this model.', 'physics-0701176-2-0-8': 'The second model is less realistic, but allows us to derive a general analytical result.', 'physics-0701176-2-1-0': '# Introduction', 'physics-0701176-2-2-0': 'There is a great need to predict the distribution of the number of hurricanes that might make landfall in the US in the next few years.', 'physics-0701176-2-2-1': 'Such predictions are of use to all the entities that are affected by hurricanes, ranging from local and national governments to insurance and reinsurance companies.', 'physics-0701176-2-2-2': 'How, then, should we make such predictions?', 'physics-0701176-2-2-3': 'There is no obvious best method.', 'physics-0701176-2-2-4': 'For instance, one might consider making a prediction based on time-series analysis of the time-series of historical landfalling hurricane numbers; one might consider making a prediction of basin hurricane numbers using time-series analysis, and convert that prediction to a prediction of landfalling hurricane numbers; one might consider trying to predict SSTs first, and convert that prediction to a prediction of landfalling numbers; or one might try and use output from a numerical model of the climate system.', 'physics-0701176-2-2-5': 'All of these are valid approaches, and each has their own pros and cons.', 'physics-0701176-2-3-0': 'In this article, we consider the idea of first predicting SST and then predicting hurricane numbers given a prediction of SST.', 'physics-0701176-2-3-1': 'There are two obvious flavours of this.', 'physics-0701176-2-3-2': ""The first is what we will call the 'direct' (or 'one-step') method, in which one regresses historical numbers of landfalling hurricanes directly onto historical SSTs, and uses the fitted regression relation to convert a prediction of future SSTs into a prediction of future hurricane numbers."", 'physics-0701176-2-3-3': ""The second is what we will call the 'indirect' (or 'two-step') method, in which one regresses basin hurricane numbers onto historical SSTs, predicts basin numbers, and then predicts landfalling numbers from basin numbers."", 'physics-0701176-2-3-4': 'In the simplest version of the indirect method one might predict landfalling numbers as a constant proportion of the number of basin hurricanes, where this proportion is estimated using historical data.', 'physics-0701176-2-4-0': 'Consideration of the direct and indirect SST-based methods motivates the question: at a theoretical level, which of these two methods is likely to work best?', 'physics-0701176-2-4-1': 'This is a statistical question about the properties of regression and proportion models.', 'physics-0701176-2-4-2': 'We consider this abstract question in the context of two simple models.', 'physics-0701176-2-4-3': 'The first model is the more realistic of the two.', 'physics-0701176-2-4-4': 'It uses observed SSTs, models the mean number of hurricanes in the basin as a linear function of SST, and models each basin hurricane as having a constant probability of making landfall.', 'physics-0701176-2-4-5': 'We run simulations that allow us to directly compare the performance of the direct and indirect methods in the context of this model.', 'physics-0701176-2-4-6': 'The second model is less realistic, but allows us to derive a general analytical result for the relative performance of the direct and indirect methods.', 'physics-0701176-2-4-7': 'In this model we represent SST, basin and landfalling hurricane numbers as being normally distributed and linearly related.', 'physics-0701176-2-5-0': ""We don't think the answer as to which of the direct or indirect methods is better is a priori obvious."", 'physics-0701176-2-5-1': 'On the one hand, the direct method has fewer parameters to estimate, which might work in its favour.', 'physics-0701176-2-5-2': 'On the other hand, the indirect method allows us to use more data by incorporating the basin hurricane numbers into the analysis.', 'physics-0701176-2-6-0': 'Section [REF] describes the methods used in the simulation study, and section [REF] describes the results from that study.', 'physics-0701176-2-6-1': 'In section [REF] we derive general analytic results for the linear-normal model.', 'physics-0701176-2-6-2': 'Finally in section [REF] we discuss our results.', 'physics-0701176-2-7-0': '# Simulation-based analysis: methods', 'physics-0701176-2-8-0': 'For our simulation study, we compare the direct and indirect methods described above as follows.', 'physics-0701176-2-9-0': '## Generating artificial basin hurricane numbers', 'physics-0701176-2-10-0': 'First, we simulate 10,000 sets of artificial basin hurricane numbers for the period 1950-2005, giving a total of 10,000 x 56 = 560,000 years of simulated hurricane numbers.', 'physics-0701176-2-10-1': 'These numbers are created by sampling from poisson distributions with mean given by: [EQUATION] where [MATH] is the observed MDR SST for each year in the period 1950-2005.', 'physics-0701176-2-10-2': 'The values of [MATH] and [MATH] are derived from model 4 in table 7 in [CITATION], in which observed basin hurricane numbers were regressed onto observed SSTs using data for 1950-2005.', 'physics-0701176-2-10-3': 'They have values of 6.25 and 5, respectively.', 'physics-0701176-2-11-0': 'The basin hurricane numbers we create by this method should contain roughly the same long-term SST driven variability as the observed basin hurricane numbers, but different numbers of hurricanes in the individual years.', 'physics-0701176-2-11-1': ""We say 'roughly' the same, because (a) the linear model we are using to relate SST to hurricane numbers is undoubtedly not exactly correct, although given the analysis in [CITATION] is certainly seems to be reasonable, and (b) the parameters of the linear model are only estimated."", 'physics-0701176-2-12-0': '## Generating artificial landfalling hurricane numbers', 'physics-0701176-2-13-0': 'Given the 10,000 sets of simulated basin hurricane numbers described above, we then create 10,000 sets of simulated landfalling hurricane numbers by applying the rule that each basin hurricane has a probability of 0.254 of making landfall (this value is taken from observed data for 1950-2005).', 'physics-0701176-2-14-0': 'The landfalling hurricane numbers we create by this method should contain roughly the same long-term SST driven variability as the observed landfalling series, but different numbers of hurricane in the individual years.', 'physics-0701176-2-14-1': 'They should also contain roughly the right dependency structure between the number of hurricanes in the basin and the number at landfall (e.g. that years with more hurricanes in the basin will tend to have more hurricanes at landfall).', 'physics-0701176-2-15-0': '## Making predictions', 'physics-0701176-2-16-0': 'We now have 10,000 sets of 56 years of artificial data for basin and landfalling hurricanes.', 'physics-0701176-2-16-1': 'This data contains a realistic representation of the SST-driven variability of hurricane numbers, and of the dependency structure between the numbers of hurricanes in the basin and at landfall, but different actual numbers of hurricanes from the observations.', 'physics-0701176-2-16-2': 'We can consider this data as 10,000 realisations of what might have occurred over the last 56 years, had the SSTs been the same, but the evolution of the atmosphere different.', 'physics-0701176-2-16-3': 'This data is a test-bed that can help us understand aspects of the predictability of landfalling hurricanes given SST.', 'physics-0701176-2-17-0': 'The observed and simulated data is illustrated in figures [REF] to [REF].', 'physics-0701176-2-17-1': 'Figure [REF] shows the observed basin data (solid black line) and the observed landfall data (solid grey line).', 'physics-0701176-2-17-2': 'The dashed black line shows the variability in the observed basin data that is explained using SSTs.', 'physics-0701176-2-17-3': 'The dotted grey line shows the variability in the observed landfall data that is explained using SSTs using the direct method, and the dotted grey line shows the variability in the landfall data that is explained using SSTs using the indirect method.', 'physics-0701176-2-18-0': 'Figures [REF] to [REF] show 4 realisations of the simulated data.', 'physics-0701176-2-18-1': 'In each figure the dotted and dashed lines are the same as in figure [REF], and show the SST driven signal.', 'physics-0701176-2-18-2': 'The solid black line then shows the simulated basin hurricane numbers and the solid grey line shows the simulated landfalling hurricane numbers.', 'physics-0701176-2-19-0': 'We test predictions of landfalling hurricane numbers using the direct method as follows:', 'physics-0701176-2-20-0': 'We test the indirect method in almost exactly the same way, except that this time we also fit a model for predicting landfalling numbers from basin numbers.', 'physics-0701176-2-21-0': '## Comparing the predictions', 'physics-0701176-2-22-0': 'We compare the direct and indirect predictions in two ways:', 'physics-0701176-2-23-0': 'We also repeat the entire calculation a number of times as a rough way to evaluate the convergence of our results.', 'physics-0701176-2-24-0': '# Simulation-based analysis: results', 'physics-0701176-2-25-0': 'We now present the results from our simulation study.', 'physics-0701176-2-25-1': 'The RMSE for the direct method is 1.61 hurricanes, while the RMSE for the indirect method is 1.58 hurricanes.', 'physics-0701176-2-25-2': 'This difference is small, but the sign of it does appear to be real: when we repeat the whole experiment a number of times, we always find that the indirect method beats the direct method.', 'physics-0701176-2-26-0': 'The indirect method beats the direct method 51.8% of the time.', 'physics-0701176-2-27-0': 'Given the design of the experiment, these results tell us how the two methods perform, on average over the whole range of SST values.', 'physics-0701176-2-27-1': ""Next year's SST, however, is likely to be warm relative to historical SSTs."", 'physics-0701176-2-27-2': 'We therefore also consider the more specific question of how the methods are likely to perform for given warm SSTs.', 'physics-0701176-2-27-3': 'Based on [CITATION], we fit a linear trend to the historical SSTs, and extrapolate this trend out to 2011.', 'physics-0701176-2-27-4': 'This then gives SST values that are warmer than anything experienced in history (27.987[MATH]C to be precise).', 'physics-0701176-2-27-5': 'We then repeat the whole analysis for predictions for this warm SST only.', 'physics-0701176-2-27-6': 'The results are more or less as before: the indirect method still wins, only this time by a slightly larger margin.', 'physics-0701176-2-27-7': 'The ratio of RMSE scores (direct divided by indirect) increases from 1.02 to 1.04.', 'physics-0701176-2-28-0': '# The Linear normal case', 'physics-0701176-2-29-0': 'We now study a slightly less realistic model, in which we take SSTs and hurricane numbers in the basin and at landfall to be normally distributed.', 'physics-0701176-2-29-1': 'These changes allow us to derive a very general result for the relative performance of the direct and indirect methods.', 'physics-0701176-2-30-0': '## The setup', 'physics-0701176-2-31-0': ""Here's how we set the problem up in this case."", 'physics-0701176-2-32-0': 'Consider two simple regression models for centred random variables [MATH] and [MATH], [EQUATION] where [MATH] and [MATH] are independent.', 'physics-0701176-2-32-1': 'Here [MATH], [MATH], [MATH], [MATH] and [MATH] are [MATH] column vectors, [MATH] and [MATH] are scalars, and [MATH] is the [MATH] identity matrix.', 'physics-0701176-2-32-2': 'We will assume [MATH] is fixed.', 'physics-0701176-2-33-0': 'In relation to the hurricane problem, [MATH] is the time-series of [MATH] years of SST values, [MATH] is the time-series of [MATH] years of basin hurricane numbers and [MATH] is the time-series of [MATH] years of landfalling hurricane numbers.', 'physics-0701176-2-33-1': 'Note that in our notation [MATH] is the whole time-series of SST, written as a vector, and similarly for [MATH] and [MATH].', 'physics-0701176-2-33-2': 'Using vector notation avoids the messy use of subscripts.', 'physics-0701176-2-33-3': 'Two immediate comments about this setup: (a) we are assuming that basin and landfalling hurricane numbers are normally distributed.', 'physics-0701176-2-33-4': ""This doesn't really make sense, since they are counts that can only take integer values: using a poisson distribution would make more sense."", 'physics-0701176-2-33-5': ""We are starting off by addressing this question for normally distributed data because it's more tractable that way; (b) we are assuming a linear relationship (with offset and slope) between basin hurricanes and landfalling hurricanes."", 'physics-0701176-2-33-6': ""This is also a little odd, since there is no reason to have an offset in this relationship: if there aren't any basin hurricanes, there can't be any landfalling hurricanes."", 'physics-0701176-2-33-7': 'The most obvious model would be that each hurricane has a constant proportion of making landfall.', 'physics-0701176-2-33-8': ""Again, we are starting off by addressing this question in a linear context because it's more tractable that way."", 'physics-0701176-2-34-0': 'We want to know about the accuracy of forecasts that we might make with the direct and indirect methods.', 'physics-0701176-2-34-1': 'This translates mathematically into saying that we want to estimate [EQUATION] where [MATH].', 'physics-0701176-2-35-0': 'The problem then boils down to measuring the quality of the estimator of [MATH] since, if [MATH] is an estimator of [MATH] then [EQUATION]', 'physics-0701176-2-35-1': 'So we now consider the direct and indirect methods for estimating [MATH].', 'physics-0701176-2-36-0': '## Direct estimator of [MATH]', 'physics-0701176-2-37-0': 'We start by considering the direct, or one-step, method.', 'physics-0701176-2-37-1': 'This means we consider the relationship between [MATH] and [MATH], ignoring [MATH].', 'physics-0701176-2-37-2': 'The usual OLS estimator for [MATH] is [EQUATION]', 'physics-0701176-2-37-3': 'What are the statistical properties of this estimator?', 'physics-0701176-2-38-0': 'In terms of mean: [EQUATION] i.e. the estimator is unbiased.', 'physics-0701176-2-39-0': 'In terms of variance [EQUATION]', 'physics-0701176-2-39-1': 'We know that [MATH], so [EQUATION]', 'physics-0701176-2-39-2': 'By equation [REF] this then gives us an expression for the performance of the direct method.', 'physics-0701176-2-40-0': '## Indirect estimator of [MATH]', 'physics-0701176-2-41-0': 'We now consider the indirect, or two-step, method.', 'physics-0701176-2-41-1': 'This means considering the relationships between [MATH] and [MATH], and [MATH] and [MATH].', 'physics-0701176-2-42-0': 'First, we consider estimating each regression separately.', 'physics-0701176-2-42-1': 'The OLS estimators for the slopes in each case are: [EQUATION]', 'physics-0701176-2-42-2': 'We now put the two models together, to create a single regression model based on the separate estimates for the two steps.', 'physics-0701176-2-42-3': 'We call the estimate of the slope of this combined model [MATH].', 'physics-0701176-2-42-4': 'Combining the expressions above, we have that: [EQUATION]', 'physics-0701176-2-42-5': 'What are the statistical properties of this estimator [MATH]?', 'physics-0701176-2-43-0': 'It is clear (by independence of [MATH] and [MATH]) that [MATH] is unbiased; [EQUATION]', 'physics-0701176-2-43-1': 'The variance is more awkward.', 'physics-0701176-2-43-2': 'Note that if [MATH] were known then [MATH] and [MATH] would be fixed constants.', 'physics-0701176-2-43-3': 'Thus, [EQUATION] and so [EQUATION] where we have used a standard relation for disaggregating the variance: [EQUATION]', 'physics-0701176-2-43-4': 'Using the facts that [EQUATION] and approximating to second order: [EQUATION] where [MATH].', 'physics-0701176-2-44-0': '## Comparing the two estimators', 'physics-0701176-2-45-0': 'We are now in a position to compare the estimators for the direct and indirect methods.', 'physics-0701176-2-45-1': 'Subtracting equation [REF] from equation [REF] gives: [EQUATION]', 'physics-0701176-2-45-2': 'The right hand side of this equation is clearly positive for [MATH].', 'physics-0701176-2-46-0': 'This indicates:', 'physics-0701176-2-47-0': '# Conclusions', 'physics-0701176-2-48-0': 'We have compared the likely performance of direct and indirect methods for predicting landfalling hurricane numbers from SST.', 'physics-0701176-2-48-1': 'The direct method is based on building a linear regression model directly from SST to landfalling hurricane numbers.', 'physics-0701176-2-48-2': 'The indirect method is based on building a regression model from SST to basin numbers, and then predicting landfalling numbers from basin numbers using a constant proportion.', 'physics-0701176-2-49-0': 'First, we compare these two methods in the context of a reasonably realistic model, using simulations.', 'physics-0701176-2-49-1': 'We find that the indirect method is better than the direct method, but that the difference is small.', 'physics-0701176-2-50-0': 'Secondly, we compare the two methods in the context of a less realistic model in which all variables are normally distributed.', 'physics-0701176-2-50-1': 'For this model we are able to derive the interesting general result that the indirect method should always be better.', 'physics-0701176-2-51-0': 'Which method should we then use in practice?', 'physics-0701176-2-51-1': 'If we had to chose one method, our results seem to imply that we should choose the indirect method, since it is more accurate.', 'physics-0701176-2-51-2': 'The simulation results suggest, however, that the performance of the two methods is likely to be very close for the values of the parameters appropriate for hurricanes in the real world.', 'physics-0701176-2-51-3': 'Given the possibility to use two methods we would use both, as alterative points of view.', 'physics-0701176-2-52-0': 'Ideally we would also be able to solve the more realistic model analytically, as we have done for the linear-normal case.', 'physics-0701176-2-52-1': 'We are working on that.'}","[['physics-0701176-1-22-0', 'physics-0701176-2-23-0'], ['physics-0701176-1-29-0', 'physics-0701176-2-51-0'], ['physics-0701176-1-4-0', 'physics-0701176-2-4-0'], ['physics-0701176-1-4-1', 'physics-0701176-2-4-1'], ['physics-0701176-1-0-0', 'physics-0701176-2-0-0'], ['physics-0701176-1-0-1', 'physics-0701176-2-0-1'], ['physics-0701176-1-0-2', 'physics-0701176-2-0-2'], ['physics-0701176-1-0-3', 'physics-0701176-2-0-3'], ['physics-0701176-1-0-4', 'physics-0701176-2-0-4'], ['physics-0701176-1-0-5', 'physics-0701176-2-0-5'], ['physics-0701176-1-10-0', 'physics-0701176-2-11-0'], ['physics-0701176-1-10-1', 'physics-0701176-2-11-1'], ['physics-0701176-1-19-0', 'physics-0701176-2-20-0'], ['physics-0701176-1-15-0', 'physics-0701176-2-16-0'], ['physics-0701176-1-15-1', 'physics-0701176-2-16-1'], 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1801.00115,"{'1801.00115-1-0-0': 'This paper discusses an attempt to develop a mathematically rigorous theory of Quantum Electrodynamics (QED).', '1801.00115-1-0-1': 'It deviates from the standard version of QED mainly in two aspects: it is assumed that the Coulomb forces are carried by transversely polarized photons, and a reducible representation of the canonical commutation and anti-commutation relations is used.', '1801.00115-1-0-2': 'Both interventions together should suffice to eliminate the mathematical inconsistencies of standard QED.', '1801.00115-1-1-0': '# Introduction', '1801.00115-1-2-0': 'In recent work [CITATION] Erik Verlinde formulated the claim that gravity is an emergent force.', '1801.00115-1-2-1': 'By this is meant that gravitational forces can be derived from other, more fundamental forces.', '1801.00115-1-2-2': 'Coulomb forces and gravitational forces have much in common.', '1801.00115-1-2-3': 'Both are inversely proportional to the square of the distance.', '1801.00115-1-2-4': 'And it is difficult to reconcile them with the aversion of physicists to action at a distance.', '1801.00115-1-2-5': 'It is therefore obvious to claim that also the Coulomb forces are emergent forces.', '1801.00115-1-2-6': 'While the claim of Verlinde is made in the context of cosmology the emergence of the Coulomb forces is formulated in the context of quantum field theory [CITATION].', '1801.00115-1-3-0': 'Two arguments are brought forward.', '1801.00115-1-3-1': 'In the common theory of Quantum Electrodynamics (QED), which includes Coulomb forces, it is possible to remove them by a simple transformation of observables [CITATION].', '1801.00115-1-3-2': 'The inverse transformation can be used to reintroduce Coulomb forces given a theory in which they are absent.', '1801.00115-1-3-3': 'The other argument is based on a proof that electrons bind with long-wavelength photons.', '1801.00115-1-3-4': 'This mechanism is discussed further on in the present paper.', '1801.00115-1-4-0': 'A theory of Quantum Electrodynamics which does not include Coulomb forces is appealing because it can avoid some of the technical problems which plague the standard version of the theory.', '1801.00115-1-4-1': 'The claim made in [CITATION] is that Coulomb attraction and repulsion are carried by transversely polarized photons.', '1801.00115-1-4-2': 'This eliminates the need for longitudinal and scalar photons and allows for building a rigorous theory of QED [CITATION].', '1801.00115-1-5-0': '# The Standard Model', '1801.00115-1-6-0': 'The Standard Model of elementary particles summarizes much of our present day understanding of the fundamental laws of physics.', '1801.00115-1-6-1': 'It is a highly effective theory.', '1801.00115-1-6-2': 'It explains almost all phenomena with an amazing degree of precision.', '1801.00115-1-6-3': ""Never the less, it cannot be the final theory of physics, in the first place because the integration with Einstein's general relativity theory is missing."", '1801.00115-1-7-0': 'The Standard Model is a quantum field theory, which implies that particles are described by fields, for instance a photon field or an electron field.', '1801.00115-1-7-1': 'Creation and annihilation operators add or remove one quantum of the corresponding field.', '1801.00115-1-7-2': 'This wording suggests that particles are created or annihilated.', '1801.00115-1-7-3': 'However, it is more careful to say that creation and annihilation operators are a tool to construct quantum fields and to describe interaction processes.', '1801.00115-1-8-0': 'Two different fields can interact with each other by the exchange of one or more quanta.', '1801.00115-1-8-1': 'Einstein proposed this mechanism to explain the photo-electric effect.', '1801.00115-1-8-2': 'This assumption is a corner stone of all quantum theories.', '1801.00115-1-8-3': 'The interesting question why the exchange of quanta occurs at seemingly random discrete moments of time is not discussed here.', '1801.00115-1-9-0': 'The prototype of a quantum field theory is Quantum Electrodynamics (QED), the relativistic quantum theory of electromagnetism.', '1801.00115-1-9-1': 'QED was extended to include all electro-weak interactions, and later on also strong interactions.', '1801.00115-1-9-2': 'Much effort goes into the exploration of further extensions.', '1801.00115-1-9-3': 'The whole construction has become an inverted pyramid resting on a few basic principles most of which were decided on in the early days of quantum mechanics.', '1801.00115-1-9-4': 'It is therefore of utmost importance that existing holes in these foundations are eliminated.', '1801.00115-1-10-0': 'It is indeed worrying that the common formulation of QED is mathematically inconsistent.', '1801.00115-1-10-1': 'A first reason for that is the perturbative approach, which involves a non-converging series expansion.', '1801.00115-1-10-2': 'It is even worse: many of the individual terms of the expansion are ill-defined.', '1801.00115-1-10-3': 'As a consequence, repair techniques are needed.', '1801.00115-1-10-4': 'An alternative is offered by non-perturbative QED.', '1801.00115-1-10-5': 'But even then not all problems disappear.', '1801.00115-1-11-0': '# Mathematical problems', '1801.00115-1-12-0': 'The technical difficulties of QED have their consequences.', '1801.00115-1-12-1': 'Certain questions cannot be treated in a reliable manner.', '1801.00115-1-12-2': 'Let me mention one.', '1801.00115-1-13-0': 'The physical vacuum differs from the mathematical vacuum.', '1801.00115-1-13-1': 'The latter is defined by a vanishing of all fields.', '1801.00115-1-13-2': 'However it is not the state of minimal energy.', '1801.00115-1-13-3': 'The interaction between different fields creates states of negative energy.', '1801.00115-1-13-4': 'A well-known example in the quantum mechanics of particles is the hydrogen atom.', '1801.00115-1-13-5': 'Its eigenstates all have negative energy while the scattering states have positive energy.', '1801.00115-1-13-6': 'By analogy one can expect that in QED there exist states of negative energy in which the electromagnetic field and the electron/positron field are bound in some way.', '1801.00115-1-13-7': 'If this expectation is correct then the state of minimal energy is the physical vacuum.', '1801.00115-1-13-8': 'No mathematical proof of its existence exists.', '1801.00115-1-13-9': 'In a non-relativistic context Lieb and coworkers (see for instance [CITATION]) prove the existence of a ground state for models consisting of particles which do not interact among themselves but do interact with the radiation field.', '1801.00115-1-13-10': 'An ultra-violet cutoff is applied to the latter.', '1801.00115-1-14-0': 'In Quantum Chromodynamics (QCD) one accepts that a symmetry-breaking phase transition occurs at low energies.', '1801.00115-1-14-1': 'Such a phase transition gives a satisfactory explanation for the confinement phenomenon.', '1801.00115-1-14-2': 'Also the Englert-Brout-Higgs mechanism, which in the Standard Model is responsible for giving mass to particles, invokes a phase transition.', '1801.00115-1-14-3': 'Because these phase transitions are continuous and symmetry-breaking the physical ground state is non-unique and is accompanied by Goldstone bosons.', '1801.00115-1-14-4': 'Mathematical proofs for these statements are missing.', '1801.00115-1-15-0': '# Reducible QED', '1801.00115-1-16-0': 'One way out to aim at a rigorous theory of Quantum Electrodynamics is to give up one of the axioms of quantum field theory.', '1801.00115-1-16-1': 'In the context of constructive quantum field theory [CITATION] evidence exists that in 4 space time dimensions the only models satisfying the Wightman axioms are free-field models.', '1801.00115-1-16-2': 'Hence some modification of the underlying assumptions is indicated.', '1801.00115-1-17-0': 'A mild modification of the accepted body of axioms is to allow for reducible representations of the Lie algebra of canonical commutation and anti-commutation relations.', '1801.00115-1-17-1': 'Non-commuting observables are at the heart of quantum mechanics, while in classical mechanics all observables mutually commute.', '1801.00115-1-17-2': 'The two theories, classical and quantum, can be made as far apart as possible by requiring that in quantum mechanics the only observables which commute with all others are the multiplications with a scalar number.', '1801.00115-1-17-3': 'If this is the case then the representation of the algebra of observables is said to be irreducible.', '1801.00115-1-17-4': 'In addition, any representation can be decomposed into irreducible representations.', '1801.00115-1-17-5': 'However, the argument that it therefore suffices to study the irreducible representations is misleading.', '1801.00115-1-17-6': 'This may be so when the decomposition is discrete and involves a small number of irreducible components.', '1801.00115-1-17-7': 'The continuous decompositions considered in what follows add a degree of complexity rather than simplify the theory.', '1801.00115-1-18-0': 'Reducible QED is studied in the work of Czachor and collaborators.', '1801.00115-1-18-1': 'See [CITATION] and references given in these papers.', '1801.00115-1-18-2': 'The main assumption is that the irreducible representations are labeled by a three-dimensional wave vector [MATH] and that the decomposition of the reducible representation is then an integral over the wave vector [MATH].', '1801.00115-1-18-3': 'The recent version of [CITATION] differs from the original version of Czachor by making explicit that for every wave vector [MATH] there is a properly normalized wave function [MATH] and for every observable [MATH] there is a local copy [MATH] such that the quantum expectation of [MATH] is given by [EQUATION]', '1801.00115-1-18-4': 'For additive quantities such as the total energy this expression is quite obvious: the total energy of the field is obtained by integrating the wave vector-dependent energy density.', '1801.00115-1-19-0': 'Reducible representations are known in quantum field theory in the context of superselection rules.', '1801.00115-1-19-1': 'In the latter case one selects a single component [MATH] which represents the state of the system.', '1801.00115-1-19-2': 'Here the quantum field is represented by all [MATH] simultaneously.', '1801.00115-1-19-3': 'The selected wave vector [MATH] is just a filter through which the quantum field is looked at.', '1801.00115-1-20-0': '# World view', '1801.00115-1-21-0': 'Historically, the first picture one had of Quantum Electrodynamics was that of ordinary space filled with quantum harmonic oscillators, one pair at each point to cope with the two polarizations of the free electromagnetic field.', '1801.00115-1-21-1': 'This picture survives here with the modification that there is one pair of quantum oscillators for each wave vector [MATH].', '1801.00115-1-21-2': 'Hence, the Euclidean space is replaced by its Fourier space.', '1801.00115-1-21-3': 'The 3 classical dimensions appear because the representation of the two-dimensional quantum harmonic oscillator is not the irreducible one but is reducible.', '1801.00115-1-22-0': 'The representation of fermionic fields such as that of electrons and positrons is also reducible.', '1801.00115-1-22-1': 'The reducing wave vector is independent of the photonic wave vector.', '1801.00115-1-22-2': 'The Hilbert space of the irreducible fermionic representation is finite-dimensional.', '1801.00115-1-22-3': 'The electron/positron field has 16 independent states for each wave vector [MATH].', '1801.00115-1-22-4': 'Its Fock space is obtained from the vacuum state by the action of 4 fermionic creation operators adding an electron or a positron, each with either spin up or spin down.', '1801.00115-1-23-0': 'An important difference between reducible QED and QED of the Standard Model concerns the superposition of photons with unequal wave vector.', '1801.00115-1-23-1': 'In the recent version of reducible QED this superposition requires another field as intermediary.', '1801.00115-1-23-2': 'For instance, expression (13) of [CITATION] [EQUATION] becomes here [EQUATION]', '1801.00115-1-23-3': 'Here, [MATH] and [MATH] are for instance the spin up respectively spin down states of an auxiliary electron field.', '1801.00115-1-23-4': 'The wave function [MATH] describes the simultaneous presence of two combinations.', '1801.00115-1-23-5': 'The former describes two photons with wave vector [MATH] combined with an electron with spin up, the latter describes two photons with wave vector [MATH] combined with an electron with spin down.', '1801.00115-1-23-6': 'In conclusion: with a single field superpositions can be made at a given wave vector.', '1801.00115-1-23-7': 'Wave functions at different values of the wave vector are always present and do no require further superposition.', '1801.00115-1-23-8': 'Entangled states require two or more distinct fields.', '1801.00115-1-24-0': 'Many expressions in reducible QED look similar to those of common QED except that the integration over wave vectors is missing.', '1801.00115-1-24-1': 'It is shifted towards the evaluation of expectation values.', '1801.00115-1-24-2': 'Commutation and anti-commutation relations look different because Dirac delta functions are replaced by cosine and sine functions.', '1801.00115-1-24-3': 'An immediate advantage is that operator-valued distribution functions are avoided and that problems caused by ultra-violet divergences are postponed.', '1801.00115-1-25-0': 'In fact, it is not clear what happens with the electromagnetic field at large energy density.', '1801.00115-1-25-1': 'High energy fields can be obtained in two ways, either by increasing the energy of individual photons or by increasing the number of photons.', '1801.00115-1-25-2': 'Recent experiments are exploring the situation.', '1801.00115-1-25-3': 'See for instance [CITATION].', '1801.00115-1-26-0': '# The radiation gauge', '1801.00115-1-27-0': 'In the recent version of reducible QED [CITATION] only transversely polarized photons occur.', '1801.00115-1-27-1': 'The longitudinal and scalar photons of the traditional theory are absent.', '1801.00115-1-27-2': 'This implies that the number of degrees of freedom of the electromagnetic field is 2 rather than 3 or 4.', '1801.00115-1-27-3': 'There is no need for the construction of Gupta [CITATION] and Bleuler [CITATION], which intends to remove the unphysical degrees of freedom.', '1801.00115-1-27-4': 'This is an important simplification, which however raises a number of questions.', '1801.00115-1-28-0': 'In a theory containing only transverse photons it is obvious to use the radiation gauge.', '1801.00115-1-28-1': 'This is the Coulomb gauge, which is often used in Solid State Physics, in absence of Coulomb forces.', '1801.00115-1-28-2': 'A drawback of using this gauge is that it is not manifestly Lorentz covariant.', '1801.00115-1-28-3': 'What one wins by using this simplifying gauge is lost at the moment one considers a Lorentz boost.', '1801.00115-1-28-4': 'Then calculations, needed to restore the radiation gauge, are rather painful.', '1801.00115-1-28-5': 'However, this is not a fundamental problem.', '1801.00115-1-29-0': 'If no gauge freedom is left, what is then the role of gauge theories?', '1801.00115-1-29-1': 'They are the unifying concept behind the different boson fields appearing in the Standard Model.', '1801.00115-1-29-2': 'The reasoning goes that a global gauge symmetry of the free fields becomes a local symmetry of the interacting fields.', '1801.00115-1-29-3': 'Total charge [MATH] is a conserved quantity also in reducible QED.', '1801.00115-1-29-4': 'The corresponding symmetry group of unitary operators [MATH] corresponds with the U(1) gauge group of the Standard Model.', '1801.00115-1-29-5': 'It multiplies the wave function of an electron with a phase factor [MATH].', '1801.00115-1-29-6': ""It is shown in [CITATION] that also in reducible QED this global symmetry group can be extended with local symmetries, where 'local' now means local in the space of wave vectors."", '1801.00115-1-29-7': 'The constant [MATH] then becomes a function of the wave vectors [MATH] of the photon and [MATH] of the electron/positron field.', '1801.00115-1-29-8': 'It suffices that [MATH] remains constant when [MATH] is replace by [MATH].', '1801.00115-1-29-9': 'Then [MATH] is still a symmetry of the Hamiltonian.', '1801.00115-1-29-10': 'It reflects that the wave vector of the electron/positron field can only change by emission or absorption of a photon.', '1801.00115-1-30-0': '# Emergence of Coulomb forces', '1801.00115-1-31-0': 'The main concern for a theory involving only transverse photons is how to explain the Coulomb forces observed in nature.', '1801.00115-1-31-1': 'The explanation given in [CITATION] is based on an analogy with long range forces which act between polarons.', '1801.00115-1-31-2': 'The polaron [CITATION] is a concept of Solid State Physics.', '1801.00115-1-31-3': 'A free electron in a dielectric crystal interacts with lattice vibrations, called phonons, and can form a bound state with them.', '1801.00115-1-31-4': 'This bound state is the polaron.', '1801.00115-1-31-5': 'Two polarons interact with each other because they share the same lattice vibrations.', '1801.00115-1-31-6': 'This interaction is long-ranged.', '1801.00115-1-32-0': 'Similarly, an electron of QED interacts with the electromagnetic field.', '1801.00115-1-32-1': 'The distinction between a hypothetical bare electron and a dressed electron, surrounded by a cloud of photons, is as old as QED itself.', '1801.00115-1-32-2': 'The effect of the electron on itself via its interaction with the electromagnetic field yields a contribution to its energy.', '1801.00115-1-32-3': 'This is called the self-energy of the electron.', '1801.00115-1-32-4': 'In an electrostatic context the self-energy is the potential energy of the charge of the electron in its own Coulomb field.', '1801.00115-1-32-5': 'For a point particle this contribution is infinite.', '1801.00115-1-32-6': 'However, in the present paper the Coulomb field is absent by assumption.', '1801.00115-1-32-7': 'Hence, the problem of this divergent energy, in its original form, disappears.', '1801.00115-1-32-8': 'The interaction between the electron/positron field and the electromagnetic field still yields a static contribution to the energy.', '1801.00115-1-32-9': 'As discussed below it decreases the total energy instead of blowing it up to +infinity.', '1801.00115-1-32-10': 'In addition, the effects of the dynamic interaction between the two fields are very complicated.', '1801.00115-1-32-11': 'For a discussion of the latter see for instance [CITATION].', '1801.00115-1-33-0': 'In reducible QED one can prove [CITATION] that an electron can form a bound state with a transversely polarized photon field.', '1801.00115-1-33-1': 'The binding energy is minus twice the kinetic energy of the photon field.', '1801.00115-1-33-2': 'This result is typical for a linear interaction of the electron field with a photon field described by a quadratic Hamiltonian.', '1801.00115-1-33-3': 'It proves that in some cases the self-energy is negative.', '1801.00115-1-33-4': 'The bound state formed in this way is similar to the polaron.', '1801.00115-1-33-5': 'In addition, one can show that the binding also exists for photons with long wavelength and with a wave vector almost parallel to that of the electron field.', '1801.00115-1-33-6': 'This is important because it makes it plausible that different regions of the electron field develop a long-range interaction, which is then the Coulomb interaction.', '1801.00115-1-33-7': 'However, a mathematical proof that the Coulomb attraction and repulsion are reproduced by this mechanism is still missing.', '1801.00115-1-34-0': '# Discussion', '1801.00115-1-35-0': 'The assumption that the Coulomb forces are emergent instead of being fundamental forces has far reaching consequences.', '1801.00115-1-35-1': 'A number of problems are eliminated or can at least be avoided.', '1801.00115-1-35-2': 'No superfluous degrees of freedom appear because only transversely polarized photons are taken into account.', '1801.00115-1-35-3': 'The problem of the divergence of the self-energy of the electron is absent.', '1801.00115-1-36-0': 'The combination of reducible QED with the assumption of emergent Coulomb forces allows for the development of a mathematically consistent theory of QED.', '1801.00115-1-36-1': 'In particular, ultraviolet divergences are not hindering because the integration over wave vectors is postponed to the moment of evaluation of quantum expectations.', '1801.00115-1-36-2': 'In this context rigorous proofs can be given of the existence of bound states due to the interaction of the electron field with long wave length photons.', '1801.00115-1-36-3': 'An analogy with the polarons of Solid State Physics makes it then plausible that Coulomb forces are carried by these transversely polarized long wavelength photons.', '1801.00115-1-37-0': 'Some experimental evidence for the present version of reducible QED is found in Solid State Physics.', '1801.00115-1-37-1': 'It is generally accepted that free electrons in metals do not experience any long range Coulomb repulsion.', '1801.00115-1-37-2': 'In the present context this is an immediate consequence of the characteristic property of metals that long wavelength photons cannot propagate inside the material.', '1801.00115-1-37-3': 'A prediction of the present version of reducible QED is that entanglement of photons with distinct wave vectors requires an ancillary field.', '1801.00115-1-37-4': 'It is not easy to test this property because in any experiment entanglement with the environment is hard to avoid.', '1801.00115-1-38-0': 'The claim that the Coulomb forces are emergent, if correct, requires significant modifications to the Standard Model.', '1801.00115-1-38-1': 'The concept of gauge theories survives in a modified form, as indicated.', '1801.00115-1-38-2': 'The gauge freedom is not any longer due to the presence of superfluous degrees of freedom but is the consequence of working in a reducible representation.', '1801.00115-1-38-3': 'What this means for the weak and strong interactions has still to be investigated.', '1801.00115-1-39-0': 'It is tempting to extrapolate the present work in the direction of quantum gravity.', '1801.00115-1-39-1': 'Technical difficulties seem treatable.', '1801.00115-1-39-2': 'The existence of long wavelength gravitational waves has been established recently.', '1801.00115-1-39-3': 'However, the existence of the graviton as the quantum of the gravitational field is still an open question.', '1801.00115-1-39-4': 'It is also hard to believe that the gravitational forces, which we experience all the time, would be carried by low energy quantum particles about which we do not know anything.'}","{'1801.00115-2-0-0': 'This paper discusses an attempt to develop a mathematically rigorous theory of Quantum Electrodynamics (QED).', '1801.00115-2-0-1': 'It deviates from the standard version of QED mainly in two aspects: it is assumed that the Coulomb forces are carried by transversely polarized photons, and a reducible representation of the canonical commutation and anti-commutation relations is used.', '1801.00115-2-0-2': 'Both interventions together should suffice to eliminate the mathematical inconsistencies of standard QED.', '1801.00115-2-1-0': '# Introduction', '1801.00115-2-2-0': 'In recent work [CITATION] Erik Verlinde formulated the claim that gravity is an emergent force.', '1801.00115-2-2-1': 'By this is meant that gravitational forces can be derived from other, more fundamental forces.', '1801.00115-2-2-2': 'Coulomb forces and gravitational forces have much in common.', '1801.00115-2-2-3': 'Both are inversely proportional to the square of the distance.', '1801.00115-2-2-4': 'And it is difficult to reconcile them with the aversion of physicists to action at a distance.', '1801.00115-2-2-5': 'It is therefore obvious to claim that also the Coulomb forces are emergent forces [CITATION].', '1801.00115-2-2-6': 'While the claim of Verlinde is made in the context of cosmology the emergence of the Coulomb forces is formulated here in the context of quantum field theory.', '1801.00115-2-3-0': 'Two arguments are brought forward.', '1801.00115-2-3-1': 'In the common theory of Quantum Electrodynamics (QED), which includes Coulomb forces, it is possible to remove them by a simple transformation of observables [CITATION].', '1801.00115-2-3-2': 'The inverse transformation can be used to reintroduce Coulomb forces given a theory in which they are absent (See Appendix [REF]).', '1801.00115-2-3-3': 'The other argument is based on a mathematical proof that electrons bind with long-wavelength photons.', '1801.00115-2-3-4': 'This mechanism is discussed further on in the present paper.', '1801.00115-2-3-5': 'Evidence for the feasibility of this kind of binding has been reported recently [CITATION] in experiments involving single spins in silicon quantum dots binding with long-wavelength microcavity photons.', '1801.00115-2-4-0': 'A theory of Quantum Electrodynamics which does not include Coulomb forces is appealing because it can avoid some of the technical problems which plague the standard version of the theory.', '1801.00115-2-4-1': 'The claim made in [CITATION] is that Coulomb attraction and repulsion are carried by transversely polarized photons.', '1801.00115-2-4-2': 'This eliminates the need for longitudinal and scalar photons and allows for building a rigorous theory of QED.', '1801.00115-2-4-3': 'In the present paper the mathematical arguments supporting this claim are presented in appendix.', '1801.00115-2-5-0': '# The Standard Model', '1801.00115-2-6-0': 'The Standard Model of elementary particles summarizes much of our present day understanding of the fundamental laws of physics.', '1801.00115-2-6-1': 'It is a highly effective theory.', '1801.00115-2-6-2': 'It explains almost all phenomena with an amazing degree of precision.', '1801.00115-2-6-3': ""Never the less, it cannot be the final theory of physics, in the first place because the integration with Einstein's general relativity theory is missing."", '1801.00115-2-7-0': 'The Standard Model is a quantum field theory, which implies that particles are described by fields, for instance a photon field or an electron field.', '1801.00115-2-7-1': 'Creation and annihilation operators add or remove one quantum of the corresponding field.', '1801.00115-2-7-2': 'This wording suggests that particles are created or annihilated.', '1801.00115-2-7-3': 'However, it is more careful to say that creation and annihilation operators are a tool to construct quantum fields and to describe interaction processes.', '1801.00115-2-8-0': 'Two different fields can interact with each other by the exchange of one or more quanta.', '1801.00115-2-8-1': 'Einstein proposed this mechanism to explain the photo-electric effect.', '1801.00115-2-8-2': 'This assumption is a corner stone of all quantum theories.', '1801.00115-2-8-3': 'The interesting question why the exchange of quanta occurs at seemingly random discrete moments of time is not discussed here.', '1801.00115-2-9-0': 'The prototype of a quantum field theory is Quantum Electrodynamics (QED), the relativistic quantum theory of electromagnetism.', '1801.00115-2-9-1': 'QED was extended to include all electro-weak interactions, and later on also strong interactions.', '1801.00115-2-9-2': 'Much effort goes into the exploration of further extensions.', '1801.00115-2-9-3': 'The whole construction has become an inverted pyramid resting on a few basic principles most of which were decided on in the early days of quantum mechanics.', '1801.00115-2-9-4': 'It is therefore of utmost importance that existing holes in these foundations are eliminated.', '1801.00115-2-10-0': 'It is indeed worrying that the common formulation of QED is mathematically inconsistent.', '1801.00115-2-10-1': 'A first reason for that is the perturbative approach, which involves a non-converging series expansion.', '1801.00115-2-10-2': 'It is even worse: many of the individual terms of the expansion are ill-defined.', '1801.00115-2-10-3': 'As a consequence, repair techniques are needed.', '1801.00115-2-10-4': 'An alternative is offered by non-perturbative QED.', '1801.00115-2-10-5': 'But even then not all problems disappear.', '1801.00115-2-11-0': '# Mathematical problems', '1801.00115-2-12-0': 'The technical difficulties of QED have their consequences.', '1801.00115-2-12-1': 'Certain questions cannot be treated in a reliable manner.', '1801.00115-2-12-2': 'Let me mention one.', '1801.00115-2-13-0': 'The physical vacuum differs from the mathematical vacuum.', '1801.00115-2-13-1': 'The latter is defined by a vanishing of all fields.', '1801.00115-2-13-2': 'However it is not the state of minimal energy.', '1801.00115-2-13-3': 'The interaction between different fields creates states of negative energy.', '1801.00115-2-13-4': 'A well-known example in the quantum mechanics of particles is the hydrogen atom.', '1801.00115-2-13-5': 'Its eigenstates all have negative energy while the scattering states have positive energy.', '1801.00115-2-13-6': 'By analogy one can expect that in QED there exist states of negative energy in which the electromagnetic field and the electron/positron field are bound in some way.', '1801.00115-2-13-7': 'If this expectation is correct then the state of minimal energy is the physical vacuum.', '1801.00115-2-13-8': 'No mathematical proof of its existence exists.', '1801.00115-2-13-9': 'In a non-relativistic context Lieb and coworkers (See for instance [CITATION]) prove the existence of a ground state for models consisting of particles which do not interact among themselves but do interact with the radiation field.', '1801.00115-2-13-10': 'An ultra-violet cutoff is applied to the latter.', '1801.00115-2-14-0': 'In Quantum Chromodynamics (QCD) one accepts that a symmetry-breaking phase transition occurs at low energies.', '1801.00115-2-14-1': 'Such a phase transition gives a satisfactory explanation for the confinement phenomenon.', '1801.00115-2-14-2': 'Also the Englert-Brout-Higgs mechanism, which in the Standard Model is responsible for giving mass to particles, invokes a phase transition.', '1801.00115-2-14-3': 'Because these phase transitions are continuous and symmetry-breaking the physical ground state is non-unique and is accompanied by Goldstone bosons.', '1801.00115-2-14-4': 'Mathematical proofs for these statements are missing.', '1801.00115-2-15-0': '# Reducible QED', '1801.00115-2-16-0': 'One way out to aim at a rigorous theory of Quantum Electrodynamics is to give up one of the axioms of quantum field theory.', '1801.00115-2-16-1': 'In the context of constructive quantum field theory [CITATION] evidence exists that in Minkowski space the only models satisfying the Wightman axioms are free-field models.', '1801.00115-2-16-2': 'Hence some modification of the underlying assumptions is indicated.', '1801.00115-2-17-0': 'A mild modification of the accepted body of axioms is to allow for reducible representations of the Lie algebra of canonical commutation and anti-commutation relations.', '1801.00115-2-17-1': 'Non-commuting observables are at the heart of quantum mechanics, while in classical mechanics all observables mutually commute.', '1801.00115-2-17-2': 'The two theories, classical and quantum, can be made as far apart as possible by requiring that in quantum mechanics the only observables which commute with all others are the multiplications with a scalar number.', '1801.00115-2-17-3': 'If this is the case then the representation of the algebra of observables is said to be irreducible.', '1801.00115-2-17-4': 'In addition, any representation can be decomposed into irreducible representations [CITATION].', '1801.00115-2-17-5': 'However, the argument that it therefore suffices to study the irreducible representations is misleading.', '1801.00115-2-17-6': 'This may be so when the decomposition is discrete and involves a small number of irreducible components.', '1801.00115-2-17-7': 'The continuous decompositions considered in what follows add a degree of complexity rather than simplify the theory.', '1801.00115-2-18-0': 'Reducible QED is studied in the work of Czachor and collaborators (See [CITATION] and references given in these papers).', '1801.00115-2-18-1': 'The main assumption is that the irreducible representations are labeled by a three-dimensional wave vector [MATH] and that the decomposition of the reducible representation is then an integral over the wave vector [MATH].', '1801.00115-2-18-2': 'The version presented here differs from the original version of Czachor by making explicit that for every wave vector [MATH] there is a properly normalized wave function [MATH] and for every observable [MATH] there is a local copy [MATH] such that the quantum expectation of [MATH] is given by [EQUATION]', '1801.00115-2-18-3': 'For additive quantities such as the total energy this expression is quite obvious: the total energy of the field is obtained by integrating the wave vector-dependent energy density.', '1801.00115-2-18-4': 'Details of this formalism are found in Appendix [REF].', '1801.00115-2-19-0': 'Reducible representations are known in quantum field theory in the context of superselection rules.', '1801.00115-2-19-1': 'In the latter case one selects a single component [MATH] which represents the state of the system.', '1801.00115-2-19-2': 'Here the quantum field is represented by all [MATH] simultaneously.', '1801.00115-2-19-3': 'The selected wave vector [MATH] is just a filter through which the quantum field is looked at.', '1801.00115-2-20-0': '# World view', '1801.00115-2-21-0': 'Historically, the first picture one had of Quantum Electrodynamics was that of ordinary space filled with quantum harmonic oscillators, one pair at each point to cope with the two polarizations of the free electromagnetic field.', '1801.00115-2-21-1': 'This picture survives here with the modification that there is one pair of quantum oscillators for each wave vector [MATH].', '1801.00115-2-21-2': 'Hence, the Euclidean space is replaced by its Fourier space.', '1801.00115-2-21-3': 'The 3 classical dimensions appear because the representation of the two-dimensional quantum harmonic oscillator is not the irreducible one but is reducible.', '1801.00115-2-22-0': 'The representation of fermionic fields such as that of electrons and positrons is also reducible.', '1801.00115-2-22-1': 'The reducing wave vector is independent of the photonic wave vector.', '1801.00115-2-22-2': 'The Hilbert space of the irreducible fermionic representation is finite-dimensional.', '1801.00115-2-22-3': 'The electron/positron field has 16 independent states for each wave vector [MATH].', '1801.00115-2-22-4': 'Its Fock space is obtained from the vacuum state by the action of 4 fermionic creation operators adding an electron or a positron, each with either spin up or spin down.', '1801.00115-2-23-0': 'An important difference between reducible QED and QED of the Standard Model concerns the superposition of photons with unequal wave vector.', '1801.00115-2-23-1': 'In the present version of reducible QED this superposition requires another field as intermediary.', '1801.00115-2-23-2': 'For instance, expression (13) of [CITATION] [EQUATION] becomes here [EQUATION]', '1801.00115-2-23-3': 'Here, [MATH] and [MATH] are for instance the spin up respectively spin down states of an auxiliary electron field.', '1801.00115-2-23-4': 'The wave function [MATH] describes the simultaneous presence of two combinations.', '1801.00115-2-23-5': 'The former describes two photons with wave vector [MATH] combined with an electron with spin up, the latter describes two photons with wave vector [MATH] combined with an electron with spin down.', '1801.00115-2-23-6': 'In conclusion: with a single field superpositions can be made at a given wave vector.', '1801.00115-2-23-7': 'Wave functions at different values of the wave vector are always present and do no require further superposition.', '1801.00115-2-23-8': 'Entangled states require two or more distinct fields.', '1801.00115-2-24-0': 'Many expressions in reducible QED look similar to those of common QED except that the integration over wave vectors is missing.', '1801.00115-2-24-1': 'It is shifted towards the evaluation of expectation values.', '1801.00115-2-24-2': 'Commutation and anti-commutation relations look different because Dirac delta functions are replaced by cosine and sine functions.', '1801.00115-2-24-3': 'An immediate advantage is that operator-valued distribution functions are avoided and that problems caused by ultra-violet divergences are postponed.', '1801.00115-2-25-0': 'In fact, it is not clear what happens with the electromagnetic field at large energy density.', '1801.00115-2-25-1': 'High energy fields can be obtained in two ways, either by increasing the energy of individual photons or by increasing the number of photons.', '1801.00115-2-25-2': 'Recent experiments are exploring the situation (See for instance [CITATION]).', '1801.00115-2-26-0': '# The radiation gauge', '1801.00115-2-27-0': 'In the present version of reducible QED only transversely polarized photons occur (See Appendix [REF], [REF]).', '1801.00115-2-27-1': 'The longitudinal and scalar photons of the traditional theory are absent.', '1801.00115-2-27-2': 'This implies that the number of degrees of freedom of the electromagnetic field is 2 rather than 3 or 4.', '1801.00115-2-27-3': 'There is no need for the construction of Gupta [CITATION] and Bleuler [CITATION], which intends to remove the nonphysical degrees of freedom.', '1801.00115-2-27-4': 'This is an important simplification, which however raises a number of questions.', '1801.00115-2-28-0': 'In a theory containing only transverse photons it is obvious to use the radiation gauge.', '1801.00115-2-28-1': 'This is the Coulomb gauge [CITATION], which is often used in Solid State Physics, in absence of Coulomb forces.', '1801.00115-2-28-2': 'A drawback of using this gauge is that it is not manifestly Lorentz covariant.', '1801.00115-2-28-3': 'What one wins by using this simplifying gauge is lost at the moment one considers a Lorentz boost.', '1801.00115-2-28-4': 'Then calculations, needed to restore the radiation gauge, are rather painful.', '1801.00115-2-28-5': 'However, this is not a fundamental problem.', '1801.00115-2-29-0': 'If no gauge freedom is left, what is then the role of gauge theories?', '1801.00115-2-29-1': 'They are the unifying concept behind the different boson fields appearing in the Standard Model.', '1801.00115-2-29-2': 'The reasoning goes that a global gauge symmetry of the free fields becomes a local symmetry of the interacting fields.', '1801.00115-2-29-3': 'Total charge [MATH] is a conserved quantity also in reducible QED.', '1801.00115-2-29-4': 'The corresponding symmetry group of unitary operators [MATH] corresponds with the U(1) gauge group of the Standard Model.', '1801.00115-2-29-5': 'It multiplies the wave function of an electron with a phase factor [MATH].', '1801.00115-2-29-6': ""It is shown in Appendix [REF] that also in reducible QED this global symmetry group can be extended with local symmetries, where 'local' now means local in the space of wave vectors."", '1801.00115-2-29-7': 'The constant [MATH] then becomes a function of the wave vectors [MATH] of the photon and [MATH] of the electron/positron field.', '1801.00115-2-29-8': 'It suffices that [MATH] remains constant when [MATH] is replace by [MATH].', '1801.00115-2-29-9': 'Then [MATH] is still a symmetry of the Hamiltonian.', '1801.00115-2-29-10': 'It reflects that the wave vector of the electron/positron field can only change by emission or absorption of a photon.', '1801.00115-2-30-0': '# Emergence of Coulomb forces', '1801.00115-2-31-0': 'The main concern for a theory involving only transverse photons is how to explain the Coulomb forces observed in nature.', '1801.00115-2-31-1': 'The explanation given in the present theory is based on an analogy with long range forces which act between polarons.', '1801.00115-2-31-2': 'The polaron [CITATION] is a concept of Solid State Physics.', '1801.00115-2-31-3': 'A free electron in a dielectric crystal interacts with lattice vibrations, called phonons, and can form a bound state with them.', '1801.00115-2-31-4': 'This bound state is the polaron.', '1801.00115-2-31-5': 'Two polarons interact with each other because they share the same lattice vibrations.', '1801.00115-2-31-6': 'This interaction is long-ranged.', '1801.00115-2-32-0': 'Similarly, an electron of QED interacts with the electromagnetic field.', '1801.00115-2-32-1': 'The distinction between a hypothetical bare electron and a dressed electron, surrounded by a cloud of photons, is as old as QED itself.', '1801.00115-2-32-2': 'The effect of the electron on itself via its interaction with the electromagnetic field yields a contribution to its energy.', '1801.00115-2-32-3': 'This is called the self-energy of the electron.', '1801.00115-2-32-4': 'In an electrostatic context the self-energy is the potential energy of the charge of the electron in its own Coulomb field.', '1801.00115-2-32-5': 'For a point particle this contribution is infinite.', '1801.00115-2-32-6': 'However, in the present paper the Coulomb field is absent by assumption.', '1801.00115-2-32-7': 'Hence, the problem of this divergent energy, in its original form, disappears.', '1801.00115-2-32-8': 'The interaction between the electron/positron field and the electromagnetic field still yields a static contribution to the energy.', '1801.00115-2-32-9': 'As discussed below it decreases the total energy instead of blowing it up to +infinity.', '1801.00115-2-32-10': 'In addition, the effects of the dynamic interaction between the two fields are very complicated.', '1801.00115-2-32-11': 'For a discussion of the latter see for instance [CITATION].', '1801.00115-2-33-0': 'In reducible QED one can prove that an electron can form a bound state with a transversely polarized photon field (See Appendix [REF]).', '1801.00115-2-33-1': 'The binding energy is minus twice the kinetic energy of the photon field.', '1801.00115-2-33-2': 'This result is typical for a linear interaction of the electron field with a photon field described by a quadratic Hamiltonian.', '1801.00115-2-33-3': 'In some cases the self-energy is negative (See Appendix [REF]).', '1801.00115-2-33-4': 'The bound state formed in this way is similar to the polaron.', '1801.00115-2-33-5': 'In addition, Appendix [REF] shows that the binding also exists for photons with long wavelength and with a wave vector almost parallel to that of the electron field.', '1801.00115-2-33-6': 'This is important because it makes it plausible that different regions of the electron field develop a long-range interaction, which is then the Coulomb interaction.', '1801.00115-2-33-7': 'However, a mathematical proof that the Coulomb attraction and repulsion are reproduced by this mechanism is still missing.', '1801.00115-2-34-0': '# Discussion', '1801.00115-2-35-0': 'The assumption that the Coulomb forces are emergent instead of being fundamental forces has far reaching consequences, some of which have been discussed above.', '1801.00115-2-35-1': 'A number of problems of the standard theory are eliminated or can at least be avoided.', '1801.00115-2-35-2': 'No superfluous degrees of freedom appear because only transversely polarized photons are taken into account.', '1801.00115-2-35-3': 'The problem of the divergence of the self-energy of the electron is absent.', '1801.00115-2-36-0': 'The combination of reducible QED with the assumption of emergent Coulomb forces allows for the development of a mathematically consistent theory of QED.', '1801.00115-2-36-1': 'In particular, ultraviolet divergences are not hindering because the integration over wave vectors is postponed to the moment of evaluation of quantum expectations.', '1801.00115-2-36-2': 'In this context rigorous proofs can be given of the existence of bound states due to the interaction of the electron field with long wave length photons.', '1801.00115-2-36-3': 'An analogy with the polarons of Solid State Physics makes it then plausible that Coulomb forces are carried by these transversely polarized long wavelength photons.', '1801.00115-2-37-0': 'Some experimental evidence for the present version of reducible QED is found in Solid State Physics.', '1801.00115-2-37-1': 'It is generally accepted that free electrons in metals do not experience any long range Coulomb repulsion.', '1801.00115-2-37-2': 'In the present context this is an immediate consequence of the characteristic property of metals that long wavelength photons cannot propagate inside the material.', '1801.00115-2-37-3': 'Evidence for the binding of the electron spin with long-range photons is given in [CITATION].', '1801.00115-2-37-4': 'A prediction of the present version of reducible QED is that entanglement of photons with distinct wave vectors requires an ancillary field.', '1801.00115-2-37-5': 'It is not easy to test this property because in any experiment entanglement with the environment is hard to avoid.', '1801.00115-2-38-0': 'The claim that the Coulomb forces are emergent, if correct, requires significant modifications to the Standard Model.', '1801.00115-2-38-1': 'The concept of gauge theories survives in a modified form, as indicated.', '1801.00115-2-38-2': 'The gauge freedom is not any longer due to the presence of superfluous degrees of freedom but is the consequence of working in a reducible representation.', '1801.00115-2-38-3': 'What this means for the weak and strong interactions has still to be investigated.', '1801.00115-2-39-0': 'It is tempting to extrapolate the present work in the direction of quantum gravity.', '1801.00115-2-39-1': 'Technical difficulties seem treatable.', '1801.00115-2-39-2': 'The existence of long wavelength gravitational waves has been established recently [CITATION].', '1801.00115-2-39-3': 'However, the existence of the graviton as the quantum of the gravitational field is still an open question.', '1801.00115-2-39-4': 'It is also hard to believe that the gravitational forces, which we experience all the time, would be carried by low energy quantum particles about which we do not know anything.', '1801.00115-2-40-0': '# Appendices', '1801.00115-2-41-0': '# Reducible quantum fields', '1801.00115-2-42-0': '## Definition', '1801.00115-2-43-0': 'Let [MATH] is a given Hilbert space, either finite dimensional or separable, and [MATH] an open subset of [MATH].', '1801.00115-2-43-1': 'In the sequel [MATH] will be either [MATH] or [MATH].', '1801.00115-2-43-2': 'Normalized elements of [MATH] are called wave functions, elements of [MATH] are called wave vectors.', '1801.00115-2-43-3': 'Maps of [MATH] into [MATH] are called quantum fields.', '1801.00115-2-44-0': 'Let [MATH] denote the linear space of continuous fields [EQUATION] where [MATH] is the annihilation operator introduced before.', '1801.00115-2-44-1': 'Its domain of definition is the subspace of [MATH] consisting of all [MATH] in [MATH] such that', '1801.00115-2-45-0': 'Physically acceptable free fields necessarily are superpositions with the vacuum field.', '1801.00115-2-45-1': 'Only then it is feasible to obtain a finite value for the expectation value of the energy operator [MATH].', '1801.00115-2-46-0': '## The classical wave equation', '1801.00115-2-47-0': 'A large class of solutions of the free wave equation [MATH] consists of functions [MATH] of the form [EQUATION] where [MATH] is a continuous function of [MATH].', '1801.00115-2-48-0': 'The so-called normalization factor [MATH] is the usual one [EQUATION] except that the constant [MATH] is inserted also here to make it dimensionless.', '1801.00115-2-48-1': 'The insertion of this normalization factor leads further on to a satisfactory physical interpretation of the profile function [MATH].', '1801.00115-2-49-0': 'The total energy of the classical field [MATH] is given by [EQUATION]', '1801.00115-2-49-1': 'From ([REF]) one obtains [EQUATION]', '1801.00115-2-49-2': 'The interpretation in the context of quantum mechanics is standard.', '1801.00115-2-49-3': 'The factor [MATH] is the density of particles with wave vector [MATH] and corresponding energy [MATH].', '1801.00115-2-50-0': 'The particle density [MATH] has the dimension of the inverse of a volume in [MATH]-space.', '1801.00115-2-50-1': 'Introduce therefore the dimensionless function [EQUATION]', '1801.00115-2-51-0': '## Correspondence principle', '1801.00115-2-52-0': 'Introduce field operators [MATH], with [MATH] in Minkowski space [MATH], defined by [EQUATION]', '1801.00115-2-52-1': 'The eigenstates [MATH], [MATH] of the harmonic oscillator belong to the domain of the r.h.s. of ([REF]), as well as all coherent states [MATH], [MATH].', '1801.00115-2-52-2': 'It is obvious to define [MATH] as the self-adjoint extension of the r.h.s. of ([REF]).', '1801.00115-2-52-3': 'The map [MATH] defines a diagonal operator [MATH] of [MATH].', '1801.00115-2-52-4': 'It is called the free field operator.', '1801.00115-2-53-0': 'The free field operators satisfy the commutation relations [EQUATION]', '1801.00115-2-54-0': '## Field operators', '1801.00115-2-55-0': 'Because the electromagnetic wave has two polarizations it is obvious to consider a 2-dimensional quantum harmonic oscillator instead of the single oscillator used in Appendix [REF] on scalar bosons.', '1801.00115-2-56-0': 'Let [MATH] and [MATH] be the annihilation operators for a photon with horizontal respectively vertical polarization.', '1801.00115-2-56-1': 'The free-field Hamiltonian of the quantized electromagnetic field [MATH] is the diagonal operator defined by [EQUATION]', '1801.00115-2-56-2': 'Field operators [MATH] are defined by [EQUATION] with polarization vectors [MATH] and [MATH] given by two rows of the rotation matrix [MATH] [EQUATION]', '1801.00115-2-56-3': 'This is of the form ([REF]) with [EQUATION] because [EQUATION] vanishes, as well as a similar expression for the vertical polarization.', '1801.00115-2-57-0': 'Finally let us calculate the commutation relations [EQUATION]', '1801.00115-2-57-1': ""For [MATH] it reduces to the d'Alembert equation [MATH], discussed in Appendix [REF]."", '1801.00115-2-57-2': 'Propagating wave solutions are of the same form as in Appendix [REF] [EQUATION] but with a dispersion relation given by the positive square root [EQUATION]', '1801.00115-2-58-0': '## Fermionic state space', '1801.00115-2-59-0': 'Let [MATH] denote the linear space of continuous fields [MATH].', '1801.00115-2-59-1': 'Like in the case of bosonic fields it is a locally convex Hausdorff space.', '1801.00115-2-59-2': 'However, because the Hilbert space [MATH] is finite-dimensional it is also a Banach space.', '1801.00115-2-59-3': 'An element [MATH] of [MATH] is said to be properly normalized if [MATH] for all [MATH].', '1801.00115-2-59-4': 'States of the fermionic quantum field theory are represented by properly normalized fields.', '1801.00115-2-60-0': 'Note that any properly normalize field [MATH] of [MATH] can be written into the form [EQUATION] where [MATH], [MATH] and [MATH] are real-valued functions of [MATH].', '1801.00115-2-60-1': 'By adopting this way of writing one tacitly assumes that [MATH] means absence of the fermion, while [MATH] means presence of the fermion.', '1801.00115-2-60-2': 'Note the analogy with single photon states as described in Appendix [REF].', '1801.00115-2-60-3': 'With this interpretation [MATH] becomes the density of the fermion field with wave vector [MATH].', '1801.00115-2-61-0': 'The Hamiltonian ([REF]) of Larmor precession defines a diagonal operator [MATH] by [MATH] with [EQUATION]', '1801.00115-2-61-1': 'A constant matrix has been added to make the Hamiltonian non-negative.', '1801.00115-2-61-2': 'This does not change the dynamics of the Larmor precession.', '1801.00115-2-61-3': 'The domain of definition of [MATH] is all of [MATH].', '1801.00115-2-62-0': 'With the help of ([REF]) the quantum expectation of the Hamiltonian becomes [EQUATION]', '1801.00115-2-62-1': 'This reveals that [MATH] is a distribution of quantum particles with dispersion relation [MATH].', '1801.00115-2-62-2': 'It is restricted by the condition that [MATH] for all [MATH].', '1801.00115-2-62-3': 'Because the energy must remain finite the distribution [MATH] should go to 0 fast enough for large values of the wave vector [MATH].', '1801.00115-2-63-0': '## Field operator', '1801.00115-2-64-0': 'Introduce now the field operator [MATH] defined by [MATH] with [EQUATION]', '1801.00115-2-64-1': 'It is tradition to decompose this field operator into so-called positive-frequency and negative-frequency operators [EQUATION]', '1801.00115-2-64-2': 'These anti-commutation relations are non-canonical.', '1801.00115-2-64-3': 'Note that [EQUATION]', '1801.00115-2-64-4': 'The integral converges provided that the density [MATH] tends fast enough either to 0 or to 1 for large values of [MATH].', '1801.00115-2-64-5': 'The expression ([REF]) is of the form ([REF]) with [EQUATION]', '1801.00115-2-64-6': 'Note that ([REF]) implies that [EQUATION]', '1801.00115-2-64-7': 'Note that the Hamiltonian is positive.', '1801.00115-2-64-8': 'It is tradition to assign negative energies to positrons and positive energies to electrons.', '1801.00115-2-64-9': 'This tradition is not followed here because it does not make sense.', '1801.00115-2-64-10': ""It is a remainder of Dirac's interpretation of positrons as holes in a sea of electrons."", '1801.00115-2-64-11': 'The alternative treatment assigns the vacuum state to one of the eigenstates of [MATH] instead of assigning a particle/anti-particle pair to the two eigenstates.', '1801.00115-2-64-12': 'The dimension of the Hilbert space goes up from 4 (the number of components of a Dirac spinor) to 16.', '1801.00115-2-64-13': 'This is meaningful because the Dirac equation considered here is an equation for field operators and not the original one which holds for classical field spinors (See ([REF]) below).', '1801.00115-2-65-0': 'Number operators [MATH] are defined by [EQUATION]', '1801.00115-2-65-1': 'They appear in the Hamiltonian [EQUATION]', '1801.00115-2-65-2': ""The field operators [MATH] satisfy Heisenberg's equations of motion [EQUATION]"", '1801.00115-2-65-3': 'The vectors [MATH] are the analogues of the polarization vectors of the photon.', '1801.00115-2-65-4': ""They are partly fixed by the requirement that the vector with components [MATH] satisfies Dirac's equation [EQUATION]"", '1801.00115-2-65-5': 'Indeed, using [EQUATION]', '1801.00115-2-65-6': 'An electron/positron field is now determined by a properly normalized field [MATH] of the form [EQUATION] whenever the integral converges.', '1801.00115-2-65-7': 'This Dirac spinor [MATH] with 4 components satisfies the Dirac equation [EQUATION]', '1801.00115-2-65-8': 'Finally note that each of the Dirac field operators, as constructed here, is not only a solution of the Klein-Gordon equation [EQUATION] but also of the partial equations [EQUATION]', '1801.00115-2-66-0': '# The Dirac current', '1801.00115-2-67-0': '## Two-point correlations', '1801.00115-2-68-0': 'Fix a properly normalized electron field [MATH].', '1801.00115-2-68-1': 'A two-point correlation function for the Dirac field operators [MATH] is defined by [EQUATION] whenever the integrals converge.', '1801.00115-2-68-2': 'Note the order of the indices [MATH].', '1801.00115-2-68-3': ""A short calculation using Dirac's equation shows that the vector [MATH] with 4 components [EQUATION]"", '1801.00115-2-68-4': 'The vector [MATH], introduced above, describes a current, which however is not yet the electric current.', '1801.00115-2-68-5': 'The components of [MATH] are real numbers.', '1801.00115-2-68-6': 'Indeed, using [MATH] one verifies that [EQUATION]', '1801.00115-2-68-7': 'Here, [MATH] is a unit of charge.', '1801.00115-2-68-8': 'The domain of definition of [MATH] consists of the fields [MATH] for which the integrals [EQUATION]', '1801.00115-2-68-9': 'This is a well-known expression for the Dirac current, adapted to the present context.', '1801.00115-2-69-0': 'The total charge [MATH] is the diagonal operator satisfying [EQUATION]', '1801.00115-2-69-1': 'One finds [EQUATION]', '1801.00115-2-69-2': 'The kinetic energy of the photon field is given by ([REF]) [EQUATION] with [EQUATION]', '1801.00115-2-69-3': 'The density of the electron field equals [EQUATION]', '1801.00115-2-70-0': '## Long-wavelength analysis', '1801.00115-2-71-0': 'Above it is shown that for a wave function of the form [EQUATION]', '1801.00115-2-71-1': 'This gives [EQUATION]', '1801.00115-2-71-2': 'Here, [MATH] is the time evolution of the interacting system.', '1801.00115-2-71-3': 'The new operators are marked with a double prime to distinguish them from the operators of the non-interacting system and those of the interacting system.', '1801.00115-2-71-4': 'The latter are denoted with a single prime.', '1801.00115-2-71-5': ""One verifies immediately that Gauss' law is satisfied [EQUATION]""}","[['1801.00115-1-22-0', '1801.00115-2-22-0'], ['1801.00115-1-22-1', '1801.00115-2-22-1'], ['1801.00115-1-22-2', '1801.00115-2-22-2'], ['1801.00115-1-22-3', '1801.00115-2-22-3'], ['1801.00115-1-22-4', '1801.00115-2-22-4'], ['1801.00115-1-38-0', '1801.00115-2-38-0'], ['1801.00115-1-38-1', 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['1801.00115-1-31-5', '1801.00115-2-31-5'], ['1801.00115-1-31-6', '1801.00115-2-31-6'], ['1801.00115-1-37-0', '1801.00115-2-37-0'], ['1801.00115-1-37-1', '1801.00115-2-37-1'], ['1801.00115-1-37-2', '1801.00115-2-37-2'], ['1801.00115-1-37-3', '1801.00115-2-37-4'], ['1801.00115-1-37-4', '1801.00115-2-37-5'], ['1801.00115-1-32-0', '1801.00115-2-32-0'], ['1801.00115-1-32-1', '1801.00115-2-32-1'], ['1801.00115-1-32-2', '1801.00115-2-32-2'], ['1801.00115-1-32-3', '1801.00115-2-32-3'], ['1801.00115-1-32-4', '1801.00115-2-32-4'], ['1801.00115-1-32-5', '1801.00115-2-32-5'], ['1801.00115-1-32-6', '1801.00115-2-32-6'], ['1801.00115-1-32-7', '1801.00115-2-32-7'], ['1801.00115-1-32-8', '1801.00115-2-32-8'], ['1801.00115-1-32-9', '1801.00115-2-32-9'], ['1801.00115-1-32-10', '1801.00115-2-32-10'], ['1801.00115-1-32-11', '1801.00115-2-32-11'], ['1801.00115-1-8-0', '1801.00115-2-8-0'], ['1801.00115-1-8-1', '1801.00115-2-8-1'], ['1801.00115-1-8-2', '1801.00115-2-8-2'], ['1801.00115-1-8-3', 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['1801.00115-1-23-3', '1801.00115-2-23-3'], ['1801.00115-1-23-4', '1801.00115-2-23-4'], ['1801.00115-1-23-5', '1801.00115-2-23-5'], ['1801.00115-1-23-6', '1801.00115-2-23-6'], ['1801.00115-1-23-7', '1801.00115-2-23-7'], ['1801.00115-1-23-8', '1801.00115-2-23-8'], ['1801.00115-1-14-0', '1801.00115-2-14-0'], ['1801.00115-1-14-1', '1801.00115-2-14-1'], ['1801.00115-1-14-2', '1801.00115-2-14-2'], ['1801.00115-1-14-3', '1801.00115-2-14-3'], ['1801.00115-1-14-4', '1801.00115-2-14-4'], ['1801.00115-1-24-0', '1801.00115-2-24-0'], ['1801.00115-1-24-1', '1801.00115-2-24-1'], ['1801.00115-1-24-2', '1801.00115-2-24-2'], ['1801.00115-1-24-3', '1801.00115-2-24-3'], ['1801.00115-1-2-0', '1801.00115-2-2-0'], ['1801.00115-1-2-1', '1801.00115-2-2-1'], ['1801.00115-1-2-2', '1801.00115-2-2-2'], ['1801.00115-1-2-3', '1801.00115-2-2-3'], ['1801.00115-1-2-4', '1801.00115-2-2-4'], ['1801.00115-1-16-0', '1801.00115-2-16-0'], ['1801.00115-1-16-2', '1801.00115-2-16-2'], ['1801.00115-1-9-0', '1801.00115-2-9-0'], ['1801.00115-1-9-1', '1801.00115-2-9-1'], ['1801.00115-1-9-2', '1801.00115-2-9-2'], ['1801.00115-1-9-3', '1801.00115-2-9-3'], ['1801.00115-1-9-4', '1801.00115-2-9-4'], ['1801.00115-1-35-2', '1801.00115-2-35-2'], ['1801.00115-1-35-3', '1801.00115-2-35-3'], ['1801.00115-1-21-0', '1801.00115-2-21-0'], ['1801.00115-1-21-1', '1801.00115-2-21-1'], ['1801.00115-1-21-2', '1801.00115-2-21-2'], ['1801.00115-1-21-3', '1801.00115-2-21-3'], ['1801.00115-1-27-1', '1801.00115-2-27-1'], ['1801.00115-1-27-2', '1801.00115-2-27-2'], ['1801.00115-1-27-4', '1801.00115-2-27-4'], ['1801.00115-1-6-0', '1801.00115-2-6-0'], ['1801.00115-1-6-1', '1801.00115-2-6-1'], ['1801.00115-1-6-2', '1801.00115-2-6-2'], ['1801.00115-1-6-3', '1801.00115-2-6-3'], ['1801.00115-1-33-1', '1801.00115-2-33-1'], ['1801.00115-1-33-2', '1801.00115-2-33-2'], ['1801.00115-1-33-4', '1801.00115-2-33-4'], ['1801.00115-1-33-6', '1801.00115-2-33-6'], ['1801.00115-1-33-7', '1801.00115-2-33-7'], ['1801.00115-1-4-0', '1801.00115-2-4-0'], ['1801.00115-1-4-1', '1801.00115-2-4-1'], ['1801.00115-1-36-0', '1801.00115-2-36-0'], ['1801.00115-1-36-1', '1801.00115-2-36-1'], ['1801.00115-1-36-2', '1801.00115-2-36-2'], ['1801.00115-1-36-3', '1801.00115-2-36-3'], ['1801.00115-1-7-0', '1801.00115-2-7-0'], ['1801.00115-1-7-1', '1801.00115-2-7-1'], ['1801.00115-1-7-2', '1801.00115-2-7-2'], ['1801.00115-1-7-3', '1801.00115-2-7-3'], ['1801.00115-1-29-0', '1801.00115-2-29-0'], ['1801.00115-1-29-1', '1801.00115-2-29-1'], ['1801.00115-1-29-2', '1801.00115-2-29-2'], ['1801.00115-1-29-3', '1801.00115-2-29-3'], ['1801.00115-1-29-4', '1801.00115-2-29-4'], ['1801.00115-1-29-5', '1801.00115-2-29-5'], ['1801.00115-1-29-7', '1801.00115-2-29-7'], ['1801.00115-1-29-8', '1801.00115-2-29-8'], ['1801.00115-1-29-9', '1801.00115-2-29-9'], ['1801.00115-1-29-10', '1801.00115-2-29-10'], ['1801.00115-1-19-0', '1801.00115-2-19-0'], ['1801.00115-1-19-1', '1801.00115-2-19-1'], ['1801.00115-1-19-2', '1801.00115-2-19-2'], ['1801.00115-1-19-3', '1801.00115-2-19-3'], ['1801.00115-1-28-0', '1801.00115-2-28-0'], ['1801.00115-1-28-2', '1801.00115-2-28-2'], ['1801.00115-1-28-3', '1801.00115-2-28-3'], ['1801.00115-1-28-4', '1801.00115-2-28-4'], ['1801.00115-1-28-5', '1801.00115-2-28-5'], ['1801.00115-1-18-2', '1801.00115-2-18-1'], ['1801.00115-1-18-4', '1801.00115-2-18-3'], ['1801.00115-1-3-2', '1801.00115-2-3-2'], ['1801.00115-1-3-3', '1801.00115-2-3-3'], ['1801.00115-1-39-2', '1801.00115-2-39-2'], ['1801.00115-1-31-1', '1801.00115-2-31-1'], ['1801.00115-1-17-4', '1801.00115-2-17-4'], ['1801.00115-1-13-9', '1801.00115-2-13-9'], ['1801.00115-1-23-1', '1801.00115-2-23-1'], ['1801.00115-1-2-5', '1801.00115-2-2-5'], ['1801.00115-1-2-6', '1801.00115-2-2-6'], ['1801.00115-1-16-1', '1801.00115-2-16-1'], ['1801.00115-1-35-1', '1801.00115-2-35-1'], ['1801.00115-1-27-3', '1801.00115-2-27-3'], ['1801.00115-1-33-0', '1801.00115-2-33-0'], ['1801.00115-1-33-5', '1801.00115-2-33-5'], ['1801.00115-1-4-2', '1801.00115-2-4-2'], ['1801.00115-1-29-6', '1801.00115-2-29-6'], ['1801.00115-1-28-1', '1801.00115-2-28-1'], ['1801.00115-1-18-3', '1801.00115-2-18-2'], ['1801.00115-1-35-0', '1801.00115-2-35-0'], ['1801.00115-1-27-0', '1801.00115-2-27-0'], ['1801.00115-1-33-3', '1801.00115-2-33-3'], ['1801.00115-1-18-0', '1801.00115-2-18-0'], ['1801.00115-1-18-1', '1801.00115-2-18-0'], ['1801.00115-1-25-0', '1801.00115-2-25-0'], ['1801.00115-1-25-1', '1801.00115-2-25-1'], ['1801.00115-1-25-2', '1801.00115-2-25-2']]","[['1801.00115-1-22-0', '1801.00115-2-22-0'], ['1801.00115-1-22-1', '1801.00115-2-22-1'], ['1801.00115-1-22-2', '1801.00115-2-22-2'], ['1801.00115-1-22-3', '1801.00115-2-22-3'], ['1801.00115-1-22-4', '1801.00115-2-22-4'], ['1801.00115-1-38-0', '1801.00115-2-38-0'], ['1801.00115-1-38-1', '1801.00115-2-38-1'], ['1801.00115-1-38-2', '1801.00115-2-38-2'], ['1801.00115-1-38-3', '1801.00115-2-38-3'], ['1801.00115-1-3-0', '1801.00115-2-3-0'], ['1801.00115-1-3-1', '1801.00115-2-3-1'], ['1801.00115-1-3-4', 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['1801.00115-1-33-3', '1801.00115-2-33-3'], ['1801.00115-1-18-0', '1801.00115-2-18-0'], ['1801.00115-1-18-1', '1801.00115-2-18-0']]","[['1801.00115-1-25-0', '1801.00115-2-25-0'], ['1801.00115-1-25-1', '1801.00115-2-25-1'], ['1801.00115-1-25-2', '1801.00115-2-25-2']]","['1801.00115-2-53-0', '1801.00115-2-64-3', '1801.00115-2-65-5', '1801.00115-2-69-1', '1801.00115-2-71-1']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1801.00115,,,
1802.09996,"{'1802.09996-1-0-0': 'The decreasing enumeration of the points of a Poisson random measure whose mean measure is Radon on [MATH] can be represented as a non-increasing function of the jump times of a standard Poisson process.', '1802.09996-1-0-1': 'This observation allows to generalize the essential idea from a well-known exact simulation algorithm for arbitrary extreme-value copulas to copulas of a more general family of max-ID distributions, with reciprocal Archimedean copulas being a particular example.', '1802.09996-1-1-0': '# Introduction', '1802.09996-1-2-0': 'A copula [MATH] is a multivariate distribution function of a random vector whose components are all uniformly distributed on [MATH], cf. [CITATION] for background.', '1802.09996-1-2-1': 'The family of reciprocal Archimedean copulas has been introduced and analyzed in [CITATION].', '1802.09996-1-2-2': 'A copula in this class has the analytical form', '1802.09996-1-3-0': 'C_F(u_1,,u_d) = _A P_d,oF( _k AF^-1(u_k))/ _A P_d,eF( _k AF^-1(u_k)),', '1802.09996-1-4-0': 'where [MATH] (resp. [MATH]) is the set of all non-empty subsets of [MATH] with odd (resp. even) cardinality.', '1802.09996-1-4-1': 'The nomenclature of this copula family is justified by some striking analogies with the well-understood family of Archimedan copulas, cf. [CITATION] for background on the latter.', '1802.09996-1-4-2': 'For instance, [CITATION] show that [MATH] is a proper copula in dimension [MATH] if and only if the parameterizing function [MATH] has the form', '1802.09996-1-5-0': 'F(t) = ( -_t^( 1-tx)^d-1(dx))=:(-(t)),', '1802.09996-1-6-0': 'with [MATH] a non-finite Radon measure on [MATH] such that [MATH], called the spectral measure.', '1802.09996-1-6-1': 'In this case, a random vector [MATH] with distribution function [MATH] has stochastic representation', '1802.09996-1-7-0': 'Y = (_k1R_kQ_1^(k),,_k1R_kQ_d^(k)),', '1802.09996-1-8-0': 'where [MATH], [MATH], are independent and uniformly distributed on the [MATH]-dimensional simplex [MATH], and, independently, [MATH] is an enumeration of the points of a Poisson random measure on [MATH] with mean measure [MATH].', '1802.09996-1-8-1': 'It is important to notice that the index [MATH] in this maximum runs through an enumeration of the points [MATH] of the Poisson random measure.', '1802.09996-1-8-2': 'This collection of points is almost surely countably infinite since the measure [MATH] is non-finite.', '1802.09996-1-8-3': 'The most prominent member of the family of reciprocal Archimedean copulas is the Galambos copula, the terminology dating back to [CITATION], which arises for the choice [MATH], cf. [CITATION].', '1802.09996-1-8-4': 'Further background on the Galambos copula can be found in [CITATION].', '1802.09996-1-8-5': 'The stochastic representation ([REF]) is difficult to simulate from due to the infinite maximum, which is why [CITATION] only propose an approximative simulation strategy.', '1802.09996-1-8-6': 'For the Galambos copula, an alternative and exact simulation strategy is developed in [CITATION], based on an idea originally due to [CITATION].', '1802.09996-1-8-7': 'Section [REF] shows how the idea of this algorithm can be generalized to arbitrary reciprocal Archimedean copulas.', '1802.09996-1-8-8': 'Section [REF] concludes.', '1802.09996-1-9-0': '# Exact simulation of reciprocal Archimedean copulas', '1802.09996-1-10-0': 'Let [MATH] be a Radon measure on [MATH], i.e. [MATH] for all [MATH], with the property that [MATH].', '1802.09996-1-10-1': 'We denote [MATH] and define a pseudo-inverse via', '1802.09996-1-11-0': 'S^-1(t):=x>0:S(x) t,t [0,u_ ].', '1802.09996-1-12-0': 'For later reference we remark that right-continuity of [MATH] implies', '1802.09996-1-13-0': 'y < S(x) S^-1(y) > x,x (0,),y [0,u_ ].', '1802.09996-1-14-0': 'To make Lemma [REF] below slightly more general than required, we explicitly allow [MATH] to be finite here, in which case [MATH] is only defined for [MATH], but for the application to reciprocal Archimedean copulas required is only the case when [MATH].', '1802.09996-1-14-1': 'For [MATH] we denote by [MATH] the Dirac measure at [MATH].', '1802.09996-1-14-2': 'We denote by [MATH] a Poisson random measure on [MATH] with mean measure [MATH], the random variable [MATH] representing the number of points [MATH] of [MATH].', '1802.09996-1-14-3': '[CITATION] is an excellent textbook for background on Poisson random measures.', '1802.09996-1-14-4': 'The most important, and characterizing, property of a Poisson random measure on a measurable space [MATH] with mean measure [MATH] is the Laplace functional formula', '1802.09996-1-15-0': 'EBig[ e^-_Ef(x)P(dx)] = (-_E ( 1-e^-f(x))(dx) ),', '1802.09996-1-16-0': 'where [MATH] is a Radon measure on [MATH], and [MATH] is a non-negative, Borel-measurable function on [MATH].', '1802.09996-1-16-1': 'Recall that if [MATH] is non-finite, [MATH] has countably many points [MATH].', '1802.09996-1-16-2': 'But if [MATH] is finite, the number of points [MATH] has a Poisson distribution with parameter [MATH].', '1802.09996-1-16-3': 'Hence, regarding notation it is convenient for us to treat both cases jointly by denoting the points of [MATH] by [MATH], possibly allowing for the value [MATH] in case of non-finite [MATH].', '1802.09996-1-16-4': 'Without loss of generality, we further enumerate the points [MATH] such that [MATH] almost surely.', '1802.09996-1-16-5': '[Stochastic representation of [MATH]] Let [MATH] be a list of independent and identically distributed exponential random variables with unit mean.', '1802.09996-1-16-6': 'Introducing the random variable', '1802.09996-1-17-0': 'N_:= & , if u_ =', '1802.09996-1-18-0': '_k 11__1++_k u_ & , else', '1802.09996-1-19-0': ',', '1802.09996-1-20-0': 'we have the distributional equality', '1802.09996-1-21-0': 'R_k_k = 1,,N d=S^-1(_1++_k)_k = 1,,N_.', '1802.09996-1-22-0': 'Define the point measure [MATH].', '1802.09996-1-22-1': 'We notice that [MATH] equals a Poisson random measure on [MATH] with mean measure the Lebesgue measure [MATH], hence [EQUATION]', '1802.09996-1-22-2': 'To verify the last equation [MATH], denote by [MATH] the Lebesgue measure on [MATH], and observe that the map [MATH] is measurable.', '1802.09996-1-22-3': 'Consider the measure [MATH] defined by [MATH], [MATH] a Borel set in [MATH] and [MATH] its pre-image under [MATH] in [MATH].', '1802.09996-1-22-4': 'Then we observe for [MATH] that [EQUATION]', '1802.09996-1-22-5': 'Consequently, [MATH] and we have established the measure-theoretic change of variable formula [EQUATION]', '1802.09996-1-22-6': 'Applying it to the function [MATH] implies [MATH].', '1802.09996-1-22-7': 'The claim now follows from uniqueness of the Laplace functional of Poisson random measure, since [MATH] was an arbitrary non-negative, Borel-measurable function.', '1802.09996-1-23-0': 'If the measure [MATH] is absolutely continuous with positive density on [MATH], the function [MATH] is continuously and strictly decreasing and [MATH] is the regular inverse.', '1802.09996-1-23-1': 'The following example sheds some light on the situation in the case of discrete measures [MATH].', '1802.09996-1-24-0': '[Discrete spectral measures] Let [MATH] with [MATH] and [MATH] with [MATH] (to guarantee that [MATH] is non-finite), and consider [MATH] on [MATH], obviously Radon on [MATH].', '1802.09996-1-24-1': 'The functions [MATH] and [MATH] in this case are given by [EQUATION]', '1802.09996-1-24-2': 'As a concrete one-parametric example, let [MATH] and [MATH], [MATH], [MATH].', '1802.09996-1-24-3': 'It is not difficult to observe that in this case the formulas above boil down to', '1802.09996-1-25-0': 'S(t) = _u t1u ,S^-1(t)=1/t ,', '1802.09996-1-26-0': 'with [MATH] and [MATH] denoting the usual floor and ceiling functions mapping [MATH] to [MATH].', '1802.09996-1-26-1': 'The limit [MATH] appearing in the definition of [MATH] is required for right-continuity and only plays a role at the countably many points [MATH].', '1802.09996-1-26-2': 'The associated distribution function [MATH] generating the [MATH]-dimensional reciprocal Archimedean copula associated with [MATH] is determined by', '1802.09996-1-27-0': '(t) = _k =1^1/t ( 1-kt)^d-1 (d=2)= 1t ( 1-t 1t +1/2),', '1802.09996-1-28-0': 'where the last equation in the bivariate case is stated explicitly for later reference.', '1802.09996-1-29-0': 'Next, we turn to the simulation of reciprocal Archimedean copulas and introduce the notation [EQUATION]', '1802.09996-1-29-1': 'It is observed that every single component of [MATH] is smaller or equal than [MATH], since the sequence [MATH] is non-increasing by our enumeration.', '1802.09996-1-29-2': 'This implies', '1802.09996-1-30-0': 'Y = Y_ = Y_M, with M:=n 1:R_n M_n.', '1802.09996-1-31-0': 'Since [MATH] is almost surely decreasing to zero and [MATH] is almost surely non-decreasing, [MATH] is almost surely finite.', '1802.09996-1-31-1': 'Consequently, in order to simulate [MATH], it is sufficient to simulate iteratively [MATH] for [MATH] until the stopping criterion [MATH] takes place, i.e. until [MATH].', '1802.09996-1-31-2': 'This is precisely the simulation idea of Algorithm 1 in [CITATION] for extreme-value copulas, see also [CITATION], enhanced to fit the scope of reciprocal Archimedean copulas as well with the help of Lemma [REF].', '1802.09996-1-31-3': 'Algorithm [REF] summarizes this strategy in pseudo code.', '1802.09996-1-31-4': 'It requires evaluation of [MATH] and of [MATH], which are the sole numerical obstacles.', '1802.09996-1-31-5': 'Notice further that we propose to simulate the random vectors [MATH] according to the well-known stochastic representation', '1802.09996-1-32-0': 'Q^(k) d= ( E_1E_1++E_d,,E_dE_1++E_d),', '1802.09996-1-33-0': 'where [MATH] are independent exponential random variables with unit mean, cf. [CITATION].', '1802.09996-1-34-0': '[Exact simulation of reciprocal Archimedean copulas] Consider a [MATH]-dimensional family of reciprocal Archimedean copulas with generator [MATH], associated with radial measure [MATH].', '1802.09996-1-34-1': 'We denote by [MATH] the survival function of its radial measure, respectively its pseudo inverse by [MATH].', '1802.09996-1-35-0': '[(3)] Return [MATH], where [MATH], [MATH].', '1802.09996-1-36-0': '[Generalization to other max-ID distributions] If the random vectors [MATH] in ([REF]) are not uniformly distributed on [MATH], but instead follow some other distribution on [MATH], Algorithm [REF] can still be used for simulation, provided one has at hand a simulation algorithm for [MATH].', '1802.09996-1-36-1': 'In this case, one breaks out of the cosmos of reciprocal Archimedean copulas.', '1802.09996-1-36-2': 'In the particular case [MATH] this generalized algorithm equals precisely [CITATION] for arbitrary extreme-value copulas.', '1802.09996-1-36-3': 'However, for other radial measures [MATH] one also breaks out of the cosmos of extreme-value copulas.', '1802.09996-1-36-4': 'The resulting distribution function of [MATH] in the general case is', '1802.09996-1-37-0': 'P(Y y) = (-EBig[S( y_1Q_1^(1),,y_dQ_d^(1)) ]),', '1802.09996-1-38-0': 'for [MATH].', '1802.09996-1-38-1': 'For [MATH] the function [MATH] is the survival function of the Radon measure [MATH], showing that the [MATH]-variate function [MATH] is again a distribution function.', '1802.09996-1-38-2': 'Multivariate distribution functions with this property are called max-ID, see [CITATION] for background.', '1802.09996-1-39-0': 'Concerning the implementation of Algorithm [REF], the biggest numerical difficulty is the evaluation of the inverse [MATH].', '1802.09996-1-39-1': 'One might be lucky to have a closed form of [MATH] available.', '1802.09996-1-39-2': 'For example, in case of the Galambos copula the function [MATH] is given by [MATH] with constant [MATH], cf. [CITATION].', '1802.09996-1-39-3': 'Interestingly, Algorithm [REF] in this particular case is different than the one derived in [CITATION], which is designed for extreme-value copulas rather than reciprocal Archimedean copulas, but which also includes the Galambos copula.', '1802.09996-1-39-4': 'The algorithm of [CITATION] is always based on the decreasing sequence [MATH], but ours on [MATH] instead.', '1802.09996-1-39-5': 'For [MATH], these two sequences agree, so the simulation algorithms coincide.', '1802.09996-1-39-6': 'For [MATH], however, they are truly different, since Algorithm [REF] always simulates the uniform law on the simplex and varies the sequence [MATH], while [CITATION] stick with the sequence [MATH] and instead vary the measure on the simplex.', '1802.09996-1-39-7': 'Example [REF] shows that discrete measures [MATH] give rise to [MATH] having (piecewise constant) closed form.', '1802.09996-1-39-8': 'As an example, scatter plots for the bivariate reciprocal Archimedean copula associated with the generator [MATH], for [MATH] in ([REF]), are depicted in Figure [REF].', '1802.09996-1-39-9': 'From our simulations, it appears as though the resulting family converges to the independence copula for [MATH], and to some limiting copula (but not the upper Frechet bound) for [MATH].', '1802.09996-1-39-10': 'Furthermore, since [MATH] maps to the discrete set [MATH], the copula assigns positive mass to the one-dimensional subsets', '1802.09996-1-40-0': 'A_k:=(F(uk ),F(1-uk)):u (0,1) [0,1]^2,k 1,', '1802.09996-1-41-0': 'of the unit square.', '1802.09996-1-41-1': 'By construction, this mass is decreasing in [MATH], and the first sets [MATH] are clearly visible in the scatter plots.', '1802.09996-1-42-0': 'In case [MATH] is not given in closed form, there are several potential ideas to evaluate [MATH] numerically, some of which are collected in the bullet points below.', '1802.09996-1-42-1': 'To this end, it is convenient to recall that [MATH] is given in terms of [MATH] by the Williamson transform inversion formula [EQUATION] with [MATH] denoting the right-hand derivative of [MATH], cf. [CITATION].', '1802.09996-1-42-2': 'In particular, [MATH] is [MATH]-monotone, which implies that the first [MATH] derivatives exist, and [MATH] is convex (so that [MATH] exists).', '1802.09996-1-42-3': 'If [MATH] has discontinuities, such as in the case of a discrete measure [MATH] like in Example [REF], [MATH] is not [MATH]-monotone, i.e. [MATH] does not exist.', '1802.09996-1-42-4': 'Here are three ideas for the required numerical evaluation of [MATH] in Algorithm [REF]:', '1802.09996-1-43-0': '# Conclusion', '1802.09996-1-44-0': 'An exact simulation algorithm for reciprocal Archimedean copulas has been presented.', '1802.09996-1-44-1': 'It was based on the concatenation of two ideas.', '1802.09996-1-44-2': 'On the one hand, we derived a convenient stochastic representation of the points of a Poisson random measure on [MATH] whose mean measure equals the spectral measure of the reciprocal Archimedean copula.', '1802.09996-1-44-3': 'On the other hand, an idea of [CITATION] has been enhanced from a simulation algorithm for extreme-value copulas to copulas of more general max-ID distributions.'}","{'1802.09996-2-0-0': 'The decreasing enumeration of the points of a Poisson random measure whose mean measure is Radon on [MATH] can be represented as a non-increasing function of the jump times of a standard Poisson process.', '1802.09996-2-0-1': 'This observation allows to generalize the essential idea from a well-known exact simulation algorithm for arbitrary extreme-value copulas to copulas of a more general family of max-infinitely divisible distributions, with reciprocal Archimedean copulas being a particular example.', '1802.09996-2-1-0': '# Introduction', '1802.09996-2-2-0': 'A copula [MATH] is a multivariate distribution function of a random vector whose components are all uniformly distributed on [MATH], cf. [CITATION] for background.', '1802.09996-2-2-1': 'The family of reciprocal Archimedean copulas has been introduced and analyzed in [CITATION].', '1802.09996-2-2-2': 'With a parameterizing univariate distribution function [MATH], a copula in this class has the analytical form', '1802.09996-2-3-0': 'C_F(u_1,,u_d) = _A P_d,oF( _k AF^-1(u_k))/ _A P_d,eF( _k AF^-1(u_k)),', '1802.09996-2-4-0': 'where [MATH] (resp. [MATH]) is the set of all non-empty subsets of [MATH] with odd (resp. even) cardinality.', '1802.09996-2-4-1': 'The nomenclature of this copula family is justified by some striking analogies with the well-understood family of Archimedean copulas, cf. [CITATION] for background on the latter.', '1802.09996-2-4-2': 'For instance, [CITATION] show that [MATH] is a proper copula in dimension [MATH] if and only if the parameterizing function [MATH] has the form', '1802.09996-2-5-0': 'F(t) = ( -_t^( 1-tx)^d-1(dx))=:(-(t)),', '1802.09996-2-6-0': 'with [MATH] a non-finite Radon measure on [MATH] such that [MATH], called the radial measure.', '1802.09996-2-6-1': 'In this case, a random vector [MATH] with distribution function [MATH] has stochastic representation', '1802.09996-2-7-0': 'Y = (_k1R_kQ_1^(k),,_k1R_kQ_d^(k)),', '1802.09996-2-8-0': 'where [MATH], [MATH], are independent and uniformly distributed on the [MATH]-dimensional simplex [MATH], and, independently, [MATH] is an enumeration of the points of a Poisson random measure on [MATH] with mean measure [MATH].', '1802.09996-2-8-1': 'It is important to notice that the index [MATH] in this maximum runs through an enumeration of the points [MATH] of the Poisson random measure.', '1802.09996-2-8-2': 'This collection of points is almost surely countably infinite since the measure [MATH] is non-finite.', '1802.09996-2-8-3': 'The most prominent member of the family of reciprocal Archimedean copulas is the Galambos copula, the terminology dating back to [CITATION], which arises for the choice [MATH], cf. [CITATION].', '1802.09996-2-8-4': 'Further background on the Galambos copula can be found in [CITATION].', '1802.09996-2-8-5': 'The stochastic representation ([REF]) is difficult to simulate from due to the infinite maximum, which is why [CITATION] only propose an approximative simulation strategy.', '1802.09996-2-8-6': 'For extreme-value copulas, a family whose intersection with reciprocal Archimedean copulas equals the Galambos copula, an alternative and exact simulation strategy is developed in [CITATION], based on an idea originally due to [CITATION].', '1802.09996-2-8-7': 'Section [REF] shows how the idea of this algorithm can be generalized to include arbitrary reciprocal Archimedean copulas.', '1802.09996-2-8-8': 'Section [REF] concludes.', '1802.09996-2-9-0': '# Exact simulation of reciprocal Archimedean copulas', '1802.09996-2-10-0': 'Let [MATH] be a Radon measure on [MATH], i.e. [MATH] for all [MATH], with the property that [MATH].', '1802.09996-2-10-1': 'We denote [MATH] and define a pseudo-inverse via', '1802.09996-2-11-0': 'S^-1(t):=x>0:S(x) t,t [0,u_ ].', '1802.09996-2-12-0': 'For later reference we remark that right-continuity of [MATH] implies', '1802.09996-2-13-0': 'y < S(x) S^-1(y) > x,x (0,),y [0,u_ ].', '1802.09996-2-14-0': 'We explicitly allow [MATH] to be finite here, in which case [MATH] is only defined for [MATH], but for the application to reciprocal Archimedean copulas only the case when [MATH] is relevant.', '1802.09996-2-14-1': 'For [MATH] we denote by [MATH] the Dirac measure at [MATH].', '1802.09996-2-14-2': 'We denote by [MATH] a Poisson random measure on [MATH] with mean measure [MATH], the random variable [MATH] representing the number of points [MATH] of [MATH].', '1802.09996-2-14-3': '[CITATION] is an excellent textbook for background on Poisson random measures.', '1802.09996-2-14-4': 'The most important, and characterizing, property of a Poisson random measure on a measurable space [MATH] with mean measure [MATH] is the Laplace functional formula', '1802.09996-2-15-0': 'EBig[ e^-_Ef(x)P(dx)] = (-_E ( 1-e^-f(x))(dx) ),', '1802.09996-2-16-0': 'where [MATH] is a Radon measure on [MATH], and [MATH] is a non-negative, Borel-measurable function on [MATH].', '1802.09996-2-16-1': 'Recall that if [MATH] is non-finite, [MATH] has countably many points [MATH].', '1802.09996-2-16-2': 'But if [MATH] is finite, the number of points [MATH] has a Poisson distribution with parameter [MATH].', '1802.09996-2-16-3': 'Hence, regarding notation it is convenient for us to treat both cases jointly by denoting the points of [MATH] by [MATH], possibly allowing for the value [MATH] in case of non-finite [MATH].', '1802.09996-2-16-4': 'Without loss of generality, we further enumerate the points [MATH] such that [MATH] almost surely.', '1802.09996-2-16-5': 'The following auxiliary result follows from ([REF]) by a change of variables from (the possibly complicated measure) [MATH] to the Lebesgue measure [MATH], resulting in a stochastic representation of [MATH] that is convenient for our purpose of simulating reciprocal Archimedean copulas.', '1802.09996-2-16-6': 'Even though this computation is presumably standard in the literature on Poisson random measures, we state it as a separate lemma and provide a proof here, because it is educational, one key ingredient for the derived simulation algorithm, and apparently lesser known in the literature on copulas and dependence modeling.', '1802.09996-2-17-0': '[Stochastic representation of [MATH]] Let [MATH] be a sequence of independent and identically distributed exponential random variables with unit mean.', '1802.09996-2-17-1': 'Introducing the random variable', '1802.09996-2-18-0': 'N_:= & if u_ = ,', '1802.09996-2-19-0': '_k 11__1++_k u_ & else,', '1802.09996-2-20-0': ',', '1802.09996-2-21-0': 'we have the distributional equality', '1802.09996-2-22-0': 'R_k_k = 1,,N d=S^-1(_1++_k)_k = 1,,N_.', '1802.09996-2-23-0': 'Define the point measure [MATH].', '1802.09996-2-23-1': 'We notice that [MATH] equals a Poisson random measure on [MATH] with mean measure the Lebesgue measure [MATH], hence [EQUATION]', '1802.09996-2-23-2': 'To verify equation [MATH], denote by [MATH] the Lebesgue measure on [MATH], and observe that the map [MATH] is measurable.', '1802.09996-2-23-3': 'Consider the measure [MATH] defined by [MATH], [MATH] a Borel set in [MATH] and [MATH] its pre-image under [MATH] in [MATH].', '1802.09996-2-23-4': 'Then we observe for [MATH] that [EQUATION]', '1802.09996-2-23-5': 'Consequently, [MATH] and we have the measure-theoretic change of variable formula [EQUATION]', '1802.09996-2-23-6': 'Applying it to the function [MATH] implies [MATH].', '1802.09996-2-23-7': 'The claim now follows from uniqueness of the Laplace functional of Poisson random measure, since [MATH] was an arbitrary non-negative, Borel-measurable function.', '1802.09996-2-24-0': '[Simulation of infinitely divisible laws on [MATH]] It is not the first time that the change of variables technique of Lemma [REF] is found useful for an application to simulation.', '1802.09996-2-24-1': 'To provide another example, [CITATION] uses essentially the same technique to represent a non-negative infinitely divisible random variable [MATH] with associated Levy measure [MATH] as', '1802.09996-2-25-0': 'X = _k =1^NR_k d= _k=1^N_S^-1(_1++_k)', '1802.09996-2-26-0': 'and discusses the possibility to simulate [MATH] based on this stochastic representation.', '1802.09996-2-27-0': 'If the radial measure [MATH] is absolutely continuous with positive density on [MATH], the function [MATH] is continuous and strictly decreasing and [MATH] is the regular inverse.', '1802.09996-2-27-1': 'The following example sheds some light on the situation in the case of discrete measures [MATH].', '1802.09996-2-28-0': '[Discrete radial measures] Let [MATH] with [MATH] and [MATH] with [MATH] (to guarantee that [MATH] is non-finite), and consider [MATH] on [MATH], obviously Radon on [MATH].', '1802.09996-2-28-1': 'The functions [MATH] and [MATH] in this case are given by [EQUATION]', '1802.09996-2-28-2': 'As a concrete one-parametric example, let [MATH] and [MATH], [MATH], [MATH].', '1802.09996-2-28-3': 'It is not difficult to observe that in this case the formulas above boil down to', '1802.09996-2-29-0': 'S(t) = _u t1u ,S^-1(t)=1/t ,', '1802.09996-2-30-0': 'with [MATH] and [MATH] denoting the usual floor and ceiling functions mapping [MATH] to [MATH].', '1802.09996-2-30-1': 'The associated distribution function [MATH] generating the [MATH]-dimensional reciprocal Archimedean copula associated with [MATH] is determined by', '1802.09996-2-31-0': '(t) = _k =1^1/t ( 1-kt)^d-1 (d=2)= 1t ( 1-t 1t +1/2),', '1802.09996-2-32-0': 'where the last equation in the bivariate case is stated explicitly for later reference.', '1802.09996-2-33-0': 'Next, we turn to the simulation of reciprocal Archimedean copulas and introduce the notation [EQUATION]', '1802.09996-2-33-1': 'It is observed that every single component of [MATH] is smaller or equal than [MATH], since the sequence [MATH] is non-increasing by our enumeration.', '1802.09996-2-33-2': 'This implies', '1802.09996-2-34-0': 'Y = Y_ = Y_M, with M:=n 1:R_n+1 M_n.', '1802.09996-2-35-0': 'Since [MATH] is almost surely decreasing to zero and [MATH] is almost surely non-decreasing, [MATH] is almost surely finite.', '1802.09996-2-35-1': 'Consequently, in order to simulate [MATH], it is sufficient to simulate iteratively [MATH] for [MATH] until the stopping criterion [MATH] takes place, i.e. until [MATH].', '1802.09996-2-35-2': 'This is precisely the simulation idea of Algorithm 1 in [CITATION] for extreme-value copulas, see also [CITATION], enhanced to fit the scope of reciprocal Archimedean copulas as well with the help of Lemma [REF].', '1802.09996-2-35-3': 'Algorithm [REF] summarizes this strategy in pseudo code.', '1802.09996-2-35-4': 'It requires evaluation of [MATH] and of [MATH], which are the sole numerical obstacles.', '1802.09996-2-35-5': 'Notice further that we propose to simulate the random vectors [MATH] according to the well-known stochastic representation', '1802.09996-2-36-0': 'Q^(k) d= ( E_1E_1++E_d,,E_dE_1++E_d),', '1802.09996-2-37-0': 'where [MATH] are independent exponential random variables with unit mean, cf. [CITATION].', '1802.09996-2-38-0': '[Exact simulation of reciprocal Archimedean copulas] Consider a [MATH]-dimensional family of reciprocal Archimedean copulas with generator [MATH], associated with radial measure [MATH].', '1802.09996-2-38-1': 'We denote by [MATH] the survival function of its radial measure, respectively its pseudo inverse by [MATH].', '1802.09996-2-39-0': '[(3)] Return [MATH], where [MATH], [MATH].', '1802.09996-2-40-0': '[Generalization to more general distributions] If the random vectors [MATH] in ([REF]) are not uniformly distributed on [MATH], but instead follow some other distribution on [MATH], Algorithm [REF] can still be used for simulation, provided one has at hand a simulation algorithm for [MATH].', '1802.09996-2-40-1': 'In this case, one breaks out of the cosmos of reciprocal Archimedean copulas.', '1802.09996-2-40-2': 'In the particular case [MATH] this generalized algorithm equals precisely [CITATION] for arbitrary extreme-value copulas.', '1802.09996-2-40-3': 'However, for other radial measures [MATH] one also breaks out of the cosmos of extreme-value copulas.', '1802.09996-2-40-4': 'The resulting distribution function of [MATH] in the general case is', '1802.09996-2-41-0': 'P(Y y) = (-EBig[S( y_1Q_1^(1),,y_dQ_d^(1)) ]),', '1802.09996-2-42-0': 'for [MATH].', '1802.09996-2-42-1': 'For [MATH] the function [MATH] is the survival function of the Radon measure [MATH], showing that the [MATH]-variate function [MATH] is again a distribution function.', '1802.09996-2-42-2': 'Multivariate distribution functions with this property are called max-infinitely divisible, see [CITATION] for background.', '1802.09996-2-43-0': 'Concerning the implementation of Algorithm [REF], the biggest numerical difficulty is the evaluation of the inverse [MATH].', '1802.09996-2-43-1': 'One might be lucky to have a closed form of [MATH] available.', '1802.09996-2-43-2': 'For example, in case of the Galambos copula the function [MATH] is given by [MATH] with constant [MATH], cf. [CITATION].', '1802.09996-2-43-3': 'Interestingly, Algorithm [REF] in this particular case is different than the one derived in [CITATION], which is designed for extreme-value copulas rather than reciprocal Archimedean copulas, but which also includes the Galambos copula.', '1802.09996-2-43-4': 'The algorithm of [CITATION] is always based on the decreasing sequence [MATH], but ours on [MATH] instead.', '1802.09996-2-43-5': 'For [MATH], these two sequences agree, so the simulation algorithms coincide.', '1802.09996-2-43-6': 'For [MATH], however, they are truly different, since Algorithm [REF] always simulates the uniform law on the simplex and varies the sequence [MATH], while [CITATION] stick with the sequence [MATH] and instead vary the measure on the simplex.', '1802.09996-2-43-7': 'Example [REF] shows that discrete measures [MATH] give rise to [MATH] having (piecewise constant) closed form.', '1802.09996-2-43-8': 'As an example, scatter plots for the bivariate reciprocal Archimedean copula associated with the generator [MATH], for [MATH] in ([REF]), are depicted in Figure [REF].', '1802.09996-2-43-9': 'From our simulations, it appears as though the resulting family converges to the independence copula for [MATH], and to some limiting copula (but not the upper Frechet bound) for [MATH].', '1802.09996-2-43-10': 'Furthermore, since [MATH] maps to the discrete set [MATH], the copula assigns positive mass to the one-dimensional subsets', '1802.09996-2-44-0': 'A_k:=(F(uk ),F(1-uk)):u (0,1) [0,1]^2,k 1,', '1802.09996-2-45-0': 'of the unit square.', '1802.09996-2-45-1': 'By construction, this mass is decreasing in [MATH], and the first sets [MATH] are clearly visible in the scatter plots.', '1802.09996-2-46-0': 'In case [MATH] is not given in closed form, it is convenient to recall that [MATH] is given in terms of [MATH] by the Williamson transform inversion formula [EQUATION] with [MATH] denoting the right-hand derivative of [MATH], cf. [CITATION].', '1802.09996-2-46-1': 'In particular, [MATH] is [MATH]-monotone, which implies that the first [MATH] derivatives exist, and [MATH] is convex (so that [MATH] exists).', '1802.09996-2-46-2': 'If [MATH] has discontinuities, such as in the case of a discrete measure [MATH] like in Example [REF], [MATH] is not [MATH]-monotone, i.e. [MATH] does not exist.', '1802.09996-2-46-3': 'Besides traditional Newton-Raphson inversion, here are two more ideas for dealing with [MATH] in Algorithm [REF]:', '1802.09996-2-47-0': '[Expected runtime in dependence on the dimension] Using the stochastic representation ([REF]) for the uniform distribution on the simplex, the simulation of the random vector [MATH] in the [MATH]-th while-loop requires to simulate [MATH] independent exponential random variables, hence has complexity order linear in [MATH].', '1802.09996-2-47-1': 'However, the number of required while-loops is random itself.', '1802.09996-2-47-2': 'In the special case [MATH], which corresponds to the Galambos copula with parameter [MATH] and [MATH], Algorithm [REF] coincides with [CITATION].', '1802.09996-2-47-3': '[CITATION], which in turn refers to [CITATION], shows that the expected number of required while-loops in this case equals [MATH], where [MATH] is a random vector with survival copula the Galambos copula in concern and all one-dimensional margins unit exponentially distributed.', '1802.09996-2-47-4': 'It follows from a computation in [CITATION] that', '1802.09996-2-48-0': '1 E = _i=1^ddi(-1)^i+1H_i d,', '1802.09996-2-49-0': 'where [MATH], [MATH], denotes the harmonic series.', '1802.09996-2-49-1': 'Hence, in this special case the complexity order of the algorithm is known explicitly as a function of the dimension [MATH] and lies somewhere between [MATH] and [MATH].', '1802.09996-2-49-2': 'Unfortunately, the proof of this result relies heavily on the fact that [MATH].', '1802.09996-2-49-3': 'In the case of general radial measure, we have [EQUATION]', '1802.09996-2-49-4': 'From this formula for [MATH], [CITATION] make use of the fact that [MATH], for [MATH], correspond to order statistics of independent samples from the uniform law on [MATH], which we cannot for general [MATH] (hence [MATH]).', '1802.09996-2-49-5': 'However, using this known Galambos case as benchmark, from the last formula we observe that [MATH] is larger than in the known benchmark case if the function [MATH] is increasing, and that it is smaller if [MATH] is decreasing.', '1802.09996-2-49-6': 'Although we cannot say a lot about the case when [MATH] is neither increasing nor decreasing, this at least provides a feeling for the effect of the choice of radial measure on the expected runtime of the algorithm.', '1802.09996-2-49-7': 'In the case when [MATH] is absolutely continuous with density [MATH] one may check whether [MATH] is increasing (resp. decreasing) by checking whether [MATH] (resp. [MATH]) for all [MATH], which is a quite intuitive condition in terms of the density.', '1802.09996-2-49-8': 'Heuristically, it says that heavier tails of the radial measure make the algorithm faster, and vice versa.', '1802.09996-2-50-0': '# Conclusion', '1802.09996-2-51-0': 'An exact simulation algorithm for reciprocal Archimedean copulas has been presented.', '1802.09996-2-51-1': 'It was based on the concatenation of two ideas.', '1802.09996-2-51-2': 'On the one hand, via a change of variables transformation the points of a Poisson random measure on [MATH], whose mean measure equals the radial measure of the reciprocal Archimedean copula, have been represented as a decreasing function of the jump times of a standard Poisson process.', '1802.09996-2-51-3': 'On the other hand, an idea of [CITATION] has been enhanced from a simulation algorithm for extreme-value copulas to copulas of more general max-infinitely divisible distributions, of which reciprocal Archimedean copulas are a particular representative.'}","[['1802.09996-1-4-0', '1802.09996-2-4-0'], ['1802.09996-1-4-2', '1802.09996-2-4-2'], ['1802.09996-1-22-0', '1802.09996-2-23-0'], ['1802.09996-1-22-1', '1802.09996-2-23-1'], ['1802.09996-1-22-3', '1802.09996-2-23-3'], ['1802.09996-1-22-4', '1802.09996-2-23-4'], ['1802.09996-1-22-6', '1802.09996-2-23-6'], ['1802.09996-1-22-7', '1802.09996-2-23-7'], ['1802.09996-1-28-0', '1802.09996-2-32-0'], ['1802.09996-1-39-0', '1802.09996-2-43-0'], ['1802.09996-1-39-1', '1802.09996-2-43-1'], ['1802.09996-1-39-2', '1802.09996-2-43-2'], ['1802.09996-1-39-3', '1802.09996-2-43-3'], ['1802.09996-1-39-4', '1802.09996-2-43-4'], ['1802.09996-1-39-5', '1802.09996-2-43-5'], ['1802.09996-1-39-6', '1802.09996-2-43-6'], ['1802.09996-1-39-7', '1802.09996-2-43-7'], ['1802.09996-1-39-8', '1802.09996-2-43-8'], ['1802.09996-1-39-9', '1802.09996-2-43-9'], ['1802.09996-1-39-10', '1802.09996-2-43-10'], ['1802.09996-1-24-1', '1802.09996-2-28-1'], ['1802.09996-1-24-2', '1802.09996-2-28-2'], 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math-0510432,"{'math-0510432-1-0-0': 'Given a closed orientable Euclidean cone 3-manifold [MATH] with cone angles [MATH] and which is not almost product, we describe the space of constant curvature cone structures on [MATH] with cone angles [MATH].', 'math-0510432-1-0-1': 'We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbolic ones and we also deduce global rigidity for Euclidean cone structures.', 'math-0510432-1-1-0': '# Introduction', 'math-0510432-1-2-0': 'Let [MATH] be a closed orientable Euclidean cone 3-manifold with cone angles [MATH].', 'math-0510432-1-2-1': 'Its singular locus [MATH] is a trivalent graph consisting of [MATH] circles and edges, that we enumerate from [MATH] to [MATH].', 'math-0510432-1-2-2': 'To those edges one can associate a multiangle [MATH], where [MATH] is the angle of the [MATH]-th circle or edge of [MATH].', 'math-0510432-1-2-3': 'Let further [MATH] denote the vector of lengths of singular circles and edges.', 'math-0510432-1-2-4': 'We are interested in the space of cone structures on [MATH] obtained by changing the multiangle.', 'math-0510432-1-3-0': 'A hyperbolic cone structure on [MATH] is a hyperbolic cone manifold [MATH] with an embedding [MATH] such that [MATH] is a retract of [MATH].', 'math-0510432-1-4-0': 'A Euclidean cone 3-manifold [MATH] is almost product if [MATH] where [MATH] is a two-dimensional cone manifold and [MATH] is a finite subgroup.', 'math-0510432-1-5-0': 'Let [MATH] be a closed orientable Euclidean cone 3-manifold with cone angles [MATH].', 'math-0510432-1-5-1': 'If [MATH] is not almost product, then for every multiangle [MATH] there exists a unique cone manifold structure of constant curvature in [MATH] on [MATH] with those cone angles.', 'math-0510432-1-6-0': 'If all cone angles of [MATH] are [MATH], then every point in [MATH] is the multiangle of a hyperbolic cone structure on [MATH].', 'math-0510432-1-7-0': 'If some of the cone angles is [MATH], then the subset [MATH] of multiangles of Euclidean cone structures is a smooth, properly embedded hypersurface that splits [MATH] into 2 connected components [MATH] and [MATH], corresponding to multiangles of spherical and hyperbolic cone structures respectively.', 'math-0510432-1-7-1': 'Furthermore, for each [MATH] the tangent space of [MATH] at [MATH] is orthogonal to the vector of singular lengths [MATH].', 'math-0510432-1-8-0': 'This theorem describes the structures corresponding to multiangles in the open cube.', 'math-0510432-1-8-1': 'We can also describe the structures corresponding to multiangles contained in some parts of the boundary.', 'math-0510432-1-8-2': 'For instance, we show that the multiangles contained in [MATH] (i.e. none of the angles is [MATH] and at least one of them is zero) are angles of hyperbolic cone structures.', 'math-0510432-1-8-3': 'However, we do not give a description of structures in all points of [MATH], as this would involve studying collapses at cone angle [MATH] and deformations and regenerations of Seifert fibered geometries.', 'math-0510432-1-9-0': 'Let [MATH] be an irreducible closed orientable [MATH]-orbifold.', 'math-0510432-1-9-1': 'If there exists a Euclidean cone structure [MATH] on [MATH] with cone angles strictly less than the orbifold angles of [MATH], then [MATH] is spherical.', 'math-0510432-1-10-0': 'To prove this corollary, we show in Lemma [REF] that the Euclidean cone structure is not almost product, using the irreducibility of [MATH].', 'math-0510432-1-10-1': 'By Theorem [REF] we can obtain a spherical cone structure on [MATH] by increasing any of the cone angles.', 'math-0510432-1-10-2': 'We are using here that [MATH] is orthogonal to the vector of singular lengths [MATH].', 'math-0510432-1-10-3': 'By Proposition [REF] the orbifold angles can be realized by a spherical cone structure.', 'math-0510432-1-10-4': '[MATH]', 'math-0510432-1-11-0': 'This corollary gives an alternative argument to the last step in the proof of the orbifold theorem in [CITATION], which is more natural from the point of view of cone manifolds.', 'math-0510432-1-11-1': 'D. Cooper and S.P. Kerckhoff have announced a different approach to the spherical uniformisation.', 'math-0510432-1-12-0': 'A special case of this theorem was proved in [CITATION], assuming that the singular locus was a knot and a technical hypothesis involving cohomology.', 'math-0510432-1-12-1': 'Even if some of the techniques of [CITATION] are used, this paper does not rely on it.', 'math-0510432-1-13-0': 'Now we describe the organization of the paper: We are interested in the rotational part of the holonomy of the Euclidean cone manifold in [MATH] (in fact its lift to [MATH]), that we denote by [MATH].', 'math-0510432-1-13-1': 'Some basic properties of this representation are studied in Section [REF].', 'math-0510432-1-13-2': 'In order to deform the structure, we view [MATH] as the stabilizer of a point in [MATH] or [MATH], thus we study the local properties of the varieties of representations in the isometries of [MATH] and [MATH] around [MATH].', 'math-0510432-1-13-3': 'This is done in Section [REF], using the cohomological results of Section [REF].', 'math-0510432-1-13-4': 'In Section [REF] we give the conditions for a deformation of the representation [MATH] to correspond to a path of hyperbolic or spherical cone manifolds.', 'math-0510432-1-13-5': 'This is applied in Section [REF] to construct paths of hyperbolic and spherical structures by deforming one of the cone angles.', 'math-0510432-1-13-6': 'The analysis of the local parametrization of the variety of representations is completed in Section [REF], where all deformations of the structure are constructed.', 'math-0510432-1-13-7': 'The global results are established in Section [REF].', 'math-0510432-1-13-8': 'Finally, in Section [REF] we illustrate the main theorem for cone manifold structures on the 3-sphere with singular locus given by the Whitehead link.'}","{'math-0510432-2-0-0': 'Given a closed orientable Euclidean cone 3-manifold [MATH] with cone angles [MATH] and which is not almost product, we describe the space of constant curvature cone structures on [MATH] with cone angles [MATH].', 'math-0510432-2-0-1': 'We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbolic ones and we also deduce global rigidity for Euclidean cone structures.', 'math-0510432-2-1-0': '# Introduction', 'math-0510432-2-2-0': 'Let [MATH] be a closed orientable Euclidean cone 3-manifold with cone angles [MATH].', 'math-0510432-2-2-1': 'Its singular locus [MATH] is a trivalent graph consisting of [MATH] circles and edges, that we enumerate from [MATH] to [MATH].', 'math-0510432-2-2-2': 'To those edges one can associate a multiangle [MATH], where [MATH] is the angle of the [MATH]-th circle or edge of [MATH].', 'math-0510432-2-2-3': 'Let further [MATH] denote the vector of lengths of singular circles and edges.', 'math-0510432-2-2-4': 'We are interested in the space of cone structures on [MATH] obtained by changing the multiangle.', 'math-0510432-2-3-0': 'A hyperbolic cone structure on [MATH] is a hyperbolic cone manifold [MATH] with an embedding [MATH] such that [MATH] is a retract of [MATH].', 'math-0510432-2-4-0': 'A Euclidean cone 3-manifold [MATH] is almost product if [MATH] where [MATH] is a two-dimensional cone manifold and [MATH] is a finite subgroup.', 'math-0510432-2-5-0': 'Let [MATH] be a closed orientable Euclidean cone 3-manifold with cone angles [MATH].', 'math-0510432-2-5-1': 'If [MATH] is not almost product, then for every multiangle [MATH] there exists a unique cone manifold structure of constant curvature in [MATH] on [MATH] with those cone angles.', 'math-0510432-2-6-0': 'If all cone angles of [MATH] are [MATH], then every point in [MATH] is the multiangle of a hyperbolic cone structure on [MATH].', 'math-0510432-2-7-0': 'If some of the cone angles is [MATH], then the subset [MATH] of multiangles of Euclidean cone structures is a smooth, properly embedded hypersurface that splits [MATH] into 2 connected components [MATH] and [MATH], corresponding to multiangles of spherical and hyperbolic cone structures respectively.', 'math-0510432-2-7-1': 'Furthermore, for each [MATH] the tangent space of [MATH] at [MATH] is orthogonal to the vector of singular lengths [MATH].', 'math-0510432-2-8-0': 'This theorem describes the structures corresponding to multiangles in the open cube.', 'math-0510432-2-8-1': 'We can also describe the structures corresponding to multiangles contained in some parts of the boundary.', 'math-0510432-2-8-2': 'For instance, we show that the multiangles contained in [MATH] (i.e. none of the angles is [MATH] and at least one of them is zero) are angles of hyperbolic cone structures.', 'math-0510432-2-8-3': 'However, we do not give a description of structures in all points of [MATH], as this would involve studying collapses at cone angle [MATH] and deformations and regenerations of Seifert fibered geometries.', 'math-0510432-2-9-0': 'Let [MATH] be an irreducible closed orientable [MATH]-orbifold.', 'math-0510432-2-9-1': 'If there exists a Euclidean cone structure [MATH] on [MATH] with cone angles strictly less than the orbifold angles of [MATH], then [MATH] is spherical.', 'math-0510432-2-10-0': 'To prove this corollary, we show in Lemma [REF] that the Euclidean cone structure is not almost product, using the irreducibility of [MATH].', 'math-0510432-2-10-1': 'By Theorem [REF] we can obtain a spherical cone structure on [MATH] by increasing any of the cone angles.', 'math-0510432-2-10-2': 'We are using here that [MATH] is orthogonal to the vector of singular lengths [MATH].', 'math-0510432-2-10-3': 'By Proposition [REF] the orbifold angles can be realized by a spherical cone structure.', 'math-0510432-2-10-4': '[MATH]', 'math-0510432-2-11-0': 'This corollary gives an alternative argument to the last step in the proof of the orbifold theorem in [CITATION], which is more natural from the point of view of cone manifolds.', 'math-0510432-2-11-1': 'D. Cooper and S.P. Kerckhoff have announced a different approach to the spherical uniformisation.', 'math-0510432-2-12-0': 'A special case of this theorem was proved in [CITATION], assuming that the singular locus was a knot and a technical hypothesis involving cohomology.', 'math-0510432-2-12-1': 'Even if some of the techniques of [CITATION] are used, this paper does not rely on it.', 'math-0510432-2-13-0': 'Now we describe the organization of the paper: We are interested in the rotational part of the holonomy of the Euclidean cone manifold in [MATH] (in fact its lift to [MATH]), that we denote by [MATH].', 'math-0510432-2-13-1': 'Some basic properties of this representation are studied in Section [REF].', 'math-0510432-2-13-2': 'In order to deform the structure, we view [MATH] as the stabilizer of a point in [MATH] or [MATH], thus we study the local properties of the varieties of representations in the isometries of [MATH] and [MATH] around [MATH].', 'math-0510432-2-13-3': 'This is done in Section [REF], using the cohomological results of Section [REF].', 'math-0510432-2-13-4': 'In Section [REF] we give the conditions for a deformation of the representation [MATH] to correspond to a path of hyperbolic or spherical cone manifolds.', 'math-0510432-2-13-5': 'This is applied in Section [REF] to construct paths of hyperbolic and spherical structures by deforming one of the cone angles.', 'math-0510432-2-13-6': 'The analysis of the local parametrization of the variety of representations is completed in Section [REF], where all deformations of the structure are constructed.', 'math-0510432-2-13-7': 'The global results are established in Section [REF].', 'math-0510432-2-13-8': 'Finally, in Section [REF] we illustrate the main theorem for cone manifold structures on the 3-sphere with singular locus given by the Whitehead link.'}","[['math-0510432-1-3-0', 'math-0510432-2-3-0'], ['math-0510432-1-10-0', 'math-0510432-2-10-0'], ['math-0510432-1-10-1', 'math-0510432-2-10-1'], 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1106.0179,"{'1106.0179-1-0-0': 'The dynamical response of metallic clusters up to [MATH] atoms is investigated using the restricted molecular dynamics simulations scheme.', '1106.0179-1-0-1': 'Exemplarily, sodium like material is considered.', '1106.0179-1-0-2': 'Correlation functions are evaluated to investigate the spatial structure of collective electron excitations and optical response of laser excited clusters.', '1106.0179-1-0-3': 'In particular, the spectrum of bi-local correlation functions shows resonances representing different modes of collective excitations inside the nano plasma.', '1106.0179-1-0-4': 'The spatial structure, the resonance energy and width of the eigenmodes have been investigated for various values of electron density, temperature, cluster size and ionization degree.', '1106.0179-1-0-5': 'Comparison with bulk properties is performed and the dispersion relation of collective excitations is discussed.', '1106.0179-1-1-0': '# Introduction', '1106.0179-1-2-0': 'Nano plasmas can now be readily produced in laser irradiated clusters, and new physical phenomena have come into focus experimentally as well as theoretically.', '1106.0179-1-2-1': 'Interactions between laser fields of [MATH] W cm[MATH] and clusters have been investigated over the last few years, see Refs. [CITATION]-[CITATION].', '1106.0179-1-2-2': 'After laser interaction, extremely large absorption rates of nearly 100%, see [CITATION], as well as x-ray radiation, see Refs. [CITATION]-[CITATION], were found.', '1106.0179-1-2-3': 'In pump-probe experiments, e.g. by Doppner et al. [CITATION] and Fennel et al. [CITATION], the absorption rate of a second laser pulse is strongly dependent on the time delay what is caused by the dynamical properties of the expanding cluster.', '1106.0179-1-2-4': 'We will discuss the dynamical response function of the electrons in a nano plasma that is responsible for scattering and absorption of electromagnetic radiation.', '1106.0179-1-3-0': 'Collective electronic excitations of the nano plasma usually are interpreted as Mie resonances of a homogeneously charged sphere.', '1106.0179-1-3-1': 'Absorption cross section experiments by Xia et al. [CITATION] show multiple resonance structures indeed.', '1106.0179-1-3-2': 'The effect of collective electron motion can also be seen in ultraviolet (UPS) and x-ray photoelectron spectroscopy (XPS) experiments, see [CITATION], which are used to detect binding energies of core level electrons in small metal clusters [CITATION].', '1106.0179-1-3-3': 'In fusion related experiments by Ditmire et al. [CITATION], Grillon et al. [CITATION], as well as Madison et al. [CITATION] ignition processes are started via irradiation of deuterium clusters.', '1106.0179-1-3-4': 'Collective effects in the optical response are discussed in the context of metallic nanoshells by Hoflich et al. [CITATION] as well as nanocavities by Maier et al. [CITATION].', '1106.0179-1-4-0': 'In theoretical calculations of finite systems, see Raitza et al. [CITATION], a more complex resonance structure was found.', '1106.0179-1-4-1': 'Earlier investigations by Reinhard et al. [CITATION] and Kull et al. [CITATION] led to comparable results.', '1106.0179-1-4-2': 'The method of resonance structure analysis using spherical harmonics is known from the discussion of giant dipole resonances of nuclei, see Reinhard et al. [CITATION].', '1106.0179-1-4-3': 'Quantum and semi-classical methods, see Refs. [CITATION]-[CITATION], respectively, were used to investigate the cluster excitation via laser fields.', '1106.0179-1-4-4': 'Collisional absorption processes in nano plasmas have been the subject of theoretical investigations by Hilse et al. [CITATION].', '1106.0179-1-4-5': 'Using density functional theory (DFT) calculations, the electronic structures of cold clusters were analyzed by Ekardt [CITATION], Kummel et al. [CITATION], Brack et al. [CITATION], as well as Krotscheck et al. [CITATION].', '1106.0179-1-4-6': 'The damping of collective electron oscillations was investigated by Ramunno et al. [CITATION] emphasizing the importance of collisional processes beside the Landau damping.', '1106.0179-1-5-0': 'In this work, molecular dynamics (MD) simulations will be used to study nano plasmas in metal clusters.', '1106.0179-1-5-1': 'Clusters consisting of 55 up to 1000 sodium like atoms are considered after short pulse laser irradiation with intensities in the order of [MATH] Wcm[MATH].', '1106.0179-1-5-2': 'Properties of the nano plasma are mainly determined by the dynamics of electrons which are bound to the cluster but ionized from the former atoms, comparable to conduction electrons in bulk systems.', '1106.0179-1-5-3': 'As already shown in earlier publications, see [CITATION], plasma parameters as known from bulk (temperature and particle density) but also the cluster size and net charge are justified for characterization since the electrons can be assumed to be in local thermal equilibrium within time scales considered here.', '1106.0179-1-5-4': 'We focus on parameter ranges where the plasma can be treated classically.', '1106.0179-1-5-5': 'Strong correlations are taken into account via collisions of all particles.', '1106.0179-1-5-6': 'Concepts that have been well established for infinite bulk systems near thermodynamic equilibrium have to be modified for applications to finite systems, e.g. clusters.', '1106.0179-1-5-7': 'In particular, we are interested in the dynamical structure factor and the response function for such finite nano plasmas.', '1106.0179-1-5-8': 'In order to bridge from finite systems to bulk plasmas, we investigate size effects, e.g. in the dynamical collision frequency.', '1106.0179-1-5-9': 'First results in this direction have been reported in Refs. [CITATION].', '1106.0179-1-6-0': 'In Sec. II, correlation functions and their relation to optical properties of homogeneous bulk plasma are introduced as far as it will be of interest to extend the approach to finite systems.', '1106.0179-1-6-1': 'Expressions will also be used for comparison with nano plasmas in the limit of large clusters.', '1106.0179-1-6-2': 'Sec. III explains the restricted molecular dynamics (RMD) scheme for the calculation of the particle trajectories from which the total and bi-local current density correlation functions are determined.', '1106.0179-1-6-3': 'Symmetries in the correlation matrix discussed in Sec. IV can be used for an improved statistics.', '1106.0179-1-6-4': 'The decomposition of the correlation matrix into eigenvectors and eigenvalues is interpreted as a decomposition into collective excitation modes.', '1106.0179-1-6-5': 'In Sec. V, the excitation modes will be characterized with respect to spherical harmonics.', '1106.0179-1-6-6': 'In the further analysis, we focus on modes with a dipole moment, which are also seen in the total current density auto-correlation function.', '1106.0179-1-6-7': 'First results for resonance frequencies and damping are presented.', '1106.0179-1-6-8': 'Regarding the dipole-like modes, the spatial structure at the selected resonance frequency will be discussed in Subsec. A and Subsec. B. Conclusion and outlook are given in Sec. VI.', '1106.0179-1-7-0': '# Linear response theory of plasmas in equilibrium', '1106.0179-1-8-0': 'Within linear response theory as derived by Kubo et al., see [CITATION], the reaction of a many-particle system to weak external perturbations can be related to the dynamical behavior of fluctuations in thermal equilibrium.', '1106.0179-1-8-1': 'Denoting the equilibrium statistical operator with [MATH], we introduce the two-time correlation function of the fluctuations [MATH] as the Kubo scalar product [EQUATION] where the time dependence is given in the Heisenberg picture.', '1106.0179-1-8-2': 'The indices [MATH] and [MATH] identify quantum observables.', '1106.0179-1-8-3': 'In particular we consider local properties so that they contain also the position [MATH].', '1106.0179-1-8-4': 'In case of [MATH], it is called auto-correlation function (ACF).', '1106.0179-1-9-0': 'In the classical case, equilibrium two-time correlation functions can be calculated according to [EQUATION] where we assumed ergodic systems - the ensemble average can be replaced by a time average.', '1106.0179-1-9-1': 'The spectrum of the equilibrium correlation function [MATH] then results from Laplace transformation.', '1106.0179-1-10-0': 'We consider an induced electron density fluctuation [MATH] at time [MATH] as the deviation from the equilibrium density distribution [MATH] due to an external potential [MATH] at times [MATH].', '1106.0179-1-10-1': 'Close to equilibrium, the correlation between the external potential and the induced density fluctuation is only dependent on the time difference [MATH].', '1106.0179-1-10-2': 'Thus, one is able to discuss its spectrum after Laplace transform.', '1106.0179-1-10-3': 'In the same way, the induced electrical current density [MATH] is related to the external electric field [MATH].', '1106.0179-1-10-4': ""Via Kubo's theory, these induced quantities, [MATH] and [MATH], can be expressed within linear response, see [CITATION], as [EQUATION]"", '1106.0179-1-10-5': 'The spectrum of the density fluctuation correlation [MATH] is related to a scalar response function.', '1106.0179-1-10-6': 'The current-density correlation [MATH] represents in general a tensor due to the directions of the current density vector.', '1106.0179-1-11-0': 'Before considering non-local response functions, we shortly mention homogeneous systems.', '1106.0179-1-11-1': 'Properties of the bulk plasmas with electron density [MATH] and inverse temperature [MATH] are only dependent on the difference of the positions [MATH].', '1106.0179-1-11-2': 'Thus, after Fourier transform of the spatial difference [MATH], the correlations are dependent on a wave vector [MATH].', '1106.0179-1-12-0': 'The dynamical structure factor is directly related to the density fluctuation correlation, as [EQUATION] with [MATH] the number of particles.', '1106.0179-1-12-1': 'For further relations to the dielectric function and the optical response of a homogeneous bulk plasma see [CITATION].', '1106.0179-1-12-2': 'Note that the density fluctuations Eq. ([REF]) as well as the density correlation function in Eq. ([REF]) can be expressed in terms of the current-density correlation function via partial integration and using the continuity equation.', '1106.0179-1-12-3': 'Thus, the dynamical structure factor is divided into a static part [MATH] and a dynamical part which is directly related to the longitudinal part of the current-density correlation function [EQUATION]', '1106.0179-1-12-4': 'It is of fundamental interest to describe the collective behavior of the system as response to external fields, in particular emission, absorption and scattering of light.', '1106.0179-1-12-5': 'In bulk systems, the wave vector and frequency dependent response function reads [EQUATION] which can be evaluated using quantum statistical approaches such as Green function theory, see [CITATION], or numerical approaches such as MD simulations, see [CITATION].', '1106.0179-1-12-6': 'As collisions are relevant in strongly correlated systems, the dynamical collision frequency [MATH] is derived and appears in a generalized Drude formula [CITATION] [EQUATION]', '1106.0179-1-12-7': 'In the classical case, the current-density correlation function has been extensively discussed in the long wavelength limit [MATH] applying MD simulations and perturbative approaches.', '1106.0179-1-12-8': 'Exemplarily, we refer to [CITATION].', '1106.0179-1-13-0': 'The state of a homogeneous one-component plasma in thermodynamic equilibrium is characterized by the nonideality parameter [MATH] and the degeneracy parameter [MATH], [MATH] is the temperature of the electrons.', '1106.0179-1-13-1': 'Considering the response function [MATH] in the long wavelength limit, a sharp peak arises at the plasmon frequency [MATH], see Eq. ([REF]).', '1106.0179-1-13-2': 'For finite wavelengths, the resonance is shifted and can be approximated by the so called Gross-Bohm plasmon dispersion for small wave numbers [MATH], see [CITATION], [MATH] with the Debye screening length [MATH].', '1106.0179-1-13-3': 'This relation has recently been revisited with respect to the relevance of collisions by Thiele et al. [CITATION].', '1106.0179-1-13-4': 'According to Eq. ([REF]), the general behavior of the response function [MATH] in the long-wavelength limit is closely related to the collision frequency which is relevant in non-ideal plasmas, see [CITATION].', '1106.0179-1-13-5': 'In the two-component plasma, a phonon mode can arise in addition to the plasmon excitations [CITATION].', '1106.0179-1-14-0': 'The response function [MATH] and the related dynamical structure factor [MATH] as well as the optical properties have been intensively investigated for electron-ion bulk systems, see Refs. [CITATION].', '1106.0179-1-14-1': 'In this work, the inhomogeneous case of finite clusters in local thermal equilibrium will be discussed.', '1106.0179-1-14-2': 'The response of inhomogeneous systems is not only dependent on the difference of the positions, but on [MATH] and [MATH] separately.', '1106.0179-1-14-3': 'Therefore, spatially resolved current density correlation functions [MATH] can not be diagonalized by spatial Fourier transform.', '1106.0179-1-14-4': 'Instead of plane waves, other basis functions have to be found in order to characterize the collective excitations of electrons.', '1106.0179-1-15-0': '# MD simulations of finite plasmas', '1106.0179-1-16-0': 'Finite plasma systems have been investigated using the restricted molecular dynamics (RMD) simulations, see Raitza et al. [CITATION].', '1106.0179-1-16-1': 'A two-component system of singly charged ions and electrons will be described using an error function pseudo potential for the interaction of particles [MATH] and [MATH] [EQUATION] where [MATH] is the charge of the [MATH]th particle.', '1106.0179-1-16-2': 'The Coulomb interaction is modified at short distances, assuming a Gaussian wave function for electrons motivated by the account of quantum effects.', '1106.0179-1-16-3': 'Considering a sodium like system, the potential parameter [MATH] nm was chosen in order to reproduce the ionization energy of [MATH] eV for solid sodium, as already discussed for MD simulations by Suraud et al. [CITATION].', '1106.0179-1-17-0': 'The velocity Verlet algorithm [CITATION] was applied to solve the classical equations of motion for electrons and ions.', '1106.0179-1-17-1': 'This method takes into account the conservation of the total energy of the finite system, as long as there is no external potential.', '1106.0179-1-17-2': 'To follow the fast electron dynamics, time steps of [MATH] fs were taken to calculate the time evolution.', '1106.0179-1-17-3': 'Contrary to bulk MD simulations no periodic boundary conditions are applied.', '1106.0179-1-18-0': 'Icosahedral arrangements of 55, 147, and 309 ions, see [CITATION], were considered as initial configuration for the ion positions.', '1106.0179-1-18-1': 'For these nearly spherically, homogeneously distributed ions, the ion density typical for solid sodium is given by an ionic next neighbor distance of [MATH] nm.', '1106.0179-1-18-2': 'In addition, randomly distributed ion configurations within a given sphere were considered for comparison and the number of ions was increased up to 1000 particles.', '1106.0179-1-18-3': 'Starting with a neutral cluster, the electrons have been positioned nearby the ions with small, randomly distributed deviations from the ion positions.', '1106.0179-1-19-0': 'To simulate experiments where clusters are excited by short pulse lasers, MD simulations are performed under the influence of an electric field, assuming a Gaussian shape and pulse duration of about 100 fs.', '1106.0179-1-19-1': 'Due to the largely increased kinetic energy of the electrons, ionization processes occur.', '1106.0179-1-19-2': 'After the laser field is switched off, the ionization degree of the cluster is determined by the number of electrons found outside the cluster radius with positive total energy, so that they can escape from the cluster.', '1106.0179-1-19-3': 'Due to ion excitation on larger time scales, a slow expansion of the positively charged cluster is obseved [CITATION], leading to Coulomb explosion experimentally.', '1106.0179-1-20-0': 'Considering the single-time properties, it was found in [CITATION] that already local thermodynamic equilibrium (LTE) is established within a few fs after the electron heating.', '1106.0179-1-20-1': 'In particular, at each time step, the momentum distribution of electrons is well described by a Maxwell distribution, and the spatial density profile agrees with a Boltzmann distribution with respect to the average potential that is determined by the actual ion configuration and the self-consistent electronic mean field.', '1106.0179-1-20-2': 'The fact that electrons are considered within sub fs time intervals, while the ion configuration remains nearly unchanged, enables us to separate the electron dynamics from the ion dynamics.', '1106.0179-1-21-0': 'Subsequently, the dynamical properties of the electron subsystem can be calculated for a frozen ionic configuration thus referring to a specific time.', '1106.0179-1-21-1': 'This is considered as an adiabatic approximation to the true dynamical properties of the electron subsystem which have to take into account the slow change in the ion configuration.', '1106.0179-1-21-2': 'More rigorously, non-stationary time dependent correlation functions have to be treated for the full charged particle system.', '1106.0179-1-22-0': 'Using the RMD simulations scheme as introduced in [CITATION], the ions are kept fixed acting as external trap potential.', '1106.0179-1-22-1': 'Starting from an initial state, the many-electron trajectory [MATH] is calculated, solving the classical equations of motion of the electrons.', '1106.0179-1-22-2': 'From this, all further physical properties of the electron subsystem inside the cluster are determined.', '1106.0179-1-22-3': 'Within RMD simulations, we consider no temporal variation of the plasma parameters that are determined by the frozen ion distribution, the electron temperature and the degree of ionization.', '1106.0179-1-22-4': 'A long-time run can be performed in order to replace the ensemble average by a temporal average.', '1106.0179-1-22-5': 'This has been successfully done for the single-time properties such as the momentum distribution and the density profile, see [CITATION] and will now be applied to the two-time correlation functions.', '1106.0179-1-23-0': 'Using classical MD simulation techniques, the results are valid for non-degenerate plasmas.', '1106.0179-1-23-1': 'This restricts the temperature range to [MATH] eV where our simulations can be compared with realistic sodium clusters.', '1106.0179-1-23-2': 'Values for the plasma parameter [MATH] can be treated since we are not confined to the weak coupling limit as, e.g., in perturbation theory.', '1106.0179-1-24-0': 'In our RMD calculations, we start from a homogeneous ion configuration (icosahedral or randomly distributed) inside the cluster at fixed ion density.', '1106.0179-1-24-1': 'In the case of random distribution, we perform averaging over different initial configurations of ions.', '1106.0179-1-24-2': 'The Langevin thermostat was used to heat the electrons at an initial stage.', '1106.0179-1-24-3': 'Hot electrons are emitted during this stage so that the cluster becomes ionized.', '1106.0179-1-24-4': 'Evaluating the trajectories of electrons, sufficient time of about 200 fs has to be allowed before a stationary ionization degree is established.', '1106.0179-1-24-5': 'Then the thermostat is switched off and a production run is performed using NVE ensemble.', '1106.0179-1-24-6': 'It is checked that the mean cluster charge [MATH] and the system temperature do not change any more.', '1106.0179-1-24-7': 'In Fig. [REF], the cluster charge [MATH] depending on cluster size [MATH] is shown.', '1106.0179-1-24-8': 'A power fit [MATH] with, for example, [MATH] and [MATH] for [MATH]1 eV shows the trend of the size dependent ionization degree.', '1106.0179-1-25-0': 'Using the trajectories of all [MATH] electrons obtained from the RMD simulations scheme, the local current density [MATH] at position [MATH] was calculated for each time step [MATH] [EQUATION] which is the sum over all electron momenta [MATH] inside a small volume [MATH] at position [MATH], where [MATH], and [MATH] for electrons found outside [MATH].', '1106.0179-1-25-1': 'The size of the volume determines the spatial resolution of the local current density [MATH].', '1106.0179-1-25-2': 'However, it must be taken sufficiently large to reduce statistical fluctuations.', '1106.0179-1-26-0': 'The bi-local correlation tensor of the normalized spatially-resolved current density is calculated according to Eq. ([REF]) as [EQUATION] with [MATH] the total current density.', '1106.0179-1-26-1': 'Typical values are [MATH] and [MATH] fs.', '1106.0179-1-26-2': 'Its Laplace transform reads [EQUATION]', '1106.0179-1-26-3': 'In the following, we restrict ourselves to the diagonal components [MATH] of this tensor, where only parallel components of the current density vectors are correlated as already introduced in Sec. II.', '1106.0179-1-26-4': 'As it will be shown in the following sections, this bi-local current-density correlation is important to understand the excitation modes of nano plasmas.', '1106.0179-1-26-5': 'The non-diagonal components of the bi-local correlation tensor are small in comparison to the diagonal components.', '1106.0179-1-26-6': 'Beside the bi-local current density correlation function considered here, the bi-local density fluctuation correlation [MATH] as well as the bi-local force correlation [MATH] are useful quantities in the context of optical properties.', '1106.0179-1-26-7': 'These correlations can be evaluated from the trajectory in a similar way and are related to the bi-local current density correlation.', '1106.0179-1-26-8': 'This will not be discussed in an upcoming paper.', '1106.0179-1-27-0': 'Because of the spherical symmetry of the cluster geometry during excitation and expansion, the volume is divided into sections [MATH] according to [MATH], [MATH], [MATH] equidistant intervals of spherical coordinates, i.e. the distance [MATH] to the center of the cluster, the inclination angle [MATH] as well as the azimuthal angle [MATH], respectively.', '1106.0179-1-27-1': 'The cluster radius [MATH] is given by the root mean square radius of ions according to [MATH].', '1106.0179-1-27-2': 'The sections are numbered by a single counter [MATH] with three independent counters according to the three coordinates: [MATH], [MATH] and [MATH].', '1106.0179-1-27-3': 'With respect to Eq. ([REF]) the bi-local correlation matrix [MATH] for the spatially resolved cluster and its Laplace transform [MATH] have been calculated.', '1106.0179-1-28-0': 'The total current density ACF can be calculated from the trajectories directly.', '1106.0179-1-28-1': 'Please note, that it can be also calculated from the bi-local current density correlation matrix [EQUATION] using the cluster volume [MATH] and the individual cell volumes [EQUATION]', '1106.0179-1-28-2': 'The consistency of these expressions has been checked throughout our explicit calculations.', '1106.0179-1-29-0': '# From bi-local correlation function to excitation modes', '1106.0179-1-30-0': 'In the following, we discuss calculations for the current-density ACF, Eq. ([REF]), and the bi-local current-density correlation spectrum [MATH].', '1106.0179-1-30-1': 'Exemplarily, we present results for the Na[MATH] cluster at electron temperature [MATH] eV, cluster charge [MATH] and ionic density [MATH] cm[MATH].', '1106.0179-1-30-2': 'Starting with a solid density cluster, these are typical parameters obtained directly after the interaction with a short pulse laser of 100 fs duration and intensity of [MATH] Wcm[MATH].', '1106.0179-1-30-3': 'Calculations of other cluster sizes will be presented in the following sections.', '1106.0179-1-31-0': 'The real part of the total current-density ACF [MATH] is shown in Fig. [REF].', '1106.0179-1-31-1': 'Three maxima are obtained.', '1106.0179-1-31-2': 'This feature differs from the bulk behavior and is interpreted as different resonances of the electron system.', '1106.0179-1-31-3': 'To investigate the origin of the different maxima as collective excitations of the nano plasma, the bi-local current-density correlation matrix was calculated as well.', '1106.0179-1-32-0': 'The following spatial symmetries in the matrix [MATH] were found [EQUATION]', '1106.0179-1-32-1': 'In our case, the [MATH] elements of the full matrix can be reduced to [MATH] independent elements due to the symmetries Eq. ([REF]) - Eq. ([REF]), thus improving statistics via averaging equal elements.', '1106.0179-1-33-0': 'Because of the different size of section volumes in spherical coordinates there are large variations in the mean number of particles in a section.', '1106.0179-1-33-1': 'Provided that we have [MATH] electrons and [MATH] sections the average number of particles in some sections can be even smaller than unity.', '1106.0179-1-33-2': 'In this case, the local current density Eq. ([REF]) is affected by strong fluctuations due to the discrete number of particles.', '1106.0179-1-33-3': 'This problem is reduced, when you consider the current [MATH] as the contribution of smaller cells will be damped.', '1106.0179-1-33-4': 'Therefore, we used the non-normalized form of the correlation function for further analysis [EQUATION] for which the following eigenproblem was solved, [EQUATION]', '1106.0179-1-33-5': 'Thus, the matrix is decomposed into eigenvectors [MATH] as well as their eigenvalues [MATH] at each frequency.', '1106.0179-1-33-6': 'The eigenvectors represent the spatial structure of the mode ([MATH]).', '1106.0179-1-33-7': 'The orthonormality condition [EQUATION] holds.', '1106.0179-1-34-0': 'For two selected frequencies, the 10 strongest eigenvalues of the Na[MATH] cluster are shown in Fig. [REF].', '1106.0179-1-34-1': 'At [MATH] fs[MATH] (black), a resonance frequency was found with one outstanding, leading eigenvalue.', '1106.0179-1-34-2': 'The second and third largest eigenvalue are of same strength, which suggests degeneracy due to the symmetry of the correlation matrix.', '1106.0179-1-34-3': 'At off-resonant frequencies, i.e. at [MATH] fs[MATH] (shaded), all eigenvalues are of the same order of magnitude.', '1106.0179-1-35-0': 'In Fig. [REF] (left) the strongest eigenvalues [MATH] of the Na[MATH] cluster are shown in dependence of frequency.', '1106.0179-1-35-1': 'They are colored according to their strength and numbered ascending with descending strength.', '1106.0179-1-35-2': 'In the shown frequency range, modes [MATH] with well defined maxima are found.', '1106.0179-1-35-3': 'The spatial oscillation structure can be identified by analyzing the eigenvectors.', '1106.0179-1-36-0': 'In Fig. [REF] (right), the spectra of eigenvalues are sorted in an alternative way, according to the spatial structure of the eigenvector which is obtained over the whole frequency range.', '1106.0179-1-36-1': 'Overall, the black solid mode is the strongest.', '1106.0179-1-36-2': 'Its resonance frequencies are also found in the total current-density ACF (indicated via vertical blue dashed lines) and are therefore of particular interest.', '1106.0179-1-36-3': 'Resonances in the total current-density ACF, shown in Fig. [REF], are only possible in the case of non-zero total current, which is caused by a dipole-like oscillation.', '1106.0179-1-36-4': 'Thus, resonances which are seen in the total current-density ACF are oscillation modes with a dipole moment.', '1106.0179-1-36-5': 'Other resonance structures, for example, are breathing modes that have no dipole moment.', '1106.0179-1-36-6': 'After characterization of the resonance structures, the dipole-like resonances will be investigated in more detail.', '1106.0179-1-37-0': '# Analysis of the collective modes', '1106.0179-1-38-0': 'The decomposition of the locally resolved current correlation matrix into eigenvalues [MATH], as shown in Fig. [REF], gives a very complex set of resonance structures in comparison to the 1D case, see [CITATION].', '1106.0179-1-38-1': 'The spatial mode structures in 1D chains were characterized by their wave number [MATH].', '1106.0179-1-38-2': 'To analyze the more complicated spatial oscillation structure of 3D clusters, a spherical Fourier decomposition of the eigenvectors into the spherical Bessel function [MATH] and spherical harmonics [MATH] was performed according to [EQUATION] where [MATH] is the spherical Fourier component with ordinal numbers [MATH].', '1106.0179-1-38-3': 'The normalization factor [MATH] as well as the wave number [MATH] are chosen in the way that the eigenvector has a root at the cluster surface.', '1106.0179-1-39-0': 'In Fig. [REF] (right), the four strongest eigenvalue modes are characterized by pairs of ordinal numbers [MATH] which determine the main angular part of the eigenvector by the spherical harmonics [MATH].', '1106.0179-1-39-1': 'The leading dipole-like mode, represented via solid black lines in Fig. [REF] (right), is characterized by the overlap of the spherical harmonic functions [MATH] and [MATH].', '1106.0179-1-39-2': 'For the Na[MATH] cluster, one can find three resonance frequencies which are identical to the ones found in the total current-density ACF.', '1106.0179-1-39-3': 'The latter are indicated by vertical dashed lines (blue online) in Fig. [REF] (right).', '1106.0179-1-40-0': 'In our investigations, we looked at other cluster parameters as well and found similar behavior.', '1106.0179-1-40-1': 'Comparisons will be made in the following chapters.', '1106.0179-1-40-2': 'For further analysis of the exication modes, we now consider a larger cluster consisting of 1000 ions.', '1106.0179-1-40-3': 'There, four pronounced dipole-like resonances were found.', '1106.0179-1-40-4': 'In Fig. [REF], the spatial structures of the current-density [MATH] is shown for the Na[MATH] cluster at the resonance frequencies of the leading dipole-like mode.', '1106.0179-1-40-5': 'The behavior is shown in the [MATH]plane at a fixed azimuthal angle [MATH] on which it does not depend.', '1106.0179-1-41-0': 'At the resonance frequency [MATH] fs, the electrons are oscillating with a current density [MATH].', '1106.0179-1-41-1': 'As shown in Fig. [REF] on the left hand side, all electrons of this mode are moving in the same direction and no nodes can be seen.', '1106.0179-1-41-2': 'Assuming a constant velocity field amplitude [MATH], the change of the current density with distance [MATH] is directly related to the density profile [MATH] of the electrons.', '1106.0179-1-42-0': 'The third and fourth mode from the left of Fig. [REF] are similar to a plane wave oscillation of electrons, but trapped inside the cluster.', '1106.0179-1-42-1': 'To identify a wave number of the plane wave oscillation, a Fourier decomposition of plane waves in [MATH]-direction was done.', '1106.0179-1-42-2': 'A maximum at [MATH] nm[MATH] and [MATH] nm[MATH], respectively, is found which identify the wavelengths of the plane wave oscillations.', '1106.0179-1-42-3': 'Only in the large cluster with 1000 ions, a plane wave oscillation with higher wavenumber was found.', '1106.0179-1-42-4': 'All other modes can be seen in smaller clusters as well.', '1106.0179-1-42-5': 'The second resonance structure from the left in Fig. [REF] looks like a mix of the first and the third resonance structure.', '1106.0179-1-43-0': 'We want to point out one further feature of the mode spectra in Fig. [REF] (right).', '1106.0179-1-43-1': 'The dashed red line represents in fact two resonance structures with exactly the same eigenvalues at all frequencies.', '1106.0179-1-43-2': 'The eigenvectors are orthogonal since they are characterized by the same spherical harmonic function [MATH] but have a phase shift in [MATH]-direction: [MATH].', '1106.0179-1-43-3': 'Further degenerations are obtained for weaker eigenvalue modes as well.', '1106.0179-1-44-0': 'All eigenvectors [MATH] are decomposed into a superposition of spherical Bessel functions [MATH] with a set of ordinal numbers [MATH].', '1106.0179-1-44-1': 'No leading ordinal number [MATH] was found, which characterizes the spatial resonance structure in [MATH] direction.', '1106.0179-1-45-0': '## Resonance frequency of the rigid oscillation', '1106.0179-1-46-0': 'The total current density ACF shown in Fig. [REF] as well as the leading eigenvalue mode in Fig. [REF] (right) show the strongest resonance at the frequency [MATH] fs[MATH].', '1106.0179-1-46-1': 'This resonance belongs to the dipole-like mode with the eigenvector shown in Fig. [REF] on the left hand side.', '1106.0179-1-46-2': 'We will now analyze this collective excitation mode in terms of a rigid oscillation.', '1106.0179-1-47-0': 'The electrons with density profile [MATH] are assumed to move nearly rigidly in the external potential [MATH] due to the fixed ions.', '1106.0179-1-47-1': 'The potential energy of the electrons due to a small shift with respect to the ions reads [EQUATION]', '1106.0179-1-47-2': 'The change of the potential energy [MATH] in [MATH] direction is due to the restoring force on the electron profile.', '1106.0179-1-47-3': 'In harmonic approximation of the equation of motion, the resonance frequency is identified as [EQUATION]', '1106.0179-1-47-4': 'For small rigid shifts [MATH], assuming radially dependent electron density profiles and external potentials in Eq. ([REF]) the integration over the angular dependence of the potential energy calculation can be executed.', '1106.0179-1-47-5': 'The resonance frequency Eq. ([REF]) is then given according to [EQUATION]', '1106.0179-1-47-6': 'As a first example for a density profile, we assume a homogeneously charged ion sphere with radius [MATH] and an electron sphere with radius [MATH].', '1106.0179-1-47-7': 'The densities of the electron and ion spheres is equal ([MATH]).', '1106.0179-1-47-8': 'Therefore, the difference of ion and electron radius is determined by the cluster charge, basically the difference of the simulated electron number [MATH] and ion number [MATH].', '1106.0179-1-47-9': 'Thus, in the case of positively charged clusters, as discussed here, the electron sphere radius is smaller than the ion radius ([MATH]).', '1106.0179-1-47-10': 'The error function potential Eq. ([REF]) was taken as electron-ion-interaction potential for the calculation of the resonance frequency, as it was used for the MD simulation as well.', '1106.0179-1-47-11': 'The resonance frequency than reads [EQUATION]', '1106.0179-1-47-12': 'In the limit of large clusters with high number of ions the resonance frequency equals the Mie frequency, [MATH].', '1106.0179-1-47-13': 'Assuming only a weak charged cluster, the sphere radii have nearly the same size ([MATH]) and the system is nearly neutral.', '1106.0179-1-47-14': 'The limit for small clusters, down to just one atom, depends strongly on the pseudopotential.', '1106.0179-1-47-15': 'In our case, the resonance frequency [MATH] is due to the oscillation of a single electron in the ionic error-function pseudo-potential Eq. ([REF]).', '1106.0179-1-48-0': 'In Fig. [REF] (left), the resonance frequency [MATH] of the dipole-like mode is shown in dependence on the size of the ion sphere.', '1106.0179-1-48-1': 'Results from MD simulations (empty circles) for Na[MATH], Na[MATH] and Na[MATH] cluster at [MATH] cm[MATH] as well as the Na[MATH] cluster at [MATH] cm[MATH] are shown.', '1106.0179-1-48-2': 'The resonance frequencies have been calculated using Eq. ([REF]) for ion densities of [MATH] cm[MATH] (solid shaded line, red online) and [MATH] cm[MATH] (solid black line).', '1106.0179-1-48-3': 'The limits of large clusters, the Mie frequency [MATH], are given as dotted lines colored according to the two densities.', '1106.0179-1-49-0': 'Additionally, the electron density profile [MATH] was deducted from MD simulations for all cluster sizes and used to derive the resonance frequency [MATH] solving Eq. ([REF]) numerically.', '1106.0179-1-49-1': 'As a result (full circles in Fig. [REF] (left)), the resonance frequency of the dipole-like mode is obtained with a deviation to the direct simulation results of less than 5%.', '1106.0179-1-49-2': 'Using homogeneously charged ion and electron spheres leads to reasonable agreement in the limits of large clusters as well as for small clusters.', '1106.0179-1-49-3': 'Taking the spatial structure of the density profile into account, there is good agreement with the direct simulation results in the intermediate cluster size regime as well.', '1106.0179-1-50-0': '## Dispersion of the plane wave mode', '1106.0179-1-51-0': 'While the Mie-like resonance, discussed in the previous subsection, is almost spherically symmetric, we obtain an increasing plane wave character of the dipole-like mode with increasing frequency.', '1106.0179-1-51-1': 'The third resonance frequency of the total current density ACF for the Na[MATH] cluster at [MATH] fs[MATH], see Fig. [REF], is mainly caused by a plane wave like eigenvector, which is similar to the eigenvector of the Na[MATH] cluster, shown in Fig. [REF] (right).', '1106.0179-1-51-2': 'This mode is discussed by Kresin et al. [CITATION] as compressional volume plasmon.', '1106.0179-1-51-3': 'Here, oscillations of electrons in opposite directions must be taken into account for the analytical calculation of the resonance frequency.', '1106.0179-1-51-4': 'We assume homogeneously charged spheres for the electrons with radius [MATH] and for the ions with radius [MATH] as it was already discussed in the previous subsection.', '1106.0179-1-51-5': 'The electron motion is treated as a hydrodynamical liquid using the Euler equation [EQUATION] where [MATH] is the spatially resolved current density of the electrons, [MATH] is the pressure of the electron gas and [MATH] is the external potential, composed of contributions from the electrons and ions.', '1106.0179-1-51-6': 'Using the following ansatz [EQUATION] we consider small perturbations in [MATH]-direction restricting ourselves to longitudinal effects.', '1106.0179-1-51-7': 'One is able to linearize the Euler equation.', '1106.0179-1-51-8': 'The system is assumed to be in LTE, described by the quantities [MATH], [MATH], [MATH] as well as [MATH].', '1106.0179-1-51-9': 'Electrons are moving in the external field of ions and in the mean field of electrons.', '1106.0179-1-51-10': 'The external potential is [EQUATION]', '1106.0179-1-51-11': 'The external potential has a equilibrium part and a perturbative part [MATH], which is mainly dependent on the linear density perturbation [MATH].', '1106.0179-1-52-0': 'Assuming Boltzmann distribution we express the ideal gas pressure of the electrons [MATH] via the electron density.', '1106.0179-1-52-1': 'Using the equation of continuity, one is able to express the Euler equation in terms of linear perturbations of the density.', '1106.0179-1-52-2': 'Thus, the equilibrium part of the external potential compensates the pressure term on the right hand side of Eq. ([REF]).', '1106.0179-1-52-3': 'Restricting ourselves to linear perturbations of the Euler equation, only the third term on the right hand side of Eq. ([REF]) remains, which is connected to the external potential.', '1106.0179-1-52-4': 'Finally, all terms of the Euler equation Eq. ([REF]) are lead back to a linear density fluctuation [MATH].', '1106.0179-1-52-5': 'Thus, one ends up with [EQUATION]', '1106.0179-1-52-6': 'This relation leads to real valued solutions for the resonance frequencies for standing waves with [MATH] only and scales with the plasma frequency [MATH].', '1106.0179-1-52-7': 'In the limit [MATH] we find [MATH] which coincides with the bulk limit.', '1106.0179-1-53-0': 'From the eigenvector of the plane wave mode, one can derive the wave number [MATH], which corresponds to [MATH].', '1106.0179-1-53-1': 'This means the dispersion of the plane wave mode is determined by the radius of the electron cloud.', '1106.0179-1-53-2': 'Results for this case are shown in Fig. [REF] (right) for different cluster sizes and are compared with the simulation data.', '1106.0179-1-53-3': 'For the cluster with 1000 ions a plane wave mode with [MATH] and [MATH] was found as well.', '1106.0179-1-53-4': 'In Fig. [REF] (right), it is marked with an empty square.', '1106.0179-1-53-5': 'Its spatial structure is shown in Fig. [REF] (right).', '1106.0179-1-53-6': 'The simulation data for the Na[MATH] cluster fit the dispersion Eq. ([REF]) as well.', '1106.0179-1-53-7': 'Deviations of the plane wave resonance for smaller clusters are caused by to the radial dependence of the electron density profile.', '1106.0179-1-54-0': '# Conclusion', '1106.0179-1-55-0': 'We have investigated collective excitation modes of a nano plasma in highly excited metal clusters.', '1106.0179-1-55-1': 'The collective excitation of electrons inside the cluster are obtained from bi-local current density correlation functions by solving the eigenvalue problem of the current-density correlation matrix.', '1106.0179-1-55-2': 'Using RMD simulations, the local current density [MATH] for excited clusters of 55 up to 1000 ions with densities of [MATH] cm[MATH] as well as [MATH] cm[MATH] and temperatures of [MATH] eV have been investigated.', '1106.0179-1-55-3': 'Pseudo-potentials of sodium were used to calculate the electron dynamics without consideration of degeneration effects any further.', '1106.0179-1-55-4': 'For the analysis of electron dynamics at lower temperatures, the inclusion of quantum effects for the calculation of the local current density [MATH] of cluster electrons is an open question at this point.', '1106.0179-1-55-5': 'It would be useful to go beyond classical description to discuss for example cold, non-excited clusters.', '1106.0179-1-56-0': 'The spectrum of dipole-like modes was investigated in more detail.', '1106.0179-1-56-1': 'Using analytical calculations, it was possible to relate the position of resonance modes in the frequency domain to their spatial mode structure.', '1106.0179-1-56-2': 'Results for the cluster size dependence of the resonance frequency have been shown.', '1106.0179-1-56-3': 'A smooth transition to the bulk behavior has been obtained.', '1106.0179-1-56-4': 'The analysis of further resonance frequencies and also other modes including breathing modes would be desirable.', '1106.0179-1-56-5': 'The width of mode resonances and the role of collision-less damping effects as well as the collision frequency need to be investigated in the future.', '1106.0179-1-56-6': 'The systematic change of the collision frequency with cluster size up to the bulk limit remains an interesting field.', '1106.0179-1-57-0': 'From RMD simulations, different collective excitations have been found in nano plasmas, including dipole-like and breathing modes.', '1106.0179-1-57-1': 'These collective excitations will influence the scattering and absorption properties of clusters, see [CITATION].', '1106.0179-1-57-2': 'Collective effects of electron motion play a role when analyzing ultraviolet (UPS) or x-ray photo-electron spectroscopy (XPS) experiments, as has been pointed out by Andersson et al. [CITATION].', '1106.0179-1-57-3': 'It is a challenge to experimentalists to confirm the occurence of different collective excitations in nano plasmas.'}","{'1106.0179-2-0-0': 'The dynamical response of metallic clusters up to [MATH] atoms is investigated using the restricted molecular dynamics simulations scheme.', '1106.0179-2-0-1': 'Exemplarily, sodium like material is considered.', '1106.0179-2-0-2': 'Correlation functions are evaluated to investigate the spatial structure of collective electron excitations and optical response of laser excited clusters.', '1106.0179-2-0-3': 'In particular, the spectrum of bi-local correlation functions shows resonances representing different modes of collective excitations inside the nano plasma.', '1106.0179-2-0-4': 'The spatial structure, the resonance energy and width of the eigenmodes have been investigated for various values of electron density, temperature, cluster size and ionization degree.', '1106.0179-2-0-5': 'Comparison with bulk properties is performed and the dispersion relation of collective excitations is discussed.', '1106.0179-2-1-0': '# Introduction', '1106.0179-2-2-0': 'Nano plasmas can now be readily produced in laser irradiated clusters, and new physical phenomena have come into focus experimentally as well as theoretically.', '1106.0179-2-2-1': 'Interactions between laser fields of [MATH] W cm[MATH] and clusters have been investigated over the last few years, see Refs. [CITATION]-[CITATION].', '1106.0179-2-2-2': 'After laser interaction, extremely large absorption rates of nearly 100%, see [CITATION], as well as x-ray radiation, see Refs. [CITATION]-[CITATION], were found.', '1106.0179-2-2-3': 'In pump-probe experiments, e.g. by Doppner et al. [CITATION] and Fennel et al. [CITATION], the absorption rate of a second laser pulse is strongly dependent on the time delay what is caused by the dynamical properties of the expanding cluster.', '1106.0179-2-2-4': 'We will discuss the dynamical response function of the electrons in a nano plasma that is responsible for scattering and absorption of electromagnetic radiation.', '1106.0179-2-3-0': 'Collective electronic excitations of the nano plasma usually are interpreted as Mie resonances of a homogeneously charged sphere.', '1106.0179-2-3-1': 'Absorption cross section experiments by Xia et al. [CITATION] show multiple resonance structures indeed.', '1106.0179-2-3-2': 'The effect of collective electron motion can also be seen in ultraviolet (UPS) and x-ray photoelectron spectroscopy (XPS) experiments, see [CITATION], which are used to detect binding energies of core level electrons in small metal clusters [CITATION].', '1106.0179-2-3-3': 'In fusion related experiments by Ditmire et al. [CITATION], Grillon et al. [CITATION], as well as Madison et al. [CITATION] ignition processes are started via irradiation of deuterium clusters.', '1106.0179-2-3-4': 'Collective effects in the optical response are discussed in the context of metallic nanoshells by Hoflich et al. [CITATION] as well as nanocavities by Maier et al. [CITATION].', '1106.0179-2-4-0': 'In theoretical calculations of finite systems, see Raitza et al. [CITATION], a more complex resonance structure was found.', '1106.0179-2-4-1': 'Earlier investigations by Reinhard et al. [CITATION] and Kull et al. [CITATION] led to comparable results.', '1106.0179-2-4-2': 'The method of resonance structure analysis using spherical harmonics is known from the discussion of giant dipole resonances of nuclei, see Reinhard et al. [CITATION].', '1106.0179-2-4-3': 'Quantum and semi-classical methods, see Refs. [CITATION]-[CITATION], respectively, were used to investigate the cluster excitation via laser fields.', '1106.0179-2-4-4': 'Collisional absorption processes in nano plasmas have been the subject of theoretical investigations by Hilse et al. [CITATION].', '1106.0179-2-4-5': 'Using density functional theory (DFT) calculations, the electronic structures of cold clusters were analyzed by Ekardt [CITATION], Kummel et al. [CITATION], Brack et al. [CITATION], as well as Krotscheck et al. [CITATION].', '1106.0179-2-4-6': 'The damping of collective electron oscillations was investigated by Ramunno et al. [CITATION] emphasizing the importance of collisional processes beside the Landau damping.', '1106.0179-2-5-0': 'In this work, molecular dynamics (MD) simulations will be used to study nano plasmas in metal clusters.', '1106.0179-2-5-1': 'Clusters consisting of 55 up to 1000 sodium like atoms are considered after short pulse laser irradiation with intensities in the order of [MATH] Wcm[MATH].', '1106.0179-2-5-2': 'Properties of the nano plasma are mainly determined by the dynamics of electrons which are bound to the cluster but ionized from the former atoms, comparable to conduction electrons in bulk systems.', '1106.0179-2-5-3': 'As already shown in earlier publications, see [CITATION], plasma parameters as known from bulk (temperature and particle density) but also the cluster size and net charge are justified for characterization since the electrons can be assumed to be in local thermal equilibrium within time scales considered here.', '1106.0179-2-5-4': 'We focus on parameter ranges where the plasma can be treated classically.', '1106.0179-2-5-5': 'Strong correlations are taken into account via collisions of all particles.', '1106.0179-2-5-6': 'Concepts that have been well established for infinite bulk systems near thermodynamic equilibrium have to be modified for applications to finite systems, e.g. clusters.', '1106.0179-2-5-7': 'In particular, we are interested in the dynamical structure factor and the response function for such finite nano plasmas.', '1106.0179-2-5-8': 'In order to bridge from finite systems to bulk plasmas, we investigate size effects, e.g. in the dynamical collision frequency.', '1106.0179-2-5-9': 'First results in this direction have been reported in Refs. [CITATION].', '1106.0179-2-6-0': 'In Sec. II, correlation functions and their relation to optical properties of homogeneous bulk plasma are introduced as far as it will be of interest to extend the approach to finite systems.', '1106.0179-2-6-1': 'Expressions will also be used for comparison with nano plasmas in the limit of large clusters.', '1106.0179-2-6-2': 'Sec. III explains the restricted molecular dynamics (RMD) scheme for the calculation of the particle trajectories from which the total and bi-local current density correlation functions are determined.', '1106.0179-2-6-3': 'Symmetries in the correlation matrix discussed in Sec. IV can be used for an improved statistics.', '1106.0179-2-6-4': 'The decomposition of the correlation matrix into eigenvectors and eigenvalues is interpreted as a decomposition into collective excitation modes.', '1106.0179-2-6-5': 'In Sec. V, the excitation modes will be characterized with respect to spherical harmonics.', '1106.0179-2-6-6': 'In the further analysis, we focus on modes with a dipole moment, which are also seen in the total current density auto-correlation function.', '1106.0179-2-6-7': 'First results for resonance frequencies and damping are presented.', '1106.0179-2-6-8': 'Regarding the dipole-like modes, the spatial structure at the selected resonance frequency will be discussed in Subsec. A and Subsec. B. Conclusion and outlook are given in Sec. VI.', '1106.0179-2-7-0': '# Linear response theory of plasmas in equilibrium', '1106.0179-2-8-0': 'Within linear response theory as derived by Kubo et al., see [CITATION], the reaction of a many-particle system to weak external perturbations can be related to the dynamical behavior of fluctuations in thermal equilibrium.', '1106.0179-2-8-1': 'Denoting the equilibrium statistical operator with [MATH], we introduce the two-time correlation function of the fluctuations [MATH] as the Kubo scalar product [EQUATION] where the time dependence is given in the Heisenberg picture.', '1106.0179-2-8-2': 'The indices [MATH] and [MATH] identify quantum observables.', '1106.0179-2-8-3': 'In particular we consider local properties so that they contain also the position [MATH].', '1106.0179-2-8-4': 'In case of [MATH], it is called auto-correlation function (ACF).', '1106.0179-2-9-0': 'In the classical case, equilibrium two-time correlation functions can be calculated according to [EQUATION] where we assumed ergodic systems - the ensemble average can be replaced by a time average.', '1106.0179-2-9-1': 'The spectrum of the equilibrium correlation function [MATH] then results from Laplace transformation.', '1106.0179-2-10-0': 'We consider an induced electron density fluctuation [MATH] at time [MATH] as the deviation from the equilibrium density distribution [MATH] due to an external potential [MATH] at times [MATH].', '1106.0179-2-10-1': 'Close to equilibrium, the correlation between the external potential and the induced density fluctuation is only dependent on the time difference [MATH].', '1106.0179-2-10-2': 'Thus, one is able to discuss its spectrum after Laplace transform.', '1106.0179-2-10-3': 'In the same way, the induced electrical current density [MATH] is related to the external electric field [MATH].', '1106.0179-2-10-4': ""Via Kubo's theory, these induced quantities, [MATH] and [MATH], can be expressed within linear response, see [CITATION], as [EQUATION]"", '1106.0179-2-10-5': 'The spectrum of the density fluctuation correlation [MATH] is related to a scalar response function.', '1106.0179-2-10-6': 'The current-density correlation [MATH] represents in general a tensor due to the directions of the current density vector.', '1106.0179-2-11-0': 'Before considering non-local response functions, we shortly mention homogeneous systems.', '1106.0179-2-11-1': 'Properties of the bulk plasmas with electron density [MATH] and inverse temperature [MATH] are only dependent on the difference of the positions [MATH].', '1106.0179-2-11-2': 'Thus, after Fourier transform of the spatial difference [MATH], the correlations are dependent on a wave vector [MATH].', '1106.0179-2-12-0': 'The dynamical structure factor is directly related to the density fluctuation correlation, as [EQUATION] with [MATH] the number of particles.', '1106.0179-2-12-1': 'For further relations to the dielectric function and the optical response of a homogeneous bulk plasma see [CITATION].', '1106.0179-2-12-2': 'Note that the density fluctuations Eq. ([REF]) as well as the density correlation function in Eq. ([REF]) can be expressed in terms of the current-density correlation function via partial integration and using the continuity equation.', '1106.0179-2-12-3': 'Thus, the dynamical structure factor is divided into a static part [MATH] and a dynamical part which is directly related to the longitudinal part of the current-density correlation function [EQUATION]', '1106.0179-2-12-4': 'It is of fundamental interest to describe the collective behavior of the system as response to external fields, in particular emission, absorption and scattering of light.', '1106.0179-2-12-5': 'In bulk systems, the wave vector and frequency dependent response function reads [EQUATION] which can be evaluated using quantum statistical approaches such as Green function theory, see [CITATION], or numerical approaches such as MD simulations, see [CITATION].', '1106.0179-2-12-6': 'As collisions are relevant in strongly correlated systems, the dynamical collision frequency [MATH] is derived and appears in a generalized Drude formula [CITATION] [EQUATION]', '1106.0179-2-12-7': 'In the classical case, the current-density correlation function has been extensively discussed in the long wavelength limit [MATH] applying MD simulations and perturbative approaches.', '1106.0179-2-12-8': 'Exemplarily, we refer to [CITATION].', '1106.0179-2-13-0': 'The state of a homogeneous one-component plasma in thermodynamic equilibrium is characterized by the nonideality parameter [MATH] and the degeneracy parameter [MATH], [MATH] is the temperature of the electrons.', '1106.0179-2-13-1': 'Considering the response function [MATH] in the long wavelength limit, a sharp peak arises at the plasmon frequency [MATH], see Eq. ([REF]).', '1106.0179-2-13-2': 'For finite wavelengths, the resonance is shifted and can be approximated by the so called Gross-Bohm plasmon dispersion for small wave numbers [MATH], see [CITATION], [MATH] with the Debye screening length [MATH].', '1106.0179-2-13-3': 'This relation has recently been revisited with respect to the relevance of collisions by Thiele et al. [CITATION].', '1106.0179-2-13-4': 'According to Eq. ([REF]), the general behavior of the response function [MATH] in the long-wavelength limit is closely related to the collision frequency which is relevant in non-ideal plasmas, see [CITATION].', '1106.0179-2-13-5': 'In the two-component plasma, a phonon mode can arise in addition to the plasmon excitations [CITATION].', '1106.0179-2-14-0': 'The response function [MATH] and the related dynamical structure factor [MATH] as well as the optical properties have been intensively investigated for electron-ion bulk systems, see Refs. [CITATION].', '1106.0179-2-14-1': 'In this work, the inhomogeneous case of finite clusters in local thermal equilibrium will be discussed.', '1106.0179-2-14-2': 'The response of inhomogeneous systems is not only dependent on the difference of the positions, but on [MATH] and [MATH] separately.', '1106.0179-2-14-3': 'Therefore, spatially resolved current density correlation functions [MATH] can not be diagonalized by spatial Fourier transform.', '1106.0179-2-14-4': 'Instead of plane waves, other basis functions have to be found in order to characterize the collective excitations of electrons.', '1106.0179-2-15-0': '# MD simulations of finite plasmas', '1106.0179-2-16-0': 'Finite plasma systems have been investigated using the restricted molecular dynamics (RMD) simulations, see Raitza et al. [CITATION].', '1106.0179-2-16-1': 'A two-component system of singly charged ions and electrons will be described using an error function pseudo potential for the interaction of particles [MATH] and [MATH] [EQUATION] where [MATH] is the charge of the [MATH]th particle.', '1106.0179-2-16-2': 'The Coulomb interaction is modified at short distances, assuming a Gaussian wave function for electrons motivated by the account of quantum effects.', '1106.0179-2-16-3': 'Considering a sodium like system, the potential parameter [MATH] nm was chosen in order to reproduce the ionization energy of [MATH] eV for solid sodium, as already discussed for MD simulations by Suraud et al. [CITATION].', '1106.0179-2-17-0': 'The velocity Verlet algorithm [CITATION] was applied to solve the classical equations of motion for electrons and ions.', '1106.0179-2-17-1': 'This method takes into account the conservation of the total energy of the finite system, as long as there is no external potential.', '1106.0179-2-17-2': 'To follow the fast electron dynamics, time steps of [MATH] fs were taken to calculate the time evolution.', '1106.0179-2-17-3': 'Contrary to bulk MD simulations no periodic boundary conditions are applied.', '1106.0179-2-18-0': 'Icosahedral arrangements of 55, 147, and 309 ions, see [CITATION], were considered as initial configuration for the ion positions.', '1106.0179-2-18-1': 'For these nearly spherically, homogeneously distributed ions, the ion density typical for solid sodium is given by an ionic next neighbor distance of [MATH] nm.', '1106.0179-2-18-2': 'In addition, randomly distributed ion configurations within a given sphere were considered for comparison and the number of ions was increased up to 1000 particles.', '1106.0179-2-18-3': 'Starting with a neutral cluster, the electrons have been positioned nearby the ions with small, randomly distributed deviations from the ion positions.', '1106.0179-2-19-0': 'To simulate experiments where clusters are excited by short pulse lasers, MD simulations are performed under the influence of an electric field, assuming a Gaussian shape and pulse duration of about 100 fs.', '1106.0179-2-19-1': 'Due to the largely increased kinetic energy of the electrons, ionization processes occur.', '1106.0179-2-19-2': 'After the laser field is switched off, the ionization degree of the cluster is determined by the number of electrons found outside the cluster radius with positive total energy, so that they can escape from the cluster.', '1106.0179-2-19-3': 'Due to ion excitation on larger time scales, a slow expansion of the positively charged cluster is obseved [CITATION], leading to Coulomb explosion experimentally.', '1106.0179-2-20-0': 'Considering the single-time properties, it was found in [CITATION] that already local thermodynamic equilibrium (LTE) is established within a few fs after the electron heating.', '1106.0179-2-20-1': 'In particular, at each time step, the momentum distribution of electrons is well described by a Maxwell distribution, and the spatial density profile agrees with a Boltzmann distribution with respect to the average potential that is determined by the actual ion configuration and the self-consistent electronic mean field.', '1106.0179-2-20-2': 'The fact that electrons are considered within sub fs time intervals, while the ion configuration remains nearly unchanged, enables us to separate the electron dynamics from the ion dynamics.', '1106.0179-2-21-0': 'Subsequently, the dynamical properties of the electron subsystem can be calculated for a frozen ionic configuration thus referring to a specific time.', '1106.0179-2-21-1': 'This is considered as an adiabatic approximation to the true dynamical properties of the electron subsystem which have to take into account the slow change in the ion configuration.', '1106.0179-2-21-2': 'More rigorously, non-stationary time dependent correlation functions have to be treated for the full charged particle system.', '1106.0179-2-22-0': 'Using the RMD simulations scheme as introduced in [CITATION], the ions are kept fixed acting as external trap potential.', '1106.0179-2-22-1': 'Starting from an initial state, the many-electron trajectory [MATH] is calculated, solving the classical equations of motion of the electrons.', '1106.0179-2-22-2': 'From this, all further physical properties of the electron subsystem inside the cluster are determined.', '1106.0179-2-22-3': 'Within RMD simulations, we consider no temporal variation of the plasma parameters that are determined by the frozen ion distribution, the electron temperature and the degree of ionization.', '1106.0179-2-22-4': 'A long-time run can be performed in order to replace the ensemble average by a temporal average.', '1106.0179-2-22-5': 'This has been successfully done for the single-time properties such as the momentum distribution and the density profile, see [CITATION] and will now be applied to the two-time correlation functions.', '1106.0179-2-23-0': 'Using classical MD simulation techniques, the results are valid for non-degenerate plasmas.', '1106.0179-2-23-1': 'This restricts the temperature range to [MATH] eV where our simulations can be compared with realistic sodium clusters.', '1106.0179-2-23-2': 'Values for the plasma parameter [MATH] can be treated since we are not confined to the weak coupling limit as, e.g., in perturbation theory.', '1106.0179-2-24-0': 'In our RMD calculations, we start from a homogeneous ion configuration (icosahedral or randomly distributed) inside the cluster at fixed ion density.', '1106.0179-2-24-1': 'In the case of random distribution, we perform averaging over different initial configurations of ions.', '1106.0179-2-24-2': 'The Langevin thermostat was used to heat the electrons at an initial stage.', '1106.0179-2-24-3': 'We have chosen the Langevin thermostat introducing a friction term with suitable sign to adjust the intended kinetic energy.', '1106.0179-2-24-4': 'Furthermore, a random source term is applied that thermalizes the system.', '1106.0179-2-24-5': 'Hot electrons are emitted during this stage so that the cluster becomes ionized.', '1106.0179-2-24-6': 'Evaluating the trajectories of electrons, sufficient time of about 200 fs has to be allowed before a stationary ionization degree is established.', '1106.0179-2-24-7': 'Then the thermostat is switched off and a data taking is performed using an ensemble at fixed number of density, volume and energy.', '1106.0179-2-24-8': 'It is checked that the mean cluster charge [MATH] and the system temperature do not change any more.', '1106.0179-2-24-9': 'In Fig. [REF], the cluster charge [MATH] depending on cluster size [MATH] is shown.', '1106.0179-2-24-10': 'A power fit [MATH] with, for example, [MATH] and [MATH] for [MATH]1 eV shows the trend of the size dependent ionization degree.', '1106.0179-2-25-0': 'Using the trajectories of all [MATH] electrons obtained from the RMD simulations scheme, the local current density [MATH] at position [MATH] was calculated for each time step [MATH] [EQUATION] which is the sum over all electron momenta [MATH] inside a small volume [MATH] at position [MATH], where [MATH], and [MATH] for electrons found outside [MATH].', '1106.0179-2-25-1': 'The size of the volume determines the spatial resolution of the local current density [MATH].', '1106.0179-2-25-2': 'However, it must be taken sufficiently large to reduce statistical fluctuations.', '1106.0179-2-26-0': 'The bi-local correlation tensor of the normalized spatially-resolved current density is calculated according to Eq. ([REF]) as [EQUATION] with [MATH] the total current density.', '1106.0179-2-26-1': 'Typical values are [MATH] and [MATH] fs.', '1106.0179-2-26-2': 'Its Laplace transform reads [EQUATION]', '1106.0179-2-26-3': 'In the following, we restrict ourselves to the diagonal components [MATH] of this tensor, where only parallel components of the current density vectors are correlated as already introduced in Sec. II.', '1106.0179-2-26-4': 'As it will be shown in the following sections, this bi-local current-density correlation is important to understand the excitation modes of nano plasmas.', '1106.0179-2-26-5': 'The non-diagonal components of the bi-local correlation tensor are small in comparison to the diagonal components.', '1106.0179-2-26-6': 'Beside the bi-local current density correlation function considered here, the bi-local density fluctuation correlation [MATH] as well as the bi-local force correlation [MATH] are useful quantities in the context of optical properties.', '1106.0179-2-26-7': 'These correlations can be evaluated from the trajectory in a similar way and are related to the bi-local current density correlation.', '1106.0179-2-26-8': 'This will be discussed in an upcoming paper.', '1106.0179-2-27-0': 'Because of the spherical symmetry of the cluster geometry during excitation and expansion, the volume is divided into sections [MATH] according to [MATH], [MATH], [MATH] equidistant intervals of spherical coordinates, i.e. the distance [MATH] to the center of the cluster, the inclination angle [MATH] as well as the azimuthal angle [MATH], respectively.', '1106.0179-2-27-1': 'The cluster radius [MATH] is given by the root mean square radius of ions according to [MATH].', '1106.0179-2-27-2': 'The sections are numbered by a single counter [MATH] with three independent counters according to the three coordinates: [MATH], [MATH] and [MATH].', '1106.0179-2-27-3': 'With respect to Eq. ([REF]) the bi-local correlation matrix [MATH] for the spatially resolved cluster and its Laplace transform [MATH] have been calculated.', '1106.0179-2-28-0': 'The total current density ACF can be calculated from the trajectories directly.', '1106.0179-2-28-1': 'Please note, that it can be also calculated from the bi-local current density correlation matrix [EQUATION] using the cluster volume [MATH] and the individual cell volumes [EQUATION]', '1106.0179-2-28-2': 'The consistency of these expressions has been checked throughout our explicit calculations.', '1106.0179-2-29-0': '# From bi-local correlation function to excitation modes', '1106.0179-2-30-0': 'In the following, we discuss calculations for the current-density ACF, Eq. ([REF]), and the bi-local current-density correlation spectrum [MATH].', '1106.0179-2-30-1': 'Exemplarily, we present results for the Na[MATH] cluster at electron temperature [MATH] eV, cluster charge [MATH] and ionic density [MATH] cm[MATH].', '1106.0179-2-30-2': 'The electrons form a nano plasma with nonideality parameter [MATH] and degeneracy parameter [MATH].', '1106.0179-2-30-3': 'Starting with a solid density cluster, these are typical parameters obtained directly after the interaction with a short pulse laser of 100 fs duration and intensity of [MATH] Wcm[MATH].', '1106.0179-2-30-4': 'Calculations of other cluster sizes will be presented in the following sections.', '1106.0179-2-31-0': 'The real part of the total current-density ACF [MATH] is shown in Fig. [REF].', '1106.0179-2-31-1': 'Three maxima are obtained.', '1106.0179-2-31-2': 'This feature differs from the bulk behavior and is interpreted as different resonances of the electron system.', '1106.0179-2-31-3': 'To investigate the origin of the different maxima as collective excitations of the nano plasma, the bi-local current-density correlation matrix was calculated as well.', '1106.0179-2-32-0': 'The following spatial symmetries in the matrix [MATH] were found [EQUATION]', '1106.0179-2-32-1': 'In our case, the [MATH] elements of the full matrix can be reduced to [MATH] independent elements due to the symmetries Eq. ([REF]) - Eq. ([REF]), thus improving statistics via averaging equal elements.', '1106.0179-2-33-0': 'Because of the different size of section volumes in spherical coordinates there are large variations in the mean number of particles in a section.', '1106.0179-2-33-1': 'Provided that we have [MATH] electrons and [MATH] sections the average number of particles in some sections can be even smaller than unity.', '1106.0179-2-33-2': 'In this case, the local current density Eq. ([REF]) is affected by strong fluctuations due to the discrete number of particles.', '1106.0179-2-33-3': 'This problem is reduced, when you consider the current [MATH] as the contribution of smaller cells will be damped.', '1106.0179-2-33-4': 'Therefore, we used the non-normalized form of the correlation function for further analysis [EQUATION] for which the following eigenproblem was solved, [EQUATION]', '1106.0179-2-33-5': 'Thus, the matrix is decomposed into eigenvectors [MATH] as well as their eigenvalues [MATH] at each frequency.', '1106.0179-2-33-6': 'The eigenvectors represent the spatial structure of the mode ([MATH]).', '1106.0179-2-33-7': 'The orthonormality condition [EQUATION] holds.', '1106.0179-2-34-0': 'For two selected frequencies, the 10 strongest eigenvalues of the Na[MATH] cluster are shown in Fig. [REF].', '1106.0179-2-34-1': 'At [MATH] fs[MATH] (black), a resonance frequency was found with one outstanding, leading eigenvalue.', '1106.0179-2-34-2': 'The second and third largest eigenvalue are of same strength, which suggests degeneracy due to the symmetry of the correlation matrix.', '1106.0179-2-34-3': 'At off-resonant frequencies, i.e. at [MATH] fs[MATH] (shaded, red online), all eigenvalues are of the same order of magnitude.', '1106.0179-2-35-0': 'In Fig. [REF] (a) the strongest eigenvalues [MATH] of the Na[MATH] cluster are shown in dependence of frequency.', '1106.0179-2-35-1': 'They are colored according to their strength and numbered ascending with descending strength.', '1106.0179-2-35-2': 'In the shown frequency range, modes [MATH] with well defined maxima are found.', '1106.0179-2-35-3': 'The spatial oscillation structure can be identified by analyzing the eigenvectors.', '1106.0179-2-36-0': 'In Fig. [REF] (b), the spectra of eigenvalues are sorted in an alternative way, according to the spatial structure of the eigenvector which is obtained over the whole frequency range.', '1106.0179-2-36-1': 'Overall, the black solid mode is the strongest.', '1106.0179-2-36-2': 'Its resonance frequencies are also found in the total current-density ACF (indicated via vertical blue dashed lines) and are therefore of particular interest.', '1106.0179-2-36-3': 'Resonances in the total current-density ACF, shown in Fig. [REF], are only possible in the case of non-zero total current, which is caused by a dipole-like oscillation.', '1106.0179-2-36-4': 'Thus, resonances which are seen in the total current-density ACF are oscillation modes with a dipole moment.', '1106.0179-2-36-5': 'Other resonance structures, for example, are breathing modes that have no dipole moment.', '1106.0179-2-36-6': 'After characterization of the resonance structures, the dipole-like resonances will be investigated in more detail.', '1106.0179-2-37-0': '# Analysis of the collective modes', '1106.0179-2-38-0': 'The decomposition of the locally resolved current correlation matrix into eigenvalues [MATH], as shown in Fig. [REF], gives a very complex set of resonance structures in comparison to the 1D case, see [CITATION].', '1106.0179-2-38-1': 'The spatial mode structures in 1D chains were characterized by their wave number [MATH].', '1106.0179-2-38-2': 'To analyze the more complicated spatial oscillation structure of 3D clusters, a spherical Fourier decomposition of the eigenvectors into the spherical Bessel function [MATH] and spherical harmonics [MATH] was performed according to [EQUATION] where [MATH] is the spherical Fourier component with ordinal numbers [MATH].', '1106.0179-2-38-3': 'The normalization factor [MATH] as well as the wave number [MATH] are chosen in the way that the eigenvector has a root at the cluster surface.', '1106.0179-2-39-0': 'In Fig. [REF] (b), the four strongest eigenvalue modes are characterized by pairs of ordinal numbers [MATH] which determine the main angular part of the eigenvector by the spherical harmonics [MATH].', '1106.0179-2-39-1': 'The leading dipole-like mode, represented via solid black lines in Fig. [REF] (b), is characterized by the overlap of the spherical harmonic functions [MATH] and [MATH].', '1106.0179-2-39-2': 'For the Na[MATH] cluster, one can find three resonance frequencies which are identical to the ones found in the total current-density ACF.', '1106.0179-2-39-3': 'The latter are indicated by vertical dashed lines (blue online) in Fig. [REF] (b).', '1106.0179-2-40-0': 'In our investigations, we looked at other cluster parameters as well and found similar behavior.', '1106.0179-2-40-1': 'Comparisons will be made in the following chapters.', '1106.0179-2-40-2': 'For further analysis of the exication modes, we now consider a larger cluster consisting of 1000 ions.', '1106.0179-2-40-3': 'There, four pronounced dipole-like resonances were found.', '1106.0179-2-40-4': 'In Fig. [REF], the spatial structures of the current-density [MATH] is shown for the Na[MATH] cluster at the resonance frequencies of the leading dipole-like mode.', '1106.0179-2-40-5': 'The behavior is shown in the [MATH]plane at a fixed azimuthal angle [MATH] on which it does not depend.', '1106.0179-2-41-0': 'At the resonance frequency [MATH] fs, the electrons are oscillating with a current density [MATH].', '1106.0179-2-41-1': 'As shown in Fig. [REF] (a), all electrons of this mode are moving in the same direction and no nodes can be seen.', '1106.0179-2-41-2': 'Assuming a constant velocity field amplitude [MATH], the change of the current density with distance [MATH] is directly related to the density profile [MATH] of the electrons.', '1106.0179-2-42-0': 'The modes in Fig. [REF] (c) and (d) are similar to a plane wave oscillation of electrons, but trapped inside the cluster.', '1106.0179-2-42-1': 'To identify a wave number of the plane wave oscillation, a Fourier decomposition of plane waves in [MATH]-direction was done.', '1106.0179-2-42-2': 'A maximum at [MATH] nm[MATH] and [MATH] nm[MATH], respectively, is found which identify the wavelengths of the plane wave oscillations.', '1106.0179-2-42-3': 'Only in the large cluster with 1000 ions, a plane wave oscillation with higher wavenumber was found.', '1106.0179-2-42-4': 'All other modes can be seen in smaller clusters as well.', '1106.0179-2-42-5': 'The resonance structure in Fig. [REF] (b) looks like a mix of the first and the third resonance structure.', '1106.0179-2-43-0': 'We want to point out one further feature of the mode spectra in Fig. [REF] (b).', '1106.0179-2-43-1': 'The dashed red line represents in fact two resonance structures with exactly the same eigenvalues at all frequencies.', '1106.0179-2-43-2': 'The eigenvectors are orthogonal since they are characterized by the same spherical harmonic function [MATH] but have a phase shift in [MATH]-direction: [MATH].', '1106.0179-2-43-3': 'Further degenerations are obtained for weaker eigenvalue modes as well.', '1106.0179-2-44-0': 'All eigenvectors [MATH] are decomposed into a superposition of spherical Bessel functions [MATH] with a set of ordinal numbers [MATH].', '1106.0179-2-44-1': 'No leading ordinal number [MATH] was found, which characterizes the spatial resonance structure in [MATH] direction.', '1106.0179-2-45-0': '## Resonance frequency of the rigid oscillation', '1106.0179-2-46-0': 'The total current density ACF shown in Fig. [REF] as well as the leading eigenvalue mode in Fig. [REF] (right) show the strongest resonance at the frequency [MATH] fs[MATH].', '1106.0179-2-46-1': 'This resonance belongs to the dipole-like mode with the eigenvector shown in Fig. [REF] on the left hand side.', '1106.0179-2-46-2': 'We will now analyze this collective excitation mode in terms of a rigid oscillation.', '1106.0179-2-47-0': 'The electrons with density profile [MATH] are assumed to move nearly rigidly in the external potential [MATH] due to the fixed ions.', '1106.0179-2-47-1': 'The potential energy of the electrons due to a small shift with respect to the ions reads [EQUATION]', '1106.0179-2-47-2': 'The change of the potential energy [MATH] in [MATH] direction is due to the restoring force on the electron profile.', '1106.0179-2-47-3': 'In harmonic approximation of the equation of motion, the resonance frequency is identified as [EQUATION]', '1106.0179-2-47-4': 'For small rigid shifts [MATH], assuming radially dependent electron density profiles and external potentials in Eq. ([REF]) the integration over the angular dependence of the potential energy calculation can be executed.', '1106.0179-2-47-5': 'The resonance frequency Eq. ([REF]) is then given according to [EQUATION]', '1106.0179-2-47-6': 'As a first example for a density profile, we assume a homogeneously charged ion sphere with radius [MATH] and an electron sphere with radius [MATH].', '1106.0179-2-47-7': 'The densities of the electron and ion spheres is equal ([MATH]).', '1106.0179-2-47-8': 'Therefore, the difference of ion and electron radius is determined by the cluster charge, basically the difference of the simulated electron number [MATH] and ion number [MATH].', '1106.0179-2-47-9': 'Thus, in the case of positively charged clusters, as discussed here, the electron sphere radius is smaller than the ion radius ([MATH]).', '1106.0179-2-47-10': 'The error function potential Eq. ([REF]) was taken as electron-ion-interaction potential for the calculation of the resonance frequency, as it was used for the MD simulation as well.', '1106.0179-2-47-11': 'The resonance frequency than reads [EQUATION]', '1106.0179-2-47-12': 'In the limit of large clusters with high number of ions the resonance frequency equals the Mie frequency, [MATH].', '1106.0179-2-47-13': 'Assuming only a weak charged cluster, the sphere radii have nearly the same size ([MATH]) and the system is nearly neutral.', '1106.0179-2-47-14': 'The limit for small clusters, down to just one atom, depends strongly on the pseudopotential.', '1106.0179-2-47-15': 'In our case, the resonance frequency [MATH] is due to the oscillation of a single electron in the ionic error-function pseudo-potential Eq. ([REF]).', '1106.0179-2-48-0': 'In Fig. [REF] (a), the resonance frequency [MATH] of the dipole-like mode is shown in dependence on the size of the ion sphere.', '1106.0179-2-48-1': 'Results from MD simulations (empty circles) for Na[MATH], Na[MATH] and Na[MATH] cluster at [MATH] cm[MATH] as well as the Na[MATH] cluster at [MATH] cm[MATH] are shown.', '1106.0179-2-48-2': 'The resonance frequencies have been calculated using Eq. ([REF]) for ion densities of [MATH] cm[MATH] (solid shaded line, red online) and [MATH] cm[MATH] (solid black line).', '1106.0179-2-48-3': 'The limits of large clusters, the Mie frequency [MATH], are given as dotted lines colored according to the two densities.', '1106.0179-2-49-0': 'Additionally, the electron density profile [MATH] was deducted from MD simulations for all cluster sizes and used to derive the resonance frequency [MATH] solving Eq. ([REF]) numerically.', '1106.0179-2-49-1': 'As a result (full circles in Fig. [REF] (a)), the resonance frequency of the dipole-like mode is obtained with a deviation to the direct simulation results of less than 5%.', '1106.0179-2-49-2': 'Using homogeneously charged ion and electron spheres leads to reasonable agreement in the limits of large clusters as well as for small clusters.', '1106.0179-2-49-3': 'Taking the spatial structure of the density profile into account, there is good agreement with the direct simulation results in the intermediate cluster size regime as well.', '1106.0179-2-50-0': '## Dispersion of the plane wave mode', '1106.0179-2-51-0': 'While the Mie-like resonance, discussed in the previous subsection, is almost spherically symmetric, we obtain an increasing plane wave character of the dipole-like mode with increasing frequency.', '1106.0179-2-51-1': 'The third resonance frequency of the total current density ACF for the Na[MATH] cluster at [MATH] fs[MATH], see Fig. [REF], is mainly caused by a plane wave like eigenvector, which is similar to the eigenvector of the Na[MATH] cluster, shown in Fig. [REF] (right).', '1106.0179-2-51-2': 'This mode is discussed by Kresin et al. [CITATION] as compressional volume plasmon.', '1106.0179-2-51-3': 'Here, oscillations of electrons in opposite directions must be taken into account for the analytical calculation of the resonance frequency.', '1106.0179-2-51-4': 'We assume homogeneously charged spheres for the electrons with radius [MATH] and for the ions with radius [MATH] as it was already discussed in the previous subsection.', '1106.0179-2-51-5': 'The electron motion is treated as a hydrodynamical liquid using the Euler equation [EQUATION] where [MATH] is the spatially resolved current density of the electrons, [MATH] is the pressure of the electron gas and [MATH] is the external potential, composed of contributions from the electrons and ions.', '1106.0179-2-51-6': 'Using the following ansatz [EQUATION] we consider small perturbations in [MATH]-direction restricting ourselves to longitudinal effects.', '1106.0179-2-51-7': 'One is able to linearize the Euler equation.', '1106.0179-2-51-8': 'The system is assumed to be in LTE, described by the quantities [MATH], [MATH], [MATH] as well as [MATH].', '1106.0179-2-51-9': 'Electrons are moving in the external field of ions and in the mean field of electrons.', '1106.0179-2-51-10': 'The external potential is [EQUATION]', '1106.0179-2-51-11': 'The external potential has a equilibrium part and a perturbative part [MATH], which is mainly dependent on the linear density perturbation [MATH].', '1106.0179-2-52-0': 'Assuming Boltzmann distribution we express the ideal gas pressure of the electrons [MATH] via the electron density.', '1106.0179-2-52-1': 'Using the equation of continuity, one is able to express the Euler equation in terms of linear perturbations of the density.', '1106.0179-2-52-2': 'Thus, the equilibrium part of the external potential compensates the pressure term on the right hand side of Eq. ([REF]).', '1106.0179-2-52-3': 'Restricting ourselves to linear perturbations of the Euler equation, only the third term on the right hand side of Eq. ([REF]) remains, which is connected to the external potential.', '1106.0179-2-52-4': 'Finally, all terms of the Euler equation Eq. ([REF]) are lead back to a linear density fluctuation [MATH].', '1106.0179-2-52-5': 'Thus, one ends up with [EQUATION]', '1106.0179-2-52-6': 'This relation leads to real valued solutions for the resonance frequencies for standing waves with [MATH] only and scales with the plasma frequency [MATH].', '1106.0179-2-52-7': 'In the limit [MATH] we find [MATH] which coincides with the bulk limit.', '1106.0179-2-53-0': 'From the eigenvector of the plane wave mode, one can derive the wave number [MATH], which corresponds to [MATH].', '1106.0179-2-53-1': 'This means the dispersion of the plane wave mode is determined by the radius of the electron cloud.', '1106.0179-2-53-2': 'Results for this case are shown in Fig. [REF] (b) for different cluster sizes and are compared with the simulation data.', '1106.0179-2-53-3': 'For the cluster with 1000 ions a plane wave mode with [MATH] and [MATH] was found as well.', '1106.0179-2-53-4': 'In Fig. [REF] (b), it is marked with an empty square.', '1106.0179-2-53-5': 'Its spatial structure is shown in Fig. [REF] (d).', '1106.0179-2-53-6': 'The simulation data for the Na[MATH] cluster fit the dispersion Eq. ([REF]) as well.', '1106.0179-2-53-7': 'Deviations of the plane wave resonance for smaller clusters are caused by to the radial dependence of the electron density profile.', '1106.0179-2-54-0': '# Conclusion', '1106.0179-2-55-0': 'We have investigated collective excitation modes of a nano plasma in highly excited metal clusters.', '1106.0179-2-55-1': 'The collective excitation of electrons inside the cluster are obtained from bi-local current density correlation functions by solving the eigenvalue problem of the current-density correlation matrix.', '1106.0179-2-55-2': 'Using RMD simulations, the local current density [MATH] for excited clusters of 55 up to 1000 ions with densities of [MATH] cm[MATH] as well as [MATH] cm[MATH] and temperatures of [MATH] eV have been investigated.', '1106.0179-2-55-3': 'Pseudo-potentials of sodium were used to calculate the electron dynamics without consideration of degeneration effects any further.', '1106.0179-2-55-4': 'For the analysis of electron dynamics at lower temperatures, the inclusion of quantum effects for the calculation of the local current density [MATH] of cluster electrons is an open question at this point.', '1106.0179-2-55-5': 'It would be useful to go beyond present classical description to discuss for example cold, non-excited clusters.', '1106.0179-2-56-0': 'The spectrum of dipole-like modes was investigated in more detail.', '1106.0179-2-56-1': 'Using analytical calculations, it was possible to relate the position of resonance modes in the frequency domain to their spatial mode structure.', '1106.0179-2-56-2': 'Results for the cluster size dependence of the resonance frequency have been shown.', '1106.0179-2-56-3': 'A smooth transition to the bulk behavior has been obtained.', '1106.0179-2-56-4': 'The analysis of further resonance frequencies and also other modes including breathing modes would be desirable.', '1106.0179-2-56-5': 'The width of mode resonances and the role of collision-less damping effects as well as the collision frequency need to be investigated in the future.', '1106.0179-2-56-6': 'The systematic change of the collision frequency with cluster size up to the bulk limit remains an interesting field.', '1106.0179-2-57-0': 'From RMD simulations, different collective excitations have been found in nano plasmas, including dipole-like and breathing modes.', '1106.0179-2-57-1': 'These collective excitations will influence the scattering and absorption properties of clusters, see [CITATION].', '1106.0179-2-57-2': 'Collective effects of electron motion play a role when analyzing ultraviolet (UPS) or x-ray photo-electron spectroscopy (XPS) experiments, as has been pointed out by Andersson et al. [CITATION].', '1106.0179-2-57-3': 'It is a challenge to experimentalists to confirm the occurence of different collective excitations in nano plasmas.'}","[['1106.0179-1-24-0', '1106.0179-2-24-0'], ['1106.0179-1-24-1', '1106.0179-2-24-1'], ['1106.0179-1-24-2', '1106.0179-2-24-2'], ['1106.0179-1-24-3', '1106.0179-2-24-5'], ['1106.0179-1-24-4', '1106.0179-2-24-6'], ['1106.0179-1-24-6', '1106.0179-2-24-8'], ['1106.0179-1-24-7', '1106.0179-2-24-9'], ['1106.0179-1-24-8', '1106.0179-2-24-10'], ['1106.0179-1-57-0', '1106.0179-2-57-0'], ['1106.0179-1-57-1', '1106.0179-2-57-1'], ['1106.0179-1-57-2', '1106.0179-2-57-2'], ['1106.0179-1-57-3', '1106.0179-2-57-3'], ['1106.0179-1-20-0', '1106.0179-2-20-0'], ['1106.0179-1-20-1', '1106.0179-2-20-1'], ['1106.0179-1-20-2', '1106.0179-2-20-2'], ['1106.0179-1-28-0', '1106.0179-2-28-0'], ['1106.0179-1-28-1', '1106.0179-2-28-1'], ['1106.0179-1-28-2', '1106.0179-2-28-2'], ['1106.0179-1-38-0', '1106.0179-2-38-0'], ['1106.0179-1-38-1', '1106.0179-2-38-1'], 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1904.03543,"{'1904.03543-1-0-0': 'Acoustic scenes are rich and redundant in their content.', '1904.03543-1-0-1': 'In this work, we present a spatio-temporal attention pooling layer coupled with a convolutional recurrent neural network to learn from patterns that are discriminative while suppressing those that are irrelevant for acoustic scene classification.', '1904.03543-1-0-2': 'The convolutional layers in this network learn invariant features from time-frequency input.', '1904.03543-1-0-3': 'The bidirectional recurrent layers are then able to encode the temporal dynamics of the resulting convolutional features.', '1904.03543-1-0-4': 'Afterwards, a two-dimensional attention mask is formed via the outer product of the spatial and temporal attention vectors learned from two designated attention layers to weigh and pool the recurrent output into a final feature vector for classification.', '1904.03543-1-0-5': 'The network is trained with between-class examples generated from between-class data augmentation.', '1904.03543-1-0-6': 'Experiments demonstrate that the proposed method not only outperforms a strong convolutional neural network baseline but also sets new state-of-the-art performance on the LITIS Rouen dataset.', '1904.03543-1-1-0': 'Index Terms: audio scene classification, attention pooling, convolutional neural network, recurrent neural network', '1904.03543-1-2-0': '# Introduction', '1904.03543-1-3-0': 'Audio scene classification (ASC) is one of the main tasks in environmental sound analysis.', '1904.03543-1-3-1': 'It allows a machine to recognize a surrounding environment based on its acoustic sounds [CITATION].', '1904.03543-1-3-2': 'One way to look at an audio scene is to consider its foreground events mixed with its background noise.', '1904.03543-1-3-3': 'Due to the complex content of audio scenes, it is challenging to classify them correctly, as classification models tend to overfit the training data.', '1904.03543-1-3-4': 'A good practice in audio scene classification is to split a long recording (e.g. 30 seconds) into short segments (a few seconds long) [CITATION].', '1904.03543-1-3-5': 'By this, we increase the data variation and a classification model can be trained more efficiently with a large set of small segments rather than a small set of the whole long recordings.', '1904.03543-1-3-6': 'Classification of long recordings is then achieved by aggregating classification results across the decomposed short segments.', '1904.03543-1-4-0': 'Similar to many other research fields, using deep learning for the ASC task has become a norm.', '1904.03543-1-4-1': 'Convolutional neural network (CNNs) [CITATION] are most commonly used deep learning techniques, thanks to their feature learning capability.', '1904.03543-1-4-2': 'Leveraging the nature of audio signals, sequential modelling with recurrent neural networks (RNNs) [CITATION] and temporal transformer networks [CITATION] have also demonstrated results on par with the convolutional counterparts.', '1904.03543-1-4-3': 'The deep learning models were trained either on time-frequency representations, such as log Mel-scale spectrograms [CITATION], or high-level features like label tree embedding features [CITATION].', '1904.03543-1-4-4': 'Mitigation of overfitting effects via feature fusion [CITATION] and model fusion [CITATION] has also been extensively explored.', '1904.03543-1-5-0': 'Given the rich content of acoustic scenes, they typically contain a lot of irrelevant and redundant information.', '1904.03543-1-5-1': 'This fact naturally gives rise to the question of how to encourage a deep learning model to automatically discover and focus on discriminative patterns and suppress irrelevant ones from the acoustic scenes for better classification.', '1904.03543-1-5-2': 'We seek to address that question in this work using an attention mechanism [CITATION].', '1904.03543-1-5-3': 'To this end, we propose a spatio-temporal attention pooling layer in combination with a convolutional recurrent neural network (CRNN), inspired by their success in the audio event detection task [CITATION].', '1904.03543-1-5-4': 'The convolutional layers of the CRNN network are used to learn invariant features from time-frequency input, whose temporal dynamics are subsequently modelled by the upper bidirectional recurrent layers.', '1904.03543-1-5-5': 'Temporal soft attention [CITATION] has usually been coupled with a recurrent layer to learn a weighting vector to combine its recurrent output vectors at different time steps into a single feature vector.', '1904.03543-1-5-6': 'However, spatial attention (i.e. attention on the feature dimension), and hence, joint spatio-temporal attention, have been left uncharted.', '1904.03543-1-5-7': 'With the proposed spatio-temporal attention layer, we aim to learn a two-dimensional attention mask for spatio-temporal pooling purpose.', '1904.03543-1-5-8': 'The rationale is that those entries of the recurrent output that are more informative should be assigned with strong weights and vice versa.', '1904.03543-1-5-9': 'We expect discriminative features to be accentuated in the induced feature vector, while irrelevant ones to be suppressed and blocked after the spatio-temporal attention pooling.', '1904.03543-1-5-10': 'In addition, we harness between-class data augmentation [CITATION] to generate between-class examples to better train the network to minimize the Kullback-Leibler (KL) divergence loss.', '1904.03543-1-6-0': '# The proposed CRNN with spatio-temporal attention pooling', '1904.03543-1-7-0': '## Input', '1904.03543-1-8-0': 'Following the common practice in ASC [CITATION], we decompose an audio snippet, which is 30 seconds long in the experimental LITIS Rouen dataset [CITATION] (cf. Section [REF]), into non-overlapping 2-second segments.', '1904.03543-1-8-1': 'It has been shown in previous works that an auxiliary channel which is created by excluding background noise from an audio recording is also helpful for the classification task, as the prominent foreground events of the scene are exposed more clearly to a network [CITATION].', '1904.03543-1-8-2': 'Hence, we create such an auxiliary channel by subtracting background noise using the minimum statistics estimation and subtraction method [CITATION].', '1904.03543-1-9-0': 'A 2-channel short audio segment is then transformed into the time-frequency domain, e.g. using log Mel spectrogram, to obtain a multi-channel image [MATH], where [MATH], [MATH], and [MATH] denote the number of frequency bins, the number of time indices, and the number of channels, respectively (cf. Section [REF] for further detail).', '1904.03543-1-10-0': '## Convolutional layers', '1904.03543-1-11-0': 'The convolutional block of the network consists of three convolutional layers followed by three max-pooling layers.', '1904.03543-1-11-1': 'For clarity, we show the configuration of the convolutional and max-pooling layers in Table [REF] alongside their corresponding resulting feature maps.', '1904.03543-1-12-0': 'For each convolutional layer, after the convolution operation, Rectified Linear Unit (ReLU) activation [CITATION] and batch normalization [CITATION] are exercised on the feature map.', '1904.03543-1-12-1': 'The number of convolutional filters of a convolutional layer is designed to be the double of its preceding layer, i.e. [MATH], in order to gain the representation power when the spectral size gets smaller and smaller after the pooling layers.', '1904.03543-1-12-2': 'Note that zero-padding (also known as SAME padding) is used during convolution to keep the temporal size unchanged (i.e. always equal to [MATH]).', '1904.03543-1-13-0': 'With the pooling kernel size [MATH] and a stride [MATH], the max pooling layers are only effective on the frequency dimension to gain spectral invariance.', '1904.03543-1-13-1': 'As a consequence, the spectral dimension is reduced from [MATH] of the original input to [MATH] after the three pooling layers, respectively.', '1904.03543-1-13-2': 'The last resulting feature map of size [MATH] is reshaped to [MATH], where [MATH], to present to the upper recurrent layers of the network which will be elaborated in the following section.', '1904.03543-1-14-0': '## Bidirectional recurrent layers with spatio-temporal attention pooling', '1904.03543-1-15-0': 'In the context of sequential modelling with recurrent layers, the convolutional output [MATH] is interpreted as a sequence of [MATH] feature vectors [MATH] where each [MATH], [MATH].', '1904.03543-1-15-1': 'A bidirectional recurrent layer then reads the sequence of convolutional feature vectors into a sequence of recurrent output vectors [MATH], where [EQUATION]', '1904.03543-1-15-2': 'Here, [MATH] represent the forward and backward hidden state vectors of size [MATH] at recurrent time step [MATH], respectively, while [MATH] indicates vector concatenation.', '1904.03543-1-15-3': '[MATH] denotes a weight matrix and [MATH] denotes bias terms.', '1904.03543-1-15-4': '[MATH] represents the hidden layer function of the recurrent layer and is realized by a Gated Recurrent Unit (GRU) [CITATION] here.', '1904.03543-1-15-5': 'We further stack multiple bidirectional GRU cells onto one another to form a deep RNN for sequential modelling as in [CITATION].', '1904.03543-1-16-0': 'The recurrent output [MATH] is of size [MATH].', '1904.03543-1-16-1': 'In order to learn a spatio-temporal attention mask to pool and reduce [MATH] into a single feature vector, we learn two attention vectors, [MATH] for temporal attention and [MATH] for spatial attention.', '1904.03543-1-16-2': 'Formally, the temporal attention weight [MATH] at the time index [MATH], [MATH], and the spatial attention weight [MATH] at the spatial index [MATH], [MATH] are computed as [EQUATION] respectively.', '1904.03543-1-16-3': 'In ([REF]) and ([REF]), [MATH] represents the column of [MATH] at the column (i.e. temporal) index [MATH] whereas [MATH] represents the row of [MATH] at the row (i.e. spatial) index [MATH].', '1904.03543-1-16-4': '[MATH] and [MATH] denote the scoring functions of the temporal and spatial attention layers and are given by [EQUATION] respectively, where [MATH] and [MATH] are the trainable weight matrices.', '1904.03543-1-16-5': 'The spatio-temporal attention mask [MATH] is then obtained as [EQUATION] where [MATH] denotes vector outer product operation.', '1904.03543-1-17-0': 'The final feature vector [MATH] is achieved via spatio-temporal attention pooling.', '1904.03543-1-17-1': 'Intuitively, element-wise multiplication between the recurrent output [MATH] and the spatio-temporal attention mask is first carried out, followed by summation over the time dimension.', '1904.03543-1-17-2': 'Formally, the [MATH]-th entry, [MATH], of [MATH] is given as [EQUATION]', '1904.03543-1-17-3': 'Inspired by [CITATION], a [MATH] activation is applied prior to the summation in ([REF]).', '1904.03543-1-17-4': 'Due to its output range [MATH], it is likely that [MATH] activation does not only suppress the irrelevant features but also enhances the informative ones in the resulting feature vector [MATH] [CITATION].', '1904.03543-1-18-0': 'Eventually, the obtained feature vector [MATH] is presented to a softmax layer to accomplish classification.', '1904.03543-1-19-0': '## Calibration with Support Vector Machine', '1904.03543-1-20-0': 'Compared to the standard softmax, Support Vector Machines (SVM) usually achieve better generalization due to their maximum margin property [CITATION].', '1904.03543-1-20-1': 'Similar to [CITATION], after training the network, we calibrate the final classifier by employing a linear SVM in replacement for the softmax layer.', '1904.03543-1-20-2': 'The trained network is used to extract feature vectors for the original training examples (without data augmentation) which are used to train the SVM classifier.', '1904.03543-1-20-3': 'During testing, the SVM classifier is subsequently used to classify those feature vectors extracted for the test examples.', '1904.03543-1-20-4': 'The raw SVM scores are also calibrated and converted into a proper posterior probability as in [CITATION].', '1904.03543-1-21-0': '# Between-class data augmentation and KL-divergence loss', '1904.03543-1-22-0': 'In deep learning, data augmentation, which is to increase the data variation by altering the property of the genuine data, is an important method to improve performance of the task at hand.', '1904.03543-1-22-1': 'Techniques like adding background noise [CITATION], pitch shifting [CITATION], and sample mixing [CITATION], have been proven to be useful for environmental sound recognition in general.', '1904.03543-1-22-2': 'Motivated by the work of Tokozume et al. [CITATION], we pursue a between-class (BC) data augmentation approach that mixes two samples of different classes with a random factor to generate BC examples for network training.', '1904.03543-1-23-0': 'Let [MATH] and [MATH] denote two samples of two different classes and let [MATH] and [MATH] denote their corresponding one-hot labels.', '1904.03543-1-23-1': 'A random factor [MATH] is then generated and used to mix the two samples and their labels to create a new BC sample [MATH] and its labels [MATH]: [EQUATION]', '1904.03543-1-23-2': 'Note that the BC label [MATH] is no longer a one-hot label but still a proper probability distribution, which represents the amplitude of the constituents [MATH] and [MATH] in the between-class sample [MATH].', '1904.03543-1-23-3': 'For example, mixing a restaurant scene and a tubestation scene with a factor [MATH] will result in a label restaurant: 0.3, tubestation: 0.7.', '1904.03543-1-23-4': ""Training a network with the BC sample [MATH], we expect the network's class probability distribution output [MATH] to be as close to the [MATH] as possible."", '1904.03543-1-23-5': 'Therefore, KL-divergence between [MATH] and [MATH] is used as the network loss: [EQUATION] where [MATH] is the number of classes.', '1904.03543-1-23-6': ""Learning with between-class examples was shown to enlarge Fisher's criterion, i.e. the ratio of the between-class distance to the within-class variance [CITATION]."", '1904.03543-1-24-0': '# Experiments', '1904.03543-1-25-0': '## LITIS-Rouen dataset', '1904.03543-1-26-0': 'The LITIS-Rouen dataset consists of 3026 examples of 19 scene categories [CITATION].', '1904.03543-1-26-1': 'Each class is specific to a location such as a train station or an open market.', '1904.03543-1-26-2': 'The audio recordings have a duration of 30 seconds and a sampling rate of 22050 Hz.', '1904.03543-1-26-3': 'The dataset has a total duration of 1500 minutes.', '1904.03543-1-26-4': 'We follow the training/testing splits in the seminal work [CITATION] and report average performances over 20 splits.', '1904.03543-1-27-0': '## Features', '1904.03543-1-28-0': 'A 2-second audio segment, sampled at [MATH] Hz, was transformed into a log Mel-scale spectrogram with [MATH] Mel-scale filters in the frequency range from 50 Hz to Nyquist rate.', '1904.03543-1-28-1': 'A frame size of 50 ms with 50% overlap was used, resulting in [MATH] frames in total.', '1904.03543-1-28-2': 'Likewise, another image was produced for the auxiliary channel (cf. Section [REF]).', '1904.03543-1-28-3': 'All in all, we obtained a multi-channel image [MATH] where [MATH] denotes the number of channels.', '1904.03543-1-29-0': 'Beside log Mel-scale spectrogram, we also studied log Gammatone spectrogram in this work.', '1904.03543-1-29-1': 'Repeating a similar feature extraction procedure using [MATH] Gammatone filters in replacement of the above-mentioned Mel-scale filters, we obtained a multi-channel log Gammatone spectrogram image for a 2-second audio segment.', '1904.03543-1-30-0': '## Network parameters', '1904.03543-1-31-0': 'The studied networks were implemented using Tensorflow framework [CITATION].', '1904.03543-1-31-1': 'We applied dropout [CITATION] to the convolutional layers described in Section [REF] with a dropout rate of [MATH].', '1904.03543-1-31-2': 'The GRU cells used to realize two bidirectional recurrent layers in Section [REF] have their hidden size [MATH], and a dropout rate of [MATH] was commonly applied to their inputs and outputs.', '1904.03543-1-31-3': 'Both the spatial and temporal attention layers have the same size of 64.', '1904.03543-1-31-4': 'The networks were trained for 500 epochs with a minibatch size of 100.', '1904.03543-1-31-5': 'Adam optimizer [CITATION] was used for network training with a learning rate of [MATH].', '1904.03543-1-32-0': 'Finally, the trade-off parameter [MATH] for the SVM classifier used for calibration was fixed at 0.1.', '1904.03543-1-33-0': '## Baseline', '1904.03543-1-34-0': 'In order to illustrate the efficiency of the recurrent layers with spatio-temporal attention pooling, we used the CNN block in Figure [REF] as a deep CNN baseline.', '1904.03543-1-34-1': 'For this baseline, global max pooling was used after the last pooling layer to derive the final feature vector for classification.', '1904.03543-1-34-2': 'Other configuration settings were the same as for the proposed CRNN with spatio-temporal attention pooling.', '1904.03543-1-35-0': '## Experimental results', '1904.03543-1-36-0': 'Table [REF] shows the classification accuracy obtained by the proposed network (referred to as Att-CRNN) and the CNN baseline.', '1904.03543-1-36-1': 'Note that the classification label of a 30-second recording was derived via aggregation of the classification results of its 2-second segments.', '1904.03543-1-36-2': 'To this end, probabilistic multiplicative fusion, followed by likelihood maximization were carried out similar to [CITATION].', '1904.03543-1-37-0': 'Overall, the proposed Att-CRNN outperforms the CNN baseline regardless of the features used, improving the accuracy on 2-second segment classification by [MATH] and [MATH] absolute using log Mel-scale and log Gammatone spectrograms, respectively.', '1904.03543-1-37-1': 'The better generalization of the proposed Att-CRNN over the CNN baseline can also be seen via patterns of their test accuracy curves during network training as shown in Figure [REF] for the first cross-validation fold.', '1904.03543-1-37-2': 'In turn, the improvements on the 2-second segment classification led to [MATH] and [MATH] absolute gains on the 30-second recordings classification using log Mel-scale and log Gammatone spectrograms, respectively.', '1904.03543-1-37-3': 'These are equivalent to a relative classification error reduction of [MATH] and [MATH], respectively.', '1904.03543-1-38-0': '## Performance comparison with state-of-the-art', '1904.03543-1-39-0': 'The experimental LITIS Rouen dataset has been extensively evaluated in literature.', '1904.03543-1-39-1': 'Table [REF] provides a comprehensive comparison between the performance obtained by the proposed Att-CRNN and the CNN baseline to those reported in previous works in terms of overall accuracy, average F1-score, and average precision.', '1904.03543-1-39-2': 'Overall, this comparison shows that our presented systems obtain better performance than all other counterparts.', '1904.03543-1-39-3': 'On the one hand, despite being simple, the CNN baseline alone performs comparably well compared to the state-of-the-art system, i.e. Temporal Transformer CNN [CITATION], likely due to the positive effect of the between-class data augmentation.', '1904.03543-1-39-4': 'On the other hand, the proposed Att-CRNN with log Mel-scale and log Gammatone spectrogram as features improves the accuracy by [MATH] and [MATH] over the state-of-the-art system, respectively.', '1904.03543-1-39-5': 'These accuracy gains are equivalent to a relative classification error reduction of [MATH] and [MATH], respectively.', '1904.03543-1-39-6': 'Combining the classification results of the Att-CRNN on both types of feature using the probabilistic multiplicative aggregation [CITATION] (i.e. Att-CRNN (fusion) in Table [REF]) further enlarges the margin up to [MATH] absolute gain on the overall accuracy, or [MATH] on relative classification error reduction.', '1904.03543-1-40-0': '# Conclusions', '1904.03543-1-41-0': 'This paper has presented an approach for audio scene classification using spatio-temporal attention pooling in combination with convolutional recurrent neural networks.', '1904.03543-1-41-1': 'The convolutional layers in this network are expected to learn invariant features from the input whose temporal dynamics are further encoded by bidirectional recurrent layers.', '1904.03543-1-41-2': 'Attention layers then learn attention weight vectors in the spatial and temporal dimensions from the recurrent output, which collectively construct a spatio-temporal attention mask able to weigh and pool the recurrent output into a single feature vector for classification.', '1904.03543-1-41-3': 'The proposed network was trained with between-class examples and KL-divergence loss.', '1904.03543-1-41-4': 'Evaluated on the LITIS Rouen dataset, the proposed method achieved good classification performance, outperforming a strong CNN baseline as well as the previously published state-of-the-art systems.'}","{'1904.03543-2-0-0': 'Acoustic scenes are rich and redundant in their content.', '1904.03543-2-0-1': 'In this work, we present a spatio-temporal attention pooling layer coupled with a convolutional recurrent neural network to learn from patterns that are discriminative while suppressing those that are irrelevant for acoustic scene classification.', '1904.03543-2-0-2': 'The convolutional layers in this network learn invariant features from time-frequency input.', '1904.03543-2-0-3': 'The bidirectional recurrent layers are then able to encode the temporal dynamics of the resulting convolutional features.', '1904.03543-2-0-4': 'Afterwards, a two-dimensional attention mask is formed via the outer product of the spatial and temporal attention vectors learned from two designated attention layers to weigh and pool the recurrent output into a final feature vector for classification.', '1904.03543-2-0-5': 'The network is trained with between-class examples generated from between-class data augmentation.', '1904.03543-2-0-6': 'Experiments demonstrate that the proposed method not only outperforms a strong convolutional neural network baseline but also sets new state-of-the-art performance on the LITIS Rouen dataset.', '1904.03543-2-1-0': 'Index Terms: audio scene classification, attention pooling, convolutional neural network, recurrent neural network', '1904.03543-2-2-0': '# Introduction', '1904.03543-2-3-0': 'Audio scene classification (ASC) is one of the main tasks in environmental sound analysis.', '1904.03543-2-3-1': 'It allows a machine to recognize a surrounding environment based on its acoustic sounds [CITATION].', '1904.03543-2-3-2': 'One way to look at an audio scene is to consider its foreground events mixed with its background noise.', '1904.03543-2-3-3': 'Due to the complex content of audio scenes, it is challenging to classify them correctly, as classification models tend to overfit the training data.', '1904.03543-2-3-4': 'A good practice in audio scene classification is to split a long recording (e.g. 30 seconds) into short segments (a few seconds long) [CITATION].', '1904.03543-2-3-5': 'By this, we increase the data variation and a classification model can be trained more efficiently with a large set of small segments rather than a small set of the whole long recordings.', '1904.03543-2-3-6': 'Classification of long recordings is then achieved by aggregating classification results across the decomposed short segments.', '1904.03543-2-4-0': 'Similar to many other research fields, using deep learning for the ASC task has become a norm.', '1904.03543-2-4-1': 'Convolutional neural network (CNNs) [CITATION] are most commonly used deep learning techniques, thanks to their feature learning capability.', '1904.03543-2-4-2': 'Leveraging the nature of audio signals, sequential modelling with recurrent neural networks (RNNs) [CITATION] and temporal transformer networks [CITATION] have also demonstrated results on par with the convolutional counterparts.', '1904.03543-2-4-3': 'The deep learning models were trained either on time-frequency representations, such as log Mel-scale spectrograms [CITATION], or high-level features like label tree embedding features [CITATION].', '1904.03543-2-4-4': 'Mitigation of overfitting effects via feature fusion [CITATION] and model fusion [CITATION] has also been extensively explored.', '1904.03543-2-5-0': 'Given the rich content of acoustic scenes, they typically contain a lot of irrelevant and redundant information.', '1904.03543-2-5-1': 'This fact naturally gives rise to the question of how to encourage a deep learning model to automatically discover and focus on discriminative patterns and suppress irrelevant ones from the acoustic scenes for better classification.', '1904.03543-2-5-2': 'We seek to address that question in this work using an attention mechanism [CITATION].', '1904.03543-2-5-3': 'To this end, we propose a spatio-temporal attention pooling layer in combination with a convolutional recurrent neural network (CRNN), inspired by their success in the audio event detection task [CITATION].', '1904.03543-2-5-4': 'The convolutional layers of the CRNN network are used to learn invariant features from time-frequency input, whose temporal dynamics are subsequently modelled by the upper bidirectional recurrent layers.', '1904.03543-2-5-5': 'Temporal soft attention [CITATION] has usually been coupled with a recurrent layer to learn a weighting vector to combine its recurrent output vectors at different time steps into a single feature vector.', '1904.03543-2-5-6': 'However, spatial attention (i.e. attention on the feature dimension), and hence, joint spatio-temporal attention, have been left uncharted.', '1904.03543-2-5-7': 'With the proposed spatio-temporal attention layer, we aim to learn a two-dimensional attention mask for spatio-temporal pooling purpose.', '1904.03543-2-5-8': 'The rationale is that those entries of the recurrent output that are more informative should be assigned with strong weights and vice versa.', '1904.03543-2-5-9': 'We expect discriminative features to be accentuated in the induced feature vector, while irrelevant ones to be suppressed and blocked after the spatio-temporal attention pooling.', '1904.03543-2-5-10': 'In addition, we harness between-class data augmentation [CITATION] to generate between-class examples to better train the network to minimize the Kullback-Leibler (KL) divergence loss.', '1904.03543-2-6-0': '# The proposed CRNN with spatio-temporal attention pooling', '1904.03543-2-7-0': '## Input', '1904.03543-2-8-0': 'Following the common practice in ASC [CITATION], we decompose an audio snippet, which is 30 seconds long in the experimental LITIS Rouen dataset [CITATION] (cf. Section [REF]), into non-overlapping 2-second segments.', '1904.03543-2-8-1': 'It has been shown in previous works that an auxiliary channel which is created by excluding background noise from an audio recording is also helpful for the classification task, as the prominent foreground events of the scene are exposed more clearly to a network [CITATION].', '1904.03543-2-8-2': 'Hence, we create such an auxiliary channel by subtracting background noise using the minimum statistics estimation and subtraction method [CITATION].', '1904.03543-2-9-0': 'A 2-channel short audio segment is then transformed into the time-frequency domain, e.g. using log Mel spectrogram, to obtain a multi-channel image [MATH], where [MATH], [MATH], and [MATH] denote the number of frequency bins, the number of time indices, and the number of channels, respectively (cf. Section [REF] for further detail).', '1904.03543-2-10-0': '## Convolutional layers', '1904.03543-2-11-0': 'The convolutional block of the network consists of three convolutional layers followed by three max-pooling layers.', '1904.03543-2-11-1': 'For clarity, we show the configuration of the convolutional and max-pooling layers in Table [REF] alongside their corresponding resulting feature maps.', '1904.03543-2-12-0': 'For each convolutional layer, after the convolution operation, Rectified Linear Unit (ReLU) activation [CITATION] and batch normalization [CITATION] are exercised on the feature map.', '1904.03543-2-12-1': 'The number of convolutional filters of a convolutional layer is designed to be the double of its preceding layer, i.e. [MATH], in order to gain the representation power when the spectral size gets smaller and smaller after the pooling layers.', '1904.03543-2-12-2': 'Note that zero-padding (also known as SAME padding) is used during convolution to keep the temporal size unchanged (i.e. always equal to [MATH]).', '1904.03543-2-13-0': 'With the pooling kernel size [MATH] and a stride [MATH], the max pooling layers are only effective on the frequency dimension to gain spectral invariance.', '1904.03543-2-13-1': 'As a consequence, the spectral dimension is reduced from [MATH] of the original input to [MATH] after the three pooling layers, respectively.', '1904.03543-2-13-2': 'The last resulting feature map of size [MATH] is reshaped to [MATH], where [MATH], to present to the upper recurrent layers of the network which will be elaborated in the following section.', '1904.03543-2-14-0': '## Bidirectional recurrent layers with spatio-temporal attention pooling', '1904.03543-2-15-0': 'In the context of sequential modelling with recurrent layers, the convolutional output [MATH] is interpreted as a sequence of [MATH] feature vectors [MATH] where each [MATH], [MATH].', '1904.03543-2-15-1': 'A bidirectional recurrent layer then reads the sequence of convolutional feature vectors into a sequence of recurrent output vectors [MATH], where [EQUATION]', '1904.03543-2-15-2': 'Here, [MATH] represent the forward and backward hidden state vectors of size [MATH] at recurrent time step [MATH], respectively, while [MATH] indicates vector concatenation.', '1904.03543-2-15-3': '[MATH] denotes a weight matrix and [MATH] denotes bias terms.', '1904.03543-2-15-4': '[MATH] represents the hidden layer function of the recurrent layer and is realized by a Gated Recurrent Unit (GRU) [CITATION] here.', '1904.03543-2-15-5': 'We further stack multiple bidirectional GRU cells onto one another to form a deep RNN for sequential modelling as in [CITATION].', '1904.03543-2-16-0': 'The recurrent output [MATH] is of size [MATH].', '1904.03543-2-16-1': 'In order to learn a spatio-temporal attention mask to pool and reduce [MATH] into a single feature vector, we learn two attention vectors, [MATH] for temporal attention and [MATH] for spatial attention.', '1904.03543-2-16-2': 'Formally, the temporal attention weight [MATH] at the time index [MATH], [MATH], and the spatial attention weight [MATH] at the spatial index [MATH], [MATH] are computed as [EQUATION] respectively.', '1904.03543-2-16-3': 'In ([REF]) and ([REF]), [MATH] represents the column of [MATH] at the column (i.e. temporal) index [MATH] whereas [MATH] represents the row of [MATH] at the row (i.e. spatial) index [MATH].', '1904.03543-2-16-4': '[MATH] and [MATH] denote the scoring functions of the temporal and spatial attention layers and are given by [EQUATION] respectively, where [MATH] and [MATH] are the trainable weight matrices and [MATH] and [MATH] are the trainable biases.', '1904.03543-2-16-5': 'The spatio-temporal attention mask [MATH] is then obtained as [EQUATION] where [MATH] denotes vector outer product operation.', '1904.03543-2-17-0': 'The final feature vector [MATH] is achieved via spatio-temporal attention pooling.', '1904.03543-2-17-1': 'Intuitively, element-wise multiplication between the recurrent output [MATH] and the spatio-temporal attention mask is first carried out, followed by summation over the time dimension.', '1904.03543-2-17-2': 'Formally, the [MATH]-th entry, [MATH], of [MATH] is given as [EQUATION]', '1904.03543-2-17-3': 'Inspired by [CITATION], a [MATH] activation is applied prior to the summation in ([REF]).', '1904.03543-2-17-4': 'Due to its output range [MATH], it is likely that [MATH] activation does not only suppress the irrelevant features but also enhances the informative ones in the resulting feature vector [MATH] [CITATION].', '1904.03543-2-18-0': 'Eventually, the obtained feature vector [MATH] is presented to a softmax layer to accomplish classification.', '1904.03543-2-19-0': '## Calibration with Support Vector Machine', '1904.03543-2-20-0': 'Compared to the standard softmax, Support Vector Machines (SVM) usually achieve better generalization due to their maximum margin property [CITATION].', '1904.03543-2-20-1': 'Similar to [CITATION], after training the network, we calibrate the final classifier by employing a linear SVM in replacement for the softmax layer.', '1904.03543-2-20-2': 'The trained network is used to extract feature vectors for the original training examples (without data augmentation) which are used to train the SVM classifier.', '1904.03543-2-20-3': 'During testing, the SVM classifier is subsequently used to classify those feature vectors extracted for the test examples.', '1904.03543-2-20-4': 'The raw SVM scores are also calibrated and converted into a proper posterior probability as in [CITATION].', '1904.03543-2-21-0': '# Between-class data augmentation and KL-divergence loss', '1904.03543-2-22-0': 'In deep learning, data augmentation, which is to increase the data variation by altering the property of the genuine data, is an important method to improve performance of the task at hand.', '1904.03543-2-22-1': 'Techniques like adding background noise [CITATION], pitch shifting [CITATION], and sample mixing [CITATION], have been proven to be useful for environmental sound recognition in general.', '1904.03543-2-22-2': 'Motivated by the work of Tokozume et al. [CITATION], we pursue a between-class (BC) data augmentation approach that mixes two samples of different classes with a random factor to generate BC examples for network training.', '1904.03543-2-23-0': 'Let [MATH] and [MATH] denote two samples of two different classes and let [MATH] and [MATH] denote their corresponding one-hot labels.', '1904.03543-2-23-1': 'A random factor [MATH] is then generated and used to mix the two samples and their labels to create a new BC sample [MATH] and its labels [MATH]: [EQUATION]', '1904.03543-2-23-2': 'Note that the BC label [MATH] is no longer a one-hot label but still a proper probability distribution, which represents the amplitude of the constituents [MATH] and [MATH] in the between-class sample [MATH].', '1904.03543-2-23-3': 'For example, mixing a restaurant scene and a tubestation scene with a factor [MATH] will result in a label restaurant: 0.3, tubestation: 0.7.', '1904.03543-2-23-4': ""Training a network with the BC sample [MATH], we expect the network's class probability distribution output [MATH] to be as close to the [MATH] as possible."", '1904.03543-2-23-5': 'Therefore, KL-divergence between [MATH] and [MATH] is used as the network loss: [EQUATION] where [MATH] is the number of classes.', '1904.03543-2-23-6': ""Learning with between-class examples was shown to enlarge Fisher's criterion, i.e. the ratio of the between-class distance to the within-class variance [CITATION]."", '1904.03543-2-24-0': '# Experiments', '1904.03543-2-25-0': '## LITIS-Rouen dataset', '1904.03543-2-26-0': 'The LITIS-Rouen dataset consists of 3026 examples of 19 scene categories [CITATION].', '1904.03543-2-26-1': 'Each class is specific to a location such as a train station or an open market.', '1904.03543-2-26-2': 'The audio recordings have a duration of 30 seconds and a sampling rate of 22050 Hz.', '1904.03543-2-26-3': 'The dataset has a total duration of 1500 minutes.', '1904.03543-2-26-4': 'We follow the training/testing splits in the seminal work [CITATION] and report average performances over 20 splits.', '1904.03543-2-27-0': '## Features', '1904.03543-2-28-0': 'A 2-second audio segment, sampled at [MATH] Hz, was transformed into a log Mel-scale spectrogram with [MATH] Mel-scale filters in the frequency range from 50 Hz to Nyquist rate.', '1904.03543-2-28-1': 'A frame size of 50 ms with 50% overlap was used, resulting in [MATH] frames in total.', '1904.03543-2-28-2': 'Likewise, another image was produced for the auxiliary channel (cf. Section [REF]).', '1904.03543-2-28-3': 'All in all, we obtained a multi-channel image [MATH] where [MATH] denotes the number of channels.', '1904.03543-2-29-0': 'Beside log Mel-scale spectrogram, we also studied log Gammatone spectrogram in this work.', '1904.03543-2-29-1': 'Repeating a similar feature extraction procedure using [MATH] Gammatone filters in replacement of the above-mentioned Mel-scale filters, we obtained a multi-channel log Gammatone spectrogram image for a 2-second audio segment.', '1904.03543-2-30-0': '## Network parameters', '1904.03543-2-31-0': 'The studied networks were implemented using Tensorflow framework [CITATION].', '1904.03543-2-31-1': 'We applied dropout [CITATION] to the convolutional layers described in Section [REF] with a dropout rate of [MATH].', '1904.03543-2-31-2': 'The GRU cells used to realize two bidirectional recurrent layers in Section [REF] have their hidden size [MATH], and a dropout rate of [MATH] was commonly applied to their inputs and outputs.', '1904.03543-2-31-3': 'Both the spatial and temporal attention layers have the same size of 64.', '1904.03543-2-31-4': 'The networks were trained for 500 epochs with a minibatch size of 100.', '1904.03543-2-31-5': 'Adam optimizer [CITATION] was used for network training with a learning rate of [MATH].', '1904.03543-2-32-0': 'Finally, the trade-off parameter [MATH] for the SVM classifier used for calibration was fixed at 0.1.', '1904.03543-2-33-0': '## Baseline', '1904.03543-2-34-0': 'In order to illustrate the efficiency of the recurrent layers with spatio-temporal attention pooling, we used the CNN block in Figure [REF] as a deep CNN baseline.', '1904.03543-2-34-1': 'For this baseline, global max pooling was used after the last pooling layer to derive the final feature vector for classification.', '1904.03543-2-34-2': 'Other configuration settings were the same as for the proposed CRNN with spatio-temporal attention pooling.', '1904.03543-2-35-0': '## Experimental results', '1904.03543-2-36-0': 'Table [REF] shows the classification accuracy obtained by the proposed network (referred to as Att-CRNN) and the CNN baseline.', '1904.03543-2-36-1': 'Note that the classification label of a 30-second recording was derived via aggregation of the classification results of its 2-second segments.', '1904.03543-2-36-2': 'To this end, probabilistic multiplicative fusion, followed by likelihood maximization were carried out similar to [CITATION].', '1904.03543-2-37-0': 'Overall, the proposed Att-CRNN outperforms the CNN baseline regardless of the features used, improving the accuracy on 2-second segment classification by [MATH] and [MATH] absolute using log Mel-scale and log Gammatone spectrograms, respectively.', '1904.03543-2-37-1': 'The better generalization of the proposed Att-CRNN over the CNN baseline can also be seen via patterns of their test accuracy curves during network training as shown in Figure [REF] for the first cross-validation fold.', '1904.03543-2-37-2': 'In turn, the improvements on the 2-second segment classification led to [MATH] and [MATH] absolute gains on the 30-second recordings classification using log Mel-scale and log Gammatone spectrograms, respectively.', '1904.03543-2-37-3': 'These are equivalent to a relative classification error reduction of [MATH] and [MATH], respectively.', '1904.03543-2-38-0': '## Performance comparison with state-of-the-art', '1904.03543-2-39-0': 'The experimental LITIS Rouen dataset has been extensively evaluated in literature.', '1904.03543-2-39-1': 'Table [REF] provides a comprehensive comparison between the performance obtained by the proposed Att-CRNN and the CNN baseline to those reported in previous works in terms of overall accuracy, average F1-score, and average precision.', '1904.03543-2-39-2': 'Overall, this comparison shows that our presented systems obtain better performance than all other counterparts.', '1904.03543-2-39-3': 'On the one hand, despite being simple, the CNN baseline alone performs comparably well compared to the state-of-the-art system, i.e. Temporal Transformer CNN [CITATION], likely due to the positive effect of the between-class data augmentation.', '1904.03543-2-39-4': 'On the other hand, the proposed Att-CRNN with log Mel-scale and log Gammatone spectrogram as features improves the accuracy by [MATH] and [MATH] over the state-of-the-art system, respectively.', '1904.03543-2-39-5': 'These accuracy gains are equivalent to a relative classification error reduction of [MATH] and [MATH], respectively.', '1904.03543-2-39-6': 'Combining the classification results of the Att-CRNN on both types of feature using the probabilistic multiplicative aggregation [CITATION] (i.e. Att-CRNN (fusion) in Table [REF]) further enlarges the margin up to [MATH] absolute gain on the overall accuracy, or [MATH] on relative classification error reduction.', '1904.03543-2-40-0': '# Conclusions', '1904.03543-2-41-0': 'This paper has presented an approach for audio scene classification using spatio-temporal attention pooling in combination with convolutional recurrent neural networks.', '1904.03543-2-41-1': 'The convolutional layers in this network are expected to learn invariant features from the input whose temporal dynamics are further encoded by bidirectional recurrent layers.', '1904.03543-2-41-2': 'Attention layers then learn attention weight vectors in the spatial and temporal dimensions from the recurrent output, which collectively construct a spatio-temporal attention mask able to weigh and pool the recurrent output into a single feature vector for classification.', '1904.03543-2-41-3': 'The proposed network was trained with between-class examples and KL-divergence loss.', '1904.03543-2-41-4': 'Evaluated on the LITIS Rouen dataset, the proposed method achieved good classification performance, outperforming a strong CNN baseline as well as the previously published state-of-the-art systems.'}","[['1904.03543-1-8-0', 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1208.4493,"{'1208.4493-1-0-0': 'We consider a manifold endowed with two different vielbeins [MATH] and [MATH] corresponding to two different metrics [MATH] and [MATH].', '1208.4493-1-0-1': 'Such a situation arises generically in bimetric or massive gravity (including the recently discussed version of de Rham, Gabadadze and Tolley), as well as in perturbative quantum gravity where one vielbein parametrizes the background space-time and the other the dynamical degrees of freedom.', '1208.4493-1-0-2': 'We determine the conditions under which the relation [MATH] can be imposed (or the ""Deser-van Nieuwenhuizen"" gauge chosen).', '1208.4493-1-0-3': 'We clarify and correct various statements which have been made about this issue.', '1208.4493-1-1-0': '# Introduction', '1208.4493-1-2-0': 'There are various situations in physics where one has to consider a manifold endowed with two different vielbein fields.', '1208.4493-1-2-1': 'Obviously, this appears to be the case in bimetric theories, theories where two different metrics are defined on the same space-time manifold [CITATION].', '1208.4493-1-2-2': 'Each of these metrics can then be described by a different vielbein.', '1208.4493-1-2-3': 'This is also true even if one of the two metrics is not dynamical.', '1208.4493-1-2-4': 'It also applies to non linear massive gravity (for recent reviews see [CITATION]), which is nothing else than a special class of bigravity, and in particular it applies to the recently introduced massive gravity theories of de Rham-Gabadadze-Tolley (dRGT in the following) [CITATION] as well as to the extension of these to the dynamical bimetric case [CITATION].', '1208.4493-1-2-5': 'A similar situation also occurs when one expands General Relativity around a fixed background metric and expresses both the background and the dynamical metric in terms of vielbeins.', '1208.4493-1-2-6': 'This is the starting point of many works dealing with quantum gravity (see e.g. [CITATION]).', '1208.4493-1-3-0': 'Considering such situations, let us define, in arbitrary [MATH] dimensions, [MATH] and [MATH] to be two bases of 1-forms obeying at every space-time point [EQUATION] or equivalently [EQUATION] where [MATH] and [MATH] are respectively the metrics associated with the vielbeins.', '1208.4493-1-3-1': 'We will also need the vectors [MATH] and [MATH], respectively dual to the 1-forms [MATH] and [MATH], that verify [EQUATION]', '1208.4493-1-3-2': 'For future use, let us rewrite the above relations (and consequences thereof) using matrix notations.', '1208.4493-1-3-3': 'We have [EQUATION] where [MATH] is the [MATH] identity matrix, [MATH] denotes the matrix transpose of the matrix [MATH], [MATH] is just [MATH] and the same relations hold between [MATH], [MATH] and [MATH] respectively.', '1208.4493-1-4-0': 'The defining relations ([REF]) and ([REF]) imply the gauge symmetry [EQUATION] with [MATH] and [MATH] Lorentz matrices.', '1208.4493-1-5-0': 'It is often convenient to ask for a ""symmetry"" condition on the vielbeins which reads [EQUATION]', '1208.4493-1-5-1': 'Notice that this condition can also be written as [MATH] and that Ref. [CITATION] uses an equivalent form which reads [MATH].', '1208.4493-1-6-0': 'In the recent discussions about massive gravity, such a condition has been used to ensure the existence of, and express, the matrix square root of [MATH] which enters in a crucial way in the definition of dRGT theory (see e.g. [CITATION]).', '1208.4493-1-6-1': 'Indeed, whenever condition ([REF]) holds, [MATH] defined as [EQUATION] verifies the defining equation of the matrix square root of [MATH] given by [EQUATION]', '1208.4493-1-6-2': 'It has also been argued by Hinterbichler and Rosen [CITATION] that, in the vielbein reformulation of dRGT theories, condition ([REF]) is obtained as a consequence of field equations.', '1208.4493-1-6-3': 'To prove this, they use a decomposition of an arbitrary matrix [MATH] (representing some unconstrained arbitrary vielbein multiplied by [MATH]) as [EQUATION] where [MATH] is a Lorentz matrix and [MATH] is a symmetric matrix.', '1208.4493-1-6-4': 'This is reminiscent of the so-called polar decomposition stating that an arbitrary invertible matrix can be written as the product of an orthogonal matrix with a symmetric matrix.', '1208.4493-1-6-5': 'However we will show that such a decomposition does not hold in general if one replaces the orthogonal matrix by a Lorentz transformation.', '1208.4493-1-6-6': 'This makes in particular the argument of Ref. [CITATION] incomplete.', '1208.4493-1-7-0': 'Furthermore, in massive gravity as well as in perturbative quantum gravity condition ([REF]) has been used as a gauge condition.', '1208.4493-1-7-1': 'In the quantum gravity context, this gauge (sometimes dubbed Deser-van Nieuwenhuizen gauge in reference to [CITATION]) has been first introduced via a gauge fixing term in the action and dealt with perturbatively [CITATION].', '1208.4493-1-7-2': 'It was then later argued that this gauge can be set ""non perturbatively"", i.e. that given a set of arbitrary vielbeins [MATH] and [MATH] that do not fullfill condition ([REF]), one can always Lorentz rotate them as in ([REF]), and ([REF]) to define a new set of vielbeins obeying this condition [CITATION] (with the consequence that the corresponding gauge would not suffer from Gribov-like ambiguities).', '1208.4493-1-7-3': 'Interestingly enough, the same statements have also been made in the context of massive gravity.', '1208.4493-1-7-4': 'Indeed, there as well the condition ([REF]) has been used ""perturbatively"" (i.e. in the case when both metrics [MATH] and [MATH] are close to one another, see e.g. [CITATION]), but it has also been argued that condition ([REF]) can be reached as a (Lorentz) gauge choice for arbitrary metrics [CITATION].', '1208.4493-1-7-5': 'This contradicts various other statements made in the literature, for example in Ref. [CITATION], where it is stated that gauge ([REF]) cannot be set beyond perturbation theory.', '1208.4493-1-7-6': 'Settling this contradiction, as we intend to do here, will also illuminate issues discussed in the previous paragraph, since, as we will show, to set ([REF]) via suitable Lorentz rotations of the vielbeins involves a decomposition similar to ([REF]).', '1208.4493-1-8-0': 'To be precise, the purpose of this note is to determine when and how the condition ([REF]) can be enforced, as well as when the decomposition ([REF]) holds.', '1208.4493-1-8-1': 'These questions, beyond their mathematical interest, are especially important for massive gravity.', '1208.4493-1-8-2': 'Indeed, one can argue that the vielbein formulation of dRGT theories has several advantages over their metric formulations.', '1208.4493-1-8-3': 'First of all, it allows a simple extraction of what plays the role of the Hamiltonian constraint [CITATION].', '1208.4493-1-8-4': 'Second, in some cases it also allows to dynamically derive the existence of the square root of [MATH] that has to be assumed or enforced by Lagrange multipliers in the metric formulation [CITATION].', '1208.4493-1-8-5': 'Finally, the frame formulation permits a simple discussion of the constraints and the counting of dynamical degrees of freedom in the Lagrangian framework [CITATION].', '1208.4493-1-8-6': 'In this formulation, relation ([REF]) plays a key role, and it is important to know whether it can be obtained by Lorentz gauge transformations, or it needs additional constraints to be imposed.', '1208.4493-1-9-0': 'This paper is organized as follows.', '1208.4493-1-9-1': 'In the next section, we will discuss necessary and sufficient conditions for ([REF]) and ([REF]) to hold.', '1208.4493-1-9-2': 'Then, in section 3, using results on matrix square roots, we will spell out sufficient conditions to achieve ([REF]) and ([REF]).', '1208.4493-1-9-3': 'In the last sections, and before concluding, we will discuss the specific cases of [MATH], [MATH], and [MATH] space-time dimensions, and in particular the cases which cannot be handled via the results of section 3.', '1208.4493-1-10-0': 'Before proceeding, let us mention a special choice for one of the metrics (say [MATH]) and the associated vielbein [MATH].', '1208.4493-1-10-1': 'This choice is made in some contexts (e.g. dRGT theories, but also perturbative quantum gravity).', '1208.4493-1-10-2': 'It amounts to first assuming that the metric [MATH] is flat and takes the canonical form [MATH], i.e. [EQUATION] and then choosing [MATH], i.e. such that (in components) [EQUATION]', '1208.4493-1-10-3': 'When the choice ([REF])-([REF]) is made, the constraint ([REF]) simply reads (labelling here space-time indices and Lorentz indices with the same set of letters) [EQUATION] stating that the vielbein [MATH] can be represented as a symmetric matrix.', '1208.4493-1-10-4': 'This choice will not be used to derive the results of this paper, but will just sometimes be considered as an example.', '1208.4493-1-11-0': '# Necessary and sufficient conditions', '1208.4493-1-12-0': 'Let us first try to set the constraint ([REF]) by using the freedom to Lorentz rotate independently the two sets of vielbeins [MATH] and [MATH].', '1208.4493-1-12-1': 'Considering two arbitrarily chosen vielbeins [MATH] and [MATH], assume that there exist two Lorentz transforms [MATH] and [MATH] such that the matrix [MATH] defined by [EQUATION] is symmetric.', '1208.4493-1-12-2': 'Defining [MATH] as the matrix of components [MATH] given by [EQUATION] (note that this definition implies that [MATH] is invertible), the above equality ([REF]) reads in matricial notations [EQUATION]', '1208.4493-1-12-3': 'Multiplying it on the right by [MATH] and on the left by [MATH] we get [EQUATION]', '1208.4493-1-12-4': 'For [MATH] to be symmetric, the matrix on the left hand side above should be symmetric, call it [MATH].', '1208.4493-1-12-5': 'Defining the Lorentz tranformation [MATH] by [MATH] we get that the invertible matrix [MATH] should be written as in Eq. ([REF]).', '1208.4493-1-12-6': 'Being a Lorentz transformation, [MATH] verifies [EQUATION]', '1208.4493-1-12-7': 'As we already stated, a decomposition such as in Eq. ([REF]) does not hold in general (in constrast to the polar decomposition).', '1208.4493-1-12-8': 'Indeed, rewriting ([REF]) as [MATH] and inserting this into ([REF]) we get after some trivial manipulation, that [MATH] and [MATH] should fullfill the necessary condition [EQUATION]', '1208.4493-1-12-9': 'Running backward the above argument it is easy to see that the above condition is also sufficient (just because the matrix defined as [MATH] will be a Lorentz transformation).', '1208.4493-1-12-10': 'Hence we have proven the following proposition.', '1208.4493-1-13-0': 'In particular, when [MATH] is given by ([REF]), we have (using relations ([REF])-([REF]) as well as definition ([REF])) [EQUATION]', '1208.4493-1-13-1': 'So if [MATH] has a square root [MATH], then (i) above holds: a square root of [MATH] being then given by [MATH].', '1208.4493-1-13-2': 'We then prove the following proposition,', '1208.4493-1-14-0': 'We first assume that [MATH] can be written as [MATH] with [MATH] a symmetric matrix.', '1208.4493-1-14-1': 'Then using this hypothesis into the first equality of ([REF]) we get [EQUATION]', '1208.4493-1-14-2': 'The matrix [MATH] being symmetric, this proves one side of the equivalence.', '1208.4493-1-14-3': 'Conversely, we assume that there exists a symmetric matrix [MATH] such that [MATH].', '1208.4493-1-14-4': 'Then [MATH] is given by [EQUATION]', '1208.4493-1-14-5': 'Noticing that the matrix [MATH] is symmetric ends the proof.', '1208.4493-1-15-0': 'Hence, gathering the above results, we have proven the following statement.', '1208.4493-1-16-0': '[Direct proof] Suppose first we have vielbeins [MATH] and [MATH] satisfying the above symmetry property.', '1208.4493-1-16-1': 'Then [EQUATION] and if we define [MATH] we get [MATH].', '1208.4493-1-16-2': 'Moreover [EQUATION] which shows that the matrix [MATH] is symmetric.', '1208.4493-1-16-3': 'Notice that this is equivalent to [MATH] symmetric.', '1208.4493-1-16-4': 'Conversely, suppose we have a real matrix [MATH] such that [MATH] and [MATH] symmetric.', '1208.4493-1-16-5': 'We start by choosing an arbitrary vielbein [MATH] for the metric [MATH] i.e. [MATH], and we denote by [MATH] its dual vector i.e. [MATH].', '1208.4493-1-16-6': 'We then define [MATH].', '1208.4493-1-16-7': 'This implies that [EQUATION]', '1208.4493-1-16-8': 'But the symmetry of [MATH] implies that [MATH] so [EQUATION] and [MATH] is a well-defined vielbein for the metric [MATH].', '1208.4493-1-16-9': 'Notice that this definition tells us [MATH].', '1208.4493-1-16-10': 'It remains to be shown that these vielbeins have the required symmetry property.', '1208.4493-1-16-11': 'We start from the symmetry of [MATH] [EQUATION] which we can rewrite [EQUATION]', '1208.4493-1-16-12': 'Multiplying by [MATH] we get [MATH] and this completes the proof.', '1208.4493-1-17-0': 'As we just showed the hypotheses (i) of Propositions [REF] and [REF] are that a certain real (invertible) matrix has a real square root.', '1208.4493-1-17-1': 'It is however well known that not all real invertible matrices have real square roots (see e.g. [CITATION]) and we will later recall what are the necessary and sufficient conditions for this to occur.', '1208.4493-1-17-2': 'In our case, though, the matrix which should have a square root is not totally arbitrary.', '1208.4493-1-17-3': 'For example, in Proposition [REF] it must be of the form [MATH].', '1208.4493-1-17-4': 'This alone does however not ensure the existence of a square root.', '1208.4493-1-17-5': ""For example, choosing [EQUATION] we get [EQUATION] which doesn't have any real square roots."", '1208.4493-1-17-6': 'Indeed, such a [MATH] diagonal matrix with four distinct eigenvalues has [MATH] square roots which are given here by [MATH].', '1208.4493-1-17-7': 'None of them is real.', '1208.4493-1-17-8': 'Hence the decomposition ([REF]) can at best hold for a restricted set of matrices.', '1208.4493-1-18-0': 'We thus see that considering the matrix [MATH] above as given by the form ([REF]) invalidates the result of Ref. [CITATION].', '1208.4493-1-18-1': 'Notice that, if one makes now the simple choice ([REF])-([REF]) (and considering equation ([REF])), our example involves a ""mismatch"" between the time directions of the two metrics [MATH] and [MATH].', '1208.4493-1-18-2': 'However, beyond perturbation theory there is no reason to think that these time directions should coincide or even be compatible.', '1208.4493-1-18-3': 'We will come back to this question later.', '1208.4493-1-18-4': 'Notice further that perturbatively, if [MATH], with [MATH] small, then to the first order in [MATH], and so the assumption (i) and (ii) of Proposition [REF] are always true perturbatively.', '1208.4493-1-19-0': '# Sufficient conditions', '1208.4493-1-20-0': 'Here, in order to formulate simple sufficient conditions allowing to obtain ([REF]) and ([REF]), we will discuss the precise relation between hypotheses (i) and (ii) of Propositions [REF] and [REF].', '1208.4493-1-20-1': 'We need to recall how square roots of real matrices are obtained.', '1208.4493-1-20-2': 'We first use the following theorem (that we quote here from Ref. [CITATION]).', '1208.4493-1-21-0': 'Let [MATH] be an invertible real square matrix (of arbitrary dimension).', '1208.4493-1-21-1': 'If [MATH] has no real negative eigenvalues, then there are precisely [MATH] real square roots of [MATH] which are polynomial functions of [MATH], where [MATH] is the number of distinct eigenvalues of [MATH] and [MATH] is the number of distinct complex conjugate eigenvalue pairs.', '1208.4493-1-21-2': 'If [MATH] has a real negative eigenvalue, then [MATH] has no real square root which is a polynomial function of [MATH].', '1208.4493-1-22-0': 'Let us first use this theorem to prove that (i) of Proposition [REF] (respectively Proposition [REF]) implies (ii) of the same proposition whenever the matrix [MATH] (respectively the matrix [MATH]) has no real negative eigenvalues.', '1208.4493-1-22-1': 'To see this, just consider a real matrix [MATH] with no negative eigenvalues, given by the product of two symmetric invertible matrices [MATH] and [MATH].', '1208.4493-1-22-2': 'By virtue of the above theorem, we know that this matrix has at least one real square root which is a polynomial function of [MATH], that we note [MATH].', '1208.4493-1-22-3': 'One then has [EQUATION] where the sum runs over a finite number of integers [MATH], and [MATH] are real numbers.', '1208.4493-1-22-4': 'Using the fact that [MATH], one then has [EQUATION] where the term [MATH]contains [MATH] factors of [MATH] and [MATH] factors of [MATH], and is a symmetric matrix.', '1208.4493-1-22-5': 'This means that that the square root [MATH] is given by the product of [MATH] by a symmetric matrix.', '1208.4493-1-22-6': 'It is enough to prove our assertion by choosing [MATH] to be given by [MATH] and [MATH] to be given by [MATH] (respectively [MATH] given by [MATH] and [MATH] to be given by [MATH]).', '1208.4493-1-22-7': 'Hence, using the above result, and Propositions [REF] and [REF] we have shown the following two propositions', '1208.4493-1-23-0': 'If [MATH] has one (or more) real negative eigenvalue, Theorem [REF] does not imply that [MATH] does not have a real square root, but just that such a square root cannot be a polynomial function of [MATH].', '1208.4493-1-23-1': 'In order to enunciate the necessary and sufficient conditions for a real matrix to have a real square root, one first needs to introduce the so-called Jordan decomposition of a matrix.', '1208.4493-1-23-2': 'It uses Jordan blocks which can be defined as [MATH] matrices wich are of the form [MATH] given by (for [MATH]) [EQUATION] where [MATH] is a complex number, and one has [MATH] for [MATH].', '1208.4493-1-23-3': 'One can then show that for an arbitrary [MATH] matrix [MATH], there exists an invertible matrix [MATH], and a matrix [MATH] such that [EQUATION] and the matrix J is a so called Jordan matrix of the form [EQUATION] where [MATH] is an integer and the matrices [MATH] are called the Jordan blocks of [MATH].', '1208.4493-1-23-4': 'For a given matrix [MATH], the number of Jordan blocks, the nature of the distinct Jordan blocks, and the number of times a given Jordan block occurs in the Jordan matrix [MATH] are uniquely determined.', '1208.4493-1-23-5': 'Moreover, the [MATH] are the eigenvalues of [MATH].', '1208.4493-1-23-6': 'One can further show that a given Jordan block [MATH] with [MATH], has precisely two upper triangular square roots, [MATH], which are in addition polynomial functions of [MATH] [CITATION].', '1208.4493-1-23-7': 'These can be used to find all the square roots (possibly complex) of a given matrix using the following theorem.', '1208.4493-1-24-0': 'Let [MATH] be a [MATH] complex matrix which has a Jordan decomposition given by ([REF])-([REF]), then all the square roots (which may include complex matrices) of [MATH] are given by the matrices [MATH], where [MATH] is an arbitrary matrix which commutes with [MATH].', '1208.4493-1-25-0': 'The Jordan blocks of a matrix also play a crucial role in the following theorem which gives the necessary and sufficient condition for a real matrix to have a real square root (see e.g. [CITATION]).', '1208.4493-1-26-0': 'Let [MATH] be an invertible real square matrix (of arbitrary dimension).', '1208.4493-1-26-1': 'The matrix [MATH] has a real square root if and only if for each of its negative eigenvalues [MATH], the number of identical Jordan block [MATH] where this eigenvalue occurs in the Jordan decomposition of the matrix [MATH] is even.', '1208.4493-1-27-0': 'In the following, we will use the above theorems to discuss in detail the cases which are not covered by our Propositions [REF] and [REF].', '1208.4493-1-27-1': 'Namely, we will ask if it possible for a matrix to fullfill condition (i) (of Propositions [REF] and [REF]) without obeying condition (ii) (of the same propositions).', '1208.4493-1-27-2': 'We will do it for various space-time dimensions, starting with the two dimensional case, which has less interest as far as gravity is concerned, but where results useful for the other cases can be derived.', '1208.4493-1-27-3': 'In this case we will also be able to give an explicit proof of the propositions of section [REF].', '1208.4493-1-28-0': '# Two dimensional case', '1208.4493-1-29-0': 'A certain number of the results derived before can easily be obtained in two dimensions by an explicit calculation.', '1208.4493-1-29-1': 'Consider first the decomposition ([REF]).', '1208.4493-1-29-2': 'We ask if an arbitrary [MATH] invertible matrix [MATH] given by [EQUATION] can be written as (beginning here with proper orthochronous Lorentz transformations) [EQUATION] where [MATH] and [MATH] (and [MATH] a real number).', '1208.4493-1-29-3': 'Expanding the matrix product in the right hand side, we obtain a system of 4 linear equations obeyed by the three coefficients [MATH] which we can use, eliminating [MATH], to get the necessary condition [MATH], which cannot hold for [MATH].', '1208.4493-1-29-4': 'This obviously shows that the decomposition ([REF]) is not always possible, as we showed in a more general way in Proposition [REF].', '1208.4493-1-30-0': 'In two dimensions, one can also explicitly show that the condition (i) of Proposition [REF] always implies the condition (ii) of the same proposition.', '1208.4493-1-30-1': 'Indeed, consider a [MATH] matrix [MATH], that is written as [MATH], with [MATH] symmetric.', '1208.4493-1-30-2': 'Let us then assume that this matrix has a square root.', '1208.4493-1-30-3': 'According to the proof of Proposition [REF], we know that if this matrix has no negative eigenvalues, it has a square root which is a product of [MATH] times a symmetric matrix.', '1208.4493-1-30-4': 'Let us study the case where it has at least one negative eigenvalue.', '1208.4493-1-30-5': 'In this case, according to Theorem [REF], it must be of the form [MATH], where [MATH] is a positive non zero number (note that such a matrix is indeed in the form [MATH]).', '1208.4493-1-30-6': 'It remains then to study all the square roots of [EQUATION]', '1208.4493-1-30-7': 'The matrix equation [MATH] is easy to solve explicitly.', '1208.4493-1-30-8': 'We obtain that a real square root [MATH] is given by any of the matrices [EQUATION] where [MATH] and [MATH] are real numbers and [MATH] is non zero.', '1208.4493-1-30-9': 'Choosing then [MATH] and [MATH] which obey the constraint [MATH] we find an infinite family of real matrix square roots of [MATH] which are written in the form of the product of [MATH] by a symmetric matrix.', '1208.4493-1-30-10': 'A similar straightforward calculation can be made to prove that hypothesis (i) of Proposition [REF] implies (ii) of the same proposition.', '1208.4493-1-30-11': 'In fact, it is easy to see that for every symmetric matrix [EQUATION] with [MATH] there exist real [MATH], [MATH] such that [EQUATION] is symmetric i.e. such that [MATH].', '1208.4493-1-30-12': 'Indeed, either [MATH] and the discriminant of the above second order polynomial equation with respect to [MATH], [MATH], is positive for large enough [MATH], or [MATH] in which case [MATH] must be non-zero and [MATH] is an obvious solution.', '1208.4493-1-30-13': 'This shows that in 2 dimensions, being able to choose zweibeins obeying ([REF]) is equivalent to the existence of a real square root of [MATH].', '1208.4493-1-31-0': '# Four dimensional case', '1208.4493-1-32-0': 'Considering here the case of [MATH] real matrices, and using Theorems [REF] and [REF], we have that the only real invertible matrices [MATH] that have at least one negative real eigenvalue and also have at least one real square root must have one of the following Jordan forms [EQUATION] where [MATH] is one of the Jordan matrices [EQUATION] where [MATH] and [MATH] are positive real numbers, [MATH] is always non zero, the same is true for [MATH] and [MATH] except in the case of [MATH] where [MATH] and [MATH] cannot vanish simultaneously.', '1208.4493-1-33-0': 'Let us first consider the case where one simply has [MATH].', '1208.4493-1-33-1': 'As will be shown below, [MATH] is an example of a real matrix which is a product of [MATH] by a symmetric matrix and has real square roots (in fact it has infinitely many, as we will see below), but which is such that none of those square roots is a product of [MATH] by a symmetric matrix.', '1208.4493-1-33-2': 'As such it shows that, in full generality, hypothesis (i) of Proposition [REF] does not imply (ii) of the same proposition (this has to be contrasted with the two dimensional case where we showed the opposite).', '1208.4493-1-33-3': 'Let us indeed find all the square roots of [MATH] using Theorem [REF].', '1208.4493-1-33-4': 'We first determine all the matrices [MATH] which commute with the Jordan matrix [MATH].', '1208.4493-1-33-5': 'It is easy to see (e.g. by explicitly computing the commutator) that those matrices are simply of the form [MATH] where [MATH] and [MATH] are arbitrary [MATH] invertible matrices.', '1208.4493-1-33-6': 'This means in turn that all the square roots (including complex square roots) of [MATH] are of the form [EQUATION]', '1208.4493-1-33-7': 'But this implies also that the [MATH] matrices [MATH] (respectively [MATH]) are square roots of the [MATH] Jordan matrices [MATH] (respectively [MATH]).', '1208.4493-1-33-8': 'Hence, using ([REF]) we obtain all real square roots of [MATH] as [EQUATION] where [MATH], [MATH], [MATH] and [MATH] are real but otherwise arbitrary (with [MATH] and [MATH] non vanishing).', '1208.4493-1-33-9': 'However, none of the above square roots is the product of [MATH] by a symmetric matrix (this would require [MATH]), which proves our assertion.', '1208.4493-1-34-0': 'The same example can a priori be applied to Proposition [REF].', '1208.4493-1-34-1': 'However, there, hypothesis (i) concerns the matrix [MATH] which, as we will see, cannot be equal to [MATH].', '1208.4493-1-34-2': 'Let us indeed consider the more general case where the matrix [MATH] is diagonalizable.', '1208.4493-1-34-3': 'This has to be the case if [MATH] is similar to [MATH] (or indeed [MATH] or [MATH]).', '1208.4493-1-34-4': 'One can show that this is a sufficient (and in fact also necessary) condition to be able to diagonalize in a common basis the matrices [MATH] and [MATH] corresponding to the symmetric bilinear forms represented by the metrics [CITATION].', '1208.4493-1-34-5': 'In this common basis, each of the diagonal matrices corresponding to [MATH] and [MATH] has only one negative eigenvalue, and hence there is no way that [MATH] can be equal or similar (in the mathematical sense) to [MATH], which has four negative eigenvalues.', '1208.4493-1-34-6': 'The same reasoning excludes [MATH] and [MATH] as admissible Jordan matrices associated to [MATH], because those matrices are diagonal and have eigenvalues which cannot all be given by products of eigenvalues of two commonly diagonal metrics with a Lorentzian signature.', '1208.4493-1-34-7': 'The only cases left over are thus those with Jordan matrices given by [MATH], [MATH], [MATH] and [MATH].', '1208.4493-1-34-8': 'We will not discuss here the case of [MATH] and [MATH] (see below), but will however study the [MATH] and [MATH] cases (which are admissible as Jordan matrices of [MATH]).', '1208.4493-1-34-9': 'In these cases, however, there exists a basis where both metrics are diagonal (as a consequence of the result we just used) and the time direction of any of the two metrics is space-like with respect to the other.', '1208.4493-1-34-10': 'This might be considered as a pathology if both metrics are dynamical, leading possibly to causality violations.', '1208.4493-1-34-11': 'However, nothing goes wrong a priori if only one metric is dynamical, even though, this has to be checked on a case by case basis.', '1208.4493-1-34-12': 'For example, in dRGT theory the light cone seen by some of the polarizations of the massive graviton is not set by the dynamical metric background alone, but also by the non dynamical metric, as exemplified e.g. in appendix A of ref. [CITATION].', '1208.4493-1-35-0': 'In the cases of matrices [MATH] which have [MATH] or [MATH] as Jordan matrices, one can find examples of matrices given by the product of [MATH] with a symmetric matrix, and having real square roots, but such that none of these square roots are the product of [MATH] with a symmetric matrix.', '1208.4493-1-35-1': 'Indeed, consider the invertible matrix [MATH] given by [EQUATION] such that [EQUATION]', '1208.4493-1-35-2': 'Applying Theorem [REF] and the same reasoning as above, all the real square roots of [MATH] are given by the matrices [EQUATION] with [MATH] and [MATH] real and [MATH] non zero .', '1208.4493-1-35-3': 'We can compute this explictly to check that these matrices can never be written as the product of [MATH] with a symmetric matrix.', '1208.4493-1-35-4': 'Indeed, looking for example at the [MATH] and [MATH] elements of the above matrix ([REF]), it can be seen that they are equal if and only if [MATH] vanishes, which never occurs for real [MATH] and [MATH].', '1208.4493-1-35-5': 'This example can be directly applied to Proposition [REF] by considering the choice ([REF]) for [MATH] (and then [MATH]), showing that in general (i) of this proposition does not imply (ii).', '1208.4493-1-36-0': 'So far, we have only discarded cases of matrices similar (in the mathematical sense) to [MATH] and [MATH], i.e. only discussed in detail diagonalizable matrices (possibly only in [MATH]).', '1208.4493-1-36-1': 'There is however a well defined criterion allowing us to do so.', '1208.4493-1-36-2': 'Indeed, one can show that a sufficient condition for [MATH] to be diagonalizable is that the light cones of [MATH] and that of [MATH] do not intersect (except at the origin) [CITATION].', '1208.4493-1-36-3': 'When it is the case, using the result we mentioned above, one can find a basis where [MATH] and [MATH] are commonly diagonal.', '1208.4493-1-36-4': 'Then, in this basis, either the time directions of the two metrics coincide and the sufficient condition of Proposition [REF] is fullfilled - one can find a basis of vierbein satisfying ([REF]) - or they do not coincide, and then one has either [MATH] or [MATH] as Jordan matrices and the counter example above applies.', '1208.4493-1-37-0': '# Three dimensional case', '1208.4493-1-38-0': 'The results obtained in the previous section can easily be extended to the case of a spacetime with 3 dimensions, which has some relevance for physics and in particular massive gravity [CITATION].', '1208.4493-1-38-1': 'In three dimensions, the only cases which are not covered by Propositions [REF] and [REF] are the cases of real invertible matrices [MATH] which have the form [EQUATION] where [MATH] and [MATH] are non zero positive real numbers, and [MATH] is an invertible matrix.', '1208.4493-1-38-2': 'Here it is also easy to find an example of the kind ([REF])-([REF]).', '1208.4493-1-38-3': 'Indeed consider now [MATH] to be given by [EQUATION]', '1208.4493-1-38-4': 'This matrix has the form of a product of [MATH] with a symmetric matrix, but none of its real square roots, given by [EQUATION] (with [MATH] and [MATH] real numbers, [MATH] non vanishing) has.', '1208.4493-1-38-5': 'Hence, one can make here similar considerations as those given at the end of the previous section.', '1208.4493-1-39-0': '# Conclusions', '1208.4493-1-40-0': 'In this note, we studied in detail the sufficient and necessary conditions for two vielbeins [MATH] and [MATH] associated with two metrics [MATH] and [MATH] defined on a given manifold to be chosen so that they obey the symmetry condition ([REF]) which has been used as a gauge condition in vielbein gravity or massive gravity.', '1208.4493-1-40-1': 'We also studied as a byproduct the necessary and sufficient condition for an arbitrary matrix [MATH] to be decomposed as in ([REF]).', '1208.4493-1-40-2': 'We showed that, in contrast to what has sometimes been claimed in the literature, the condition ([REF]) and the decomposition ([REF]) cannot be achieved in general but require some extra assumptions related to the existence and properties of square roots of matrices.', '1208.4493-1-40-3': 'These assumptions are gathered in Propositions 1 to 5 of the present work.', '1208.4493-1-40-4': 'We also showed that, in the 4 dimensional case, it is enough to assume that the light cones of the two metrics do not intersect and that the metrics share the same time direction (in the sense given at the end of section [REF]), in order to satisfy a sufficient condition for ([REF]) to be true.'}","{'1208.4493-2-0-0': '# Introduction', '1208.4493-2-1-0': 'There are various situations in physics where one has to consider a manifold endowed with two different vielbein fields.', '1208.4493-2-1-1': 'Obviously, this appears to be the case in bimetric theories, theories where two different metrics are defined on the same space-time manifold [CITATION].', '1208.4493-2-1-2': 'Each of these metrics can then be described by a different vielbein.', '1208.4493-2-1-3': 'This is also true even if one of the two metrics is not dynamical.', '1208.4493-2-1-4': 'It also applies to non linear massive gravity (for recent reviews see [CITATION]), which is nothing else than a special class of bigravity, and in particular it applies to the recently introduced massive gravity theories of de Rham-Gabadadze-Tolley (dRGT in the following) [CITATION] as well as to the extension of these to the dynamical bimetric case [CITATION].', '1208.4493-2-1-5': 'A similar situation also occurs when one expands General Relativity around a fixed background metric and expresses both the background and the dynamical metric in terms of vielbeins.', '1208.4493-2-1-6': 'This is the starting point of many works dealing with quantum gravity (see e.g. [CITATION]).', '1208.4493-2-2-0': 'Considering such situations, let us define, in arbitrary [MATH] dimensions, [MATH] and [MATH] to be two bases of 1-forms obeying at every space-time point [EQUATION] or equivalently [EQUATION] where [MATH] and [MATH] are respectively the metrics associated with the vielbeins.', '1208.4493-2-2-1': 'We will also need the vectors [MATH] and [MATH], respectively dual to the 1-forms [MATH] and [MATH], that verify [EQUATION]', '1208.4493-2-2-2': 'For future use, let us rewrite the above relations (and consequences thereof) using matrix notations.', '1208.4493-2-2-3': 'We have [EQUATION] where [MATH] is the [MATH] identity matrix, [MATH] denotes the matrix transpose of the matrix [MATH], [MATH] is just [MATH] and the same relations hold between [MATH], [MATH] and [MATH] respectively.', '1208.4493-2-3-0': 'The defining relations ([REF]) and ([REF]) imply the gauge symmetry [EQUATION] with [MATH] and [MATH] Lorentz matrices.', '1208.4493-2-4-0': 'It is often convenient to ask for a ""symmetry"" condition on the vielbeins which reads [EQUATION]', '1208.4493-2-4-1': 'Notice that this condition can also be written as [MATH] and that Ref. [CITATION] uses an equivalent form which reads [MATH].', '1208.4493-2-5-0': 'In the recent discussions about massive gravity, such a condition has been used to ensure the existence of, and express, the matrix square root of [MATH] which enters in a crucial way in the definition of dRGT theory (see e.g. [CITATION]).', '1208.4493-2-5-1': 'Indeed, whenever condition ([REF]) holds, [MATH] defined as [EQUATION] verifies the defining equation of the matrix square root of [MATH] given by [EQUATION]', '1208.4493-2-5-2': 'It has also been argued by Hinterbichler and Rosen [CITATION] that, in the vielbein reformulation of dRGT theories, condition ([REF]) is obtained as a consequence of field equations.', '1208.4493-2-5-3': 'To prove this, they use a decomposition of an arbitrary matrix [MATH] (representing some unconstrained arbitrary vielbein multiplied by [MATH]) as [EQUATION] where [MATH] is a Lorentz matrix and [MATH] is a symmetric matrix.', '1208.4493-2-5-4': 'This is reminiscent of the so-called polar decomposition stating that an arbitrary invertible matrix can be written as the product of an orthogonal matrix with a symmetric matrix.', '1208.4493-2-5-5': 'However we will show that such a decomposition does not hold in general if one replaces the orthogonal matrix by a Lorentz transformation.', '1208.4493-2-5-6': 'This makes in particular the argument of Ref. [CITATION] incomplete.', '1208.4493-2-6-0': 'Furthermore, in massive gravity as well as in perturbative quantum gravity condition ([REF]) has been used as a gauge condition.', '1208.4493-2-6-1': 'In the quantum gravity context, this gauge (sometimes dubbed Deser-van Nieuwenhuizen gauge in reference to [CITATION]) has been first introduced via a gauge fixing term in the action and dealt with perturbatively [CITATION].', '1208.4493-2-6-2': 'It was then later argued that this gauge can be set ""non perturbatively"", i.e. that given a set of arbitrary vielbeins [MATH] and [MATH] that do not fullfill condition ([REF]), one can always Lorentz rotate them as in ([REF]), and ([REF]) to define a new set of vielbeins obeying this condition [CITATION] (with the consequence that the corresponding gauge would not suffer from Gribov-like ambiguities).', '1208.4493-2-6-3': 'Interestingly enough, the same statements have also been made in the context of massive gravity.', '1208.4493-2-6-4': 'Indeed, there as well the condition ([REF]) has been used ""perturbatively"" (i.e. in the case when both metrics [MATH] and [MATH] are close to one another, see e.g. [CITATION]), but it has also been argued that condition ([REF]) can be reached as a (Lorentz) gauge choice for arbitrary metrics [CITATION].', '1208.4493-2-6-5': 'This contradicts various other statements made in the literature, for example in Ref. [CITATION], where it is stated that gauge ([REF]) cannot be set beyond perturbation theory.', '1208.4493-2-6-6': 'Settling this contradiction, as we intend to do here, will also illuminate issues discussed in the previous paragraph, since, as we will show, to set ([REF]) via suitable Lorentz rotations of the vielbeins involves a decomposition similar to ([REF]).', '1208.4493-2-7-0': 'To be precise, the purpose of this note is to determine when and how the condition ([REF]) can be enforced, as well as when the decomposition ([REF]) holds.', '1208.4493-2-7-1': 'These questions, beyond their mathematical interest, are especially important for massive gravity.', '1208.4493-2-7-2': 'Indeed, one can argue that the vielbein formulation of dRGT theories has several advantages over their metric formulations.', '1208.4493-2-7-3': 'First of all, it allows a simple extraction of what plays the role of the Hamiltonian constraint [CITATION].', '1208.4493-2-7-4': 'Second, in some cases it also allows to dynamically derive the existence of the square root of [MATH] that has to be assumed or enforced by Lagrange multipliers in the metric formulation [CITATION].', '1208.4493-2-7-5': 'Finally, the frame formulation permits a simple discussion of the constraints and the counting of dynamical degrees of freedom in the Lagrangian framework [CITATION].', '1208.4493-2-7-6': 'In this formulation, relation ([REF]) plays a key role, and it is important to know whether it can be obtained by Lorentz gauge transformations, or it needs additional constraints to be imposed.', '1208.4493-2-8-0': 'This paper is organized as follows.', '1208.4493-2-8-1': 'In the next section, we will discuss necessary and sufficient conditions for ([REF]) and ([REF]) to hold.', '1208.4493-2-8-2': 'Then, in section 3, using results on matrix square roots, we will spell out sufficient conditions to achieve ([REF]) and ([REF]).', '1208.4493-2-8-3': 'In the next sections we will discuss the specific cases of [MATH], [MATH], and [MATH] space-time dimensions, and in particular some examples clarifying the results of section 3 as well as some left over cases.', '1208.4493-2-8-4': 'Finally we will quickly look at the stability of these conditions with respect to the dynamics of the system, i.e. we will discuss whether they are preserved under time evolution in some particular theories, and we will point out some consequences for massive gravity.', '1208.4493-2-9-0': 'Before proceeding, let us mention a special choice for one of the metrics (say [MATH]) and the associated vielbein [MATH].', '1208.4493-2-9-1': 'This choice is made in some contexts (e.g. dRGT theories, but also perturbative quantum gravity).', '1208.4493-2-9-2': 'It amounts to first assuming that the metric [MATH] is flat and takes the canonical form [MATH], i.e. [EQUATION] and then choosing [MATH], i.e. such that (in components) [EQUATION]', '1208.4493-2-9-3': 'When the choice ([REF])-([REF]) is made, the constraint ([REF]) simply reads (labelling here space-time indices and Lorentz indices with the same set of letters) [EQUATION] stating that the vielbein [MATH] can be represented as a symmetric matrix.', '1208.4493-2-9-4': 'This choice will not be used to derive the results of this paper, but will just sometimes be considered as an example.', '1208.4493-2-10-0': '# Necessary and sufficient conditions', '1208.4493-2-11-0': 'Let us first try to set the constraint ([REF]) by using the freedom to Lorentz rotate independently the two sets of vielbeins [MATH] and [MATH].', '1208.4493-2-11-1': 'Considering two arbitrarily chosen vielbeins [MATH] and [MATH], assume that there exist two Lorentz transformations [MATH] and [MATH] such that the matrix [MATH] defined by [EQUATION] is symmetric.', '1208.4493-2-11-2': 'Defining [MATH] as the matrix of components [MATH] given by [EQUATION] (note that this definition implies that [MATH] is invertible), the above equality ([REF]) reads in matricial notations [EQUATION]', '1208.4493-2-11-3': 'Multiplying it on the right by [MATH] and on the left by [MATH] we get [EQUATION]', '1208.4493-2-11-4': 'For [MATH] to be symmetric, the matrix on the left hand side above should be symmetric, call it [MATH].', '1208.4493-2-11-5': 'Defining the Lorentz transformation [MATH] by [MATH] we get that the invertible matrix [MATH] should be written as in Eq. ([REF]).', '1208.4493-2-11-6': 'Being a Lorentz transformation, [MATH] verifies [EQUATION]', '1208.4493-2-11-7': 'As we already stated, a decomposition such as in Eq. ([REF]) does not hold in general (in constrast to the polar decomposition).', '1208.4493-2-11-8': 'Indeed, rewriting ([REF]) as [MATH] and inserting this into ([REF]) we get after some trivial manipulation, that [MATH] and [MATH] should fullfill the necessary condition [EQUATION]', '1208.4493-2-11-9': 'Running backward the above argument it is easy to see that the above condition is also sufficient (just because the matrix defined as [MATH] will be a Lorentz transformation).', '1208.4493-2-11-10': 'Hence we have proven the following proposition.', '1208.4493-2-12-0': 'In particular, when [MATH] is given by ([REF]), we have (using relations ([REF])-([REF]) as well as definition ([REF])) [EQUATION]', '1208.4493-2-12-1': 'So if [MATH] has a square root [MATH], then (i) above holds: a square root of [MATH] being then given by [MATH].', '1208.4493-2-12-2': 'We then prove the following proposition,', '1208.4493-2-13-0': 'We first assume that [MATH] can be written as [MATH] with [MATH] a symmetric matrix.', '1208.4493-2-13-1': 'Then using this hypothesis into the first equality of ([REF]) we get [EQUATION]', '1208.4493-2-13-2': 'The matrix [MATH] being symmetric, this proves one side of the equivalence.', '1208.4493-2-13-3': 'Conversely, we assume that there exists a symmetric matrix [MATH] such that [MATH].', '1208.4493-2-13-4': 'Then [MATH] is given by [EQUATION]', '1208.4493-2-13-5': 'Noticing that the matrix [MATH] is symmetric ends the proof.', '1208.4493-2-14-0': 'Hence, gathering the above results, we have proven the following statement.', '1208.4493-2-15-0': '[Direct proof] Suppose first we have vielbeins [MATH] and [MATH] satisfying the above symmetry property.', '1208.4493-2-15-1': 'Then [EQUATION] and if we define [MATH] we get [MATH].', '1208.4493-2-15-2': 'Moreover [EQUATION] which shows that the matrix [MATH] is symmetric.', '1208.4493-2-15-3': 'Notice that this is equivalent to [MATH] symmetric.', '1208.4493-2-15-4': 'Conversely, suppose we have a real matrix [MATH] such that [MATH] and [MATH] symmetric.', '1208.4493-2-15-5': 'We start by choosing an arbitrary vielbein [MATH] for the metric [MATH] i.e. [MATH], and we denote by [MATH] its dual vector i.e. [MATH].', '1208.4493-2-15-6': 'We then define [MATH].', '1208.4493-2-15-7': 'This implies that [EQUATION]', '1208.4493-2-15-8': 'But the symmetry of [MATH] implies that [MATH] so [EQUATION] and [MATH] is a well-defined vielbein for the metric [MATH].', '1208.4493-2-15-9': 'Notice that this definition tells us [MATH].', '1208.4493-2-15-10': 'It remains to be shown that these vielbeins have the required symmetry property.', '1208.4493-2-15-11': 'We start from the symmetry of [MATH] [EQUATION] which we can rewrite [EQUATION]', '1208.4493-2-15-12': 'Multiplying by [MATH] we get [MATH] and this completes the proof.', '1208.4493-2-16-0': 'As we just showed the hypotheses (i) of Propositions [REF] and [REF] are that a certain real (invertible) matrix has a real square root.', '1208.4493-2-16-1': 'It is however well known that not all real invertible matrices have real square roots (see e.g. [CITATION]) and we will later recall what are the necessary and sufficient conditions for this to occur.', '1208.4493-2-16-2': 'In our case, though, the matrix which should have a square root is not totally arbitrary.', '1208.4493-2-16-3': 'For example, in Proposition [REF] it must be of the form [MATH].', '1208.4493-2-16-4': 'This alone does however not ensure the existence of a square root.', '1208.4493-2-16-5': ""For example, choosing [EQUATION] we get [EQUATION] which doesn't have any real square roots."", '1208.4493-2-16-6': 'Indeed, such a [MATH] diagonal matrix with four distinct eigenvalues has [MATH] square roots which are given here by [MATH].', '1208.4493-2-16-7': 'None of them is real.', '1208.4493-2-16-8': 'Hence the decomposition ([REF]) can at best hold for a restricted set of matrices.', '1208.4493-2-17-0': 'We thus see that considering the matrix [MATH] above as given by the form ([REF]) invalidates the result of Ref. [CITATION].', '1208.4493-2-17-1': 'Notice that, if one makes now the simple choice ([REF])-([REF]) (and considering equation ([REF])), our example involves a ""mismatch"" between the time directions of the two metrics [MATH] and [MATH].', '1208.4493-2-17-2': 'However, beyond perturbation theory there is no reason to think that these time directions should coincide or even be compatible.', '1208.4493-2-17-3': 'We will come back to this question later.', '1208.4493-2-17-4': 'Notice further that perturbatively, if [MATH], with [MATH] small, then to the first order in [MATH], and so the assumptions (i) and (ii) of Proposition [REF] are always true perturbatively.', '1208.4493-2-18-0': '# Sufficient conditions', '1208.4493-2-19-0': 'Here, in order to formulate simple sufficient conditions allowing to obtain ([REF]) and ([REF]), we will discuss the precise relation between hypotheses (i) and (ii) of Propositions [REF] and [REF].', '1208.4493-2-19-1': 'We need to recall how square roots of real matrices are obtained.', '1208.4493-2-19-2': 'We first use the following theorem (that we quote here from Ref. [CITATION]).', '1208.4493-2-20-0': 'Let [MATH] be an invertible real square matrix (of arbitrary dimension).', '1208.4493-2-20-1': 'If [MATH] has no real negative eigenvalues, then there are precisely [MATH] real square roots of [MATH] which are polynomial functions of [MATH], where [MATH] is the number of distinct eigenvalues of [MATH] and [MATH] is the number of distinct complex conjugate eigenvalue pairs.', '1208.4493-2-20-2': 'If [MATH] has a real negative eigenvalue, then [MATH] has no real square root which is a polynomial function of [MATH].', '1208.4493-2-21-0': 'Let us first use this theorem to prove that (i) of Proposition [REF] (respectively Proposition [REF]) implies (ii) of the same proposition whenever the matrix [MATH] (respectively the matrix [MATH]) has no real negative eigenvalues.', '1208.4493-2-21-1': 'To see this, just consider a real matrix [MATH] with no negative eigenvalues, given by the product of two symmetric invertible matrices [MATH] and [MATH].', '1208.4493-2-21-2': 'By virtue of the above theorem, we know that this matrix has at least one real square root which is a polynomial function of [MATH], that we note [MATH].', '1208.4493-2-21-3': 'One then has [EQUATION] where the sum runs over a finite number of integers [MATH], and [MATH] are real numbers.', '1208.4493-2-21-4': 'Using the fact that [MATH], one then has [EQUATION] where the term [MATH]contains [MATH] factors of [MATH] and [MATH] factors of [MATH], and is a symmetric matrix.', '1208.4493-2-21-5': 'This means that that the square root [MATH] is given by the product of [MATH] by a symmetric matrix.', '1208.4493-2-21-6': 'It is enough to prove our assertion by choosing [MATH] to be given by [MATH] and [MATH] to be given by [MATH] (respectively [MATH] given by [MATH] and [MATH] to be given by [MATH]).', '1208.4493-2-21-7': 'Hence, using the above result, and Propositions [REF] and [REF] we have shown the following two propositions', '1208.4493-2-22-0': 'If [MATH] has one (or more) real negative eigenvalue, Theorem [REF] does not imply that [MATH] does not have a real square root, but just that such a square root cannot be a polynomial function of [MATH].', '1208.4493-2-22-1': 'In order to enunciate the necessary and sufficient conditions for a real matrix to have a real square root, one first needs to introduce the so-called Jordan decomposition of a matrix.', '1208.4493-2-22-2': 'It uses Jordan blocks which can be defined as [MATH] matrices wich are of the form [MATH] given by (for [MATH]) [EQUATION] where [MATH] is a complex number, and one has [MATH] for [MATH].', '1208.4493-2-22-3': 'One can then show that for an arbitrary [MATH] matrix [MATH], there exists an invertible matrix [MATH] (possibly complex), and a matrix [MATH] such that [EQUATION] and the matrix J is a so called Jordan matrix of the form [EQUATION] where [MATH] is an integer and the matrices [MATH] are called the Jordan blocks of [MATH].', '1208.4493-2-22-4': 'For a given matrix [MATH], the number of Jordan blocks, the nature of the distinct Jordan blocks, and the number of times a given Jordan block occurs in the Jordan matrix [MATH] are uniquely determined.', '1208.4493-2-22-5': 'Moreover, the [MATH] are the eigenvalues of [MATH].', '1208.4493-2-22-6': 'One can further show that a given Jordan block [MATH] with [MATH], has precisely two upper triangular square roots, [MATH], which are in addition polynomial functions of [MATH] [CITATION].', '1208.4493-2-22-7': 'These can be used to find all the square roots (possibly complex) of a given matrix using the following theorem.', '1208.4493-2-23-0': 'Let [MATH] be a [MATH] complex matrix which has a Jordan decomposition given by ([REF])-([REF]), then all the square roots (which may include complex matrices) of [MATH] are given by the matrices [MATH], where [MATH] is an arbitrary matrix which commutes with [MATH].', '1208.4493-2-24-0': 'The Jordan blocks of a matrix also play a crucial role in the following theorem which gives the necessary and sufficient condition for a real matrix to have a real square root (see e.g. [CITATION]).', '1208.4493-2-25-0': 'Let [MATH] be an invertible real square matrix (of arbitrary dimension).', '1208.4493-2-25-1': 'The matrix [MATH] has a real square root if and only if for each of its negative eigenvalues [MATH], the number of identical Jordan block [MATH] where this eigenvalue occurs in the Jordan decomposition of the matrix [MATH] is even.', '1208.4493-2-26-0': 'In the following, we will use the above theorems to discuss in detail the cases which are not covered by our Propositions [REF] and [REF].', '1208.4493-2-26-1': 'Namely, we will ask if it possible for a matrix to fullfill condition (i) (of Propositions [REF] and [REF]) without obeying condition (ii) (of the same propositions).', '1208.4493-2-26-2': 'We will do it for various space-time dimensions, starting with the two dimensional case, which has less interest as far as gravity is concerned, but where results useful for the other cases can be derived.', '1208.4493-2-26-3': 'In this case we will also be able to give an explicit proof of the propositions of section [REF].', '1208.4493-2-27-0': '# Two dimensional case', '1208.4493-2-28-0': 'A certain number of the results derived before can easily be obtained in two dimensions by an explicit calculation.', '1208.4493-2-28-1': 'Consider first the decomposition ([REF]).', '1208.4493-2-28-2': 'We ask if an arbitrary [MATH] invertible matrix [MATH] given by [EQUATION] can be written as (beginning here with proper orthochronous Lorentz transformations) [EQUATION] where [MATH] and [MATH] (and [MATH] a real number).', '1208.4493-2-28-3': 'Expanding the matrix product in the right hand side, we obtain a system of 4 linear equations obeyed by the three coefficients [MATH] which we can use, eliminating [MATH], to get the necessary condition [MATH], which cannot hold for [MATH].', '1208.4493-2-28-4': 'This obviously shows that the decomposition ([REF]) is not always possible, as we showed in a more general way in Proposition [REF].', '1208.4493-2-29-0': 'In two dimensions, one can also explicitly show that the condition (i) of Proposition [REF] always implies the condition (ii) of the same proposition.', '1208.4493-2-29-1': 'Indeed, consider a [MATH] matrix [MATH], that is written as [MATH], with [MATH] symmetric.', '1208.4493-2-29-2': 'Let us then assume that this matrix has a square root.', '1208.4493-2-29-3': 'According to the proof of Proposition [REF], we know that if this matrix has no negative eigenvalues, it has a square root which is a product of [MATH] times a symmetric matrix.', '1208.4493-2-29-4': 'Let us study the case where it has at least one negative eigenvalue.', '1208.4493-2-29-5': 'In this case, according to Theorem [REF], it must be of the form [MATH], where [MATH] is a positive non zero number (note that such a matrix is indeed in the form [MATH]).', '1208.4493-2-29-6': 'It remains then to study all the square roots of [EQUATION]', '1208.4493-2-29-7': 'The matrix equation [MATH] is easy to solve explicitly.', '1208.4493-2-29-8': 'We obtain that a real square root [MATH] is given by any of the matrices [EQUATION] where [MATH] and [MATH] are real numbers and [MATH] is non zero.', '1208.4493-2-29-9': 'Choosing then [MATH] and [MATH] which obey the constraint [MATH] we find an infinite family of real matrix square roots of [MATH] which are written in the form of the product of [MATH] by a symmetric matrix.', '1208.4493-2-29-10': 'A similar straightforward calculation can be made to prove that hypothesis (i) of Proposition [REF] implies (ii) of the same proposition.', '1208.4493-2-29-11': 'In fact, it is easy to see that for every symmetric matrix [EQUATION] with [MATH] there exist real [MATH], [MATH] such that [EQUATION] is symmetric i.e. such that [MATH].', '1208.4493-2-29-12': 'Indeed, either [MATH] and the discriminant of the above second order polynomial equation with respect to [MATH], [MATH], is positive for large enough [MATH], or [MATH] in which case [MATH] must be non-zero and [MATH] is an obvious solution.', '1208.4493-2-29-13': 'This shows that in 2 dimensions, being able to choose zweibeins obeying ([REF]) is equivalent to the existence of a real square root of [MATH].', '1208.4493-2-30-0': '# Three dimensional case', '1208.4493-2-31-0': 'The results obtained in the previous section can be extended to the case of a spacetime with 3 dimensions, which has some relevance for physics and in particular massive gravity [CITATION].', '1208.4493-2-31-1': 'In three dimensions, the only cases which are not covered by Propositions [REF] and [REF] are the cases of real invertible matrices [MATH] which have the form [EQUATION] where [MATH] and [MATH] are non zero positive real numbers, and [MATH] is an invertible matrix.', '1208.4493-2-31-2': 'Notice that because [MATH], [MATH] and [MATH] are real, [MATH] may also be assumed to be real.', '1208.4493-2-31-3': 'Before going any further, notice that one can find [MATH] matrices [MATH], in the form [MATH] with [MATH] symmetric, having real square roots, but such that none of these square roots is the product of [MATH] by a symmetric matrix.', '1208.4493-2-31-4': 'Indeed consider [MATH] to be given by [EQUATION]', '1208.4493-2-31-5': 'This matrix has the form of a product of [MATH] with a symmetric matrix, but none of its real square roots, given by [EQUATION] (with [MATH] and [MATH] real numbers, [MATH] non vanishing) has the same form.', '1208.4493-2-31-6': 'However, this example does not apply to the cases of interest here because [MATH] does not have the correct signature: instead of being of signature [MATH] as e.g. a matrix of the form [MATH], it is negative definite.', '1208.4493-2-32-0': 'In contrast we are going to show that (i) of Proposition [REF] (respectively Proposition [REF]) implies (ii) of the same proposition whenever the matrix [MATH] (respectively the matrix [MATH]) is of the form ([REF]).', '1208.4493-2-32-1': 'In order to do that let us assume (for the same reason as in section [REF]) that [MATH] with [MATH] and [MATH] two symmetric matrices of [MATH] signature.', '1208.4493-2-32-2': 'The fact that [MATH] is symmetric implies that [MATH] commutes with [MATH] and thus it must be of the form [EQUATION] with [MATH] a symmetric two by two matrix and [MATH] a real number such that [MATH].', '1208.4493-2-32-3': 'Since [MATH] is of [MATH] signature, it is obvious that [MATH] cannot be negative definite.', '1208.4493-2-32-4': 'From the fact that [MATH] has [MATH] signature we can infer that [MATH] also has the same signature.', '1208.4493-2-32-5': 'But [EQUATION] and thus [MATH] cannot be positive definite either.', '1208.4493-2-32-6': 'We therefore necessarily conclude that [MATH] must have [MATH] signature and that [MATH].', '1208.4493-2-32-7': 'This means that there exists a two by two invertible matrix [MATH] such that [EQUATION]', '1208.4493-2-32-8': 'Now let us define [EQUATION]', '1208.4493-2-32-9': 'This matrix clearly commutes with [MATH] and if we further define [EQUATION] we can see that [MATH] and thus [MATH] is a real square root of [MATH].', '1208.4493-2-32-10': 'Furthermore it is easy to see using ([REF]), ([REF]) and ([REF]) that [EQUATION] is symmetric.', '1208.4493-2-32-11': 'This provides a constructive proof of our statement.', '1208.4493-2-33-0': '# Four dimensional case', '1208.4493-2-34-0': 'Considering here the case of [MATH] real matrices, and using Theorems [REF] and [REF], we have that the only real invertible matrices [MATH] that have at least one negative real eigenvalue and also have at least one real square root must have one of the following Jordan forms [EQUATION] where [MATH] is one of the Jordan matrices [EQUATION] where [MATH] and [MATH] are positive real numbers, [MATH] and [MATH] are always non zero, and [MATH] can only vanish in the case of [MATH].', '1208.4493-2-34-1': 'Because [MATH] is real, the invertible matrix [MATH] may be chosen to be real in the [MATH], [MATH], [MATH] and [MATH] cases.', '1208.4493-2-34-2': 'The case of [MATH] is a bit more tricky, but we can also assume [MATH] to be real as long as we replace the Jordan matrix [MATH] by its real counterpart [EQUATION]', '1208.4493-2-34-3': 'We will show here that results similar to the ones obtained above in the [MATH] and [MATH] cases hold for [MATH] whenever [MATH] is of the form ([REF]) and [MATH] with [MATH] and [MATH] two symmetric matrices of Lorentzian signature.', '1208.4493-2-34-4': 'We will look in turn at the different cases for what concerns [MATH].', '1208.4493-2-34-5': 'Consider first the case where the matrix [MATH] is diagonalizable over [MATH].', '1208.4493-2-34-6': 'One can show that this is a sufficient (and in fact also necessary) condition to be able to diagonalize (in the sense of forms) in a common basis the matrices [MATH] and [MATH] corresponding to two symmetric bilinear forms [CITATION].', '1208.4493-2-34-7': 'In this common basis, each of the diagonal matrices corresponding to [MATH] and [MATH] has only one negative eigenvalue, and hence there is no way that [MATH] can be equal or similar (in the mathematical sense) to [MATH], which has four negative eigenvalues.', '1208.4493-2-34-8': 'This excludes the [MATH] case from the start.', '1208.4493-2-35-0': 'The discussion of the [MATH] case proceeds along the same lines as in the [MATH] case.', '1208.4493-2-35-1': 'The fact that [MATH] is symmetric implies that [MATH] commutes with [MATH] and thus it must be of the form [EQUATION] with [MATH] and [MATH] symmetric two by two matrices such that [MATH].', '1208.4493-2-35-2': 'Notice that [MATH] must be diagonal whenever [MATH].', '1208.4493-2-35-3': 'Since [MATH] is of [MATH] signature, it is obvious that [MATH] and [MATH] cannot be negative definite.', '1208.4493-2-35-4': 'From the fact that [MATH] has [MATH] signature we can infer that [MATH] also has the same signature.', '1208.4493-2-35-5': 'But [EQUATION] and thus [MATH] cannot be positive definite either.', '1208.4493-2-35-6': 'We therefore necessarily get that [MATH] must have [MATH] signature and that [MATH] must be positive definite.', '1208.4493-2-35-7': 'In particular this means that there exist two by two invertible matrices [MATH] and [MATH] such that [EQUATION] and whenever [MATH], we can further assume that [MATH] is diagonal (this is because [MATH] is then diagonal and positive definite).', '1208.4493-2-35-8': 'Now let us define [EQUATION]', '1208.4493-2-35-9': 'This matrix clearly commutes with [MATH] and if we further define [EQUATION] we can see that [MATH] and thus [MATH] is a real square root of [MATH].', '1208.4493-2-35-10': 'Analogously to what has been done in the previous section, using ([REF]), ([REF]) and ([REF]), it is also easy to see that [EQUATION] is symmetric.', '1208.4493-2-35-11': 'This shows, as in the [MATH] case, that whenever [MATH] and hypothesis (i) of Proposition [REF] (respectively Proposition [REF]) is verified, hypothesis (ii) of the same proposition is also verified.', '1208.4493-2-36-0': 'The three remaining cases ([MATH] and [MATH]) actually never occur as long as we assume that [MATH] is the product of two symmetric matrices of Lorentzian signature ([MATH]), as we now show.', '1208.4493-2-36-1': 'In the [MATH] case, it is easier to work with the real Jordan form of [MATH] i.e. [MATH].', '1208.4493-2-36-2': 'In order to understand the implications of the symmetry of [MATH] we need to introduce the matrix [EQUATION]', '1208.4493-2-36-3': 'Then it is easy to see that, given the particular form of [MATH], the symmetry of [MATH] implies that [MATH] commutes with [MATH].', '1208.4493-2-36-4': 'Therefore [EQUATION] with [MATH] a symmetric two by two matrix and [MATH], [MATH] real numbers such that [MATH].', '1208.4493-2-36-5': 'Since the signature of [MATH] is [MATH] and [MATH] (which is the opposite of the determinant of the [MATH] lower block in the right matrix above), [MATH] must be positive definite.', '1208.4493-2-36-6': 'But we also know that the signature of [MATH] is [MATH] and since [EQUATION] [MATH] cannot be positive definite and we have a contradiction.', '1208.4493-2-36-7': 'This proves by reductio ad absurdum that the [MATH] case cannot occur in this context.', '1208.4493-2-36-8': 'A similar argument works for the [MATH] case.', '1208.4493-2-36-9': 'Indeed the symmetry of [MATH] again implies that [MATH] commutes with [MATH].', '1208.4493-2-36-10': 'Therefore [EQUATION] with [MATH] a symmetric two by two matrix and [MATH], [MATH] real numbers such that [MATH].', '1208.4493-2-36-11': 'Since the signature of [MATH] is [MATH] and [MATH], [MATH] must be positive definite.', '1208.4493-2-36-12': 'But, with a similar argument as in the above case, we know that [MATH] cannot be positive definite and we again stumble upon a contradiction.', '1208.4493-2-36-13': 'Finally the [MATH] case can be handled in the same manner.', '1208.4493-2-36-14': 'Introducing [EQUATION] we can express the symmetry of [MATH] as the fact that [MATH] commutes with [MATH].', '1208.4493-2-36-15': 'This in turn means that [EQUATION] with [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] real numbers such that [MATH].', '1208.4493-2-36-16': 'But [MATH] which is incompatible with the Lorentzian signature of [MATH] and this excludes the last case.', '1208.4493-2-37-0': 'This lengthy discussion has shown that (i) of Proposition [REF] (respectively Proposition [REF]) implies (ii) of the same proposition whenever the matrix [MATH] (respectively the matrix [MATH]) is of the form ([REF]).', '1208.4493-2-38-0': 'In this section (as well as the previous two) we have therefore shown that (at least up to dimension [MATH]) hypotheses (ii) of Propositions [REF] and [REF] are superfluous.', '1208.4493-2-38-1': 'To summarize, we have proven the following two propositions.', '1208.4493-2-39-0': 'We expect that these results continue to hold in higher dimensions even though we do not have a dimension independent proof.', '1208.4493-2-40-0': '# Time evolution and application to ghost-free massive gravity', '1208.4493-2-41-0': 'Now that we have discussed the different necessary and sufficient conditions for ([REF]) to hold, we may ask ourselves if these conditions are preserved through time evolution.', '1208.4493-2-41-1': 'It is easy to see that there is no general answer to this question i.e. it depends on the theory.', '1208.4493-2-41-2': 'Consider for example the case of a bimetric theory where the two metrics are not coupled to each other (or just very weakly).', '1208.4493-2-41-3': 'The action of such a theory in four dimensions is given by [EQUATION]', '1208.4493-2-41-4': 'It is easy to see that in some coordinate patch a solution to the equations of motion of this theory is simply given by [EQUATION]', '1208.4493-2-41-5': 'This solution corresponds to Minkowski space-time for [MATH] and de Sitter space-time for [MATH].', '1208.4493-2-41-6': 'In particular, whatever the time coordinate [MATH] [EQUATION]', '1208.4493-2-41-7': 'At [MATH] this matrix reduces to [MATH] and admits a real square root [MATH] such that [MATH] is symmetric.', '1208.4493-2-41-8': 'For instance [EQUATION] clearly verifies the above conditions.', '1208.4493-2-41-9': 'This means that on the [MATH] hypersurface, one may choose vierbeins obeying condition ([REF]).', '1208.4493-2-41-10': 'However as soon as [MATH] this condition ceases to be true as [MATH] does not even admit a real square root anymore.', '1208.4493-2-41-11': 'Thus in the above theory, condition ([REF]) is not preserved under time evolution.', '1208.4493-2-42-0': 'In contrast, let us consider the recently proposed dRGT theory [CITATION].', '1208.4493-2-42-1': 'We first note that in the metric formulation of this theory, one assumes the existence of a real square root of [MATH] (where [MATH] is a dynamical metric and [MATH] a non-dynamical one); then, according to proposition [REF], this mere assumption is equivalent to assuming the existence of vierbeins verifying condition ([REF]).', '1208.4493-2-42-2': 'On the other hand, in the vielbein formulation of dRGT theory, it has been shown in [CITATION] (see also [CITATION]) that, at least for some region of parameter space, condition ([REF]) is imposed by the equations of motion and is therefore preserved under time evolution.', '1208.4493-2-42-3': 'When this is the case, the propositions proven in this work then also imply that the existence of the matrix square root of [MATH] is dynamically imposed.', '1208.4493-2-43-0': '# Conclusions', '1208.4493-2-44-0': 'In this note, we studied in detail the sufficient and necessary conditions for two vielbeins [MATH] and [MATH] associated with two metrics [MATH] and [MATH] defined on a given manifold to be chosen so that they obey the symmetry condition ([REF]) which has been used as a gauge condition in vielbein gravity or massive gravity.', '1208.4493-2-44-1': 'We also studied as a byproduct the necessary and sufficient condition for an arbitrary matrix [MATH] to be decomposed as in ([REF]).', '1208.4493-2-44-2': 'We showed that, in contrast to what has sometimes been claimed in the literature, the condition ([REF]) and the decomposition ([REF]) cannot be achieved in general but require some extra assumptions related to the existence and properties of square roots of matrices.', '1208.4493-2-44-3': 'These assumptions are gathered in Propositions 1 to 7 of the present work.', '1208.4493-2-44-4': 'An example where this result is particularly relevant is dRGT massive gravity.', '1208.4493-2-44-5': 'Indeed, this theory has been considered in two different frameworks: the first one uses two metrics [MATH] and [MATH] in such a way that the mass term involves the symmetric polynomials of [MATH] [CITATION], while the second one relies on two vielbeins [MATH] and [MATH] and the mass term is polynomial in these 1-forms [CITATION].', '1208.4493-2-44-6': 'A consequence of our results is that, in general, these two formulations are not equivalent.', '1208.4493-2-44-7': 'They become so only when condition ([REF]) is satisfied.', '1208.4493-2-44-8': 'In a region of parameter space it has been shown in [CITATION] that the above condition holds as a consequence of the equations of motion, and thus the equivalence is true dynamically.', '1208.4493-2-44-9': 'In the complementary parameter space region however, this is not true in general and it is even possible that the real square-root [MATH] does not exist.', '1208.4493-2-45-0': 'We also showed that, in general, in the 4 dimensional case, it is enough to assume that the matrix [MATH] admits a real square root, in order to satisfy a sufficient condition for ([REF]) to be true.', '1208.4493-2-45-1': ""However, for general theories with two metrics, this assumption may be violated dynamically as can be seen explicitly from the example of two decoupled metrics obeying Einstein's equations.""}","[['1208.4493-1-15-0', '1208.4493-2-14-0'], ['1208.4493-1-9-0', '1208.4493-2-8-0'], ['1208.4493-1-9-1', '1208.4493-2-8-1'], ['1208.4493-1-9-2', '1208.4493-2-8-2'], ['1208.4493-1-21-0', '1208.4493-2-20-0'], ['1208.4493-1-21-1', '1208.4493-2-20-1'], ['1208.4493-1-21-2', '1208.4493-2-20-2'], ['1208.4493-1-22-0', '1208.4493-2-21-0'], ['1208.4493-1-22-1', '1208.4493-2-21-1'], ['1208.4493-1-22-2', '1208.4493-2-21-2'], ['1208.4493-1-22-3', '1208.4493-2-21-3'], ['1208.4493-1-22-4', '1208.4493-2-21-4'], ['1208.4493-1-22-5', '1208.4493-2-21-5'], ['1208.4493-1-22-6', '1208.4493-2-21-6'], ['1208.4493-1-22-7', '1208.4493-2-21-7'], ['1208.4493-1-17-0', '1208.4493-2-16-0'], ['1208.4493-1-17-1', '1208.4493-2-16-1'], ['1208.4493-1-17-2', '1208.4493-2-16-2'], ['1208.4493-1-17-3', '1208.4493-2-16-3'], ['1208.4493-1-17-4', '1208.4493-2-16-4'], ['1208.4493-1-17-5', '1208.4493-2-16-5'], ['1208.4493-1-17-6', '1208.4493-2-16-6'], ['1208.4493-1-17-7', '1208.4493-2-16-7'], 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['1208.4493-1-7-2', '1208.4493-2-6-2'], ['1208.4493-1-7-3', '1208.4493-2-6-3'], ['1208.4493-1-7-4', '1208.4493-2-6-4'], ['1208.4493-1-7-5', '1208.4493-2-6-5'], ['1208.4493-1-7-6', '1208.4493-2-6-6'], ['1208.4493-1-2-0', '1208.4493-2-1-0'], ['1208.4493-1-2-1', '1208.4493-2-1-1'], ['1208.4493-1-2-2', '1208.4493-2-1-2'], ['1208.4493-1-2-3', '1208.4493-2-1-3'], ['1208.4493-1-2-4', '1208.4493-2-1-4'], ['1208.4493-1-2-5', '1208.4493-2-1-5'], ['1208.4493-1-2-6', '1208.4493-2-1-6'], ['1208.4493-1-14-0', '1208.4493-2-13-0'], ['1208.4493-1-14-1', '1208.4493-2-13-1'], ['1208.4493-1-14-2', '1208.4493-2-13-2'], ['1208.4493-1-14-3', '1208.4493-2-13-3'], ['1208.4493-1-14-4', '1208.4493-2-13-4'], ['1208.4493-1-14-5', '1208.4493-2-13-5'], ['1208.4493-1-13-0', '1208.4493-2-12-0'], ['1208.4493-1-13-1', '1208.4493-2-12-1'], ['1208.4493-1-4-0', '1208.4493-2-3-0'], ['1208.4493-1-18-0', '1208.4493-2-17-0'], ['1208.4493-1-18-1', '1208.4493-2-17-1'], ['1208.4493-1-18-2', '1208.4493-2-17-2'], ['1208.4493-1-18-3', '1208.4493-2-17-3'], ['1208.4493-1-25-0', '1208.4493-2-24-0'], ['1208.4493-1-27-0', '1208.4493-2-26-0'], ['1208.4493-1-27-1', '1208.4493-2-26-1'], ['1208.4493-1-27-2', '1208.4493-2-26-2'], ['1208.4493-1-27-3', '1208.4493-2-26-3'], ['1208.4493-1-20-0', '1208.4493-2-19-0'], ['1208.4493-1-20-1', '1208.4493-2-19-1'], ['1208.4493-1-20-2', '1208.4493-2-19-2'], ['1208.4493-1-16-0', '1208.4493-2-15-0'], ['1208.4493-1-16-1', '1208.4493-2-15-1'], ['1208.4493-1-16-2', '1208.4493-2-15-2'], ['1208.4493-1-16-3', '1208.4493-2-15-3'], ['1208.4493-1-16-4', '1208.4493-2-15-4'], ['1208.4493-1-16-5', '1208.4493-2-15-5'], ['1208.4493-1-16-6', '1208.4493-2-15-6'], ['1208.4493-1-16-7', '1208.4493-2-15-7'], ['1208.4493-1-16-8', '1208.4493-2-15-8'], ['1208.4493-1-16-9', '1208.4493-2-15-9'], ['1208.4493-1-16-10', '1208.4493-2-15-10'], ['1208.4493-1-16-11', '1208.4493-2-15-11'], ['1208.4493-1-16-12', '1208.4493-2-15-12'], ['1208.4493-1-3-0', '1208.4493-2-2-0'], ['1208.4493-1-3-1', 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['1208.4493-1-40-4', '1208.4493-2-45-0'], ['1208.4493-1-34-2', '1208.4493-2-34-5'], ['1208.4493-1-34-7', '1208.4493-2-34-2']]","[['1208.4493-1-38-0', '1208.4493-2-31-0'], ['1208.4493-1-38-1', '1208.4493-2-31-1'], ['1208.4493-1-38-3', '1208.4493-2-31-4'], ['1208.4493-1-38-4', '1208.4493-2-31-5']]","['1208.4493-1-13-2', '1208.4493-2-12-2']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1208.4493,,,
astro-ph-9810425,"{'astro-ph-9810425-1-0-0': 'A numerical scheme is proposed for the solution of the three-dimensional radiative transfer equation with variable optical depth.', 'astro-ph-9810425-1-0-1': 'We show that time-dependent ray tracing is an attractive choice for simulations of astrophysical ionization fronts, particularly when one is interested in covering a wide range of optical depths within a 3D clumpy medium.', 'astro-ph-9810425-1-0-2': 'Our approach combines the explicit advection of radiation variables with the implicit solution of local rate equations given the radiation field at each point.', 'astro-ph-9810425-1-0-3': 'Our scheme is well suited to the solution of problems for which line transfer is not important, and could, in principle, be extended to those situations also.', 'astro-ph-9810425-1-0-4': 'This scheme allows us to calculate the propagation of supersonic ionization fronts into an inhomogeneous medium.', 'astro-ph-9810425-1-0-5': 'The approach can be easily implemented on a single workstation and also should be fully parallelizable.', 'astro-ph-9810425-1-1-0': '# Introduction', 'astro-ph-9810425-1-2-0': 'Understanding the effect of the radiation field on the thermal state of interstellar and intergalactic gas is important for many areas of astrophysics, and in particular for star and galaxy formation.', 'astro-ph-9810425-1-2-1': 'Of special interest for cosmological structure formation are the epoch of reionization of the Universe (Gnedin Ostriker 1997), the effects of self-shielding on the formation of disk and dwarf galaxies (Navarro Steinmetz 1997, Kepner et al. 1997) and the absorption properties of Ly[MATH] clouds [CITATION].', 'astro-ph-9810425-1-2-2': 'While our knowledge of the physics of large-scale structure and galaxy formation has benefited significantly from numerical N-body and gas-dynamical models, there is very little that has been done to include radiative transfer (RT) into these simulations.', 'astro-ph-9810425-1-2-3': 'Challenges seem to abound, not least of all, the fact that the intensity of radiation in general is a function of seven independent variables (three spatial coordinates, two angles, frequency and time).', 'astro-ph-9810425-1-2-4': 'While for many applications it has been possible to reduce the dimensionality (e.g. to build realistic stellar atmosphere models), the clumpy state of the interstellar or intergalactic medium does not provide any spatial symmetries.', 'astro-ph-9810425-1-2-5': 'Moreover, coupled equations of radiation hydrodynamics (RHD) have very complicated structure, and are often of mixed advection-diffusion type which makes it very difficult to solve them numerically.', 'astro-ph-9810425-1-2-6': 'Besides that, the radiation field in optically thin regions usually evolves at the speed of light, yielding an enormous gap of many orders of magnitude between the characteristic time-scales for a system.', 'astro-ph-9810425-1-3-0': 'One way to avoid the latter problem is to solve all equations on the fluid-flow time-scale.', 'astro-ph-9810425-1-3-1': 'While there are arguments which seem to preserve causality in such an approach [CITATION], even then the numerical solution is an incredibly difficult challenge [CITATION].', 'astro-ph-9810425-1-3-2': 'On the other hand, several astrophysical problems allow one to follow the system of interest on a radiation propagation time-scale, without imposing a prohibitively large number of time steps.', 'astro-ph-9810425-1-3-3': 'In the context of cosmological RT, we would like to resolve the characteristic distance between the sources of reionization.', 'astro-ph-9810425-1-3-4': 'Cold dark matter (CDM) cosmologies predict the collapse of the first baryonic objects as early as [MATH], with the typical Jeans mass of order [MATH] (see Haiman Loeb 1997, and references therein).', 'astro-ph-9810425-1-3-5': 'The corresponding comoving scale of fragmenting clouds is [MATH].', 'astro-ph-9810425-1-3-6': 'If stellar sources reside in primordial globular clusters of this mass, then for [MATH] the average separation between these objects is [MATH] (comoving).', 'astro-ph-9810425-1-3-7': 'For explicit schemes the Courant condition imposes a time-step [EQUATION]', 'astro-ph-9810425-1-3-8': 'The evolution from [MATH] to [MATH] takes [MATH] (assuming density parameter [MATH], and defining the Hubble constant to be [MATH]).', 'astro-ph-9810425-1-3-9': 'Substituting [MATH] and [MATH] into eq. ([REF]) yields the total number of time-steps in the range [MATH] over the entire course of evolution.', 'astro-ph-9810425-1-3-10': 'In other words, to resolve reionization by [MATH] stellar clusters, we need to compute [MATH] few tens of thousands of time-steps on average.', 'astro-ph-9810425-1-3-11': 'For [MATH] clusters the number of steps required will be ten times smaller.', 'astro-ph-9810425-1-3-12': 'A [MATH] grid will result in computational boxes of several [MATH] on a side.', 'astro-ph-9810425-1-3-13': 'A full cosmological radiative transfer simulation with boxes at least this big and for all the required timesteps has hardly been feasible in the past.', 'astro-ph-9810425-1-4-0': 'This is not the only challenge we face.', 'astro-ph-9810425-1-4-1': 'Since any two points can affect each other via the radiation field, even for a monochromatic problem, we must describe the propagation of the radiation field anisotropies in the full five-dimensional space.', 'astro-ph-9810425-1-4-2': 'Standard steady-state RT solvers, which have been widely used in stellar atmosphere models, are not efficient in this case.', 'astro-ph-9810425-1-4-3': 'Non-local thermodynamic equilibrium (NLTE) steady-state radiative transport relies on obtaining the numerical solution via an iterative process for the whole computational region at once, and is usually effective only for very simplified geometries.', 'astro-ph-9810425-1-4-4': 'Any refinement of the discretization grid and/or increase in the number of atomic rate equations to compute NLTE effects will necessarily result in an exponential increase in the number of iterations required to achieve the same accuracy.', 'astro-ph-9810425-1-4-5': 'On the other hand, the 3D solution of the steady-state transfer equation in the absence of any spatial symmetries can often be obtained with Monte Carlo methods [CITATION].', 'astro-ph-9810425-1-4-6': 'However, these methods demonstrate very slow convergence at higher resolutions and are hardly applicable if one is interested in following a time-dependent system.', 'astro-ph-9810425-1-5-0': 'The change in the degree of ionization in a low-density environment occurs on a radiation propagation time-scale [MATH].', 'astro-ph-9810425-1-5-1': 'To track ionization fronts (I-fronts) in this regime, it is best to apply a high-resolution shock-capturing scheme similar to those originally developed in fluid dynamics.', 'astro-ph-9810425-1-5-2': 'One possible approach is the direct numerical solution of the monochromatic photon Boltzmann equation in the 5-dimensional phase space (Razoumov, in preparation).', 'astro-ph-9810425-1-5-3': 'To allow for a trade-off between calculational speed (plus memory usage) and accuracy, a more conventional approach is to truncate the system of angle-averaged radiation moment equations at a fixed moment and to use some closure scheme to reconstruct the angle dependence of the intensity at each point in 3D space.', 'astro-ph-9810425-1-5-4': 'The method of variable Eddington factors first introduced by Auer Mihalas (1970) has been shown to produce very accurate closure for time-dependent problems in both 2D [CITATION] and 3D [CITATION].', 'astro-ph-9810425-1-5-5': 'However, to the best of our knowledge, all schemes employed so far for calculating the time-dependent variable Eddington factor were based on a steady-state reconstruction of the radiation field through all of the computational region at once, given the thermal state of material and level populations at each point.', 'astro-ph-9810425-1-5-6': 'Since advection (or spatial transport) of moments is still followed on the time-scale of typical changes in the ionizational balance of the system, this approximation certainly provides physically valid results, assuming that the reconstruction is being performed often enough.', 'astro-ph-9810425-1-5-7': 'However, the steady-steady closure relies on the iterative solution of a large system of non-linear equations, which becomes an exceedingly difficult problem, from the computational point of view, as one moves to higher spatial and angular resolution and to the inclusion of more complicated microphysics.', 'astro-ph-9810425-1-6-0': 'The goal of the present paper is to demonstrate that in the cosmological context it is possible and practical to solve the whole RT problem on a radiation propagation time-scale [MATH] - as opposed to the fluid-flow time-scale - and we present a simple technique which gives an accurate solution for the angle-dependent intensity in three spatial dimensions.', 'astro-ph-9810425-1-6-1': 'The scheme can track discontinuities accurately in 3D and is stable up to the Courant number of unity.', 'astro-ph-9810425-1-6-2': 'Since all advection of radiation variables is being done at [MATH], the scheme is well tailored to the numerical study of the propagation of I-fronts into a non-homogeneous medium (Razoumov, in preparation).', 'astro-ph-9810425-1-7-0': 'This paper is organized as follows.', 'astro-ph-9810425-1-7-1': 'In Section [REF] we briefly review the state of numerical RT in the study of reionization.', 'astro-ph-9810425-1-7-2': 'We then concentrate on methods for 5D numerical advection.', 'astro-ph-9810425-1-7-3': 'In Section [REF] we describe our numerical algorithm and we present the results of numerical tests in Section [REF].', 'astro-ph-9810425-1-7-4': 'Finally, in Section [REF] we discuss the next steps towards a realistic 3D RT simulation.', 'astro-ph-9810425-1-8-0': '# Formulation of the problem', 'astro-ph-9810425-1-9-0': 'It is believed that light from the first baryonic objects the first generation of stars and quasars at [MATH] led to a phase-like transition in the ionizational state of the Universe.', 'astro-ph-9810425-1-9-1': 'This process of reionization significantly affected the subsequent evolution of structure formation (Couchman Rees 1986).', 'astro-ph-9810425-1-9-2': ""In detail reionization did not happen at a single epoch, with details of 'pre-heating', percolation, helium ionization and other physical processes having been studied in great detail over the last decade (some recent contributions include Madau et al. 1997, Haiman Loeb 1997, Gnedin Ostriker 1997, Shapiro et al. 1998, Tajiri Umemura 1998)."", 'astro-ph-9810425-1-10-0': 'It now seems clear that the full solution of the problem requires a detailed treatment of the effects of RT.', 'astro-ph-9810425-1-10-1': 'To complicate matters, by the time of the first star formation, the small-scale density inhomogeneities have entered the non-linear regime [CITATION], and the medium was filled with clumpy structures.', 'astro-ph-9810425-1-10-2': 'The success of cosmological N-body and hydrodynamical models in quantifying the growth of these objects (e.g., Zhang et al. 1998) suggests that the next step will be to include the effects of global energy exchange by radiation.', 'astro-ph-9810425-1-10-3': 'Indeed, there is a need for time-dependent 3D RT models as numerical tools for understanding the effect of inhomogeneities in the dynamical evolution of the interstellar/intergalactic medium.', 'astro-ph-9810425-1-10-4': 'For instance, the ability of gas to cool down and form structures depends crucially on the ionizational state of a whole array of different chemical elements, which in turn directly depends on the local energy density of the radiation field.', 'astro-ph-9810425-1-11-0': 'The hydrogen component of the Universe is most likely ionized by photons just above the Lyman limit, because (1) the cross-section of photoionization drops as [MATH] at higher frequencies, and (2) the medium will be dominated by softer photons, even in the case of quasar reionization (when ionizing photons come mostly from diffuse Hii regions).', 'astro-ph-9810425-1-11-1': 'Therefore, we argue that monochromatic transfer will be a fairly good approximation in our models.', 'astro-ph-9810425-1-12-0': 'Several research groups have made significant progress in simulating 3D inhomogeneous reionization.', 'astro-ph-9810425-1-12-1': 'Recently, Umemura et al. (1998) calculated reionization from [MATH] to [MATH], solving the 3D steady-state RT equation along with the time-dependent ionization rate equations for hydrogen and helium.', 'astro-ph-9810425-1-12-2': 'The radiation field was integrated along spatial dimensions using the method of short characteristics [CITATION].', 'astro-ph-9810425-1-12-3': 'The steady-state solution implies the assumption that the radiation field adjusts instantaneously to any changes in the ionization profile.', 'astro-ph-9810425-1-12-4': 'One draw-back of this approach, however, is in low-density voids where there are probably enough Lyman photons to ionize every hydrogen atom, so that the velocity of I-fronts is simply equal to the speed of light.', 'astro-ph-9810425-1-12-5': 'Then the rate equations still have to be solved on the radiation propagation timescale.', 'astro-ph-9810425-1-12-6': 'Besides, implicit techniques in the presence of inhomogeneities will become exponentially complicated, if we want to solve time-dependent rate equations for multiple chemical species.', 'astro-ph-9810425-1-13-0': 'If reionization by quasars alone is ruled out (Madau 1998, however see Haiman Loeb 1998), then I-fronts will be caused by Lyman photons from low-luminosity stellar sources at high redshifts.', 'astro-ph-9810425-1-13-1': 'In this case the pressure gradient across the ionization zone is more likely to become important before the front is slowed down by the finite recombination time.', 'astro-ph-9810425-1-13-2': 'Norman et al. (1998) outline a scheme for incorporating inhomogeneous reionization with hydrodynamical effects which arise when the speed of I-fronts drops close to the local sound speed.', 'astro-ph-9810425-1-13-3': 'In their method the default time-step is dictated by the speed of the atomic processes, and the radiation field is reconstructed through an elliptic solver (a quasi-static approximation).', 'astro-ph-9810425-1-14-0': 'In the present paper we ignore hydrodynamical effects, concentrating on an efficient method to track supersonic I-fronts.', 'astro-ph-9810425-1-14-1': 'Our approach is to solve the time-dependent RT coupled with an implicit local solver for the rate equations.', 'astro-ph-9810425-1-14-2': 'This method gives the correct speed of front propagation and it also quickly converges to a steady-state solution for equilibrium systems.', 'astro-ph-9810425-1-14-3': 'However, we should note that until a detailed comparison is made between explicit advection (at the speed of light) and the implicit reconstruction (through an elliptic solver), it is difficult to judge which approach works best in simulating inhomogeneous reionization in detail.', 'astro-ph-9810425-1-15-0': 'Although radiation propagates with the speed of light and the intensity of radiation depends on five spatial variables, plus frequency and time, the RT equation is inherently simpler than the equations of compressible hydrodynamics, since its advection part is strictly linear.', 'astro-ph-9810425-1-15-1': 'Non-linearities are usually introduced when we are trying to reduce the dimensionality of the problem.', 'astro-ph-9810425-1-15-2': 'Much of the difficulty, thus, comes from inability to get decent numerical resolution in the 5D (or 6D with frequency) space with present-day computers.', 'astro-ph-9810425-1-16-0': 'In the current work we have attempted to develop an efficient method to describe the anisotropies in the monochromatic radiation field propagating through an inhomogeneous medium, which we now describe.', 'astro-ph-9810425-1-17-0': '[]The numerical technique', 'astro-ph-9810425-1-18-0': '## Time-dependent ray tracing', 'astro-ph-9810425-1-19-0': 'The RT equation (without cosmological terms) reads [EQUATION] where [MATH] is the intensity of radiation in direction [MATH] and [MATH] and [MATH] are the local emissivity and opacity.', 'astro-ph-9810425-1-19-1': 'In a static medium integration over all directions and frequencies inside some range [MATH], with corresponding weights, yields the system of moment equations, the first two of which are [EQUATION]', 'astro-ph-9810425-1-19-2': 'Here we have introduced the frequency-averaged opacities [EQUATION]', 'astro-ph-9810425-1-19-3': 'Here [MATH], [MATH] and [MATH] are the first three moments (zero, first and second rank tensors, respectively) of the specific intensity [MATH].', 'astro-ph-9810425-1-20-0': 'The basic idea of our technique is to use an upwind monotonic scheme to propagate 1D wavefronts along a large number of rays in 3D at the speed of light.', 'astro-ph-9810425-1-20-1': 'Following Stone Mihalas (1992), we apply an operator split explicit-implicit scheme, in which advection of radiation variables is treated explicitly and the atomic and molecular rate equations are solved implicitly and separately at each point.', 'astro-ph-9810425-1-20-2': ""The whole computational volume is covered uniformly with rays, which makes this scheme geometrically similar to the 'method of long characteristics' discussed in Stone, Mihalas Norman (1992)."", 'astro-ph-9810425-1-20-3': 'Since the advection part on the left-hand side of eq. ([REF]) is strictly linear, the simplest way to propagate intensities is just to shift wavefronts by one grid zone at each time step accounting for sources and sinks of radiation.', 'astro-ph-9810425-1-20-4': 'Alternatively, one could use the third-order-accurate piecewise parabolic advection (PPA) of Stone Mihalas (1992).', 'astro-ph-9810425-1-20-5': 'In either case we can track sharp discontinuities in 1D with very little numerical diffusion, and, therefore, our approach is well suited to the calculation of I-fronts.', 'astro-ph-9810425-1-21-0': 'At each new time step, we project 1D intensities onto a 3D grid to reconstruct the mean energy density at each point.', 'astro-ph-9810425-1-21-1': 'This is the most demanding operation from the computational point of view, since at each of our [MATH] points we have to deal with the angular dependence of the radiation field.', 'astro-ph-9810425-1-21-2': 'When this update is done, we solve the matter-radiation interaction equations implicitly to compute the local level populations.', 'astro-ph-9810425-1-21-3': 'This gives us the 3D distributions of emissivity and opacity which are then mapped back to the rays and used in the advection scheme at the next time-step.', 'astro-ph-9810425-1-21-4': 'This simple scheme which we will refer to as time-dependent ray tracing can be used as a stand-alone solver, or as a closure scheme for the system of moment equations through the use of variable Eddington factors (as in Stone et al. 1992).', 'astro-ph-9810425-1-22-0': 'In the absence of any sinks and sources of radiation, the intensity is conserved exactly along each ray.', 'astro-ph-9810425-1-22-1': 'Since the number of rays does not vary with time, our method guarantees exact conservation of the radiation energy in 3D.', 'astro-ph-9810425-1-23-0': 'One advantage of the use of radiation moments is that the advection mechanism is essentially reduced to 3D, and it is relatively straightforward to implement the multi-dimensional conservation scheme for the linear advection part of the moment equations ([REF]) and ([REF]).', 'astro-ph-9810425-1-23-1': 'Then one could use a much denser spatial grid for the solution of the moment equations, and a relatively course grid for the angular reconstruction of the intensity of radiation via ray tracing.', 'astro-ph-9810425-1-23-2': 'In practice, however, we have found that the mismatch between the spatial resolutions of the moment solver and of the ray tracing usually leads to numerical instabilities.', 'astro-ph-9810425-1-23-3': 'In what follows, we consider ray tracing only as a stand-alone solver, and we will describe the merging of the moment solver and the ray tracing elsewhere (Razoumov, in preparation).', 'astro-ph-9810425-1-24-0': 'The most obvious draw-back of time-dependent ray tracing is the need to use a large number of rays to obtain an accurate description of the radiation field.', 'astro-ph-9810425-1-24-1': 'However, efficient placing of the rays can significantly reduce the number of operations.', 'astro-ph-9810425-1-24-2': 'Ultimately we are interested in getting a solution for the mean radiation energy density and material properties on a 3D rectangular grid.', 'astro-ph-9810425-1-24-3': 'Instead of shooting rays though each grid node in 3D, we choose to cover the whole computational volume with a separate grid of rays which is uniform both in space and in angular directions.', 'astro-ph-9810425-1-24-4': 'To reduce the Poisson error associated with a Monte-Carlo integration, we use high-order interpolation techniques to exchange data between the two grids.', 'astro-ph-9810425-1-24-5': 'Since the rays do not pass exactly through 3D grid points, we use four rays in any given direction to compute the angular-dependent intensity.', 'astro-ph-9810425-1-24-6': 'At each point on our 3D rectangular mesh we assume a piece-wise linear dependence of the intensity [MATH] on two angles, [MATH] and [MATH].', 'astro-ph-9810425-1-24-7': 'We then integrate the intensity over [MATH] with appropriate weights to get scalar quantities at each point.', 'astro-ph-9810425-1-24-8': 'The quadrature terms for the integration are modified to allow for the non-orthogonal angular grid and are calculated once, at the beginning of the simulation.', 'astro-ph-9810425-1-25-0': 'Another - perhaps, a better - way of coupling angular and spatial variations of the intensity may be an extension of the Spherical Harmonics Discrete Ordinate Method [CITATION].', 'astro-ph-9810425-1-25-1': 'For steady-state transfer problems, instead of storing the radiation field, this method stores the source function as a spherical harmonic series at each point.', 'astro-ph-9810425-1-25-2': 'Although the direct implementation of this technique for time-dependent problems is probably not realistic, due to the lookback time (i.e. the finite speed of light propagation), the spherical harmonic representation of the radiation field might require less storage and might result in smoother angular dependence as compared with a pure ray tracing approach.', 'astro-ph-9810425-1-26-0': '## Local chemistry equations', 'astro-ph-9810425-1-27-0': 'Abel et al. (1997) presented a detailed method for computing non-equilibrium primordial chemistry.', 'astro-ph-9810425-1-27-1': 'Since in our calculation all advection of radiation variables is performed explicitly, we can solve NLTE rate equations separately at each point.', 'astro-ph-9810425-1-27-2': 'This makes it very easy to implement an implicit solver for all atomic and molecular processes.', 'astro-ph-9810425-1-28-0': 'Following the primordial chemistry recipe of Abel et al. (1997) and Anninos et al. (1997), rate equations for nine species (H, H[MATH], H[MATH], H[MATH], H[MATH], He, He[MATH], He[MATH] and e[MATH]) have been included into the model.', 'astro-ph-9810425-1-28-1': 'The resulting six equations can be stored in the compact form: [EQUATION] with indices corresponding to individual species.', 'astro-ph-9810425-1-28-2': 'To demonstrate the capabilities of explicit advection, we have here reduced this set of equations to just photoionization and radiative recombination in a pure hydrogen medium.', 'astro-ph-9810425-1-28-3': 'The time evolution of the degree of ionization is given simply by [EQUATION] with emissivity and opacity [EQUATION]', 'astro-ph-9810425-1-28-4': ""The full recombination coefficient [EQUATION] is the sum of recombination coefficients to the ground state ([MATH]) and to all levels above the ground state ([MATH], the 'case B' recombination coefficient), [MATH] is the frequency just above the Lyman limit, and we assume that recombinations in Lyman lines occur on a short timescale compared to [MATH]."", 'astro-ph-9810425-1-28-5': 'This simple notation ensures radiation energy conservation in eq. ([REF]) for pure scattering of Lyman continuum photons (i.e., when [MATH]).', 'astro-ph-9810425-1-29-0': '# Tests', 'astro-ph-9810425-1-30-0': 'For all of our test runs, except the study of the shadow behind a neutral clump in Sec. ([REF]), we set up a numerical grid with dimensions [MATH].', 'astro-ph-9810425-1-30-1': 'The angular resolution [MATH] was chosen to match the equivalent resolution of [MATH] data points for 5D advection.', 'astro-ph-9810425-1-30-2': 'There are [MATH] rays passing in the immediate neighbourhood ([MATH]th of the total computational volume) of each 3D grid cell, each ray containing [MATH] grid nodes.', 'astro-ph-9810425-1-30-3': 'Thus, in 5D we obtain the equivalent resolution of [MATH] data points.', 'astro-ph-9810425-1-31-0': '## An isolated Stromgren sphere in the presence of a density gradient', 'astro-ph-9810425-1-32-0': 'The simplest test possible is that of a single, isolated Stromgren sphere expanding around a point source of ionizing radiation.', 'astro-ph-9810425-1-32-1': 'In the context of quasar reionization for a homogeneous medium the exact solution was given by Shapiro (1986).', 'astro-ph-9810425-1-32-2': 'We consider this test so easy to pass that the results would prove uninformative.', 'astro-ph-9810425-1-32-3': 'Instead, we put a point source of radiation into a density gradient along one of the principal axes of the cube.', 'astro-ph-9810425-1-32-4': 'In the absence of diffuse radiation from Hii regions ([MATH]) the only ionizing photons come directly from the source in the centre, in which case the shape of the ionized bubble would be a simple superposition of Stromgren spheres with radii [MATH] varying with the azimuthal angle [MATH] and given by the classical solution [CITATION] [EQUATION] where [MATH] is the photon production rate of the central source.', 'astro-ph-9810425-1-32-5': 'For an exponential density gradient along the y-axis [EQUATION] ([MATH], [MATH] being the hydrogen densities on the opposite faces of the cube), the equilibrium Stromgren radius [MATH] is given by a simple equation [EQUATION] where', 'astro-ph-9810425-1-33-0': '[EQUATION].', 'astro-ph-9810425-1-34-0': 'In Fig. [REF] we plot a time sequence of models with ionization by a central source without scattering of Lyman photons ([MATH]) and with complete scattering ([MATH]).', 'astro-ph-9810425-1-34-1': 'The numerical solution without scattering appears to be very close to the exact one.', 'astro-ph-9810425-1-34-2': 'The sharp transition layer between the ionized and the neutral regions in the low optical depth regime indicates that, indeed, the scheme introduces very little numerical diffusion even when extended to 3D.', 'astro-ph-9810425-1-34-3': 'There is slight asymmetry between the left- and the right-hand sides of the I-front, which is left in the algorithm intentionally to demonstrate the effect of the finite angular resolution (Fig. [REF]).', 'astro-ph-9810425-1-35-0': 'Ideally, one would like to place more rays going through individual point sources of radiation.', 'astro-ph-9810425-1-35-1': 'However, in a dynamically changing environment the number and positions of sources will vary with time.', 'astro-ph-9810425-1-35-2': 'There is nothing in our scheme that precludes allocation of new rays as new sources appear in the volume.', 'astro-ph-9810425-1-35-3': 'As long as all rays employed so far are kept in the model, we can guarantee the exact conservation of energy.', 'astro-ph-9810425-1-35-4': 'Methods could probably be developed to remove rays when ionizing sources shut off.', 'astro-ph-9810425-1-35-5': 'Another and, in our mind, more promising way to deal with evolving point sources is to use the 5D solution of the monochromatic photon Boltzmann equation separately for those sources, and the ray tracing model outlined in this paper for the background radiation.', 'astro-ph-9810425-1-36-0': '## Ionization in the presence of a UV background', 'astro-ph-9810425-1-37-0': 'The uniform coverage of the whole volume with rays implies that extended sources of radiation will be represented statistically much better than point sources.', 'astro-ph-9810425-1-37-1': 'A simple test mimicing the evolution of dense clouds in the presence of ionizing radiation is to enclose the computational region in an isotropic bath of photons.', 'astro-ph-9810425-1-37-2': 'The simplest way to accomplish this is just to set up a uniform, isotropically glowing boundary at the edges of the cube at [MATH].', 'astro-ph-9810425-1-37-3': 'An effective demonstration of time-dependent ray tracing would be its ability to deal with any distribution of state variables within the simulation volume.', 'astro-ph-9810425-1-37-4': ""For this test, we set up a density condensation shaped as the acronym for 'radiation hydrodynamics' (RHD), with a density [MATH] times that of the ambient homogeneous medium."", 'astro-ph-9810425-1-37-5': 'Fig. [REF] shows the result of this run.', 'astro-ph-9810425-1-37-6': 'Most of the low-density environment is ionized on the radiation propagation timescale.', 'astro-ph-9810425-1-37-7': 'It takes much longer for ionizing photons to penetrate into the dense regions.', 'astro-ph-9810425-1-37-8': 'Whether these regions can be ionized on a timescale of interest, depends on the ratio of the recombination timescale to the flux of background radiation.', 'astro-ph-9810425-1-37-9': 'During partial ionization one can easily notice shadows in between the clouds.', 'astro-ph-9810425-1-37-10': ""One can also see ionization 'eating in' to the neutral zone, e.g. in the disappearance of the serifs on the letters at late times."", 'astro-ph-9810425-1-37-11': 'Note that the width of the ionization fronts does not usually exceed one grid zone (Fig. [REF]).', 'astro-ph-9810425-1-38-0': '## Diffuse radiation from Hii regions: shadows behind neutral clouds', 'astro-ph-9810425-1-39-0': 'Part of the ionizing radiation at high redshifts comes in the form of hydrogen Lyman continuum photons from recombinations in diffuse ionized regions.', 'astro-ph-9810425-1-39-1': 'The following test simulating the formation of shadow regions behind dense clouds at the resolution [MATH] was adapted from Canto, Steffen Shapiro (1998).', 'astro-ph-9810425-1-39-2': 'A neutral clump of radius [MATH] is being illuminated by a parallel flux [MATH] of stellar ionizing photons from one side.', 'astro-ph-9810425-1-39-3': 'A shadow behind the clump is being photoionized by secondary recombination photons from the surrounding Hii region (Fig. [REF]).', 'astro-ph-9810425-1-39-4': 'Neglecting hydrodynamical effects, the width [MATH] of the shadow region can be estimated using a simple two-stream approximation [CITATION]: [EQUATION] and the dimensionless parameter [MATH] is defined as [EQUATION]', 'astro-ph-9810425-1-39-5': 'For [MATH], recombination Lyman continuum photons from the illuminated region will eventually photoionize the shadow completely.', 'astro-ph-9810425-1-39-6': 'For [MATH], radiative losses through low-energy cascade recombination photons will stop the I-front, forming a neutral cylinder behind the dense clump.', 'astro-ph-9810425-1-39-7': 'Strictly speaking, equations ([REF])-([REF]) are valid only for a shadow completely photoionized by secondary photons, and should be viewed as an approximation to I-fronts driven by secondary photons.', 'astro-ph-9810425-1-39-8': 'In Fig. [REF] we plot the radius [MATH] of the shadow neutral region as a function of [MATH] in our 3D numerical models.', 'astro-ph-9810425-1-39-9': 'Due to the finite accuracy of our simulations we take the width of the neutral core of the shadow at a non-zero level [MATH], instead of the exact outer boundary of the neutral region at [MATH].', 'astro-ph-9810425-1-39-10': 'This approximation essentially shifts the position of the curve in Fig. [REF] along the [MATH]-axis, giving slightly different value for the flux of scattered photons.', 'astro-ph-9810425-1-39-11': 'For convenience, we also plot a sequence of solutions from equations ([REF])-([REF]) for different fluxes of scattered radiation (which are, in turn, proportional to [MATH]).', 'astro-ph-9810425-1-39-12': 'We find fairly close correspondence between the shape of the curve in our models and the analytic solution, within the limits of the simplifying assumptions made for the analytic solution.', 'astro-ph-9810425-1-40-0': '## Diffuse radiation from Hii regions: ionization of a central void', 'astro-ph-9810425-1-41-0': 'To demonstrate the ability of our scheme to handle more complicated situations, we also set up a model with ionization of a central low-density void by secondary, recombination photons.', 'astro-ph-9810425-1-41-1': 'The void region is surrounded by two nested cubes with opposite faces open.', 'astro-ph-9810425-1-41-2': 'The walls of the cubes are set to be [MATH] denser than the rest of the medium, and the ionizing UV flux is introduced at all faces of the computational volume.', 'astro-ph-9810425-1-42-0': 'Similar to the test problem of Section [REF], if [MATH], then the medium will be ionized completely, since there is a constant flux of primordial ionizing photons.', 'astro-ph-9810425-1-42-1': 'The speed of ionization depends on the values of [MATH], [MATH], [MATH] and [MATH].', 'astro-ph-9810425-1-42-2': 'Note, however, that if [MATH] is too high, the I-front will be very slow, since a large portion of the original ionizing photons are scattered back.', 'astro-ph-9810425-1-42-3': 'On the other hand, if [MATH] is too low, the I-front will propagate much faster in those regions where ionization is driven by primordial photons, but in shadowed regions there will be too few recombination photons.', 'astro-ph-9810425-1-42-4': 'Thus, it seems that the speed of ionization of the central void will be the highest at some intermediate [MATH].', 'astro-ph-9810425-1-42-5': 'In Fig. [REF] we demonstrate ionization of the void region for models with complete scattering and with no scattering at all.', 'astro-ph-9810425-1-43-0': 'As expected, for the no-scattering model ([MATH]) the central region remains neutral, since there is no direct path for the ionizing photons.', 'astro-ph-9810425-1-43-1': 'However, for the model which includes scattering ([MATH]), the central region eventually becomes ionized.', 'astro-ph-9810425-1-43-2': 'This demonstrates that our scheme is able to deal with re-scattering of the ionizing photons.', 'astro-ph-9810425-1-43-3': 'Beyond this simple test case, there are many astrophysical situations where progress can be made with our method.', 'astro-ph-9810425-1-43-4': 'For example, analytic solutions are often used, which are steady-state, and which assume a sharp boundary between the neutral and ionized zones.', 'astro-ph-9810425-1-43-5': 'Using our numerical techniques it should be possible to follow general systems with complex density inhomogeneities and regions of partial ionization.', 'astro-ph-9810425-1-44-0': '# Conclusions', 'astro-ph-9810425-1-45-0': 'Ultimately, the choice of numerical technique in radiative transport depends on the type of problem one is trying to solve.', 'astro-ph-9810425-1-45-1': 'The starting point in our discussion is that the photoionization time-scale in the low optical depth regime ([MATH]) is of order [MATH], suggesting that explicit advection might be a faster method in covering at least these regions.', 'astro-ph-9810425-1-45-2': 'We have demonstrated that numerical solution on a time-scale [MATH], in three dimensions, is possible with existing desktop hardware.', 'astro-ph-9810425-1-45-3': 'Compared to the elliptic-type solvers on the fluid-flow time-scale or the time-scale of atomic processes, explicit radiative advection produces very accurate results without the need to solve a large system of coupled non-linear elliptic equations.', 'astro-ph-9810425-1-45-4': 'The computing requirements with explicit advection grow linearly with the inclusion of new atomic and molecular rate equations, which is certainly not the case for quasi-static solvers.', 'astro-ph-9810425-1-45-5': 'Although it is feasible that the development of multigrid techniques for elliptic equations might actually approach similar scaling.', 'astro-ph-9810425-1-46-0': 'Using equation ([REF]) we can see that the entire history of reionization can be modeled with [MATH]-[MATH] time-steps (depending on the required resolution), which makes explicit advection an attractive choice for these calculations.', 'astro-ph-9810425-1-46-1': 'However, the efficiency of the explicit radiative solver has still to be explored.', 'astro-ph-9810425-1-46-2': 'Future work should include a detailed comparison between explicit advection and implicit reconstruction (through an elliptic solver), to demonstrate which method works best for calculating inhomogeneous reionization.', 'astro-ph-9810425-1-47-0': 'As we have demonstrated here, for certain problems, including the propagation of supersonic I-fronts, the Courant condition does not seem to impose prohibitively small time steps.', 'astro-ph-9810425-1-47-1': 'In this case the biggest challenge is to accurately describe anisotropies in the radiation field, i.e. to solve for inhomogeneous advection in the 5D phase space, in the presence of non-uniform sources and sinks of radiation.', 'astro-ph-9810425-1-47-2': 'Strictly speaking, the storage of one variable at, say, [MATH] data points requires about [MATH]GB of memory, which stretches the capabilities of top-end desktop workstations.', 'astro-ph-9810425-1-47-3': 'One attractive possibility for future exploration is to directly solve the monochromatic photon Boltzmann equation in 5D.', 'astro-ph-9810425-1-47-4': 'To demonstrate the feasibility of the numerical solution, however, among different methods, we here chose to concentrate on simple ray tracing at the speed of light.', 'astro-ph-9810425-1-47-5': 'The numerical approach we have used is completely conservative and produces very little numerical dissipation.', 'astro-ph-9810425-1-48-0': 'The global exchange of energy via the radiation field at a fixed wavelength is probably one of the easiest problems to solve in numerical RT.', 'astro-ph-9810425-1-48-1': 'It seems likely that in the cosmological context, a few years from now, with progress in computer technology, this problem will be routinely solved in three spatial dimensions with the sort of resolution obtained in modern, state-of-the-art hydrodynamical simulations ([MATH]-[MATH] three-dimensional data points).', 'astro-ph-9810425-1-49-0': 'On the other hand, the full solution of the RHD equations, retaining all [MATH] terms, is a much more complicated problem.', 'astro-ph-9810425-1-49-1': 'In this case one deals with frequency-dependent RT, and issues such as line transfer, broadening effects and spatial motions within the simulation box become important.', 'astro-ph-9810425-1-49-2': 'Nevertheless, we want to conclude that with a reasonable expenditure of computational resources, of the type available today, it is possible to numerically model many different aspects of the full 3D radiative transfer problem.', 'astro-ph-9810425-1-49-3': 'And we feel that the methods described here represent a significant and realizable step towards the goal of full cosmological RHD.', 'astro-ph-9810425-1-50-0': '# ACKNOWLEDGMENTS', 'astro-ph-9810425-1-51-0': 'We wish to thank Taishi Nakamoto for providing us with the results of reionization models from the University of Tsukuba ahead of publication.', 'astro-ph-9810425-1-51-1': 'A. R. would like to thank Jason R. Auman for numerous enlighting discussions, Gregory G. Fahlman for constant encouragement on this project, as well as Randall J. LeVeque for help with numerical methods for multidimensional conservation laws.', 'astro-ph-9810425-1-51-2': 'This work was supported by the Natural Sciences and Engineering Research Council of Canada.'}","{'astro-ph-9810425-2-0-0': 'A numerical scheme is proposed for the solution of the three-dimensional radiative transfer equation with variable optical depth.', 'astro-ph-9810425-2-0-1': 'We show that time-dependent ray tracing is an attractive choice for simulations of astrophysical ionization fronts, particularly when one is interested in covering a wide range of optical depths within a 3D clumpy medium.', 'astro-ph-9810425-2-0-2': 'Our approach combines the explicit advection of radiation variables with the implicit solution of local rate equations given the radiation field at each point.', 'astro-ph-9810425-2-0-3': 'Our scheme is well suited to the solution of problems for which line transfer is not important, and could, in principle, be extended to those situations also.', 'astro-ph-9810425-2-0-4': 'This scheme allows us to calculate the propagation of supersonic ionization fronts into an inhomogeneous medium.', 'astro-ph-9810425-2-0-5': 'The approach can be easily implemented on a single workstation and also should be fully parallelizable.', 'astro-ph-9810425-2-1-0': '# Introduction', 'astro-ph-9810425-2-2-0': 'Understanding the effect of the radiation field on the thermal state of interstellar and intergalactic gas is important for many areas of astrophysics, and in particular for star and galaxy formation.', 'astro-ph-9810425-2-2-1': 'Of special interest for cosmological structure formation are the epoch of reionization of the Universe (Gnedin Ostriker 1997), the effects of self-shielding on the formation of disk and dwarf galaxies (Navarro Steinmetz 1997, Kepner et al. 1997) and the absorption properties of Ly[MATH] clouds [CITATION].', 'astro-ph-9810425-2-2-2': 'While our knowledge of the physics of large-scale structure and galaxy formation has benefited significantly from numerical N-body and gas-dynamical models, there is very little that has been done to include radiative transfer (RT) into these simulations.', 'astro-ph-9810425-2-2-3': 'Challenges seem to abound, not least of all, the fact that the intensity of radiation in general is a function of seven independent variables (three spatial coordinates, two angles, frequency and time).', 'astro-ph-9810425-2-2-4': 'While for many applications it has been possible to reduce the dimensionality (e.g. to build realistic stellar atmosphere models), the clumpy state of the interstellar or intergalactic medium does not provide any spatial symmetries.', 'astro-ph-9810425-2-2-5': 'Moreover, coupled equations of radiation hydrodynamics (RHD) have very complicated structure, and are often of mixed advection-diffusion type which makes it very difficult to solve them numerically.', 'astro-ph-9810425-2-2-6': 'Besides that, the radiation field in optically thin regions usually evolves at the speed of light, yielding an enormous gap of many orders of magnitude between the characteristic time-scales for a system.', 'astro-ph-9810425-2-3-0': 'One way to avoid the latter problem is to solve all equations on the fluid-flow time-scale.', 'astro-ph-9810425-2-3-1': 'While there are arguments which seem to preserve causality in such an approach [CITATION], even then the numerical solution is an incredibly difficult challenge [CITATION].', 'astro-ph-9810425-2-3-2': 'On the other hand, several astrophysical problems allow one to follow the system of interest on a radiation propagation time-scale, without imposing a prohibitively large number of time steps.', 'astro-ph-9810425-2-3-3': 'In the context of cosmological RT, we would like to resolve the characteristic distance between the sources of reionization.', 'astro-ph-9810425-2-3-4': 'Cold dark matter (CDM) cosmologies predict the collapse of the first baryonic objects as early as [MATH], with the typical Jeans mass of order [MATH] (see Haiman Loeb 1997, and references therein).', 'astro-ph-9810425-2-3-5': 'The corresponding comoving scale of fragmenting clouds is [MATH].', 'astro-ph-9810425-2-3-6': 'If stellar sources reside in primordial globular clusters of this mass, then for [MATH] the average separation between these objects is [MATH] (comoving).', 'astro-ph-9810425-2-3-7': 'For explicit schemes the Courant condition imposes a time-step [EQUATION]', 'astro-ph-9810425-2-3-8': 'The evolution from [MATH] to [MATH] takes [MATH] (assuming density parameter [MATH], and defining the Hubble constant to be [MATH]).', 'astro-ph-9810425-2-3-9': 'Substituting [MATH] and [MATH] into eq. ([REF]) yields the total number of time-steps in the range [MATH] over the entire course of evolution.', 'astro-ph-9810425-2-3-10': 'In other words, to resolve reionization by [MATH] stellar clusters, we need to compute [MATH] few tens of thousands of time-steps on average.', 'astro-ph-9810425-2-3-11': 'For [MATH] clusters the number of steps required will be ten times smaller.', 'astro-ph-9810425-2-3-12': 'A [MATH] grid will result in computational boxes of several [MATH] on a side.', 'astro-ph-9810425-2-3-13': 'A full cosmological radiative transfer simulation with boxes at least this big and for all the required timesteps has hardly been feasible in the past.', 'astro-ph-9810425-2-4-0': 'This is not the only challenge we face.', 'astro-ph-9810425-2-4-1': 'Since any two points can affect each other via the radiation field, even for a monochromatic problem, we must describe the propagation of the radiation field anisotropies in the full five-dimensional space.', 'astro-ph-9810425-2-4-2': 'Standard steady-state RT solvers, which have been widely used in stellar atmosphere models, are not efficient in this case.', 'astro-ph-9810425-2-4-3': 'Non-local thermodynamic equilibrium (NLTE) steady-state radiative transport relies on obtaining the numerical solution via an iterative process for the whole computational region at once, and is usually effective only for very simplified geometries.', 'astro-ph-9810425-2-4-4': 'Any refinement of the discretization grid and/or increase in the number of atomic rate equations to compute NLTE effects will necessarily result in an exponential increase in the number of iterations required to achieve the same accuracy.', 'astro-ph-9810425-2-4-5': 'On the other hand, the 3D solution of the steady-state transfer equation in the absence of any spatial symmetries can often be obtained with Monte Carlo methods [CITATION].', 'astro-ph-9810425-2-4-6': 'However, these methods demonstrate very slow convergence at higher resolutions and are hardly applicable if one is interested in following a time-dependent system.', 'astro-ph-9810425-2-5-0': 'The change in the degree of ionization in a low-density environment occurs on a radiation propagation time-scale [MATH].', 'astro-ph-9810425-2-5-1': 'To track ionization fronts (I-fronts) in this regime, it is best to apply a high-resolution shock-capturing scheme similar to those originally developed in fluid dynamics.', 'astro-ph-9810425-2-5-2': 'One possible approach is the direct numerical solution of the monochromatic photon Boltzmann equation in the 5-dimensional phase space (Razoumov, in preparation).', 'astro-ph-9810425-2-5-3': 'To allow for a trade-off between calculational speed (plus memory usage) and accuracy, a more conventional approach is to truncate the system of angle-averaged radiation moment equations at a fixed moment and to use some closure scheme to reconstruct the angle dependence of the intensity at each point in 3D space.', 'astro-ph-9810425-2-5-4': 'The method of variable Eddington factors first introduced by Auer Mihalas (1970) has been shown to produce very accurate closure for time-dependent problems in both 2D [CITATION] and 3D [CITATION].', 'astro-ph-9810425-2-5-5': 'However, to the best of our knowledge, all schemes employed so far for calculating the time-dependent variable Eddington factor were based on a steady-state reconstruction of the radiation field through all of the computational region at once, given the thermal state of material and level populations at each point.', 'astro-ph-9810425-2-5-6': 'Since advection (or spatial transport) of moments is still followed on the time-scale of typical changes in the ionizational balance of the system, this approximation certainly provides physically valid results, assuming that the reconstruction is being performed often enough.', 'astro-ph-9810425-2-5-7': 'However, the steady-steady closure relies on the iterative solution of a large system of non-linear equations, which becomes an exceedingly difficult problem, from the computational point of view, as one moves to higher spatial and angular resolution and to the inclusion of more complicated microphysics.', 'astro-ph-9810425-2-6-0': 'The goal of the present paper is to demonstrate that in the cosmological context it is possible and practical to solve the whole RT problem on a radiation propagation time-scale [MATH] - as opposed to the fluid-flow time-scale - and we present a simple technique which gives an accurate solution for the angle-dependent intensity in three spatial dimensions.', 'astro-ph-9810425-2-6-1': 'The scheme can track discontinuities accurately in 3D and is stable up to the Courant number of unity.', 'astro-ph-9810425-2-6-2': 'Since all advection of radiation variables is being done at [MATH], the scheme is well tailored to the numerical study of the propagation of I-fronts into a non-homogeneous medium (Razoumov, in preparation).', 'astro-ph-9810425-2-7-0': 'This paper is organized as follows.', 'astro-ph-9810425-2-7-1': 'In Section [REF] we briefly review the state of numerical RT in the study of reionization.', 'astro-ph-9810425-2-7-2': 'We then concentrate on methods for 5D numerical advection.', 'astro-ph-9810425-2-7-3': 'In Section [REF] we describe our numerical algorithm and we present the results of numerical tests in Section [REF].', 'astro-ph-9810425-2-7-4': 'Finally, in Section [REF] we discuss the next steps towards a realistic 3D RT simulation.', 'astro-ph-9810425-2-8-0': '# Formulation of the problem', 'astro-ph-9810425-2-9-0': 'It is believed that light from the first baryonic objects the first generation of stars and quasars at [MATH] led to a phase-like transition in the ionizational state of the Universe.', 'astro-ph-9810425-2-9-1': 'This process of reionization significantly affected the subsequent evolution of structure formation (Couchman Rees 1986).', 'astro-ph-9810425-2-9-2': ""In detail reionization did not happen at a single epoch, with details of 'pre-heating', percolation, helium ionization and other physical processes having been studied in great detail over the last decade (some recent contributions include Madau et al. 1997, Haiman Loeb 1997, Gnedin Ostriker 1997, Shapiro et al. 1998, Tajiri Umemura 1998)."", 'astro-ph-9810425-2-10-0': 'It now seems clear that the full solution of the problem requires a detailed treatment of the effects of RT.', 'astro-ph-9810425-2-10-1': 'To complicate matters, by the time of the first star formation, the small-scale density inhomogeneities have entered the non-linear regime [CITATION], and the medium was filled with clumpy structures.', 'astro-ph-9810425-2-10-2': 'The success of cosmological N-body and hydrodynamical models in quantifying the growth of these objects (e.g., Zhang et al. 1998) suggests that the next step will be to include the effects of global energy exchange by radiation.', 'astro-ph-9810425-2-10-3': 'Indeed, there is a need for time-dependent 3D RT models as numerical tools for understanding the effect of inhomogeneities in the dynamical evolution of the interstellar/intergalactic medium.', 'astro-ph-9810425-2-10-4': 'For instance, the ability of gas to cool down and form structures depends crucially on the ionizational state of a whole array of different chemical elements, which in turn directly depends on the local energy density of the radiation field.', 'astro-ph-9810425-2-11-0': 'The hydrogen component of the Universe is most likely ionized by photons just above the Lyman limit, because (1) the cross-section of photoionization drops as [MATH] at higher frequencies, and (2) the medium will be dominated by softer photons, even in the case of quasar reionization (when ionizing photons come mostly from diffuse Hii regions).', 'astro-ph-9810425-2-11-1': 'Therefore, we argue that monochromatic transfer will be a fairly good approximation in our models.', 'astro-ph-9810425-2-12-0': 'Several research groups have made significant progress in simulating 3D inhomogeneous reionization.', 'astro-ph-9810425-2-12-1': 'Recently, Umemura et al. (1998) calculated reionization from [MATH] to [MATH], solving the 3D steady-state RT equation along with the time-dependent ionization rate equations for hydrogen and helium.', 'astro-ph-9810425-2-12-2': 'The radiation field was integrated along spatial dimensions using the method of short characteristics [CITATION].', 'astro-ph-9810425-2-12-3': 'The steady-state solution implies the assumption that the radiation field adjusts instantaneously to any changes in the ionization profile.', 'astro-ph-9810425-2-12-4': 'One draw-back of this approach, however, is in low-density voids where there are probably enough Lyman photons to ionize every hydrogen atom, so that the velocity of I-fronts is simply equal to the speed of light.', 'astro-ph-9810425-2-12-5': 'Then the rate equations still have to be solved on the radiation propagation timescale.', 'astro-ph-9810425-2-12-6': 'Besides, implicit techniques in the presence of inhomogeneities will become exponentially complicated, if we want to solve time-dependent rate equations for multiple chemical species.', 'astro-ph-9810425-2-13-0': 'If reionization by quasars alone is ruled out (Madau 1998, however see Haiman Loeb 1998), then I-fronts will be caused by Lyman photons from low-luminosity stellar sources at high redshifts.', 'astro-ph-9810425-2-13-1': 'In this case the pressure gradient across the ionization zone is more likely to become important before the front is slowed down by the finite recombination time.', 'astro-ph-9810425-2-13-2': 'Norman et al. (1998) outline a scheme for incorporating inhomogeneous reionization with hydrodynamical effects which arise when the speed of I-fronts drops close to the local sound speed.', 'astro-ph-9810425-2-13-3': 'In their method the default time-step is dictated by the speed of the atomic processes, and the radiation field is reconstructed through an elliptic solver (a quasi-static approximation).', 'astro-ph-9810425-2-14-0': 'In the present paper we ignore hydrodynamical effects, concentrating on an efficient method to track supersonic I-fronts.', 'astro-ph-9810425-2-14-1': 'Our approach is to solve the time-dependent RT coupled with an implicit local solver for the rate equations.', 'astro-ph-9810425-2-14-2': 'This method gives the correct speed of front propagation and it also quickly converges to a steady-state solution for equilibrium systems.', 'astro-ph-9810425-2-14-3': 'However, we should note that until a detailed comparison is made between explicit advection (at the speed of light) and the implicit reconstruction (through an elliptic solver), it is difficult to judge which approach works best in simulating inhomogeneous reionization in detail.', 'astro-ph-9810425-2-15-0': 'Although radiation propagates with the speed of light and the intensity of radiation depends on five spatial variables, plus frequency and time, the RT equation is inherently simpler than the equations of compressible hydrodynamics, since its advection part is strictly linear.', 'astro-ph-9810425-2-15-1': 'Non-linearities are usually introduced when we are trying to reduce the dimensionality of the problem.', 'astro-ph-9810425-2-15-2': 'Much of the difficulty, thus, comes from inability to get decent numerical resolution in the 5D (or 6D with frequency) space with present-day computers.', 'astro-ph-9810425-2-16-0': 'In the current work we have attempted to develop an efficient method to describe the anisotropies in the monochromatic radiation field propagating through an inhomogeneous medium, which we now describe.', 'astro-ph-9810425-2-17-0': '[]The numerical technique', 'astro-ph-9810425-2-18-0': '## Time-dependent ray tracing', 'astro-ph-9810425-2-19-0': 'The RT equation (without cosmological terms) reads [EQUATION] where [MATH] is the intensity of radiation in direction [MATH] and [MATH] and [MATH] are the local emissivity and opacity.', 'astro-ph-9810425-2-19-1': 'In a static medium integration over all directions and frequencies inside some range [MATH], with corresponding weights, yields the system of moment equations, the first two of which are [EQUATION]', 'astro-ph-9810425-2-19-2': 'Here we have introduced the frequency-averaged opacities [EQUATION]', 'astro-ph-9810425-2-19-3': 'Here [MATH], [MATH] and [MATH] are the first three moments (zero, first and second rank tensors, respectively) of the specific intensity [MATH].', 'astro-ph-9810425-2-20-0': 'The basic idea of our technique is to use an upwind monotonic scheme to propagate 1D wavefronts along a large number of rays in 3D at the speed of light.', 'astro-ph-9810425-2-20-1': 'Following Stone Mihalas (1992), we apply an operator split explicit-implicit scheme, in which advection of radiation variables is treated explicitly and the atomic and molecular rate equations are solved implicitly and separately at each point.', 'astro-ph-9810425-2-20-2': ""The whole computational volume is covered uniformly with rays, which makes this scheme geometrically similar to the 'method of long characteristics' discussed in Stone, Mihalas Norman (1992)."", 'astro-ph-9810425-2-20-3': 'Since the advection part on the left-hand side of eq. ([REF]) is strictly linear, the simplest way to propagate intensities is just to shift wavefronts by one grid zone at each time step accounting for sources and sinks of radiation.', 'astro-ph-9810425-2-20-4': 'Alternatively, one could use the third-order-accurate piecewise parabolic advection (PPA) of Stone Mihalas (1992).', 'astro-ph-9810425-2-20-5': 'In either case we can track sharp discontinuities in 1D with very little numerical diffusion, and, therefore, our approach is well suited to the calculation of I-fronts.', 'astro-ph-9810425-2-21-0': 'At each new time step, we project 1D intensities onto a 3D grid to reconstruct the mean energy density at each point.', 'astro-ph-9810425-2-21-1': 'This is the most demanding operation from the computational point of view, since at each of our [MATH] points we have to deal with the angular dependence of the radiation field.', 'astro-ph-9810425-2-21-2': 'When this update is done, we solve the matter-radiation interaction equations implicitly to compute the local level populations.', 'astro-ph-9810425-2-21-3': 'This gives us the 3D distributions of emissivity and opacity which are then mapped back to the rays and used in the advection scheme at the next time-step.', 'astro-ph-9810425-2-21-4': 'This simple scheme which we will refer to as time-dependent ray tracing can be used as a stand-alone solver, or as a closure scheme for the system of moment equations through the use of variable Eddington factors (as in Stone et al. 1992).', 'astro-ph-9810425-2-22-0': 'In the absence of any sinks and sources of radiation, the intensity is conserved exactly along each ray.', 'astro-ph-9810425-2-22-1': 'Since the number of rays does not vary with time, our method guarantees exact conservation of the radiation energy in 3D.', 'astro-ph-9810425-2-23-0': 'One advantage of the use of radiation moments is that the advection mechanism is essentially reduced to 3D, and it is relatively straightforward to implement the multi-dimensional conservation scheme for the linear advection part of the moment equations ([REF]) and ([REF]).', 'astro-ph-9810425-2-23-1': 'Then one could use a much denser spatial grid for the solution of the moment equations, and a relatively course grid for the angular reconstruction of the intensity of radiation via ray tracing.', 'astro-ph-9810425-2-23-2': 'In practice, however, we have found that the mismatch between the spatial resolutions of the moment solver and of the ray tracing usually leads to numerical instabilities.', 'astro-ph-9810425-2-23-3': 'In what follows, we consider ray tracing only as a stand-alone solver, and we will describe the merging of the moment solver and the ray tracing elsewhere (Razoumov, in preparation).', 'astro-ph-9810425-2-24-0': 'The most obvious draw-back of time-dependent ray tracing is the need to use a large number of rays to obtain an accurate description of the radiation field.', 'astro-ph-9810425-2-24-1': 'However, efficient placing of the rays can significantly reduce the number of operations.', 'astro-ph-9810425-2-24-2': 'Ultimately we are interested in getting a solution for the mean radiation energy density and material properties on a 3D rectangular grid.', 'astro-ph-9810425-2-24-3': 'Instead of shooting rays though each grid node in 3D, we choose to cover the whole computational volume with a separate grid of rays which is uniform both in space and in angular directions.', 'astro-ph-9810425-2-24-4': 'To reduce the Poisson error associated with a Monte-Carlo integration, we use high-order interpolation techniques to exchange data between the two grids.', 'astro-ph-9810425-2-24-5': 'Since the rays do not pass exactly through 3D grid points, we use four rays in any given direction to compute the angular-dependent intensity.', 'astro-ph-9810425-2-24-6': 'At each point on our 3D rectangular mesh we assume a piece-wise linear dependence of the intensity [MATH] on two angles, [MATH] and [MATH].', 'astro-ph-9810425-2-24-7': 'We then integrate the intensity over [MATH] with appropriate weights to get scalar quantities at each point.', 'astro-ph-9810425-2-24-8': 'The quadrature terms for the integration are modified to allow for the non-orthogonal angular grid and are calculated once, at the beginning of the simulation.', 'astro-ph-9810425-2-25-0': 'Another - perhaps, a better - way of coupling angular and spatial variations of the intensity may be an extension of the Spherical Harmonics Discrete Ordinate Method [CITATION].', 'astro-ph-9810425-2-25-1': 'For steady-state transfer problems, instead of storing the radiation field, this method stores the source function as a spherical harmonic series at each point.', 'astro-ph-9810425-2-25-2': 'Although the direct implementation of this technique for time-dependent problems is probably not realistic, due to the lookback time (i.e. the finite speed of light propagation), the spherical harmonic representation of the radiation field might require less storage and might result in smoother angular dependence as compared with a pure ray tracing approach.', 'astro-ph-9810425-2-26-0': '## Local chemistry equations', 'astro-ph-9810425-2-27-0': 'Abel et al. (1997) presented a detailed method for computing non-equilibrium primordial chemistry.', 'astro-ph-9810425-2-27-1': 'Since in our calculation all advection of radiation variables is performed explicitly, we can solve NLTE rate equations separately at each point.', 'astro-ph-9810425-2-27-2': 'This makes it very easy to implement an implicit solver for all atomic and molecular processes.', 'astro-ph-9810425-2-28-0': 'Following the primordial chemistry recipe of Abel et al. (1997) and Anninos et al. (1997), rate equations for nine species (H, H[MATH], H[MATH], H[MATH], H[MATH], He, He[MATH], He[MATH] and e[MATH]) have been included into the model.', 'astro-ph-9810425-2-28-1': 'The resulting six equations can be stored in the compact form: [EQUATION] with indices corresponding to individual species.', 'astro-ph-9810425-2-28-2': 'To demonstrate the capabilities of explicit advection, we have here reduced this set of equations to just photoionization and radiative recombination in a pure hydrogen medium.', 'astro-ph-9810425-2-28-3': 'The time evolution of the degree of ionization is given simply by [EQUATION] with emissivity and opacity [EQUATION]', 'astro-ph-9810425-2-28-4': ""The full recombination coefficient [EQUATION] is the sum of recombination coefficients to the ground state ([MATH]) and to all levels above the ground state ([MATH], the 'case B' recombination coefficient), [MATH] is the frequency just above the Lyman limit, and we assume that recombinations in Lyman lines occur on a short timescale compared to [MATH]."", 'astro-ph-9810425-2-28-5': 'This simple notation ensures radiation energy conservation in eq. ([REF]) for pure scattering of Lyman continuum photons (i.e., when [MATH]).', 'astro-ph-9810425-2-29-0': '# Tests', 'astro-ph-9810425-2-30-0': 'For all of our test runs, except the study of the shadow behind a neutral clump in Sec. ([REF]), we set up a numerical grid with dimensions [MATH].', 'astro-ph-9810425-2-30-1': 'The angular resolution [MATH] was chosen to match the equivalent resolution of [MATH] data points for 5D advection.', 'astro-ph-9810425-2-30-2': 'There are [MATH] rays passing in the immediate neighbourhood ([MATH]th of the total computational volume) of each 3D grid cell, each ray containing [MATH] grid nodes.', 'astro-ph-9810425-2-30-3': 'Thus, in 5D we obtain the equivalent resolution of [MATH] data points.', 'astro-ph-9810425-2-31-0': '## An isolated Stromgren sphere in the presence of a density gradient', 'astro-ph-9810425-2-32-0': 'The simplest test possible is that of a single, isolated Stromgren sphere expanding around a point source of ionizing radiation.', 'astro-ph-9810425-2-32-1': 'In the context of quasar reionization for a homogeneous medium the exact solution was given by Shapiro (1986).', 'astro-ph-9810425-2-32-2': 'We consider this test so easy to pass that the results would prove uninformative.', 'astro-ph-9810425-2-32-3': 'Instead, we put a point source of radiation into a density gradient along one of the principal axes of the cube.', 'astro-ph-9810425-2-32-4': 'In the absence of diffuse radiation from Hii regions ([MATH]) the only ionizing photons come directly from the source in the centre, in which case the shape of the ionized bubble would be a simple superposition of Stromgren spheres with radii [MATH] varying with the azimuthal angle [MATH] and given by the classical solution [CITATION] [EQUATION] where [MATH] is the photon production rate of the central source.', 'astro-ph-9810425-2-32-5': 'For an exponential density gradient along the y-axis [EQUATION] ([MATH], [MATH] being the hydrogen densities on the opposite faces of the cube), the equilibrium Stromgren radius [MATH] is given by a simple equation [EQUATION] where', 'astro-ph-9810425-2-33-0': '[EQUATION].', 'astro-ph-9810425-2-34-0': 'In Fig. [REF] we plot a time sequence of models with ionization by a central source without scattering of Lyman photons ([MATH]) and with complete scattering ([MATH]).', 'astro-ph-9810425-2-34-1': 'The numerical solution without scattering appears to be very close to the exact one.', 'astro-ph-9810425-2-34-2': 'The sharp transition layer between the ionized and the neutral regions in the low optical depth regime indicates that, indeed, the scheme introduces very little numerical diffusion even when extended to 3D.', 'astro-ph-9810425-2-34-3': 'There is slight asymmetry between the left- and the right-hand sides of the I-front, which is left in the algorithm intentionally to demonstrate the effect of the finite angular resolution (Fig. [REF]).', 'astro-ph-9810425-2-35-0': 'Ideally, one would like to place more rays going through individual point sources of radiation.', 'astro-ph-9810425-2-35-1': 'However, in a dynamically changing environment the number and positions of sources will vary with time.', 'astro-ph-9810425-2-35-2': 'There is nothing in our scheme that precludes allocation of new rays as new sources appear in the volume.', 'astro-ph-9810425-2-35-3': 'As long as all rays employed so far are kept in the model, we can guarantee the exact conservation of energy.', 'astro-ph-9810425-2-35-4': 'Methods could probably be developed to remove rays when ionizing sources shut off.', 'astro-ph-9810425-2-35-5': 'Another and, in our mind, more promising way to deal with evolving point sources is to use the 5D solution of the monochromatic photon Boltzmann equation separately for those sources, and the ray tracing model outlined in this paper for the background radiation.', 'astro-ph-9810425-2-36-0': '## Ionization in the presence of a UV background', 'astro-ph-9810425-2-37-0': 'The uniform coverage of the whole volume with rays implies that extended sources of radiation will be represented statistically much better than point sources.', 'astro-ph-9810425-2-37-1': 'A simple test mimicing the evolution of dense clouds in the presence of ionizing radiation is to enclose the computational region in an isotropic bath of photons.', 'astro-ph-9810425-2-37-2': 'The simplest way to accomplish this is just to set up a uniform, isotropically glowing boundary at the edges of the cube at [MATH].', 'astro-ph-9810425-2-37-3': 'An effective demonstration of time-dependent ray tracing would be its ability to deal with any distribution of state variables within the simulation volume.', 'astro-ph-9810425-2-37-4': ""For this test, we set up a density condensation shaped as the acronym for 'radiation hydrodynamics' (RHD), with a density [MATH] times that of the ambient homogeneous medium."", 'astro-ph-9810425-2-37-5': 'Fig. [REF] shows the result of this run.', 'astro-ph-9810425-2-37-6': 'Most of the low-density environment is ionized on the radiation propagation timescale.', 'astro-ph-9810425-2-37-7': 'It takes much longer for ionizing photons to penetrate into the dense regions.', 'astro-ph-9810425-2-37-8': 'Whether these regions can be ionized on a timescale of interest, depends on the ratio of the recombination timescale to the flux of background radiation.', 'astro-ph-9810425-2-37-9': 'During partial ionization one can easily notice shadows in between the clouds.', 'astro-ph-9810425-2-37-10': ""One can also see ionization 'eating in' to the neutral zone, e.g. in the disappearance of the serifs on the letters at late times."", 'astro-ph-9810425-2-37-11': 'Note that the width of the ionization fronts does not usually exceed one grid zone (Fig. [REF]).', 'astro-ph-9810425-2-38-0': '## Diffuse radiation from Hii regions: shadows behind neutral clouds', 'astro-ph-9810425-2-39-0': 'Part of the ionizing radiation at high redshifts comes in the form of hydrogen Lyman continuum photons from recombinations in diffuse ionized regions.', 'astro-ph-9810425-2-39-1': 'The following test simulating the formation of shadow regions behind dense clouds at the resolution [MATH] was adapted from Canto, Steffen Shapiro (1998).', 'astro-ph-9810425-2-39-2': 'A neutral clump of radius [MATH] is being illuminated by a parallel flux [MATH] of stellar ionizing photons from one side.', 'astro-ph-9810425-2-39-3': 'A shadow behind the clump is being photoionized by secondary recombination photons from the surrounding Hii region (Fig. [REF]).', 'astro-ph-9810425-2-39-4': 'Neglecting hydrodynamical effects, the width [MATH] of the shadow region can be estimated using a simple two-stream approximation [CITATION]: [EQUATION] and the dimensionless parameter [MATH] is defined as [EQUATION]', 'astro-ph-9810425-2-39-5': 'For [MATH], recombination Lyman continuum photons from the illuminated region will eventually photoionize the shadow completely.', 'astro-ph-9810425-2-39-6': 'For [MATH], radiative losses through low-energy cascade recombination photons will stop the I-front, forming a neutral cylinder behind the dense clump.', 'astro-ph-9810425-2-39-7': 'Strictly speaking, equations ([REF])-([REF]) are valid only for a shadow completely photoionized by secondary photons, and should be viewed as an approximation to I-fronts driven by secondary photons.', 'astro-ph-9810425-2-39-8': 'In Fig. [REF] we plot the radius [MATH] of the shadow neutral region as a function of [MATH] in our 3D numerical models.', 'astro-ph-9810425-2-39-9': 'Due to the finite accuracy of our simulations we take the width of the neutral core of the shadow at a non-zero level [MATH], instead of the exact outer boundary of the neutral region at [MATH].', 'astro-ph-9810425-2-39-10': 'This approximation essentially shifts the position of the curve in Fig. [REF] along the [MATH]-axis, giving slightly different value for the flux of scattered photons.', 'astro-ph-9810425-2-39-11': 'For convenience, we also plot a sequence of solutions from equations ([REF])-([REF]) for different fluxes of scattered radiation (which are, in turn, proportional to [MATH]).', 'astro-ph-9810425-2-39-12': 'We find fairly close correspondence between the shape of the curve in our models and the analytic solution, within the limits of the simplifying assumptions made for the analytic solution.', 'astro-ph-9810425-2-40-0': '## Diffuse radiation from Hii regions: ionization of a central void', 'astro-ph-9810425-2-41-0': 'To demonstrate the ability of our scheme to handle more complicated situations, we also set up a model with ionization of a central low-density void by secondary, recombination photons.', 'astro-ph-9810425-2-41-1': 'The void region is surrounded by two nested cubes with opposite faces open.', 'astro-ph-9810425-2-41-2': 'The walls of the cubes are set to be [MATH] denser than the rest of the medium, and the ionizing UV flux is introduced at all faces of the computational volume.', 'astro-ph-9810425-2-42-0': 'Similar to the test problem of Section [REF], if [MATH], then the medium will be ionized completely, since there is a constant flux of primordial ionizing photons.', 'astro-ph-9810425-2-42-1': 'The speed of ionization depends on the values of [MATH], [MATH], [MATH] and [MATH].', 'astro-ph-9810425-2-42-2': 'Note, however, that if [MATH] is too high, the I-front will be very slow, since a large portion of the original ionizing photons are scattered back.', 'astro-ph-9810425-2-42-3': 'On the other hand, if [MATH] is too low, the I-front will propagate much faster in those regions where ionization is driven by primordial photons, but in shadowed regions there will be too few recombination photons.', 'astro-ph-9810425-2-42-4': 'Thus, it seems that the speed of ionization of the central void will be the highest at some intermediate [MATH].', 'astro-ph-9810425-2-42-5': 'In Fig. [REF] we demonstrate ionization of the void region for models with complete scattering and with no scattering at all.', 'astro-ph-9810425-2-43-0': 'As expected, for the no-scattering model ([MATH]) the central region remains neutral, since there is no direct path for the ionizing photons.', 'astro-ph-9810425-2-43-1': 'However, for the model which includes scattering ([MATH]), the central region eventually becomes ionized.', 'astro-ph-9810425-2-43-2': 'This demonstrates that our scheme is able to deal with re-scattering of the ionizing photons.', 'astro-ph-9810425-2-43-3': 'Beyond this simple test case, there are many astrophysical situations where progress can be made with our method.', 'astro-ph-9810425-2-43-4': 'For example, analytic solutions are often used, which are steady-state, and which assume a sharp boundary between the neutral and ionized zones.', 'astro-ph-9810425-2-43-5': 'Using our numerical techniques it should be possible to follow general systems with complex density inhomogeneities and regions of partial ionization.', 'astro-ph-9810425-2-44-0': '# Conclusions', 'astro-ph-9810425-2-45-0': 'Ultimately, the choice of numerical technique in radiative transport depends on the type of problem one is trying to solve.', 'astro-ph-9810425-2-45-1': 'The starting point in our discussion is that the photoionization time-scale in the low optical depth regime ([MATH]) is of order [MATH], suggesting that explicit advection might be a faster method in covering at least these regions.', 'astro-ph-9810425-2-45-2': 'We have demonstrated that numerical solution on a time-scale [MATH], in three dimensions, is possible with existing desktop hardware.', 'astro-ph-9810425-2-45-3': 'Compared to the elliptic-type solvers on the fluid-flow time-scale or the time-scale of atomic processes, explicit radiative advection produces very accurate results without the need to solve a large system of coupled non-linear elliptic equations.', 'astro-ph-9810425-2-45-4': 'The computing requirements with explicit advection grow linearly with the inclusion of new atomic and molecular rate equations, which is certainly not the case for quasi-static solvers.', 'astro-ph-9810425-2-45-5': 'Although it is feasible that the development of multigrid techniques for elliptic equations might actually approach similar scaling.', 'astro-ph-9810425-2-46-0': 'Using equation ([REF]) we can see that the entire history of reionization can be modeled with [MATH]-[MATH] time-steps (depending on the required resolution), which makes explicit advection an attractive choice for these calculations.', 'astro-ph-9810425-2-46-1': 'However, the efficiency of the explicit radiative solver has still to be explored.', 'astro-ph-9810425-2-46-2': 'Future work should include a detailed comparison between explicit advection and implicit reconstruction (through an elliptic solver), to demonstrate which method works best for calculating inhomogeneous reionization.', 'astro-ph-9810425-2-47-0': 'As we have demonstrated here, for certain problems, including the propagation of supersonic I-fronts, the Courant condition does not seem to impose prohibitively small time steps.', 'astro-ph-9810425-2-47-1': 'In this case the biggest challenge is to accurately describe anisotropies in the radiation field, i.e. to solve for inhomogeneous advection in the 5D phase space, in the presence of non-uniform sources and sinks of radiation.', 'astro-ph-9810425-2-47-2': 'Strictly speaking, the storage of one variable at, say, [MATH] data points requires about [MATH]GB of memory, which stretches the capabilities of top-end desktop workstations.', 'astro-ph-9810425-2-47-3': 'One attractive possibility for future exploration is to directly solve the monochromatic photon Boltzmann equation in 5D.', 'astro-ph-9810425-2-47-4': 'To demonstrate the feasibility of the numerical solution, however, among different methods, we here chose to concentrate on simple ray tracing at the speed of light.', 'astro-ph-9810425-2-47-5': 'The numerical approach we have used is completely conservative and produces very little numerical dissipation.', 'astro-ph-9810425-2-48-0': 'The global exchange of energy via the radiation field at a fixed wavelength is probably one of the easiest problems to solve in numerical RT.', 'astro-ph-9810425-2-48-1': 'It seems likely that in the cosmological context, a few years from now, with progress in computer technology, this problem will be routinely solved in three spatial dimensions with the sort of resolution obtained in modern, state-of-the-art hydrodynamical simulations ([MATH]-[MATH] three-dimensional data points).', 'astro-ph-9810425-2-49-0': 'On the other hand, the full solution of the RHD equations, retaining all [MATH] terms, is a much more complicated problem.', 'astro-ph-9810425-2-49-1': 'In this case one deals with frequency-dependent RT, and issues such as line transfer, broadening effects and spatial motions within the simulation box become important.', 'astro-ph-9810425-2-49-2': 'Nevertheless, we want to conclude that with a reasonable expenditure of computational resources, of the type available today, it is possible to numerically model many different aspects of the full 3D radiative transfer problem.', 'astro-ph-9810425-2-49-3': 'And we feel that the methods described here represent a significant and realizable step towards the goal of full cosmological RHD.', 'astro-ph-9810425-2-50-0': '# ACKNOWLEDGMENTS', 'astro-ph-9810425-2-51-0': 'We wish to thank Taishi Nakamoto for providing us with the results of reionization models from the University of Tsukuba ahead of publication.', 'astro-ph-9810425-2-51-1': 'A. R. would like to thank Jason R. Auman for numerous enlighting discussions, Gregory G. Fahlman for constant encouragement on this project, as well as Randall J. LeVeque for help with numerical methods for multidimensional conservation laws.', 'astro-ph-9810425-2-51-2': 'This work was supported by the Natural Sciences and Engineering Research Council of Canada.'}","[['astro-ph-9810425-1-43-0', 'astro-ph-9810425-2-43-0'], ['astro-ph-9810425-1-43-1', 'astro-ph-9810425-2-43-1'], ['astro-ph-9810425-1-43-2', 'astro-ph-9810425-2-43-2'], ['astro-ph-9810425-1-43-3', 'astro-ph-9810425-2-43-3'], ['astro-ph-9810425-1-43-4', 'astro-ph-9810425-2-43-4'], ['astro-ph-9810425-1-43-5', 'astro-ph-9810425-2-43-5'], ['astro-ph-9810425-1-51-0', 'astro-ph-9810425-2-51-0'], ['astro-ph-9810425-1-51-1', 'astro-ph-9810425-2-51-1'], ['astro-ph-9810425-1-51-2', 'astro-ph-9810425-2-51-2'], ['astro-ph-9810425-1-6-0', 'astro-ph-9810425-2-6-0'], ['astro-ph-9810425-1-6-1', 'astro-ph-9810425-2-6-1'], ['astro-ph-9810425-1-6-2', 'astro-ph-9810425-2-6-2'], ['astro-ph-9810425-1-32-0', 'astro-ph-9810425-2-32-0'], ['astro-ph-9810425-1-32-1', 'astro-ph-9810425-2-32-1'], 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'astro-ph-9810425-3-61-0'], ['astro-ph-9810425-2-42-5', 'astro-ph-9810425-3-66-0'], ['astro-ph-9810425-2-28-3', 'astro-ph-9810425-3-34-0'], ['astro-ph-9810425-2-20-3', 'astro-ph-9810425-3-30-0']]",[],"[['astro-ph-9810425-2-41-2', 'astro-ph-9810425-3-64-2'], ['astro-ph-9810425-2-34-0', 'astro-ph-9810425-3-55-2'], ['astro-ph-9810425-2-34-1', 'astro-ph-9810425-3-55-3'], ['astro-ph-9810425-2-45-2', 'astro-ph-9810425-3-69-0'], ['astro-ph-9810425-2-45-4', 'astro-ph-9810425-3-69-3'], ['astro-ph-9810425-2-45-5', 'astro-ph-9810425-3-69-3'], ['astro-ph-9810425-2-23-3', 'astro-ph-9810425-3-44-3'], ['astro-ph-9810425-2-6-2', 'astro-ph-9810425-3-6-2'], ['astro-ph-9810425-2-43-3', 'astro-ph-9810425-3-67-3'], ['astro-ph-9810425-2-4-0', 'astro-ph-9810425-3-4-0'], ['astro-ph-9810425-2-3-10', 'astro-ph-9810425-3-3-10'], ['astro-ph-9810425-2-3-11', 'astro-ph-9810425-3-3-10'], ['astro-ph-9810425-2-24-6', 'astro-ph-9810425-3-25-0'], ['astro-ph-9810425-2-24-7', 'astro-ph-9810425-3-28-0'], ['astro-ph-9810425-2-24-8', 'astro-ph-9810425-3-29-0'], ['astro-ph-9810425-2-39-2', 'astro-ph-9810425-3-59-2'], ['astro-ph-9810425-2-39-12', 'astro-ph-9810425-3-62-0'], ['astro-ph-9810425-2-28-2', 'astro-ph-9810425-3-33-0'], ['astro-ph-9810425-2-20-0', 'astro-ph-9810425-3-19-4'], ['astro-ph-9810425-2-20-4', 'astro-ph-9810425-3-30-2']]",[],"['astro-ph-9810425-1-17-0', 'astro-ph-9810425-1-33-0', 'astro-ph-9810425-2-17-0', 'astro-ph-9810425-2-33-0', 'astro-ph-9810425-3-17-0', 'astro-ph-9810425-3-23-0', 'astro-ph-9810425-3-26-0', 'astro-ph-9810425-3-27-0', 'astro-ph-9810425-3-38-0', 'astro-ph-9810425-3-54-0']","{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}",https://arxiv.org/abs/astro-ph/9810425,"{'astro-ph-9810425-3-0-0': 'A numerical scheme is proposed for the solution of the three-dimensional radiative transfer equation with variable optical depth.', 'astro-ph-9810425-3-0-1': 'We show that time-dependent ray tracing is an attractive choice for simulations of astrophysical ionization fronts, particularly when one is interested in covering a wide range of optical depths within a 3D clumpy medium.', 'astro-ph-9810425-3-0-2': 'Our approach combines the explicit advection of radiation variables with the implicit solution of local rate equations given the radiation field at each point.', 'astro-ph-9810425-3-0-3': 'Our scheme is well suited to the solution of problems for which line transfer is not important, and could, in principle, be extended to those situations also.', 'astro-ph-9810425-3-0-4': 'This scheme allows us to calculate the propagation of supersonic ionization fronts into an inhomogeneous medium.', 'astro-ph-9810425-3-0-5': 'The approach can be easily implemented on a single workstation and also should be fully parallelizable.', 'astro-ph-9810425-3-1-0': '# Introduction', 'astro-ph-9810425-3-2-0': 'Understanding the effect of the radiation field on the thermal state of interstellar and intergalactic gas is important for many areas of astrophysics, and in particular for star and galaxy formation.', 'astro-ph-9810425-3-2-1': 'Of special interest for cosmological structure formation are the epoch of reionization of the Universe (Gnedin Ostriker 1997, Miralda-Escude et al. 1998), the effects of self-shielding on the formation of disk and dwarf galaxies (Navarro Steinmetz 1997, Kepner et al. 1997) and the absorption properties of Ly[MATH] clouds (Katz et al. 1996, Meiksin 1994).', 'astro-ph-9810425-3-2-2': 'While our knowledge of the physics of large-scale structure and galaxy formation has benefited significantly from numerical N-body and gas-dynamical models (see, e.g., Zhang et al. 1998, and references therein), there is very little that has been done to include radiative transfer (RT) into these simulations.', 'astro-ph-9810425-3-2-3': 'Challenges seem to abound, not least of all, the fact that the intensity of radiation in general is a function of seven independent variables (three spatial coordinates, two angles, frequency and time).', 'astro-ph-9810425-3-2-4': 'While for many applications it has been possible to reduce the dimensionality (e.g. to build realistic stellar atmosphere models), the clumpy state of the interstellar or intergalactic medium does not provide any spatial symmetries.', 'astro-ph-9810425-3-2-5': 'Moreover, coupled equations of radiation hydrodynamics (RHD) have very complicated structure, and are often of mixed advection-diffusion type which makes it very difficult to solve them numerically.', 'astro-ph-9810425-3-2-6': 'Besides that, the radiation field in optically thin regions usually evolves at the speed of light, yielding an enormous gap of many orders of magnitude between the characteristic time-scales for a system.', 'astro-ph-9810425-3-3-0': 'One way to avoid the latter problem is to solve all equations on the fluid-flow time-scale.', 'astro-ph-9810425-3-3-1': 'While there are arguments which seem to preserve causality in such an approach [CITATION], even then the numerical solution is an incredibly difficult challenge [CITATION].', 'astro-ph-9810425-3-3-2': 'On the other hand, several astrophysical problems allow one to follow the system of interest on a radiation propagation time-scale, without imposing a prohibitively large number of time steps.', 'astro-ph-9810425-3-3-3': 'In the context of cosmological RT, we would like to resolve the characteristic distance between the sources of reionization.', 'astro-ph-9810425-3-3-4': 'Cold dark matter (CDM) cosmologies predict the collapse of the first baryonic objects as early as [MATH], with the typical Jeans mass of order [MATH] (see Haiman Loeb 1997, and references therein).', 'astro-ph-9810425-3-3-5': 'The corresponding comoving scale of fragmenting clouds is [MATH].', 'astro-ph-9810425-3-3-6': 'If stellar sources reside in primordial globular clusters of this mass, then for [MATH] the average separation between these objects is [MATH] (comoving).', 'astro-ph-9810425-3-3-7': 'For explicit schemes the Courant condition imposes a time-step [EQUATION]', 'astro-ph-9810425-3-3-8': 'Thus evolution from [MATH] to [MATH] takes [MATH] (assuming density parameter [MATH], and defining the Hubble constant to be [MATH]).', 'astro-ph-9810425-3-3-9': 'Substituting [MATH] and [MATH] into eq. ([REF]) yields the total number of time-steps in the range [MATH] over the entire course of evolution.', 'astro-ph-9810425-3-3-10': 'In other words, to resolve reionization by [MATH] stellar clusters, we need to compute [MATH] few tens of thousands of time-steps on average, and for [MATH] clusters the number of steps required will be ten times smaller.', 'astro-ph-9810425-3-3-11': 'A [MATH] grid with the required resolution will result in computational boxes of several [MATH] on a side.', 'astro-ph-9810425-3-3-12': 'A full cosmological radiative transfer simulation with boxes at least this big and for all the required timesteps has not been feasible in the past.', 'astro-ph-9810425-3-4-0': 'This is far from the only challenge we face.', 'astro-ph-9810425-3-4-1': 'Since any two points can affect each other via the radiation field, even for a monochromatic problem, we must describe the propagation of the radiation field anisotropies in the full five-dimensional space.', 'astro-ph-9810425-3-4-2': 'Standard steady-state RT solvers, which have been widely used in stellar atmosphere models, are not efficient in this case.', 'astro-ph-9810425-3-4-3': 'Non-local thermodynamic equilibrium (NLTE) steady-state radiative transport relies on obtaining the numerical solution via an iterative process for the whole computational region at once, and is usually effective only for very simplified geometries.', 'astro-ph-9810425-3-4-4': 'Any refinement of the discretization grid and/or increase in the number of atomic rate equations to compute NLTE effects will necessarily result in an exponential increase in the number of iterations required to achieve the same accuracy.', 'astro-ph-9810425-3-4-5': 'On the other hand, the 3D solution of the steady-state transfer equation in the absence of any spatial symmetries can often be obtained with Monte Carlo methods [CITATION].', 'astro-ph-9810425-3-4-6': 'However, these methods demonstrate very slow convergence at higher resolutions and are hardly applicable if one is interested in following a time-dependent system.', 'astro-ph-9810425-3-5-0': 'The change in the degree of ionization in a low-density environment occurs on a radiation propagation time-scale [MATH].', 'astro-ph-9810425-3-5-1': 'To track ionization fronts (I-fronts) in this regime, it is best to apply a high-resolution shock-capturing scheme similar to those originally developed in fluid dynamics.', 'astro-ph-9810425-3-5-2': 'One possible approach is the direct numerical solution of the monochromatic photon Boltzmann equation in the 5-dimensional phase space (Razoumov, in preparation).', 'astro-ph-9810425-3-5-3': 'To allow for a trade-off between calculational speed (plus memory usage) and accuracy, a more conventional approach is to truncate the system of angle-averaged radiation moment equations at a fixed moment and to use some closure scheme to reconstruct the angle dependence of the intensity at each point in 3D space.', 'astro-ph-9810425-3-5-4': 'The method of variable Eddington factors first introduced by Auer Mihalas (1970) has been shown to produce very accurate closure for time-dependent problems in both 2D [CITATION] and 3D [CITATION].', 'astro-ph-9810425-3-5-5': 'However, to the best of our knowledge, all schemes employed so far for calculating the time-dependent variable Eddington factor were based on a steady-state reconstruction of the radiation field through all of the computational region at once, given the thermal state of material and level populations at each point.', 'astro-ph-9810425-3-5-6': 'Since advection (or spatial transport) of moments is still followed on the time-scale of typical changes in the ionizational balance of the system, this approximation certainly provides physically valid results, assuming that the reconstruction is being performed often enough.', 'astro-ph-9810425-3-5-7': 'However, the steady-steady closure relies on the iterative solution of a large system of non-linear equations, which becomes an exceedingly difficult problem, from the computational point of view, as one moves to higher spatial and angular resolution and to the inclusion of more complicated microphysics.', 'astro-ph-9810425-3-6-0': 'The goal of the present paper is to demonstrate that in the cosmological context it is possible and practical to solve the whole RT problem on a radiation propagation time-scale [MATH] - as opposed to the fluid-flow time-scale - and we present a simple technique which gives an accurate solution for the angle-dependent intensity in three spatial dimensions.', 'astro-ph-9810425-3-6-1': 'The scheme can track discontinuities accurately in 3D and is stable up to the Courant number of unity.', 'astro-ph-9810425-3-6-2': 'Since all advection of radiation variables is being done at [MATH], the scheme is well tailored to the numerical study of the propagation of I-fronts into a non-homogeneous medium with any optical depth, and gives very accurate results for scattering.', 'astro-ph-9810425-3-6-3': 'Rather than solve the proper chemistry equations applicable to cosmological structures, we adopt a simple toy model described below.', 'astro-ph-9810425-3-6-4': 'Similarly, in the current paper we do not make any attempt to model the thermal properties of the gas, concentrating just on efficient multidimensional advection techniques.', 'astro-ph-9810425-3-7-0': 'This paper is organized as follows.', 'astro-ph-9810425-3-7-1': 'In Section [REF] we briefly review the state of numerical RT in the study of reionization.', 'astro-ph-9810425-3-7-2': 'We then concentrate on methods for 5D numerical advection.', 'astro-ph-9810425-3-7-3': 'In Section [REF] we describe our numerical algorithm and we present the results of numerical tests in Section [REF].', 'astro-ph-9810425-3-7-4': 'Finally, in Section [REF] we discuss the next steps towards a realistic 3D RT simulation.', 'astro-ph-9810425-3-8-0': '# Formulation of the problem', 'astro-ph-9810425-3-9-0': 'It is believed that light from the first baryonic objects the first generation of stars and quasars at [MATH] led to a phase-like transition in the ionizational state of the Universe.', 'astro-ph-9810425-3-9-1': 'This process of reionization significantly affected the subsequent evolution of structure formation (Couchman Rees 1986).', 'astro-ph-9810425-3-9-2': ""In detail reionization did not happen at a single epoch, with details of 'pre-heating', percolation, helium ionization and other physical processes having been studied in great detail over the last decade (some recent contributions include Madau et al. 1997, Haiman Loeb 1997, Gnedin Ostriker 1997, Shapiro et al. 1998, Tajiri Umemura 1998)."", 'astro-ph-9810425-3-10-0': 'It now seems clear that the full solution of the problem requires a detailed treatment of the effects of RT.', 'astro-ph-9810425-3-10-1': 'To complicate matters, by the time of the first star formation, the small-scale density inhomogeneities had entered the non-linear regime [CITATION], and the medium was filled with clumpy structures.', 'astro-ph-9810425-3-10-2': 'The success of cosmological N-body and hydrodynamical models in quantifying the growth of these objects (e.g., Zhang et al. 1998) suggests that the next step will be to include the effects of global energy exchange by radiation.', 'astro-ph-9810425-3-10-3': 'Indeed, there is a need for time-dependent 3D RT models as numerical tools for understanding the effect of inhomogeneities in the dynamical evolution of the interstellar/intergalactic medium.', 'astro-ph-9810425-3-10-4': 'For instance, the ability of gas to cool down and form structures depends crucially on the ionizational state of a whole array of different chemical elements, which in turn directly depends on the local energy density of the radiation field.', 'astro-ph-9810425-3-11-0': 'The hydrogen component of the Universe is most likely ionized by photons just above the Lyman limit, because (1) the cross-section of photoionization drops as [MATH] at higher frequencies, and (2) the medium will be dominated by softer photons, even in the case of quasar reionization (when ionizing photons come mostly from diffuse Hii regions).', 'astro-ph-9810425-3-11-1': 'Therefore, we argue that either monochromatic or frequency-averaged transfer will be a fairly good approximation in our models.', 'astro-ph-9810425-3-12-0': 'Recently, the problem of simulating 3D inhomogeneous reionization with realistic radiative transfer has attracted considerable interest in the scientific community.', 'astro-ph-9810425-3-12-1': 'Umemura et al. (1998) calculated reionization from [MATH] to [MATH], solving the 3D steady-state RT equation along with the time-dependent ionization rate equations for hydrogen and helium.', 'astro-ph-9810425-3-12-2': 'The radiation field was integrated along spatial dimensions using the method of short characteristics [CITATION].', 'astro-ph-9810425-3-12-3': 'The steady-state solution implies the assumption that the radiation field adjusts instantaneously to any changes in the ionization profile.', 'astro-ph-9810425-3-12-4': 'One draw-back of this approach, however, is in low-density voids where there are probably enough Lyman photons to ionize every hydrogen atom, so that the velocity of I-fronts is simply equal to the speed of light.', 'astro-ph-9810425-3-12-5': 'Then the rate equations still have to be solved on the radiation propagation timescale.', 'astro-ph-9810425-3-12-6': 'Besides, implicit techniques in the presence of inhomogeneities will become exponentially complicated, if we want to solve time-dependent rate equations for multiple chemical species.', 'astro-ph-9810425-3-13-0': 'Norman et al. (1998) and Abel et al. (1998) present a scheme for solving the cosmological radiative transfer problem by decomposing the total radiation field into two parts: highly anisotropic direct ionizing radiation from point sources such as quasars and stellar clusters, and the diffuse component from recombinations in the photoionized gas.', 'astro-ph-9810425-3-13-1': 'In their method the direct ionizing radiation is being attenuated along a small number of rays, each of which is forced to pass through one of the few point sources within the simulation volume.', 'astro-ph-9810425-3-13-2': 'The diffuse part of the radiation field is found with a separate technique which can benefit from the nearly isotropic form of this component, for instance, through the use of the diffusion approximation.', 'astro-ph-9810425-3-13-3': 'Both solutions are obtained neglecting the time dependent term in the radiative transfer equation, with the default time-step dictated by the speed of the atomic processes.', 'astro-ph-9810425-3-14-0': 'If reionization by quasars alone is ruled out (Madau 1998, however see Haiman Loeb 1998), then I-fronts will be caused by Lyman photons from low-luminosity stellar sources at high redshifts.', 'astro-ph-9810425-3-14-1': 'In this case the pressure gradient across the ionization zone is more likely to become important before the front is slowed down by the finite recombination time.', 'astro-ph-9810425-3-14-2': 'In the present paper we ignore hydrodynamical effects, concentrating on an efficient method to track supersonic I-fronts.', 'astro-ph-9810425-3-14-3': 'Our approach is to solve the time-dependent RT coupled with an implicit local solver for the rate equations.', 'astro-ph-9810425-3-14-4': 'This method gives the correct speed of front propagation and it also quickly converges to a steady-state solution for equilibrium systems.', 'astro-ph-9810425-3-14-5': 'However, we should note that until a detailed comparison is made between explicit advection (at the speed of light) and the implicit reconstruction (through an elliptic solver), it is difficult to judge which approach works best in simulating inhomogeneous reionization in detail.', 'astro-ph-9810425-3-15-0': 'Although radiation propagates with the speed of light and the intensity of radiation depends on five spatial variables, plus frequency and time, the RT equation is inherently simpler than the equations of compressible hydrodynamics, since its advection part is strictly linear.', 'astro-ph-9810425-3-15-1': 'Non-linearities are usually introduced when we are trying to reduce the dimensionality of the problem.', 'astro-ph-9810425-3-15-2': 'Much of the difficulty thus comes from inability to get decent numerical resolution in the 5D (or 6D with frequency) space with present-day computers.', 'astro-ph-9810425-3-16-0': 'In the current work we have attempted to develop an efficient method to describe the anisotropies in the monochromatic radiation field propagating through an inhomogeneous medium, which we now describe.', 'astro-ph-9810425-3-17-0': '[]The numerical technique', 'astro-ph-9810425-3-18-0': 'The classical RT equation (without cosmological terms) reads [EQUATION] where [MATH] is the intensity of radiation in direction [MATH] and [MATH] and [MATH] are the local emissivity and opacity.', 'astro-ph-9810425-3-18-1': 'This equation is valid for those problems in which the light-crossing time across the computational volume [MATH] is much smaller than the Hubble time, and we can neglect the redshift effects within the simulation volume.', 'astro-ph-9810425-3-18-2': 'With proper boundary conditions one can easily account for the Doppler shift in the background radiation.', 'astro-ph-9810425-3-19-0': 'The basic idea of our technique is to solve eq. ([REF]) directly for the angle-dependent intensity [MATH] at each point.', 'astro-ph-9810425-3-19-1': 'The total radiation field at each point is divided into the direct (from ionizing sources) and diffuse (due to recombinations in the gas) components: [EQUATION] where [MATH] are the three indices in a rectangular grid.', 'astro-ph-9810425-3-19-2': 'The energy density [MATH] due to direct photons coming from point sources of ionization can be easily calculated on the 3D grid via summation over all sources (assuming that there are not too many of them within the volume): [EQUATION] is the optical depth, [MATH] is the physical distance between the current point and the source and [MATH] is the age of the source.', 'astro-ph-9810425-3-19-3': 'Note that the rate of emission of photons [MATH] can be modified to allow for variability of sources on short timescales.', 'astro-ph-9810425-3-19-4': 'For the diffuse component we use an upwind monotonic scheme to propagate 1D wavefronts [MATH] along a large number of rays in 3D at the speed of light.', 'astro-ph-9810425-3-19-5': 'Following Stone Mihalas (1992), we apply an operator split explicit-implicit scheme, in which advection of radiation variables is treated explicitly and the atomic and molecular rate equations are solved implicitly and separately at each point.', 'astro-ph-9810425-3-19-6': 'Unlike Abel et al. (1998), we use rays to track the diffuse component of the radiation field and the direct ionizing background radiation (streaming into the computational volume).', 'astro-ph-9810425-3-19-7': 'Since we need to draw rays essentially through every grid point in the 3D volume, at first glance this approach appears to have very large memory requirements.', 'astro-ph-9810425-3-19-8': 'However, efficient placing of the rays can significantly reduce the computational effort.', 'astro-ph-9810425-3-20-0': '## A uniform and isotropic grid of rays', 'astro-ph-9810425-3-21-0': 'At each time-step we are interested in getting a solution for the mean radiation energy density and material properties on a 3D [MATH] rectangular grid.', 'astro-ph-9810425-3-21-1': 'Instead of shooting rays though each grid node in 3D, we choose to cover the whole computational volume with a separate grid of rays which is uniform both in space and in angular directions.', 'astro-ph-9810425-3-21-2': 'Assuming that the computational volume corresponds to the range [MATH], we first construct a 2D rectangular base grid containing [MATH] nodes with coordinates with the origin at the centre of the cube', 'astro-ph-9810425-3-22-0': '[EQUATION] where the [MATH] factor ensures that the entire volume is covered with rays, and we shoot rays normal to the base grid through all of its grid points.', 'astro-ph-9810425-3-22-1': 'Note that the separation between 2D nodes is allowed to be larger than [MATH].', 'astro-ph-9810425-3-22-2': 'To cover the whole volume with rays, we then rotate the base grid by an angle [MATH] around the [MATH]-axis and by [MATH] around the [MATH]-axis, where the rotation angles are discretized to mimic an isotropic distribution of rays (with fewer azimuthal angles close to the poles)', 'astro-ph-9810425-3-23-0': '[EQUATION]', 'astro-ph-9810425-3-23-1': 'Only those rays which pass through the volume (and not all of them do!)', 'astro-ph-9810425-3-23-2': 'are stored in memory.', 'astro-ph-9810425-3-23-3': 'Let the total number of all possible orientations of the base grid be [EQUATION]', 'astro-ph-9810425-3-23-4': 'The resulting number of rays is significantly smaller than [MATH].', 'astro-ph-9810425-3-23-5': 'In fact, with [MATH] and [MATH]), we only require [MATH] rays.', 'astro-ph-9810425-3-24-0': 'Now, that we have the grid of rays and the 3D rectangular mesh, we have to specify the rules of interpolation between them.', 'astro-ph-9810425-3-24-1': 'Before we start our simulations, for each 3D grid point [MATH] we also store an array of the closest four rays going through its neighbourhood in the direction [MATH].', 'astro-ph-9810425-3-24-2': 'Since the rays do not pass exactly through 3D grid points, we use the values of the intensity on the four closest rays in that direction to compute the angular-dependent intensity.', 'astro-ph-9810425-3-24-3': 'Assume that for the point [MATH] the distances to the four closest rays going in the direction [MATH] are [MATH], [MATH], [MATH] and [MATH], respectively.', 'astro-ph-9810425-3-24-4': 'We project the point [MATH] onto these rays and read the values of the intensities, which we write as [MATH], [MATH], [MATH] and [MATH].', 'astro-ph-9810425-3-24-5': 'We then calculate the intensity at [MATH] in the direction [MATH] according to [EQUATION] where the weights [MATH] for [MATH] are set to zero if [MATH] rays were found in the immediate neighbourhood of [MATH].', 'astro-ph-9810425-3-24-6': 'This might be the case close to the edges of the computational volume; we shall comment more on this while discussing the boundary conditions.', 'astro-ph-9810425-3-24-7': ""The form of eq. ([REF]) was chosen specifically because: (1) if a ray labeled [MATH] happens to pass exactly through the point [MATH], then [MATH]; (2) if all four rays encompass the point, [MATH]; and (3) if [MATH] happens to be far from all four rays, then the resulting intensity will just be an average of the four [MATH]'s."", 'astro-ph-9810425-3-25-0': 'At each point on our 3D rectangular mesh we assume a piece-wise linear dependence of the intensity [MATH] on two angles, [MATH] and [MATH], [EQUATION] within a spherical rectangular element (or a spherical triangle adjacent to either of the poles) bounded by the angles [MATH], [MATH] in [MATH] and [MATH] and [MATH] in [MATH], with the rectangular grid defined as', 'astro-ph-9810425-3-26-0': '[EQUATION] and', 'astro-ph-9810425-3-27-0': '[EQUATION].', 'astro-ph-9810425-3-28-0': 'We then integrate the intensity over [MATH] with appropriate weights to get the scalar radiation energy density at each point [MATH] of the rectangular spherical grid [EQUATION] where the quadrature terms for the integration', 'astro-ph-9810425-3-29-0': '[EQUATION] are modified to allow for the non-orthogonal angular grid (eq. [REF]).', 'astro-ph-9810425-3-30-0': 'Since the advection part on the left-hand side of eq. ([REF]) is strictly linear, the simplest way to propagate intensities is just to shift wavefronts by one grid zone at each time-step, accounting for sources and sinks of radiation.', 'astro-ph-9810425-3-30-1': 'Assuming that all discretization points along each ray are strictly equidistant, the intensity at a point [MATH] is updated simply as [EQUATION]', 'astro-ph-9810425-3-30-2': 'Alternatively, one could take special care of the length of each ray segment contained within a particular 3D grid cell, and use a scheme similar to the third-order-accurate piecewise parabolic advection method (PPA) of Stone Mihalas (1992).', 'astro-ph-9810425-3-30-3': 'In either case we can track sharp discontinuities in 1D with very little numerical diffusion, and, therefore, our approach is well suited to the calculation of I-fronts.', 'astro-ph-9810425-3-31-0': '## Local chemistry equations', 'astro-ph-9810425-3-32-0': 'Since in our calculation all advection of radiation variables is performed explicitly, we can solve NLTE rate equations separately at each point.', 'astro-ph-9810425-3-32-1': 'This makes it relatively easy to implement an implicit solver for all atomic and molecular processes.', 'astro-ph-9810425-3-33-0': 'To demonstrate the capabilities of explicit advection, instead of solving the proper chemistry equations for multiple species with primordial chemical composition, we have here adopted a simple toy model with just photoionization and radiative recombination in a pure hydrogen medium.', 'astro-ph-9810425-3-33-1': 'The implicit solution of possibly stiff rate equations described below can be implemented in a similar manner for more realistic chemistry models.', 'astro-ph-9810425-3-34-0': 'The time evolution of the degree of ionization [MATH] is given simply by [EQUATION] with opacity [EQUATION]', 'astro-ph-9810425-3-34-1': 'Here [MATH] is the photoionization coefficient, [MATH] the energy density of ionizing radiation and [MATH] the recombination coefficient.', 'astro-ph-9810425-3-34-2': 'The correct expression for the total emissivity [MATH] of the gas can be obtained by considering conservation of the thermal energy density [MATH] for matter: [EQUATION] where all [MATH] symbols represent the change of variables during one time step.', 'astro-ph-9810425-3-34-3': ""The full recombination coefficient [EQUATION] is the sum of recombination coefficients to the ground state ([MATH]) and to all levels above the ground state ([MATH], the 'case B' recombination coefficient), [MATH] is the frequency just above the Lyman limit, and we assume that recombinations in Lyman lines occur on a short timescale compared to [MATH]."", 'astro-ph-9810425-3-34-4': 'Similarly, the full emissivity (or gas energy loss through recombinations) is [EQUATION] where [MATH].', 'astro-ph-9810425-3-34-5': 'This simple notation ensures radiation energy conservation in eq. ([REF]) for pure scattering of Lyman continuum photons (i.e., when [MATH]).', 'astro-ph-9810425-3-34-6': 'Eq. ([REF]) does not account properly for the number of photons entering the volume, so that a large photoionization coefficient [MATH] might lead to overproduction of ions.', 'astro-ph-9810425-3-34-7': 'To compensate for this, the number of photoionizations inside a 3D grid cell per unit time is not allowed to be larger than the number of photons actually absorbed inside this cell, i.e. [EQUATION]', 'astro-ph-9810425-3-34-8': 'Eq. ([REF],[REF]) are solved separately at each point, given the local radiation energy density [MATH].', 'astro-ph-9810425-3-34-9': 'Discretization of eq. ([REF]) in time yields [EQUATION] where [MATH] is just the right-hand side of eq. ([REF]) and [MATH] for stability.', 'astro-ph-9810425-3-34-10': ""This equation can almost always be solved via Newton's method for small enough [MATH]."", 'astro-ph-9810425-3-34-11': 'Linearizing eq. ([REF]), we get the [MATH]-th approximation to the value of [MATH] at time [MATH]: [EQUATION] which can then be iterated.', 'astro-ph-9810425-3-35-0': '## The algorithm', 'astro-ph-9810425-3-36-0': 'We start calculations by specifying the initial conditions (temperature, degree of ionization, and in the simplest cases no radiation field inside the volume) and boundary conditions (the intensity of radiation entering the volume - all outward flux at the edges can freely escape the computational box).', 'astro-ph-9810425-3-36-1': 'The inward flux at the boundaries is isotropic within [MATH] and is simply', 'astro-ph-9810425-3-37-0': '[EQUATION] where [MATH] is the average background radiation energy density.', 'astro-ph-9810425-3-37-1': 'Since each ray within the volume starts and ends at the sides of the volume, we automatically have boundary conditions for each of the 1D advection problems.', 'astro-ph-9810425-3-37-2': 'The density field is kept static for all tests in this study, but since the radiation field is being evolved explicitly at the speed of light, one could easily evolve the underlying density distribution on a much bigger fluid flow timescale if desired.', 'astro-ph-9810425-3-38-0': 'The course of the algorithm at each time step can be divided into the following steps:', 'astro-ph-9810425-3-39-0': 'At the beginning of each time-step we advect 1D intensities according to eq. ([REF]) along each ray.', 'astro-ph-9810425-3-39-1': 'The numerical resolution along each ray is simply set to the resolution [MATH] of the 3D rectangular mesh, so that along the ray [MATH] the point [MATH] has a coordinate', 'astro-ph-9810425-3-40-0': '[EQUATION] where [MATH] is the length of the ray segment inside the cube.', 'astro-ph-9810425-3-40-1': 'Note that one can have much coarser 1D grids along rays, speeding up the advection but sacrificing both spatial and angular resolution.', 'astro-ph-9810425-3-40-2': 'The advected intensities [MATH] are then projected onto a 3D grid to reconstruct the mean energy density at each point using eq. ([REF] - [REF]).', 'astro-ph-9810425-3-40-3': 'This operation is one of the most demanding from the computational point of view, since at each of our [MATH] points we have to deal with the angular dependence of the radiation field.', 'astro-ph-9810425-3-40-4': 'When this update is done, we solve the matter-radiation interaction equations implicitly to compute the local level populations (eq. ([REF])).', 'astro-ph-9810425-3-40-5': 'This gives us the 3D distributions of emissivity and opacity (eq. ([REF] - [REF])) which are then mapped back to the rays and used in the advection scheme at the next time-step.', 'astro-ph-9810425-3-41-0': 'This simple scheme which we will refer to as time-dependent ray tracing can be used as a stand-alone solver, or as a closure scheme for the system of moment equations through the use of variable Eddington factors (as in Stone et al. 1992).', 'astro-ph-9810425-3-41-1': 'In the absence of any sinks and sources of radiation, the intensity is conserved exactly along each ray.', 'astro-ph-9810425-3-41-2': 'Since the number of rays does not vary with time, our method guarantees exact conservation of the radiation energy in 3D.', 'astro-ph-9810425-3-42-0': 'Note that if I-fronts do not propagate fast, for instance, in the cosmological context where the evolution on the light-crossing timescale will normally require many thousand time steps, it is possible to update the radiation energy density [MATH] once every few tens or few hundred time steps, while still evolving the intensities and solving all rate equations properly at the speed of light.', 'astro-ph-9810425-3-42-1': 'We found that in practice this shortcut leads to an increase in speed by a factor of 10 or even higher without any loss of accuracy.', 'astro-ph-9810425-3-42-2': 'However, it is important to compute the energy density properly along the edges of I-fronts to guarantee the correct rate of growth of ionized regions.', 'astro-ph-9810425-3-43-0': 'To further accelerate this computation, we use adaptive time stepping to put higher time resolution on the 3D cells in the vicinity of I-fronts.', 'astro-ph-9810425-3-43-1': 'Since wavefronts cannot propagate further than one grid zone during one time step, only those cells which just experienced large change in [MATH] and their immediate adjacent neighbours need a proper update of [MATH].', 'astro-ph-9810425-3-43-2': 'Depending on the width of I-fronts, typically only a few percent of all 3D cells require the new, exact value of [MATH].', 'astro-ph-9810425-3-44-0': 'One advantage of the use of angle-averaged moments of radiation is that the advection mechanism is essentially reduced to 3D, and it is relatively straightforward to implement the multi-dimensional conservation scheme for the linear advection part of the moment equations.', 'astro-ph-9810425-3-44-1': 'Then one could use a much denser spatial grid for the solution of the moment equations, and a relatively course grid for the angular reconstruction of the intensity of radiation via ray tracing.', 'astro-ph-9810425-3-44-2': 'In practice, however, we have found that the mismatch between the spatial resolutions of the moment solver and of the ray tracing usually leads to numerical instabilities.', 'astro-ph-9810425-3-44-3': 'In what follows, we consider ray tracing only as a stand-alone solver.', 'astro-ph-9810425-3-45-0': 'Another - perhaps, a better - way of coupling angular and spatial variations of the intensity may be an extension of the Spherical Harmonics Discrete Ordinate Method [CITATION].', 'astro-ph-9810425-3-45-1': 'For steady-state transfer problems, instead of storing the radiation field, this method keeps track of the source function as a spherical harmonic series at each point.', 'astro-ph-9810425-3-45-2': 'Although the direct implementation of this technique for time-dependent problems is probably not realistic, due to the lookback time (i.e. the finite speed of light propagation), the spherical harmonic representation of the radiation field might require less storage and might result in smoother angular dependence as compared with a pure ray tracing approach.', 'astro-ph-9810425-3-46-0': '# Tests', 'astro-ph-9810425-3-47-0': 'For all of our test runs, except the study of the shadow behind a neutral clump in Sec. [REF], we set up a numerical grid with dimensions [MATH].', 'astro-ph-9810425-3-47-1': 'The angular resolution [MATH] was chosen to match the equivalent resolution of [MATH] data points for 5D advection.', 'astro-ph-9810425-3-47-2': 'There are [MATH] rays passing in the immediate neighbourhood ([MATH]th of the total computational volume) of each 3D grid cell, each ray containing [MATH] grid nodes.', 'astro-ph-9810425-3-47-3': 'Thus, in 5D we obtain the equivalent resolution of [MATH] data points.', 'astro-ph-9810425-3-48-0': 'Also, for all simulations we take a [MATH] (comoving) cosmological volume, scaled down to [MATH].', 'astro-ph-9810425-3-48-1': 'All densities (from low-density optically thin ambient gas to dense clumps) fall in the range [MATH], assuming a Hubble constant of [MATH].', 'astro-ph-9810425-3-48-2': 'This low density contrast is chosen to cover the typical range encountered in cosmological hydrodynamics, and is ideally suited to demonstrate transient features during patchy ionization.', 'astro-ph-9810425-3-48-3': 'The absorption coefficients are taken to mimic complete self-shielding (assumed to be [MATH]>[MATH][MATH]) at neutral hydrogen column densities higher than [MATH], except where we probe different regimes, specifically in Sec. [REF] where [MATH] corresponds to [MATH], and Sec. [REF] where [MATH] corresponds to [MATH].', 'astro-ph-9810425-3-49-0': '## An isolated spherical expanding I-front', 'astro-ph-9810425-3-50-0': 'A simple problem that would test the ability of the method to track I-fronts properly is that of a single, isolated Stromgren sphere expanding around a source of ionizing radiation (see, e.g., Abel et al. 1998).', 'astro-ph-9810425-3-50-1': 'One difficulty of the current approach is that rays are drawn in a way to cover the whole computational volume uniformly and isotropically.', 'astro-ph-9810425-3-50-2': 'For a single point source of radiation this would mean that only a small number of rays pass through its neighbourhood.', 'astro-ph-9810425-3-50-3': 'Although intensities along individual rays are strictly conserved, there is no guarantee that the energy density has the right value far from a source of a specified luminosity.', 'astro-ph-9810425-3-51-0': 'At time [MATH] we turn on a point source, which starts to blow an expanding [MATH] bubble around it.', 'astro-ph-9810425-3-51-1': 'Since our algorithm conserves 1D intensities exactly, independently of spatial or angular resolution, the I-front must propagate at the right speed, which can be obtained analytically by equating the number of direct ionizing photons to the flux of neutral atoms crossing the front.', 'astro-ph-9810425-3-51-2': 'In the absence of radiative recombinations ([MATH]), the [MATH] bubble grows indefinitely with the radius [EQUATION] where [MATH].', 'astro-ph-9810425-3-51-3': 'Parameters used for this calculation are the diffuse neutral gas density [MATH] (at [MATH]) and the central source luminosity [MATH] (all emitted in photons above the hydrogen Lyman limit).', 'astro-ph-9810425-3-51-4': 'The comparison between the numerical speed of the I-front and the exact solution from eq. ([REF]) is plotted in Fig. 1.', 'astro-ph-9810425-3-51-5': 'The difference between the two always stays within one grid zone.', 'astro-ph-9810425-3-52-0': '## An isolated Stromgren sphere in the presence of a density gradient', 'astro-ph-9810425-3-53-0': 'For this test we put a point source of radiation into a density gradient along one of the principal axes of the cube.', 'astro-ph-9810425-3-53-1': 'In the absence of diffuse radiation from Hii regions ([MATH]) the only ionizing photons come directly from the source in the centre, in which case the shape of the ionized bubble would be a simple superposition of Stromgren spheres with radii [MATH] varying with the azimuthal angle [MATH] and given by the classical solution [CITATION] [EQUATION] where [MATH] is the photon production rate of the central source.', 'astro-ph-9810425-3-53-2': 'For an exponential density gradient along the y-axis [EQUATION] ([MATH], [MATH] being the hydrogen densities on the opposite faces of the cube), the equilibrium Stromgren radius [MATH] is given by a simple equation [EQUATION] where', 'astro-ph-9810425-3-54-0': '[EQUATION].', 'astro-ph-9810425-3-55-0': 'We take the physical densities on the opposite faces of the volume to be [MATH] and [MATH], and the luminosity is [MATH].', 'astro-ph-9810425-3-55-1': 'Since we do not solve any realistic atomic rate equations in this paper, we take the value of the total hydrogen recombination rate from Hummer (1994) for some fiducial temperature ([MATH]).', 'astro-ph-9810425-3-55-2': 'In Fig. 2 we plot a time sequence of models, for ionization by a central source, with no scattering of Lyman photons (i.e. [MATH]).', 'astro-ph-9810425-3-55-3': 'The numerical solution at [MATH] appears to be very close to the exact one for an equilibrium Stromgren sphere.', 'astro-ph-9810425-3-55-4': 'The sharp transition layer between the ionized and the neutral regions in the high optical depth regime indicates that, indeed, the scheme introduces very little numerical diffusion even when extended to 3D.', 'astro-ph-9810425-3-56-0': '## Ionization in the presence of a UV background', 'astro-ph-9810425-3-57-0': 'The uniform coverage of the whole volume with rays implies that extended sources of radiation will be represented statistically much better than point sources.', 'astro-ph-9810425-3-57-1': 'A simple test mimicking the evolution of dense clouds in the presence of ionizing radiation is to enclose the computational region in an isotropic bath of photons.', 'astro-ph-9810425-3-57-2': 'The simplest way to accomplish this is just to set up a uniform, isotropically glowing boundary at the edges of the cube at [MATH].', 'astro-ph-9810425-3-57-3': 'An effective demonstration of time-dependent ray tracing would be its ability to deal with any distribution of state variables within the simulation volume.', 'astro-ph-9810425-3-57-4': ""For this test, we set up a density condensation shaped as the acronym for 'radiation hydrodynamics' (RHD), with a density [MATH] times that of the ambient homogeneous medium."", 'astro-ph-9810425-3-57-5': 'The ambient medium has a constant density of [MATH], and the energy density of the background radiation is [MATH].', 'astro-ph-9810425-3-57-6': 'Fig. 4 shows the result of this run.', 'astro-ph-9810425-3-57-7': 'Most of the low-density environment is ionized on the radiation propagation timescale.', 'astro-ph-9810425-3-57-8': 'It takes somewhat longer for ionizing photons to penetrate into the dense regions.', 'astro-ph-9810425-3-57-9': 'Whether these regions can be ionized on a timescale of interest, depends on the ratio of the recombination timescale to the flux of background radiation.', 'astro-ph-9810425-3-57-10': ""One can easily see ionization 'eating in' to the neutral zone, e.g. in the disappearance of the serifs on the letters at late times."", 'astro-ph-9810425-3-57-11': 'Note that the width of the ionization fronts does not usually exceed one grid zone (Fig. 5).', 'astro-ph-9810425-3-58-0': '## Diffuse radiation from Hii regions: shadows behind neutral clouds', 'astro-ph-9810425-3-59-0': 'Part of the ionizing radiation at high redshifts comes in the form of hydrogen Lyman continuum photons from recombinations in diffuse ionized regions.', 'astro-ph-9810425-3-59-1': 'The following test, simulating the formation of shadow regions behind dense clouds at the resolution [MATH], was adapted from Canto, Steffen Shapiro (1998).', 'astro-ph-9810425-3-59-2': ""A neutral clump of radius [MATH] (comoving) is being illuminated by a parallel flux [MATH] of stellar ionizing photons (just above [MATH]) from one side; here [MATH] is Planck's constant."", 'astro-ph-9810425-3-59-3': 'A shadow behind the clump is being photoionized by secondary recombination photons from the surrounding Hii region (Fig. 6) of physical density [MATH].', 'astro-ph-9810425-3-59-4': 'Neglecting hydrodynamical effects, the width [MATH] of the shadow region can be estimated using a simple two-stream approximation [CITATION]: [EQUATION] and the dimensionless parameter [MATH] is defined as [EQUATION]', 'astro-ph-9810425-3-59-5': 'For [MATH], recombination Lyman continuum photons from the illuminated region will eventually photoionize the shadow completely.', 'astro-ph-9810425-3-59-6': 'For [MATH], radiative losses through low-energy cascade recombination photons will stop the I-front, forming a neutral cylinder behind the dense clump.', 'astro-ph-9810425-3-59-7': 'Strictly speaking, equations ([REF])-([REF]) are valid only for a shadow completely photoionized by secondary photons, and should be viewed as an approximation to I-fronts driven by scattering.', 'astro-ph-9810425-3-60-0': 'For this run we modify boundary conditions to include recombination photons originating outside the box.', 'astro-ph-9810425-3-60-1': 'Each of the rays - starting on any face except the upper side of the volume (which goes through the neutral shadow) - carries the additional intensity of [MATH], where the angular brackets denote the average throughout the currently photoionized gas inside the box.', 'astro-ph-9810425-3-61-0': 'In Fig. 7 we plot the radius [MATH] of the shadow neutral region as a function of [MATH] in our 3D numerical models.', 'astro-ph-9810425-3-61-1': 'To play with the size of the neutral core, we fix the total recombination coefficient but we vary the portion of recombinations into the ground state (which would produce more Lyman continuum photons capable of ionizing the medium).', 'astro-ph-9810425-3-61-2': 'The width of the I-front driven by secondary photons depends on the assumed opacity (the optical depth of [MATH] corresponding to the column density of [MATH]).', 'astro-ph-9810425-3-61-3': 'This low opacity was chosen to reach equilibrium quicker - equilibrium itself does not depend on the opacity - but it cannot be too low otherwise there will be an unnecessary spread of the transition zone in the I-front over many grid cells.', 'astro-ph-9810425-3-62-0': 'We find a remarkably good agreement between the results of our models and the analytic solution, taking into account that ionization of the shadow is due to scattering in the medium.', 'astro-ph-9810425-3-63-0': '## Diffuse radiation from Hii regions: ionization of a central void', 'astro-ph-9810425-3-64-0': 'To demonstrate the ability of our scheme to handle scattering in more complicated situations, we also set up a model with ionization of a central low-density void by secondary, recombination photons.', 'astro-ph-9810425-3-64-1': 'The void region is surrounded by two nested cubes with opposite faces open.', 'astro-ph-9810425-3-64-2': 'The walls of the cubes are set to be much denser than the rest of the medium, to screen completely the central void from direct ionizing photons, and the ionizing UV flux is introduced at all faces of the computational volume.', 'astro-ph-9810425-3-64-3': 'The total hydrogen recombination coefficient [MATH] is again taken for the temperature [MATH].', 'astro-ph-9810425-3-64-4': 'Similar to Sec. 4.4, we vary the amount of scattering in the medium by changing the fraction [MATH] of atoms recombining into the ground state.', 'astro-ph-9810425-3-64-5': 'In reality at [MATH] the value [MATH] [CITATION] gives a solution in between our extreme values of [MATH].', 'astro-ph-9810425-3-65-0': 'Similar to the test problem of Section [REF], if [MATH], then the medium will be ionized completely, since there is a constant flux of primordial ionizing photons.', 'astro-ph-9810425-3-65-1': 'The speed of ionization depends on the values of [MATH], [MATH], [MATH] and [MATH].', 'astro-ph-9810425-3-65-2': 'Note, however, that if [MATH] is too high, the I-front will be very slow, since a large portion of the original ionizing photons are scattered back.', 'astro-ph-9810425-3-65-3': 'On the other hand, if [MATH] is too low, the I-front will propagate much faster in those regions where ionization is driven by primordial photons, but in shadowed regions there will be too few recombination photons.', 'astro-ph-9810425-3-65-4': 'Thus, it seems that the speed of ionization of the central void will be the highest at some intermediate [MATH].', 'astro-ph-9810425-3-66-0': 'In Fig. 8 we demonstrate ionization of the void region for models with complete scattering and with no scattering at all.', 'astro-ph-9810425-3-66-1': 'The parameters used for this model are [MATH] and the diffuse neutral gas density [MATH].', 'astro-ph-9810425-3-67-0': 'As expected, for the no-scattering model ([MATH]) the central region remains neutral, since there is no direct path for the ionizing photons.', 'astro-ph-9810425-3-67-1': 'However, for the model which includes scattering ([MATH]), at least part of the central region becomes ionized.', 'astro-ph-9810425-3-67-2': 'This demonstrates that our scheme is perfectly capable of dealing with re-scattering of the ionizing photons.', 'astro-ph-9810425-3-67-3': 'Beyond this simple test case, there are many astrophysical situations where progress can be made via numerical radiative transfer.', 'astro-ph-9810425-3-67-4': 'For example, analytic solutions are often used, which are steady-state, and which assume a sharp boundary between the neutral and ionized zones.', 'astro-ph-9810425-3-67-5': 'Using our numerical techniques it should be possible to follow general systems with complex density inhomogeneities as well as regions of partial ionization.', 'astro-ph-9810425-3-68-0': '# Conclusions', 'astro-ph-9810425-3-69-0': 'In this paper we have demonstrated that, with existing desktop hardware, it is possible to model cosmological inhomogeneous reionization on a light-crossing time [MATH] in three spatial dimensions.', 'astro-ph-9810425-3-69-1': 'Since the photoionization time-scale in the low optical depth regime ([MATH]) is of order of the light-crossing time [MATH], explicit advection might be a faster method in covering at least these regions.', 'astro-ph-9810425-3-69-2': 'Compared to the elliptic-type solvers on the fluid-flow time-scale or the time-scale of atomic processes, explicit radiative advection produces very accurate results without the need to solve a large system of coupled non-linear elliptic equations.', 'astro-ph-9810425-3-69-3': 'The computing requirements with explicit advection grow linearly with the inclusion of new atomic and molecular rate equations, which is certainly not the case for quasi-static solvers (although it is feasible that the development of multigrid techniques for elliptic equations might actually approach similar scaling).', 'astro-ph-9810425-3-70-0': 'Using eq. ([REF]) we can see that the entire history of reionization can be modeled with [MATH]-[MATH] time-steps (depending on the required resolution), which makes explicit advection an attractive choice for these calculations.', 'astro-ph-9810425-3-70-1': 'However, the efficiency of the explicit radiative solver has still to be explored.', 'astro-ph-9810425-3-70-2': 'Future work should include a detailed comparison between explicit advection and implicit reconstruction (through an elliptic solver), to demonstrate which method works best for calculating inhomogeneous reionization.', 'astro-ph-9810425-3-71-0': 'As we have demonstrated here, for certain problems, including the propagation of supersonic I-fronts, the Courant condition does not seem to impose prohibitively small time-steps.', 'astro-ph-9810425-3-71-1': 'In this case the biggest challenge is to accurately describe anisotropies in the radiation field, i.e. to solve for inhomogeneous advection in the 5D phase space, in the presence of non-uniform sources and sinks of radiation.', 'astro-ph-9810425-3-71-2': 'Strictly speaking, the storage of one variable at, say, [MATH] data points requires about [MATH]GB of memory, which stretches the capabilities of top-end desktop workstations.', 'astro-ph-9810425-3-71-3': 'One attractive possibility for future exploration is to directly solve the monochromatic photon Boltzmann equation in 5D.', 'astro-ph-9810425-3-71-4': 'To demonstrate the feasibility of the numerical solution, however, among different methods, we here chose to concentrate on simple ray tracing at the speed of light.', 'astro-ph-9810425-3-71-5': 'The numerical approach we have used is completely conservative and produces very little numerical dissipation.', 'astro-ph-9810425-3-72-0': 'We want to conclude that despite the high dimensionality of the problem, with a reasonable expenditure of computational resources, of the type available today, it is possible to numerically model many different aspects of the full 3D radiative transfer problem.', 'astro-ph-9810425-3-72-1': 'Furthermore we feel that the methods described here represent a significant and realizable step towards the goal of full cosmological RHD.', 'astro-ph-9810425-3-73-0': '# ACKNOWLEDGMENTS', 'astro-ph-9810425-3-74-0': 'We wish to thank Taishi Nakamoto for providing us with the results of reionization models from the University of Tsukuba ahead of publication.', 'astro-ph-9810425-3-74-1': 'A.R. would like to thank Jason R. Auman for numerous enlightening discussions, and Gregory G. Fahlman for constant encouragement on this project, as well as Randall J. LeVeque for help with numerical methods for multidimensional conservation laws.', 'astro-ph-9810425-3-74-2': 'This work was supported by the Natural Sciences and Engineering Research Council of Canada.'}",,
math-0205050,"{'math-0205050-1-0-0': '# Introduction', 'math-0205050-1-1-0': ""It is an interesting problem to define 'good' models of Shimura varieties over the ring of integers of the reflex field, or over its completion at some prime ideal."", 'math-0205050-1-1-1': 'For Shimura varieties of PEL type, which can be described as moduli spaces of abelian varieties, it is desirable to define such a model by a moduli problem, too.', 'math-0205050-1-2-0': 'In their book [CITATION], Rapoport and Zink define such models in the case of parahoric level structures.', 'math-0205050-1-2-1': 'They also define the so-called local models which etale-locally around each point of the special fibre are isomorphic to the model of the Shimura variety, but which are defined purely in terms of linear algebra.', 'math-0205050-1-2-2': 'They provide a very useful tool to investigate local properties of the corresponding models.', 'math-0205050-1-3-0': 'Rapoport and Zink conjectured that these models are flat, which is a property a good model certainly should have.', 'math-0205050-1-3-1': 'The conjecture is true for Shimura varieties associated to unitary or symplectic groups that split over an unramified extension of [MATH] (see [CITATION], [CITATION]).', 'math-0205050-1-3-2': 'But Pappas [CITATION] showed that it is in general false if the group splits only after a ramified base change.', 'math-0205050-1-3-3': 'Often the models are not even topologically flat, i. e. the closure of the generic fibre is not even set-theoretically equal to the whole model.', 'math-0205050-1-3-4': 'We follow the terminology of Pappas and Rapoport and call the local model associated to these models of the Shimura variety the naive local model.', 'math-0205050-1-4-0': 'For groups of the form [MATH], where [MATH] is a totally ramified extension, Pappas and Rapoport [CITATION] then defined a new local model, which they call the local model.', 'math-0205050-1-4-1': 'In the case of level structures corresponding to a maximal parahoric subgroup, it is essentially defined as the closure of the generic fibre in the old local model, so it is automatically flat.', 'math-0205050-1-4-2': 'Pappas and Rapoport showed that this new model has several good properties; for example its special fibre is normal, with Cohen-Macaulay singularities, and over a finite extension of [MATH] admits a resolution of singularities.', 'math-0205050-1-4-3': 'The definition of the local model in the maximal parahoric case gives rise to a definition of a new local model in the general parahoric case.', 'math-0205050-1-4-4': 'It is not obvious anymore that these general local models are flat, too, and it is the purpose of this note to show that they are topologically flat.', 'math-0205050-1-4-5': 'Together with the methods used in [CITATION] (Frobenius splitting of Schubert varieties in the affine flag variety), one can then infer flatness.', 'math-0205050-1-5-0': 'Let [MATH] be a (possibly ramified) finite extension.', 'math-0205050-1-5-1': 'Then the local model associated to [MATH] is flat over [MATH].', 'math-0205050-1-6-0': 'In the final section of this article we show that the local model for groups of the form [MATH], [MATH] totally ramified, is topologically flat, too.', 'math-0205050-1-6-1': '(In this case the naive local model and the local model coincide topologically.)', 'math-0205050-1-6-2': 'But here it is more difficult to deduce the flatness from the topological flatness, and we can only make a conjecture in this direction.', 'math-0205050-1-6-3': 'On the other hand, it seems reasonable to expect that in this case the naive local model is flat itself (and thus coincides with the local model).', 'math-0205050-1-7-0': 'In a subsequent article [CITATION] Pappas and Rapoport define yet another model, which they call the canonical model.', 'math-0205050-1-7-1': 'Loosely speaking, it is the image of a certain morphism from a twisted product of unramified local models to the naive local model.', 'math-0205050-1-7-2': 'In particular it is always flat, because the unramified local models are flat, and in addition has other good properties.', 'math-0205050-1-7-3': 'Nevertheless the flatness question for the original local model remains interesting; if the local model is flat, it must coincide with the canonical local model.', 'math-0205050-1-7-4': ""On the one hand this shows that the special fibre of the canonical local model has the 'correct' combinatorial description, on the other hand one gets an interesting 'resolution' of the local model."", 'math-0205050-1-8-0': 'I would like to thank T. Haines for several fruitful conversations on this topic; in particular the key lemma [REF] came up in a discussion with him.', 'math-0205050-1-8-1': 'I am also grateful to G. Pappas, M. Rapoport and T. Wedhorn for many helpful remarks.', 'math-0205050-1-8-2': 'The main part of this article was written during my stay at the Institute for Advanced Study in Princeton.', 'math-0205050-1-8-3': 'I would like to thank the Institute for providing a great working environment, and the Deutsche Forschungsgemeinschaft and the National Science Foundation (grant no.', 'math-0205050-1-8-4': 'DMS 97-29992) for their support.', 'math-0205050-1-9-0': '# Definitions', 'math-0205050-1-10-0': 'We will use the notation of Pappas and Rapoport [CITATION].', 'math-0205050-1-10-1': 'Let us repeat part of it:', 'math-0205050-1-11-0': 'Let [MATH] be a field, complete with respect to a non-archimedean valuation.', 'math-0205050-1-11-1': 'Let [MATH] denote its ring of integers, [MATH] a uniformizer.', 'math-0205050-1-11-2': 'We assume that the residue class field is perfect.', 'math-0205050-1-11-3': 'We fix a separable closure [MATH] of [MATH].', 'math-0205050-1-12-0': 'Let [MATH] be a totally ramified extension of degree [MATH] inside [MATH].', 'math-0205050-1-12-1': 'Let [MATH] be the ring of integers of [MATH], and let [MATH] be a uniformizer which is the root of an Eisenstein polynomial [MATH].', 'math-0205050-1-12-2': 'For each embedding [MATH] we choose an integer [MATH].', 'math-0205050-1-12-3': 'Associated to these data we have the reflex field [MATH], a finite extension of [MATH] contained in [MATH], which is defined by [MATH].', 'math-0205050-1-13-0': 'Further, let [MATH], [MATH], and denote the canonical basis by [MATH].', 'math-0205050-1-14-0': 'Choose [MATH].', 'math-0205050-1-15-0': ""Let us recall the definition of the 'naive' local model [MATH] (of course [MATH] depends on [MATH], not just on [MATH], so this is an abuse of notation which nevertheless seems useful):"", 'math-0205050-1-16-0': 'It is defined over the ring of integers [MATH] of the reflex field [MATH], and its [MATH]-valued points are the isomorphism classes of commutative diagrams', 'math-0205050-1-17-0': '_i_0, S & & _i_1,S & & & & _i_m-1,S & ^& _i_0,S', 'math-0205050-1-18-0': '& & & & &&& &', 'math-0205050-1-19-0': 'F_0 & & F_1 & & & & F_m-1 && F_0', 'math-0205050-1-20-0': 'where [MATH] is [MATH], and where the [MATH] are [MATH]-submodules.', 'math-0205050-1-20-1': ""We require that locally on [MATH], the [MATH] are direct summands as [MATH]-modules and that we have the following identity of polynomials ('determinant condition'): [EQUATION]."", 'math-0205050-1-21-0': 'Pappas and Rapoport show that this naive local model is almost never flat, even if [MATH] consists of only one element.', 'math-0205050-1-21-1': 'In this case, i. e. when [MATH] consists of only one element, they define a new local model [MATH] as the scheme-theoretic closure of the generic fibre in the naive local model, and show that this new local model has several good properties.', 'math-0205050-1-21-2': ""In particular, its special fibre is reduced, and can be described as a union of Schubert varieties with the 'right' index set."", 'math-0205050-1-22-0': 'Based on this, they define a new Iwahori type local model [MATH] as the closed subscheme of the naive local model such that all projections to the parahoric local model map to the new local model.', 'math-0205050-1-22-1': 'Below we will show that this local model is topologically flat.', 'math-0205050-1-22-2': 'Furthermore, using the technique of Frobenius splittings, one can see that its special fibre is reduced.', 'math-0205050-1-22-3': 'Thus [MATH] is flat.', 'math-0205050-1-23-0': '# A certain map between local models', 'math-0205050-1-24-0': 'It is clear that in the definition of the local model, we have not used the fact that the sequence [MATH] is a lattice chain - in fact we could make a completely analogous definition for any sequence of free [MATH]-modules.', 'math-0205050-1-24-1': ""(Compare [CITATION], 'general schemes of compatible subspaces'.)"", 'math-0205050-1-25-0': 'In particular, consider the following situation: Choose a partition [MATH].', 'math-0205050-1-25-1': 'We can then decompose the lattices [MATH] as [EQUATION] where [MATH].', 'math-0205050-1-25-2': 'Here the [MATH] denote the canonical [MATH]-basis of [MATH].', 'math-0205050-1-26-0': 'Now choose [MATH] such that [MATH].', 'math-0205050-1-27-0': 'This gives rise to a decomposition of our lattice chain.', 'math-0205050-1-27-1': 'More precisely for each [MATH] we get a sequence [EQUATION] of free [MATH]-modules.', 'math-0205050-1-27-2': 'The difference between these sequences and the original lattice chain (apart from the rank of the lattices) is that now some of the maps may be the identity.', 'math-0205050-1-27-3': ""Nevertheless, we can define a 'local model' [MATH]."", 'math-0205050-1-27-4': ""It is isomorphic to some local model in the original sense (because as far as the local model is concerned, we can just omit the 'identity steps')."", 'math-0205050-1-28-0': 'The reflex field associated to a tuple [MATH] will in general be different from the one belonging to [MATH].', 'math-0205050-1-28-1': 'Thus the [MATH] will in general not be defined over the same ring as the naive local model [MATH] associated to the [MATH].', 'math-0205050-1-28-2': 'To simplify the situation, we make the following additional assumption: whenever [MATH], we have [MATH] for all [MATH].', 'math-0205050-1-28-3': 'Under this assumption, all [MATH] will be defined over [MATH], or even over a smaller ring.', 'math-0205050-1-28-4': 'If necessary we apply a base change, and in the following we consider all the [MATH] as [MATH]-schemes.', 'math-0205050-1-29-0': 'We then have a canonical map [EQUATION]', 'math-0205050-1-30-0': '# The case [MATH]', 'math-0205050-1-31-0': 'In this section, we will look at the (trivial) case where [MATH].', 'math-0205050-1-31-1': 'We will use the following observations in section [REF].', 'math-0205050-1-32-0': 'Let us give a description of the local model corresponding to a maximal parahoric subgroup (the standard local model in the sense of [CITATION], section 2).', 'math-0205050-1-32-1': '(Actually, in this case this coincides with the Iwahori type local model.)', 'math-0205050-1-33-0': 'So, we have integers [MATH], [MATH].', 'math-0205050-1-33-1': 'Let us assume, for notational convenience, that the [MATH] are in descending order, such that they are completely determined by [MATH].', 'math-0205050-1-34-0': 'The local model in this case is just [MATH], where [MATH] is the corresponding reflex field.', 'math-0205050-1-34-1': 'The point in the special fibre is the subspace [MATH] corresponding to the [MATH]-matrix [EQUATION] the operator [MATH] has Jordan type [MATH].', 'math-0205050-1-34-2': 'Let us denote the subspace corresponding to the [MATH]-valued point of the local model by [MATH].', 'math-0205050-1-34-3': 'The description of this subspace in terms of matrices is the following.', 'math-0205050-1-34-4': 'We have a matrix [EQUATION] such that [EQUATION] for some matrix [MATH] with characteristic polynomial [EQUATION].', 'math-0205050-1-34-5': 'Furthermore, the reduction of [MATH] modulo [MATH] is [MATH].', 'math-0205050-1-35-0': 'Example.', 'math-0205050-1-35-1': 'Assume that [MATH].', 'math-0205050-1-35-2': 'Now, if we let [MATH] and choose the [MATH] such that the resulting characteristic polynomial is [MATH], we get [EQUATION]', 'math-0205050-1-36-0': '# Lifting of points', 'math-0205050-1-37-0': 'The local model [MATH] is topologically flat, i. e. the generic points of the irreducible components can be lifted to the special fibre.', 'math-0205050-1-38-0': 'Proof.', 'math-0205050-1-38-1': 'To simplify notation, we consider only the Iwahori case.', 'math-0205050-1-38-2': 'We can embed the special fibre of [MATH] into the affine flag variety for [MATH] in the standard way.', 'math-0205050-1-39-0': 'By the definition of the local model and since the special fibers of the maximal-parahoric new local models have the right stratification, the special fibre is the union of the [MATH]-permissible alcoves.', 'math-0205050-1-39-1': 'Here [MATH] is the dual partition to [MATH].', 'math-0205050-1-40-0': 'By recent work of Haines and Ngo [CITATION] we know that in this case (as in the minuscule case), the set of [MATH]-permissible alcoves coincides with the set of [MATH]-admissible alcoves.', 'math-0205050-1-40-1': 'Thus the maximal elements (with respect to the Bruhat order), which correspond to the irreducible components of the special fibre, are just the conjugates of [MATH] under the finite Weyl group, i.e. the permutations of [MATH] under [MATH].', 'math-0205050-1-41-0': 'Choose a permutation [MATH] of [MATH] and write [MATH].', 'math-0205050-1-42-0': 'A key observation is the following lemma which came up in a discussion with T. Haines.', 'math-0205050-1-43-0': 'Let [MATH] be a discrete valuation ring, let [MATH] be the closed and [MATH] the generic point of [MATH].', 'math-0205050-1-43-1': 'Let [MATH] be an [MATH]-scheme of finite type, and [MATH] a closed point of the special fibre which lies in a unique irreducible component [MATH] of [MATH].', 'math-0205050-1-44-0': 'Furthermore, assume that [MATH] is irreducible, that [MATH], and that [MATH] can be lifted to a closed point of [MATH].', 'math-0205050-1-45-0': 'Then the generic point of [MATH] can be lifted to [MATH].', 'math-0205050-1-46-0': 'Proof.', 'math-0205050-1-46-1': 'Denote by [MATH] the scheme-theoretic closure of [MATH] in [MATH].', 'math-0205050-1-46-2': 'Then [MATH] is flat over [MATH], and [MATH].', 'math-0205050-1-46-3': 'Furthermore, [MATH] (see [CITATION], theorem 15.1), and [MATH], since [MATH] can be lifted to a closed point of [MATH].', 'math-0205050-1-47-0': 'Thus [MATH], which implies that [MATH].', 'math-0205050-1-47-1': '[MATH]', 'math-0205050-1-48-0': 'Although the lemma is easy to prove, it reduces the combinatorial difficulty of our task enormously.', 'math-0205050-1-49-0': 'The lemma shows that it really is enough to show that one suitably chosen closed point of the special fibre can be lifted to the generic fibre.', 'math-0205050-1-49-1': 'We will choose the point [MATH], where each [MATH] is given by the [MATH]-matrix [EQUATION].', 'math-0205050-1-50-0': 'It is clear that [MATH] lies in the stratum corresponding to [MATH].', 'math-0205050-1-50-1': '(Look at the Jordan type of [MATH].)', 'math-0205050-1-51-0': 'Now how can we lift this point to the generic fibre?', 'math-0205050-1-51-1': 'We will describe two different methods to do this.', 'math-0205050-1-51-2': 'The first one is more elementary, the second one more conceptual.', 'math-0205050-1-52-0': '## First method (elementary)', 'math-0205050-1-53-0': 'To simplify the notation, let us assume that the [MATH] are in descending order.', 'math-0205050-1-53-1': 'We will build the corresponding matrix (over [MATH]) by putting together several matrices of the form [MATH].', 'math-0205050-1-53-2': 'More precisely, consider the [MATH]-valued point where each subspace [MATH] is given by the same matrix [EQUATION].', 'math-0205050-1-54-0': 'Note that the coefficients of this matrix are indeed contained in [MATH].', 'math-0205050-1-54-1': 'Clearly, the reduction of [MATH] is [MATH].', 'math-0205050-1-54-2': 'We have to check that', 'math-0205050-1-55-0': 'But both conditions are clearly satisfied, since the [MATH] satisfy the corresponding conditions and since there are no interactions between the blocks.', 'math-0205050-1-55-1': 'Let us make that more precise.', 'math-0205050-1-56-0': 'To check the first condition, fix any [MATH].', 'math-0205050-1-56-1': 'Note that [MATH] is [MATH]-invariant: this means that there is a matrix [MATH] such that [EQUATION] and we can simply take [MATH].', 'math-0205050-1-56-2': 'Now let us check the determinant condition; the characteristic polynomial of [MATH] is just the product of the characteristic polynomials of the [MATH], so we get [EQUATION] as it should be.', 'math-0205050-1-57-0': 'The second condition is easily checked, too.', 'math-0205050-1-57-1': 'We just have to observe that [EQUATION].', 'math-0205050-1-58-0': 'Thus we have indeed found a lifting of our point to the generic fibre.', 'math-0205050-1-59-0': '## Second method (conceptual)', 'math-0205050-1-60-0': 'In this section, we will describe the [MATH]-valued point constructed in the previous section as a morphism from a product of trivial (i.e. [MATH]) local models to the given local model, which comes from the situation studied in section [REF].', 'math-0205050-1-61-0': 'As partition of [MATH], we will choose the partition into singleton sets: [MATH], [MATH].', 'math-0205050-1-61-1': 'Furthermore we choose [MATH] such that [EQUATION].', 'math-0205050-1-61-2': 'It is easy to see that such [MATH] exist and that they are uniquely determined.', 'math-0205050-1-61-3': 'Furthermore, whenever [MATH], we have [MATH] for all [MATH].', 'math-0205050-1-61-4': 'Thus all [MATH] are defined over a subring of [MATH], and we denote the base change to [MATH] by [MATH].', 'math-0205050-1-62-0': 'Now consider the map [EQUATION] associated to these data.', 'math-0205050-1-62-1': 'All the [MATH] are just isomorphic to [MATH], so their product is [MATH] again.', 'math-0205050-1-63-0': 'It is not hard to check is that the image of the closed point under this map is the point described above.', 'math-0205050-1-64-0': '# The case of the symplectic group', 'math-0205050-1-65-0': 'Finally, let us consider the question of topological flatness of the local model for the symplectic group.', 'math-0205050-1-65-1': 'Let us repeat, with slight notational modifications, the definition of the naive local model for the symplectic group given in [CITATION]; cf. also [CITATION].', 'math-0205050-1-65-2': 'To simplify the notation, we consider only the Iwahori case.', 'math-0205050-1-66-0': 'Consider a totally ramified extension [MATH] of degree [MATH] as before.', 'math-0205050-1-66-1': 'Let [MATH] with basis [MATH], and denote by [MATH] the standard symplectic pairing, i.e. [EQUATION].', 'math-0205050-1-66-2': 'Let [MATH] be an [MATH]-generator of the inverse different [MATH] (if [MATH] is tamely ramified over [MATH], we can take [MATH]).', 'math-0205050-1-66-3': 'Let [MATH].', 'math-0205050-1-66-4': 'This is a non-degenerate alternating form on [MATH] with values in [MATH].', 'math-0205050-1-67-0': 'The standard lattice chain [MATH] is self-dual with respect to [MATH].', 'math-0205050-1-68-0': 'Now let [MATH] for all [MATH].', 'math-0205050-1-68-1': 'Associated to [MATH], [MATH] and [MATH] we have the naive local model for [MATH].', 'math-0205050-1-68-2': 'In this case the reflex field is [MATH] and [MATH] is just [MATH].', 'math-0205050-1-69-0': ""The naive local model for the symplectic group is the closed subscheme of [MATH] consisting of 'self-dual' families of subspaces: [EQUATION]."", 'math-0205050-1-69-1': 'Here [MATH] denotes the [MATH]-dual [MATH], and the isomorphism [MATH] is the one given by the pairing.', 'math-0205050-1-70-0': 'Similarly, we define the local model [MATH] as the closed subscheme of [MATH] consisting of self-dual families of subspaces: [EQUATION].', 'math-0205050-1-70-1': 'Actually, we know that in this case the two local models for [MATH] coincide topologically.', 'math-0205050-1-70-2': 'Namely, it is enough to check this in the maximal parahoric case where the special fibre is irreducible, so that one just has to see that the generic and special fibres have the same dimension.', 'math-0205050-1-70-3': 'Of course this implies that [MATH] and [MATH] are topologically the same, too.', 'math-0205050-1-70-4': 'So in order to prove the topological flatness, we can work with either one.', 'math-0205050-1-71-0': 'The local model [MATH] is topologically flat, i.e. the generic points of the irreducible components of the special fibre of [MATH] can be lifted to the generic fibre.', 'math-0205050-1-71-1': 'The same holds for [MATH].', 'math-0205050-1-72-0': 'Proof.', 'math-0205050-1-72-1': 'The irreducible components of [MATH] correspond to the conjugates of [MATH] under the Weyl group of the symplectic group (considered as a subgroup of [MATH]).', 'math-0205050-1-72-2': '(Here we use the fact that the set of permissible alcoves for the symplectic group coincides with the intersection of the set of admissible alcoves for the general linear group with the standard apartment for the symplectic group: [MATH]; see [CITATION]).', 'math-0205050-1-72-3': 'As in the linear case, it is enough to show that for each irreducible component there is one (suitably chosen) point that can be lifted to the generic fibre.', 'math-0205050-1-73-0': 'In this case we cannot hope to find the lifting as the image of a map of a product of trivial local models for the symplectic group, because decomposing the lattice chain as we did in the linear case would not preserve the pairing.', 'math-0205050-1-73-1': 'Nevertheless the proof of the proposition is even simpler in this case, because the [MATH] we have to deal with is so special.', 'math-0205050-1-74-0': 'We consider the symplectic local model as a closed subscheme of the linear local model.', 'math-0205050-1-74-1': 'Let [MATH] be a conjugate of [MATH] under the Weyl group of the symplectic group.', 'math-0205050-1-74-2': 'It is enough to show that we can lift the point where each [MATH] is given by the [MATH]-matrix [EQUATION] (which obviously lies inside [MATH]) to the generic fibre (of [MATH]).', 'math-0205050-1-75-0': 'Let us first lift this point to a point in the generic fibre of the linear local model [MATH].', 'math-0205050-1-75-1': 'Afterwards we will show that the lifting actually lies in the symplectic model.', 'math-0205050-1-76-0': 'Since all the [MATH] are either [MATH] or [MATH], we can just lift this point by exactly the same matrix over [MATH]!', 'math-0205050-1-76-1': 'It is clear that these matrices do describe a point in the (linear) local model over [MATH], in particular that the determinant condition is satisfied.', 'math-0205050-1-77-0': 'But since [MATH] is not an arbitrary permutation of [MATH], but one under the Weyl group of the symplectic group, it can never happen that [MATH] and [MATH] are both 1.', 'math-0205050-1-77-1': ""In other words, if in the matrix above there is a unit matrix somewhere in rows [MATH], the columns [MATH] will entirely consist of 0's."", 'math-0205050-1-77-2': 'Taking into account that all the [MATH] are given by the same matrix, it is then clear that this point lies in the symplectic local model.', 'math-0205050-1-77-3': '[MATH]', 'math-0205050-1-78-0': 'We cannot prove that [MATH] (or [MATH]) is flat.', 'math-0205050-1-78-1': 'But since [MATH] is topologically flat, we are led to the', 'math-0205050-1-79-0': 'The naive local model [MATH] is flat.', 'math-0205050-1-80-0': 'To prove this conjecture, it would be sufficient to show that the special fibre is reduced.', 'math-0205050-1-80-1': 'This would follow if one could identify it with a union of Schubert varieties in some affine flag variety, and by the usual method using Frobenius splittings, it would be enough to do this for the local models which correspond to a maximal parahoric subgroup.', 'math-0205050-1-80-2': 'But even that seems to be a difficult problem in commutative algebra.', 'math-0205050-1-81-0': 'Ulrich Gortz', 'math-0205050-1-82-0': 'Mathematisches Institut', 'math-0205050-1-83-0': 'der Universitat zu Koln', 'math-0205050-1-84-0': 'Weyertal 86-90', 'math-0205050-1-85-0': '50931 Koln, Germany', 'math-0205050-1-86-0': 'ugoertz@mi.uni-koeln.de'}","{'math-0205050-2-0-0': 'Local models are schemes defined in terms of linear algebra which can be used to study the local structure of integral models of certain Shimura varieties, with parahoric level structure.', 'math-0205050-2-0-1': 'We investigate the local models for groups of the form [MATH] and [MATH] where [MATH] is a totally ramified extension, as defined by Pappas and Rapoport, and show that they are topologically flat.', 'math-0205050-2-0-2': 'In the linear case, flatness can be deduced from this.', 'math-0205050-2-1-0': '# Introduction', 'math-0205050-2-2-0': 'U. Gortz, Topological flatness of local models U. Gortz, Topological flatness of local models', 'math-0205050-2-3-0': ""It is an interesting problem to define 'good' models of Shimura varieties over the ring of integers of the reflex field, or over its completion at some prime ideal."", 'math-0205050-2-3-1': 'For Shimura varieties of PEL type, which can be described as moduli spaces of abelian varieties, it is desirable to define such a model by a moduli problem, too.', 'math-0205050-2-4-0': 'In their book [CITATION], Rapoport and Zink define such models in the case of parahoric level structures.', 'math-0205050-2-4-1': 'They also define the so-called local models which etale-locally around each point of the special fibre are isomorphic to the model of the Shimura variety, but which are defined purely in terms of linear algebra.', 'math-0205050-2-4-2': 'They provide a very useful tool to investigate local properties of the corresponding models.', 'math-0205050-2-5-0': 'Rapoport and Zink conjectured that these models are flat, which is a property a good model certainly should have.', 'math-0205050-2-5-1': 'The conjecture is true for Shimura varieties associated to unitary or symplectic groups that split over an unramified extension of [MATH] (see [CITATION], [CITATION]).', 'math-0205050-2-5-2': 'But Pappas [CITATION] showed that it is in general false if the group splits only after a ramified base change.', 'math-0205050-2-5-3': 'Often the models are not even topologically flat, i. e. the closure of the generic fibre is not even set-theoretically equal to the whole model.', 'math-0205050-2-5-4': 'We follow the terminology of Pappas and Rapoport and call the local model associated to these models of the Shimura variety the naive local model.', 'math-0205050-2-6-0': 'For groups of the form [MATH], where [MATH] is a totally ramified extension, Pappas and Rapoport [CITATION] then defined a new local model, which they call the local model.', 'math-0205050-2-6-1': 'In the case of level structures corresponding to a maximal parahoric subgroup, it is essentially defined as the closure of the generic fibre in the old local model, so it is automatically flat.', 'math-0205050-2-6-2': 'Pappas and Rapoport showed that this new model has several good properties; for example its special fibre is normal, with Cohen-Macaulay singularities, and over a finite extension of [MATH] admits a resolution of singularities.', 'math-0205050-2-6-3': 'The definition of the local model in the maximal parahoric case gives rise to a definition of a new local model in the general parahoric case.', 'math-0205050-2-6-4': 'It is not obvious anymore that these general local models are flat, too, and it is the purpose of this note to show that they are topologically flat.', 'math-0205050-2-6-5': 'Together with the methods used in [CITATION] (Frobenius splitting of Schubert varieties in the affine flag variety), one can then infer flatness.', 'math-0205050-2-7-0': 'Let [MATH] be a (possibly ramified) finite extension.', 'math-0205050-2-7-1': 'Then the local model associated to [MATH] is flat over [MATH].', 'math-0205050-2-8-0': 'An essential ingredient of the proof is a combinatorial result of Haines and Ngo about the so-called [MATH]-admissible and [MATH]-permissible sets (see below for further details).', 'math-0205050-2-8-1': 'This result, as it stands, relates to the Iwahori case.', 'math-0205050-2-8-2': 'In section [REF] we show how it can be used to prove the corresponding result in the general parahoric case.', 'math-0205050-2-9-0': 'In section [REF] we show that the local model for groups of the form [MATH], [MATH] totally ramified, is topologically flat, too.', 'math-0205050-2-9-1': '(In this case the naive local model and the local model coincide topologically.)', 'math-0205050-2-9-2': 'But here it is more difficult to deduce the flatness from the topological flatness, and we can only make a conjecture in this direction.', 'math-0205050-2-9-3': 'On the other hand, it seems reasonable to expect that in this case the naive local model is flat itself (and thus coincides with the local model).', 'math-0205050-2-10-0': 'In a subsequent article [CITATION] Pappas and Rapoport define yet another model, which they call the canonical model.', 'math-0205050-2-10-1': 'Loosely speaking, it is the image of a certain morphism from a twisted product of unramified local models to the naive local model.', 'math-0205050-2-10-2': 'In particular it is always flat, because the unramified local models are flat, and in addition has other good properties.', 'math-0205050-2-10-3': 'Nevertheless the flatness question for the original local model remains interesting; if the local model is flat, it must coincide with the canonical local model.', 'math-0205050-2-10-4': ""On the one hand this shows that the special fibre of the canonical local model has the 'correct' combinatorial description, on the other hand one gets an interesting 'resolution' of the local model."", 'math-0205050-2-11-0': 'The results of Pappas and Rapoport together with the combinatorial results of Haines and Ngo, and in section [REF], respectively, can be used to give a shorter (but less elementary) proof of the topological flatness, as came out of discussions with Haines.', 'math-0205050-2-11-1': 'See the remark after proposition [REF] for an outline.', 'math-0205050-2-12-0': 'I would like to thank T. Haines for many fruitful conversations on this topic.', 'math-0205050-2-12-1': 'I am also grateful to G. Pappas, M. Rapoport and T. Wedhorn for their helpful remarks.', 'math-0205050-2-12-2': 'The main part of this article was written during my stay at the Institute for Advanced Study in Princeton.', 'math-0205050-2-12-3': 'I would like to thank the Institute for providing a great working environment.', 'math-0205050-2-12-4': 'The support of the Deutsche Forschungsgemeinschaft and the National Science Foundation (grant no.', 'math-0205050-2-12-5': 'DMS 97-29992) is gratefully acknowledged.', 'math-0205050-2-13-0': '# Definitions', 'math-0205050-2-14-0': 'We will use the notation of Pappas and Rapoport [CITATION].', 'math-0205050-2-14-1': 'Let us repeat part of it:', 'math-0205050-2-15-0': 'Let [MATH] be a field, complete with respect to a non-archimedean valuation.', 'math-0205050-2-15-1': 'Let [MATH] denote its ring of integers, [MATH] a uniformizer.', 'math-0205050-2-15-2': 'We assume that the residue class field is perfect.', 'math-0205050-2-15-3': 'We fix a separable closure [MATH] of [MATH].', 'math-0205050-2-16-0': 'Let [MATH] be a totally ramified extension of degree [MATH] inside [MATH].', 'math-0205050-2-16-1': 'Let [MATH] be the ring of integers of [MATH], and let [MATH] be a uniformizer which is the root of an Eisenstein polynomial [MATH].', 'math-0205050-2-16-2': 'For each embedding [MATH] we choose an integer [MATH].', 'math-0205050-2-16-3': 'Associated to these data we have the reflex field [MATH], a finite extension of [MATH] contained in [MATH], which is defined by [MATH].', 'math-0205050-2-17-0': 'Further, let [MATH], [MATH], and denote the canonical basis by [MATH].', 'math-0205050-2-18-0': 'Choose [MATH].', 'math-0205050-2-19-0': ""Let us recall the definition of the 'naive' local model [MATH] (of course [MATH] depends on [MATH], not just on [MATH], so this is an abuse of notation which nevertheless seems useful):"", 'math-0205050-2-20-0': 'It is defined over the ring of integers [MATH] of the reflex field [MATH], and its [MATH]-valued points are the isomorphism classes of commutative diagrams', 'math-0205050-2-21-0': '_i_0, S & & _i_1,S & & & & _i_m-1,S & ^& _i_0,S', 'math-0205050-2-22-0': '& & & & &&& &', 'math-0205050-2-23-0': 'F_0 & & F_1 & & & & F_m-1 && F_0', 'math-0205050-2-24-0': 'where [MATH] is [MATH], and where the [MATH] are [MATH]-submodules.', 'math-0205050-2-24-1': ""We require that locally on [MATH], the [MATH] are direct summands as [MATH]-modules and that we have the following identity of polynomials ('determinant condition'): [EQUATION]."", 'math-0205050-2-25-0': 'Pappas and Rapoport show that this naive local model is almost never flat, even if [MATH] consists of only one element.', 'math-0205050-2-25-1': 'In this case, i. e. when [MATH] consists of only one element, they define a new local model [MATH] as the scheme-theoretic closure of the generic fibre in the naive local model, and show that this new local model has several good properties.', 'math-0205050-2-25-2': ""In particular, its special fibre is reduced, and can be described as a union of Schubert varieties with the 'right' index set."", 'math-0205050-2-26-0': 'Based on this, they define a new Iwahori type local model [MATH] as the closed subscheme of the naive local model such that all projections to the parahoric local model map to the new local model.', 'math-0205050-2-26-1': 'Below we will show that this local model is topologically flat.', 'math-0205050-2-26-2': 'Furthermore, using the technique of Frobenius splittings, one can see that its special fibre is reduced.', 'math-0205050-2-26-3': 'Thus [MATH] is flat.', 'math-0205050-2-27-0': '# A certain map between local models', 'math-0205050-2-28-0': 'It is clear that in the definition of the local model, we have not used the fact that the sequence [MATH] is a lattice chain - in fact we could make a completely analogous definition for any sequence of free [MATH]-modules.', 'math-0205050-2-28-1': ""(Compare [CITATION], 'general schemes of compatible subspaces'.)"", 'math-0205050-2-29-0': 'In particular, consider the following situation: Choose a partition [MATH].', 'math-0205050-2-29-1': 'We can then decompose the lattices [MATH] as [EQUATION] where [MATH].', 'math-0205050-2-29-2': 'Here the [MATH] denote the canonical [MATH]-basis of [MATH].', 'math-0205050-2-30-0': 'Now choose [MATH] such that [MATH].', 'math-0205050-2-31-0': 'This gives rise to a decomposition of our lattice chain.', 'math-0205050-2-31-1': 'More precisely for each [MATH] we get a sequence [EQUATION] of free [MATH]-modules.', 'math-0205050-2-31-2': 'The difference between these sequences and the original lattice chain (apart from the rank of the lattices) is that now some of the maps may be the identity.', 'math-0205050-2-31-3': ""Nevertheless, we can define a 'local model' [MATH]."", 'math-0205050-2-31-4': ""It is isomorphic to some local model in the original sense (because as far as the local model is concerned, we can just omit the 'identity steps')."", 'math-0205050-2-32-0': 'The reflex field associated to a tuple [MATH] will in general be different from the one belonging to [MATH].', 'math-0205050-2-32-1': 'Thus the [MATH] will in general not be defined over the same ring as the naive local model [MATH] associated to the [MATH].', 'math-0205050-2-32-2': 'To simplify the situation, we make the following additional assumption: whenever [MATH], we have [MATH] for all [MATH].', 'math-0205050-2-32-3': 'Under this assumption, all [MATH] will be defined over [MATH], or even over a smaller ring.', 'math-0205050-2-32-4': 'If necessary we apply a base change, and in the following we consider all the [MATH] as [MATH]-schemes.', 'math-0205050-2-33-0': 'We then have a canonical map [EQUATION]', 'math-0205050-2-34-0': '# The case [MATH]', 'math-0205050-2-35-0': 'In this section, we will look at the (trivial) case where [MATH].', 'math-0205050-2-35-1': 'We will use the following observations in section [REF].', 'math-0205050-2-36-0': 'Let us give a description of the local model corresponding to a maximal parahoric subgroup (the standard local model in the sense of [CITATION], section 2).', 'math-0205050-2-36-1': '(Actually, in this case this coincides with the Iwahori type local model.)', 'math-0205050-2-37-0': 'So, we have integers [MATH], [MATH].', 'math-0205050-2-37-1': 'Let us assume, for notational convenience, that the [MATH] are in descending order, such that they are completely determined by [MATH].', 'math-0205050-2-38-0': 'The local model in this case is just [MATH], where [MATH] is the corresponding reflex field.', 'math-0205050-2-38-1': 'The point in the special fibre is the subspace [MATH] corresponding to the [MATH]-matrix [EQUATION] the operator [MATH] has Jordan type [MATH].', 'math-0205050-2-38-2': 'Let us denote the subspace corresponding to the [MATH]-valued point of the local model by [MATH].', 'math-0205050-2-38-3': 'The description of this subspace in terms of matrices is the following.', 'math-0205050-2-38-4': 'We have a matrix [EQUATION] such that [EQUATION] for some matrix [MATH] with characteristic polynomial [EQUATION].', 'math-0205050-2-38-5': 'Furthermore, the reduction of [MATH] modulo [MATH] is [MATH].', 'math-0205050-2-39-0': 'Example.', 'math-0205050-2-39-1': 'Assume that [MATH].', 'math-0205050-2-39-2': 'Now, if we let [MATH] and choose the [MATH] such that the resulting characteristic polynomial is [MATH], we get [EQUATION]', 'math-0205050-2-40-0': '# Lifting of points', 'math-0205050-2-41-0': 'The local model [MATH] is topologically flat, i. e. the generic points of the irreducible components can be lifted to the special fibre.', 'math-0205050-2-42-0': 'Remark.', 'math-0205050-2-42-1': 'The following easier, though less elementary, proof resulted from discussions with Haines.', 'math-0205050-2-42-2': 'It uses the theory of the splitting model developed by Pappas and Rapoport.', 'math-0205050-2-43-0': 'In [CITATION], Pappas and Rapoport define the splitting model, which is a twisted product of unramified local models, and which maps to the local model [MATH].', 'math-0205050-2-43-1': 'By definition, the canonical local model [MATH] is the image of this morphism.', 'math-0205050-2-44-0': 'It is not hard to see that the special fibre of [MATH] consists of the Schubert cells corresponding to the elements of the [MATH]-admissible set.', 'math-0205050-2-44-1': 'By the theorem of Haines and Ngo (and the generalization in section [REF], respectively), this set coincides with the [MATH]-permissible set, which parametrizes the Schubert cells in the special fibre of [MATH].', 'math-0205050-2-45-0': 'Since the splitting model is flat by [CITATION], the canonical local model is flat, and thus the local model [MATH] is topologically flat.', 'math-0205050-2-46-0': 'We now give a more direct proof of the proposition.', 'math-0205050-2-47-0': 'Proof.', 'math-0205050-2-47-1': 'In this section, we consider only the Iwahori case.', 'math-0205050-2-47-2': 'It will be obvious though, that the proof carries over to the general parahoric case without any problems once the results of Haines and Ngo are generalized correspondingly.', 'math-0205050-2-47-3': 'In section [REF] we will give a proof of the more general statement which we need.', 'math-0205050-2-48-0': 'We can embed the special fibre of [MATH] into the affine flag variety for [MATH] in the standard way, see [CITATION].', 'math-0205050-2-49-0': 'By the definition of the local model and since the special fibers of the maximal-parahoric new local models have the right stratification, the special fibre is the union of the Schubert cells corresponding to the [MATH]-permissible alcoves.', 'math-0205050-2-49-1': 'Here [MATH] is the dual partition to [MATH].', 'math-0205050-2-50-0': 'In order to show that the local model associated to some dominant coweight [MATH] is topologically flat, we need to have a good understanding of the set of irreducible components of its special fibre.', 'math-0205050-2-50-1': 'This amounts to a combinatorial problem in the extended affine Weyl group, namely to relating the so-called [MATH]-permissible and [MATH]-admissible sets.', 'math-0205050-2-51-0': 'By recent work of Haines and Ngo [CITATION] we know that in this case (as in the minuscule case), the set of [MATH]-permissible alcoves coincides with the set of [MATH]-admissible alcoves.', 'math-0205050-2-51-1': 'Thus the maximal elements (with respect to the Bruhat order), which correspond to the irreducible components of the special fibre, are just the conjugates of [MATH] under the finite Weyl group, i.e. the permutations of [MATH] under [MATH].', 'math-0205050-2-52-0': 'Choose a permutation [MATH] of [MATH] and write [MATH].', 'math-0205050-2-53-0': 'A key observation is the following lemma which came up in a discussion with T. Haines.', 'math-0205050-2-54-0': 'Let [MATH] be a discrete valuation ring, let [MATH] be the closed and [MATH] the generic point of [MATH].', 'math-0205050-2-54-1': 'Let [MATH] be an [MATH]-scheme of finite type, and [MATH] a closed point of the special fibre which lies in a unique irreducible component [MATH] of [MATH].', 'math-0205050-2-55-0': 'Furthermore, assume that [MATH] is irreducible, that [MATH], and that [MATH] can be lifted to a closed point of [MATH].', 'math-0205050-2-56-0': 'Then the generic point of [MATH] can be lifted to [MATH].', 'math-0205050-2-57-0': 'Proof.', 'math-0205050-2-57-1': 'Denote by [MATH] the scheme-theoretic closure of [MATH] in [MATH].', 'math-0205050-2-57-2': 'Then [MATH] is flat over [MATH], and [MATH].', 'math-0205050-2-57-3': 'Furthermore, [MATH] (see [CITATION], theorem 15.1), and [MATH], since [MATH] can be lifted to a closed point of [MATH].', 'math-0205050-2-58-0': 'Thus [MATH], which implies that [MATH].', 'math-0205050-2-58-1': '[MATH]', 'math-0205050-2-59-0': 'Although the lemma is easy to prove, it reduces the combinatorial difficulty of our task enormously.', 'math-0205050-2-60-0': 'The lemma shows that it really is enough to show that one suitably chosen closed point of the special fibre can be lifted to the generic fibre.', 'math-0205050-2-60-1': 'We will choose the point [MATH], where each [MATH] is given by the [MATH]-matrix [EQUATION].', 'math-0205050-2-61-0': 'It is clear that [MATH] lies in the stratum corresponding to [MATH].', 'math-0205050-2-61-1': '(Look at the Jordan type of [MATH].)', 'math-0205050-2-62-0': 'Now how can we lift this point to the generic fibre?', 'math-0205050-2-62-1': 'We will describe two different methods to do this.', 'math-0205050-2-62-2': 'The first one is more elementary, the second one more conceptual.', 'math-0205050-2-63-0': '## First method (elementary)', 'math-0205050-2-64-0': 'To simplify the notation, let us assume that the [MATH] are in descending order.', 'math-0205050-2-64-1': 'We will build the corresponding matrix (over [MATH]) by putting together several matrices of the form [MATH].', 'math-0205050-2-64-2': 'More precisely, consider the [MATH]-valued point where each subspace [MATH] is given by the same matrix [EQUATION].', 'math-0205050-2-65-0': 'Note that the coefficients of this matrix are indeed contained in [MATH].', 'math-0205050-2-65-1': 'Clearly, the reduction of [MATH] is [MATH].', 'math-0205050-2-65-2': 'We have to check that', 'math-0205050-2-66-0': 'But both conditions are clearly satisfied, since the [MATH] satisfy the corresponding conditions and since there are no interactions between the blocks.', 'math-0205050-2-66-1': 'Let us make that more precise.', 'math-0205050-2-67-0': 'To check the first condition, fix any [MATH].', 'math-0205050-2-67-1': 'Note that [MATH] is [MATH]-invariant: this means that there is a matrix [MATH] such that [EQUATION] and we can simply take [MATH].', 'math-0205050-2-67-2': 'Now let us check the determinant condition; the characteristic polynomial of [MATH] is just the product of the characteristic polynomials of the [MATH], so we get [EQUATION] as it should be.', 'math-0205050-2-68-0': 'The second condition is easily checked, too.', 'math-0205050-2-68-1': 'We just have to observe that [EQUATION].', 'math-0205050-2-69-0': 'Thus we have indeed found a lifting of our point to the generic fibre.', 'math-0205050-2-70-0': '## Second method (conceptual)', 'math-0205050-2-71-0': 'In this section, we will describe the [MATH]-valued point constructed in the previous section as a morphism from a product of trivial (i.e. [MATH]) local models to the given local model, which comes from the situation studied in section [REF].', 'math-0205050-2-72-0': 'As partition of [MATH], we will choose the partition into singleton sets: [MATH], [MATH].', 'math-0205050-2-72-1': 'Furthermore we choose [MATH] such that [EQUATION].', 'math-0205050-2-72-2': 'It is easy to see that such [MATH] exist and that they are uniquely determined.', 'math-0205050-2-72-3': 'Furthermore, whenever [MATH], we have [MATH] for all [MATH].', 'math-0205050-2-72-4': 'Thus all [MATH] are defined over a subring of [MATH], and we denote the base change to [MATH] by [MATH].', 'math-0205050-2-73-0': 'Now consider the map [EQUATION] associated to these data.', 'math-0205050-2-73-1': 'All the [MATH] are just isomorphic to [MATH], so their product is [MATH] again.', 'math-0205050-2-74-0': 'It is not hard to check is that the image of the closed point under this map is the point described above.', 'math-0205050-2-75-0': '# The case of the symplectic group', 'math-0205050-2-76-0': 'Finally, let us consider the question of topological flatness of the local model for the symplectic group.', 'math-0205050-2-76-1': 'Let us repeat, with slight notational modifications, the definition of the naive local model for the symplectic group given in [CITATION]; cf. also [CITATION].', 'math-0205050-2-76-2': 'To simplify the notation, we consider only the Iwahori case.', 'math-0205050-2-77-0': 'Consider a totally ramified extension [MATH] of degree [MATH] as before.', 'math-0205050-2-77-1': 'Let [MATH] with basis [MATH], and denote by [MATH] the standard symplectic pairing, i.e. [EQUATION].', 'math-0205050-2-77-2': 'Let [MATH] be an [MATH]-generator of the inverse different [MATH] (if [MATH] is tamely ramified over [MATH], we can take [MATH]).', 'math-0205050-2-77-3': 'Let [MATH].', 'math-0205050-2-77-4': 'This is a non-degenerate alternating form on [MATH] with values in [MATH].', 'math-0205050-2-78-0': 'The standard lattice chain [MATH] is self-dual with respect to [MATH].', 'math-0205050-2-79-0': 'Now let [MATH] for all [MATH].', 'math-0205050-2-79-1': 'Associated to [MATH], [MATH] and [MATH] we have the naive local model for [MATH].', 'math-0205050-2-79-2': 'In this case the reflex field is [MATH] and [MATH] is just [MATH].', 'math-0205050-2-80-0': ""The naive local model for the symplectic group is the closed subscheme of [MATH] consisting of 'self-dual' families of subspaces: [EQUATION]."", 'math-0205050-2-80-1': 'Here [MATH] denotes the [MATH]-dual [MATH], and the isomorphism [MATH] is the one given by the pairing.', 'math-0205050-2-81-0': 'Similarly, we define the local model [MATH] as the closed subscheme of [MATH] consisting of self-dual families of subspaces: [EQUATION].', 'math-0205050-2-81-1': 'Actually, we know that in this case the two local models for [MATH] coincide topologically.', 'math-0205050-2-81-2': 'Namely, it is enough to check this in the maximal parahoric case where the special fibre is irreducible, so that one just has to see that the generic and special fibres have the same dimension.', 'math-0205050-2-81-3': 'Of course this implies that [MATH] and [MATH] are topologically the same, too.', 'math-0205050-2-81-4': 'So in order to prove the topological flatness, we can work with either one.', 'math-0205050-2-82-0': 'The local model [MATH] is topologically flat, i.e. the generic points of the irreducible components of the special fibre of [MATH] can be lifted to the generic fibre.', 'math-0205050-2-82-1': 'The same holds for [MATH].', 'math-0205050-2-83-0': 'Proof.', 'math-0205050-2-83-1': 'Since the symplectic local model is defined as a subscheme of the linear local model, the strata of the special fibre of [MATH] correspond to the intersection [MATH] (in the Iwahori case).', 'math-0205050-2-84-0': 'The results of Haines and Ngo [CITATION] show that this set coincides with the [MATH]-admissible set for the symplectic group, and thus the irreducible components of the special fibre of the local model correspond to the conjugates of [MATH] under the Weyl group of the symplectic group (considered as a subgroup of [MATH]).', 'math-0205050-2-85-0': ""In section [REF] we generalize Haines' and Ngo's results to the parahoric case."", 'math-0205050-2-85-1': 'Since this is the only difference between the Iwahori case and the general parahoric case, until the end of this section we will assume that we are in the Iwahori case, to simplify notation.', 'math-0205050-2-86-0': 'As in the linear case, it is enough to show that for each irreducible component there is one (suitably chosen) point that can be lifted to the generic fibre.', 'math-0205050-2-87-0': 'In this case we cannot hope to find the lifting as the image of a map of a product of trivial local models for the symplectic group, because decomposing the lattice chain as we did in the linear case would not preserve the pairing.', 'math-0205050-2-87-1': 'Nevertheless the proof of the proposition is even simpler in this case, because the [MATH] we have to deal with is so special.', 'math-0205050-2-88-0': 'We consider the symplectic local model as a closed subscheme of the linear local model.', 'math-0205050-2-88-1': 'Let [MATH] be a conjugate of [MATH] under the Weyl group of the symplectic group.', 'math-0205050-2-88-2': 'It is enough to show that we can lift the point where each [MATH] is given by the [MATH]-matrix [EQUATION] (which obviously lies inside [MATH]) to the generic fibre (of [MATH]).', 'math-0205050-2-89-0': 'Let us first lift this point to a point in the generic fibre of the linear local model [MATH].', 'math-0205050-2-89-1': 'Afterwards we will show that the lifting actually lies in the symplectic model.', 'math-0205050-2-90-0': 'Since all the [MATH] are either [MATH] or [MATH], we can just lift this point by exactly the same matrix over [MATH]!', 'math-0205050-2-90-1': 'It is clear that these matrices do describe a point in the (linear) local model over [MATH], in particular that the determinant condition is satisfied.', 'math-0205050-2-91-0': 'But since [MATH] is not an arbitrary permutation of [MATH], but one under the Weyl group of the symplectic group, it can never happen that [MATH] and [MATH] are both 1.', 'math-0205050-2-91-1': ""In other words, if in the matrix above there is a unit matrix somewhere in rows [MATH], the columns [MATH] will entirely consist of 0's."", 'math-0205050-2-91-2': 'Taking into account that all the [MATH] are given by the same matrix, it is then clear that this point lies in the symplectic local model.', 'math-0205050-2-91-3': '[MATH]', 'math-0205050-2-92-0': 'We cannot prove that [MATH] (or [MATH]) is flat.', 'math-0205050-2-92-1': 'But since [MATH] is topologically flat, we are led to the', 'math-0205050-2-93-0': 'The naive local model [MATH] is flat.', 'math-0205050-2-94-0': 'To prove this conjecture, it would be sufficient to show that the special fibre is reduced.', 'math-0205050-2-94-1': 'This would follow if one could identify it with a union of Schubert varieties in some affine flag variety, and by the usual method using Frobenius splittings, it would be enough to do this for the local models which correspond to a maximal parahoric subgroup.', 'math-0205050-2-94-2': 'But even that seems to be a difficult problem in commutative algebra.', 'math-0205050-2-95-0': '# The parahoric case', 'math-0205050-2-96-0': 'In this section we will provide the generalization of the results of Haines and Ngo about the [MATH]-admissible and the [MATH]-permissible sets which is needed to prove the topological flatness in the general parahoric case.', 'math-0205050-2-97-0': 'Although we are interested only in the cases of [MATH] and [MATH], the [MATH]-admissible and [MATH]-permissible sets can be defined for any split connected reductive group.', 'math-0205050-2-97-1': 'We denote by [MATH] the extended affine Weyl group, and by [MATH] its subgroup of elements of length zero.', 'math-0205050-2-97-2': 'In other words, [MATH] is the stabilizer of the base alcove.', 'math-0205050-2-97-3': 'Then [MATH] is the semi-direct product of the affine Weyl group [MATH] and [MATH].', 'math-0205050-2-98-0': 'We fix a dominant coweight [MATH], and denote by [MATH] the unique element of [MATH] such that [MATH].', 'math-0205050-2-99-0': 'We denote by [MATH] the closure of the base alcove, and by [MATH] the convex hull of the translates of [MATH] under the finite Weyl group.', 'math-0205050-2-100-0': '(Kottwitz-Rapoport)', 'math-0205050-2-101-0': 'i) The [MATH]-permissible set [MATH] is the set of elements [MATH] such that [MATH] for all [MATH].', 'math-0205050-2-102-0': 'ii) The [MATH]-admissible set [MATH] is the set of [MATH] such that there exists [MATH] with [MATH].', 'math-0205050-2-103-0': 'It was shown by Kottwitz and Rapoport that these two sets coincide for [MATH] or [MATH] and minuscule [MATH], and that the [MATH]-admissible set is always contained in the [MATH]-permissible set.', 'math-0205050-2-104-0': 'i) If [MATH] is a dominant coweight for [MATH], then [MATH].', 'math-0205050-2-105-0': 'ii) If [MATH] is a multiple of the dominant minuscule coweight [MATH] for [MATH], then [MATH].', 'math-0205050-2-106-0': '(Haines and Ngo also show that in general the [MATH]-admissible and the [MATH]-permissible set do not coincide.)', 'math-0205050-2-107-0': 'It is clear that the set of irreducible components of the special fibre of [MATH] is exactly the set of maximal elements (with respect to the Bruhat order) of the [MATH]-permissible set.', 'math-0205050-2-107-1': 'The theorem says that these maximal elements are just the conjugates of [MATH] under the finite Weyl group.', 'math-0205050-2-108-0': 'The theorem as it stands relates to the Iwahori case.', 'math-0205050-2-108-1': 'To prove topological flatness in the general parahoric case, we need a generalized version which covers the parahoric case, too.', 'math-0205050-2-108-2': 'Clearly the Iwahori case is the most difficult among all the parahoric cases, and as we will show, the general parahoric case can be deduced from the Iwahori case relatively easily.', 'math-0205050-2-109-0': '## [MATH]', 'math-0205050-2-110-0': 'We use the notation of [CITATION].', 'math-0205050-2-110-1': 'Let us recall part of it:', 'math-0205050-2-111-0': 'Let [MATH] be a non-empty subset of [MATH], and denote by [MATH] its inverse image under the projection [MATH].', 'math-0205050-2-111-1': 'A family [MATH], [MATH], is called a face of type [MATH] if it satisfies the following conditions:', 'math-0205050-2-112-0': '[MATH] for all [MATH], [MATH] for all [MATH], [MATH], [MATH] for all [MATH].', 'math-0205050-2-113-0': 'We denote the set of faces of type [MATH] by [MATH].', 'math-0205050-2-114-0': 'Clearly, a face of type [MATH] is simply an alcove.', 'math-0205050-2-114-1': 'We denote by [MATH] the standard alcove [MATH], [MATH], [MATH], as well as the corresponding face of type [MATH], [MATH].', 'math-0205050-2-115-0': 'The extended affine Weyl group [MATH] acts transitively on the set [MATH].', 'math-0205050-2-115-1': 'Taking [MATH] as a base point, we identify [MATH] with the set [MATH], where [MATH] is the stabilizer of [MATH] in [MATH].', 'math-0205050-2-116-0': 'If [MATH] is a non-empty subset, and [MATH] its inverse image under the projection [MATH], we have a [MATH]-equivariant surjection [MATH], defined by [MATH].', 'math-0205050-2-117-0': 'We can now adapt the definition above to the parahoric case:', 'math-0205050-2-118-0': '(Kottwitz-Rapoport)', 'math-0205050-2-119-0': 'i) The [MATH]-permissible set [MATH] is the set of elements [MATH] such that for all [MATH], [MATH].', 'math-0205050-2-120-0': 'ii) The [MATH]-admissible set [MATH] is the image of [MATH] under the surjection [MATH].', 'math-0205050-2-121-0': 'Let [MATH] be a non-empty subset.', 'math-0205050-2-121-1': 'Then the restriction of the map [MATH] to [MATH] is a surjection [MATH].', 'math-0205050-2-122-0': 'Proof.', 'math-0205050-2-122-1': 'It is clearly enough to prove the proposition for [MATH].', 'math-0205050-2-122-2': ""Given [MATH] we can then fill in the missing [MATH]'s step by step, and thus are reduced to the following lemma."", 'math-0205050-2-122-3': '[MATH]', 'math-0205050-2-123-0': 'Let [MATH], [MATH] be integers, [MATH], and let [MATH].', 'math-0205050-2-123-1': 'Assume that [MATH], [MATH], [MATH] is minuscule and [MATH].', 'math-0205050-2-124-0': 'Then there exists [MATH] such that [MATH], [MATH] and [MATH] are minuscule and [MATH].', 'math-0205050-2-125-0': 'Proof.', 'math-0205050-2-125-1': 'For [MATH] we have [MATH] if and only if [MATH], where [MATH] is dominant.', 'math-0205050-2-126-0': 'We denote the standard basis vectors of [MATH] by [MATH].', 'math-0205050-2-126-1': 'For [MATH], let [MATH] be the unique integer with [MATH], [MATH].', 'math-0205050-2-126-2': 'We define [MATH] to be the basis vector [MATH].', 'math-0205050-2-127-0': 'We are looking for [MATH], [MATH], such that [MATH] with [EQUATION] satisfies the conditions above, i. e. such that', 'math-0205050-2-128-0': '[MATH] is minuscule, i. e. [MATH]', 'math-0205050-2-129-0': '(The other two conditions are satisfied automatically.)', 'math-0205050-2-130-0': 'Let [MATH] be the unique integer in [MATH] such that [MATH].', 'math-0205050-2-131-0': '1st case.', 'math-0205050-2-131-1': '[MATH]', 'math-0205050-2-132-0': 'In this case we can simply choose [MATH].', 'math-0205050-2-133-0': '2nd case.', 'math-0205050-2-133-1': '[MATH]', 'math-0205050-2-134-0': 'This case is more complicated.', 'math-0205050-2-134-1': 'Let [MATH] be such that [MATH] is dominant, and among all such [MATH], choose [MATH] with [MATH] maximal.', 'math-0205050-2-135-0': 'Let [EQUATION] and [EQUATION]', 'math-0205050-2-135-1': 'Let [MATH].', 'math-0205050-2-135-2': 'Obviously this implies [MATH].', 'math-0205050-2-135-3': 'So all that remains to show is that [MATH] with this choice of [MATH].', 'math-0205050-2-136-0': 'By the definition of [MATH] and [MATH], [MATH] is dominant, too.', 'math-0205050-2-136-1': 'Thus it is enough to show that [MATH].', 'math-0205050-2-137-0': 'If [MATH], then this is clear.', 'math-0205050-2-137-1': 'So let us now consider the case [MATH].', 'math-0205050-2-138-0': 'We write [MATH].', 'math-0205050-2-138-1': 'We have to show that [EQUATION] for [MATH].', 'math-0205050-2-138-2': '(It is clear that for [MATH] we have equality since [MATH].)', 'math-0205050-2-139-0': '- For [MATH] and for [MATH], we have [EQUATION] because [MATH].', 'math-0205050-2-140-0': '- Let [MATH].', 'math-0205050-2-141-0': 'We have [MATH], and for all [MATH] we have [MATH] and thus [MATH].', 'math-0205050-2-141-1': 'Since furthermore [MATH], we see that [EQUATION].', 'math-0205050-2-142-0': 'So we have [EQUATION] - Finally, consider [MATH].', 'math-0205050-2-143-0': 'We know that [MATH], since [MATH], and we want to show that for [MATH] we even have [MATH] here.', 'math-0205050-2-143-1': 'For [MATH] this is certainly true.', 'math-0205050-2-144-0': 'Now suppose we had [MATH] for some [MATH], [MATH].', 'math-0205050-2-144-1': 'This implies [EQUATION] and thus [MATH] (because for [MATH] all [MATH] are equal).', 'math-0205050-2-144-2': 'But then we get [EQUATION] which is a contradiction.', 'math-0205050-2-145-0': 'The lemma is proved.', 'math-0205050-2-145-1': '[MATH]', 'math-0205050-2-146-0': 'Let [MATH] be a dominant coweight for [MATH], and let [MATH] be the inverse image of a non-empty subset [MATH].', 'math-0205050-2-146-1': 'Then [MATH].', 'math-0205050-2-147-0': 'Proof.', 'math-0205050-2-147-1': 'The [MATH]-admissible set is always contained in the [MATH]-permissible set.', 'math-0205050-2-147-2': 'Since we know that in the Iwahori case the two sets coincide, and because we have surjections [MATH] (by the proposition) and [MATH] (obvious), the corollary follows.', 'math-0205050-2-147-3': '[MATH]', 'math-0205050-2-148-0': '## [MATH]', 'math-0205050-2-149-0': 'Now let [MATH].', 'math-0205050-2-149-1': 'Since the proofs for the symplectic group are based on reductions to the linear case, we use a subscript [MATH] to denote data corresponding to the symplectic group; notation without subscript refers to the [MATH]-case, as in the previous section.', 'math-0205050-2-150-0': 'We denote by [MATH] the automorphism given by [MATH].', 'math-0205050-2-151-0': ""This automorphism acts on the root system of [MATH], and the '[MATH]-invariant part', denoted [MATH], is the root system of [MATH]."", 'math-0205050-2-151-1': 'See [CITATION], 9, 10.', 'math-0205050-2-151-2': 'In particular, the extended affine Weyl group [MATH] for the general symplectic group is a subgroup of the extended affine Weyl group for the general linear group.', 'math-0205050-2-151-3': 'Furthermore, by [CITATION] Proposition 9.6, the Bruhat order on [MATH] is inherited from the Bruhat order on [MATH].', 'math-0205050-2-152-0': ""The vectors [MATH], [MATH], serve as 'vertices' of the base alcove for the symplectic group."", 'math-0205050-2-153-0': 'Let [MATH] be a dominant coweight for [MATH].', 'math-0205050-2-153-1': 'We can consider the [MATH]-admissible set [MATH] and the [MATH]-permissible set [MATH] as subsets of [MATH].', 'math-0205050-2-153-2': 'We have [EQUATION]', 'math-0205050-2-153-3': 'Here we denote by [MATH] the convex hull of the translates of [MATH] under the finite Weyl group (of [MATH]).', 'math-0205050-2-154-0': 'Now let [MATH] be a non-empty symmetric subset, i.e. a non-empty subset such that its inverse image [MATH] under the projection [MATH] satisfies [MATH].', 'math-0205050-2-154-1': '(Obviously there is a one-to-one correspondence between symmetric subsets of [MATH] and subsets of [MATH].)', 'math-0205050-2-154-2': 'We describe the set of [MATH]-faces of type [MATH] as a subset of [MATH]: A [MATH]-face of type [MATH] is a face [MATH] for [MATH] such that there exists [MATH] with [EQUATION] for all [MATH].', 'math-0205050-2-154-3': 'Clearly a [MATH]-face of type [MATH] is simply an alcove.', 'math-0205050-2-154-4': 'We denote the set of [MATH]-faces of type [MATH] by [MATH].', 'math-0205050-2-155-0': 'We obtain a commutative diagram', 'math-0205050-2-156-0': 'W_G / W_G,I & ^& F_G, I', 'math-0205050-2-157-0': '& &', 'math-0205050-2-158-0': 'W / W_I & ^& F_I.', 'math-0205050-2-159-0': 'Here [MATH] is the stabilizer of [MATH] in [MATH].', 'math-0205050-2-160-0': 'We can now define parahoric versions of the admissible and permissible sets.', 'math-0205050-2-161-0': 'i) The [MATH]-permissible set [MATH] is the set of elements [MATH] such that for all [MATH], [MATH].', 'math-0205050-2-162-0': 'ii) The [MATH]-admissible set [MATH] is the image of [MATH] under the surjection [MATH].', 'math-0205050-2-163-0': 'It turns out, however, that the set which naturally describes the strata occuring in the special fibre of a local model is not the [MATH]-permissible set but the intersection [MATH].', 'math-0205050-2-163-1': 'The goal of this section is to show that actually [EQUATION].', 'math-0205050-2-164-0': 'The key point is to show that the natural map [MATH] is surjective.', 'math-0205050-2-165-0': 'If [MATH] is a non-empty symmetric subset, we have a [MATH]-equivariant surjection [MATH].', 'math-0205050-2-166-0': 'Now suppose [MATH] are non-empty symmetric subsets, such that [MATH] for some [MATH] with [MATH].', 'math-0205050-2-166-1': '(For an integer [MATH] we denote by [MATH] its class in [MATH].)', 'math-0205050-2-166-2': 'Let [MATH] be the smallest integer in [MATH] which is greater than [MATH].', 'math-0205050-2-167-0': '([CITATION], Lemma 10.3) In the situation above we have a bijection [EQUATION].', 'math-0205050-2-168-0': 'Let [MATH] be a positive multiple of the dominant minuscule coweight [MATH] for [MATH].', 'math-0205050-2-168-1': 'Let [MATH] be a non-empty symmetric subset.', 'math-0205050-2-168-2': 'Then the natural map [MATH] is surjective.', 'math-0205050-2-169-0': 'Proof.', 'math-0205050-2-169-1': 'It is clearly enough to consider [MATH] as in the lemma.', 'math-0205050-2-169-2': 'Let [MATH].', 'math-0205050-2-169-3': 'We would like to extend this face to a [MATH]-face of type [MATH] by defining suitable [MATH].', 'math-0205050-2-169-4': 'By the lemma above and lemma [REF], we find a [MATH]-face [MATH] of type [MATH] which maps to [MATH] under [MATH] and such that [MATH].', 'math-0205050-2-170-0': 'We have to show that then [MATH] automatically holds, too.', 'math-0205050-2-170-1': 'Now [MATH] is of the form [MATH] for some [MATH], and [EQUATION] since [MATH] is a [MATH]-face.', 'math-0205050-2-171-0': 'Since [MATH] has this special form, for a dominant coweight [MATH] we have [MATH] if and only if [MATH] for all [MATH] (and [MATH]).', 'math-0205050-2-171-1': 'If this holds for some [MATH], it clearly holds for [MATH] as well, so we are done.', 'math-0205050-2-171-2': '[MATH]', 'math-0205050-2-172-0': 'Let [MATH] be a positive multiple of the dominant coweight for [MATH], and let [MATH] be a non-empty symmetric subset.', 'math-0205050-2-172-1': 'Then [MATH].', 'math-0205050-2-173-0': 'Proof.', 'math-0205050-2-173-1': 'First, we have an inclusion [MATH].', 'math-0205050-2-173-2': ""This is theorem 10.1 in [CITATION] for [MATH], and it is easy to see that this is a 'vertex-by-vertex' proof, i.e. it works for arbitrary I."", 'math-0205050-2-174-0': 'Now recall that in addition we know that in any case the [MATH]-admissible set is contained in the [MATH]-permissible set.', 'math-0205050-2-175-0': 'Since [MATH] and [MATH] both coincide with the image of [MATH], the corollary follows.', 'math-0205050-2-175-1': '[MATH]'}","[['math-0205050-1-24-0', 'math-0205050-2-28-0'], ['math-0205050-1-24-1', 'math-0205050-2-28-1'], ['math-0205050-1-3-0', 'math-0205050-2-5-0'], ['math-0205050-1-3-1', 'math-0205050-2-5-1'], ['math-0205050-1-3-2', 'math-0205050-2-5-2'], ['math-0205050-1-3-3', 'math-0205050-2-5-3'], ['math-0205050-1-3-4', 'math-0205050-2-5-4'], ['math-0205050-1-39-1', 'math-0205050-2-49-1'], ['math-0205050-1-6-1', 'math-0205050-2-9-1'], ['math-0205050-1-6-2', 'math-0205050-2-9-2'], 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1509.04181,"{'1509.04181-1-0-0': 'The [MATH] and [MATH] contributions recently calculated in Phys.', '1509.04181-1-0-1': 'Rev. D 88, 114001 (2013) are revised taking into account asymptotics prediction based on QCD.', '1509.04181-1-1-0': 'The [MATH] and [MATH] reactions allow the comparative study of the production of scalar [MATH] and [MATH] and tensor [MATH] and [MATH] mesons.', '1509.04181-1-1-1': 'While the last are well-known P-wave [MATH] states, there are many indications that the light scalar mesons are four quark states.', '1509.04181-1-2-0': 'Recently the cross-sections of these reactions were calculated in Ref. [CITATION].', '1509.04181-1-2-1': 'The [MATH] and [MATH] are [MATH] pb at 1.5 GeV and [MATH] pb at [MATH] GeV, while the [MATH] and [MATH] were estimated several pb at [MATH] GeV.', '1509.04181-1-3-0': 'The [MATH] is given by Eq. (9) of Ref. [CITATION]: [EQUATION]', '1509.04181-1-3-1': 'Here [MATH] are inverse propagators of vector mesons V, any of [MATH]: [EQUATION] the forms of widths [MATH] are taken from Ref. [CITATION].', '1509.04181-1-4-0': 'The [MATH] width is [CITATION] [EQUATION]', '1509.04181-1-4-1': 'In Ref. [CITATION] the asymptotics at [MATH] was not agreed with QCD asymptotics of [MATH], to within logarithms equal to [MATH], where [MATH] are charges of [MATH], [MATH], and [MATH] quarks.', '1509.04181-1-4-2': 'In our case Eq. ([REF]) the asymptotics is [CITATION] [EQUATION] where [EQUATION].', '1509.04181-1-5-0': 'So this agreement requires [MATH], i.e. radial vector excitations contribution cancels the contribution of vector bound states [MATH] and [MATH] in the leading order: [EQUATION]', '1509.04181-1-5-1': 'Note the same condition was obtained in Ref. [CITATION] after taking into account the QCD based asymptotics of the [MATH] amplitude.', '1509.04181-1-6-0': 'Denoting as in Ref. [CITATION] [MATH] and [MATH] the requirement Eq. ([REF]) may be rewritten as [EQUATION]', '1509.04181-1-6-1': 'The parameters of vector mesons ([MATH], [MATH] and coupling constants) are not well-established.', '1509.04181-1-6-2': 'We will use the same ones as in Ref. [CITATION], they are close to values obtained in Ref. [CITATION].', '1509.04181-1-6-3': 'The estimations were calculated in Ref. [CITATION] for the case [MATH], now we change these relations to fulfill Eq. ([REF]).', '1509.04181-1-7-0': 'Let us take the masses of radial excitations as in Ref. [CITATION]: [EQUATION] and consider three pairs [MATH] describing different possible regimes and not far from ones chosen in Ref. [CITATION].', '1509.04181-1-8-0': 'In case [MATH] the weight of [MATH] and [MATH] excitations contribution is small for [MATH] GeV in comparison with [MATH] and [MATH] contribution.', '1509.04181-1-8-1': 'The [MATH] is about [MATH] pb for these constants, see Fig. [REF]a.', '1509.04181-1-8-2': 'It is very hard to observe such a cross section on the current experimental level, and the background situation should be treated accurately in this case.', '1509.04181-1-9-0': 'In cases [MATH] and [MATH] cross section is large enough to be observed, see Fig. [REF]a.', '1509.04181-1-9-1': 'The background is much smaller than signal, see Ref. [CITATION].', '1509.04181-1-10-0': 'The [MATH] treatment is similar [CITATION]: [EQUATION] and [EQUATION]', '1509.04181-1-10-1': 'The QCD asymptotics requires [EQUATION]', '1509.04181-1-10-2': 'Denoting [MATH] and [MATH] this condition may be rewritten as [EQUATION]', '1509.04181-1-10-3': 'The pairs [MATH] provide three variants shown in Fig. [REF]b.', '1509.04181-1-11-0': 'Recently the [MATH] cross section was measured at BABAR at [MATH] GeV: [MATH] fb [CITATION].', '1509.04181-1-11-1': 'To compare our results with this one, we assumed that the imaginary parts of [MATH], [MATH] and [MATH] tend to constants at [MATH] and take this constants as values at [MATH] GeV.', '1509.04181-1-11-2': 'Taking into account that [MATH], we obtain for the considered [MATH] variants [MATH] fb, [MATH] fb and [MATH] fb correspondingly.', '1509.04181-1-11-3': 'As this estimate is only a guide, one can conclude that the considered variants do not contradict the BABAR result.'}","{'1509.04181-2-0-0': 'The predictions for transition form factors [MATH] and [MATH] and corresponding [MATH] and [MATH] cross sections are obtained for the energy region up to [MATH] GeV.', '1509.04181-2-0-1': 'These predictions are coordinated with the recent Belle data on the [MATH] transition.', '1509.04181-2-0-2': 'It is shown that the QCD asymptotics of the amplitudes of the reactions [MATH] and [MATH] can be reached only by taking into account a compensation of contributions of [MATH], [MATH] with contributions of their radial excitations.', '1509.04181-2-0-3': 'The relation [MATH], obtained with the help of the [MATH] model of [MATH] and [MATH] and QCD, is used to determine the [MATH] at high energies.', '1509.04181-2-0-4': 'Recent BABAR measurement of the [MATH] cross section at [MATH] GeV gives hope for detailed investigation of the [MATH] and [MATH] transition form factors at high energy region.', '1509.04181-2-1-0': '# Introduction', '1509.04181-2-2-0': 'The [MATH] and [MATH] transition form factors manifest themselves in different reactions.', '1509.04181-2-2-1': 'In [MATH] processes they are in the time-like region of [MATH], while in [MATH] reactions these form factors are in the space-like region of [MATH].', '1509.04181-2-2-2': 'These reactions are connected with each other, so progress in the investigation of one reaction can shed light of the others.', '1509.04181-2-3-0': 'Recently [MATH] transition form factor in the space-like region have been measured at Belle for [MATH] up to [MATH] GeV[MATH] [CITATION].', '1509.04181-2-3-1': 'Besides, BABAR collaboration measured the [MATH] cross section at [MATH] GeV [CITATION], unfortunately, with large error.', '1509.04181-2-3-2': 'The last result shows that it is possible to investigate the [MATH] and [MATH] transition form factors in the time-like region at extremely high energy ([MATH] GeV[MATH] and higher) in detail.', '1509.04181-2-4-0': 'It will be shown below that these transition form factors are of interest in the whole accessible [MATH] region.', '1509.04181-2-4-1': 'The point is that the QCD asymptotics can be reached only by taking into account cancellation of contributions of [MATH] and [MATH] with contributions of their radial excitations.', '1509.04181-2-5-0': 'The Belle data on the [MATH] transition form factor was analyzed in Ref. [CITATION].', '1509.04181-2-5-1': 'In comparison to that paper, we take into account [MATH] and [MATH] contribution in the [MATH] leg (the [MATH] and [MATH] were taken into account in Ref. [CITATION]).', '1509.04181-2-5-2': 'This leads to modification of the Eq. (10) of the Ref. [CITATION] only.', '1509.04181-2-6-0': 'The [MATH] cross sections were estimated in Ref. [CITATION] under assumptions [MATH] and [MATH].', '1509.04181-2-6-1': 'The recent experimental data, the requirement of QCD asymptotics and the naive quark model relations taken into account allow to make prediction on quite a different level.', '1509.04181-2-7-0': 'In Sec. [REF] the [MATH] cross section is presented.', '1509.04181-2-7-1': 'It is shown that the cross section should be larger than the Ref. [CITATION] estimation, it could be measured at the energies [MATH] GeV at modern colliders, for example, at VEPP-2000.', '1509.04181-2-7-2': 'Our estimation of the cross section at the BABAR energy is smaller than the BABAR result.', '1509.04181-2-8-0': 'In Sec. [REF] we present the [MATH] cross section.', '1509.04181-2-8-1': 'In the [MATH] model for [MATH] and [MATH], [MATH] and [MATH], the [MATH] and [MATH] transition form factors are related as [MATH] at high [MATH] (in agreement with QCD), one could estimate the [MATH] cross section at high energies using this relation.', '1509.04181-2-9-0': 'In Sec. [REF] the brief conclusion is presented and the perspectives are discussed.', '1509.04181-2-10-0': '# The [MATH] and [MATH] cross sections', '1509.04181-2-11-0': '## The [MATH]', '1509.04181-2-12-0': 'It is known that in the reaction [MATH] tensor mesons are produced mainly by the photons with the opposite helicity states.', '1509.04181-2-12-1': 'The effective Lagrangian in this case is [EQUATION] where [MATH] is a photon field and [MATH] is a tensor [MATH] field.', '1509.04181-2-12-2': 'For [MATH] transition [EQUATION] where [MATH] is the transition form factor.', '1509.04181-2-13-0': 'In the frame of Generalized Vector Dominance Model (GVDM) we assume that the effective Lagrangian of the reaction [MATH] is [CITATION]: [EQUATION] where [MATH].', '1509.04181-2-13-1': 'It is assumed that [MATH] and other constants with crossed vector mesons are suppressed due to small overlap of the spatial wave functions of [MATH] and similar for [MATH]: [EQUATION]', '1509.04181-2-13-2': 'Assuming the GVDM mechanism [MATH] one obtains [CITATION] [EQUATION]', '1509.04181-2-13-3': 'Here [MATH] are inverse propagators of vector mesons V: [EQUATION] the forms of widths [MATH] are described below.', '1509.04181-2-13-4': 'The constants [MATH] are related to [MATH] widths as usual: [EQUATION]', '1509.04181-2-13-5': 'Let us denote [MATH] and similar for [MATH], [MATH], [MATH].', '1509.04181-2-13-6': 'In the [MATH] model for [MATH] [EQUATION]', '1509.04181-2-13-7': 'The [MATH] width is [CITATION] [EQUATION] [MATH] and the factor [MATH] takes into account [MATH] contributions.', '1509.04181-2-14-0': 'The QCD asymptotics of [MATH] at [MATH] is equal to [MATH] to within logarithms, where [MATH] are charges of [MATH], [MATH], and [MATH] quarks.', '1509.04181-2-14-1': 'Due to asymptotic freedom at [MATH] imaginary part of [MATH] vanishes [EQUATION] and [EQUATION]', '1509.04181-2-14-2': 'It means that [EQUATION] so [EQUATION] should be equal to zero, i.e. radial vector excitations contribution compensates the contribution of vector bound state [MATH] in the leading order: [EQUATION]', '1509.04181-2-14-3': 'Note the analogous condition was obtained in Refs. [CITATION] after taking into account the QCD based asymptotics of the [MATH] amplitude.', '1509.04181-2-15-0': 'Denoting like in Ref. [CITATION] [MATH] and [MATH] the requirement Eq. ([REF]) may be rewritten as [EQUATION]', '1509.04181-2-15-1': 'In these terms Eqs. ([REF]) and ([REF]) give [EQUATION]', '1509.04181-2-15-2': 'Since [MATH] contributions are suppressed, we neglect the difference between [MATH] and [MATH] in [MATH] to simplify the formula for transition form factor: [EQUATION]', '1509.04181-2-15-3': 'The parameters of vector radial excitations ([MATH], [MATH] and coupling constants) are not well-established.', '1509.04181-2-15-4': 'For the aims of this work in region [MATH] GeV we use propagators in the form Eq. ([REF]) with constant widths with parameters taken from Ref. [CITATION]: [EQUATION]', '1509.04181-2-15-5': 'Note one can use more complicated forms for [MATH] from Ref. [CITATION].', '1509.04181-2-16-0': 'To find the [MATH] pair we need one more source of information.', '1509.04181-2-16-1': 'Recently the Belle data on the [MATH] transition form factor [CITATION] were analyzed in Ref. [CITATION] without taking [MATH] into account, i.e. [MATH] in terms of this work.', '1509.04181-2-16-2': 'To take [MATH] into account it is enough to modify Eq. (10) of that paper: [EQUATION]', '1509.04181-2-16-3': 'This gives two minimums of the [MATH] function, constructed for the Belle data: [MATH] and [MATH].', '1509.04181-2-16-4': 'So then we will consider three Cases,', '1509.04181-2-17-0': 'Case 1: [MATH] - best minimum of the [MATH] function ([MATH]);', '1509.04181-2-18-0': 'Case 2: [MATH] - second minimum of the [MATH] function ([MATH]);[EQUATION]', '1509.04181-2-18-1': 'In Case 1 the [MATH] contribution is small, it is not far from Case 3.', '1509.04181-2-18-2': 'The [MATH] value in Case 2 is worse than in Cases 1 and 3, but since our consideration should be treated as a guide, this minimum is also interesting for investigation.', '1509.04181-2-19-0': 'All these three Cases are shown on Fig. [REF].', '1509.04181-2-19-1': 'In Case 3 (Fig. [REF]b) the [MATH] peak is larger than in Fig. [REF]a at [MATH] times.', '1509.04181-2-20-0': 'The [MATH] cross section is shown on Fig. [REF] for both minimums.', '1509.04181-2-20-1': 'One can see that Cases with small and essential [MATH] contribution differ a lot: in Case 1 the cross section peak is [MATH] pb, in the second one it is two orders of magnitude higher.', '1509.04181-2-20-2': 'In both Cases this cross section could be measured on the modern accelerators.', '1509.04181-2-21-0': 'Note that maximum of the [MATH] cross section was estimated [MATH] pb in Ref. [CITATION].', '1509.04181-2-21-1': 'The peak [MATH] nb in Case 2 is sensitive to the [MATH] width that is not well established.', '1509.04181-2-21-2': 'If [MATH] was [MATH] MeV (twice as much), the peak would reduce to [MATH] nb.', '1509.04181-2-22-0': 'One can estimate relations between [MATH] couplings to vector pairs using values obtained in Ref. [CITATION] [EQUATION] and get [EQUATION]', '1509.04181-2-22-1': 'Recently the [MATH] cross section was measured at BABAR at [MATH] GeV: [MATH] fb [CITATION].', '1509.04181-2-23-0': 'Taking into account [MATH] and Eq. ([REF]), [MATH] at [MATH], we find [MATH] fb for [MATH] and [MATH] fb for [MATH].', '1509.04181-2-23-1': 'The elaboration of the pioneer measurement of [MATH] is important for the [MATH] transition form factor investigations.', '1509.04181-2-24-0': '## The [MATH]', '1509.04181-2-25-0': 'The [MATH] treatment is similar [CITATION]: [EQUATION] and [EQUATION]', '1509.04181-2-25-1': 'The QCD asymptotics requires [EQUATION] here Eqs. ([REF]) were used.', '1509.04181-2-25-2': 'Denoting [MATH] and [MATH] the condition Eq. ([REF]) may be rewritten as [EQUATION]', '1509.04181-2-25-3': 'Note that in the naive [MATH] model for [MATH] [EQUATION]', '1509.04181-2-25-4': 'Together with Eqs. ([REF]) it means that [EQUATION]', '1509.04181-2-25-5': 'Eqs. ([REF]) lead to [EQUATION] with transition form factor [EQUATION]', '1509.04181-2-25-6': 'QCD also reqiures at high [MATH] [EQUATION]', '1509.04181-2-25-7': ""It's easy to see that our [MATH] consideration, Eqs. ([REF]), ([REF]), ([REF]) and ([REF]), agrees with Eq. ([REF])."", '1509.04181-2-26-0': 'It is natural that our consideration also provides the relation [EQUATION] well-known in the [MATH] model, which is in perfect agreement with the data, see Eqs. ([REF]) and ([REF]).', '1509.04181-2-27-0': 'Note that using the above-mentioned BABAR measurement [MATH] fb, [MATH] and Eq. ([REF]) one can obtain [EQUATION]', '1509.04181-2-27-1': 'On the Fig. [REF] we show the [MATH] with sets [MATH] as in Eqs. ([REF]), see Eq. ([REF]).', '1509.04181-2-27-2': 'We take [CITATION] [EQUATION]', '1509.04181-2-27-3': 'In our approach [MATH] relations are the same as Eq. ([REF]).', '1509.04181-2-28-0': '# Conclusion', '1509.04181-2-29-0': 'It is shown that the QCD asymptotics of the amplitudes of the reactions [MATH] and [MATH] can be reached only by taking into account a compensation of contributions of [MATH], [MATH] with contributions of their radial excitations.', '1509.04181-2-29-1': 'It is shown also that the [MATH] and [MATH] excitation contribution to the [MATH] and [MATH] cross sections is essential and allows the experimental investigation of [MATH] and [MATH] processes, for example, at VEPP-2000 collider.', '1509.04181-2-29-2': 'Note that the best channel to study the [MATH] production is the [MATH] because [MATH] [CITATION] and the background is expected to be small.', '1509.04181-2-30-0': 'At the BABAR energy [MATH] GeV our estimation of the [MATH] cross section is much less than the average value presented in [CITATION], but the experimental error is half of the average value, so no conclusions concerning this disagreement could be done now.', '1509.04181-2-30-1': 'Anyway, the pioneer BABAR measurement gives hope that the detailed investigation of the [MATH] and [MATH] transition form factors at high energy region is possible.', '1509.04181-2-31-0': 'Emphasize that the results obtained in Sec. II do not represent a precise prediction, they should be treated as a guide.'}","[['1509.04181-1-10-0', '1509.04181-2-25-0'], ['1509.04181-1-6-0', '1509.04181-2-15-0'], ['1509.04181-1-6-1', '1509.04181-2-15-3'], ['1509.04181-1-5-1', '1509.04181-2-14-3'], ['1509.04181-1-3-1', '1509.04181-2-13-3'], ['1509.04181-1-10-1', '1509.04181-2-25-1'], ['1509.04181-1-10-2', '1509.04181-2-25-2'], ['1509.04181-1-5-0', '1509.04181-2-14-2'], ['1509.04181-1-11-0', '1509.04181-2-22-1'], ['1509.04181-1-11-2', '1509.04181-2-23-0']]","[['1509.04181-1-10-0', '1509.04181-2-25-0']]","[['1509.04181-1-6-0', '1509.04181-2-15-0'], ['1509.04181-1-6-1', '1509.04181-2-15-3'], ['1509.04181-1-5-1', '1509.04181-2-14-3']]",[],"[['1509.04181-1-3-1', '1509.04181-2-13-3'], ['1509.04181-1-10-1', '1509.04181-2-25-1'], ['1509.04181-1-10-2', '1509.04181-2-25-2'], ['1509.04181-1-5-0', '1509.04181-2-14-2']]","[['1509.04181-1-11-0', '1509.04181-2-22-1'], ['1509.04181-1-11-2', '1509.04181-2-23-0']]","['1509.04181-1-3-0', '1509.04181-1-4-0', '1509.04181-1-4-1', '1509.04181-1-4-2', '1509.04181-2-16-4', '1509.04181-2-25-7']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1509.04181,,,
0812.4765,"{'0812.4765-1-0-0': 'We show that any entropy solution [MATH] of a convection diffusion equation [MATH] in [MATH] belongs to [MATH].', '0812.4765-1-0-1': 'The proof does not use the uniqueness of the solution.', '0812.4765-1-1-0': '# The problem, and main result', '0812.4765-1-2-0': 'Convection diffusion equations appear in a large class of problems, and have been widely studied.', '0812.4765-1-2-1': 'We consider in the sequel only equations under conservative form: [EQUATION] so that we can give some sense to [REF] in the distributional sense.', '0812.4765-1-2-2': 'In this paper, we consider entropy solutions of [REF] that do not take into account any boundary condition, or condition for [MATH].', '0812.4765-1-3-0': 'The proof does not use a [MATH]-contraction principle (see e.g. Alt Luckaus [CITATION] or Otto [CITATION]), so that it can be applied in case where uniqueness is not insured, like for example complex spatial coupling of different conservation laws as in [CITATION], or for cases where uniqueness fails because of boundary conditions or conditions at [MATH], as it will be stressed in the sequel.', '0812.4765-1-4-0': 'Let us now state the required assumptions on the data.', '0812.4765-1-4-1': 'Let [MATH] be an open subset of [MATH]), and let [MATH] be a positive real value or [MATH].', '0812.4765-1-4-2': '[EQUATION]', '0812.4765-1-4-3': 'One has to make the following assumption on the source term: [EQUATION]', '0812.4765-1-4-4': 'In the sequel, [MATH] (resp. [MATH]) denotes [MATH] (resp. [MATH]), and [MATH] is the function defined by [EQUATION].', '0812.4765-1-5-0': 'We consider entropy weak solutions of [REF], as in the famous work of Kruzkov [CITATION] for hyperbolic equations.', '0812.4765-1-5-1': 'This notion can be extended to degenerated parabolic equations, as noticed by Carrillo [CITATION].', '0812.4765-1-5-2': 'This leads to the following definition of entropy weak solution:', '0812.4765-1-6-0': 'A function [MATH] is said to be an entropy weak solution if:', '0812.4765-1-7-0': '[MATH], [MATH], [MATH], [MATH], [MATH], [EQUATION]', '0812.4765-1-7-1': 'Suppose first that [MATH] is a continuous function, then the fact that any weak solution [MATH] is an entropy weak solution is just based on a convexity inequality, and on the fact that [MATH] for all [MATH].', '0812.4765-1-7-2': 'More details are available in [CITATION] (see also [CITATION]).', '0812.4765-1-8-0': 'The fact that an entropy weak solution [MATH] is a weak solution is obvious if [MATH] belongs to [MATH] (consider [MATH]).', '0812.4765-1-9-0': 'Suppose now that [MATH] only belongs to [MATH].', '0812.4765-1-9-1': 'Let [MATH], then for all [MATH], one has [EQUATION] which added to [REF] yields: [MATH], [EQUATION]', '0812.4765-1-9-2': 'One will now let [MATH] tend to [MATH] in [REF].', '0812.4765-1-9-3': 'Suppose that [MATH], then [EQUATION] and the dominated convergence theorem gives: [MATH], [EQUATION]', '0812.4765-1-9-4': 'The same way, one has: [MATH], [EQUATION]', '0812.4765-1-9-5': 'Letting [MATH] tend to [MATH], one gets: [MATH], [EQUATION]', '0812.4765-1-9-6': 'This insures that: [MATH], [EQUATION]', '0812.4765-1-9-7': 'It is now easy to check that [REF] still holds for [MATH], and so this achieves the proof of propostion [REF]', '0812.4765-1-10-0': 'In the case where [MATH], the point 2 of definition [REF] can be replaced by [EQUATION] and one can remove the assumption [MATH] in [REF].', '0812.4765-1-10-1': 'Actually, in such a case, Kruzkov entropies [MATH] are sufficient to obtain the time continuity.', '0812.4765-1-10-2': 'The assumptions [MATH] and [MATH] will only be useful to insure [MATH] belongs to [MATH] in order to recover the regular convex entropies, which are necessary to treat the parabolic case, as it was shown in the work of Carrillo [CITATION].', '0812.4765-1-11-0': 'The definition [REF] does not take into account any boundary condition, or condition at [MATH].', '0812.4765-1-11-1': 'This lack of regularity can lead to non-uniqueness cases, as the one shown in the book of Friedman [CITATION] (also available in the one of Smoller [CITATION]): the very simple problem [EQUATION] admits multiple classical solutions if one does not ask some condition for large [MATH] like e.g. [MATH].', '0812.4765-1-11-2': 'Indeed, it is easy to check that [EQUATION] is a classical solution of [REF].', '0812.4765-1-11-3': 'So [MATH] is a weak solution of [REF], and thus an entropy weak solution thanks to proposition [REF].', '0812.4765-1-11-4': 'It also belongs to [MATH], thanks to its regularity.', '0812.4765-1-12-0': 'In the following theorem, we claim that any entropy solution is time continuous with respect with the time variable, at least locally with respect to the space variable.', '0812.4765-1-13-0': 'Let [MATH] be a entropy solution in the sense of definition [REF], then there exists [MATH] such that [MATH] a.e. on [MATH] and fulfilling [EQUATION].', '0812.4765-1-13-1': 'Furthermore, if there exists [MATH] and a neighborhood [MATH] of [MATH] in [MATH] such that [EQUATION] then we have: [EQUATION]', '0812.4765-1-14-0': '# Essential continuity for [MATH]', '0812.4765-1-15-0': 'In this section, we give a simple way to prove the classical result stated in proposition [REF].', '0812.4765-1-16-0': 'One says that [MATH] is a right-Lebesgue point if there exists [MATH] in [MATH] such that for all compact subset [MATH] of [MATH], [EQUATION].', '0812.4765-1-16-1': 'We denote by [MATH] the set of right-Lebesgue points.', '0812.4765-1-17-0': 'It is well known that [MATH] and that [MATH] (in the [MATH]-sense) a.e. in [MATH].', '0812.4765-1-17-1': 'In the sequel, we will prove that [MATH], and that [MATH] belongs to [MATH].', '0812.4765-1-17-2': 'We begin by considering the essential continuity for the initial time [MATH].', '0812.4765-1-18-0': 'The limit as [MATH] tends to [MATH], [MATH] can be seen as an essential limit, as it is done in lemma 7.41 in the book of Malek et al. [CITATION] in the case of a purely hyperbolic problem, or by Otto [CITATION] in the case of a non strongly degenerated parabolic equation.', '0812.4765-1-18-1': 'See also the paper of Blanchard and Porretta [CITATION] for the case of renormalized solutions for degenerate parabolic equations.', '0812.4765-1-19-0': 'First, notice that for all [MATH], and for all [MATH], [MATH] is also a right-hand side Lebesgue point of [MATH].', '0812.4765-1-19-1': 'Indeed, if [MATH] denotes a compact subset of [MATH], one has for a.e [MATH] [EQUATION] and so, for all [MATH], [EQUATION]', '0812.4765-1-19-2': 'Let [MATH], and [MATH], one denotes [EQUATION].', '0812.4765-1-20-0': 'Let [MATH], and let [MATH] be such that [MATH].', '0812.4765-1-20-1': 'Let [MATH], with [MATH] and [MATH].', '0812.4765-1-20-2': 'One denotes [MATH].', '0812.4765-1-20-3': 'The function [MATH] belongs to [MATH].', '0812.4765-1-21-0': 'Taking [MATH] and [MATH] in ([REF]), an integrating with respect to [MATH] yields: [EQUATION] where all the gradient are considered with respect to [MATH], and not [MATH].', '0812.4765-1-22-0': 'One has [EQUATION] then, since [MATH] for all [MATH] in [MATH], using [EQUATION] we obtain [EQUATION]', '0812.4765-1-22-1': 'For all [MATH], [EQUATION] and then, one can let [MATH] tend to [MATH] in ([REF]), so that ([REF]) implies: [EQUATION] where [MATH] belongs to [MATH] for all [MATH].', '0812.4765-1-22-2': 'Since [MATH] is dense in [MATH], one can let in a first step [MATH] tend to [MATH], so that [MATH] vanishes: [EQUATION]', '0812.4765-1-22-3': 'One can now let [MATH] tend to [MATH], and using the fact that [MATH] belongs to [MATH], and that [MATH] is compactly supported in [MATH], one gets: [EQUATION].', '0812.4765-1-22-4': 'This achieves the proof of proposition [REF].', '0812.4765-1-23-0': '# Time continuity for any [MATH]', '0812.4765-1-24-0': 'In this section, we want to prove the following proposition:', '0812.4765-1-25-0': 'In the sequel, we still denote by [MATH] the representative defined using the right Lebesgue points introduced in definition [REF].', '0812.4765-1-25-1': 'Proving the essential continuity for every [MATH] is easy.', '0812.4765-1-25-2': 'Indeed, if one replaces [MATH] by [MATH] in ([REF]), and then if one lets [MATH] tend to [MATH], one gets: [EQUATION]', '0812.4765-1-25-3': 'One can thus apply the proposition [REF] with [MATH] instead of [MATH], and [MATH] instead of [MATH]: [MATH], [EQUATION].', '0812.4765-1-25-4': 'We will prove the uniform continuity of [MATH] from [MATH] to [MATH] for all [MATH].', '0812.4765-1-25-5': 'This will give as a direct consequence that [MATH] and [MATH].', '0812.4765-1-26-0': 'Suppose that [REF], [REF] hold.', '0812.4765-1-26-1': 'Let [MATH] belong to [MATH], let [MATH] belong to [MATH], and let [MATH] be two entropy solutions associated to the choice of [MATH] and initial data [MATH] for [MATH] and [MATH] and initial data [MATH] for [MATH] in definition [REF].', '0812.4765-1-26-2': 'We now have all the tools for the proof of proposition [REF].', '0812.4765-1-27-0': 'Proof of proposition [REF]', '0812.4765-1-28-0': 'Let [MATH], let [MATH], and [MATH] such that [MATH] (this is the case of almost every [MATH]).', '0812.4765-1-28-1': 'Let [MATH], let [MATH].', '0812.4765-1-29-0': 'Taking [MATH], [MATH], [MATH] in [REF], and letting [MATH] tend to [MATH] yields: [EQUATION]', '0812.4765-1-29-1': 'We deduce from [REF] that [EQUATION] and since [MATH] and [MATH] belong to [MATH], one can claim that: [EQUATION]', '0812.4765-1-29-2': 'One can now use proposition [REF] in [REF], so that we get that [EQUATION] which is the [MATH]-space for measure of density [MATH] w.r.t. Lebesgue measure.', '0812.4765-1-29-3': 'We deduce that, for all [MATH], [MATH] is uniformly continuous from [MATH] to [MATH], and this insures that [MATH].', '0812.4765-1-29-4': 'This holds for any [MATH], and so we can claim that [MATH].', '0812.4765-1-29-5': '[MATH]', '0812.4765-1-30-0': 'It remains to prove the last part of theorem [REF] by considering some test functions [MATH] instead of [MATH].', '0812.4765-1-30-1': 'We will need some additional regularity on the solution: [EQUATION] [REF] gives the uniform (w.r.t. [MATH]) local equiintegrability of [MATH] (and so of [MATH]) on a neighborhood of [MATH].', '0812.4765-1-30-2': 'We deduce, using [MATH] that [MATH].', '0812.4765-1-31-0': 'End of the proof of theorem [REF]', '0812.4765-1-32-0': 'Suppose that [REF],[REF],[REF],[REF] hold, then thanks to proposition [REF], there exists a weak solution [MATH].', '0812.4765-1-33-0': 'For [MATH], [MATH], [MATH], there exists [MATH] such that: [MATH], [MATH], [EQUATION].', '0812.4765-1-33-1': 'Let [MATH] be a compact subset of [MATH].', '0812.4765-1-33-2': 'Then there exists [MATH] such that [MATH] for all [MATH], and [MATH] if [MATH].', '0812.4765-1-33-3': 'Let [MATH] and let [MATH] such that: [EQUATION]', '0812.4765-1-33-4': 'Suppose that [REF] holds.', '0812.4765-1-33-5': 'For [MATH] small enough, one has [MATH] and then, for all [MATH], for all [MATH], [EQUATION] where [MATH] denotes [MATH], and [MATH].', '0812.4765-1-33-6': 'Since [MATH] tends to [MATH] as [MATH] tends to [MATH], there exists [MATH] such that: [EQUATION]', '0812.4765-1-33-7': 'Suppose now that [MATH] has been chosen such that [REF] holds.', '0812.4765-1-33-8': 'The function [MATH] belongs to [MATH], and then there exists [MATH] such that :[MATH], [MATH], [EQUATION]', '0812.4765-1-33-9': 'Adding [REF] and [REF] shows that for all [MATH] in [MATH], for all [MATH], [EQUATION]', '0812.4765-1-33-10': 'So [MATH] is uniformly continuous from [MATH] to [MATH], and then [EQUATION] [MATH]', '0812.4765-1-34-0': 'To conclude this paper, let us give a counter-example to the time continuity in the case where the entropy criterion is not fulfilled for t=0.', '0812.4765-1-34-1': 'Consider the Burgers equation, in the one dimensional case, leading to the following initial value problem.', '0812.4765-1-34-2': '[EQUATION]', '0812.4765-1-34-3': 'Problem [REF] admits [MATH] as unique entropy solution in the sense of definition [REF].', '0812.4765-1-35-0': 'We define [EQUATION].', '0812.4765-1-35-1': 'Then it is easy to check that:', '0812.4765-1-36-0': 'Thanks to [REF], [MATH] is a weak solution of [REF], and an entropy criterion [REF] is fulfilled only for [MATH].', '0812.4765-1-36-1': 'The fact that the entropy criterion fails for [MATH], and that the flux [MATH] is not bounded (see [CITATION]) allows the function [MATH] to be discontinuous at [MATH].', '0812.4765-1-36-2': 'Indeed, for all [MATH], [EQUATION]'}","{'0812.4765-2-0-0': 'We show that any entropy solution [MATH] of a convection diffusion equation [MATH] in [MATH] belongs to [MATH].', '0812.4765-2-0-1': 'The proof does not use the uniqueness of the solution.', '0812.4765-2-0-2': 'Mathematical Subject Classification: 35L65, 35B65, 35K65 Keywords: Entropy solution, time continuity, scalar conservation laws', '0812.4765-2-1-0': '# The problem, and main result', '0812.4765-2-2-0': 'Convection diffusion equations appear in a large class of problems, and have been widely studied.', '0812.4765-2-2-1': 'We consider in the sequel only equations under conservative form: [EQUATION] so that we can give some sense to [REF] in the distributional sense.', '0812.4765-2-2-2': 'In this paper, we consider entropy solutions of [REF] that do not take into account any boundary condition, or condition for [MATH].', '0812.4765-2-3-0': 'The proof does not use a [MATH]-contraction principle (see e.g. Alt Luckaus [CITATION] or Otto [CITATION]), so that it can be applied in case where uniqueness is not insured, like for example complex spatial coupling of different conservation laws as in [CITATION], or for cases where uniqueness fails because of boundary conditions or conditions at [MATH], as it will be stressed in the sequel.', '0812.4765-2-4-0': 'Let us now state the required assumptions on the data.', '0812.4765-2-5-0': 'Let [MATH] be an open subset of [MATH]), and let [MATH] be a positive real value or [MATH].', '0812.4765-2-5-1': '[EQUATION]', '0812.4765-2-5-2': 'One has to make the following assumption on the source term: [EQUATION]', '0812.4765-2-5-3': 'In the sequel, [MATH] (resp. [MATH]) denotes [MATH] (resp. [MATH]), and [MATH] is the function defined by [EQUATION].', '0812.4765-2-6-0': 'We consider entropy weak solutions of [REF], as in the famous work of Kruzkov [CITATION] for hyperbolic equations.', '0812.4765-2-6-1': 'This notion can be extended to degenerated parabolic equations, as noticed by Carrillo [CITATION].', '0812.4765-2-6-2': 'This leads to the following definition of entropy weak solution:', '0812.4765-2-7-0': 'A function [MATH] is said to be an entropy weak solution if:', '0812.4765-2-8-0': '[MATH], [MATH], [MATH], [MATH], [MATH], [EQUATION]', '0812.4765-2-8-1': 'Suppose first that [MATH] is a continuous function, then the fact that any weak solution [MATH] is an entropy weak solution is just based on a convexity inequality, and on the fact that [MATH] for all [MATH].', '0812.4765-2-8-2': 'More details are available in [CITATION] (see also [CITATION]).', '0812.4765-2-9-0': 'The fact that an entropy weak solution [MATH] is a weak solution is obvious if [MATH] belongs to [MATH] (consider [MATH]).', '0812.4765-2-10-0': 'Suppose now that [MATH] only belongs to [MATH].', '0812.4765-2-10-1': 'Let [MATH], then for all [MATH], one has [EQUATION] which added to [REF] yields: [MATH], [EQUATION]', '0812.4765-2-10-2': 'One will now let [MATH] tend to [MATH] in [REF].', '0812.4765-2-10-3': 'Suppose that [MATH], then [EQUATION] and the dominated convergence theorem gives: [MATH], [EQUATION]', '0812.4765-2-10-4': 'The same way, one has: [MATH], [EQUATION]', '0812.4765-2-10-5': 'Letting [MATH] tend to [MATH], one gets: [MATH], [EQUATION]', '0812.4765-2-10-6': 'This insures that: [MATH], [EQUATION]', '0812.4765-2-10-7': 'It is now easy to check that [REF] still holds for [MATH], and so this achieves the proof of propostion [REF]', '0812.4765-2-11-0': 'In the case where [MATH], the point 2 of definition [REF] can be replaced by [EQUATION] and one can remove the assumption [MATH] in [REF].', '0812.4765-2-11-1': 'Actually, in such a case, Kruzkov entropies [MATH] are sufficient to obtain the time continuity.', '0812.4765-2-11-2': 'The assumptions [MATH] and [MATH] will only be useful to insure [MATH] belongs to [MATH] in order to recover the regular convex entropies, which are necessary to treat the parabolic case, as it was shown in the work of Carrillo [CITATION].', '0812.4765-2-12-0': 'The definition [REF] does not take into account any boundary condition, or condition at [MATH].', '0812.4765-2-12-1': 'This lack of regularity can lead to non-uniqueness cases, as the one shown in the book of Friedman [CITATION] (also available in the one of Smoller [CITATION]): the very simple problem [EQUATION] admits multiple classical solutions if one does not ask some condition for large [MATH] like e.g. [MATH].', '0812.4765-2-12-2': 'Indeed, it is easy to check that [EQUATION] is a classical solution of [REF].', '0812.4765-2-12-3': 'So [MATH] is a weak solution of [REF], and thus an entropy weak solution thanks to proposition [REF].', '0812.4765-2-12-4': 'It also belongs to [MATH], thanks to its regularity.', '0812.4765-2-13-0': 'Let us give another example, proposed by Michel Pierre [CITATION].', '0812.4765-2-13-1': 'We now consider the problem [EQUATION] which admits the constant function equal to [MATH] as unique smooth solution.', '0812.4765-2-13-2': 'A non-smooth solution to the problem [REF] can be built as follows.', '0812.4765-2-13-3': 'Denote by [MATH] the fundamental solution of the heat equation in the one-dimensional case: [EQUATION] then [MATH] also satisfies the heat equation in the distributional sense.', '0812.4765-2-13-4': 'The function [MATH], given by [EQUATION] satisfies [MATH] for all [MATH], belongs to [MATH] but is not continuous in [MATH].', '0812.4765-2-13-5': 'Indeed, one has [EQUATION].', '0812.4765-2-13-6': 'The function [MATH] belongs to [MATH], then there exists a unique [MATH] solution to the problem [EQUATION].', '0812.4765-2-13-7': 'Defining [MATH], then [MATH] is a solution to the problem [REF] which is not the trivial solution since it is not regular.', '0812.4765-2-13-8': 'Nevertheless, [MATH] is a weak solution to the problem and thus a entropy weak solution thanks to proposition [REF].', '0812.4765-2-13-9': 'Thanks to its regularity, it clearly appears that [MATH] belongs to [MATH].', '0812.4765-2-14-0': 'In the following theorem, we claim that any entropy solution is time continuous with respect with the time variable, at least locally with respect to the space variable.', '0812.4765-2-15-0': 'Let [MATH] be a entropy solution in the sense of definition [REF], then there exists [MATH] such that [MATH] a.e. on [MATH] and fulfilling [EQUATION].', '0812.4765-2-15-1': 'Furthermore, if there exists [MATH] and a neighborhood [MATH] of [MATH] in [MATH] such that [EQUATION] then we have: [EQUATION]', '0812.4765-2-16-0': '# Essential continuity for [MATH]', '0812.4765-2-17-0': 'In this section, we give a simple way to prove the classical result stated in proposition [REF].', '0812.4765-2-18-0': 'One says that [MATH] is a right-Lebesgue point if there exists [MATH] in [MATH] such that for all compact subset [MATH] of [MATH], [EQUATION].', '0812.4765-2-18-1': 'We denote by [MATH] the set of right-Lebesgue points.', '0812.4765-2-19-0': 'It is well known that [MATH] and that [MATH] (in the [MATH]-sense) a.e. in [MATH].', '0812.4765-2-19-1': 'In the sequel, we will prove that [MATH], and that [MATH] belongs to [MATH].', '0812.4765-2-19-2': 'We begin by considering the essential continuity for the initial time [MATH].', '0812.4765-2-20-0': 'The limit as [MATH] tends to [MATH], [MATH] can be seen as an essential limit, as it is done in lemma 7.41 in the book of Malek et al. [CITATION] in the case of a purely hyperbolic problem, or by Otto [CITATION] in the case of a non strongly degenerated parabolic equation.', '0812.4765-2-20-1': 'See also the paper of Blanchard and Porretta [CITATION] for the case of renormalized solutions for degenerate parabolic equations.', '0812.4765-2-21-0': 'First, notice that for all [MATH], and for all [MATH], [MATH] is also a right-hand side Lebesgue point of [MATH].', '0812.4765-2-21-1': 'Indeed, if [MATH] denotes a compact subset of [MATH], one has for a.e [MATH] [EQUATION] and so, for all [MATH], [EQUATION]', '0812.4765-2-21-2': 'Let [MATH], and [MATH], one denotes [EQUATION].', '0812.4765-2-22-0': 'Let [MATH], and let [MATH] be such that [MATH].', '0812.4765-2-22-1': 'Let [MATH], with [MATH] and [MATH].', '0812.4765-2-22-2': 'One denotes [MATH].', '0812.4765-2-22-3': 'The function [MATH] belongs to [MATH].', '0812.4765-2-23-0': 'Taking [MATH] and [MATH] in ([REF]), an integrating with respect to [MATH] yields: [EQUATION] where all the gradient are considered with respect to [MATH], and not [MATH].', '0812.4765-2-24-0': 'One has [EQUATION] then, since [MATH] for all [MATH] in [MATH], using [EQUATION] we obtain [EQUATION]', '0812.4765-2-24-1': 'For all [MATH], [EQUATION] and then, one can let [MATH] tend to [MATH] in ([REF]), so that ([REF]) implies: [EQUATION] where [MATH] belongs to [MATH] for all [MATH].', '0812.4765-2-24-2': 'Since [MATH] is dense in [MATH], one can let in a first step [MATH] tend to [MATH], so that [MATH] vanishes: [EQUATION]', '0812.4765-2-24-3': 'One can now let [MATH] tend to [MATH], and using the fact that [MATH] belongs to [MATH], and that [MATH] is compactly supported in [MATH], one gets: [EQUATION].', '0812.4765-2-24-4': 'This achieves the proof of proposition [REF].', '0812.4765-2-25-0': '# Time continuity for any [MATH]', '0812.4765-2-26-0': 'In this section, we want to prove the following proposition:', '0812.4765-2-27-0': 'In the sequel, we still denote by [MATH] the representative defined using the right Lebesgue points introduced in definition [REF].', '0812.4765-2-28-0': 'Proving the essential continuity for every [MATH] is easy.', '0812.4765-2-28-1': 'Indeed, if one replaces [MATH] by [MATH] in ([REF]), and then if one lets [MATH] tend to [MATH], one gets: [EQUATION]', '0812.4765-2-28-2': 'One can thus apply the proposition [REF] with [MATH] instead of [MATH], and [MATH] instead of [MATH]: [MATH], [EQUATION].', '0812.4765-2-28-3': 'We will prove the uniform continuity of [MATH] from [MATH] to [MATH] for all [MATH].', '0812.4765-2-28-4': 'This will give as a direct consequence that [MATH] and [MATH].', '0812.4765-2-29-0': 'Suppose that [REF], [REF] hold.', '0812.4765-2-29-1': 'Let [MATH] belong to [MATH], let [MATH] belong to [MATH], and let [MATH] be two entropy solutions associated to the choice of [MATH] and initial data [MATH] for [MATH] and [MATH] and initial data [MATH] for [MATH] in definition [REF].', '0812.4765-2-29-2': 'We now have all the tools for the proof of proposition [REF].', '0812.4765-2-30-0': 'Proof of proposition [REF]', '0812.4765-2-31-0': 'Let [MATH], let [MATH], and [MATH] such that [MATH] (this is the case of almost every [MATH]).', '0812.4765-2-31-1': 'Let [MATH], let [MATH].', '0812.4765-2-32-0': 'Taking [MATH], [MATH], [MATH] in [REF], and letting [MATH] tend to [MATH] yields: [EQUATION]', '0812.4765-2-32-1': 'We deduce from [REF] that [EQUATION] and since [MATH] and [MATH] belong to [MATH], one can claim that: [EQUATION]', '0812.4765-2-32-2': 'One can now use proposition [REF] in [REF], so that we get that [EQUATION] which is the [MATH]-space for measure of density [MATH] w.r.t. Lebesgue measure.', '0812.4765-2-32-3': 'We deduce that, for all [MATH], [MATH] is uniformly continuous from [MATH] to [MATH], and this insures that [MATH].', '0812.4765-2-32-4': 'This holds for any [MATH], and so we can claim that [MATH].', '0812.4765-2-32-5': '[MATH]', '0812.4765-2-33-0': 'It remains to prove the last part of theorem [REF] by considering some test functions [MATH] instead of [MATH].', '0812.4765-2-33-1': 'We will need some additional regularity on the solution: [EQUATION] [REF] gives the uniform (w.r.t. [MATH]) local equiintegrability of [MATH] (and so of [MATH]) on a neighborhood of [MATH].', '0812.4765-2-33-2': 'We deduce, using [MATH] that [MATH].', '0812.4765-2-34-0': 'End of the proof of theorem [REF]', '0812.4765-2-35-0': 'Suppose that [REF],[REF],[REF],[REF] hold, then thanks to proposition [REF], there exists a weak solution [MATH].', '0812.4765-2-36-0': 'For [MATH], [MATH], [MATH], there exists [MATH] such that: [MATH], [MATH], [EQUATION].', '0812.4765-2-36-1': 'Let [MATH] be a compact subset of [MATH].', '0812.4765-2-36-2': 'Then there exists [MATH] such that [MATH] for all [MATH], and [MATH] if [MATH].', '0812.4765-2-36-3': 'Let [MATH] and let [MATH] such that: [EQUATION]', '0812.4765-2-36-4': 'Suppose that [REF] holds.', '0812.4765-2-36-5': 'For [MATH] small enough, one has [MATH] and then, for all [MATH], for all [MATH], [EQUATION] where [MATH] denotes [MATH], and [MATH].', '0812.4765-2-36-6': 'Since [MATH] tends to [MATH] as [MATH] tends to [MATH], there exists [MATH] such that: [EQUATION]', '0812.4765-2-36-7': 'Suppose now that [MATH] has been chosen such that [REF] holds.', '0812.4765-2-36-8': 'The function [MATH] belongs to [MATH], and then there exists [MATH] such that :[MATH], [MATH], [EQUATION]', '0812.4765-2-36-9': 'Adding [REF] and [REF] shows that for all [MATH] in [MATH], for all [MATH], [EQUATION]', '0812.4765-2-36-10': 'So [MATH] is uniformly continuous from [MATH] to [MATH], and then [EQUATION] [MATH]', '0812.4765-2-37-0': 'To conclude this paper, let us give a counter-example to the time continuity in the case where the entropy criterion is not fulfilled for t=0.', '0812.4765-2-37-1': 'Consider the Burgers equation, in the one dimensional case, leading to the following initial value problem.', '0812.4765-2-37-2': '[EQUATION]', '0812.4765-2-37-3': 'Problem [REF] admits [MATH] as unique entropy solution in the sense of definition [REF].', '0812.4765-2-38-0': 'We define [EQUATION].', '0812.4765-2-38-1': 'Then it is easy to check that:', '0812.4765-2-39-0': 'Thanks to [REF], [MATH] is a weak solution of [REF], and an entropy criterion [REF] is fulfilled only for [MATH].', '0812.4765-2-39-1': 'The fact that the entropy criterion fails for [MATH], and that the flux [MATH] is not bounded (see [CITATION]) allows the function [MATH] to be discontinuous at [MATH].', '0812.4765-2-39-2': 'Indeed, for all [MATH], [EQUATION]'}","[['0812.4765-1-13-0', '0812.4765-2-15-0'], ['0812.4765-1-13-1', '0812.4765-2-15-1'], ['0812.4765-1-4-1', '0812.4765-2-5-0'], ['0812.4765-1-4-3', '0812.4765-2-5-2'], ['0812.4765-1-4-4', '0812.4765-2-5-3'], ['0812.4765-1-10-0', '0812.4765-2-11-0'], ['0812.4765-1-10-1', '0812.4765-2-11-1'], ['0812.4765-1-10-2', '0812.4765-2-11-2'], ['0812.4765-1-36-0', '0812.4765-2-39-0'], ['0812.4765-1-36-1', '0812.4765-2-39-1'], ['0812.4765-1-36-2', '0812.4765-2-39-2'], ['0812.4765-1-12-0', '0812.4765-2-14-0'], ['0812.4765-1-26-0', '0812.4765-2-29-0'], ['0812.4765-1-26-1', '0812.4765-2-29-1'], ['0812.4765-1-26-2', 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['0812.4765-1-11-4', '0812.4765-2-12-4'], ['0812.4765-1-21-0', '0812.4765-2-23-0'], ['0812.4765-1-19-0', '0812.4765-2-21-0'], ['0812.4765-1-19-1', '0812.4765-2-21-1'], ['0812.4765-1-19-2', '0812.4765-2-21-2'], ['0812.4765-1-3-0', '0812.4765-2-3-0'], ['0812.4765-1-30-0', '0812.4765-2-33-0'], ['0812.4765-1-30-1', '0812.4765-2-33-1'], ['0812.4765-1-30-2', '0812.4765-2-33-2'], ['0812.4765-1-8-0', '0812.4765-2-9-0'], ['0812.4765-1-0-0', '0812.4765-2-0-0'], ['0812.4765-1-0-1', '0812.4765-2-0-1'], ['0812.4765-1-5-0', '0812.4765-2-6-0'], ['0812.4765-1-5-1', '0812.4765-2-6-1'], ['0812.4765-1-9-0', '0812.4765-2-10-0'], ['0812.4765-1-9-1', '0812.4765-2-10-1'], ['0812.4765-1-9-2', '0812.4765-2-10-2'], ['0812.4765-1-9-3', '0812.4765-2-10-3'], ['0812.4765-1-9-4', '0812.4765-2-10-4'], ['0812.4765-1-9-5', '0812.4765-2-10-5'], ['0812.4765-1-9-6', '0812.4765-2-10-6'], ['0812.4765-1-9-7', '0812.4765-2-10-7'], ['0812.4765-1-2-0', '0812.4765-2-2-0'], ['0812.4765-1-2-1', '0812.4765-2-2-1'], ['0812.4765-1-2-2', '0812.4765-2-2-2'], ['0812.4765-1-29-0', '0812.4765-2-32-0'], ['0812.4765-1-29-1', '0812.4765-2-32-1'], ['0812.4765-1-29-2', '0812.4765-2-32-2'], ['0812.4765-1-29-3', '0812.4765-2-32-3'], ['0812.4765-1-29-4', '0812.4765-2-32-4'], ['0812.4765-1-25-0', '0812.4765-2-27-0'], ['0812.4765-1-25-1', '0812.4765-2-28-0'], ['0812.4765-1-25-2', '0812.4765-2-28-1'], ['0812.4765-1-25-3', '0812.4765-2-28-2'], ['0812.4765-1-25-4', '0812.4765-2-28-3'], ['0812.4765-1-25-5', '0812.4765-2-28-4']]","[['0812.4765-1-13-0', '0812.4765-2-15-0'], ['0812.4765-1-13-1', '0812.4765-2-15-1'], ['0812.4765-1-4-1', '0812.4765-2-5-0'], ['0812.4765-1-4-3', '0812.4765-2-5-2'], ['0812.4765-1-4-4', '0812.4765-2-5-3'], ['0812.4765-1-10-0', '0812.4765-2-11-0'], ['0812.4765-1-10-1', '0812.4765-2-11-1'], ['0812.4765-1-10-2', '0812.4765-2-11-2'], ['0812.4765-1-36-0', '0812.4765-2-39-0'], ['0812.4765-1-36-1', '0812.4765-2-39-1'], ['0812.4765-1-36-2', '0812.4765-2-39-2'], ['0812.4765-1-12-0', '0812.4765-2-14-0'], ['0812.4765-1-26-0', '0812.4765-2-29-0'], ['0812.4765-1-26-1', '0812.4765-2-29-1'], ['0812.4765-1-26-2', '0812.4765-2-29-2'], ['0812.4765-1-18-0', '0812.4765-2-20-0'], ['0812.4765-1-18-1', '0812.4765-2-20-1'], ['0812.4765-1-34-0', '0812.4765-2-37-0'], ['0812.4765-1-34-1', '0812.4765-2-37-1'], ['0812.4765-1-34-3', '0812.4765-2-37-3'], ['0812.4765-1-7-1', '0812.4765-2-8-1'], ['0812.4765-1-7-2', '0812.4765-2-8-2'], ['0812.4765-1-28-0', '0812.4765-2-31-0'], ['0812.4765-1-22-0', '0812.4765-2-24-0'], ['0812.4765-1-22-1', '0812.4765-2-24-1'], ['0812.4765-1-22-2', '0812.4765-2-24-2'], ['0812.4765-1-22-3', '0812.4765-2-24-3'], ['0812.4765-1-22-4', '0812.4765-2-24-4'], ['0812.4765-1-16-0', '0812.4765-2-18-0'], ['0812.4765-1-16-1', '0812.4765-2-18-1'], ['0812.4765-1-15-0', '0812.4765-2-17-0'], ['0812.4765-1-17-0', '0812.4765-2-19-0'], ['0812.4765-1-17-1', '0812.4765-2-19-1'], ['0812.4765-1-17-2', '0812.4765-2-19-2'], ['0812.4765-1-11-0', '0812.4765-2-12-0'], ['0812.4765-1-11-1', '0812.4765-2-12-1'], ['0812.4765-1-11-2', '0812.4765-2-12-2'], ['0812.4765-1-11-3', '0812.4765-2-12-3'], ['0812.4765-1-11-4', '0812.4765-2-12-4'], ['0812.4765-1-21-0', '0812.4765-2-23-0'], ['0812.4765-1-19-0', '0812.4765-2-21-0'], ['0812.4765-1-19-1', '0812.4765-2-21-1'], ['0812.4765-1-19-2', '0812.4765-2-21-2'], ['0812.4765-1-3-0', '0812.4765-2-3-0'], ['0812.4765-1-30-0', '0812.4765-2-33-0'], ['0812.4765-1-30-1', '0812.4765-2-33-1'], ['0812.4765-1-30-2', '0812.4765-2-33-2'], ['0812.4765-1-8-0', '0812.4765-2-9-0'], ['0812.4765-1-0-0', '0812.4765-2-0-0'], ['0812.4765-1-0-1', '0812.4765-2-0-1'], ['0812.4765-1-5-0', '0812.4765-2-6-0'], ['0812.4765-1-5-1', '0812.4765-2-6-1'], ['0812.4765-1-9-0', '0812.4765-2-10-0'], ['0812.4765-1-9-1', '0812.4765-2-10-1'], ['0812.4765-1-9-2', '0812.4765-2-10-2'], ['0812.4765-1-9-3', '0812.4765-2-10-3'], ['0812.4765-1-9-4', '0812.4765-2-10-4'], ['0812.4765-1-9-5', '0812.4765-2-10-5'], ['0812.4765-1-9-6', '0812.4765-2-10-6'], 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'0812.4765-1-33-4', '0812.4765-1-33-5', '0812.4765-1-33-6', '0812.4765-1-33-7', '0812.4765-1-33-8', '0812.4765-1-33-9', '0812.4765-1-33-10', '0812.4765-1-34-2', '0812.4765-1-35-0', '0812.4765-1-35-1', '0812.4765-2-4-0', '0812.4765-2-5-1', '0812.4765-2-6-2', '0812.4765-2-7-0', '0812.4765-2-8-0', '0812.4765-2-22-0', '0812.4765-2-22-1', '0812.4765-2-22-2', '0812.4765-2-22-3', '0812.4765-2-26-0', '0812.4765-2-30-0', '0812.4765-2-31-1', '0812.4765-2-32-5', '0812.4765-2-34-0', '0812.4765-2-35-0', '0812.4765-2-36-0', '0812.4765-2-36-1', '0812.4765-2-36-2', '0812.4765-2-36-3', '0812.4765-2-36-4', '0812.4765-2-36-5', '0812.4765-2-36-6', '0812.4765-2-36-7', '0812.4765-2-36-8', '0812.4765-2-36-9', '0812.4765-2-36-10', '0812.4765-2-37-2', '0812.4765-2-38-0', '0812.4765-2-38-1']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/0812.4765,,,
1503.05281,"{'1503.05281-1-0-0': 'ABSTRACT Inspired by the [MATH] non-linear massive gravity, we propose a new kind of modified gravity model, namely [MATH] non-linear massive gravity, by adding the dRGT mass term reformulated in the vierbein formalism, to the [MATH] theory.', '1503.05281-1-0-1': 'We then investigate the cosmological evolution of [MATH] massive gravity, and constrain it by using the latest observational data.', '1503.05281-1-0-2': 'We find that it slightly favors a crossing of the phantom divide line from the quintessence-like phase ([MATH]) to the phantom-like one ([MATH]) as redshift decreases.', '1503.05281-1-1-0': '# Introduction', '1503.05281-1-2-0': ""Einstein's theory of general relativity (GR) has achieved great success in explaining various gravitational phenomena since its birth, and till today it can pass the Solar System's tests to a very high precision."", '1503.05281-1-2-1': 'However, on scales larger than the Solar System, there are some astrophysical observations having not yet been fully understood or explained in the context of GR.', '1503.05281-1-2-2': 'Maybe the most urgent among them are the presence of an invisible (dark) matter component in our universe, first discovered by Zwicky in the [MATH]s [CITATION], as well as the late-time acceleration of our universe confirmed by various cosmological observations [CITATION].', '1503.05281-1-2-3': 'Although by extending the standard model of particle physics, there exist various promising candidates for dark matter (see e.g. reviews [CITATION]), so far there has been no natural and satisfactory explanation for the cosmic acceleration by introducing some kinds of mysterious cosmological fluids with negative pressure called dark energy (DE), since they all suffer from the fine-tuning [CITATION] and/or the cosmic coincidence problems [CITATION].', '1503.05281-1-2-4': 'Unless all of these problems are eventually solved on the particle physics side, the severity of the phenomenological problems challenges GR as the ultimate and complete theory of gravity.', '1503.05281-1-2-5': 'Furthermore, difficulties arise when trying to search for a quantum theory of gravity, since GR is not renormalizable [CITATION] and thus cannot make meaningful physical predictions.', '1503.05281-1-2-6': ""It is, therefore, reasonable to investigate the possibility of modifying Einstein's gravity theory."", '1503.05281-1-3-0': 'Among the modified gravity theories, [MATH] theory (see e.g. [CITATION] for recent reviews) serves as one of the most straightforward and popular generalizations of GR by extending the Ricci scalar [MATH] in the Einstein-Hilbert action to a general function [MATH].', '1503.05281-1-3-1': 'It has been shown that [MATH] gravity can explain the present cosmic acceleration without the need of DE [CITATION].', '1503.05281-1-3-2': 'But in general, the resulting field equations of motion are [MATH]th order in [MATH] gravity.', '1503.05281-1-3-3': 'This feature makes [MATH] theory quite hard to analyse.', '1503.05281-1-3-4': 'In comparison with [MATH] gravity, there is another kind of modified gravity, namely [MATH] theory, which has been extensively explored in the literature (see e.g. [CITATION]).', '1503.05281-1-3-5': 'Here [MATH] is the torsion scalar constructed from the curvatureless Weizenbock connection [CITATION].', '1503.05281-1-3-6': 'It is well known that [MATH] theory is based on the teleparallel space-time, in which vierbein field is used as the basic dynamical variable instead of metric in GR.', '1503.05281-1-3-7': 'One advantage of [MATH] theory in contrast to [MATH] theory is that, the field equations arising from [MATH] theory are [MATH]nd order, and thus simpler than [MATH] case.', '1503.05281-1-4-0': 'On the other hand, as a classical field theory, GR propagates a non-linear self-interacting massless spin-[MATH] field.', '1503.05281-1-4-1': 'Therefore, another way to modify GR is to give a tiny mass to the graviton.', '1503.05281-1-4-2': 'This class of theories is known as massive gravity in the literature.', '1503.05281-1-4-3': 'Actually, massive gravity has a long and elusive history.', '1503.05281-1-4-4': 'The first attempt to give a mass to graviton dates back to the year of 1939 when Fierz and Pauli proposed a unique ghost-free mass term to gravity in the linear perturbation level [CITATION].', '1503.05281-1-4-5': 'The linearised massive gravity propagates [MATH] degrees of freedom (DoF).', '1503.05281-1-4-6': 'And then, in 1970s, van Dam-Veltman-Zakharov (vDVZ) discontinuity was found.', '1503.05281-1-4-7': 'It was shown that, when the mass of graviton [MATH], Fierz-Pauli theory would not converge to massless case [CITATION], thus giving different predictions than those GR.', '1503.05281-1-4-8': 'Almost at the same time, Vainshtein [CITATION] showed that the linear approximation loses its validity around massive sources below the Vainshtein radius and non-linear effects should be considered.', '1503.05281-1-4-9': 'The vDVZ discontinuity can then be avoided by this well-known Vainshtein screening mechanism.', '1503.05281-1-4-10': 'However, it was claimed that any non-linear extensions of Fierz-Pauli theory will generally have [MATH] DoF, with the extra DoF being a Boulware-Deser (BD) ghost [CITATION].', '1503.05281-1-4-11': 'The construction of a well-defined non-linear theory of massive gravity without the BD ghost has been a challenging problem over decades.', '1503.05281-1-4-12': 'Recently, it was shown that BD ghost may be removed order-by-order in a careful construction of mass term up to a certain decoupling limit [CITATION].', '1503.05281-1-4-13': 'The full non-linear massive gravity was resummed by de Rham, Gabadadze and Tolley (dRGT) in [CITATION], and the absence of BD ghost was further confirmed by using Hamiltonian constraint analysis in [CITATION].', '1503.05281-1-4-14': 'Although dRGT theory exists accelerating solutions [CITATION], it suffers from some instabilities at the cosmological perturbation level [CITATION].', '1503.05281-1-4-15': 'Since then dRGT theory has been extended in various approaches, e.g. the mass varying massive gravity [CITATION] and the quasi-dilaton massive gravity [CITATION].', '1503.05281-1-5-0': 'Recently, a new extension of massive gravity, namely [MATH] non-linear massive gravity, has been proposed [CITATION].', '1503.05281-1-5-1': 'It is shown that this model not only shares no linear instabilities around a cosmological background, but can also describe the inflation and the recently cosmic acceleration in a unified framework [CITATION].', '1503.05281-1-5-2': 'Inspired by the [MATH] non-linear massive gravity, we propose a new kind of modified gravity theory by adding the non-linear dRGT mass term formulated in the vierbein formalism, to the [MATH] theory.', '1503.05281-1-5-3': 'We might call this theory [MATH] non-linear massive gravity, or simply [MATH] massive gravity.', '1503.05281-1-5-4': 'One merit of our model compared with the [MATH] massive gravity is that, the field equations arising from [MATH] massive gravity are [MATH]nd order, and thus much easier to analyse.', '1503.05281-1-5-5': 'Moreover, [MATH] massive gravity, as well as dRGT theory, is generally constructed in the metric formulation.', '1503.05281-1-5-6': 'However, in the metric formalism, the mass term is difficult to analyse, since it contains the square root of the metric.', '1503.05281-1-5-7': 'Another merit of our theory is that, in teleparallel space-time, we use vierbein instead of metric as the basic dynamical variable.', '1503.05281-1-5-8': 'Since vierbein can be viewed as the square root of the metric in some sense, the vierbein formalism in our theory can greatly simplify the calculation.', '1503.05281-1-6-0': 'After presenting the action of our model, we then investigate the cosmological evolution of [MATH] massive gravity.', '1503.05281-1-6-1': 'The rest of this paper is organized as follows.', '1503.05281-1-6-2': 'In Sec. [REF], we briefly review the [MATH] gravity.', '1503.05281-1-6-3': 'In Sec. [REF], we add the ghost-free dRGT mass term reformulated in the vierbein formalism to the [MATH] sector, thus giving the action of [MATH] massive gravity.', '1503.05281-1-6-4': 'The evolutionary equations for the flat Friedmann-Robertson-Walker (FRW) cosmology are given as well.', '1503.05281-1-6-5': 'In Sec. [REF], we focus on two concrete examples, namely the power-law and exponential types of [MATH] massive gravity and investigate their cosmological evolution.', '1503.05281-1-6-6': 'We constrain the model parameters by the recent cosmological data and study the phantom crossing behaviour.', '1503.05281-1-6-7': 'In Sec. [REF], we give some concluding remarks.', '1503.05281-1-7-0': '# [MATH] f(T) gravity', '1503.05281-1-8-0': '[MATH] gravity, which is a generalization of the teleparallel gravity originally proposed by Einstein [CITATION], has received great interest in the literature recently.', '1503.05281-1-8-1': 'Following [CITATION], we now briefly review the [MATH] theory.', '1503.05281-1-8-2': '[MATH] gravity is built upon teleparallel space-time.', '1503.05281-1-8-3': 'In teleparallel space-time, the basic dynamical quantity is a vierbein field [MATH], with Latin indices [MATH], and [MATH], Greek indices [MATH], and [MATH] coordinate bases.', '1503.05281-1-8-4': 'We also note that the Einstein summation notation for the indices is used throughout this paper.', '1503.05281-1-8-5': 'The vierbein is an orthonormal basis for the tangent space at each point [MATH] of the manifold, namely [MATH], with [MATH].', '1503.05281-1-8-6': 'The metric tensor can then be expressed in the dual vierbein [MATH] as [EQUATION]', '1503.05281-1-8-7': 'Rather than using the torsionless Levi-Civita connection in general relativity (GR), teleparallel space-time uses the curvatureless Weitzenbock connection [MATH], which is defined by [EQUATION]', '1503.05281-1-8-8': 'Note that the lower indices [MATH] and [MATH] are not symmetric in general, thus the torsion tensor is non-zero in the teleparallel space-time.', '1503.05281-1-8-9': 'The Weitzenbock torsion tensor is defined by [EQUATION]', '1503.05281-1-8-10': 'In teleparallel gravity, the gravitational action is given by the torsion scalar instead of the the Ricci scalar in GR.', '1503.05281-1-8-11': 'The torsion scalar is basically the square of the Weitzenbock torsion tensor, and is defined by [EQUATION] with the tensor [MATH] given by [EQUATION]', '1503.05281-1-8-12': 'In [MATH] gravity, the gravitational field is driven by a Lagrangian density [MATH], with [MATH] a function of [MATH], and the action reads [EQUATION] where [MATH], [MATH], with [MATH] the determinant of the metric [MATH] and [MATH] the Newtonian constant.', '1503.05281-1-8-13': 'Note that we have used the units in which the speed of light [MATH], and the reduced Planck constant [MATH].', '1503.05281-1-8-14': 'Here, [MATH] is the matter part of the action, and [MATH] denotes all matter fields collectively.', '1503.05281-1-8-15': 'If we set [MATH], then the action gives an equivalent description of the space-time as GR.', '1503.05281-1-9-0': '# [MATH] f(T) non-linear massive gravity', '1503.05281-1-10-0': 'In this section, we will add the dRGT mass term reformulated in the vierbein form to the [MATH] sector, thus giving the action of [MATH] massive gravity.', '1503.05281-1-10-1': 'We then derive the evolutionary equations for the flat FRW cosmology.', '1503.05281-1-11-0': '## The action', '1503.05281-1-12-0': 'Since teleparallel gravity equals to GR at the field equations level, here we give an equivalent description of dRGT massive theory by adding the mass term to teleparallel gravity.', '1503.05281-1-12-1': 'The mass term firstly proposed in dRGT theory, can be greatly simplified when transforming to the vierbein formalism.', '1503.05281-1-12-2': 'We refer to [CITATION] for a detailed derivation.', '1503.05281-1-12-3': 'By adding the mass term to teleparallel gravity, one will get the teleparallel version of ghost-free dRGT massive gravity as [EQUATION] where [MATH] are free parameters, [MATH] denotes the matrix of vierbein [MATH] and [MATH] are the so-called elementary symmetric polynomials [CITATION].', '1503.05281-1-12-4': 'For an arbitrary [MATH] matrix [MATH], the first few of [MATH] can be written in the form [CITATION] [EQUATION] with [MATH] the trace of the matrix [MATH].', '1503.05281-1-12-5': 'By generalizing the torsion scalar [MATH] to a function [MATH] in the action [REF], we have the [MATH] non-linear massive gravity as [EQUATION]', '1503.05281-1-12-6': 'Note that we have absorbed the cosmological constant arising from the mass term to the [MATH] sector.', '1503.05281-1-12-7': 'So, there are only three free parameters [MATH], [MATH], and [MATH] in the mass term.', '1503.05281-1-12-8': 'We also note that when [MATH], the action [REF] is just the teleparallel gravity plus a dRGT mass term reformulated in the vierbein formalism, thus being equivalent to the original dRGT theory at the field equations level.', '1503.05281-1-12-9': 'So, if [MATH], then the BD ghost will not show up in the action [REF].', '1503.05281-1-12-10': 'However, if [MATH] is a non-trivial function of [MATH], then additional DoF will be introduced.', '1503.05281-1-12-11': 'Therefore, we should perform Hamiltonian analysis to confirm that the action [REF] is ghost-free.', '1503.05281-1-12-12': 'In this paper we focus on the cosmological behaviour of [MATH] non-linear massive gravity, and the ghost problem or other instabilities that our theory may suffer from, will be considered in our future work.', '1503.05281-1-13-0': '## Equation of motion', '1503.05281-1-14-0': 'We now consider a spatially flat FRW space-time [EQUATION] where [MATH] is the scale factor and [MATH] is the lapse function.', '1503.05281-1-14-1': 'This metric arises from the diagonal dual vierbein [EQUATION] through Eq. [REF].', '1503.05281-1-14-2': 'Then we can easily get [MATH], [MATH], and hence [EQUATION]', '1503.05281-1-14-3': 'The torsion scalar is also obtained to be [MATH], with a prime denotes [MATH].', '1503.05281-1-15-0': 'After some algebra, action [REF] reduces to [EQUATION]', '1503.05281-1-15-1': 'For further convenience, we define some variables here, namely [EQUATION]', '1503.05281-1-15-2': 'Now we can get the field equations by varying action [REF] with respect to [MATH] and [MATH], respectively, [EQUATION] where a dot denotes [MATH], [MATH], [MATH], and [MATH], [MATH] are the energy density and pressure of all perfect fluids of generic matter, respectively.', '1503.05281-1-15-3': 'Notice that we do not need to know the explicit form of [MATH] in the above derivation.', '1503.05281-1-15-4': 'As is well known, the energy-momentum tensor of a perfect fluid takes a diagonal form in the comoving coordinates, namely [EQUATION]', '1503.05281-1-15-5': 'And from the standard definition of energy-momentum tensor [EQUATION] we can easily read off [MATH] and [MATH] as [EQUATION]', '1503.05281-1-15-6': 'From Eq. [REF] and Eq. [REF], we can get the corresponding energy conservation equation for matter, [EQUATION]', '1503.05281-1-15-7': 'We can recast Eq. [REF] and Eq. [REF] in the similar form as in the GR case by introducing the energy density and pressure of the effective DE as [EQUATION]', '1503.05281-1-15-8': 'Then we get the modified Friedmann equations as [EQUATION] in which [MATH], [MATH].', '1503.05281-1-15-9': 'From Eq. [REF] and Eq. [REF], one can obtain the corresponding energy conservation equation for DE as [EQUATION]', '1503.05281-1-15-10': 'Combining Eq. [REF] and Eq. [REF] yields [EQUATION] which is the usual conservation equation for total energy.', '1503.05281-1-15-11': 'We see that matter and DE are interacted through the graviton.', '1503.05281-1-15-12': 'We define the equation-of-state (EoS) parameter of the effective DE as [EQUATION]', '1503.05281-1-16-0': '# Observational constraints', '1503.05281-1-17-0': 'Here we are interested in the late-time behavior of [MATH] massive gravity.', '1503.05281-1-17-1': 'In this section, we study two specific functionals of [MATH] as concrete examples.', '1503.05281-1-17-2': 'These two examples will be constrained by the recent cosmological data.', '1503.05281-1-17-3': 'We then study the evolution of the effective EoS parameter of DE.', '1503.05281-1-18-0': '## General set-up', '1503.05281-1-19-0': 'Since we are interested in the late-time universe, we ignore the radiation component and only consider the pressureless matter, whose energy density is [MATH] and pressure is [MATH].', '1503.05281-1-19-1': 'Furthermore, we also set [MATH], since at the background level, the terms [MATH] and [MATH] in Eq. [REF] are indistinguishable from the space curvature and pressureless matter terms, respectively.', '1503.05281-1-19-2': 'Having these in mind, and by using Eq. [REF] and Eq. [REF], the effective EoS parameter of DE [REF] can be further simplified to [EQUATION] where the fractional density of the pressureless matter and the graviton are given by [EQUATION] respectively.', '1503.05281-1-19-3': 'Note that we always use a subscript ""[MATH]"" to denote the present value of corresponding quantity.', '1503.05281-1-19-4': 'So [MATH] and [MATH], with the redshift [MATH] defined as [MATH].', '1503.05281-1-19-5': 'Consequently, we find the expression for the dimensionless Hubble parameter [MATH], namely [EQUATION]', '1503.05281-1-19-6': 'Note that [MATH] appears in both sides of this equation, and implicitly in [MATH] and [MATH].', '1503.05281-1-20-0': 'We now briefly review the cosmological data and fitting methodology used in constraining the model parameters.', '1503.05281-1-20-1': 'We will perform a joint analysis of the Type Ia Supernovae (SNIa), the baryonic acoustic oscillation (BAO) and the cosmic microwave background (CMB) data to break the degeneracy between the model parameters.', '1503.05281-1-20-2': 'For SNIa, we use the Union2.1 dataset [CITATION] which consists of 580 data points.', '1503.05281-1-20-3': 'These data are given in terms of the distance modulus [MATH].', '1503.05281-1-20-4': 'By definition, the theoretical distance modulus is given by [EQUATION] where [MATH] with [MATH] the Hubble constant [MATH] in units of [MATH].', '1503.05281-1-20-5': 'Here, the luminosity distance can be calculated as [EQUATION] in which [MATH] denotes the model parameters.', '1503.05281-1-20-6': 'Consequently, the [MATH] from 580 Union2.1 SNIa is given by [EQUATION] where [MATH] is the corresponding [MATH] error.', '1503.05281-1-20-7': 'Following [CITATION], we marginalize over [MATH] by expanding the [MATH] with respect to [MATH] as [EQUATION] where [EQUATION].', '1503.05281-1-20-8': 'Eq. ([REF]) has a minimum for [MATH] at [EQUATION]', '1503.05281-1-20-9': 'Since [MATH] (up to a constant), we can instead minimize [MATH] which is independent of [MATH].', '1503.05281-1-20-10': 'For the data of CMB and BAO, we use the shift parameter [MATH] from CMB, and the distance parameter [MATH] from the measurement of the BAO peak in the distribution of SDSS luminous red galaxies, since they are model-independent and contain the main information of the observations of CMB and BAO, respectively (see e.g. [CITATION]).', '1503.05281-1-20-11': 'The shift parameter [MATH] of CMB is defined by [CITATION] [EQUATION] where the redshift of recombination [MATH] is determined to be [MATH] by the Planck 2015 data [CITATION].', '1503.05281-1-20-12': 'On the other hand, the Planck 2015 data have also determined the observed value of shift parameter [MATH] to be [MATH] [CITATION].', '1503.05281-1-20-13': 'So, the [MATH] for CMB is [EQUATION]', '1503.05281-1-20-14': 'The distance parameter [MATH] of the measurement of the BAO peak in the distribution of SDSS luminous red galaxies [CITATION] is given by [EQUATION] where [MATH].', '1503.05281-1-20-15': 'In [CITATION], the value of [MATH] has been determined to be [MATH].', '1503.05281-1-20-16': 'Here the scalar spectral index [MATH] is taken to be [MATH] by the Planck 2015 data [CITATION].', '1503.05281-1-20-17': 'And the corresponding [MATH] for BAO is [EQUATION]', '1503.05281-1-20-18': 'So, the total [MATH] is given by [EQUATION]', '1503.05281-1-20-19': 'Then we can minimize the total [MATH] to get the best-fit values of model parameters.', '1503.05281-1-20-20': 'The [MATH] and [MATH] confidence levels are determined by [MATH] and [MATH], respectively, if there are 3 free model parameters.', '1503.05281-1-21-0': '## The power-law case', '1503.05281-1-22-0': 'In this subsection, we consider the power-law functional of [MATH] first introduced by Bengochea et al. in [CITATION], which reads [EQUATION] where [MATH] and [MATH] are both constants.', '1503.05281-1-22-1': 'Demanding Eq. [REF] to be satisfied at redshift [MATH], we have [EQUATION]', '1503.05281-1-22-2': 'Substituting Eq. [REF] back into Eq. [REF], the corresponding background evolution reads [EQUATION]', '1503.05281-1-22-3': 'We note that, when [MATH], this model corresponds to [MATH]CDM model in fact.', '1503.05281-1-22-4': 'There are 3 free parameters in this model, namely [MATH], [MATH], and [MATH].', '1503.05281-1-22-5': 'By minimizing the corresponding total [MATH] in Eq. ([REF]), we find the best-fit parameters [MATH], [MATH], and [MATH], while [MATH].', '1503.05281-1-22-6': 'In Fig. [REF], we present the corresponding [MATH] and [MATH] confidence level contours for the power-law [MATH] massive gravity in the [MATH], the [MATH] and the [MATH] planes, respectively.', '1503.05281-1-22-7': 'We find that [MATH]CDM model (corresponding to [MATH]) is still consistent with the observations at the [MATH] confidence level.', '1503.05281-1-22-8': 'In Fig. [REF], we present the evolutionary curve of [MATH] with the best-fit values of model parameters.', '1503.05281-1-22-9': 'Apparently, the power-law case shows a phantom crossing behavior with the crossing of phantom divide line occurring at redshift [MATH].', '1503.05281-1-23-0': '## The exponential case', '1503.05281-1-24-0': 'In this subsection, we consider the exponential functional of [MATH] first introduced by Linder in [CITATION], which reads [EQUATION] where [MATH] and [MATH] are both constants.', '1503.05281-1-24-1': 'Demanding Eq. [REF] to be satisfied at redshift [MATH], we have [EQUATION]', '1503.05281-1-24-2': 'Substituting Eq. [REF] back into Eq. [REF], the corresponding background evolution reads [EQUATION]', '1503.05281-1-24-3': 'We note that, when [MATH], this model corresponds to [MATH]CDM model in fact.', '1503.05281-1-24-4': 'There are 3 free parameters in this model, namely [MATH], [MATH], and [MATH].', '1503.05281-1-24-5': 'By minimizing the corresponding total [MATH] in Eq. ([REF]), we find the best-fit parameters [MATH], [MATH], and [MATH], while [MATH].', '1503.05281-1-24-6': 'In Fig. [REF], we present the corresponding [MATH] and [MATH] confidence level contours for the exponential [MATH] massive gravity in the [MATH], the [MATH] and the [MATH] planes, respectively.', '1503.05281-1-24-7': 'We find that [MATH]CDM model (corresponding to [MATH]) is still consistent with the observations at the [MATH] confidence level.', '1503.05281-1-24-8': 'In Fig. [REF], we present the evolutionary curve of [MATH] with the best-fit values of model parameters.', '1503.05281-1-24-9': 'Apparently, the exponential case shows a phantom crossing behavior with the crossing of phantom divide line occurring at redshift [MATH].', '1503.05281-1-25-0': '# Conclusions', '1503.05281-1-26-0': 'In this paper, we extend [MATH] theory and dRGT massive gravity to a new kind of modified gravity model, namely [MATH] non-linear massive gravity, by adding the dRGT mass term to [MATH] theory.', '1503.05281-1-26-1': 'This mass term is formulated in the vierbein formalism to agree with the teleparallel space-time.', '1503.05281-1-26-2': 'Since the resulting field equations are [MATH]nd order, and the mass term does not contain the square root of the metric when using vierbein formalism, this theory is easier to analyse than [MATH] non-linear massive gravity.', '1503.05281-1-26-3': 'Besides, thanks to the rich structure of [MATH] sector and massive graviton, we argue that this theory can also unify the early inflation and late-time acceleration in a consistent framework.', '1503.05281-1-27-0': 'We then investigate the cosmological evolution of [MATH] non-linear massive gravity.', '1503.05281-1-27-1': 'In particular, we study the power-law and exponential cases of [MATH] massive gravity as two toy models.', '1503.05281-1-27-2': 'We then perform a joint constraint on the model parameters by the recent data of SNIa, CMB and BAO.', '1503.05281-1-27-3': 'We find that the power-law and exponential [MATH] massive gravity are consistent with these cosmological observations.', '1503.05281-1-27-4': 'Furthermore, we explore the evolution of the effective EoS parameter of DE, and find that it can realize the crossing of the phantom divide line from the quintessence-like phase ([MATH]) to the phantom-like one ([MATH]) by using the best-fit parameters obtained from the above cosmological constraints.', '1503.05281-1-27-5': 'We note here that the recent data shows great possibility that the EoS parameter of DE crosses the phantom divide line from the quintessence-like phase to the phantom-like phase as the redshift [MATH] decreases in the near past [CITATION].', '1503.05281-1-27-6': 'Although there exit some complicated specific [MATH] models to realize the phantom crossing behavior [CITATION], in general, especially in the original power-law and exponential [MATH] gravity, phantom crossing is impossible [CITATION].', '1503.05281-1-27-7': 'So, our results are of interest.', '1503.05281-1-28-0': 'We thank Savvas Nesseris for helpful discussion and providing his Mathematica code for data constraint, which greatly improves our work.', '1503.05281-1-28-1': 'We are grateful to Jing Liu, Xiao-Peng Yan, Ya-Nan Zhou, Xiao-Bo Zou, and Hong-Yu Li for kind help and discussions.', '1503.05281-1-28-2': 'This work was supported in part by NSFC under Grants No. 11175016 and No. 10905005, as well as NCET under Grant No. NCET-11-0790.'}","{'1503.05281-2-0-0': 'ABSTRACT Inspired by the [MATH] non-linear massive gravity, we propose a new kind of modified gravity model, namely [MATH] non-linear massive gravity, by adding the dRGT mass term reformulated in the vierbein formalism, to the [MATH] theory.', '1503.05281-2-0-1': 'We then investigate the cosmological evolution of [MATH] massive gravity, and constrain it by using the latest observational data.', '1503.05281-2-0-2': 'We find that it slightly favors a crossing of the phantom divide line from the quintessence-like phase ([MATH]) to the phantom-like one ([MATH]) as redshift decreases.', '1503.05281-2-1-0': '# Introduction', '1503.05281-2-2-0': ""Einstein's theory of general relativity (GR) has achieved great success in explaining various gravitational phenomena since its birth, and till today it can pass the Solar System's tests to a very high precision."", '1503.05281-2-2-1': 'However, on scales larger than the Solar System, there are some astrophysical observations having not yet been fully understood or explained in the context of GR.', '1503.05281-2-2-2': 'Maybe the most urgent among them are the presence of an invisible (dark) matter component in our universe, first discovered by Zwicky in the [MATH]s [CITATION], as well as the late-time acceleration of our universe confirmed by various cosmological observations [CITATION].', '1503.05281-2-2-3': 'Although by extending the standard model of particle physics, there exist various promising candidates for dark matter (see e.g. reviews [CITATION]), so far there has been no natural and satisfactory explanation for the cosmic acceleration by introducing some kinds of mysterious cosmological fluids with negative pressure called dark energy (DE), since they all suffer from the fine-tuning [CITATION] and/or the cosmic coincidence problems [CITATION].', '1503.05281-2-2-4': 'Unless all of these problems are eventually solved on the particle physics side, the severity of the phenomenological problems challenges GR as the ultimate and complete theory of gravity.', '1503.05281-2-2-5': 'Furthermore, difficulties arise when trying to search for a quantum theory of gravity, since GR is not renormalizable [CITATION] and thus cannot make meaningful physical predictions.', '1503.05281-2-2-6': ""It is, therefore, reasonable to investigate the possibility of modifying Einstein's gravity theory."", '1503.05281-2-3-0': 'Among the modified gravity theories, [MATH] theory (see e.g. [CITATION] for recent reviews) serves as one of the most straightforward and popular generalizations of GR by extending the Ricci scalar [MATH] in the Einstein-Hilbert action to a general function [MATH].', '1503.05281-2-3-1': 'It has been shown that [MATH] gravity can explain the present cosmic acceleration without the need of DE [CITATION].', '1503.05281-2-3-2': 'But in general, the resulting field equations of motion are [MATH]th order in [MATH] gravity.', '1503.05281-2-3-3': 'This feature makes [MATH] theory quite hard to analyse.', '1503.05281-2-3-4': 'In comparison with [MATH] gravity, there is another kind of modified gravity, namely [MATH] theory, which has been extensively explored in the literature (see e.g. [CITATION]).', '1503.05281-2-3-5': 'Here [MATH] is the torsion scalar constructed from the curvatureless Weitzenbock connection [CITATION].', '1503.05281-2-3-6': 'It is well known that [MATH] theory is based on the teleparallel space-time, in which vierbein field is used as the basic dynamical variable instead of metric in GR.', '1503.05281-2-3-7': 'One advantage of [MATH] theory in contrast to [MATH] theory is that, the field equations arising from [MATH] theory are [MATH]nd order, and thus simpler than [MATH] case.', '1503.05281-2-4-0': 'On the other hand, as a classical field theory, GR propagates a non-linear self-interacting massless spin-[MATH] field.', '1503.05281-2-4-1': 'Therefore, another way to modify GR is to give a tiny mass to the graviton.', '1503.05281-2-4-2': 'This class of theories is known as massive gravity in the literature.', '1503.05281-2-4-3': 'Actually, massive gravity has a long and elusive history.', '1503.05281-2-4-4': 'The first attempt to give a mass to graviton dates back to the year of 1939 when Fierz and Pauli proposed a unique ghost-free mass term to gravity in the linear perturbation level [CITATION].', '1503.05281-2-4-5': 'The linearised massive gravity propagates [MATH] degrees of freedom (DoF).', '1503.05281-2-4-6': 'And then, in 1970s, van Dam-Veltman-Zakharov (vDVZ) discontinuity was found.', '1503.05281-2-4-7': 'It was shown that, when the mass of graviton [MATH], Fierz-Pauli theory would not converge to massless case [CITATION], thus giving different predictions than those GR.', '1503.05281-2-4-8': 'Almost at the same time, Vainshtein [CITATION] showed that the linear approximation loses its validity around massive sources below the Vainshtein radius and non-linear effects should be considered.', '1503.05281-2-4-9': 'The vDVZ discontinuity can then be avoided by this well-known Vainshtein screening mechanism.', '1503.05281-2-4-10': 'However, it was claimed that any non-linear extensions of Fierz-Pauli theory will generally have [MATH] DoF, with the extra DoF being a Boulware-Deser (BD) ghost [CITATION].', '1503.05281-2-4-11': 'The construction of a well-defined non-linear theory of massive gravity without the BD ghost has been a challenging problem over decades.', '1503.05281-2-4-12': 'Recently, it was shown that BD ghost may be removed order-by-order in a careful construction of mass term up to a certain decoupling limit [CITATION].', '1503.05281-2-4-13': 'The full non-linear massive gravity was resummed by de Rham, Gabadadze and Tolley (dRGT) in [CITATION], and the absence of BD ghost was further confirmed by using Hamiltonian constraint analysis in [CITATION].', '1503.05281-2-4-14': 'Although dRGT theory exists accelerating solutions [CITATION], it suffers from some instabilities at the cosmological perturbation level [CITATION].', '1503.05281-2-4-15': 'Since then dRGT theory has been extended in various approaches, e.g. the mass varying massive gravity [CITATION] and the quasi-dilaton massive gravity [CITATION].', '1503.05281-2-5-0': 'Recently, a new extension of massive gravity, namely [MATH] non-linear massive gravity, has been proposed [CITATION].', '1503.05281-2-5-1': 'It is shown that this model not only shares no linear instabilities around a cosmological background, but can also describe the inflation and the recently cosmic acceleration in a unified framework [CITATION].', '1503.05281-2-5-2': 'Inspired by the [MATH] non-linear massive gravity, we propose a new kind of modified gravity theory by adding the non-linear dRGT mass term formulated in the vierbein formalism, to the [MATH] theory.', '1503.05281-2-5-3': 'We might call this theory [MATH] non-linear massive gravity, or simply [MATH] massive gravity.', '1503.05281-2-5-4': 'One merit of our model compared with the [MATH] massive gravity is that, the field equations arising from [MATH] massive gravity are [MATH]nd order, and thus much easier to analyse.', '1503.05281-2-5-5': 'Moreover, [MATH] massive gravity, as well as dRGT theory, is generally constructed in the metric formulation.', '1503.05281-2-5-6': 'However, in the metric formalism, the mass term is difficult to analyse, since it contains the square root of the metric.', '1503.05281-2-5-7': 'Another merit of our theory is that, in teleparallel space-time, we use vierbein instead of metric as the basic dynamical variable.', '1503.05281-2-5-8': 'Since vierbein can be viewed as the square root of the metric in some sense, the vierbein formalism in our theory can greatly simplify the calculation.', '1503.05281-2-6-0': 'After presenting the action of our model, we then investigate the cosmological evolution of [MATH] massive gravity.', '1503.05281-2-6-1': 'The rest of this paper is organized as follows.', '1503.05281-2-6-2': 'In Sec. [REF], we briefly review the [MATH] gravity.', '1503.05281-2-6-3': 'In Sec. [REF], we add the ghost-free dRGT mass term reformulated in the vierbein formalism to the [MATH] sector, thus giving the action of [MATH] massive gravity.', '1503.05281-2-6-4': 'The evolutionary equations for the flat Friedmann-Robertson-Walker (FRW) cosmology are given as well.', '1503.05281-2-6-5': 'In Sec. [REF], we focus on two concrete examples, namely the power-law and exponential types of [MATH] massive gravity and investigate their cosmological evolution.', '1503.05281-2-6-6': 'We constrain the model parameters by the recent cosmological data and study the phantom crossing behavior.', '1503.05281-2-6-7': 'In Sec. [REF], we give some concluding remarks.', '1503.05281-2-7-0': '# [MATH] f(T) gravity', '1503.05281-2-8-0': '[MATH] gravity, which is a generalization of the teleparallel gravity originally proposed by Einstein [CITATION], has received great interest in the literature recently.', '1503.05281-2-8-1': 'Following [CITATION], we now briefly review the [MATH] theory.', '1503.05281-2-8-2': '[MATH] gravity is built upon teleparallel space-time.', '1503.05281-2-8-3': 'In teleparallel space-time, the basic dynamical quantity is a vierbein field [MATH], with Latin indices [MATH], and [MATH], Greek indices [MATH], and [MATH] coordinate bases.', '1503.05281-2-8-4': 'We also note that the Einstein summation notation for the indices is used throughout this paper.', '1503.05281-2-8-5': 'The vierbein is an orthonormal basis for the tangent space at each point [MATH] of the manifold, namely [MATH], with [MATH].', '1503.05281-2-8-6': 'The metric tensor can then be expressed in the dual vierbein [MATH] as [EQUATION]', '1503.05281-2-8-7': 'Rather than using the torsionless Levi-Civita connection in general relativity (GR), teleparallel space-time uses the curvatureless Weitzenbock connection [MATH], which is defined by [EQUATION]', '1503.05281-2-8-8': 'Note that the lower indices [MATH] and [MATH] are not symmetric in general, thus the torsion tensor is non-zero in the teleparallel space-time.', '1503.05281-2-8-9': 'The Weitzenbock torsion tensor is defined by [EQUATION]', '1503.05281-2-8-10': 'In teleparallel gravity, the gravitational action is given by the torsion scalar instead of the the Ricci scalar in GR.', '1503.05281-2-8-11': 'The torsion scalar is basically the square of the Weitzenbock torsion tensor, and is defined by [EQUATION] with the tensor [MATH] given by [EQUATION]', '1503.05281-2-8-12': 'In [MATH] gravity, the gravitational field is driven by a Lagrangian density [MATH], with [MATH] a function of [MATH], and the action reads [EQUATION] where [MATH], [MATH], with [MATH] the determinant of the metric [MATH] and [MATH] the Newtonian constant.', '1503.05281-2-8-13': 'Note that we have used the units in which the speed of light [MATH], and the reduced Planck constant [MATH].', '1503.05281-2-8-14': 'Here, [MATH] is the matter part of the action, and [MATH] denotes all matter fields collectively.', '1503.05281-2-8-15': 'If we set [MATH], then the action gives an equivalent description of the space-time as GR.', '1503.05281-2-9-0': '# [MATH] f(T) non-linear massive gravity', '1503.05281-2-10-0': 'In this section, we will add the dRGT mass term reformulated in the vierbein form to the [MATH] sector, thus giving the action of [MATH] massive gravity.', '1503.05281-2-10-1': 'We then derive the evolutionary equations for the flat FRW cosmology.', '1503.05281-2-11-0': '## The action', '1503.05281-2-12-0': 'Since teleparallel gravity equals to GR at the field equations level, here we give an equivalent description of dRGT massive theory by adding the mass term to teleparallel gravity.', '1503.05281-2-12-1': 'The mass term firstly proposed in dRGT theory, can be greatly simplified when transforming to the vierbein formalism.', '1503.05281-2-12-2': 'We refer to [CITATION] for a detailed derivation.', '1503.05281-2-12-3': 'By adding the mass term to teleparallel gravity, one will get the teleparallel version of ghost-free dRGT massive gravity as [EQUATION] where [MATH] are free parameters, [MATH] denotes the matrix of vierbein [MATH] and [MATH] are the so-called elementary symmetric polynomials [CITATION].', '1503.05281-2-12-4': 'For an arbitrary [MATH] matrix [MATH], the first few of [MATH] can be written in the form [CITATION] [EQUATION] with [MATH] the trace of the matrix [MATH].', '1503.05281-2-12-5': 'By generalizing the torsion scalar [MATH] to a function [MATH] in the action [REF], we have the [MATH] non-linear massive gravity as [EQUATION]', '1503.05281-2-12-6': 'Note that we have absorbed the cosmological constant arising from the mass term to the [MATH] sector.', '1503.05281-2-12-7': 'So, there are only three free parameters [MATH], [MATH], and [MATH] in the mass term.', '1503.05281-2-12-8': 'We also note that when [MATH], the action [REF] is just the teleparallel gravity plus a dRGT mass term reformulated in the vierbein formalism, thus being equivalent to the original dRGT theory at the field equations level.', '1503.05281-2-12-9': 'So, if [MATH], then the BD ghost will not show up in the action [REF].', '1503.05281-2-12-10': 'However, if [MATH] is a non-trivial function of [MATH], then additional DoF will be introduced.', '1503.05281-2-12-11': 'Therefore, we should perform Hamiltonian analysis to confirm that the action [REF] is ghost-free.', '1503.05281-2-12-12': 'In this paper we focus on the cosmological behavior of [MATH] non-linear massive gravity, and the ghost problem or other instabilities that our theory may suffer from, will be considered in our future work.', '1503.05281-2-13-0': '## Equation of motion', '1503.05281-2-14-0': 'We now consider a spatially flat FRW space-time [EQUATION] where [MATH] is the scale factor and [MATH] is the lapse function.', '1503.05281-2-14-1': 'This metric arises from the diagonal dual vierbein [EQUATION] through Eq. [REF].', '1503.05281-2-14-2': 'Then we can easily get [MATH], [MATH], and hence [EQUATION]', '1503.05281-2-14-3': 'The torsion scalar is also obtained to be [MATH], where a prime denotes [MATH].', '1503.05281-2-15-0': 'After some algebra, action [REF] reduces to [EQUATION]', '1503.05281-2-15-1': 'For further convenience, we define some variables here, namely [EQUATION]', '1503.05281-2-15-2': 'Now we can get the field equations by varying action [REF] with respect to [MATH] and [MATH], respectively, [EQUATION] where a dot denotes [MATH], [MATH], [MATH], and [MATH], [MATH] are the energy density and pressure of all perfect fluids of generic matter, respectively.', '1503.05281-2-15-3': 'Notice that we do not need to know the explicit form of [MATH] in the above derivation.', '1503.05281-2-15-4': 'As is well known, the energy-momentum tensor of a perfect fluid takes a diagonal form in the comoving coordinates, namely [EQUATION]', '1503.05281-2-15-5': 'And from the standard definition of energy-momentum tensor [EQUATION] we can easily read off [MATH] and [MATH] as [EQUATION]', '1503.05281-2-15-6': 'From Eqs. [REF] and [REF], we can get the corresponding energy conservation equation for matter, [EQUATION]', '1503.05281-2-15-7': 'We can recast Eqs. [REF] and [REF] in the similar form as in the GR case by introducing the energy density and pressure of the effective DE as [EQUATION]', '1503.05281-2-15-8': 'Then we get the modified Friedmann equations as [EQUATION] in which [MATH], [MATH].', '1503.05281-2-15-9': 'From Eqs. [REF] and [REF], one can obtain the corresponding energy conservation equation for DE as [EQUATION]', '1503.05281-2-15-10': 'Combining Eqs. [REF] and [REF] yields [EQUATION] which is the usual conservation equation for total energy.', '1503.05281-2-15-11': 'We see that matter and DE are interacted through the graviton.', '1503.05281-2-15-12': 'We define the equation-of-state (EoS) parameter of the effective DE as [EQUATION]', '1503.05281-2-16-0': '# Observational constraints', '1503.05281-2-17-0': 'Here we are interested in the late-time behavior of [MATH] massive gravity.', '1503.05281-2-17-1': 'In this section, we study two specific functionals of [MATH] as concrete examples.', '1503.05281-2-17-2': 'These two examples will be constrained by the recent cosmological data.', '1503.05281-2-17-3': 'We then study the evolution of the effective EoS parameter of DE.', '1503.05281-2-18-0': '## General set-up', '1503.05281-2-19-0': 'Since we are interested in the late-time universe, we ignore the radiation component and only consider the pressureless matter, whose energy density is [MATH] and pressure is [MATH].', '1503.05281-2-19-1': 'Furthermore, we also set [MATH], since at the background level, the terms [MATH] and [MATH] in Eq. [REF] are indistinguishable from the space curvature and pressureless matter terms, respectively.', '1503.05281-2-19-2': 'Having these in mind, and by using Eqs. [REF] and [REF], the effective EoS parameter of DE [REF] can be further simplified to [EQUATION] where the fractional density of the pressureless matter and the graviton are given by [EQUATION] respectively.', '1503.05281-2-19-3': 'Note that we always use a subscript ""[MATH]"" to denote the present value of corresponding quantity.', '1503.05281-2-19-4': 'So [MATH] and [MATH], with the redshift [MATH] defined as [MATH].', '1503.05281-2-19-5': 'Consequently, we find the expression for the dimensionless Hubble parameter [MATH], namely [EQUATION]', '1503.05281-2-19-6': 'Note that [MATH] appears in both sides of this equation, and implicitly in [MATH] and [MATH].', '1503.05281-2-20-0': 'We now briefly review the cosmological data and fitting methodology used in constraining the model parameters.', '1503.05281-2-20-1': 'We will perform a joint analysis of the Type Ia Supernovae (SNIa), the baryonic acoustic oscillation (BAO) and the cosmic microwave background (CMB) data to break the degeneracy between the model parameters.', '1503.05281-2-20-2': 'For SNIa, we use the Union2.1 dataset [CITATION] which consists of 580 data points.', '1503.05281-2-20-3': 'These data are given in terms of the distance modulus [MATH].', '1503.05281-2-20-4': 'By definition, the theoretical distance modulus is given by [EQUATION] where [MATH] with [MATH] the Hubble constant [MATH] in units of [MATH].', '1503.05281-2-20-5': 'Here, the luminosity distance can be calculated as [EQUATION] in which [MATH] denotes the model parameters.', '1503.05281-2-20-6': 'Consequently, the [MATH] from 580 Union2.1 SNIa is given by [EQUATION] where [MATH] is the corresponding [MATH] error.', '1503.05281-2-20-7': 'Following [CITATION], we marginalize over [MATH] by expanding the [MATH] with respect to [MATH] as [EQUATION] where [EQUATION].', '1503.05281-2-20-8': 'Eq. ([REF]) has a minimum for [MATH] at [EQUATION]', '1503.05281-2-20-9': 'Since [MATH] (up to a constant), we can instead minimize [MATH] which is independent of [MATH].', '1503.05281-2-20-10': 'For the data of CMB and BAO, we use the shift parameter [MATH] from CMB, and the distance parameter [MATH] from the measurement of the BAO peak in the distribution of SDSS luminous red galaxies, since they are model-independent and contain the main information of the observations of CMB and BAO, respectively (see e.g. [CITATION]).', '1503.05281-2-20-11': 'The shift parameter [MATH] of CMB is defined by [CITATION] [EQUATION] where the redshift of recombination [MATH] is determined to be [MATH] by the Planck 2015 data [CITATION].', '1503.05281-2-20-12': 'On the other hand, the Planck 2015 data have also determined the observed value of shift parameter [MATH] to be [MATH] [CITATION].', '1503.05281-2-20-13': 'So, the [MATH] for CMB is [EQUATION]', '1503.05281-2-20-14': 'The distance parameter [MATH] of the measurement of the BAO peak in the distribution of SDSS luminous red galaxies [CITATION] is given by [EQUATION] where [MATH].', '1503.05281-2-20-15': 'In [CITATION], the value of [MATH] has been determined to be [MATH].', '1503.05281-2-20-16': 'Here the scalar spectral index [MATH] is taken to be [MATH] by the Planck 2015 data [CITATION].', '1503.05281-2-20-17': 'And the corresponding [MATH] for BAO is [EQUATION]', '1503.05281-2-20-18': 'So, the total [MATH] is given by [EQUATION]', '1503.05281-2-20-19': 'Then we can minimize the total [MATH] to get the best-fit values of model parameters.', '1503.05281-2-20-20': 'The [MATH] and [MATH] confidence levels are determined by [MATH] and [MATH], respectively, if there are 3 free model parameters.', '1503.05281-2-21-0': '## The power-law case', '1503.05281-2-22-0': 'In this subsection, we consider the power-law functional of [MATH] first introduced by Bengochea et al. in [CITATION] (see also [CITATION]), which reads [EQUATION] where [MATH] and [MATH] are both constants.', '1503.05281-2-22-1': 'Demanding Eq. [REF] to be satisfied at redshift [MATH], we have [EQUATION]', '1503.05281-2-22-2': 'Substituting Eq. [REF] back into Eq. [REF], the corresponding background evolution reads [EQUATION]', '1503.05281-2-22-3': 'We note that, when [MATH], this model corresponds to [MATH]CDM model in fact.', '1503.05281-2-22-4': 'There are 3 free parameters in this model, namely [MATH], [MATH], and [MATH].', '1503.05281-2-22-5': 'By minimizing the corresponding total [MATH] in Eq. ([REF]), we find the best-fit parameters [MATH], [MATH], and [MATH], while [MATH].', '1503.05281-2-22-6': 'In Fig. [REF], we present the corresponding [MATH] and [MATH] confidence level contours for the power-law [MATH] massive gravity in the [MATH], the [MATH] and the [MATH] planes, respectively.', '1503.05281-2-22-7': 'We find that [MATH]CDM model (corresponding to [MATH]) is still consistent with the observations at the [MATH] confidence level.', '1503.05281-2-22-8': 'In Fig. [REF], we present the evolutionary curve of [MATH] with the best-fit values of model parameters.', '1503.05281-2-22-9': 'Apparently, the power-law case shows a phantom crossing behavior with the crossing of phantom divide line occurring at redshift [MATH].', '1503.05281-2-23-0': '## The exponential case', '1503.05281-2-24-0': 'In this subsection, we consider the exponential functional of [MATH] first introduced by Linder in [CITATION], which reads [EQUATION] where [MATH] and [MATH] are both constants.', '1503.05281-2-24-1': 'Demanding Eq. [REF] to be satisfied at redshift [MATH], we have [EQUATION]', '1503.05281-2-24-2': 'Substituting Eq. [REF] back into Eq. [REF], the corresponding background evolution reads [EQUATION]', '1503.05281-2-24-3': 'We note that, when [MATH], this model corresponds to [MATH]CDM model in fact.', '1503.05281-2-24-4': 'There are 3 free parameters in this model, namely [MATH], [MATH], and [MATH].', '1503.05281-2-24-5': 'By minimizing the corresponding total [MATH] in Eq. ([REF]), we find the best-fit parameters [MATH], [MATH], and [MATH], while [MATH].', '1503.05281-2-24-6': 'In Fig. [REF], we present the corresponding [MATH] and [MATH] confidence level contours for the exponential [MATH] massive gravity in the [MATH], the [MATH] and the [MATH] planes, respectively.', '1503.05281-2-24-7': 'We find that [MATH]CDM model (corresponding to [MATH]) is still consistent with the observations at the [MATH] confidence level.', '1503.05281-2-24-8': 'In Fig. [REF], we present the evolutionary curve of [MATH] with the best-fit values of model parameters.', '1503.05281-2-24-9': 'Apparently, the exponential case shows a phantom crossing behavior with the crossing of phantom divide line occurring at redshift [MATH].', '1503.05281-2-25-0': '# Conclusions', '1503.05281-2-26-0': 'In this paper, we extend [MATH] theory and dRGT massive gravity to a new kind of modified gravity model, namely [MATH] non-linear massive gravity, by adding the dRGT mass term to [MATH] theory.', '1503.05281-2-26-1': 'This mass term is formulated in the vierbein formalism to agree with the teleparallel space-time.', '1503.05281-2-26-2': 'Since the resulting field equations are [MATH]nd order, and the mass term does not contain the square root of the metric when using vierbein formalism, this theory is easier to analyse than [MATH] non-linear massive gravity.', '1503.05281-2-26-3': 'Besides, thanks to the rich structure of [MATH] sector and massive graviton, it is natural to expect that this theory could also unify the early inflation and late-time acceleration in a consistent framework, and we leave this issue to the future works.', '1503.05281-2-27-0': 'We then investigate the cosmological evolution of [MATH] non-linear massive gravity.', '1503.05281-2-27-1': 'In particular, we study the power-law and exponential cases of [MATH] massive gravity as two toy models.', '1503.05281-2-27-2': 'We then perform a joint constraint on the model parameters by the recent data of SNIa, CMB and BAO.', '1503.05281-2-27-3': 'We find that the power-law and exponential [MATH] massive gravity are consistent with these cosmological observations.', '1503.05281-2-27-4': 'Furthermore, we explore the evolution of the effective EoS parameter of DE, and find that it can realize the crossing of the phantom divide line from the quintessence-like phase ([MATH]) to the phantom-like one ([MATH]) by using the best-fit parameters obtained from the above cosmological constraints.', '1503.05281-2-27-5': 'We note here that the recent data shows great possibility that the EoS parameter of DE crosses the phantom divide line from the quintessence-like phase to the phantom-like phase as the redshift [MATH] decreases in the near past [CITATION].', '1503.05281-2-27-6': 'Although there exist some complicated specific [MATH] models to realize the phantom crossing behavior [CITATION], in general, especially in the original power-law and exponential [MATH] gravity, phantom crossing is impossible [CITATION].', '1503.05281-2-27-7': 'So, our results are of interest.', '1503.05281-2-28-0': 'We thank Savvas Nesseris for helpful discussion and providing his Mathematica code for data constraint, which greatly improves our work.', '1503.05281-2-28-1': 'We are grateful to Jing Liu, Xiao-Peng Yan, Ya-Nan Zhou, Xiao-Bo Zou, and Hong-Yu Li for kind help and discussions.', '1503.05281-2-28-2': 'This work was supported in part by NSFC under Grants No. 11175016 and No. 10905005, as well as NCET under Grant No. NCET-11-0790.'}","[['1503.05281-1-17-0', '1503.05281-2-17-0'], ['1503.05281-1-17-1', '1503.05281-2-17-1'], ['1503.05281-1-17-2', '1503.05281-2-17-2'], ['1503.05281-1-17-3', '1503.05281-2-17-3'], ['1503.05281-1-5-0', '1503.05281-2-5-0'], ['1503.05281-1-5-1', '1503.05281-2-5-1'], ['1503.05281-1-5-2', '1503.05281-2-5-2'], ['1503.05281-1-5-3', '1503.05281-2-5-3'], ['1503.05281-1-5-4', '1503.05281-2-5-4'], ['1503.05281-1-5-5', '1503.05281-2-5-5'], ['1503.05281-1-5-6', 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'1503.05281-2-19-2']]",[],"[['1503.05281-1-26-3', '1503.05281-2-26-3']]",[],"['1503.05281-1-28-2', '1503.05281-2-28-2']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1503.05281,,,
1310.5834,"{'1310.5834-1-0-0': 'Commutation of multidimensional vector fields leads to nonlinear PDEs arising in various problems of mathematical physics and are intensively studied in recent literature.', '1310.5834-1-0-1': 'This report is aiming to solve the scattering and inverse scattering problem of the Pavlov equation, one of the prototypical integrable systems of vector fields, and to justify an existence theorem for global bounded solutions of the associated Cauchy problem with large data.', '1310.5834-1-1-0': '# Introduction', '1310.5834-1-2-0': 'The Pavlov equation [EQUATION] is a generalization to the dispersionless Kadomtsev-Petviashvili (dKP) equation [EQUATION] by studying hydrodynamic chains and associated commuting flows [CITATION], [CITATION].', '1310.5834-1-2-1': 'It can also be derived by imposing symmetries to the Einstein-Weyl equations of a three-dimensional manifold with a conformal structure of signature [MATH].', '1310.5834-1-2-2': 'The integrability of the Pavlov equation can be seen by possessing a Lax representation [MATH] of the 2-dimensional vector fields [EQUATION] with the spectral variable [MATH] [CITATION], [CITATION], [CITATION].', '1310.5834-1-3-0': 'The Pavlov equation has similar features of the dKP equation, the heavenly equation, and the 2ddToda equation in that the potentials appear as vector field terms in the associated Lax operators.', '1310.5834-1-3-1': '[CITATION].', '1310.5834-1-3-2': 'These equations are systems of hydrodynamic type and can be treated as dispersionless limits of integrable systems in general [CITATION].', '1310.5834-1-3-3': 'They arise in various problems of mathematical physics and are intensively studied [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION] etc.', '1310.5834-1-3-4': ""For example, a major method for producing solutions, Tsarev's generalized hodograph method [CITATION], is through a finite-component reduction and the Riemann invariants of hydrodynamic type integrable equations."", '1310.5834-1-3-5': 'In particular, this method was used in [CITATION], [CITATION] to construct solutions for the Pavlov equation.', '1310.5834-1-3-6': 'Other elegant integration schemes and a nonlinear [MATH]-dressing through the quasiclassical process were provided in [CITATION], [CITATION], and special solutions were found in [CITATION].', '1310.5834-1-3-7': 'However, disadvantages of these approaches include: (1) only produce local solutions; (2) cannot effectively determine the domain of existence or the solution space; (3) cannot prescribe initial data; (4) usually require some smallness data constraints.', '1310.5834-1-4-0': 'Manakov and Santini have made important contribution in studying the inverse scattering problem, at a formal level, of integrable systems of vector fields [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION].', '1310.5834-1-4-1': 'More precisely, using the Fourier theory and the ring properties of the eigenfunctions, they formulated the direct problem as an integrable equation of real analytic eigenfunctions with slowly decaying kernels [CITATION]; assuming the existence, up to real line, of the complex eigenfunction of the Lax operator, they observed that the boundary values satisfy a shifted Riemann-Hilbert problem [CITATION].', '1310.5834-1-4-2': 'Besides, various approaches had been proposed to investigate the longtime behavior of solutions of the associated nonlinear PDE [CITATION], [CITATION], and to construct exact implicit solutions [CITATION].', '1310.5834-1-5-0': 'Our main result is', '1310.5834-1-6-0': 'Theorem 7.', '1310.5834-1-6-1': 'Suppose that [MATH] is a Schwartz function with compact support in [MATH].', '1310.5834-1-6-2': 'Then there exists a unique real function [MATH] such that [MATH], [MATH], [MATH] and satisfies the Pavlov equation ([REF]) weakly.', '1310.5834-1-7-0': 'Here weak sense means that the Lax pair exists globally.', '1310.5834-1-7-1': 'We prove the therorem via investigating the inverse scattering problem of the Lax pair ([REF]), ([REF]).', '1310.5834-1-7-2': 'Observe that', '1310.5834-1-8-0': 'Hence if the support of [MATH] is located inside the strip [MATH], [MATH], then [MATH] is holomorphic in [MATH] outside a small neighbourhood of [MATH] and [EQUATION] which leads us to consider the hypothetical formulas ([REF]).', '1310.5834-1-8-1': 'Therefore the direct problem of the Pavlov equation reduces to the study of the scattering function [MATH] and the shifted Riemann-Hilbert problem [EQUATION]', '1310.5834-1-8-2': 'The unique solvability of the shifted Riemann-Hilbert problem has been proved via the Fredholm theorem under the assumption that [MATH] is compactly supported and H[MATH]lder continuous [CITATION].', '1310.5834-1-8-3': 'Our difficulties are removing the compactly supported restriction (Note that the associated integral operator for ([REF]) has a kernel which decays slowly at [MATH]!)', '1310.5834-1-8-4': 'and deriving uniform and sufficiently rapid [MATH]-decaying a priori estimates of the solutions [MATH] up to the some order of [MATH]- and [MATH]-derivatives.', '1310.5834-1-8-5': 'More precisely, we prove', '1310.5834-1-9-0': 'Theorem 1.', '1310.5834-1-9-1': 'Suppose that [MATH] is a Schwartz function with compact support in [MATH].', '1310.5834-1-9-2': 'Then, for [MATH], [MATH], we have the uniform estimates [EQUATION]', '1310.5834-1-10-0': 'Combining these asymptotic estimates on boundary with interior estimates, i.e., Calderon-Zygmund estimates for the Beltrami type Lax equations associated to a series of changes of variables for [MATH], we prove', '1310.5834-1-11-0': 'Theorem 2 [MATH] 3.', '1310.5834-1-11-1': 'Suppose that [MATH] is a Schwartz function with compact support in [MATH].', '1310.5834-1-11-2': 'Then, for [MATH], there exists a unique continuous eigenfunction [MATH] satisfying [EQUATION]', '1310.5834-1-11-3': 'Moreover, [MATH] is holomorphic in [MATH], [MATH] exist for [MATH], [MATH], and [EQUATION]', '1310.5834-1-11-4': 'Here [MATH] is the eigenfunction to [MATH] normalized at [MATH].', '1310.5834-1-12-0': 'We turn to the inverse problem of the Pavlov equation.', '1310.5834-1-12-1': 'In view of ([REF]) and the Plemelj formula, the key for the inverse problem is to establish the unique solvability of the scalar nonlinear Riemann-Hilbert problem on the real line', '1310.5834-1-13-0': '+^-_R(,)=x-y-^2 t+12i_R(,)(,)=-2i^-_I(,).', '1310.5834-1-13-1': 'In order to achieve a global solvability for [MATH] and its derivatives with large scattering data [MATH], a Newtonian iteration scheme is adopted.', '1310.5834-1-13-2': 'So the main task is to derive an [MATH]-confined estimate for initial data [MATH], where [EQUATION] and boundness estimates for the linearized operator of [MATH] and its inverse.', '1310.5834-1-13-3': 'We obtain these estimates in [MATH] by using ([REF]), estimates ([REF]), and proving a Fredholm alternative theorem of the linearized operator of [MATH].', '1310.5834-1-13-4': 'However, due to the nonlinearity and the dressing properties of the Lax operators ([MATH]-dependency of Lax operators [MATH] and [MATH]), these estimates are not uniform with respect to [MATH] and [MATH].', '1310.5834-1-13-5': 'On the other hand, the [MATH]-confined estimate is determined by the property [EQUATION] and does not crucially depend on the size of scattering data [MATH] fortunately.', '1310.5834-1-13-6': 'Hence one can iterate short time existence and estimates and a continuation principle to derive global sovability for large data.', '1310.5834-1-14-0': 'Base on the solvability of the nonlinear Riemann-Hilbert problem ([REF]), dressing properties of the Lax operators, and the asymptotic estimates for [MATH], [MATH], and [MATH], one can utilize the Cauchy integral formula to derive the Lax pair.', '1310.5834-1-15-0': 'Theorem 6.', '1310.5834-1-15-1': 'Suppose that [MATH] is a Schwartz function with compact support in [MATH].', '1310.5834-1-15-2': 'Let', '1310.5834-1-16-0': 'v(x,y,t)=12i_R((x,y,t,),)d,', '1310.5834-1-17-0': '(x,y,t,)=x-y-^2 t+12i_R((x,y,t,),) C ^.', '1310.5834-1-17-1': 'Then [MATH], [MATH], [MATH], [MATH], and [EQUATION] for [MATH].', '1310.5834-1-18-0': 'Taking cross derivatives of the Lax pair ([REF]), we then prove Theorem 7.', '1310.5834-1-18-1': 'Our final remark is, despite our solution [MATH] is uniform bounded, [MATH] and [MATH] could blow up.', '1310.5834-1-18-2': 'More precisely, we prove that there exist [MATH] such that [MATH] satisfies the Pavlov equation for each [MATH] and [MATH] in [MATH], [MATH].', '1310.5834-1-19-0': 'The plan of this paper is as follows.', '1310.5834-1-19-1': 'In Section [REF], we study properties of the real eigenfunction [MATH] and the shift function [MATH].', '1310.5834-1-19-2': 'In Section [REF], the shifted Riemann-Hilbert problem ([REF]) is investigated and Theorem 1 is justified.', '1310.5834-1-19-3': 'The complex eigenfunction [MATH] is obtained and its boundary properties are characterized in Section [REF].', '1310.5834-1-19-4': 'In Section [REF], the scattering data is defined and its analytic and algebraic constraints are provided.', '1310.5834-1-19-5': 'Section [REF] and [REF] are devoted to the inverse problem.', '1310.5834-1-19-6': 'Namely, we use the Newtonian iteration method to solve the nonlinear Riemann-Hilbert problem ([REF]) in Section [REF], show the existence of the Lax pair, and derive global bounded solutions of the Cauchy problem of the Pavlov equation with large data in Section [REF].'}","{'1310.5834-2-0-0': 'Commutation of multidimensional vector fields leads to integrable nonlinear dispersionless PDEs arising in various problems of mathematical physics and intensively studied in the recent literature.', '1310.5834-2-0-1': 'This report is aiming to solve the scattering and inverse scattering problem for integrable dispersionless PDEs, recently introduced just at a formal level, concentrating on the prototypical example of the Pavlov equation, and to justify an existence theorem for global bounded solutions of the associated Cauchy problem with small data.', '1310.5834-2-1-0': '# Introduction', '1310.5834-2-2-0': 'Integrable soliton equations, like the Korteweg - de Vries [CITATION], the Nonlinear Scrhodinger [CITATION] equations and their integrable [MATH] dimensional generalizations, the Kadomtsev - Petviashvili [CITATION] and Davey - Stewartson [CITATION] equations respectively, play a key role in the study of waves propagating in weakly nonlinear and dispersive media.', '1310.5834-2-2-1': 'The Inverse Spectral Transform (IST) method, introduced by Gardner, Green, Kruskal and Miura [CITATION], is the spectral method allowing one to solve the Cauchy problem for such PDEs, predicting that a localized disturbance evolves into a number of soliton pulses + radiation, and solitons arise as an exact balance between nonlinearity and dispersion [CITATION],[CITATION],[CITATION],[CITATION].', '1310.5834-2-2-2': 'There is another important class of integrable PDEs, the so-called dispersionless PDEs (dPDEs), or PDEs of hydrodynamic type, arising in various problems of Mathematical Physics and intensively studied in the recent literature (see, f.i., in the multidimensional context, [CITATION]).', '1310.5834-2-2-3': 'The class of integrable dPDEs includes relevant examples, like the dispersionless Kadomtsev - Petviashvili (dKP) equation [CITATION],[CITATION],[CITATION], describing the evolution of weakly nonlinear, nearly one-dimensional waves in Nature, in the absence of dispersion and dissipation [CITATION], [CITATION], [CITATION], [CITATION], the first and second heavenly equations of Plebanski [CITATION], relevant in complex gravity, and the dispersionless 2D Toda (or Boyer-Finley) equation [CITATION], whose elliptic and hyperbolic versions are relevant in twistor theory [CITATION] as integrable Einstein - Weyl geometries [CITATION], and in the ideal Hele-Shaw problem [CITATION].', '1310.5834-2-3-0': 'Since integrable dPDEs arise from the condition of commutation [MATH] of pairs of one-parameter families of vector fields, implying the existence of common zero energy eigenfunctions (elements of the common kernel): [EQUATION] they can be in an arbitrary number of dimensions [CITATION], unlike the soliton PDEs.', '1310.5834-2-3-1': 'In addition, due to the lack of dispersion, these multidimensional PDEs may or may not exhibit a gradient catastrophe at finite time.', '1310.5834-2-3-2': 'To investigate integrable dPDEs, a novel IST for vector fields, significantly different from that of soliton PDEs, has been recently constructed in [CITATION], just at a formal level, i) to solve their Cauchy problem, ii) obtain the longtime behavior of solutions, iii) costruct distinguished classes of exact implicit solutions, iv) establish if, due to the lack of dispersion, the nonlinearity of the dPDE is ""strong enough"" to cause the gradient catastrophe of localized multidimensional disturbances, and v) to study analytically the breaking mechanism [CITATION].', '1310.5834-2-4-0': 'It is important to remark that this novel IST is based on some critical assumptions, like existence of analytic eigenfunctions.', '1310.5834-2-4-1': 'In soliton theory we know that, in contrast with 1+1 systems, the relevant eigenfunctions for many 2+1 PDEs (like KPtwo) are not analytic [CITATION], and the inverse problem is formulated as a [MATH]-problem.', '1310.5834-2-4-2': 'But the methods used in soliton theory for proving the existence of the relevant eigenfunctions fail in the dispersionless case, since the corresponding operators are unbounded.', '1310.5834-2-4-3': 'In addition, since the Lax operators are vector fields, the kernel space is a ring, and the inverse problem is intrinsically nonlinear.', '1310.5834-2-4-4': 'Al last, the dispersionless theory lacks of explicit regular localized solutions (solitons or lumps do not exist), and gradient catastrophes of different nature may occur at finite time.', '1310.5834-2-5-0': 'For all these reasons, it is clearly important to make the IST for vector fields rigorous (even more important than for the case of soliton PDEs); and this is the main goal of this work.', '1310.5834-2-6-0': 'To do that, we choose, as illustrative example, the simplest integrable nonlinear dPDE available in the literature, the so-called Pavlov equation [CITATION], [CITATION], [CITATION] [EQUATION] arising in the study of integrable hydrodynamic chains [CITATION], and in Differential Geometry as a particular example of Einstein - Weyl metric [CITATION].', '1310.5834-2-6-1': 'It was first derived in [CITATION] as a conformal symmetry of the second heavenly equation.', '1310.5834-2-7-0': ""As it was pointed out to the authors [CITATION], the terms [MATH] in equation ([REF]) are in common (up to the interchange of [MATH] and [MATH]) with the zero pressure Prandtl's equation for the potential [MATH] [CITATION]: [EQUATION]"", '1310.5834-2-7-1': ""The main difference between these two equations is that the friction term of the Prandtl's equation is replaced by the diffraction term of the Pavlov equation."", '1310.5834-2-7-2': ""While the zero-pressure Prandtl's equation with suitable boundary conditions gives rise to blow-up at finite time [CITATION]."", '1310.5834-2-7-3': 'We prove in this paper that localized and sufficiently small initial data for Pavlov equation remain smooth at all times.', '1310.5834-2-8-0': ""The inviscid Prandtl's equation [EQUATION] can be linearized using some partial Legendre transformation, and it also shows formation of singularities at finite time (unpublished result by V.E. Kuznetsov [CITATION])."", '1310.5834-2-9-0': 'Equation ([REF]) arises as the commutativity condition ([REF]) of the following pair of vector fields [CITATION] [EQUATION] and is the [MATH] reduction of the following integrable system of dispersionless PDEs [CITATION] [EQUATION] describing the most general integrable Einstein - Weyl metric [CITATION], [CITATION].', '1310.5834-2-9-1': 'This system reduces instead, for [MATH], to the celebrated dKP equation: [EQUATION] the simplest prototype integrable model for the study of wave breaking in multidimensions [CITATION],[CITATION].', '1310.5834-2-10-0': 'Let us point out that, although the linearized versions of the Pavlov and dKP equations coincide, the formal IST predicts a regular dynamics for the Pavlov equation, and the gradient catastrophe at finite time for the dKP equation.', '1310.5834-2-11-0': 'In our paper we prove the following result:', '1310.5834-2-12-0': 'Suppose that [MATH] is a Schwartz function with compact support and satisfies a small norm condition (see Definition [REF]).', '1310.5834-2-12-1': 'Then the IST method provides us with a real function [MATH] such that [MATH], the functions [MATH], [MATH], [MATH], [MATH], [MATH], [MATH] lie in [MATH] and satisfy the Pavlov equation ([REF]).', '1310.5834-2-13-0': 'The behavior of [MATH] at [MATH] requires an extra investigation.', '1310.5834-2-14-0': 'Since the realization of the scheme described above requires a rather big amount of technical work, including estimates on the behavior of the integral equations kernels, to make our text more transparent, we moved the proofs of the analytic estimates to the last section of our paper.', '1310.5834-2-15-0': 'The authors would like to dedicate this paper to the memory of S. V. Manakov who successfully devoted the last period of his life to the construction of the IST method for vector fields, and to its applications to the theory of integrable dispersionless PDEs in multidimensions.', '1310.5834-2-16-0': '# The Inverse Scattering Transform: a short summary', '1310.5834-2-17-0': 'We find it convenient to summarize here the basic formal steps associated with this novel IST for the vector field [MATH] in ([REF]), allowing one to solve the Cauchy problem for the Pavlov equation [CITATION], whose rigorous aspects will be investigated in the following sections.', '1310.5834-2-18-0': 'The Direct Problem In our paper we always assume that [MATH] is a real-valued function.', '1310.5834-2-18-1': 'In analogy with the IST for KPone equation (whose Lax operator in the non-stationary Schrodinger operator, see [CITATION], [CITATION]), we make essential use of two sets of eigenfunctions - the real Jost eigenfunctions [MATH], [MATH], and the complex-analytic in [MATH] ones: [MATH], [MATH]; [MATH], [MATH] [EQUATION]', '1310.5834-2-18-2': 'The direct spectral transform consists of two steps', '1310.5834-2-19-0': 'Step 1.', '1310.5834-2-19-1': 'For real [MATH], all eigenfunctions of [MATH] have the following property: they are constant on the trajectories of the following ODE: [EQUATION] defining the characteristics of [MATH].', '1310.5834-2-19-2': 'Indeed, if the potential [MATH] is sufficiently regular and well-localized, the solution of the Cauchy problem [MATH] for the ODE ([REF]) exists unique globally in the (time) variable [MATH], with the following free particle asymptotic behavior [EQUATION]', '1310.5834-2-19-3': 'The asymptotic positions [MATH] are obviously constant when the point [MATH] moves along trajectories.', '1310.5834-2-19-4': 'Therefore [MATH] are solutions of the vector field equation [EQUATION].', '1310.5834-2-19-5': 'Due to ([REF]) we have [EQUATION] therefore they coincide with the real Jost eigenfunctions [EQUATION]', '1310.5834-2-19-6': 'Denote by [MATH] the classical time-scattering datum, connecting the asymptotic behavior of the solutions at [MATH] and at [MATH] [EQUATION] therefore [EQUATION]', '1310.5834-2-19-7': 'Step 2.', '1310.5834-2-19-8': ""The problem of existence for complex (analytic) eigenfunctions [MATH] of a vector field is usually highly nontrivial, and in all previous works by Manakov and Santini was only postulated and motivated by the analyticity properties of the Green's functions of the undressed vector fields."", '1310.5834-2-19-9': 'In our paper we present a proof based on the following observation:', '1310.5834-2-20-0': 'For [MATH] , by the change of variables [MATH], [MATH], the Lax equation [MATH] can be transformed into a linear Beltrami equation and can be solved.', '1310.5834-2-20-1': 'Moreover, we do not have to assume, at this stage, that the potential [MATH] has small norm.', '1310.5834-2-21-0': 'We show below that the limiting functions [MATH], [MATH] are also well-defined.', '1310.5834-2-21-1': 'Both real Jost eigenfunctions [MATH] enumerate the trajectories of our vector field, therefore any eigenfunction of [MATH] for [MATH] can be represented as a function either of [MATH] or [MATH], and we have: [EQUATION] defining the spectral data [MATH].', '1310.5834-2-22-0': 'Assuming that the small [MATH] behaviour be sufficiently good, we see that, for [MATH], the eigenfunction [MATH] is almost constant on the trajectories of the vector field [MATH]; these trajectories are straight lines [MATH] outside the support of [MATH] and connect the lines [MATH] and [MATH] as they go from [MATH] to [MATH] (see Fig [REF]).', '1310.5834-2-23-0': 'Assume now that [MATH], [MATH]; then [MATH] is holomorphic in [MATH] outside a small neighbourhood of [MATH]: [MATH] and, due to the almost constant behavior on the trajectories: [EQUATION]', '1310.5834-2-23-1': 'In the limit [MATH] we have [EQUATION] therefore equation ([REF]) implies [EQUATION].', '1310.5834-2-23-2': 'Hence the spectral data [MATH] of the Pavlov equation satisfy the shifted Riemann-Hilbert (RH) problem [EQUATION]', '1310.5834-2-23-3': 'Equation ([REF]) defines the spectral data [MATH] in terms of the scattering data [MATH].', '1310.5834-2-23-4': 'No small norm assumption is required also at this step.', '1310.5834-2-24-0': 'Evolution of the spectral data.', '1310.5834-2-24-1': 'The evolution of the scattering and spectral data, following from the asymptotics ([REF]) and ([REF]), is given by the explicit formula [CITATION]: [EQUATION] implying that, from the eigenfunctions [MATH] of [MATH], one can constructs the common eigenfunctions [MATH] of [MATH] and [MATH] through the formulas [EQUATION] connected through equations [EQUATION]', '1310.5834-2-24-2': 'The inverse problem The reconstruction of the real eigenfunction [MATH] at time [MATH] from the spectral data [MATH] is provided by the solution of the nonlinear integral equation [EQUATION] where [MATH] and [MATH] are the real and imaginary parts of [MATH] , and [MATH] is the Hilbert transform operator wrt [MATH] [EQUATION]', '1310.5834-2-24-3': 'We remark that, since [MATH] is analytic wrt [MATH] in the lower half-plane, its real and imaginary parts satisfy the relation [MATH].', '1310.5834-2-24-4': 'Equation ([REF]) expresses the fact that the RHS of ([REF]) for [MATH] is the boundary value of a function analytic in [MATH] in the lower half-plane.', '1310.5834-2-25-0': 'Once [MATH] is reconstructed from [MATH] solving the nonlinear integral equation ([REF]), equation ([REF]) gives [MATH], and [MATH] is finally reconstructed from: [EQUATION] or, better, as we shall see, from [EQUATION]', '1310.5834-2-25-1': 'The main difficulty associated with the direct problem is in the proof of the existence of the analytic eigenfunction and of its limits on the real [MATH] axis from above and below.', '1310.5834-2-25-2': 'While such a proof can be made in the Pavlov case, see subsection [REF], in the dKP case the existence of the analytic eigenfunctions is proven, at the moment, only sufficiently far from the real [MATH] axis [CITATION].', '1310.5834-2-25-3': 'We also remark that, soon after the formulation of the direct problem through the RH problem ([REF]) [CITATION], an alternative integral equation, obtained taking the Fourier transform of ([REF]), was also suggested [CITATION],[CITATION].', '1310.5834-2-25-4': 'It turns out that, while the construction of the spectral data from the scattering data through the RH problem with shift ([REF]) does not present difficulties, the construction that makes use of the integral equation in Fourier space requires additional effort, due to the bad behavior of its kernel, and will not be considered in this paper.', '1310.5834-2-26-0': 'A second inverse problem, a nonlinear RH (NRH) problem on the real line, was also introduced at a formal level [CITATION], and intensively used i) to study the longtime behavior of the solutions of the target dPDE [CITATION]; ii) to detect if a localized initial disturbance evolving according to such a PDE goes through a gradient catastrophe at finite time (f.i., no gradient catastrophe for the second heavenly equation [CITATION] and for the Pavlov equation [CITATION] was found, while a gradient catastrophe was indeed found for the dKP [CITATION] and for the dispersionless 2D Toda [CITATION] equations); iii) to investigate analytically the wave breaking mechanism of such multidimensional waves [CITATION]; iv) to construct classes of RH data giving rise to exactly solvable NRH problems, and to distinguished exact implicit solutions of the dispersionless PDEs through an algorithmic approach [CITATION]; v) to detect integrable differential reductions of the associated hierarchy of PDEs [CITATION], like the Dunajski interpolating equation [MATH] [CITATION], an integrable PDE interpolating between the dKP and the Pavlov equations, corresponding to the reduction [MATH] of system ([REF]).', '1310.5834-2-26-1': 'The rigorous aspects of such a NRH inverse problem, as well as the connections with the above inverse problem, will also be investigated in a subsequent paper.', '1310.5834-2-27-0': '# Direct spectral transform', '1310.5834-2-28-0': '## The real eigenfunction', '1310.5834-2-29-0': 'Throughout this paper, [EQUATION]', '1310.5834-2-29-1': 'We shall also use the Sobolev spaces with additional weights: [EQUATION].', '1310.5834-2-29-2': 'For all positive [MATH] these norms are equivalent, but in some situations it is necessary to choose an appropriate [MATH] to guaranty the contraction property for our integral operators.', '1310.5834-2-30-0': 'In our paper we assume that the potential [MATH] has compact support in [MATH], [MATH].', '1310.5834-2-30-1': 'We expect that these constraints are not critical and can be weakened (for example, it should be enough to assume that the potential decays sufficiently fast as [MATH]), but it may require a serious additional analytic work.', '1310.5834-2-30-2': 'To be more precise, let [MATH], [MATH] be a pair of positive numbers, [MATH], and [MATH] such that [EQUATION]', '1310.5834-2-30-3': 'The real eigenfunctions [MATH] for the Pavlov equation are defined by the solution of the boundary value problem: for each fixed [MATH], [CITATION] [EQUATION] where [EQUATION]', '1310.5834-2-30-4': 'Suppose [MATH] satisfying ([REF]).', '1310.5834-2-30-5': 'The real eigenfunctions [MATH] exists uniquely and [MATH] are smooth bounded functions.', '1310.5834-2-31-0': 'The solvability and uniqueness of the boundary value problem of the first order partial differential equation ([REF]), ([REF]) can be derived by solving the ordinary differential equation [EQUATION] or, equivalently, [EQUATION] where [EQUATION].', '1310.5834-2-32-0': 'Using the Picard iteration method on the integral equation defining the solution (see, for example, [CITATION]) [EQUATION] one shows that [MATH] are smooth functions, [MATH] are also bounded.', '1310.5834-2-32-1': 'Here we used the fact that [MATH] are constant in [MATH] in the regions [MATH], [MATH] due to the compact support of [MATH].', '1310.5834-2-33-0': 'We see, that [EQUATION] is also a regular function and the map [MATH] is regularly invertible for all [MATH].', '1310.5834-2-33-1': 'We do not require the small norm assumption at this step.', '1310.5834-2-34-0': 'For simplicity and convenience, we will use the following agreement: [MATH] denotes a constant, possibly dependent of [MATH], but independent of [MATH], [MATH], [MATH], and [MATH] throughout this paper.', '1310.5834-2-34-1': 'To construct the spectral data from the scattering data by solving the shifted Riemann-Hilbert problem, it is necessary to control the behavior of the scattering data and its derivatives for large [MATH].', '1310.5834-2-34-2': 'For solving the inverse problem we also need some estimates for large [MATH] and [MATH].', '1310.5834-2-35-0': 'Suppose [MATH] such, that [MATH] for [MATH].', '1310.5834-2-35-1': 'Let us define the following constants [MATH], [MATH], [MATH], [MATH]: [EQUATION]', '1310.5834-2-35-2': 'Then we have the following estimates on the scattering data: [EQUATION]', '1310.5834-2-35-3': 'Moreover, if [MATH], [EQUATION]', '1310.5834-2-35-4': 'The proof of Proposition [REF] is moved to the last Section.', '1310.5834-2-35-5': 'It is rather straightforward and is based on some standard estimates from the ODE theory,', '1310.5834-2-36-0': 'A potential [MATH] satisfies the small norm condition if the following inequalities are fulfilled:', '1310.5834-2-37-0': '[MATH], [MATH], [MATH], [MATH].', '1310.5834-2-38-0': 'The meaning of the combinations of constants arising in this definition will be explained later.', '1310.5834-2-39-0': 'Suppose [MATH] satisfying ([REF]) and [MATH] is sufficiently large.', '1310.5834-2-39-1': 'Let us introduce new variables [EQUATION].', '1310.5834-2-39-2': 'Then, for sufficiently small [MATH], the function [EQUATION] has the following properties:', '1310.5834-2-40-0': 'It vanishes outside the interval [MATH].', '1310.5834-2-40-1': 'It is smooth in both variables [MATH], [MATH].', '1310.5834-2-41-0': 'As a corollary we obtain that there exists a collection of positive constants [MATH], such that [EQUATION]', '1310.5834-2-41-1': 'As usual, we move the proof to the last section of our paper.', '1310.5834-2-42-0': '## The complex eigenfunction', '1310.5834-2-43-0': 'In this section, we prove that there exists a unique eigenfunction [MATH] for each [MATH].', '1310.5834-2-43-1': 'Moreover, [MATH] is holomorphic in [MATH], its boundary values on [MATH], denoted as [MATH], are well-defined and can be characterized by the shifted Riemann-Hilbert problem ([REF]).', '1310.5834-2-44-0': 'For [MATH], we introduce the following complex notations: [EQUATION]', '1310.5834-2-44-1': 'So [MATH] for each [MATH].', '1310.5834-2-45-0': 'For [MATH] and [MATH], there exist a unique continuous eigenfunction [MATH] and a positive function [MATH] such that', '1310.5834-2-46-0': 'z^-1(z,z,), z=x-y, W^2,p(dzdz), where p-2<()', '1310.5834-2-47-0': 'and [EQUATION]', '1310.5834-2-47-1': 'Moreover, [MATH] is holomorphic for [MATH], and [EQUATION]', '1310.5834-2-47-2': 'If [MATH] we have [EQUATION]', '1310.5834-2-47-3': 'Equation ([REF]) takes the following form: [EQUATION] or equivalently [EQUATION] where [EQUATION]', '1310.5834-2-47-4': 'The function [MATH] is real-valued, therefore [EQUATION]', '1310.5834-2-47-5': 'Using the representation [EQUATION] and the Zygmund-Calderon operator theory, it is easy to show that for any fixed [MATH] there exist [MATH] and [MATH] such that for [MATH] the norm of the operator [EQUATION] is smaller than 1 [CITATION].', '1310.5834-2-47-6': 'Then we can write [CITATION]: [EQUATION] where [MATH] satisfies the following equation: [EQUATION]', '1310.5834-2-47-7': 'This equation is uniquely solvable in the spaces [MATH], [MATH].', '1310.5834-2-47-8': ""Therefore [MATH] is a decaying at infinity continuous function by Sobolev's theorem, and [EQUATION]."", '1310.5834-2-47-9': 'We also have: [EQUATION] therefore for all regular points of the map [MATH] the Jacobian is positive, and the number of preimages is the same for all regular points.', '1310.5834-2-47-10': 'It means that the number of preimages is the same for all regular points.', '1310.5834-2-47-11': 'This map is one-to-one at infinity, therefore it is invertible and we can use [MATH] as a global coordinate on the [MATH]-plane.', '1310.5834-2-47-12': 'In this coordinate all solutions of ([REF]) are functions holomorphic in [MATH] (see Chapter II in [CITATION]).', '1310.5834-2-47-13': ""So Liouville's theorem implies that asymptotics ([REF]) fixes the solution uniquely."", '1310.5834-2-48-0': 'Let us show that [MATH] is holomorphic in [MATH] outside the real line.', '1310.5834-2-48-1': 'Differentiating ([REF]) by [MATH] we obtain [EQUATION] and [EQUATION]', '1310.5834-2-48-2': ""Therefore [MATH] is a regular holomorphic function in [MATH] decaying at infinity, and by Liouville's theorem [MATH]."", '1310.5834-2-49-0': ""The reality condition ([REF]) follows from applying Liouville's theorem and the reality conditions [MATH]."", '1310.5834-2-50-0': 'Let [MATH].', '1310.5834-2-50-1': 'Taking into account, that [MATH] we see, that [EQUATION] and [EQUATION] but [EQUATION] therefore [EQUATION].', '1310.5834-2-51-0': 'Starting from this point we will work with the Jost eigenfunction [MATH] only; therefore we shall denote it simply by [MATH], omitting the subscript: [EQUATION]', '1310.5834-2-51-1': 'Suppose [MATH] satisfying ([REF]).', '1310.5834-2-51-2': 'The complex eigenfunction [MATH] has continuous extensions on [MATH].', '1310.5834-2-51-3': 'Moreover, denote the limits on both sides of [MATH] as [MATH], then [MATH], [EQUATION] where [MATH] is characterized by the Riemann-Hilbert problem with the shift function [MATH].', '1310.5834-2-51-4': '[EQUATION]', '1310.5834-2-51-5': 'As before, we move the proof to the last Section.', '1310.5834-2-52-0': 'Theorem [REF] implies [EQUATION] and, due to ([REF]) [EQUATION]', '1310.5834-2-53-0': '## The shifted Riemann-Hilbert problem', '1310.5834-2-54-0': 'In subsection [REF] the following characterization for the boundary value of the complex eigenfunction [EQUATION] was justified.', '1310.5834-2-54-1': 'Here [MATH] satisfies the shifted Riemann-Hilbert problem ([REF]).', '1310.5834-2-55-0': 'The problem ([REF]) can be converted into the following linear equation [CITATION] [EQUATION] where [EQUATION]', '1310.5834-2-55-1': 'Under the assumptions that the mapping [MATH] be invertible for all [MATH], that [MATH] decay sufficiently fast for any fixed [MATH] and be H[MATH]lder continuous, the unique solvability of [MATH] is proven in [CITATION] by showing a Fredholm alternative for ([REF]).', '1310.5834-2-55-2': 'Also this step does not require the small norm assumption.', '1310.5834-2-56-0': 'Our goal in this section is to obtain some analytic estimates on the spectral data [MATH], including the large [MATH]-asymptotic estimates, which are important in characterizing the complex eigenfunction and are indispensable for solving the inverse problem.', '1310.5834-2-57-0': 'To simplify the calculations we shall use the following agreement in Lemmas [REF]-[REF]: we omit the [MATH]-dependence in all formulas.', '1310.5834-2-57-1': 'It is convenient to denote: [EQUATION]', '1310.5834-2-57-2': 'It is natural to solve the integral equation ([REF]) iteratively.', '1310.5834-2-57-3': 'Therefore we have to estimate the norm of [MATH], [MATH].', '1310.5834-2-58-0': 'Assume that the scattering data [MATH], [MATH] satisfy the following estimates:', '1310.5834-2-59-0': '[MATH] is 2 times continuously differentiable in [MATH].', '1310.5834-2-59-1': '[MATH].', '1310.5834-2-59-2': '[MATH].', '1310.5834-2-59-3': '[MATH].', '1310.5834-2-59-4': '[MATH].', '1310.5834-2-60-0': 'Then we have the following estimate [EQUATION].', '1310.5834-2-60-1': 'Assume that, in addition, the scattering data [MATH], [MATH] satisfy the following extra estimates:', '1310.5834-2-61-0': '[MATH] is 3 times continuously differentiable in [MATH].', '1310.5834-2-61-1': '[MATH].', '1310.5834-2-61-2': '[MATH].', '1310.5834-2-62-0': 'Then [MATH] maps the space [MATH] into the space [MATH].', '1310.5834-2-62-1': 'Moreover, if [MATH], then [EQUATION].', '1310.5834-2-63-0': 'The proof of this Lemma is moved to the last Section.', '1310.5834-2-64-0': 'We also require some estimates on the function [MATH]', '1310.5834-2-65-0': 'Assume that the scattering data satisfy the same estimates as in Lemma [REF] and', '1310.5834-2-66-0': '[MATH].', '1310.5834-2-66-1': '[MATH].', '1310.5834-2-67-0': 'Then we have:', '1310.5834-2-68-0': '[MATH].', '1310.5834-2-68-1': '[MATH].', '1310.5834-2-69-0': 'Moreover, if [MATH] has compact support: [MATH] for [MATH], then [EQUATION]', '1310.5834-2-69-1': 'The proof of this Lemma is moved to the last Section.', '1310.5834-2-70-0': 'Combining the estimates from Lemmas [REF]-[REF] we obtain the following:', '1310.5834-2-71-0': 'Assume that the potential [MATH] satisfy the small norm constraints formulated in the Definition [REF].', '1310.5834-2-71-1': 'Then we have [EQUATION]', '1310.5834-2-71-2': 'We show below, that this property guaranties the unique solvability of the inverse problem.', '1310.5834-2-72-0': 'Equation ([REF]) can be written in the short form: [EQUATION]', '1310.5834-2-72-1': 'If [MATH], it can solved iteratively and [EQUATION].', '1310.5834-2-72-2': 'By Lemma [REF], Condition 3, the small norm conditions list means exactly that [EQUATION] therefore [EQUATION].', '1310.5834-2-73-0': 'By differentiating equation ([REF]) with respect to [MATH], we obtain: [EQUATION] and [EQUATION].', '1310.5834-2-73-1': 'By Lemma [REF], in the small norm case [EQUATION].', '1310.5834-2-73-2': 'By Lemma [REF] [EQUATION].', '1310.5834-2-73-3': 'Therefore [EQUATION].', '1310.5834-2-74-0': 'The solution of the inverse problem also requires some estimates on [MATH] and its derivatives at [MATH].', '1310.5834-2-74-1': 'Let us show that, at large [MATH], the leading term of the asymptotic behavior is determined by the linear part of ([REF]).', '1310.5834-2-75-0': 'More precisely,', '1310.5834-2-76-0': 'If [MATH] and [MATH] for [MATH], then, for [MATH], we have the following estimates:', '1310.5834-2-77-0': '[MATH].', '1310.5834-2-77-1': 'For every sufficiently large [MATH], operator [MATH] maps the space [MATH] into the space [MATH] and there exists a constant [MATH] such that [EQUATION]', '1310.5834-2-77-2': 'To prove this Lemma it is sufficient to compare formulas ([REF])-([REF]) with the estimates from Lemma [REF].', '1310.5834-2-78-0': 'Using the same approach, it is possible to prove analogous estimates for all derivatives; in particular, there exists a constant [MATH] such that [EQUATION]', '1310.5834-2-78-1': 'Assume that [MATH] and [MATH] for [MATH].', '1310.5834-2-78-2': 'Then.', '1310.5834-2-78-3': 'for [MATH], we have the following estimates [EQUATION]', '1310.5834-2-78-4': 'If, in addition, [MATH] satisfies the compact support condition ([REF]), i.e. [MATH] for [MATH], then [EQUATION]', '1310.5834-2-78-5': 'Using the same approach, it is possible to prove that in the compact support case [EQUATION].', '1310.5834-2-79-0': 'Proof of Proposition [REF].', '1310.5834-2-80-0': 'From ([REF]), Proposition [REF], Lemmas [REF], [REF] and the formula [MATH] it follows immediately, that [EQUATION].', '1310.5834-2-81-0': 'Suppose [MATH] with compact support and [MATH] is small.', '1310.5834-2-81-1': 'Consider a curve in the [MATH]-plane: [EQUATION].', '1310.5834-2-81-2': 'Then for fixed [MATH] and [MATH], we have, as [MATH], [EQUATION]', '1310.5834-2-81-3': 'Suppose the support of [MATH] is contained in [MATH].', '1310.5834-2-81-4': 'Therefore the support of [MATH] lies in the area [MATH], [MATH].', '1310.5834-2-82-0': 'Outside this area [MATH], hence [MATH] is holomorphic in [MATH] in the complex plane outside the cut [MATH] on the real line.', '1310.5834-2-82-1': 'Therefore, [EQUATION] and [EQUATION].', '1310.5834-2-82-2': 'It follows that, for [MATH], [EQUATION]', '1310.5834-2-82-3': 'Therefore ([REF]) follows if [MATH].', '1310.5834-2-83-0': '# The inverse problem', '1310.5834-2-84-0': '## The reconstruction of the real eigenfunction', '1310.5834-2-85-0': 'Assume that the spectral data [MATH] are given, where [MATH].', '1310.5834-2-85-1': 'Let us recall that [MATH] is assumed to be analytic in [MATH] the lower half-plane, or equivalently [EQUATION] where [MATH] denotes the Hilbert transform wrt. the variable [MATH].', '1310.5834-2-86-0': 'Our current aim is to construct the common eigenfunctions of the Lax pair for the Pavlov equation starting from the spectral data.', '1310.5834-2-86-1': 'By the Plemelj (Sokhotski) formula [CITATION], Theorem [REF], and [REF], we have [EQUATION]', '1310.5834-2-86-2': 'Therefore, keeping in mind the time evolution ([REF]) of the spectral data and the definition ([REF]) of the common eigenfunctions of the vector field Lax Pair, the nonlinear integral equation of the inverse problem reads: [EQUATION]', '1310.5834-2-86-3': 'The solution of the inverse problem consists of two steps:', '1310.5834-2-87-0': '(Global solvability for the IST equation ([REF]) - part 1.)', '1310.5834-2-87-1': 'Suppose that the spectral data [MATH] satisfy the following constraints', '1310.5834-2-88-0': '[MATH], [MATH] are well-defined continuous functions.', '1310.5834-2-89-0': '[EQUATION]', '1310.5834-2-89-1': 'There exists a positive constant [MATH] such that [EQUATION]', '1310.5834-2-89-2': 'Then, for all [MATH], [MATH], equation ([REF]) has a unique solution [MATH] such that [MATH], where [MATH].', '1310.5834-2-90-0': 'The proof is based on the standard iteration procedure for contracting nonlinear maps.', '1310.5834-2-90-1': 'Equation ([REF]) is equivalent to [EQUATION] where [MATH] is defined by [EQUATION] or equivalently, [EQUATION].', '1310.5834-2-90-2': 'From the constraints on the spectral data it immediately follows that the maps [EQUATION] map all measurable functions of [MATH] into the space [MATH]; moreover the image of the map is located inside the ball of radius [MATH].', '1310.5834-2-90-3': '[MATH] is a unitary operator in the space [MATH]; therefore, for any measurable function [MATH], we know that [MATH], and [MATH].', '1310.5834-2-91-0': 'Let us check that operator [MATH] is a contraction.', '1310.5834-2-91-1': 'Let [MATH] be a measurable function, [MATH].', '1310.5834-2-91-2': 'We have [EQUATION]', '1310.5834-2-91-3': 'We know, that [EQUATION] therefore [EQUATION].', '1310.5834-2-91-4': 'Hence the iteration procedure: [EQUATION] perfectly converges in [MATH].', '1310.5834-2-92-0': 'Let us check now that the functions constructed above have the Jost property.', '1310.5834-2-92-1': 'Namely:', '1310.5834-2-93-0': 'Assume that the spectral data [MATH] satisfy the same constraints as in Theorem [REF], and', '1310.5834-2-94-0': 'For each [MATH] the function [MATH] is holomorphic in [MATH] in the lower half-plane.', '1310.5834-2-94-1': '[MATH], [MATH], [MATH] are well-defined continuous functions.', '1310.5834-2-94-2': 'There exists a positive constant [MATH] such, that [EQUATION]', '1310.5834-2-94-3': 'For any [MATH], there exists a positive constant [MATH]) such that, for all [MATH] such that [MATH] [EQUATION]', '1310.5834-2-94-4': 'Then, for the functions constructed in Theorem [REF] with fixed [MATH], [MATH], we have [EQUATION]', '1310.5834-2-94-5': 'Let us point out that all conditions from Theorem [REF] holds for the spectral data constructed in the framework of the direct spectral transform (we assume again that our Cauchy data [MATH] have compact support).', '1310.5834-2-94-6': 'Almost all of them were proved above, and the proof of the remaining ones are rather standard.', '1310.5834-2-94-7': 'Let us check, for example, ([REF])', '1310.5834-2-95-0': 'Let [MATH] be sufficiently large.', '1310.5834-2-95-1': 'We have 2 regions.', '1310.5834-2-96-0': 'Let [MATH].', '1310.5834-2-96-1': 'Then the second condition immediately follows from ([REF]) Let [MATH].', '1310.5834-2-96-2': 'From Proposition [REF] and ([REF]) we obtain that there exists a constant [MATH] such that [EQUATION]', '1310.5834-2-96-3': 'If [MATH], then we can use the same estimates as in Proposition [REF], and [EQUATION]', '1310.5834-2-96-4': 'It completes the proof.', '1310.5834-2-97-0': 'One can consider equation ([REF]) without assuming that the spectral data [MATH] is holomorphic in [MATH] in the lower half-plane (or, equivalently, we do not assume that equation ([REF]) is fulfilled).', '1310.5834-2-97-1': 'In this situation the function [MATH] will be also an eigenfunction for the Pavlov Lax operators [MATH], [MATH] for some [MATH] (see Theorem [REF]), but the normalization of this eigenfunction will be different from([REF]).', '1310.5834-2-98-0': 'We also require to study the linearized version of equation ([REF]).', '1310.5834-2-99-0': 'Suppose that the scattering data [MATH] satisfy the same constraints as in Theorem [REF] (which are fulfilled if [MATH] was constructed through the small norm Cauchy data [MATH]).', '1310.5834-2-99-1': 'Then, for [MATH], [MATH] the integral equation [EQUATION] admits a unique solution [MATH] such that [EQUATION]', '1310.5834-2-99-2': 'The Hilbert transform is a unitary operator in [MATH] and the norm of the Hilbert transform in [MATH], [MATH] is equal to [MATH] (see [CITATION]); therefore for [MATH] or [MATH] one has [EQUATION]', '1310.5834-2-99-3': 'Therefore [MATH] is an invertible map on [MATH] and the norm of the inverse operator in [MATH] is not greater than 2: [EQUATION] and the estimate ([REF]) follows.', '1310.5834-2-100-0': 'Below we use the following simple corollary of the Sobolev embedding theorem:', '1310.5834-2-101-0': 'Let [MATH] be an element of [MATH], [MATH].', '1310.5834-2-101-1': 'Then [MATH] is a continuous function and [EQUATION] (Global solvability for the IST equation ([REF]) - part 2.)', '1310.5834-2-101-2': 'Suppose that [MATH] satisfies the same constraints as in Theorem [REF] and, in addition, [EQUATION]', '1310.5834-2-101-3': 'Let us denote: [EQUATION]', '1310.5834-2-101-4': 'Then:', '1310.5834-2-102-0': 'For all [MATH], [MATH] the function [MATH] lies in the space [MATH] and continuously depends on [MATH] as an element of [MATH].', '1310.5834-2-102-1': 'The norm of [MATH] in the space [MATH] is uniformly bounded in [MATH] (but the [MATH]-norm may be unbounded).', '1310.5834-2-102-2': 'For all [MATH], [MATH] the following derivatives of [MATH]: [EQUATION] are well-defined as elements of the space [MATH], and [MATH], continuously depend on [MATH] and are uniformly bounded in [MATH].', '1310.5834-2-103-0': 'To construct [MATH], it is convenient to run the iteration procedure ([REF]), simultaneously for [MATH] and [MATH]: [EQUATION] where [MATH] is the Hilbert transform with respect to [MATH], [EQUATION]', '1310.5834-2-103-1': 'In any compact area in the [MATH] space the function [MATH] is bounded in [MATH] uniformly in [MATH].', '1310.5834-2-103-2': 'If [MATH], then for all [MATH], [MATH].', '1310.5834-2-103-3': 'Therefore by Lemma [REF] the [MATH] convergence of [MATH] implies the [MATH] convergence of [MATH] and the convergence of [MATH] in [MATH].', '1310.5834-2-104-0': 'By taking derivatives of both sides of ([REF]), we obtain the linearized integral equation by: [EQUATION]', '1310.5834-2-104-1': 'In terms of [MATH], equations ([REF])-([REF]) take the form: [EQUATION] where [MATH], and [EQUATION]', '1310.5834-2-104-2': 'From ([REF]) it follows that [MATH].', '1310.5834-2-104-3': 'Therefore the existence of [MATH], [MATH], [MATH] such that [MATH], [MATH], [MATH] follows from Lemma [REF].', '1310.5834-2-105-0': 'For the second derivatives of the wave function we have: [EQUATION] where [EQUATION].', '1310.5834-2-106-0': 'From ([REF]) and the properties of the first derivatives we obtain that [MATH], [MATH], [MATH], [MATH] belong to [MATH]; therefore equations ([REF]) are uniquely solvable in [MATH].', '1310.5834-2-107-0': 'Taking into account that [MATH] is continuous in [MATH] as an element of [MATH], we obtain that all coefficients of the linear equations are continuous in [MATH].', '1310.5834-2-107-1': 'This implies that the solutions are also continuous.', '1310.5834-2-108-0': '## Eigenfunctions of the Lax equation and the Cauchy problem', '1310.5834-2-109-0': '(Global solvability for small initial data) Suppose [MATH] satisfying ([REF]) and the sufficiently small condition from Definition [REF].', '1310.5834-2-109-1': 'Let [MATH] be the solution of the nonlinear inverse problem ([REF]) obtained in Theorem [REF] with the data [MATH] constructed from [MATH] through the direct problem.', '1310.5834-2-109-2': 'Define [EQUATION]', '1310.5834-2-109-3': 'Then', '1310.5834-2-110-0': 'v(x,y,t)=v(x,y,t),', '1310.5834-2-111-0': 'v,v_x,v_y,v_xx,v_xy,v_xt,v_yy C(RtimesRtimes R^+)L^(RtimesRtimes R^+),', '1310.5834-2-112-0': 'Assume, in addition, that for [MATH] we have estimates from Proposition [REF].', '1310.5834-2-112-1': 'Then, for all [MATH], the function [MATH] is well-defined and continuous in all variables.', '1310.5834-2-112-2': 'Function [MATH] satisfies the Lax equations in the space [MATH].', '1310.5834-2-112-3': 'More precisely, for each [MATH], functions [MATH], [MATH] are well-defined elements of [MATH] and [EQUATION] for almost all [MATH].', '1310.5834-2-112-4': 'Let us define a pair of functions [MATH], [MATH] by [EQUATION]', '1310.5834-2-112-5': 'Then, for each [MATH], [MATH], these functions admit natural analytic continuation in [MATH] to the lower half-plane [MATH] and the upper half-plane [MATH] respectively.', '1310.5834-2-112-6': 'Denote by [MATH], [MATH] the function, coinciding with the analytic continuation of [MATH] for [MATH] and with the analytic continuation of [MATH] for [MATH].', '1310.5834-2-112-7': 'Then we have the following integral representation: [EQUATION]', '1310.5834-2-112-8': 'Denote by [MATH] the regular part of the wave function: [MATH].', '1310.5834-2-113-0': 'Then for each fixed [MATH] we have: [EQUATION] and for any [MATH] the analytic wave function [MATH] satisfies the Lax pair [EQUATION] identically in [MATH].', '1310.5834-2-114-0': 'For [MATH] the function [MATH], constructed in terms of the inverse spectral transforms via ([REF]), coincides with the Cauchy data [MATH] for the direct spectral transform: [EQUATION]', '1310.5834-2-114-1': 'The reality condition ([REF]) follows from the fact that the inverse scattering equation ([REF]) is real for real [MATH], and ([REF]) has real coefficients.', '1310.5834-2-115-0': 'By differentiating ([REF]) we obtain', '1310.5834-2-116-0': 'v_(x,y,t)=-1_R__-I(,) _ d,', '1310.5834-2-117-0': 'v_(x,y,t)=-1_R[ __-I(,) __ + __-I(,) _] d,', '1310.5834-2-118-0': '[MATH].', '1310.5834-2-118-1': 'Using the properties ([REF]), it follows that the only integral requiring regularization is the integral for [MATH].', '1310.5834-2-118-2': 'This means that, for [MATH], the function [MATH] may be discontinuous.', '1310.5834-2-119-0': 'Let [MATH].', '1310.5834-2-119-1': 'We have [EQUATION] and [MATH] is a bounded function of [MATH], therefore the convergence of integral immediately follows from Proposition [REF].', '1310.5834-2-120-0': 'To calculate [MATH], [MATH] we use the following simple formula.', '1310.5834-2-120-1': 'Let [MATH] be a function such that [MATH], [MATH], [MATH].', '1310.5834-2-120-2': 'Then [EQUATION]', '1310.5834-2-120-3': 'Applying L to ([REF]) we obtain:', '1310.5834-2-121-0': 'Lv_x - L(_-R(,)) + (_y + _x +v_x_x) H_[_-I(,))]=', '1310.5834-2-122-0': '=v_x - __-R(,) L+ H_[L_-I(,))]+1 _R _x _-I(,)) d=v_x - __-R(,) L+ H_[__-I(,)) L]-v_x.', '1310.5834-2-122-1': 'We obtain that [MATH] and solves the homogeneous equation; therefore, by Lemma [REF], it is a zero element of [MATH].', '1310.5834-2-123-0': 'Analogously,', '1310.5834-2-124-0': 'Mv_x -v_y - __-R(,) M+ H_[__-I(,)) M]+', '1310.5834-2-125-0': '+1_R _x _-I(,)) d+ 1_R _x _-I(,)) d+ v_x1_R _x _-I(,)) d= __-R(,) M+ H_[__-I(,)) M]+', '1310.5834-2-126-0': '+v_x-v_y-v_x- v_x^2+ 1_R _x _-I(,)) d Taking into account that [EQUATION] we obtain that [MATH] and [EQUATION] therefore [EQUATION]', '1310.5834-2-126-1': 'This property is exactly equivalent to the inverse problem equation ([REF]).', '1310.5834-2-126-2': 'From ([REF]) it follows that [EQUATION] in [MATH].', '1310.5834-2-126-3': 'The standard solution of the Riemann factorization problem in terms of the Cauchy integral immediately gives us ([REF]).', '1310.5834-2-127-0': 'Combining Theorem [REF] and the H[MATH]lder inequality, we obtain ([REF]) for fixed [MATH].', '1310.5834-2-128-0': 'Finally, restricting ([REF]) to [MATH], ([REF]) yields ([REF]) and ([REF]).', '1310.5834-2-128-1': 'So [MATH], [MATH].', '1310.5834-2-128-2': 'Comparing ([REF]), ([REF]) with ([REF]), ([REF]) we obtain ([REF]).', '1310.5834-2-129-0': 'Suppose [MATH] with compact support and satisfies the sufficiently small condition from Definition [REF].', '1310.5834-2-129-1': 'Then the Cauchy problem of the Pavlov equation [EQUATION] admits a real solution [MATH] such that [MATH].', '1310.5834-2-130-0': 'Applying Proposition [REF] and computing the compatibility of the Lax pair ([REF]) and ([REF]), we obtain [EQUATION].', '1310.5834-2-130-1': 'Hence we obtain ([REF]) by ([REF]).', '1310.5834-2-131-0': '# Summary of the results and concluding remarks', '1310.5834-2-132-0': 'We have shown that the direct problem consists of the following steps:', '1310.5834-2-133-0': 'These two steps do not require small norm assumptions.', '1310.5834-2-134-0': 'The inverse problem consists of the following two steps:', '1310.5834-2-135-0': 'The following remark is important.', '1310.5834-2-136-0': 'A careful reader may notice that the above basic steps do not involve explicitly the analytic eigenfunctions; therefore, strictly speaking, the Cauchy problem for the Pavlov equation can be solved without introducing them.', '1310.5834-2-136-1': 'However, their existence pervades the whole IST.', '1310.5834-2-136-2': 'Indeed, not only it is crucial in motivating the shifted Riemann problem ([REF]) of the direct problem, but it is also equivalent to the nonlinear integral equation ([REF]) of the inverse problem.', '1310.5834-2-137-0': '# The analytic estimates', '1310.5834-2-138-0': 'In this section we present the proofs of some of the analytical estimates we use in our paper.', '1310.5834-2-139-0': 'Proof of Proposition [REF].', '1310.5834-2-140-0': ""The main tool for proving these estimates in the Gronwall's inequality."", '1310.5834-2-140-1': 'By definition, [EQUATION] where [MATH] denotes the solution of the vector field ODE: [EQUATION] with the boundary condition: [EQUATION].', '1310.5834-2-140-2': 'Therefore: [EQUATION] and [EQUATION]', '1310.5834-2-140-3': 'The function [MATH] satisfies the linearized equation: [EQUATION] with the boundary value [EQUATION].', '1310.5834-2-140-4': 'Equation ([REF]) can be written as: [EQUATION] therefore [EQUATION] and [EQUATION] which automatically implies the necessary estimate on [MATH].', '1310.5834-2-141-0': 'The next step is to estimate the solutions of the equation for [MATH] [EQUATION] with the boundary condition: [EQUATION].', '1310.5834-2-141-1': 'We have an inhomogeneous linear equation; therefore we can use the standard estimate: [EQUATION] which implies the estimate on [MATH].', '1310.5834-2-141-2': 'Equation for [MATH] has the form [EQUATION] with the boundary condition: [EQUATION].', '1310.5834-2-141-3': 'Again we can estimate the function [MATH] as product of the integral of the modulus of the inhomogeneous term times the exponent of the modulus of the homogeneous coefficient: [EQUATION] which implies the estimate on [MATH].', '1310.5834-2-142-0': 'Let us denote [MATH].', '1310.5834-2-142-1': 'Equations ([REF]), ([REF]) can be interpreted as ODEs for the functions [MATH], [MATH] in the Hilbert space [MATH].', '1310.5834-2-142-2': 'We obtain: [EQUATION]', '1310.5834-2-142-3': 'We see that [EQUATION]', '1310.5834-2-142-4': 'We assume now, that [MATH].', '1310.5834-2-142-5': 'We know, that [EQUATION] for a fixed [MATH], [MATH], but [MATH], therefore [EQUATION] which completes the proof.', '1310.5834-2-143-0': 'Proof of Proposition [REF].', '1310.5834-2-144-0': 'Due to Definition [REF], it is sufficient to prove the Lemma for [MATH].', '1310.5834-2-144-1': 'Thus we always assume [MATH] in the following proof.', '1310.5834-2-144-2': 'The cases [MATH] and [MATH] are completely analogous, therefore we assume now that [MATH].', '1310.5834-2-145-0': 'Let us rewrite the definition of the scattering data using [MATH] and [MATH] as new coordinates on the [MATH]-plane.', '1310.5834-2-145-1': 'The [MATH]-coordinate is expressed through [MATH], [MATH] using the following formulas: [EQUATION]', '1310.5834-2-145-2': 'From the implicit function theorem, this map can be inverted with respect to [MATH]: [EQUATION] where [EQUATION] or, equivalently [EQUATION].', '1310.5834-2-145-3': 'We see, that [EQUATION].', '1310.5834-2-145-4': 'Let us denote [EQUATION].', '1310.5834-2-145-5': 'Taking into account that [EQUATION] we obtain [EQUATION]', '1310.5834-2-145-6': 'For [MATH] the right-hand side of ([REF]) is smooth in [MATH], [MATH].', '1310.5834-2-145-7': 'We solve this equation in the finite interval [MATH]; therefore [MATH] smoothly depends on the parameters.', '1310.5834-2-145-8': 'It is easy to check that, for [MATH], the right-hand side of ([REF]) is identical to 0, therefore [MATH] in the region [MATH].', '1310.5834-2-146-0': 'Expanding ([REF]) at [MATH] we obtain: [EQUATION] therefore [EQUATION].', '1310.5834-2-146-1': ""From the Hadamard's lemma it follows, that [EQUATION] is a regular function of [MATH], [MATH] for sufficiently small [MATH]."", '1310.5834-2-146-2': 'We proved the first part.', '1310.5834-2-147-0': 'To prove the corollary, let us point out that, in the new variables, [EQUATION].', '1310.5834-2-147-1': 'Therefore any differentiation of the scattering data with respect to [MATH], [MATH] increases the order of zero with respect to [MATH] at the point [MATH] by one.', '1310.5834-2-147-2': 'Taking into account that [EQUATION] we finish the proof.', '1310.5834-2-148-0': 'Proof of Theorem [REF].', '1310.5834-2-149-0': 'To prove the Theorem, let us make an appropriate change of variables.', '1310.5834-2-149-1': 'It will be done in 5 steps.', '1310.5834-2-150-0': 'Step 1 : Consider a point [MATH].', '1310.5834-2-150-1': 'Denote by [MATH] the solution of the ordinary differential equation [EQUATION] with the boundary condition [EQUATION]', '1310.5834-2-150-2': 'The first change of variables [MATH] is defined by: [EQUATION]', '1310.5834-2-150-3': 'Of course the map is discontinuous on the line [MATH], and [EQUATION]', '1310.5834-2-150-4': 'In the new variables we have [EQUATION] where [EQUATION]', '1310.5834-2-150-5': 'Moreover, there exists a pair of positive constants [MATH], [MATH] such that: [EQUATION]', '1310.5834-2-150-6': 'Step 2 : To investigate the boundary behaviors of the complex eigenfunction, we observe that, for [MATH], [MATH], it is natural to conjecture that [MATH] is almost constant on the trajectories of the vector field [EQUATION]', '1310.5834-2-150-7': 'These trajectories are defined by ([REF]) and ([REF]).', '1310.5834-2-150-8': 'Hence, if [EQUATION] then [EQUATION] where [MATH] is defined by Definition [REF] (see the proof of Lemma [REF]).', '1310.5834-2-151-0': 'Recall that [MATH].', '1310.5834-2-151-1': 'Assume that the support of [MATH] is located inside the strip [MATH], [MATH].', '1310.5834-2-151-2': 'Then [MATH] is holomorphic in [MATH] outside a small neighbourhood of the real line and we have [EQUATION]', '1310.5834-2-151-3': 'Consider the Riemann-Hilbert problem with shift ([REF]), or, via function [EQUATION] w(+()+i0,_R) = w( R,', '1310.5834-2-152-0': 'w(z) = z + o(1) as z.', '1310.5834-2-152-1': 'Then the hypothetical formulas for [MATH], [MATH] read:', '1310.5834-2-153-0': '^-(x,y,_R)=w(_-(x,y,_R)-i0,_R)= w(_+(x,y,_R)+i0,_R)', '1310.5834-2-154-0': '^+(x,y,_R)=^-(x,y,_R).', '1310.5834-2-155-0': 'Step 3 : Assume [MATH] from now on.', '1310.5834-2-155-1': 'Let us use the following rescaling: [MATH] [EQUATION]', '1310.5834-2-155-2': 'In the new variables [EQUATION]', '1310.5834-2-155-3': 'Step 4 : Let us define a new complex variable [MATH], [MATH] by [EQUATION] where [MATH] is the solution of the shifted Riemann-Hilbert problem ([REF]) (existence of the solution is proved in [CITATION]).', '1310.5834-2-155-4': 'Note that the composition [MATH] is continuous by the property: [EQUATION]', '1310.5834-2-155-5': 'Consequently, ([REF]) takes the form [EQUATION] where [EQUATION] the support of [MATH] has area of order [MATH].', '1310.5834-2-156-0': 'It is natural to consider Beltrami equation ([REF]) in the space [MATH] where [MATH] is sufficiently small.', '1310.5834-2-156-1': 'Again we can write [EQUATION] where [EQUATION]', '1310.5834-2-156-2': 'Taking into account ([REF]) we see that [EQUATION]', '1310.5834-2-156-3': 'Using the estimates from [CITATION] we see, that [EQUATION] and [MATH] uniformly converges to [MATH].', '1310.5834-2-157-0': 'Step 5 : Consider the function [MATH] on the line [MATH].', '1310.5834-2-157-1': 'We see, that [EQUATION] therefore [EQUATION].', '1310.5834-2-157-2': 'On this line [EQUATION] therefore [EQUATION].', '1310.5834-2-157-3': 'The proof is completed.', '1310.5834-2-158-0': 'Proof of Lemma [REF].', '1310.5834-2-159-0': 'To start with, let us point out that [EQUATION] where [EQUATION].', '1310.5834-2-159-1': 'Therefore [EQUATION].', '1310.5834-2-159-2': 'We see that, if the corresponding derivatives exist, [EQUATION] and [EQUATION]', '1310.5834-2-159-3': 'We know that [EQUATION]', '1310.5834-2-159-4': 'From ([REF]) we see, that [EQUATION] and [EQUATION].', '1310.5834-2-159-5': 'Let us estimate now [MATH].', '1310.5834-2-159-6': 'We have [MATH].', '1310.5834-2-159-7': 'We assumed that [MATH]; therefore [EQUATION] and [EQUATION].', '1310.5834-2-159-8': 'To estimate [MATH], we use the Holder inequality [EQUATION].', '1310.5834-2-159-9': 'Combining estimates for [MATH], [MATH], [MATH] we complete the proof of the first part.', '1310.5834-2-160-0': 'To prove the second part, we use the standard estimate: [EQUATION].', '1310.5834-2-160-1': 'We have: [EQUATION] where [EQUATION].', '1310.5834-2-160-2': 'From ([REF]), we see that [EQUATION].', '1310.5834-2-160-3': 'Let us introduce the following notation: [EQUATION].', '1310.5834-2-160-4': 'Let us estimate [MATH].', '1310.5834-2-160-5': 'We have: [EQUATION]', '1310.5834-2-160-6': 'By definition, [EQUATION].', '1310.5834-2-160-7': 'Therefore [EQUATION].', '1310.5834-2-160-8': 'Finally we obtain [EQUATION]', '1310.5834-2-160-9': 'Proof of Lemma [REF].', '1310.5834-2-161-0': 'We have [EQUATION] where [EQUATION] or, equivalently, [EQUATION] where [MATH] denotes the inversion of the function [MATH] with respect to [MATH]: [EQUATION].', '1310.5834-2-161-1': 'Let us denote: [EQUATION].', '1310.5834-2-161-2': 'We have: [EQUATION] where [EQUATION] and [MATH] can be estimated using the Holder inequality [EQUATION].', '1310.5834-2-161-3': 'We obtained: [EQUATION].', '1310.5834-2-161-4': 'Similarly: [EQUATION].', '1310.5834-2-161-5': 'We have [EQUATION]', '1310.5834-2-161-6': 'We assumed that [MATH]; therefore [EQUATION].', '1310.5834-2-161-7': 'Therefore [EQUATION].', '1310.5834-2-161-8': 'Similarly: [EQUATION].', '1310.5834-2-161-9': 'Let us proof the second part.', '1310.5834-2-162-0': 'Assume that [MATH].', '1310.5834-2-162-1': 'Then [EQUATION].', '1310.5834-2-162-2': 'We have: [EQUATION].', '1310.5834-2-162-3': 'Consider now the case [MATH].', '1310.5834-2-162-4': 'Then [EQUATION].', '1310.5834-2-162-5': 'For a finite support function [EQUATION].', '1310.5834-2-162-6': 'The proof of the second formula is absolutely the same, but we take into account ([REF]).', '1310.5834-2-163-0': 'Proof of Theorem [REF].', '1310.5834-2-164-0': 'In this part we always assume that [MATH] is fixed, [MATH] is an arbitrary fixed positive constant, [MATH], [MATH] is sufficiently large (more precisely, [MATH]), [MATH].', '1310.5834-2-165-0': 'This proof consists of 3 steps:', '1310.5834-2-166-0': 'We show that it is sufficient to obtain some [MATH] estimates on [MATH] and [MATH].', '1310.5834-2-166-1': 'We show, that it is sufficient to estimate the first iteration of [MATH] and [MATH] in [MATH].', '1310.5834-2-166-2': 'We estimate the first iteration of [MATH] and [MATH] in [MATH] for [MATH].', '1310.5834-2-167-0': 'Step 1.', '1310.5834-2-168-0': 'From Lemma [REF] it follows, that it is sufficient to prove the following: [EQUATION] uniformly in [MATH] in the interval [MATH].', '1310.5834-2-169-0': 'Step 2.', '1310.5834-2-170-0': 'Let us recall that we use the following iteration procedure: [EQUATION] where [EQUATION]', '1310.5834-2-170-1': 'It is convenient to write: [EQUATION]', '1310.5834-2-170-2': 'From ([REF]), ([REF]) we immediately obtain that there exists a constant [MATH] such that [EQUATION]', '1310.5834-2-170-3': 'Using the same arguments as in Theorem [REF] we immediately obtain [EQUATION]', '1310.5834-2-170-4': 'From ([REF]) we immediately obtain that there exists [MATH] such that [EQUATION]', '1310.5834-2-170-5': 'We also know that [EQUATION]', '1310.5834-2-170-6': 'Combining all these estimates we obtain: [EQUATION]', '1310.5834-2-170-7': 'Therefore, to prove Theorem [REF], it is sufficient to show that [EQUATION] uniformly in [MATH] for [MATH], where [MATH] is an arbitrary positive constant.', '1310.5834-2-171-0': 'Step 3.', '1310.5834-2-172-0': 'The proof of both estimates are absolutely similar; moreover the second one is a little easier from a technical point of view.', '1310.5834-2-172-1': 'Let us estimate [MATH]: [EQUATION].', '1310.5834-2-172-2': 'It is convenient to represent [MATH] as a sum of three functions: [EQUATION]', '1310.5834-2-172-3': 'From ([REF]) it follows immediately that there exists a constant [MATH] such that [EQUATION]', '1310.5834-2-172-4': 'If [MATH], [MATH] then [MATH], and [EQUATION]', '1310.5834-2-172-5': 'Let us denote by [MATH] the function: [EQUATION].', '1310.5834-2-172-6': 'We have shown that [EQUATION] therefore it is sufficient to estimate [MATH].', '1310.5834-2-172-7': 'We have [EQUATION]', '1310.5834-2-172-8': 'For sufficiently large [MATH] and [MATH] we have [EQUATION] and [EQUATION]', '1310.5834-2-172-9': 'Let us estimate the [MATH]-norm of [MATH] on the interval [MATH].', '1310.5834-2-172-10': 'We have [EQUATION] where [EQUATION]', '1310.5834-2-172-11': 'From ([REF]) it follows that [EQUATION]', '1310.5834-2-172-12': 'For [MATH] we have [EQUATION]', '1310.5834-2-172-13': 'To complete the proof, we have to estimate [MATH] in the interval [MATH].', '1310.5834-2-173-0': 'For [MATH] the function [MATH] is holomorphic in [MATH] in the lower half-plane; therefore [EQUATION] and [EQUATION] where [EQUATION]', '1310.5834-2-173-1': 'If [MATH], then [EQUATION] where [EQUATION]', '1310.5834-2-173-2': 'We see, that [EQUATION]', '1310.5834-2-173-3': 'Denote: [EQUATION] where [EQUATION]', '1310.5834-2-173-4': 'Analogously, [EQUATION]', '1310.5834-2-173-5': 'We have shown that there exist positive constants [MATH], [MATH] such that [EQUATION]', '1310.5834-2-173-6': 'Analogously, there exists a constant [MATH] such that [EQUATION] ([MATH], [MATH] decay at [MATH] as [MATH], therefore we have no logarithmic terms).', '1310.5834-2-174-0': 'The proof is completed.'}","[['1310.5834-1-0-0', '1310.5834-2-0-0'], ['1310.5834-1-0-1', '1310.5834-2-0-1'], ['1310.5834-1-6-1', '1310.5834-2-12-0'], ['1310.5834-1-6-2', '1310.5834-2-12-1'], 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'1310.5834-2-160-5', '1310.5834-2-160-6', '1310.5834-2-160-7', '1310.5834-2-161-3', '1310.5834-2-161-4', '1310.5834-2-161-5', '1310.5834-2-161-7', '1310.5834-2-161-8', '1310.5834-2-162-0', '1310.5834-2-162-1', '1310.5834-2-162-2', '1310.5834-2-162-4', '1310.5834-2-163-0', '1310.5834-2-165-0', '1310.5834-2-167-0', '1310.5834-2-169-0', '1310.5834-2-171-0', '1310.5834-2-172-4', '1310.5834-2-172-7', '1310.5834-2-173-0', '1310.5834-2-173-1', '1310.5834-2-173-2', '1310.5834-2-173-3', '1310.5834-2-173-4', '1310.5834-2-173-5', '1310.5834-2-173-6', '1310.5834-2-174-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1310.5834,,,
1606.06490,"{'1606.06490-1-0-0': 'Under the assumption of outdated channel state information (CSI) at the source, we study the blocklength-limited throughput (BL-throughput) of a two-hop relaying network.', '1606.06490-1-0-1': 'We introduce SNR weights to let the source choose a relatively lower coding rate based on the weighted outdated SNR.', '1606.06490-1-0-2': 'Both a fixed weight scheme and a dynamic weight scheme are studied.', '1606.06490-1-0-3': 'We derive the expected BL-throughput of a transmission period based on the corresponding outdated CSI and compare it with the ergodic BL-throughput over channel fading.', '1606.06490-1-0-4': 'In particular, we prove that under the dynamic SNR weight scheme the BL-throughput of an upcoming transmission is concave in the coding rate and quasi-concave in the SNR weights.', '1606.06490-1-0-5': 'In addition, under the fixed SNR weight scheme we show that the average BL-throughput over time is quasi-concave in these weights.', '1606.06490-1-0-6': 'Through numerical investigations, we show the appropriateness of our theoretical model.', '1606.06490-1-0-7': 'In addition, we study the performance difference between the dynamic and the fixed weight schemes.', '1606.06490-1-0-8': 'We show numerically that relaying outperforms direct transmission under the finite blocklength regime for cases where both systems have similar performance with respect to Shannon capacity.', '1606.06490-1-0-9': 'These results indicate that as the blocklength decreases, relaying has a increasing performance advantage over direct transmission despite its inherent reduction in the transmit times.', '1606.06490-1-1-0': 'Finite blocklength regime, decode-and-forward, relaying, blocklength-limited throughput, outdated CSI.', '1606.06490-1-2-0': '# Introduction', '1606.06490-1-3-0': 'Relaying [CITATION] is well known as an efficient way in wireless communications to mitigate fading by exploiting spatial diversity and providing better channel quality.', '1606.06490-1-3-1': 'Specifically, two-hop decode-and-forward (DF) relaying protocols significantly improve the throughput and quality of service [CITATION].', '1606.06490-1-3-2': ""However, all the above studies of the advantages of relaying are under the ideal assumption of communicating arbitrarily reliably at Shannon's channel capacity, i.e., coding is assumed to be performed using a block with an infinite length."", '1606.06490-1-4-0': 'In the finite blocklength regime, the transmission (at a rate lower than the Shannon limit) becomes no longer arbitrarily reliably.', '1606.06490-1-4-1': 'Especially when the blocklength is short, the error probability (due to noise) is significant even if the rate is selected below the Shannon limit.', '1606.06490-1-4-2': 'Taking this into account, an accurate approximation of the achievable coding rate under the finite blocklength assumption for an AWGN channel was derived in [CITATION] for a single-hop transmission system.', '1606.06490-1-4-3': 'The result in [CITATION] shows that the performance loss due to finite blocklength effects is considerable and and increases as the blocklength decreases.', '1606.06490-1-4-4': 'Subsequently, these initial work with AWGN was extended to Gilbert-Elliott channels [CITATION], quasi-static fading channels [CITATION], [CITATION], quasi-static fading channels with retransmissions [CITATION], spectrum sharing networks [CITATION] as well as transmissions with packet scheduling [CITATION].', '1606.06490-1-4-5': 'However, all these works only focus on single-hop systems.', '1606.06490-1-5-0': 'Considering a two-hop decode-and-forward network, relaying provides a power gain as it halves the distance but at the same time halves the blocklength of the transmission (if equal time division is considered).', '1606.06490-1-5-1': 'Thus, the result in [CITATION], namely that the loss due to finite blocklength effects increases as the blocklength decreases, actually points to a trade-off introduced by relaying between increasing the received signal strength by halving the distance vs. finite blocklength loss due to shorter time spans.', '1606.06490-1-5-2': 'This motivates us to study the relaying performance in the finite blocklength regime.', '1606.06490-1-5-3': 'In our previous work [CITATION] and [CITATION], we addressed in general analytical performance models for relaying with finite blocklengths under the scenario of static channels and average CSI.', '1606.06490-1-5-4': 'We investigated the related blocklength-limited throughput (BL-throughput) and showed that the BL-throughput is quasi-concave in the overall error probability of relaying.', '1606.06490-1-5-5': 'The work in [CITATION] and [CITATION] is further generalized in [CITATION] into a scenario of quasi-static Rayleigh channels while still only average CSI is available at the source.', '1606.06490-1-6-0': 'In this work, we extend our previous studies [CITATION] to a fading scenario with instantaneous CSI at the source.', '1606.06490-1-6-1': 'More importantly, the feedback of instantaneous CSI is assumed to be subject to delays, i.e., resulting for example from the feedback delay.', '1606.06490-1-6-2': 'Under the above assumption, we investigate the blocklength-limited performance of relaying.', '1606.06490-1-6-3': 'Different from the average CSI scenario considered in [CITATION], with the instantaneous CSI the source is able to adjust the coding rate accordingly for each transmission period.', '1606.06490-1-6-4': 'In addition, as the instantaneous CSI is outdated, it is possible that the outdated CSI is higher than the real/exact CSI.', '1606.06490-1-6-5': 'Hence, determining the coding rate simply based on the outdated CSI likely results in an error.', '1606.06490-1-6-6': 'At the same time, the effect of finite blocklengths also contributes to errors.', '1606.06490-1-6-7': 'Thus, the analysis of the blocklength-limited performance of such a relaying system (with finite blocklengths and outdated CSI) becomes interesting and also challenging.', '1606.06490-1-6-8': 'To the best of our knowledge, this has not been studied so far.', '1606.06490-1-7-0': 'To address the above issue, in this work we introduce two SNR weights for these two links/hops of relaying.', '1606.06490-1-7-1': 'We propose the source to choose the coding rate based on weighted outdated SNRs (weighted by these SNR weights).', '1606.06490-1-7-2': 'This operation reduces the coding rate (determined by the source) and therefore reduces the expected error probability.', '1606.06490-1-7-3': 'Under the proposed operation, we derive the expected BL-throughput of the upcoming transmission period (based on the current outdated CSI) as well as the ergodic BL-throughput of relaying while a fixed SNR weight scheme and a dynamic SNR weight scheme are considered.', '1606.06490-1-7-4': 'In particular, we prove that under a dynamic SNR weight scheme the BL-throughput of an upcoming transmission is concave in the coding rate and quasi-concave in SNR weights.', '1606.06490-1-7-5': 'In addition, under the fixed SNR weight scheme we show that the BL-throughput over time is quasi-concave in SNR weights.', '1606.06490-1-7-6': 'Through numerical investigations, we show the appropriateness of our theoretical model.', '1606.06490-1-7-7': 'In addition, we show that there is no big difference between the fixed weight scheme and the dynamic weight scheme if there is a strong correlation between the outdated CSI and instantaneous channel state.', '1606.06490-1-7-8': 'Otherwise, dynamic weight scheme significantly outperforms the fixed scheme, especially for high SNR scenarios.', '1606.06490-1-7-9': 'Moreover, under the FBL regime, in particular for rather short blocklengths, relaying is more beneficial in comparison to direct transmission.', '1606.06490-1-7-10': 'In other words, increasing the received signal strength by relaying is more important than the time loss incurred through halving the blocklength.', '1606.06490-1-7-11': 'More importantly, this performance advantage of relaying is more significant for outdated CSI scenario than the perfect CSI scenario.', '1606.06490-1-8-0': 'The rest of the paper is organized as follows.', '1606.06490-1-8-1': 'Section [REF] describes the system model and briefly introduces the background theory regarding the finite blocklength regime.', '1606.06490-1-8-2': 'In Section [REF], we derive the blocklength-limited performance of relaying with outdated CSI.', '1606.06490-1-8-3': 'Section [REF] presents our numerical results.', '1606.06490-1-8-4': 'Finally, we conclude our work in Section [REF].', '1606.06490-1-9-0': '# System Model', '1606.06490-1-10-0': '## System Description', '1606.06490-1-11-0': 'We consider a simple relaying scenario with a source [MATH], a destination [MATH] and a decode-and-forward (DF) relay [MATH] as schematically shown in Fig. [REF].', '1606.06490-1-12-0': 'The entire system operates in a slotted fashion where time is divided into transmission periods of length [MATH] (symbols).', '1606.06490-1-12-1': 'Each transmission period contains two frames (each frame with length [MATH]), which are referred to as backhaul frame and relaying frame.', '1606.06490-1-12-2': 'In the backhaul frame, the source sends a data block to the relay.', '1606.06490-1-12-3': 'Afterwards, if the relay decodes the block successfully, it forwards the block to the destination in the subsequent relaying frame.', '1606.06490-1-13-0': 'Channels are assumed to experience a time-varying random fading due to moving objects in the multi-path environment.', '1606.06490-1-13-1': 'We assume a correlated, slow-fading Rayleigh-distributed process with Jakes power spectrum density [CITATION].', '1606.06490-1-13-2': 'Although the fading process is assumed to be correlated, consecutive transmission periods are assumed to have independent fading gains due to the large time duration in between.', '1606.06490-1-13-3': 'In other words, the channel fading is constant during the duration of each transmission period, i.e., satisfying a block-fading model.', '1606.06490-1-14-0': 'During a transmission period, first a backhaul frame is employed, followed by a relaying frame.', '1606.06490-1-14-1': 'Considering a transmission period [MATH], channels of the backhual link and the relaying link are denoted by [MATH] and [MATH].', '1606.06490-1-14-2': 'In addition, [MATH] and [MATH] are assumed to be independent and identically distributed (i.i.d.).', '1606.06490-1-14-3': 'The received SNR at the relay of the backhaul frame and the received SNR at the destination of the relaying frame are denoted by [MATH] and [MATH].', '1606.06490-1-14-4': 'Hence, we have [MATH], where [MATH] is the average SNR of link [MATH] (either the backhaul link or the relaying link).', '1606.06490-1-14-5': 'Moreover, the source is assumed to have outdated instantaneous CSI [MATH] and [MATH] of the two links, which is obtained by sampling the channel [MATH] symbols prior to each upcoming transmission period.', '1606.06490-1-14-6': 'According to the sequence of the backhaul frame and relaying frame in each transmission period, the time gaps (between sampling the channel and transmitting a packet by the channel) of a backhaul frame and a relaying frame are [MATH] and [MATH].', '1606.06490-1-14-7': 'Based on the results of [CITATION], [MATH] conditioned on [MATH] follows a complex Gaussian distribution: [MATH], where [MATH] are correlation coefficients for the backhaul link and the relaying link between the outdated CSI and the exact CSI.', '1606.06490-1-14-8': ""Under the assumption of a Jakes' model, [MATH] and [MATH], where [MATH] and [MATH] stands for the Doppler frequency experienced on the backhaul link and the relaying link."", '1606.06490-1-14-9': 'In addition, [MATH] denotes the zero-order Bessel function of the first kind.', '1606.06490-1-14-10': 'Based on the outdated CSI [MATH], the outdated SNRs are given by [MATH].', '1606.06490-1-15-0': '## Blocklength-Limited Performance of a Single-Hop Transmission Scenario with Perfect CSI For the real additive white Gaussian noise (AWGN) channel, [8, Theorem 54] derives an accurate approximation of the coding rate of a single-hop transmission system.', '1606.06490-1-15-1': 'With blocklength [MATH], block error probability [MATH] and SNR [MATH], the coding rate (in bits per channel use) is given by: [MATH], where [MATH] is the inverse Q-function, and as usual, the Q-function is given by [MATH].', '1606.06490-1-15-2': 'In addition, [MATH] is the channel dispersion of a real Gaussian channel which is given by [MATH].', '1606.06490-1-16-0': 'Under a quasi-static fading channel model, each channel state is assumed to be static during a transmission period.', '1606.06490-1-16-1': 'For example, in each transmission period a quasi-static fading channel with fading coefficient [MATH] can be viewed as an AWGN channel with channel gain [MATH].', '1606.06490-1-16-2': 'Therefore, the above result with a real AWGN channel can be reasonably extended to a complex quasi-static fading channel model in [CITATION]: With a received SNR [MATH], the coding rate of a transmission period (in bits per channel use) is given by: [EQUATION] where [MATH] is the Shannon capacity function of a complex channel with received SNR [MATH]: [MATH].', '1606.06490-1-17-0': 'In addition, the channel dispersion of a complex Gaussian channel is twice the one of a real Gaussian channel: [MATH].', '1606.06490-1-18-0': 'Then, for a single hop transmission under a quasi-static fading channel, with blocklength [MATH] and coding rate [MATH] (during a transmission period), the decoding (block) error probability at the receiver is given by: [EQUATION]', '1606.06490-1-18-1': 'Considering the channel fading, the expected/average error probability over channel fading is given by [CITATION]: [EQUATION] where [MATH] is the expectation over the distribution of channel SNR [MATH].', '1606.06490-1-19-0': 'In the remainder of the paper, we investigate the considered relaying system in the finite blocklength regime by applying the above approximations.', '1606.06490-1-19-1': 'As these approximations have been shown to be accurate for a sufficiently large value of [MATH] [CITATION], for simplicity we will assume them to be equal in our analysis and numerical evaluation where we consider a sufficiently large value of [MATH] at each hop of relaying.', '1606.06490-1-20-0': '# Blocklength-Limited Performance of Relaying with Outdated CSI', '1606.06490-1-21-0': 'As the CSI is outdated, if the source determines the coding rate directly based on it, it is likely that the exact channel SNR is lower than the outdated one.', '1606.06490-1-21-1': 'This introduces a significant additional error probability.', '1606.06490-1-21-2': 'Therefore, we introduce SNR weights, i.e., SNR back-offs, to let the source choose a relatively lower coding rate which is obtained by the weighted outdated SNR.', '1606.06490-1-21-3': 'In the following, we consider two different schemes of determining the SNR weights.', '1606.06490-1-22-0': '## Dynamic SNR Weight Scheme', '1606.06490-1-23-0': 'Under the dynamic scheme, the source determines two appropriate SNR weights for the backhaul link and the relaying link of each transmission period based on the instantaneous outdated CSI.', '1606.06490-1-23-1': 'Denote the determined weights at transmission period [MATH] of either the backhaul link or the relaying link by [MATH]), where [MATH].', '1606.06490-1-23-2': 'As the performance of relaying is subject to the bottleneck link of the system (which is either the backhaul link or the relaying link), the coding rate [MATH] of transmission period [MATH] could therefore be obtained by: [MATH], where [MATH] is a constant parameter (a given error probability target of the bottleneck link).', '1606.06490-1-23-3': 'Hence, with the exact SNRs [MATH] and [MATH] the overall error probability of relaying during transmission period [MATH] is given by: [EQUATION] where [MATH].', '1606.06490-1-23-4': 'Based on [REF], we immediately have the expected overall error probability conditioned on the outdated CSI [MATH].', '1606.06490-1-23-5': 'It is the expected value of [REF] over the channel fading: [EQUATION]', '1606.06490-1-23-6': 'In [REF], [MATH], [MATH] are the expected error probability of either the backhaul link and the relaying link, which are with expected values based on the outdated channel SNRs [MATH].', '1606.06490-1-23-7': 'According to [CITATION], with correlation coefficients [MATH], the conditional probability density function (PDF) of the SNR of link [MATH] during transmission period [MATH] is given by: [EQUATION] where [MATH] is the modified Bessel function of the first kind.', '1606.06490-1-24-0': 'Then, [MATH], [MATH] can be obtained by averaging [MATH] over the above PDF.', '1606.06490-1-24-1': 'We provide the derivation of [MATH] in [REF], where [MATH] and [MATH].', '1606.06490-1-25-0': 'Under the two-hop relaying scenario, if the coding rate at each hop during transmission period [MATH] is [MATH], the (source-to-destination) equivalent coding rate during the period is actually [MATH].', '1606.06490-1-25-1': 'Therefore, the expected BL-throughput of relaying during transmission period [MATH], i.e., the expected effectively transmitted information (the number of correctly received bits at the destination) per channel use, is given by: [EQUATION]', '1606.06490-1-25-2': 'So far, we derived the expected BL-throughput of relaying for an upcoming transmission period [MATH] based on the current outdated CSI (the feedbacked CSI for period [MATH]).', '1606.06490-1-25-3': 'We then have the following proposition:', '1606.06490-1-26-0': 'Consider a relaying scenario with correlated and slow-fading Rayleigh channels where only outdated CSI are available at the source.', '1606.06490-1-26-1': 'If the coding rate is determined by the weighted SNR while the SNR weights for transmission period [MATH] satisfy [MATH], [MATH] the expected BL-throughput of the upcoming transmission period [MATH] is quasi-concave in the coding rate [MATH].', '1606.06490-1-27-0': 'See Appendix A.', '1606.06490-1-28-0': 'Recall that the coding rate is chosen by the source based on [MATH].', '1606.06490-1-28-1': 'According to [REF], the coding rate is strictly increasing in [MATH] and therefore increasing in [MATH] or [MATH].', '1606.06490-1-28-2': 'Combining this with Proposition 1, we have an important corollary:', '1606.06490-1-29-0': 'Consider a relaying scenario with correlated and slow-fading Rayleigh channels where only outdated CSI are available at the source.', '1606.06490-1-29-1': 'If the coding rate is determined by the weighted SNR while these SNR weights for transmission period [MATH] satisfy [MATH], [MATH] the expected BL-throughput of the upcoming transmission period [MATH] is quasi-concave in either [MATH] or [MATH].', '1606.06490-1-30-0': 'See Appendix B.', '1606.06490-1-31-0': 'Finally, based on the expected BL-throughput for each transmission period, the ergodic BL-throughput of relaying, which is actually the expectation value of [MATH] over the distribution of [MATH], can be obtained by: [MATH].', '1606.06490-1-31-1': ""Due to the channels' Rayleigh fading behavior, [MATH] is: LLL"", '1606.06490-1-32-0': 'C_BL = E_r_i [ C_BL,i(r_i) ] = & _0^ _0^ C_BL,i( R( _1,i_1,_2,i_2 ,^*,m) ) e^ - _1_1 - _2_2d_1d_2', '1606.06490-1-33-0': '=& 1_1_2_0^__1,i_1_2_2^ C_BL,i( R(_ 1,i_1,^*,m) ) e^ - _1_1 - _2_2d_2d_1', '1606.06490-1-34-0': '& + 1_1_2_0^__2_2_1_1^ C_BL,i( R(_2,i_2,^*,m) ) e^ - _1_1 - _2_2d_1d_2 .', '1606.06490-1-35-0': '## Fixed SNR Weight Scheme', '1606.06490-1-36-0': 'In this subsection, we consider the scheme with fixed SNR weights.', '1606.06490-1-36-1': 'We denote the weights by [MATH] and [MATH].', '1606.06490-1-36-2': 'Hence, once [MATH] and [MATH] are determined by the system initialization, they will be constant during all transmission periods.', '1606.06490-1-36-3': 'It should be mentioned that under the fixed weight scheme the coding rate is not fixed as the coding rate is influenced by both the weight and the instantaneous outdated CSI.', '1606.06490-1-36-4': 'In particular, the coding rate [MATH] of transmission period [MATH] is obtained by: [MATH].', '1606.06490-1-36-5': 'According to [REF], the coding rate is strictly increasing in [MATH] and therefore monotonically increasing in [MATH] and [MATH].', '1606.06490-1-36-6': 'In addition, [REF] also holds for the fixed SNR weight scheme (while the expressions of [MATH] under dynamic and fixed schemes are different).', '1606.06490-1-36-7': 'Finally, under the fixed SNR weight scheme [MATH] the BL-throughput over time is given by: LLL', '1606.06490-1-37-0': 'C_BL = E_r_i [ C_BL,i(r_i) ] = & 1_1_2_0^__1_1_2_2^ C_BL,i( R(_1_1,^*,m) ) e^ - _1_1 - _2_2d_2d_1', '1606.06490-1-38-0': '& + 1_1_2_0^__2_2_1_1^ C_BL,i( R(_2_2,^*,m) ) e^ - _1_1 - _2_2d_1d_2.', '1606.06490-1-39-0': 'For the fixed SNR weight scheme, we have the following Proposition:', '1606.06490-1-40-0': 'Consider a relaying scenario with correlated and slow-fading Rayleigh channels where only outdated SNRs are available at the source.', '1606.06490-1-40-1': 'If the coding rate is determined by the weighted SNR while these SNR weights are constant over time and satisfy [MATH], then [MATH] the BL-throughput is quasi-concave in either [MATH] or [MATH].', '1606.06490-1-41-0': 'See Appendix C.', '1606.06490-1-42-0': 'So far, both the dynamic weight scheme and the fixed weight scheme have been discussed.', '1606.06490-1-42-1': 'It should be mentioned that Proposition 1 and Proposition 2 indicate: The expected BL-throughput of a transmission period can be optimized based on the current outdated CSI.', '1606.06490-1-42-2': 'In other words, the expected BL-throughput can be optimized per transmission period by choosing appropriate SNR weights.', '1606.06490-1-42-3': 'Therefore, the dynamic weight scheme definitely outperforms the scheme with fixed weight.', '1606.06490-1-42-4': 'Nevertheless, the cost of this advantage is complexity, i.e., the source under the dynamic scheme needs to search the optimal weights for each transmission period while the one under the fixed scheme only needs to substitute the fixed weights and the outdated CSI to [MATH].', '1606.06490-1-42-5': 'Moreover, Proposition 3 indicates that the BL-throughput under the fixed weight scheme can also be optimized by determining SNR weights.', '1606.06490-1-42-6': 'Then, straightforward questions arise: What is the performance difference between these two schemes?', '1606.06490-1-42-7': 'If the gain of searching and adapting the weights per transmission period is significant?', '1606.06490-1-42-8': 'We will provide a numerical analysis to compare them in the next section.', '1606.06490-1-43-0': '# Numerical Results and Discussion', '1606.06490-1-44-0': 'In this section, we first present numerical results to illustrate our analytical model.', '1606.06490-1-44-1': 'Subsequently, we compare the dynamic weight scheme with the fixed weight scheme.', '1606.06490-1-44-2': 'Later on, we evaluate the studied relaying system while the direct transmission scheme is treated as a comparison scheme.', '1606.06490-1-44-3': 'During the numerical analysis, we consider the following parameterization of the system model: Firstly, in the simulation we consider an outdoor urban scenario and the distances of the backhaul, relaying and direct links are set to 100 [MATH], 100 [MATH] and 200 [MATH].', '1606.06490-1-44-4': 'Secondly, we utilize the well-known COST [CITATION] model (which is a commonly-used model in LTE) for calculating the path-loss while the center frequency is set to 2 [MATH].', '1606.06490-1-44-5': 'Thirdly, we set the transmit power [MATH] to 35 [MATH] (we vary the transmit power in Fig. [REF] and Fig. 7) and noise power to -90 [MATH], respectively.', '1606.06490-1-44-6': 'Lastly, the blocklength at each hop of relaying is set to 300 (symbols) while the choice of this ([MATH]) is motivated by [8, Fig. 2] where the relative difference of the approximate and the exact achievable rates is less than 2% for the cases with [MATH] 100.', '1606.06490-1-45-0': '## Appropriateness of Our Analytical Model', '1606.06490-1-46-0': 'In this subsection, we show the appropriateness of our analytical model based on numerical results We first study under the dynamic SNR weight scheme the relationship between the expected BL-throughput of an upcoming transmission period (based on the corresponding outdated CSI) and the SNR weights.', '1606.06490-1-46-1': 'The result is shown in Fig. [REF].', '1606.06490-1-47-0': 'First of all, the figure illustrates our analytical model that the BL-throughput of a transmission period [MATH] (conditioned on the outdated CSI) is quasi-concave in SNR weights [MATH] and [MATH].', '1606.06490-1-47-1': 'Hence, by choosing appropriate values for [MATH] and [MATH] the BL-throughput of transmission period [MATH] can be optimized.', '1606.06490-1-47-2': 'Secondly, the figure also shows that the expected BL-throughput of the upcoming period is actually subject to both [MATH] and [MATH].', '1606.06490-1-47-3': 'For example, if [MATH] is chosen correctly, then the choice of [MATH] can be arbitrary big (not arbitrary small).', '1606.06490-1-47-4': 'The explanation is as follows.', '1606.06490-1-47-5': 'The case (BL-throughput being only influenced by [MATH]) corresponds to the situation where the bottleneck link (for determining the coding rate) is the backhaul link.', '1606.06490-1-47-6': 'At the same time, [MATH] (the SNR weight of the relaying link) does not influence the SNR of the backhaul link and therefore does not influence the coding rate.', '1606.06490-1-47-7': ""In other words, there is no impact of one link's weight on the BL-throughput of the upcoming transmission period if this link is not the bottleneck link."", '1606.06490-1-47-8': 'It should be mentioned that reducing the weight of a link likely makes this link become the bottleneck link.', '1606.06490-1-47-9': 'Then, the BL-throughput of the upcoming transmission period is influenced by this weight.', '1606.06490-1-48-0': 'Secondly, we consider the fixed weight scheme in Fig. [REF] where we show the relationship between the BL-throughput [MATH] and the SNR weights [MATH] and [MATH].', '1606.06490-1-49-0': 'The figure matches well with our analytical model (Proposition 3) that [MATH] is quasi-concave in [MATH] or [MATH].', '1606.06490-1-50-0': '## Relaying Performance Investigation', '1606.06490-1-51-0': '### Dynamic SNR weight scheme vs. fixed SNR weight scheme', '1606.06490-1-52-0': 'The comparison between these two weight schemes is shown in Fig. [REF]', '1606.06490-1-53-0': 'where we consider three scenarios with different channel correlation coefficients.', '1606.06490-1-53-1': 'We observe that there is no big difference between the dynamic scheme and the fixed scheme if the correlation between the channel estimate and the real channel state is large, e.g., channels vary extremely slow or/and the transmission period are extremely short.', '1606.06490-1-53-2': 'Otherwise, the dynamic scheme outperforms the fixed one.', '1606.06490-1-53-3': 'Moreover, especially for the high SNR range does the dynamic scheme outperform the static scheme.', '1606.06490-1-54-0': '### Convergence: Relaying with finite blocklengths vs. Relaying with infinite blocklengths', '1606.06490-1-55-0': 'We show the convergence comparison in Fig. [REF] where we vary channel correlation coefficients.', '1606.06490-1-56-0': 'In the figure, the maximal BL-throughput and the maximal outage capacity (based on the Shannon capacity) under the outdated CSI scenario are obtained by choosing optimal [MATH] and [MATH] for each transmission period [MATH].', '1606.06490-1-56-1': 'We find that the performance loss (due to a finite blocklength) in a relaying network is considerable under the perfect CSI scenario but is negligible under the outdated CSI scenario.', '1606.06490-1-56-2': 'In other words, relaying is more efficient under the outdated CSI scenario than the perfect CSI scenario.', '1606.06490-1-57-0': '### Relaying vs. direct transmission', '1606.06490-1-58-0': 'Further, we show in Fig. [REF] the ergodic BL-throughput under the fixed SNR weight scheme while the performance of direct transmission is also plotted as a comparison scheme.', '1606.06490-1-59-0': 'To make a fair comparison, we have the following assumptions in the numerical analysis: i.', '1606.06490-1-59-1': 'During each transmission period, relaying and direct transmission have the same (equivalent) coding rate and have the same resource consumption, i.e., the blocklength of direct transmission is twice as large as the blocklength at each hop of relaying.', '1606.06490-1-59-2': 'ii.', '1606.06490-1-59-3': 'The channel correlation coefficient of direct transmission equals the one of the backhaul link of relaying, i.e., [MATH].', '1606.06490-1-59-4': 'Therefore, the delay by which the CSI feedback of the second hop of relaying subjected is longer than direct transmission.', '1606.06490-1-59-5': 'iv.', '1606.06490-1-59-6': 'By adjusting the transmit power of direct transmission we make relaying and direct transmission have the same performance in the infinite blocklength regime, i.e., [MATH], where [MATH] is the expected SNR of direct transmission.', '1606.06490-1-59-7': 'Under the above setup, we compare relaying and direct transmission.', '1606.06490-1-59-8': 'We show the result in Fig. [REF].', '1606.06490-1-59-9': 'Although having similar performance in the infinite blocklength regime, shorter blocklength at each hop and more outdated CSI (of the second hop), relaying outperforms direct transmission in the finite blocklength regime for both the perfect CSI case and the outdated CSI case.', '1606.06490-1-59-10': 'This is actually the performance advantage of relaying (which we also observed in our previous work [CITATION] where the CSI is not outdated).', '1606.06490-1-59-11': 'More importantly, in this work we find that under the outdated CSI scenario this performance advantage of relaying is more significant than under the perfect CSI scenario.', '1606.06490-1-60-0': '### An explanation of the performance advantage of relaying', '1606.06490-1-61-0': 'An explanation of relaying outperforming direct transmission in the finite blocklength regime in term of the BL-throughput is provided as follows.', '1606.06490-1-61-1': 'As we know, finite blocklength introduces a performance loss (comparing the BL-throughput with the Shannon capacity).', '1606.06490-1-61-2': 'More importantly, this performance gap is influenced by the SNR.', '1606.06490-1-61-3': 'If we call the ratio of the performance loss over the Shannon capacity ""performance loss ratio"", i.e., [MATH], this performance loss ratio is also influenced by the SNR.', '1606.06490-1-61-4': 'In the following figure,', '1606.06490-1-62-0': 'we show the relationship between the performance loss ratio and the SNR for a single-hop link under an AWGN channel.', '1606.06490-1-62-1': 'Based on the figure, we find that increasing the SNR significantly reduces the performance loss ratio.', '1606.06490-1-62-2': 'As relaying has a significantly higher SNR at each hop in comparison to the direct link, relaying significantly mitigates the performance loss due to a finite blocklength.', '1606.06490-1-62-3': 'As a result, although relaying halves the blocklength, it shows a performance advantage over direct transmission in the finite blocklength regime.', '1606.06490-1-62-4': 'This explanation can also be extended to the considered quasi-static channels: Relaying has a low performance loss ratio for each channel state, therefore the average performance loss ratio over channel fading of relaying is lower than the direct transmission.', '1606.06490-1-62-5': 'As a result, although having similar Shannon capacity, the BL-throughput of relaying is higher than direct transmission.', '1606.06490-1-63-0': '# Conclusion', '1606.06490-1-64-0': 'In this work, we studied the finite blocklength performance of relaying under outdated CSI at the source.', '1606.06490-1-64-1': 'We showed that under the dynamic SNR weight scheme the instantaneous BL-throughput of a transmission period is concave in the coding rate and quasi-concave in SNR weights.', '1606.06490-1-64-2': 'In addition, under the fixed SNR weight scheme we showed that the average BL-throughput over time is quasi-concave in the SNR weights.', '1606.06490-1-64-3': 'By numerical analysis, we showed the appropriateness of our analytical model.', '1606.06490-1-64-4': 'Moreover, we concluded a set of guidelines for the design of efficient relaying systems (in the finite blocklength regime) from numerical analysis.', '1606.06490-1-64-5': 'Firstly, for a given scenario the performance can be optimized by adjusting the SNR weights.', '1606.06490-1-64-6': 'Secondly, the distinction in fixed and dynamic weight scheduling matters a lot whenever the correlation is weak and the SNR is high.', '1606.06490-1-64-7': 'More importantly, we showed that the lower the blocklength the more relaying pays off in comparison to direct transmission, and that this is especially relevant for the design of low latency systems.', '1606.06490-1-64-8': 'This performance advantage is due to the fact that relaying has a higher SNR at each hop (than the direct link) which is more relevant for the system performance than the halving of the transmission periods per single hop.', '1606.06490-1-65-0': 'In our previous work [CITATION] of relaying with the average CSI, we observed the performance advantage of a relaying system where a spatial diversity is exploited, i.e., the destination receives signals from both the source and the relay.', '1606.06490-1-65-1': 'Therefore, the performance advantage of relaying shown in this paper (where the signal from the source directly to the destination is not considered) further indicates that even without exploiting spatial diversity relaying still pays off in the finite blocklength regime due to having a higher SNR at each hop.', '1606.06490-1-65-2': 'Combining the results of this paper with our previous work [CITATION], we finally conclude that applying relaying is a promising way to improve the performance of low latency systems, especially for latency critical applications e.g., haptic communication, automation and control applications and cyber physical systems, where blocklengths are significantly short.', '1606.06490-1-66-0': '# Proof of the Theorem 1', '1606.06490-1-67-0': 'Based on [REF] and [REF], we immediately have [MATH].', '1606.06490-1-67-1': 'Therefore, the first and second derivatives of [MATH] with respect [MATH] to are given by: [EQUATION]', '1606.06490-1-67-2': 'In the following, we prove Theorem 1 by showing [MATH].', '1606.06490-1-68-0': 'Recall that [MATH].', '1606.06490-1-68-1': 'According to [REF], we have: [EQUATION]', '1606.06490-1-68-2': 'Based on [REF], we have: [EQUATION]', '1606.06490-1-68-3': 'According to the distribution of [MATH] conditioned [MATH], we immediately obtain the median of the distribution, which is [MATH].', '1606.06490-1-68-4': 'In other words, the following equation holds: [EQUATION]', '1606.06490-1-68-5': 'As [MATH], the following inequality holds: [MATH].', '1606.06490-1-68-6': 'Hence, [MATH] and therefore [MATH] if [MATH].', '1606.06490-1-68-7': 'In other words, [MATH] and [MATH].', '1606.06490-1-69-0': 'Considering equality [REF], we have: [EQUATION]', '1606.06490-1-69-1': 'So far, it has been shown that [EQUATION]', '1606.06490-1-69-2': 'According to [REF], the error probability of a link is higher than 0.5 only if the coding rate is higher than the Shannon capacity.', '1606.06490-1-69-3': 'Recall that the coding rate chosen by the source satisfies [MATH].', '1606.06490-1-69-4': 'This makes the expected error probability of each single link (during transmission period [MATH]) be lower than 0.5, i.e., [MATH].', '1606.06490-1-69-5': 'Based on [REF], we have: [EQUATION]', '1606.06490-1-69-6': 'Hence, [MATH] if [MATH].', '1606.06490-1-69-7': 'For large enough [MATH], we have: [EQUATION]', '1606.06490-1-69-8': 'Obviously, [MATH] holds if [MATH].', '1606.06490-1-69-9': 'Note that [MATH] and recall that we consider [MATH] with a practical value i.e., is significantly longer than 100.', '1606.06490-1-69-10': 'Hence, [MATH] is concave in [MATH].', '1606.06490-1-70-0': '# Proof of the Proposition 2', '1606.06490-1-71-0': 'The coding rate for transmission period [MATH] is determined based by the outdated CSI of the bottleneck link, i.e., [MATH].', '1606.06490-1-71-1': 'According to [REF], we learn that [MATH] is strictly increasing in [MATH].', '1606.06490-1-71-2': 'Hence, when the source determines the coding rate [MATH], high values of [MATH] and [MATH] lead to a big [MATH].', '1606.06490-1-71-3': 'In other words, [MATH] is monotonically increasing in [MATH].', '1606.06490-1-72-0': '[MATH], [MATH] and [MATH], we have [MATH], where [MATH].', '1606.06490-1-73-0': 'Based on Proposition 1, [MATH] is concave in [MATH],', '1606.06490-1-74-0': '[MATH].', '1606.06490-1-75-0': '[MATH] is quasi-concave in [MATH], where [MATH] and [MATH].', '1606.06490-1-76-0': '# Proof of the Proposition 3', '1606.06490-1-77-0': 'According to the proof of Proposition 2, [MATH], [MATH] is monotonically increasing in [MATH].', '1606.06490-1-78-0': '[MATH], [MATH], [MATH] and [MATH], we have [MATH], where [MATH].', '1606.06490-1-79-0': 'As shown in Proposition 1, [MATH] is concave in [MATH].', '1606.06490-1-79-1': 'Hence, [MATH] is concave in [MATH].', '1606.06490-1-80-0': '[MATH].', '1606.06490-1-81-0': '[MATH] is quasi-concave in [MATH], where [MATH] and [MATH].'}","{'1606.06490-2-0-0': 'Under the assumption of outdated channel state information (CSI) at the source, we consider the finite blocklength (FBL) throughput of a two-hop relaying system.', '1606.06490-2-0-1': 'Previous work has considered this setting so far only for the infinite blocklength case, where decoding can be arbitrarily reliable as long as operating below the Shannon limit.', '1606.06490-2-0-2': 'In contrast, in the FBL regime residual decoding errors can not be avoided even when transmitting below the Shannon limit.', '1606.06490-2-0-3': 'This makes the scheduling problem at the source more vulnerable to transmission errors, where we investigate the trade-off between the choice of so called scheduling weights to avoid transmission errors and the resulting coding rate.', '1606.06490-2-0-4': 'We show that the corresponding maximization of the throughput under a reliability constraint can be solved efficiently by iterative algorithms.', '1606.06490-2-0-5': 'Nevertheless, the optimal solution requires a recomputation of the scheduling weights prior to each transmission.', '1606.06490-2-0-6': 'Thus, we also study heuristics relying on choosing the scheduling weights only once.', '1606.06490-2-0-7': 'Through numerical analysis, we first provide insights on the structure of the throughout under different scheduling weights and channel correlation coefficients.', '1606.06490-2-0-8': 'We then turn to the comparison of the optimal scheduling with the heuristic and show that the performance gap between them is only significant for relay systems with high average signal-to-noise ratios (SNR) on the backhaul and relaying link.', '1606.06490-2-0-9': 'In particular, the optimal scheduling scheme provides most value in case that the data transmission is subject to strict reliability constraints, justifying the significant additional computational burden.', '1606.06490-2-1-0': 'decode-and-forward, finite blocklength, optimal scheduling, outdated CSI, relaying.', '1606.06490-2-2-0': '# Introduction', '1606.06490-2-3-0': 'In wireless communications, relaying [CITATION] is well known as an efficient way to mitigate fading by exploiting spatial diversity and providing better channel quality.', '1606.06490-2-3-1': 'Specifically, two-hop decode-and-forward (DF) relaying protocols significantly improve the throughput and quality of service [CITATION].', '1606.06490-2-3-2': ""However, typically these studies on the advantage of relaying are under the ideal assumption of communicating arbitrarily reliable at Shannon's channel capacity, i.e., code words are assumed to be infinitely long."", '1606.06490-2-4-0': 'In the finite blocklength regime, the data transmission is no longer arbitrarily reliable.', '1606.06490-2-4-1': 'Especially when the blocklength is short, the error probability (due to noise) becomes significant even if the rate is selected below the Shannon limit.', '1606.06490-2-4-2': 'Taking this into account, an accurate approximation of the achievable coding rate under the finite blocklength assumption for an additive white Gaussian noise (AWGN) channel was derived in [CITATION] for a single-hop transmission system.', '1606.06490-2-4-3': 'Subsequently, the initial work for AWGN channels was extended to Gilbert-Elliott channels [CITATION] as well as quasi-static fading channels [CITATION].', '1606.06490-2-4-4': 'It is shown in these works that the finite blocklength performance of a single-hop transmission is determined by the coding rate, error probability and blocklength.', '1606.06490-2-4-5': 'In particular, the performance loss due to the additional decoding errors at finite blocklength is considerable and increases as the blocklength decreases.', '1606.06490-2-4-6': 'Also, if the channel and the blocklength are given, the error probability of the single-hop transmission is strictly increasing in the coding rate.', '1606.06490-2-4-7': ""In our own previous work [CITATION], we extended Polyanskiy's model [CITATION] of single-hop transmission to a two-hop DF relaying network, where the relay halves the distance to provide a power gain but at the same time also halves the blocklength of the transmission."", '1606.06490-2-4-8': 'Subsequently, we provided a general analytical model of the finite blocklength performance under static/quasi-static channels in [CITATION] while assuming the transmitter to have only average CSI.', '1606.06490-2-4-9': 'More recently, the throughput of a relaying network with finite blocklengths and queuing constraints was studied in [CITATION] under the perfect CSI assumption.', '1606.06490-2-5-0': 'In practical relay systems (as for instance specified by the LTE standard) CSI feedback mechanisms are usually implemented, i.e., allowing the receiver to instantaneously estimate and feedback the CSI to the transmitter.', '1606.06490-2-5-1': 'However, typically there exists a delay between the instant of sampling the channel and the point in time when this CSI sample is received by the transmitter making the CSI feedback delayed and therefore outdated.', '1606.06490-2-5-2': 'The performance analysis and optimization of relaying systems operating on outdated CSI have been widely discussed in the infinite blocklength (IBL) regime.', '1606.06490-2-5-3': 'In [CITATION], the probability of an outage event (defined as the event when the coding rate is higher than the Shannon capacity) of a DF relaying network is studied under the outdated CSI relaying scenario.', '1606.06490-2-5-4': 'Protocols are designed in [CITATION] for a relay system operating based on outdated CSI to optimally trade-off outage, delay, and throughput.', '1606.06490-2-5-5': 'For multi-relay scenarios with outdated CSI, optimal relay selection algorithms [CITATION] are proposed to minimize the outage probability.', '1606.06490-2-5-6': 'However, these works generally ignore the impact of transmitting under finite blocklength restrictions, which introduces further subtleties in addition to the imperfect channel knowledge.', '1606.06490-2-6-0': 'In this paper, we thus study the finite blocklength performance of a relaying network assuming the source to have only outdated CSI.', '1606.06490-2-6-1': 'Different from the average CSI scenario considered in [CITATION], based on the provided CSI the source is able to adjust the coding rate per frame.', '1606.06490-2-6-2': 'However, due to the outdated CSI, scheduling the coding rate in this way can result in more frequent transmission errors.', '1606.06490-2-6-3': 'We hence study the optimal scheduling of the coding rate in the relay system with outdated CSI under a reliability constraint for the data transmission.', '1606.06490-2-6-4': 'As objective function we focus on the maximization of the FBL throughput.', '1606.06490-2-6-5': 'Solving this optimal scheduling problem requires the source to choose the coding rate based on scheduling weights, i.e. factors by which the outdated channel SNRs are rescaled.', '1606.06490-2-6-6': 'We contribute by first deriving a model for the FBL throughput of the relaying system operating with outdated CSI.', '1606.06490-2-6-7': 'Next, we propose an optimal scheduling scheme that maximizes the FBL throughput.', '1606.06490-2-6-8': 'We show that the objective function of the scheduling problem is concave in the coding rate and quasi-concave in the scheduling weights.', '1606.06490-2-6-9': 'Therefore, the optimal scheduling problem can be solved efficiently by iterative methods.', '1606.06490-2-6-10': 'Nevertheless, to mitigate the computational complexity, we also consider a sub-optimal scheduling scheme, where fixed scheduling weights are applied per frame.', '1606.06490-2-6-11': 'We refer to this scheme as constant heuristic and study the problem of choosing the constant scheduling weights which maximize the average FBL throughput over time.', '1606.06490-2-6-12': 'We finally perform numerical evaluations and show that the optimal scheme outperforms the best constant heuristic, especially when the reliability constraint is strict and/or the average SNR is high.', '1606.06490-2-6-13': 'Surprisingly, we find that the channel correlation has only a marginal impact on the performance gap between the two schemes.', '1606.06490-2-7-0': 'The rest of the paper is organized as follows.', '1606.06490-2-7-1': 'Section [REF] describes the system model and briefly reviews the background regarding the finite blocklength regime.', '1606.06490-2-7-2': 'In Section [REF], we first derive the finite blocklength performance model of the considered relaying scenario with outdated CSI.', '1606.06490-2-7-3': 'Afterwords, we state the optimization problem of interest and provide the theoretical insights that lead to the optimal solution.', '1606.06490-2-7-4': 'We then turn to the constant heuristic and provide an optimal solution for choosing the fixed scheduling weight.', '1606.06490-2-7-5': 'In Section [REF] we then present our numerical results.', '1606.06490-2-7-6': 'Finally, we conclude our work in Section [REF].', '1606.06490-2-8-0': '# System Model', '1606.06490-2-9-0': 'We consider a straightforward scenario with a source [MATH], a destination [MATH] and a relay [MATH] as schematically shown in Figure [REF].', '1606.06490-2-10-0': 'The relay is assumed to work under a DF principle.', '1606.06490-2-10-1': 'The entire system operates in a slotted fashion where time is divided into frames of length [MATH] symbols, as shown in Figure [REF].', '1606.06490-2-11-0': 'Each frame consists of two parts, the initialization part and the transmission part.', '1606.06490-2-11-1': 'For the initialization part a certain amount of symbols are spent for acquiring the CSI.', '1606.06490-2-11-2': 'During this part, messages are exchanged to essentially obtain the CSI of the backhaul and relaying link at the source node.', '1606.06490-2-11-3': 'We assume this part to have a duration of [MATH] symbols, without specifying closer the exact system operation.', '1606.06490-2-11-4': 'The second part of each frame is the transmission part containing two phases, which are the backhaul phase (of length [MATH]) and the relaying phase (of length [MATH]).', '1606.06490-2-11-5': 'During the backhaul phase, the source sends a data block to the relay.', '1606.06490-2-11-6': 'Then, if the relay decodes the block successfully, it forwards the block to the destination in the subsequent relaying phase.', '1606.06490-2-11-7': 'Overall, we assume a setting where the initialization part takes a significantly longer amount of time than a single data transmission phase, i.e. [MATH].', '1606.06490-2-11-8': 'This is motivated for example by rather short (but important) payload packets for which a reliable transmission is crucial, which justifies the acquisition of the CSI upfront.', '1606.06490-2-12-0': 'Channels are assumed to experience a time-varying Rayleigh-distributed random fading.', '1606.06490-2-12-1': 'As both the backhaul phase and the relaying phase are short, we assume that the channel state is constant during each phase.', '1606.06490-2-12-2': 'However, the channel states in different frames are assumed to be independent.', '1606.06490-2-12-3': ""Considering a frame [MATH], the channel's complex states of the backhaul link and the relaying link are denoted by [MATH] and [MATH] and are assumed to be independent and identically distributed (i.i.d.)."", '1606.06490-2-12-4': 'The received SNR at the relay of the backhaul phase and the received SNR at the destination of the relaying phase are denoted by [MATH] and [MATH].', '1606.06490-2-12-5': 'Hence, we have [MATH], where [MATH] is the average SNR of link [MATH] (either the backhaul link or the relaying link).', '1606.06490-2-12-6': 'Recall that we assume the source to acquire the instantaneous CSI by sampling the channel [MATH] symbols prior to the backhaul phase and [MATH] symbols prior to the relaying phase.', '1606.06490-2-12-7': 'Thus, due to the time-varying nature of the fading, the sampled channel coefficients, denoted by [MATH], differ from the actual instantaneous channel coefficients [MATH] that the data packet will experience.', '1606.06490-2-12-8': 'We adopt the widely-used Jakes model for the relation between [MATH] and [MATH] [CITATION]: [EQUATION] where [MATH] is a complex Gaussian random variable, i.e., [MATH].', '1606.06490-2-12-9': 'In addition, [MATH] are channel correlation coefficients.', '1606.06490-2-12-10': 'Taking the frame sequence into account, we thus obtain [MATH] and [MATH], where [MATH] and [MATH] stand for the Doppler frequency experienced on the backhaul link and the relaying link.', '1606.06490-2-12-11': 'In addition, [MATH] denotes the zero-order Bessel function of the first kind [CITATION].', '1606.06490-2-12-12': 'Based on the outdated CSI [MATH], the outdated SNRs are given by [MATH].', '1606.06490-2-12-13': 'Thus, the instantaneous channel SNRs [MATH] become now random variables conditioned on the outdated SNRs [MATH].', '1606.06490-2-12-14': 'The conditional probability density function (PDF) of the instantaneous SNRs of link [MATH] during frame [MATH] thus results to [CITATION]: [EQUATION] where [MATH] is the zero-order modified Bessel function of the first kind.', '1606.06490-2-12-15': 'We further denote by [MATH] the median of the instantaneous SNR [MATH], for which the following equation holds: [EQUATION]', '1606.06490-2-12-16': 'Due to [REF] the median of the distribution of the instantaneous channel [MATH] is [MATH], thus we have [MATH].', '1606.06490-2-13-0': '## Finite Blocklength Error Model under Perfect CSI', '1606.06490-2-14-0': 'For the real additive white Gaussian noise (AWGN) channel [8, Theorem 54] derives an accurate approximation of the coding rate of a single-hop transmission system.', '1606.06490-2-14-1': 'With blocklength [MATH], block error probability [MATH] and SNR [MATH], the coding rate (in bits per channel use) is given by [MATH], where [MATH] is the inverse of the Q-function given by [MATH].', '1606.06490-2-14-2': 'In addition, [MATH] is the channel dispersion of a real Gaussian channel which is given by [MATH].', '1606.06490-2-15-0': 'Under a quasi-static fading channel model, each channel state is assumed to be static during a frame, i.e., in each frame a quasi-static fading channel with fading coefficient [MATH] can be viewed as an AWGN channel with channel gain [MATH].', '1606.06490-2-15-1': 'Therefore, the above result of the real AWGN channel has been extended to a complex quasi-static fading channel model[CITATION]: For a received SNR [MATH], the coding rate of a frame (in bits per channel use) is given by: [EQUATION] where [MATH] is the Shannon capacity function of a complex channel with received SNR [MATH] : [MATH].', '1606.06490-2-15-2': 'In addition, the channel dispersion of a complex Gaussian channel is twice the one of a real Gaussian channel: [MATH].', '1606.06490-2-16-0': 'Then, for a single-hop transmission under a quasi-static fading channel, with blocklength [MATH] and coding rate [MATH], the decoding (block) error probability at the receiver is given by: [EQUATION]', '1606.06490-2-16-1': 'Considering the channel fading, the expected/average error probability is given by [CITATION]: [EQUATION]', '1606.06490-2-16-2': 'In the remainder of the paper, we investigate the considered relaying system in the finite blocklength regime by applying the above approximations.', '1606.06490-2-16-3': 'As these approximations have been shown to be accurate for a sufficiently large blocklength [MATH] [CITATION], for simplicity we will assume them to hold in equality in our analysis and numerical evaluation conditioned on the assumption of a sufficiently large value of [MATH] at each hop.', '1606.06490-2-17-0': '# Maximizing the FBL Throughput under Reliability Constraints', '1606.06490-2-18-0': 'As discussed in the previous section, the source has outdated channel state information that it can rely on for scheduling the data transmission in the relay system.', '1606.06490-2-18-1': 'In this section, we address thus the problem of how to optimally schedule the coding rate based on the inaccurate outdated CSI such that the throughput of the relay system is maximized.', '1606.06490-2-18-2': 'We restrict this scheduling problem to a reliability constraint such that for each data transmission a target error probability [MATH] must be met.', '1606.06490-2-18-3': 'Such a scheduling problem is justified by current discussions around industrial wireless communication systems, where small payload packets need to be transmitted within a bounded time interval while keeping a (stochastic) reliability guarantee.', '1606.06490-2-18-4': 'In the following, we first develop a throughput model of the relaying system with respect to the finite blocklength assumption, building on Section [REF].', '1606.06490-2-18-5': 'Subsequently, the mathematical statement of the optimization problem is provided.', '1606.06490-2-18-6': 'We then turn to the solution, providing both an optimal solution as well as a low-complexity heuristic.', '1606.06490-2-19-0': '## FBL Throughput Model for Relay Systems', '1606.06490-2-20-0': 'Assuming [MATH] is the scheduled coding rate for frame [MATH] with instantaneous SNRs [MATH] and [MATH], the overall error probability of the relaying system during frame [MATH] is: [EQUATION] where [MATH].', '1606.06490-2-20-1': 'Based on [REF], we immediately have the expected overall error probability conditioned on the outdated CSI [MATH].', '1606.06490-2-20-2': 'It is the expected value of [REF] over the conditioned channel fading distribution: [EQUATION]', '1606.06490-2-20-3': 'In [REF], [MATH], [MATH] are the expected error probabilities of either the backhaul link or the relaying link.', '1606.06490-2-21-0': 'Then, by averaging [MATH] over the conditional PDF in [REF], [MATH], [MATH] is given by [REF], where [MATH] and [MATH].', '1606.06490-2-22-0': 'Notice that for the relaying system considered, the (source-to-destination) equivalent coding rate during each frame [MATH] is actually [MATH].', '1606.06490-2-22-1': 'Therefore, the expected FBL throughput of relaying during frame [MATH], i.e., the expected effectively transmitted information (number of correctly received bits at the destination) per channel use, is given by: [EQUATION]', '1606.06490-2-22-2': 'The above [MATH] is the expected FBL throughput of relaying for an upcoming frame [MATH] based on a scheduled coding rate.', '1606.06490-2-22-3': 'By marginalizing over all possible channel states for both links, we finally end up with the average FBL throughput of relaying: [MATH].', '1606.06490-2-22-4': 'Note in particular that this throughput depends on the scheduled coding rate [MATH] which itself can be based on the information at hand of the source, i.e. the outdated SNRs [MATH] and [MATH].', '1606.06490-2-23-0': '## Optimal Scheduling', '1606.06490-2-24-0': 'Recall that we are interested in scenarios with reliability constraints, i.e., the (expected/average) error probability of each link should be lower than a threshold [MATH] of practical interest, e.g., [MATH].', '1606.06490-2-24-1': 'If the source schedules the coding rate directly based on the outdated CSI, it is likely that the instantaneous SNR is lower.', '1606.06490-2-24-2': 'This can introduce a significant source of block errors while generally leading to a higher coding rate in case of successful transmissions, i.e. we face a typical trade-off.', '1606.06490-2-24-3': 'To study this trade-off, we introduce weights, i.e., SNR back-offs, to let the source choose a relatively lower coding rate obtained by scaling the outdated SNR.', '1606.06490-2-24-4': 'Denote these weights for frame [MATH] for the backhaul link by [MATH] and for the relaying link by [MATH], where [MATH].', '1606.06490-2-24-5': 'Recall that the performance of the two-hop relaying system is subject to the bottleneck link which can be either the backhaul or the relaying link.', '1606.06490-2-24-6': 'Thus, for a given selection of the weights [MATH] the coding rate [MATH] of frame [MATH] is determined based on the bottleneck link: [MATH].', '1606.06490-2-25-0': 'Our aim is to determine - per frame - the optimal scheduling weights (of the backhaul link and the relaying link) for coding rate scheduling which maximizes the average FBL throughput while guaranteeing the reliability of transmissions.', '1606.06490-2-25-1': 'Therefore, the optimization problem actually equals to maximize the expected FBL throughput per frame by solving the following optimization problem: [EQUATION]', '1606.06490-2-25-2': 'For this optimization problem, note that the space of feasible solutions for the scheduling weights is restricted in the following way: We are interested in reliable transmission, i.e. we restrict the transmission to the reliability constraint [MATH].', '1606.06490-2-25-3': 'Then, according to [REF] we have [MATH], which results thus in the space [MATH].', '1606.06490-2-26-0': 'Under this constraint, the following proposition can be shown with respect to the scheduling of the weights for the considered relay system:', '1606.06490-2-27-0': 'For a relay network operating on outdated CSI, if the coding rate for frame [MATH] is scheduled according to [MATH], [MATH], the expected FBL throughput of the upcoming frame [MATH], [MATH], is concave in the coding rate [MATH].', '1606.06490-2-28-0': 'See Appendix A.', '1606.06490-2-29-0': 'Recall that the coding rate is chosen by the source based on [MATH].', '1606.06490-2-29-1': 'Due to [REF], the coding rate is strictly increasing in [MATH] and therefore increasing in [MATH] or [MATH].', '1606.06490-2-29-2': 'In combination with Proposition 1, we thus obtain an important corollary regarding the optimal scheduling of the system:', '1606.06490-2-30-0': 'For a relay network operating on outdated CSI, if the coding rate for frame [MATH] is scheduled according to [MATH], [MATH], [MATH] the expected FBL throughput of frame [MATH] is quasi-concave in [MATH] in the region [MATH] and quasi-concave in [MATH] in the region [MATH].', '1606.06490-2-31-0': 'See Appendix B.', '1606.06490-2-32-0': 'According to Corollary 1, the expected FBL throughput of frame [MATH] can be optimized by applying quasi-convex optimization techniques, e.g., backtracking line search, to obtain the optimal weights for determining the coding rate.', '1606.06490-2-32-1': 'Nevertheless, this can be computationally heavy, as this optimization step needs to be conducted prior to each data transmission.', '1606.06490-2-32-2': 'Note in this context that the smaller the reliability requirement is, the smaller is also the search space of the scheduling weights, making it more likely that for a given instance the optimal solution is on the boundary of the feasible set.', '1606.06490-2-32-3': 'Still, in order to reach the optimal system performance, some computations need to be executed prior to each frame.', '1606.06490-2-33-0': '## Constant Weight Heuristic', '1606.06490-2-34-0': 'To further reduce the computational complexity, in this section we consider scheduling schemes where the weight is not adapted per frame.', '1606.06490-2-34-1': 'Once the scheduling weights are determined at system initialization (depending on the average SNR and the correlation coefficients) they remain constant during all frames.', '1606.06490-2-34-2': 'We are interested in determining the constant heuristic with the best performance.', '1606.06490-2-35-0': 'Denote these constant weights by [MATH] and [MATH] for the backhaul and relaying link.', '1606.06490-2-35-1': 'Then, the coding rate for frame [MATH] under the constant weight scheme is subject to the instantaneous SNR and the constant weights.', '1606.06490-2-35-2': 'As a result, obviously the coding rate is not constant over different frames.', '1606.06490-2-35-3': 'In particular, the coding rate [MATH] of frame [MATH] is obtained by: [MATH].', '1606.06490-2-35-4': 'According to [REF], the coding rate [MATH] is strictly increasing in [MATH] and therefore monotonically increasing in [MATH] and [MATH].', '1606.06490-2-35-5': 'Thus, under this constant weight scheme, the average FBL throughput can be determined by: LLL', '1606.06490-2-36-0': '&_FBL (_1,_2) = E_r_i [ C_FBL(r_i) ]', '1606.06490-2-37-0': '= & _0^ _0^ C_FBL( R( _1_1,_2_2 ,_th, m) ) e^ - _1_1 - _2_2d_1d_2', '1606.06490-2-38-0': '=& 1_1_2_0^__1_1_2_2^ C_FBL( R(_1_1,_th, m) ) e^ - _1_1 - _2_2d_2d_1', '1606.06490-2-39-0': '& + 1_1_2_0^__2_2_1_1^ C_FBL( R(_2_2,_th, m) ) e^ - _1_1 - _2_2d_1d_2.', '1606.06490-2-40-0': 'Under the best constant heuristic, the aim is to maximize the average FBL throughput while the constraint is to guarantee the average error probability over time .', '1606.06490-2-40-1': 'Then, the resulting optimization problem is given by: [EQUATION]', '1606.06490-2-40-2': 'As we assume [MATH], we have [MATH].', '1606.06490-2-40-3': 'Therefore, the feasible set of [MATH] is [MATH] under the case [MATH] and covers a subset of [MATH] when [MATH].', '1606.06490-2-41-0': 'Denote [MATH] and [MATH] as the solution to the above optimization problem, i.e. they are the optimal, constant scheduling weights.', '1606.06490-2-41-1': 'We then have the following proposition:', '1606.06490-2-42-0': 'Considering a relay network operating on outdated CSI with constant scheduling weights, if the coding rate of each frame [MATH] is scheduled according to [MATH], then the average FBL throughput [MATH] is quasi-concave in [MATH] in the region [MATH] and quasi-concave in [MATH] in the region [MATH].', '1606.06490-2-43-0': 'See Appendix C.', '1606.06490-2-44-0': 'According to Proposition 2, [REF] can be efficiently solved by applying quasi-convex optimization techniques.', '1606.06490-2-44-1': 'For a relay system with a certain set of average SNRs and correlation coefficients as well as a given reliability constraint, we obtain a unique pair of fixed scheduling weights.', '1606.06490-2-44-2': 'Note that these fixed weights are then strictly applied per frame, leading to a varying coding rate that maximizes the long-term average FBL throughput (under the assumption of using fixed weights).', '1606.06490-2-44-3': 'This reduces drastically the computational complexity, but leads to an inferior system performance in comparison to the optimal scheduling scheme with adaptive scheduling weights, i.e. the optimal solution presented in Section [REF].', '1606.06490-2-45-0': '# Numerical Evaluation and Discussion', '1606.06490-2-46-0': 'In this section, we present some numerical results regrading the throughput maximization in relay systems.', '1606.06490-2-46-1': 'We consider in particular two issues: Initially, we study several aspects of the quasi-convexity of the FBL throughput with respect to the scheduling weights.', '1606.06490-2-46-2': 'In particular, we are interested in the sharpness of the optimum.', '1606.06490-2-46-3': 'This investigation is important for practical system design, as it clarifies the potential cost of non-optimal weight selection.', '1606.06490-2-46-4': 'After clarifying these issues, we move to a more general performance investigation.', '1606.06490-2-46-5': 'Here, we are especially interested in the performance comparison between the optimal scheduling scheme (with changing scheduling weights per frame) and the low-complexity best constant heuristic.', '1606.06490-2-47-0': 'From a methodological point of view, all following numerical results are based on simulations.', '1606.06490-2-47-1': 'We consider a basic scenario for these simulations with the following parameterization: We assume an urban outdoor scenario where the distances of the backhaul and relaying link are both set to 100 [MATH].', '1606.06490-2-47-2': 'For channel propagation, we utilize the well-known COST [CITATION] model (which is a commonly-used model for urban scenarios) for calculating the path loss.', '1606.06490-2-47-3': 'The center frequency is set to to 2 [MATH] while the transmit power [MATH] is selected to 35 [MATH] (we vary the transmit power in Figures [REF] and [REF]) considering a noise power of -90 [MATH], respectively.', '1606.06490-2-47-4': 'Lastly, the blocklength at each hop of relaying is set to [MATH] symbols.', '1606.06490-2-47-5': 'Recall that the channel correlation coefficients [MATH] and [MATH] of the backhaul and relaying links are subject to the settings of [MATH], [MATH] and the Doppler frequency.', '1606.06490-2-47-6': 'In particular, [MATH] as [MATH], i.e., the CSI of the relaying phase is more delayed.', '1606.06490-2-47-7': ""In the simulation, we don't set a fixed value for either the length of initialization phase [MATH] or the Doppler frequency."", '1606.06490-2-47-8': 'Instead, we consider different setups of [MATH] and [MATH], which corresponds to different settings of [MATH] and the Doppler frequency as [MATH] is fixed, while [MATH] holds for all setups.', '1606.06490-2-48-0': '## Quasi-Convexity of the FBL Throughput', '1606.06490-2-49-0': 'In this subsection, we consider numerical results regarding the quasi-convexity of the average FBL throughput.', '1606.06490-2-49-1': 'We first study the relationship between the expected FBL throughput of an upcoming frame and the choice of scheduling weights (based on the corresponding outdated CSI) in case of the optimal scheduling scheme that adapts the weights per frame.', '1606.06490-2-49-2': 'In order to do so, we fix the outdated CSI and generate realizations of the corresponding instantaneous channel states.', '1606.06490-2-49-3': 'Then, we study the expected FBL throughput (the expectation/average over all realizations) by varying the scheduling weights.', '1606.06490-2-49-4': 'The results are shown in Figure [REF].', '1606.06490-2-50-0': 'First of all, the figure illustrates that the FBL throughput per frame is quasi-concave in the scheduling weights [MATH] and [MATH].', '1606.06490-2-50-1': 'Hence, by choosing appropriate values for [MATH] and [MATH] the FBL throughput can be optimized.', '1606.06490-2-50-2': 'Secondly, the figure also shows that the expected FBL throughput of the upcoming frame is actually subject to both scheduling weights [MATH] and [MATH] in general.', '1606.06490-2-50-3': 'However, if [MATH] is chosen optimally, then the choice of [MATH] can be arbitrarily large (but not arbitrarily small).', '1606.06490-2-50-4': 'This stems essentially from the fact how the scheduling weights influence the bottleneck link.', '1606.06490-2-50-5': 'The case ""FBL throughput being only influenced by [MATH]"" corresponds to the situation where the bottleneck link (for determining the coding rate) is the backhaul link.', '1606.06490-2-50-6': 'At the same time, as long as [MATH] (the scheduling weight of the relaying link) is not set to a very small value, the impact on the SNR of the backhaul link is considerably small and therefore does not influence the coding rate.', '1606.06490-2-50-7': ""In other words, there is no impact of a link's scheduling weight on the FBL throughput of the upcoming frame if this link is not the bottleneck link."", '1606.06490-2-50-8': 'Obviously, reducing the scheduling weight of a link likely makes this link become the bottleneck link eventually.', '1606.06490-2-50-9': 'As a consequence of this dependence between [MATH] and [MATH], we observe that there are multiple solutions maximizing the FBL throughput surface for the considered channel setting.', '1606.06490-2-51-0': 'We next study the quasi-convexity of the optimal scheduling for scenarios with different channel correlation setups.', '1606.06490-2-51-1': 'The results are shown in Figure [REF] where we fix [MATH] to 0.7, i.e., make the backhaul link the bottleneck and vary [MATH].', '1606.06490-2-51-2': 'The figure reveals that a stronger channel correlation results in a higher optimal FBL throughput.', '1606.06490-2-51-3': 'More importantly, this higher maximum is achieved by a bigger scheduling weight.', '1606.06490-2-51-4': 'In other words, a strong channel correlation allows us to set the scheduling weight more aggressively, leading to a higher coding rate and a higher FBL throughput.', '1606.06490-2-52-0': 'We now turn to the constant heuristic, where the scheduling weight is determined once for the given system and then left constant for each frame.', '1606.06490-2-52-1': 'In Figure [REF], we show the relationship between the average FBL throughput [MATH] and the constant scheduling weights [MATH] and [MATH] while generating many different outdated channel instances and the corresponding instantaneous channel state realizations.', '1606.06490-2-53-0': 'Firstly, the figure confirms again our analytical insight (Proposition 2), i.e. [MATH] is quasi-concave in [MATH] or [MATH].', '1606.06490-2-53-1': 'In addition, we observe that a near-optimal FBL throughput is achieved for a large set of different scheduling weights, e.g., a small error of the optimal solution does not change the average FBL throughput too much.', '1606.06490-2-53-2': 'Hence, the FBL throughput in the case of the constant scheduling weights is somewhat robust to an erroneous choice of the weights.', '1606.06490-2-53-3': 'Similar to the optimal scheduling, we further study the average FBL throughput of the best constant heuristic with different channel correlation coefficients.', '1606.06490-2-53-4': 'The results are provided in Figure [REF].', '1606.06490-2-53-5': 'It is shown that under the best constant heuristic scheme a strong channel correlation also introduces a higher FBL throughput attained for larger scheduling weights.', '1606.06490-2-53-6': 'Nevertheless, note that the throughput difference is smaller when comparing the throughput for small and large channel correlations in case of the constant scheduling weights in comparison to the optimal, adaptive choice of the scheduling weights per frame (Figure [REF]).', '1606.06490-2-54-0': 'We conclude the discussion regarding the quasi-convexity by summarizing the following guidelines for choosing the scheduling weights: Firstly, in general the optimal weights are lower than [MATH] even for channels with high correlation coefficients.', '1606.06490-2-54-1': 'Secondly, in comparison to a weak channel correlation, a strong one allows us to set a relatively bigger scheduling weight.', '1606.06490-2-54-2': 'Thirdly, it is important to have an accurate characterization of the channel correlation, otherwise the FBL throughput can be significantly reduced.', '1606.06490-2-54-3': 'In particular, the optimal scheduling scheme is more sensitive to an inaccurate knowledge of the channel correlations than the constant weight heuristic.', '1606.06490-2-54-4': 'Furthermore, a low error probability constraint leads to a small feasible set for choosing the scheduling weights.', '1606.06490-2-54-5': 'Finally, it appears that for constant weight scheduling, the choice of the weights is less sensitive to wrong choices, especially if these choices end up being too large.', '1606.06490-2-54-6': 'In the case of the optimal scheduling, this only applies to cases where one of the link weights is set optimally.', '1606.06490-2-55-0': '## Optimal vs. Constant Scheduling', '1606.06490-2-56-0': 'In this subsection, we focus on investigating the performance gap between the two schemes presented in Section [REF] and [REF] for a set of variable parameters with respect to the SNR, reliability constraint and channel correlation coefficient.', '1606.06490-2-56-1': 'To start with, we show the FBL throughput of the two schemes versus the average channel SNR while considering different settings of the reliability threshold [MATH].', '1606.06490-2-57-0': 'The results are shown in Figure [REF], where the average FBL throughputs are based on the optimal/sub-optimal choice of coding rate under either the optimal scheduling or the best constant heuristic.', '1606.06490-2-57-1': 'Firstly, we observe that the lower the reliability threshold is, the lower the optimal average FBL throughput is.', '1606.06490-2-57-2': 'Secondly, a higher SNR also increases the gap between the optimal scheduling and the constant heuristic.', '1606.06490-2-57-3': 'Lastly, a lower reliability constraint [MATH] leads to a significantly bigger gap between the two schemes.', '1606.06490-2-57-4': 'For instance, the gap is quite big under the constraint [MATH] while it is small when [MATH].', '1606.06490-2-57-5': 'This suggests that it only pays off to spend the additional computational complexity for the optimal scheduling scheme in case of a high reliability constraint (i.e. a rather low requirement on the error probability).', '1606.06490-2-57-6': 'In case of a rather low reliability constraint, there is no big difference between the two scheduling schemes.', '1606.06490-2-57-7': 'This is essentially due to the fact that in case of a high reliability constraint the constant heuristic needs to select a proportionally lower value for the scheduling weight to fulfill the reliability constraint even in cases where the instantaneous channel state drops significantly below the outdated CSI.', '1606.06490-2-57-8': 'In case of the optimal scheduling, this can be compensated for by frame-specific scheduling of the weights.', '1606.06490-2-58-0': 'Finally, we show in Figure [REF] the average FBL throughput of the two different schemes regarding different channel correlation coefficients while the reliability constraint is fixed at [MATH].', '1606.06490-2-59-0': 'We observe that there is a big loss in comparison to the Shannon capacity, even for the FBL throughput with a strong channel correlation.', '1606.06490-2-59-1': 'In addition, a stronger channel correlation introduces a higher FBL throughputs for both the optimal scheduling and the constant heuristic.', '1606.06490-2-59-2': 'Furthermore, in the high SNR region the performance gap between the two schemes is less influenced by the channel correlation, e.g., at point SNR=25 dB the gap between the two schemes of the case [MATH] is quite similar to the gaps under the other two cases.', '1606.06490-2-59-3': 'This is due to the fact that a strong channel correlation makes the outdated CSI more accurate, which reduces the importance of choosing the best scheduling weight.', '1606.06490-2-59-4': 'As a result, the performance gap between the optimal and the heuristic scheduling schemes is relatively constant.', '1606.06490-2-59-5': 'On the other hand, in the low SNR region the gap between the two schemes is slightly bigger for the case with a strong channel correlation.', '1606.06490-2-60-0': 'Combining the insights from Figure [REF] and Figure [REF], we can conclude that the performance gap between the proposed optimal and heuristic scheduling schemes mainly depends on the error probability threshold and the average channel SNR, while it is only marginally influenced by the channel correlation coefficients.', '1606.06490-2-60-1': 'Thus, it is perhaps only worth to spend the computational complexity of the optimal scheme in case of high reliability constraints and a rather high average SNR.', '1606.06490-2-61-0': '# Conclusion', '1606.06490-2-62-0': 'In this work, we study the finite blocklength performance of relaying with outdated CSI.', '1606.06490-2-62-1': 'Both an optimal and an low-complexity sub-optimal scheduling scheme are proposed to maximize the FBL throughput while satisfying a reliability constraint regarding the data transmission.', '1606.06490-2-62-2': 'We show that in both cases an optimal performance can be obtained by exploiting the quasi-concavity of the FBL throughput with respect to scheduling weights.', '1606.06490-2-62-3': 'By numerical analysis, we conclude a set of guidelines for the design of efficient relaying systems in the FBL regime.', '1606.06490-2-62-4': 'Firstly, it is important to have accurate channel correlation information, otherwise the inaccurate channel correlation coefficients can reduce the throughput.', '1606.06490-2-62-5': 'In particular, the optimal scheme is more sensitive regarding the accuracy of the channel correlation.', '1606.06490-2-62-6': 'Secondly, the optimal scheme is more sensitive to erroneous selection of the scheduling weights in comparison to the constant scheme.', '1606.06490-2-62-7': 'Thus, in practice a precise computation of the scheduling weights in case of the optimal scheme needs to be performed which nevertheless only pays off in certain scenarios.', '1606.06490-2-62-8': 'For the constant scheme, a less accurate computation of the optimal weights leads already to a satisfactory performance in particular if the scheduling weights are chosen rather too large versus too small.', '1606.06490-2-62-9': 'Thirdly, the performance gap between the proposed two schemes depends mainly on the reliability constraint regarding the data transmissions.', '1606.06490-2-62-10': 'The stricter this constraint is, the more does the optimal scheme outperform the constant scheme.', '1606.06490-2-62-11': 'Finally, the performance gap between the two schemes is less influenced by the channel correlation coefficients.', '1606.06490-2-63-0': '# Proof of Proposition 1', '1606.06490-2-64-0': 'Based on [REF] and [REF], we immediately have [MATH].', '1606.06490-2-64-1': 'Therefore, the first and second derivatives of [MATH] with respect to [MATH] are given by: [EQUATION]', '1606.06490-2-64-2': 'In the following, we prove Proposition 1 by showing [MATH].', '1606.06490-2-65-0': 'Recall that [MATH].', '1606.06490-2-65-1': 'According to [REF], we have: [EQUATION]', '1606.06490-2-65-2': 'Based on [REF], we have: [EQUATION]', '1606.06490-2-65-3': 'As [MATH], the following inequality holds: [MATH].', '1606.06490-2-65-4': 'Hence, [MATH] and therefore [MATH] during the intervals [MATH] and [MATH].', '1606.06490-2-65-5': 'As [MATH], hence we have: [MATH] and [MATH].', '1606.06490-2-66-0': 'Considering equation [REF], we have: [EQUATION]', '1606.06490-2-66-1': 'So far, it has been shown that [EQUATION]', '1606.06490-2-66-2': 'According to [REF], the error probability of a link is higher than 0.5 only if the coding rate is higher than the Shannon capacity.', '1606.06490-2-66-3': 'Recall that the coding rate chosen by the source satisfies [MATH].', '1606.06490-2-66-4': 'This makes the expected error probability of each single link (during frame [MATH]) be lower than 0.5, i.e., [MATH].', '1606.06490-2-66-5': 'Based on [REF], we have: [EQUATION]', '1606.06490-2-66-6': 'Hence, [MATH] if [MATH].', '1606.06490-2-67-0': 'According to the Cauchy–Schwarz inequality, we have [MATH].', '1606.06490-2-68-0': 'Hence, we have:', '1606.06490-2-69-0': '[MATH],', '1606.06490-2-70-0': 'where [MATH] and [MATH].', '1606.06490-2-70-1': 'There exists a positive constant [MATH], which makes [MATH].', '1606.06490-2-70-2': 'Same to the discussion in [REF] and [REF], it holds that:', '1606.06490-2-71-0': '[MATH].', '1606.06490-2-71-1': 'Hence, [MATH].', '1606.06490-2-71-2': 'As [MATH], [MATH] is concave in [MATH].', '1606.06490-2-72-0': '# Proof of Corollary 1', '1606.06490-2-73-0': 'The coding rate of frame [MATH] is scheduled based on the outdated CSI of the bottleneck link, i.e., [MATH].', '1606.06490-2-73-1': 'According to [REF], we know that [MATH] is strictly increasing in [MATH].', '1606.06490-2-73-2': 'Hence, when the source schedules the coding rate [MATH], high values of [MATH] and [MATH] lead to a big [MATH].', '1606.06490-2-73-3': 'In other words, [MATH] is monotonically increasing in [MATH].', '1606.06490-2-74-0': '[MATH], [MATH] and [MATH], we have [MATH], where [MATH].', '1606.06490-2-75-0': 'Based on Proposition 1, [MATH] is concave in [MATH],', '1606.06490-2-76-0': '[MATH].', '1606.06490-2-77-0': '[MATH] is quasi-concave in [MATH], where [MATH] and [MATH].', '1606.06490-2-78-0': '# Proof of Proposition 2', '1606.06490-2-79-0': 'According to the proof of Corollary 1, [MATH], [MATH] is monotonically increasing in [MATH].', '1606.06490-2-80-0': '[MATH], [MATH], [MATH] and [MATH], we have [MATH], where [MATH].', '1606.06490-2-81-0': 'As shown in Proposition 1, [MATH] is concave in [MATH].', '1606.06490-2-81-1': 'Hence, [MATH] is concave in [MATH].', '1606.06490-2-82-0': '[MATH].', '1606.06490-2-83-0': '[MATH] is quasi-concave in [MATH], where [MATH] and [MATH].'}","[['1606.06490-1-71-3', '1606.06490-2-73-3'], ['1606.06490-1-69-1', '1606.06490-2-66-1'], ['1606.06490-1-69-2', '1606.06490-2-66-2'], ['1606.06490-1-69-3', '1606.06490-2-66-3'], ['1606.06490-1-69-5', '1606.06490-2-66-5'], ['1606.06490-1-15-2', '1606.06490-2-14-2'], ['1606.06490-1-23-4', '1606.06490-2-20-1'], ['1606.06490-1-3-1', '1606.06490-2-3-1'], ['1606.06490-1-67-0', '1606.06490-2-64-0'], ['1606.06490-1-28-0', '1606.06490-2-29-0'], ['1606.06490-1-14-4', '1606.06490-2-12-5'], ['1606.06490-1-14-10', '1606.06490-2-12-12'], ['1606.06490-1-8-0', '1606.06490-2-7-0'], 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'1606.06490-1-61-4', '1606.06490-1-68-0', '1606.06490-1-69-6', '1606.06490-1-72-0', '1606.06490-1-73-0', '1606.06490-1-74-0', '1606.06490-1-75-0', '1606.06490-1-78-0', '1606.06490-1-80-0', '1606.06490-1-81-0', '1606.06490-2-1-0', '1606.06490-2-21-0', '1606.06490-2-26-0', '1606.06490-2-28-0', '1606.06490-2-29-2', '1606.06490-2-31-0', '1606.06490-2-36-0', '1606.06490-2-37-0', '1606.06490-2-38-0', '1606.06490-2-39-0', '1606.06490-2-41-1', '1606.06490-2-43-0', '1606.06490-2-65-0', '1606.06490-2-65-1', '1606.06490-2-65-2', '1606.06490-2-65-3', '1606.06490-2-65-4', '1606.06490-2-65-5', '1606.06490-2-66-6', '1606.06490-2-67-0', '1606.06490-2-68-0', '1606.06490-2-69-0', '1606.06490-2-70-0', '1606.06490-2-70-1', '1606.06490-2-70-2', '1606.06490-2-71-0', '1606.06490-2-71-1', '1606.06490-2-71-2', '1606.06490-2-74-0', '1606.06490-2-75-0', '1606.06490-2-76-0', '1606.06490-2-77-0', '1606.06490-2-80-0', '1606.06490-2-82-0', '1606.06490-2-83-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1606.06490,,,
0803.2583,"{'0803.2583-1-0-0': 'We establish, for a generically big Hermitian line bundle, the convergence of truncated Harder-Narasimhan polygons and the uniform continuity of the limit.', '0803.2583-1-0-1': 'As an application, we generalize several results of Moriwaki concerning arithmetically big Hermitian line bundles to the case of continuous metrics.', '0803.2583-1-0-2': 'Furthermore, we prove a conjecture of him asserting that the arithmetic volume function is actually a limit instead of a sup-limit.', '0803.2583-1-1-0': '# Introduction', '0803.2583-1-2-0': 'Let [MATH] be a number field and [MATH] be its integer ring.', '0803.2583-1-2-1': 'Let [MATH] be a projective arithmetic variety of total dimension [MATH] over [MATH].', '0803.2583-1-2-2': 'For any Hermitian line bundle [MATH] on [MATH], the arithmetic volume of [MATH] introduced by Moriwaki (see [CITATION]) is [EQUATION]where [MATH].', '0803.2583-1-2-3': 'The Hermitian line bundle [MATH] is said to be arithmetically big if [MATH].', '0803.2583-1-2-4': 'The notion of arithmetic bigness had been firstly introduced by Moriwaki [CITATION] 2 in a different form.', '0803.2583-1-2-5': 'Recently he himself ([CITATION] Theorem 4.5) and Yuan ([CITATION] Corollary 2.4) have proved that the arithmetic bigness in [CITATION] is actually equivalent to the strict positivity of the arithmetic volume function [REF], at least in the smooth metric case.', '0803.2583-1-2-6': 'In [CITATION], Moriwaki has proved the continuity of [REF] with respect to [MATH] and then deduced some comparisons to arithmetic intersection numbers (loc. cit.', '0803.2583-1-2-7': 'Theorem 6.2).', '0803.2583-1-3-0': 'Note that the volume function [REF] is an arithmetic analogue of the classical volume function for a line bundle on a projective variety: if [MATH] is a line bundle on a projective variety [MATH] of dimension [MATH] defined over a field [MATH], the volume of [MATH] is [EQUATION]', '0803.2583-1-3-1': 'Similarly, [MATH] is said to be big if [MATH].', '0803.2583-1-3-2': ""After Fujita's approximation theorem (see [CITATION], and [CITATION] for positive characteristic case), the sup-limit in [REF] is in fact a limit (see [CITATION] 11.4.7)."", '0803.2583-1-4-0': 'During a presentation at Institut de Mathematiques de Jussieu, Moriwaki has conjectured that, in arithmetic case, the sequence [MATH] also converges.', '0803.2583-1-4-1': 'In other words, one has [EQUATION].', '0803.2583-1-4-2': ""The strategy proposed by him is to develop an analogue of Fujita's approximation theorem in arithmetic setting (see [CITATION] Remark 5.7)."", '0803.2583-1-5-0': ""In this article, we prove Moriwaki's conjecture by establishing a convergence result of Harder-Narasimhan polygons (Theorem [REF]), which is a generalization of the author's previous work [CITATION] where the main tool is the Harder-Narasimhan filtration (indexed by [MATH]) of a Hermitian vector bundle on [MATH] and its associated Borel measure."", '0803.2583-1-5-1': 'To apply the convergence of polygons, the main idea is to compare [MATH], defined as the logarithm of number of effective points in [MATH], to the positive degree [MATH], which is the maximal value of the Harder-Narasimhan polygon of [MATH].', '0803.2583-1-5-2': 'Here [MATH] denotes a Hermitian vector bundle on [MATH].', '0803.2583-1-5-3': 'We show that the arithmetic volume function coincides with the limit of normalized positive degrees and therefore prove the conjecture.', '0803.2583-1-6-0': 'In [CITATION] and [CITATION], the volume function and the arithmetic bigness have been studied under the smoothness assumption for metrics of [MATH].', '0803.2583-1-6-1': 'The important (analytical) technic used by both authors is the estimation of the distortion function, which has already appeared in [CITATION].', '0803.2583-1-6-2': 'The approach in the present work, which is similar to that in [CITATION], relies on purely algebraic arguments and allows to study the arithmetic bigness without regularity condition: the continuity of metrics suffices.', '0803.2583-1-6-3': 'We also establish an explicit link between the volume function and some geometric invariants of [MATH] such as asymptotic slopes, which permits us to prove that [MATH] is big if and only if the norm of the smallest non-zero section of [MATH] decreases exponentially when [MATH] tends to infinity.', '0803.2583-1-6-4': 'This result is analogous to Theorem 4.5 of [CITATION] or Corollary 2.4 (1)[MATH](4) of [CITATION] except that we avoid using analytical methods and hence relax the smoothness condition to continuity.', '0803.2583-1-7-0': 'In our approach, the arithmetic volume function can be interpreted as the limit of maximal values of Harder-Narasimhan polygons.', '0803.2583-1-7-1': ""Inspired by Moriwaki's work [CITATION], we shall establish the uniform continuity for limit of truncated Harder-Narasimhan polygons (Theorem [REF])."", '0803.2583-1-7-2': 'This result refines considerably loc. cit.', '0803.2583-1-7-3': 'Theorem 5.4, and holds in the general setting without regularity conditions for any generically big Hermitian line bundles.', '0803.2583-1-8-0': 'The rest of this article is organized as follows.', '0803.2583-1-8-1': 'We fist recall some notation in Arakelov geometry in the second section.', '0803.2583-1-8-2': 'In the third section, we introduce the notion of positive degree for a Hermitian vector bundle on [MATH] and we compare it to the logarithm of the number of effective elements.', '0803.2583-1-8-3': 'The main tool is the Riemann-Roch inequality on [MATH] due to Gillet and Soule [CITATION].', '0803.2583-1-8-4': 'In the fourth section, we establish the convergence of the measures associated to suitably filtered section algebra of a big line bundle (Theorem [REF]).', '0803.2583-1-8-5': 'We show in the fifth section that the arithmetic bigness of [MATH] implies the classical one of [MATH], which is a generalization of a result of Yuan [CITATION].', '0803.2583-1-8-6': 'By the convergence result in the fourth section, we are able to prove that the volume of [MATH] coincides with the limit of normalized positive degrees, and therefore the sup-limit in [REF] is in fact a limit (Theorem [REF]).', '0803.2583-1-8-7': 'Here we also need the comparison result in the third section.', '0803.2583-1-8-8': 'Finally, we prove that the arithmetic bigness is equivalent to the positivity of asymptotic maximal slope (Theorem [REF]).', '0803.2583-1-8-9': 'In the last section, we establish the continuity of the limit of truncated polygons.', '0803.2583-1-9-0': 'I would like to thank J.-B. Bost for stimulating suggestions and helpful comments, also for having pointed out an error in a previous version of this article.', '0803.2583-1-9-1': 'I am also thankful to A. Moriwaki and C. Soule for discussions.', '0803.2583-1-10-0': '# Notation and reminders', '0803.2583-1-11-0': 'Throughout this article, we fix a number field [MATH] and denote by [MATH] its algebraic integer ring, and by [MATH] its discriminant.', '0803.2583-1-11-1': 'By (projective) arithmetic variety we mean an integral projective flat [MATH]-scheme.', '0803.2583-1-12-0': '## Hermitian vector bundles', '0803.2583-1-13-0': 'If [MATH] is an arithmetic variety, one calls Hermitian vector bundle on [MATH] any pair [MATH] where [MATH] is a locally free [MATH]-module, and for any embedding [MATH], [MATH] is a continuous Hermitian norm on [MATH].', '0803.2583-1-13-1': 'One requires in addition that the metrics [MATH] are invariant by the action of complex conjugation.', '0803.2583-1-13-2': 'The rank of [MATH] is just that of [MATH].', '0803.2583-1-13-3': 'If [MATH], one says that [MATH] is a Hermitian line bundle.', '0803.2583-1-13-4': 'Note that [MATH] is itself an arithmetic variety.', '0803.2583-1-13-5': 'A Hermitian vector bundle on [MATH] is just a projective [MATH]-module equipped with Hermitian norms which are invariant under complex conjugation.', '0803.2583-1-13-6': 'Let [MATH] be a real number.', '0803.2583-1-13-7': 'Denote by [MATH] the Hermitian line bundle [EQUATION] where [MATH], [MATH] being the unit of [MATH].', '0803.2583-1-14-0': '## Arakelov degree, slope and Harder-Narasimhan polygon', '0803.2583-1-15-0': 'Several invariants are naturally defined for Hermitian vector bundles on [MATH], notably the Arakelov degree, which leads to other arithmetic invariants (cf. [CITATION]).', '0803.2583-1-15-1': 'If [MATH] is a Hermitian vector bundle of rank [MATH] on [MATH], the Arakelov degree of [MATH] is defined as the real number [EQUATION] where [MATH] is an element in [MATH] which forms a basis of [MATH].', '0803.2583-1-15-2': 'This definition does not depend on the choice of [MATH].', '0803.2583-1-15-3': 'If [MATH] is non-zero, the slope of [MATH] is defined to be the quotient [MATH].', '0803.2583-1-15-4': 'The maximal slope of [MATH] is the maximal value of slopes of all non-zero Hermitian subbundles of [MATH].', '0803.2583-1-15-5': 'The minimal slope of [MATH] is the minimal value of slopes of all non-zero Hermitian quotients of [MATH].', '0803.2583-1-15-6': 'We say that [MATH] is semistable if [MATH].', '0803.2583-1-16-0': 'Recall that the Harder-Narasimhan polygon [MATH] is by definition the concave function defined on [MATH] whose graph is the convex hull of points of the form [MATH], where [MATH] runs over all Hermitian subbundles of [MATH].', '0803.2583-1-16-1': 'By works of Stuhler [CITATION] and Grayson [CITATION], this polygon can be determined from the Harder-Narasimhan flag of [MATH], which is the only flag [EQUATION] such that the subquotients [MATH] are all semistable, and verifies [EQUATION]', '0803.2583-1-16-2': 'In fact, the vertices of [MATH] are just [MATH].', '0803.2583-1-17-0': 'For details about Hermitian vector bundles on [MATH], see [CITATION].', '0803.2583-1-18-0': '## Reminder on Borel measures', '0803.2583-1-19-0': 'Denote by [MATH] the space of all continuous functions of compact support on [MATH].', '0803.2583-1-19-1': 'Recall that a Borel measure on [MATH] is just a positive linear functional on [MATH], where the word ""positive"" means that the linear functional sends a positive function to a positive number.', '0803.2583-1-19-2': 'One says that a sequence [MATH] of Borel measures on [MATH] converges vaguely to the Borel measure [MATH] if, for any [MATH], the sequence of integrals [MATH] converges to [MATH].', '0803.2583-1-19-3': 'This is also equivalent to the convergence of integrals for any [MATH], the space of all smooth functions of compact support on [MATH].', '0803.2583-1-20-0': 'Let [MATH] be a Borel probability measure on [MATH].', '0803.2583-1-20-1': 'If [MATH], we denote by [MATH] the Borel measure such that [MATH].', '0803.2583-1-20-2': 'If [MATH], let [MATH] be the Borel measure such that [MATH].', '0803.2583-1-21-0': 'If [MATH] is a Borel probability measure on [MATH] whose support is bounded from above, we denote by [MATH] the Legendre transformation (see [CITATION] II 2.2) of the function [MATH].', '0803.2583-1-21-1': 'It is a concave function on [MATH] which takes value [MATH] at the origin.', '0803.2583-1-21-2': 'If [MATH] is a linear combination of Dirac measures, then [MATH] is a polygon (that is to say, concave and piecewise linear).', '0803.2583-1-21-3': 'An alternative definition of [MATH] is, if we denote by [MATH], then [MATH].', '0803.2583-1-21-4': 'One has [MATH] and [MATH].', '0803.2583-1-22-0': 'If [MATH] and [MATH] are two Borel probability measures on [MATH], we use the symbol [MATH] or [MATH] to denote the following condition:', '0803.2583-1-23-0': 'for any increasing and bounded function [MATH], [MATH].', '0803.2583-1-24-0': 'It defines an order on the set of all Borel probability measures on [MATH].', '0803.2583-1-24-1': 'If in addition [MATH] and [MATH] are of support bounded from above, then [MATH].', '0803.2583-1-25-0': '## Filtered spaces Let [MATH] be a field and [MATH] be a vector space of finite rank over [MATH].', '0803.2583-1-25-1': 'We call filtration of [MATH] any family [MATH] of subspaces of [MATH] subject to the following conditions [1)] for all [MATH] such that [MATH], one has [MATH], [MATH] for [MATH] sufficiently positive, [MATH] for [MATH] sufficiently negative, the function [MATH] is left continuous.', '0803.2583-1-26-0': 'Such filtration corresponds to a flag [EQUATION] together with a strictly increasing real sequence [MATH] describing the points where the function [MATH] is discontinuous.', '0803.2583-1-27-0': 'We define a function [MATH] as follows: [EQUATION].', '0803.2583-1-27-1': 'This function actually takes values in [MATH], and is finite on [MATH].', '0803.2583-1-28-0': 'If [MATH] is non-zero, the filtered space [MATH] defines a Borel probability measure [MATH] which is a linear combination of Dirac measures: [EQUATION].', '0803.2583-1-28-1': 'Note that the support of [MATH] is just [MATH].', '0803.2583-1-28-2': 'We define [MATH] and [MATH].', '0803.2583-1-28-3': 'Denote by [MATH] the polygon [MATH].', '0803.2583-1-28-4': 'If [MATH], by convention we define [MATH] as the zero measure.', '0803.2583-1-29-0': 'If [MATH] is an exact sequence of filtered vector spaces, where [MATH], then the following equality holds (cf. [CITATION] Proposition 1.2.5): [EQUATION]', '0803.2583-1-29-1': 'If [MATH] is a non-zero Hermitian vector bundle on [MATH], then the Harder-Narasimhan flag [REF] and the successive slope [REF] defines a filtration of [MATH], called the Harder-Narasimhan filtration.', '0803.2583-1-29-2': 'We denote by [MATH] the Borel measure associated to this filtration, called the measure associated to the Hermitian vector bundle [MATH].', '0803.2583-1-29-3': 'One has the following relations: [EQUATION]', '0803.2583-1-29-4': 'For details about filtered spaces and their measures and polygons, see [CITATION] 1.2.', '0803.2583-1-30-0': '## Slope inequality and its measure form', '0803.2583-1-31-0': 'For any maximal ideal [MATH] of [MATH], denote by [MATH] the completion of [MATH] with respect to the [MATH]-adic valuation [MATH] on [MATH], and by [MATH] be the [MATH]-adic absolute value such that [MATH].', '0803.2583-1-32-0': 'Let [MATH] and [MATH] be two Hermitian vector bundles on [MATH].', '0803.2583-1-32-1': 'Let [MATH] be a non-zero [MATH]-linear homomorphism.', '0803.2583-1-32-2': 'For any maximal ideal [MATH] of [MATH], let [MATH] be the norm of the linear mapping [MATH].', '0803.2583-1-32-3': 'Similarly, for any embedding [MATH], let [MATH] be the norm of [MATH].', '0803.2583-1-32-4': 'The height of [MATH] is then defined as [EQUATION]', '0803.2583-1-32-5': 'Recall the slope inequality as follows (cf. [CITATION] Proposition 4.3):', '0803.2583-1-33-0': 'The following estimation generalizing [CITATION] Corollary 2.2.6 is an application of the slope inequality.', '0803.2583-1-34-0': 'We equip [MATH] and [MATH] with Harder-Narasimhan filtrations.', '0803.2583-1-34-1': 'The slope inequality implies that [MATH] for any [MATH] (see [CITATION] Proposition 2.2.4).', '0803.2583-1-34-2': 'Let [MATH] be the image of [MATH], equipped with induced filtration.', '0803.2583-1-34-3': 'By [CITATION] Corollary 2.2.6, [MATH].', '0803.2583-1-34-4': 'By [REF], [MATH], so the proposition is proved.', '0803.2583-1-35-0': '# Positive degree and number of effective elements', '0803.2583-1-36-0': 'Let [MATH] be a Hermitian vector bundle on [MATH].', '0803.2583-1-36-1': 'Define [EQUATION] which is the logarithm of the number of effective elements in [MATH].', '0803.2583-1-36-2': 'Note that if [MATH] is a short exact sequence of Hermitian vector bundles, then [MATH].', '0803.2583-1-37-0': 'In this section, we define an invariant of [MATH], suggested by J.-B. Bost, which is called the positive degree: [EQUATION].', '0803.2583-1-37-1': 'If [MATH] is non-zero, define the positive slope of [MATH] as [MATH].', '0803.2583-1-37-2': 'Using the Riemann-Roch inequality established by Gillet and Soule [CITATION], we shall compare [MATH] to [MATH].', '0803.2583-1-38-0': '## Reminder on dualizing bundle and Riemann-Roch inequality', '0803.2583-1-39-0': 'Denote by [MATH] the arithmetic dualizing bundle on [MATH]: it is a Hermitian line bundle on [MATH] whose underlying [MATH]-module is [MATH].', '0803.2583-1-39-1': 'This [MATH]-module is generated by the trace map [MATH] up to torsion.', '0803.2583-1-39-2': 'We choose Hermitian metrics on [MATH] such that [MATH] for any embedding [MATH].', '0803.2583-1-39-3': 'The arithmetic degree of [MATH] is [MATH], where [MATH] is the discriminant of [MATH] over [MATH].', '0803.2583-1-40-0': 'We recall below a result in [CITATION], which should be considered as an arithmetic analogue of classical Riemann-Roch formula for vector bundles on a smooth projective curve.', '0803.2583-1-41-0': '## Comparison of [MATH] and [MATH]', '0803.2583-1-42-0': 'Proposition [REF] below is a comparison between [MATH] and [MATH].', '0803.2583-1-42-1': 'The following lemma, which is consequences of the Riemann-Roch inequality [REF], is needed for the proof.', '0803.2583-1-43-0': 'Let [MATH] be a non-zero Hermitian vector bundle on [MATH].', '0803.2583-1-43-1': '[1)] If [MATH], then [MATH].', '0803.2583-1-43-2': 'If [MATH], then [MATH].', '0803.2583-1-43-3': 'If [MATH], then [MATH].', '0803.2583-1-44-0': '1) Assume that [MATH] has an effective section.', '0803.2583-1-44-1': 'There then exists a homomorphism [MATH] whose height is negative or zero.', '0803.2583-1-44-2': 'By slope inequality, we obtain [MATH].', '0803.2583-1-45-0': '2) Since [MATH], we have [MATH].', '0803.2583-1-45-1': 'By 1), [MATH].', '0803.2583-1-45-2': 'Thus the desired inequality results from [REF].', '0803.2583-1-46-0': '3) Let [MATH] with [MATH].', '0803.2583-1-46-1': 'Then [MATH].', '0803.2583-1-46-2': 'By 2), [MATH].', '0803.2583-1-46-3': 'Since [MATH], [MATH].', '0803.2583-1-46-4': 'So we obtain [MATH].', '0803.2583-1-46-5': 'Moreover, [REF] implies [MATH].', '0803.2583-1-46-6': 'Therefore, we always have [MATH].', '0803.2583-1-46-7': 'Since [MATH] is arbitrary, we obtain the desired inequality.', '0803.2583-1-47-0': 'Let the Harder-Narasimhan flag of [MATH] be as in [REF].', '0803.2583-1-47-1': 'For any integer [MATH] such that [MATH], let [MATH].', '0803.2583-1-47-2': 'Let [MATH] be the first index in [MATH] such that [MATH]; if such index does not exist, let [MATH].', '0803.2583-1-47-3': 'By definition, [MATH].', '0803.2583-1-47-4': 'Note that, if [MATH], then [MATH].', '0803.2583-1-47-5': 'Therefore we always have [MATH] and hence [MATH].', '0803.2583-1-48-0': 'If [MATH], then [MATH].', '0803.2583-1-48-1': 'Otherwise [MATH] and by Lemma [REF] 3), we obtain [EQUATION]', '0803.2583-1-49-0': '# Asymptotic polygon of a big line bundle', '0803.2583-1-50-0': 'Let [MATH] be a field and [MATH] be an integral graded [MATH]-algebra such that, for [MATH] sufficiently positive, [MATH] is non-zero and has finite rank.', '0803.2583-1-50-1': 'Let [MATH] be a mapping such that [MATH].', '0803.2583-1-50-2': 'Assume that each vector space [MATH] is equipped with an [MATH]-filtration [MATH] such that [MATH] is [MATH]-quasi-filtered (cf. [CITATION] 3.2.1).', '0803.2583-1-50-3': 'In other words, we assume that there exists [MATH] such that, for any integer [MATH] and all homogeneous elements [MATH] in [MATH] of degree [MATH] in [MATH], one has [EQUATION].', '0803.2583-1-50-4': 'We suppose in addition that [MATH].', '0803.2583-1-50-5': 'Recall below some results in [CITATION] (Proposition 3.2.4 and Theorem 3.4.3).', '0803.2583-1-51-0': 'In this section, we shall generalize the second assertion of Proposition [REF] to the case where the algebra [MATH] is given by global sections of tensor power of a big line bundle on a projective variety.', '0803.2583-1-52-0': '## Convergence of measures Let [MATH] be an integral projective scheme of dimension [MATH] defined over [MATH] and [MATH] be a big invertible [MATH]-module: recall that an invertible [MATH]-module [MATH] is said to be big if its volume, defined as [EQUATION] is strictly positive.', '0803.2583-1-53-0': 'With the above notation, if [MATH], then the sequence of measures [MATH] converges vaguely to a probability measure on [MATH].', '0803.2583-1-54-0': 'For any integer [MATH], denote by [MATH] the measure [MATH].', '0803.2583-1-54-1': 'Since [MATH] is big, for sufficiently positive [MATH], [MATH], and hence [MATH] is a probability measure.', '0803.2583-1-54-2': 'In addition, Proposition [REF] 1) implies that the supports of the measures [MATH] are uniformly bounded from above.', '0803.2583-1-54-3': ""After Fujita's approximation theorem (cf. [CITATION], see also [CITATION] Ch. 11), the volume function [MATH] is in fact a limit: [EQUATION]."", '0803.2583-1-54-4': 'Furthermore, for any real number [MATH], [MATH], there exists an integer [MATH] together with a finitely generated sub-[MATH]-algebra [MATH] of [MATH] which is generated by elements in [MATH] and such that [EQUATION].', '0803.2583-1-54-5': 'The graded [MATH]-algebra [MATH], equipped with induced filtrations, is [MATH]-quasi-filtered.', '0803.2583-1-54-6': 'Therefore Proposition [REF] 2) is valid for [MATH].', '0803.2583-1-54-7': 'In other words, If we denote by [MATH] the Borel measure [MATH], then the sequence of measures [MATH] converges vaguely to a Borel probability measure [MATH] on [MATH].', '0803.2583-1-54-8': 'In particular, for any function [MATH], the sequence of integrals [MATH] is a Cauchy sequence.', '0803.2583-1-54-9': 'This assertion is also true when [MATH] is a continuous function on [MATH] whose support is bounded from below: the supports of the measures [MATH] are uniformly bounded from above.', '0803.2583-1-54-10': 'The exact sequence [MATH] implies that [EQUATION].', '0803.2583-1-54-11': 'Therefore, for any bounded Borel function [MATH], one has [EQUATION]', '0803.2583-1-54-12': 'Hence, for any bounded continuous function [MATH] satisfying [MATH], there exists [MATH] such that, for any [MATH], [EQUATION]', '0803.2583-1-54-13': 'Let [MATH] be a smooth function on [MATH] whose support is compact.', '0803.2583-1-54-14': 'We choose two increasing continuous functions [MATH] and [MATH] such that [MATH] and that the supports of them are bounded from below.', '0803.2583-1-54-15': 'Let [MATH] suffciently large such that, for any [MATH], one has [MATH].', '0803.2583-1-54-16': 'We choose, for such [MATH], a non-zero element [MATH].', '0803.2583-1-54-17': 'For any [MATH] and any [MATH], let [MATH], [MATH] and denote by [MATH], [MATH], where [MATH] and [MATH] are equipped with the induced filtrations.', '0803.2583-1-54-18': 'As the algebra [MATH] is [MATH]-quasi-filtered, we obtain, by [CITATION] Lemma 1.2.6, [MATH], where [EQUATION].', '0803.2583-1-54-19': 'This implies [EQUATION]', '0803.2583-1-54-20': 'In particular, [EQUATION]', '0803.2583-1-54-21': 'As [MATH], [MATH].', '0803.2583-1-54-22': 'Moreover, [MATH].', '0803.2583-1-54-23': 'By [CITATION] Lemma 1.2.10, we obtain [EQUATION].', '0803.2583-1-54-24': 'Therefore, [EQUATION].', '0803.2583-1-54-25': 'By [REF], [MATH].', '0803.2583-1-54-26': 'Note that [EQUATION].', '0803.2583-1-54-27': 'So [EQUATION].', '0803.2583-1-54-28': 'Hence [EQUATION].', '0803.2583-1-54-29': 'According to [REF], we obtain that there exists [MATH] such that, for all integers [MATH] and [MATH] such that [MATH], [MATH], one has [EQUATION] which implies that the sequence [MATH] converges in [MATH].', '0803.2583-1-55-0': 'Let [MATH] be the linear functional defined by [MATH].', '0803.2583-1-55-1': 'It extends in a unique way to a continuous linear functional on [MATH].', '0803.2583-1-55-2': 'Furthermore, it is positive, and so defines a Borel measure [MATH] on [MATH].', '0803.2583-1-55-3': 'Finally, by [REF], [MATH].', '0803.2583-1-55-4': 'Since [MATH] is arbitrary, [MATH] is a probability measure.', '0803.2583-1-56-0': '## Convergence of maximal values of polygons', '0803.2583-1-57-0': 'If [MATH] is a Borel probability measure on [MATH] and [MATH], denote by [MATH] the Borel probability measure on [MATH] such that, for any [MATH], [EQUATION].', '0803.2583-1-57-1': 'The measure [MATH] is called the truncation of [MATH] at [MATH].', '0803.2583-1-57-2': 'The truncation operator preserves the order ""[MATH]"".', '0803.2583-1-58-0': 'Assume that the support of [MATH] is bounded from above.', '0803.2583-1-58-1': 'The truncation of [MATH] at [MATH] modifies the ""polygon"" [MATH] only on the part with derivative [MATH].', '0803.2583-1-58-2': 'More precisely, one has [EQUATION].', '0803.2583-1-58-3': 'In particular, if [MATH], then [EQUATION]', '0803.2583-1-58-4': 'The following proposition shows that given a vague convergence sequence of Borel probability measures, almost all truncations preserve vague limit.', '0803.2583-1-59-0': 'Let [MATH] be the set of all points [MATH] in [MATH] such that [MATH] has a strictly positive mass for the measure [MATH].', '0803.2583-1-59-1': 'It is a countable set.', '0803.2583-1-59-2': 'Then by [CITATION] IV 5 [MATH] Proposition 22, for any real number [MATH] outside [MATH], [MATH] converges vaguely to [MATH].', '0803.2583-1-60-0': 'Under the assumption of Theorem [REF], the sequence [EQUATION] converges in [MATH].', '0803.2583-1-61-0': 'For [MATH], denote by [MATH].', '0803.2583-1-61-1': 'By Theorem [REF], the sequence [MATH] converges vaguely to a Borel probability measure [MATH].', '0803.2583-1-61-2': 'Let [MATH] be a number such that [MATH] converges vaguely to [MATH].', '0803.2583-1-61-3': 'Note that the supports of [MATH] are uniformly bounded.', '0803.2583-1-61-4': 'So [MATH] converges uniformly to [MATH] (see [CITATION] Proposition 1.2.9).', '0803.2583-1-61-5': 'By [REF], [MATH] converges to [MATH].', '0803.2583-1-62-0': 'If [MATH] is a finite dimensional filtered vector space over [MATH], we shall use the expression [MATH] to denote [MATH].', '0803.2583-1-62-1': 'With this notation, the assertion of Corollary [REF] becomes: [MATH] exists in [MATH].', '0803.2583-1-63-0': 'Assume that [MATH] and [MATH] are two Borel probability measures whose supports are bounded from above.', '0803.2583-1-63-1': 'Let [MATH] and [MATH].', '0803.2583-1-63-2': 'Then [EQUATION]', '0803.2583-1-63-3': 'After truncation at [MATH] we may assume that the supports of [MATH] and [MATH] are contained in [MATH].', '0803.2583-1-63-4': 'In this case [MATH] and hence [MATH].', '0803.2583-1-63-5': 'Since [EQUATION] we obtain [REF].', '0803.2583-1-64-0': 'Under the assumption of Theorem [REF], one has [EQUATION].', '0803.2583-1-64-1': 'Furthermore, in this case, the inequality [MATH] holds.', '0803.2583-1-65-0': 'For any filtered vector space [MATH], [MATH] if and only if [MATH], and in this case one always has [MATH].', '0803.2583-1-65-1': 'Therefore the second assertion is true.', '0803.2583-1-65-2': 'Furthermore, this also implies [EQUATION].', '0803.2583-1-65-3': 'It suffices then to prove the converse implication.', '0803.2583-1-65-4': 'Assume that [MATH] is a real number such that [MATH].', '0803.2583-1-65-5': 'Choose sufficiently large [MATH] such that [MATH] for any [MATH] and such that there exists a non-zero [MATH] satisfying [MATH].', '0803.2583-1-65-6': 'Since the algebra [MATH] is [MATH]-quasi-filtered, [MATH].', '0803.2583-1-65-7': ""By Fujita's approximation theorem, there exists an integer [MATH] divisible by [MATH] and a subalgebra [MATH] of [MATH] generated by a finite number of elements in [MATH] and such that [MATH]."", '0803.2583-1-65-8': 'By possible enlargement of [MATH] we may assume that [MATH] contains [MATH].', '0803.2583-1-65-9': 'By Lemma [REF], [MATH] implies [MATH].', '0803.2583-1-65-10': 'Therefore, we reduce our problem to the case where [1)] [MATH] is an algebra of finite type generated by [MATH], there exists [MATH], [MATH] such that [MATH] with [MATH], [MATH].', '0803.2583-1-66-0': ""Furthermore, by Noether's normalization theorem, we may assume that [MATH] is an algebra of polynomials, where [MATH] coincides with the element in condition 2)."", '0803.2583-1-66-1': 'Note that [EQUATION]', '0803.2583-1-66-2': 'Let [MATH] such that [MATH] for any [MATH].', '0803.2583-1-66-3': 'We obtain from [REF] that [MATH] as soon as [MATH].', '0803.2583-1-66-4': 'For [MATH], let [EQUATION] and [EQUATION].', '0803.2583-1-66-5': 'Thus [MATH], which implies [MATH] by Lemma [REF].', '0803.2583-1-67-0': '# Volume function as a limit and arithmetic bigness', '0803.2583-1-68-0': 'Let [MATH] be an arithmetic variety of dimension [MATH] and [MATH] be a Hermitian line bundle on [MATH].', '0803.2583-1-68-1': 'Denote by [MATH] and [MATH].', '0803.2583-1-68-2': 'Using the convergence result established in the previous section, we shall prove that the volume function is in fact a limit of normalized positive degrees.', '0803.2583-1-68-3': 'We also give a criterion of arithmetic bigness by the positivity of asymptotic maximal slope.', '0803.2583-1-69-0': '## Volume function and asymptotic positive degree', '0803.2583-1-70-0': 'For any [MATH], we choose a Hermitian vector bundle [MATH] whose underlying [MATH]-module is [MATH] and such that [EQUATION].', '0803.2583-1-70-1': 'Denote by [MATH] the rank of [MATH].', '0803.2583-1-70-2': 'One has [MATH].', '0803.2583-1-70-3': 'For any [MATH], define [EQUATION].', '0803.2583-1-70-4': 'Recall that the arithmetic volume function of [MATH] defined by Moriwaki (cf. [CITATION]) is [EQUATION] and [MATH] is said to be big if and only if [MATH] (cf. [CITATION] Theorem 4.5 and [CITATION] Corollary 2.4).', '0803.2583-1-71-0': 'In the following, we give an alternative proof of a result of Morkwaki and Yuan without regularity condition.', '0803.2583-1-72-0': 'For any integer [MATH], we choose two Hermitian vector bundles [MATH] and [MATH] such that [EQUATION] where [MATH] is the rank of [MATH].', '0803.2583-1-72-1': 'This is always possible due to an argument of John and Lowner ellipsoid, see [CITATION] definition-theorem 2.4.', '0803.2583-1-72-2': 'Thus [MATH].', '0803.2583-1-72-3': 'Furthermore, by [CITATION] Corollay 2.2.9, [MATH].', '0803.2583-1-72-4': 'By [REF], we obtain [EQUATION].', '0803.2583-1-72-5': 'Furthermore, [MATH].', '0803.2583-1-72-6': 'Hence [EQUATION].', '0803.2583-1-72-7': 'Since [MATH], we obtain [EQUATION] and therefore [MATH].', '0803.2583-1-72-8': 'If [MATH] is big, then [MATH], and hence [MATH] for [MATH] sufficiently positive.', '0803.2583-1-72-9': 'Combining with the fact that [EQUATION] we obtain [MATH].', '0803.2583-1-73-0': 'The following equalities hold: [EQUATION] where the positive slope [MATH] was defined in [REF].', '0803.2583-1-74-0': 'We first consider the case where [MATH] is big.', '0803.2583-1-74-1': 'The graded algebra [MATH] equipped with Harder-Narasimhan filtrations is quasi-filtered for a function of logarithmic increasing speed at infinity (see [CITATION] 4.1.3).', '0803.2583-1-74-2': 'Therefore Corollary [REF] shows that the sequence [MATH] converges in [MATH].', '0803.2583-1-74-3': 'Note that [MATH].', '0803.2583-1-74-4': 'So the last limit in [REF] exists.', '0803.2583-1-74-5': 'Furthermore, [MATH] is big on [MATH], so [EQUATION] which implies the existence of the third limit in [REF] and the last equality.', '0803.2583-1-74-6': 'Thus the existence of the first limit and the second equality follow from [REF].', '0803.2583-1-75-0': 'When [MATH] is not big, since [EQUATION] the last term in [REF] vanishes.', '0803.2583-1-75-1': 'This implies the vanishing of the second limit in [REF].', '0803.2583-1-75-2': 'Also by [REF], we obtain the vanishing of the first limit.', '0803.2583-1-76-0': 'The following relations hold: [EQUATION]', '0803.2583-1-76-1': 'The inequality is a consequence of Theorem [REF] and the comparison [MATH].', '0803.2583-1-76-2': 'Here [MATH] is an arbitrary Hermitian vector bundle on [MATH].', '0803.2583-1-76-3': 'The equality follows from a classical result which compares Arakelov degree and Euler-Poincare characteristic (see [CITATION] 4.1.4 for a proof).', '0803.2583-1-76-4': 'Attention: in [CITATION], the author has adopted the convention [MATH].', '0803.2583-1-77-0': 'Corollary [REF] is a generalization of [CITATION] Theorem 6.2 to continuous metrics case.', '0803.2583-1-78-0': '## A criterion of arithmetic bigness', '0803.2583-1-79-0': 'We shall prove that the bigness of [MATH] is equivalent to the positivity of the asymptotic maximal slope of [MATH].', '0803.2583-1-79-1': 'This result is strongly analogous to Theorem 4.5 of [CITATION].', '0803.2583-1-79-2': 'In fact, by a result of Borek [CITATION] (see also [CITATION] Proposition 3.3.1), which reformulate Minkowski\'s First Theorem, the maximal slope of a Hermitian vector bundle on [MATH] is ""close"" to the opposite of the logarithm of its first minimum.', '0803.2583-1-79-3': 'So the positivity of the asymptotic maximal slope is equivalent to the existence of (exponentially) small section when [MATH] goes to infinity.', '0803.2583-1-80-0': '[MATH] is big if and only if [MATH].', '0803.2583-1-80-1': 'Furthermore, the following inequality holds: [EQUATION].', '0803.2583-1-81-0': 'Since both conditions imply the bigness of [MATH], we may assume that [MATH] is big.', '0803.2583-1-81-1': 'Let [MATH] equipped with Harder-Narasimhan filtrations.', '0803.2583-1-81-2': 'One has [EQUATION].', '0803.2583-1-81-3': 'Therefore, the assertion follows from Theorems [REF] and [REF].', '0803.2583-1-82-0': 'After [CITATION] Proposition 3.3.1, for any non-zero Hermitian vector bundle [MATH] on [MATH], one has [EQUATION]', '0803.2583-1-82-1': 'Therefore, by [REF], the bigness of [MATH] is equivalent to each of the following conditions: [1)] [MATH] is big, and there exists [MATH] such that, when [MATH] is sufficiently large, [MATH] has a global section [MATH] satisfying [MATH] for any [MATH].', '0803.2583-1-82-2': '[MATH] is big, and there exists an integer [MATH] such that [MATH] has a global section [MATH] satisfying [MATH] for any [MATH].', '0803.2583-1-83-0': 'Thus we recover a result of Moriwaki ([CITATION] Theorem 4.5 (1)[MATH](2)).', '0803.2583-1-84-0': 'Assume [MATH] is big.', '0803.2583-1-84-1': 'Then there exists a Hermitian line bundle [MATH] on [MATH] such that [MATH] is arithmetically big.', '0803.2583-1-85-0': '# Continuity of truncated asymptotic polygon', '0803.2583-1-86-0': 'Let us keep the notation of [REF] and assume that [MATH] is big on [MATH].', '0803.2583-1-86-1': 'For any integer [MATH], denote by [MATH] the dilated measure [MATH].', '0803.2583-1-86-2': 'Recall that in [REF] we have actually established the followint result:', '0803.2583-1-87-0': 'Let [MATH] be as in the proposition above.', '0803.2583-1-87-1': 'For any [MATH], denote by [MATH] the concave function [MATH] on [MATH].', '0803.2583-1-87-2': 'The following property of [MATH] results from the definition:', '0803.2583-1-88-0': 'By definition [MATH].', '0803.2583-1-88-1': 'Using [MATH], we obtain the desired equality.', '0803.2583-1-89-0': 'We deduce from the previous proposition the equality [MATH], which has been initially proved by Moriwaki ([CITATION] Proposition 4.7).', '0803.2583-1-90-0': 'The main purpose of this section is to establish the following continuity result, which is a generalization of the continuity of the arithmetic volume function proved by Moriwaki (cf. [CITATION] Theorem 5.4).', '0803.2583-1-91-0': 'Assume [MATH] is a Hermitian line bundle on [MATH].', '0803.2583-1-91-1': 'Then, for all but countably many [MATH], the sequence of functions [MATH] converges uniformly to [MATH].', '0803.2583-1-92-0': '[[CITATION] Theorem 5.4] With the assumption of Theorem [REF], one has [EQUATION].', '0803.2583-1-93-0': 'In order to prove Theorem [REF], we need the following lemma.', '0803.2583-1-94-0': 'Let [MATH] be an arbitrary Hermitian line bundle on [MATH].', '0803.2583-1-94-1': 'If [MATH] is arithmetically big, then there exists an integer [MATH] such that [MATH] is arithmetically big and has at least one non-zero effective global section, that is, a non-zero section [MATH] such that [MATH] for any embedding [MATH].', '0803.2583-1-95-0': 'As [MATH] is arithmetically big, we obtain that [MATH] is big on [MATH].', '0803.2583-1-95-1': 'Therefore, there exists an integer [MATH] such that [MATH] is big on [MATH] and [MATH].', '0803.2583-1-95-2': 'Pick an arbitrary non-zero section [MATH] and let [MATH].', '0803.2583-1-95-3': 'After Theorem [REF] (see also Remark [REF]), there exists [MATH] such that [MATH] has a section [MATH] such that [MATH] for any [MATH].', '0803.2583-1-95-4': 'Let [MATH].', '0803.2583-1-95-5': 'Then [MATH] is a non-zero strictly effective section of [MATH].', '0803.2583-1-95-6': 'Furthermore, [MATH] is arithmetically big since it is generically big and has a strictly effective section.', '0803.2583-1-96-0': '[Proof of Theorem [REF]] After Corollary [REF], we may assume that [MATH] is arithmetically big.', '0803.2583-1-96-1': 'Let [MATH] be an integer such that [MATH] is arithmetically big and has a non-zero effective section [MATH] (cf. Lemma [REF]).', '0803.2583-1-96-2': 'For any integers [MATH] and [MATH] such that [MATH], [MATH], let [MATH] be the homomorphism defined by the multiplication by [MATH].', '0803.2583-1-96-3': 'Since [MATH] is effective, [MATH].', '0803.2583-1-96-4': 'Let [EQUATION].', '0803.2583-1-96-5': 'Note that [EQUATION].', '0803.2583-1-96-6': 'Denote by [MATH] this limit.', '0803.2583-1-96-7': 'Let [MATH] be the measure associated to [MATH].', '0803.2583-1-96-8': 'Let [MATH].', '0803.2583-1-96-9': 'After Proposition [REF], one has [MATH], or equivalently [EQUATION]', '0803.2583-1-96-10': 'As [MATH] is big, the sequence of measures [MATH] converges vaguely to a Borel probability measure [MATH].', '0803.2583-1-96-11': 'By truncation and then by passing [MATH], we obtain from [REF] that for all but countably many [MATH], [EQUATION] where we have used the trivial estimation [MATH].', '0803.2583-1-97-0': 'Now we apply Lemma [REF] on the dual Hermitian line bundle [MATH] and obtain that there exists an integer [MATH] and an effective section [MATH] of [MATH].', '0803.2583-1-97-1': 'Consider now the homomorphism [MATH] induced by multiplication by [MATH].', '0803.2583-1-97-2': 'Its height is negative.', '0803.2583-1-97-3': 'Let [EQUATION].', '0803.2583-1-97-4': 'When [MATH] tends to infinity, [MATH] converges to [EQUATION].', '0803.2583-1-97-5': 'By a similar argument as above, we obtain that for all but countably many [MATH], [EQUATION]', '0803.2583-1-97-6': 'We obtain from [REF] and [REF] the following estimation of polygons _p^-1P((T_(p+r)/p )^())(_pt)P((T_1p_p)^())(t)', '0803.2583-1-98-0': 'P((T_1p_p)^())(t)_p P((T_(p-q)/p)^())(t/_p),& 0t_p,', '0803.2583-1-99-0': '_p P((T_(p-q)/p)^())(1)+(t-_p),&_pt1.', '0803.2583-1-100-0': 'Finally, since [MATH] (which is a consequence of the continuity of the geometric volume function), combined with the fact that both [MATH] and [MATH] converge vaguely to [MATH] when [MATH], we obtain, for all but countably many [MATH], the uniform convergence of [MATH] to [MATH].'}","{'0803.2583-2-0-0': 'We establish, for a generically big Hermitian line bundle, the convergence of truncated Harder-Narasimhan polygons and the uniform continuity of the limit.', '0803.2583-2-0-1': 'As applications, we prove a conjecture of Moriwaki asserting that the arithmetic volume function is actually a limit instead of a sup-limit, and we show how to compute the asymptotic polygon of a Hermitian line bundle, by using the arithmetic volume function.', '0803.2583-2-1-0': '# Introduction', '0803.2583-2-2-0': 'Let [MATH] be a number field and [MATH] be its integer ring.', '0803.2583-2-2-1': 'Let [MATH] be a projective arithmetic variety of total dimension [MATH] over [MATH].', '0803.2583-2-2-2': 'For any Hermitian line bundle [MATH] on [MATH], the arithmetic volume of [MATH] introduced by Moriwaki (see [CITATION]) is [EQUATION]where [MATH].', '0803.2583-2-2-3': 'The Hermitian line bundle [MATH] is said to be arithmetically big if [MATH].', '0803.2583-2-2-4': 'The notion of arithmetic bigness had been firstly introduced by Moriwaki [CITATION] 2 in a different form.', '0803.2583-2-2-5': 'Recently he himself ([CITATION] Theorem 4.5) and Yuan ([CITATION] Corollary 2.4) have proved that the arithmetic bigness in [CITATION] is actually equivalent to the strict positivity of the arithmetic volume function [REF].', '0803.2583-2-2-6': 'In [CITATION], Moriwaki has proved the continuity of [REF] with respect to [MATH] and then deduced some comparisons to arithmetic intersection numbers (loc. cit.', '0803.2583-2-2-7': 'Theorem 6.2).', '0803.2583-2-3-0': 'Note that the volume function [REF] is an arithmetic analogue of the classical volume function for a line bundle on a projective variety: if [MATH] is a line bundle on a projective variety [MATH] of dimension [MATH] defined over a field [MATH], the volume of [MATH] is [EQUATION]', '0803.2583-2-3-1': 'Similarly, [MATH] is said to be big if [MATH].', '0803.2583-2-3-2': ""After Fujita's approximation theorem (see [CITATION], and [CITATION] for positive characteristic case), the sup-limit in [REF] is in fact a limit (see [CITATION] 11.4.7)."", '0803.2583-2-4-0': 'During a presentation at Institut de Mathematiques de Jussieu, Moriwaki has conjectured that, in arithmetic case, the sequence [MATH] also converges.', '0803.2583-2-4-1': 'In other words, one has [EQUATION].', '0803.2583-2-4-2': ""The strategy proposed by him is to develop an analogue of Fujita's approximation theorem in arithmetic setting (see [CITATION] Remark 5.7)."", '0803.2583-2-5-0': ""In this article, we prove Moriwaki's conjecture by establishing a convergence result of Harder-Narasimhan polygons (Theorem [REF]), which is a generalization of the author's previous work [CITATION] where the main tool is the Harder-Narasimhan filtration (indexed by [MATH]) of a Hermitian vector bundle on [MATH] and its associated Borel measure."", '0803.2583-2-5-1': 'To apply the convergence of polygons, the main idea is to compare [MATH], defined as the logarithm of the number of effective points in [MATH], to the positive degree [MATH], which is the maximal value of the Harder-Narasimhan polygon of [MATH].', '0803.2583-2-5-2': 'Here [MATH] denotes a Hermitian vector bundle on [MATH].', '0803.2583-2-5-3': 'We show that the arithmetic volume function coincides with the limit of normalized positive degrees and therefore prove the conjecture.', '0803.2583-2-6-0': 'In [CITATION] and [CITATION], the important (analytical) technic used by both authors is the estimation of the distortion function, which has already appeared in [CITATION].', '0803.2583-2-6-1': 'The approach in the present work, which is similar to that in [CITATION], relies on purely algebraic arguments.', '0803.2583-2-6-2': 'We also establish an explicit link between the volume function and some geometric invariants of [MATH] such as asymptotic slopes, which permits us to prove that [MATH] is big if and only if the norm of the smallest non-zero section of [MATH] decreases exponentially when [MATH] tends to infinity.', '0803.2583-2-6-3': 'This result is analogous to Theorem 4.5 of [CITATION] or Corollary 2.4 (1)[MATH](4) of [CITATION] except that we avoid using analytical methods.', '0803.2583-2-7-0': 'In our approach, the arithmetic volume function can be interpreted as the limit of maximal values of Harder-Narasimhan polygons.', '0803.2583-2-7-1': ""Inspired by Moriwaki's work [CITATION], we shall establish the uniform continuity for limit of truncated Harder-Narasimhan polygons (Theorem [REF])."", '0803.2583-2-7-2': 'This result refines loc. cit.', '0803.2583-2-7-3': 'Theorem 5.4.', '0803.2583-2-7-4': 'Furthermore, we show that the asymptotic polygon can be calculated from the volume function of the Hermitian line bundle twisted by pull-backs of Hermitian line bundles on [MATH].', '0803.2583-2-8-0': 'Our method works also in function field case.', '0803.2583-2-8-1': 'It establishes an explicit link between the geometric volume function and some classical geometry such as semistability and Harder-Narasimhan filtration.', '0803.2583-2-8-2': 'This generalizes for example a work of Wolfe [CITATION] (see also [CITATION] Example 2.12) concerning volume function on ruled varieties over curves.', '0803.2583-2-8-3': 'Moreover, recent results in [CITATION] show that at least in function field case, the asymptotic polygon is ""differentiable"" with respect to the line bundle, and there may be a ""measure-valued intersection product"" from which we recover arithmetic invariants by integration.', '0803.2583-2-9-0': 'The rest of this article is organized as follows.', '0803.2583-2-9-1': 'We fist recall some notation in Arakelov geometry in the second section.', '0803.2583-2-9-2': 'In the third section, we introduce the notion of positive degree for a Hermitian vector bundle on [MATH] and we compare it to the logarithm of the number of effective elements.', '0803.2583-2-9-3': 'The main tool is the Riemann-Roch inequality on [MATH] due to Gillet and Soule [CITATION].', '0803.2583-2-9-4': 'In the fourth section, we establish the convergence of the measures associated to suitably filtered section algebra of a big line bundle (Theorem [REF]).', '0803.2583-2-9-5': 'We show in the fifth section that the arithmetic bigness of [MATH] implies the classical one of [MATH], which is a generalization of a result of Yuan [CITATION].', '0803.2583-2-9-6': 'By the convergence result in the fourth section, we are able to prove that the volume of [MATH] coincides with the limit of normalized positive degrees, and therefore the sup-limit in [REF] is in fact a limit (Theorem [REF]).', '0803.2583-2-9-7': 'Here we also need the comparison result in the third section.', '0803.2583-2-9-8': 'Finally, we prove that the arithmetic bigness is equivalent to the positivity of asymptotic maximal slope (Theorem [REF]).', '0803.2583-2-9-9': 'In the sixth section, we establish the continuity of the limit of truncated polygons.', '0803.2583-2-9-10': 'Then we show in the seventh section how to compute the asymptotic polygon.'}","[['0803.2583-1-4-0', '0803.2583-2-4-0'], ['0803.2583-1-4-1', '0803.2583-2-4-1'], ['0803.2583-1-4-2', '0803.2583-2-4-2'], ['0803.2583-1-3-0', '0803.2583-2-3-0'], ['0803.2583-1-3-1', '0803.2583-2-3-1'], ['0803.2583-1-3-2', '0803.2583-2-3-2'], ['0803.2583-1-0-0', '0803.2583-2-0-0'], ['0803.2583-2-3-0', '0803.2583-3-3-0'], ['0803.2583-2-3-1', '0803.2583-3-3-1'], ['0803.2583-2-3-2', '0803.2583-3-3-2'], ['0803.2583-2-9-0', '0803.2583-3-9-0'], ['0803.2583-2-9-1', '0803.2583-3-9-1'], ['0803.2583-2-9-2', '0803.2583-3-9-2'], ['0803.2583-2-9-3', '0803.2583-3-9-3'], ['0803.2583-2-9-4', '0803.2583-3-9-4'], ['0803.2583-2-9-5', '0803.2583-3-9-5'], ['0803.2583-2-9-6', '0803.2583-3-9-6'], ['0803.2583-2-9-7', '0803.2583-3-9-7'], ['0803.2583-2-9-8', '0803.2583-3-9-8'], ['0803.2583-2-9-9', '0803.2583-3-9-9'], ['0803.2583-2-9-10', '0803.2583-3-9-10'], ['0803.2583-2-4-0', '0803.2583-3-4-0'], ['0803.2583-2-4-1', '0803.2583-3-4-1'], ['0803.2583-2-4-2', '0803.2583-3-4-2'], ['0803.2583-2-5-0', '0803.2583-3-5-0'], ['0803.2583-2-5-1', '0803.2583-3-5-1'], ['0803.2583-2-5-2', '0803.2583-3-5-2'], ['0803.2583-2-5-3', '0803.2583-3-5-3'], ['0803.2583-2-7-0', '0803.2583-3-7-0'], ['0803.2583-2-7-1', '0803.2583-3-7-1'], ['0803.2583-2-7-2', '0803.2583-3-7-2'], ['0803.2583-2-7-4', '0803.2583-3-7-4'], ['0803.2583-2-8-0', '0803.2583-3-8-0'], ['0803.2583-2-8-1', '0803.2583-3-8-1'], ['0803.2583-2-8-2', '0803.2583-3-8-2'], ['0803.2583-2-8-3', '0803.2583-3-8-3'], ['0803.2583-2-2-0', '0803.2583-3-2-0'], ['0803.2583-2-2-1', '0803.2583-3-2-1'], ['0803.2583-2-2-2', '0803.2583-3-2-2'], ['0803.2583-2-2-3', '0803.2583-3-2-3'], ['0803.2583-2-2-4', '0803.2583-3-2-4'], ['0803.2583-2-2-5', '0803.2583-3-2-5'], ['0803.2583-2-2-6', '0803.2583-3-2-6'], ['0803.2583-2-6-0', '0803.2583-3-6-0'], ['0803.2583-2-6-1', '0803.2583-3-6-1'], ['0803.2583-2-6-2', '0803.2583-3-6-2'], ['0803.2583-2-6-3', '0803.2583-3-6-3'], 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['0803.2583-1-6-2', '0803.2583-2-6-1'], ['0803.2583-1-6-3', '0803.2583-2-6-2'], ['0803.2583-1-6-4', '0803.2583-2-6-3'], ['0803.2583-1-7-0', '0803.2583-2-7-0'], ['0803.2583-1-7-1', '0803.2583-2-7-1'], ['0803.2583-1-7-2', '0803.2583-2-7-2']]","[['0803.2583-1-4-0', '0803.2583-2-4-0'], ['0803.2583-1-4-1', '0803.2583-2-4-1'], ['0803.2583-1-4-2', '0803.2583-2-4-2'], ['0803.2583-1-3-0', '0803.2583-2-3-0'], ['0803.2583-1-3-1', '0803.2583-2-3-1'], ['0803.2583-1-3-2', '0803.2583-2-3-2'], ['0803.2583-1-0-0', '0803.2583-2-0-0'], ['0803.2583-2-3-0', '0803.2583-3-3-0'], ['0803.2583-2-3-1', '0803.2583-3-3-1'], ['0803.2583-2-3-2', '0803.2583-3-3-2'], ['0803.2583-2-9-0', '0803.2583-3-9-0'], ['0803.2583-2-9-1', '0803.2583-3-9-1'], ['0803.2583-2-9-2', '0803.2583-3-9-2'], ['0803.2583-2-9-3', '0803.2583-3-9-3'], ['0803.2583-2-9-4', '0803.2583-3-9-4'], ['0803.2583-2-9-5', '0803.2583-3-9-5'], ['0803.2583-2-9-6', '0803.2583-3-9-6'], ['0803.2583-2-9-7', '0803.2583-3-9-7'], ['0803.2583-2-9-8', '0803.2583-3-9-8'], ['0803.2583-2-9-9', '0803.2583-3-9-9'], ['0803.2583-2-9-10', '0803.2583-3-9-10'], ['0803.2583-2-4-0', '0803.2583-3-4-0'], ['0803.2583-2-4-1', '0803.2583-3-4-1'], ['0803.2583-2-4-2', '0803.2583-3-4-2'], ['0803.2583-2-5-0', '0803.2583-3-5-0'], ['0803.2583-2-5-1', '0803.2583-3-5-1'], ['0803.2583-2-5-2', '0803.2583-3-5-2'], ['0803.2583-2-5-3', '0803.2583-3-5-3'], ['0803.2583-2-7-0', '0803.2583-3-7-0'], ['0803.2583-2-7-1', '0803.2583-3-7-1'], ['0803.2583-2-7-2', '0803.2583-3-7-2'], ['0803.2583-2-7-4', '0803.2583-3-7-4'], ['0803.2583-2-8-0', '0803.2583-3-8-0'], ['0803.2583-2-8-1', '0803.2583-3-8-1'], ['0803.2583-2-8-2', '0803.2583-3-8-2'], ['0803.2583-2-8-3', '0803.2583-3-8-3'], ['0803.2583-2-2-0', '0803.2583-3-2-0'], ['0803.2583-2-2-1', '0803.2583-3-2-1'], ['0803.2583-2-2-2', '0803.2583-3-2-2'], ['0803.2583-2-2-3', '0803.2583-3-2-3'], ['0803.2583-2-2-4', '0803.2583-3-2-4'], ['0803.2583-2-2-5', '0803.2583-3-2-5'], ['0803.2583-2-2-6', '0803.2583-3-2-6'], ['0803.2583-2-6-0', '0803.2583-3-6-0'], ['0803.2583-2-6-1', '0803.2583-3-6-1'], ['0803.2583-2-6-2', '0803.2583-3-6-2'], ['0803.2583-2-6-3', '0803.2583-3-6-3'], ['0803.2583-2-0-0', '0803.2583-3-0-0'], ['0803.2583-2-0-1', '0803.2583-3-0-1'], ['0803.2583-1-8-0', '0803.2583-2-9-0'], ['0803.2583-1-8-1', '0803.2583-2-9-1'], ['0803.2583-1-8-2', '0803.2583-2-9-2'], ['0803.2583-1-8-3', '0803.2583-2-9-3'], ['0803.2583-1-8-4', '0803.2583-2-9-4'], ['0803.2583-1-8-5', '0803.2583-2-9-5'], ['0803.2583-1-8-6', '0803.2583-2-9-6'], ['0803.2583-1-8-7', '0803.2583-2-9-7'], ['0803.2583-1-8-8', '0803.2583-2-9-8'], ['0803.2583-1-2-0', '0803.2583-2-2-0'], ['0803.2583-1-2-1', '0803.2583-2-2-1'], ['0803.2583-1-2-2', '0803.2583-2-2-2'], ['0803.2583-1-2-3', '0803.2583-2-2-3'], ['0803.2583-1-2-4', '0803.2583-2-2-4'], ['0803.2583-1-2-6', '0803.2583-2-2-6'], ['0803.2583-1-5-0', '0803.2583-2-5-0'], ['0803.2583-1-5-2', '0803.2583-2-5-2'], ['0803.2583-1-5-3', '0803.2583-2-5-3']]","[['0803.2583-1-8-9', '0803.2583-2-9-9'], ['0803.2583-1-2-5', '0803.2583-2-2-5'], ['0803.2583-1-5-1', '0803.2583-2-5-1']]",[],"[['0803.2583-1-0-2', '0803.2583-2-0-1']]","[['0803.2583-1-6-1', '0803.2583-2-6-0'], ['0803.2583-1-6-2', '0803.2583-2-6-1'], ['0803.2583-1-6-3', '0803.2583-2-6-2'], ['0803.2583-1-6-4', '0803.2583-2-6-3'], ['0803.2583-1-7-0', '0803.2583-2-7-0'], ['0803.2583-1-7-1', '0803.2583-2-7-1'], ['0803.2583-1-7-2', '0803.2583-2-7-2']]","['0803.2583-1-2-7', '0803.2583-1-9-0', '0803.2583-1-17-0', '0803.2583-1-22-0', '0803.2583-1-23-0', '0803.2583-1-32-5', '0803.2583-1-34-3', '0803.2583-1-37-0', '0803.2583-1-43-0', '0803.2583-1-43-1', '0803.2583-1-43-2', '0803.2583-1-43-3', '0803.2583-1-45-0', '0803.2583-1-45-1', '0803.2583-1-45-2', '0803.2583-1-46-0', '0803.2583-1-46-1', '0803.2583-1-46-2', '0803.2583-1-46-3', '0803.2583-1-47-3', '0803.2583-1-48-0', '0803.2583-1-48-1', '0803.2583-1-54-19', '0803.2583-1-54-20', '0803.2583-1-54-21', '0803.2583-1-54-22', '0803.2583-1-54-23', '0803.2583-1-54-24', '0803.2583-1-54-25', '0803.2583-1-54-26', '0803.2583-1-54-27', '0803.2583-1-54-28', '0803.2583-1-55-3', '0803.2583-1-63-1', '0803.2583-1-63-2', '0803.2583-1-66-1', '0803.2583-1-66-4', '0803.2583-1-70-2', '0803.2583-1-72-2', '0803.2583-1-72-3', '0803.2583-1-72-5', '0803.2583-1-72-6', '0803.2583-1-74-3', '0803.2583-1-81-2', '0803.2583-1-86-2', '0803.2583-1-87-2', '0803.2583-1-88-0', '0803.2583-1-88-1', '0803.2583-1-95-4', '0803.2583-1-96-4', '0803.2583-1-96-5', '0803.2583-1-96-8', '0803.2583-1-97-3', '0803.2583-1-97-6', '0803.2583-1-98-0', '0803.2583-1-99-0', '0803.2583-2-2-7', '0803.2583-2-7-3', '0803.2583-3-2-7', '0803.2583-3-7-3']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/0803.2583,"{'0803.2583-3-0-0': 'We establish, for a generically big Hermitian line bundle, the convergence of truncated Harder-Narasimhan polygons and the uniform continuity of the limit.', '0803.2583-3-0-1': 'As applications, we prove a conjecture of Moriwaki asserting that the arithmetic volume function is actually a limit instead of a sup-limit, and we show how to compute the asymptotic polygon of a Hermitian line bundle, by using the arithmetic volume function.', '0803.2583-3-1-0': '# Introduction', '0803.2583-3-2-0': 'Let [MATH] be a number field and [MATH] be its integer ring.', '0803.2583-3-2-1': 'Let [MATH] be a projective arithmetic variety of total dimension [MATH] over [MATH].', '0803.2583-3-2-2': 'For any Hermitian line bundle [MATH] on [MATH], the arithmetic volume of [MATH] introduced by Moriwaki (see [CITATION]) is [EQUATION]where [MATH].', '0803.2583-3-2-3': 'The Hermitian line bundle [MATH] is said to be arithmetically big if [MATH].', '0803.2583-3-2-4': 'The notion of arithmetic bigness had been firstly introduced by Moriwaki [CITATION] 2 in a different form.', '0803.2583-3-2-5': 'Recently he himself ([CITATION] Theorem 4.5) and Yuan ([CITATION] Corollary 2.4) have proved that the arithmetic bigness in [CITATION] is actually equivalent to the strict positivity of the arithmetic volume function [REF].', '0803.2583-3-2-6': 'In [CITATION], Moriwaki has proved the continuity of [REF] with respect to [MATH] and then deduced some comparisons to arithmetic intersection numbers (loc. cit.', '0803.2583-3-2-7': 'Theorem 6.2).', '0803.2583-3-3-0': 'Note that the volume function [REF] is an arithmetic analogue of the classical volume function for a line bundle on a projective variety: if [MATH] is a line bundle on a projective variety [MATH] of dimension [MATH] defined over a field [MATH], the volume of [MATH] is [EQUATION]', '0803.2583-3-3-1': 'Similarly, [MATH] is said to be big if [MATH].', '0803.2583-3-3-2': ""After Fujita's approximation theorem (see [CITATION], and [CITATION] for positive characteristic case), the sup-limit in [REF] is in fact a limit (see [CITATION] 11.4.7)."", '0803.2583-3-4-0': 'During a presentation at Institut de Mathematiques de Jussieu, Moriwaki has conjectured that, in arithmetic case, the sequence [MATH] also converges.', '0803.2583-3-4-1': 'In other words, one has [EQUATION].', '0803.2583-3-4-2': ""The strategy proposed by him is to develop an analogue of Fujita's approximation theorem in arithmetic setting (see [CITATION] Remark 5.7)."", '0803.2583-3-5-0': ""In this article, we prove Moriwaki's conjecture by establishing a convergence result of Harder-Narasimhan polygons (Theorem [REF]), which is a generalization of the author's previous work [CITATION] where the main tool is the Harder-Narasimhan filtration (indexed by [MATH]) of a Hermitian vector bundle on [MATH] and its associated Borel measure."", '0803.2583-3-5-1': 'To apply the convergence of polygons, the main idea is to compare [MATH], defined as the logarithm of the number of effective points in [MATH], to the positive degree [MATH], which is the maximal value of the Harder-Narasimhan polygon of [MATH].', '0803.2583-3-5-2': 'Here [MATH] denotes a Hermitian vector bundle on [MATH].', '0803.2583-3-5-3': 'We show that the arithmetic volume function coincides with the limit of normalized positive degrees and therefore prove the conjecture.', '0803.2583-3-6-0': 'In [CITATION] and [CITATION], the important (analytical) technic used by both authors is the estimation of the distortion function, which has already appeared in [CITATION].', '0803.2583-3-6-1': 'The approach in the present work, which is similar to that in [CITATION], relies on purely algebraic arguments.', '0803.2583-3-6-2': 'We also establish an explicit link between the volume function and some geometric invariants of [MATH] such as asymptotic slopes, which permits us to prove that [MATH] is big if and only if the norm of the smallest non-zero section of [MATH] decreases exponentially when [MATH] tends to infinity.', '0803.2583-3-6-3': 'This result is analogous to Theorem 4.5 of [CITATION] or Corollary 2.4 (1)[MATH](4) of [CITATION] except that we avoid using analytical methods.', '0803.2583-3-7-0': 'In our approach, the arithmetic volume function can be interpreted as the limit of maximal values of Harder-Narasimhan polygons.', '0803.2583-3-7-1': ""Inspired by Moriwaki's work [CITATION], we shall establish the uniform continuity for limit of truncated Harder-Narasimhan polygons (Theorem [REF])."", '0803.2583-3-7-2': 'This result refines loc. cit.', '0803.2583-3-7-3': 'Theorem 5.4.', '0803.2583-3-7-4': 'Furthermore, we show that the asymptotic polygon can be calculated from the volume function of the Hermitian line bundle twisted by pull-backs of Hermitian line bundles on [MATH].', '0803.2583-3-8-0': 'Our method works also in function field case.', '0803.2583-3-8-1': 'It establishes an explicit link between the geometric volume function and some classical geometry such as semistability and Harder-Narasimhan filtration.', '0803.2583-3-8-2': 'This generalizes for example a work of Wolfe [CITATION] (see also [CITATION] Example 2.12) concerning volume function on ruled varieties over curves.', '0803.2583-3-8-3': 'Moreover, recent results in [CITATION] show that at least in function field case, the asymptotic polygon is ""differentiable"" with respect to the line bundle, and there may be a ""measure-valued intersection product"" from which we recover arithmetic invariants by integration.', '0803.2583-3-9-0': 'The rest of this article is organized as follows.', '0803.2583-3-9-1': 'We fist recall some notation in Arakelov geometry in the second section.', '0803.2583-3-9-2': 'In the third section, we introduce the notion of positive degree for a Hermitian vector bundle on [MATH] and we compare it to the logarithm of the number of effective elements.', '0803.2583-3-9-3': 'The main tool is the Riemann-Roch inequality on [MATH] due to Gillet and Soule [CITATION].', '0803.2583-3-9-4': 'In the fourth section, we establish the convergence of the measures associated to suitably filtered section algebra of a big line bundle (Theorem [REF]).', '0803.2583-3-9-5': 'We show in the fifth section that the arithmetic bigness of [MATH] implies the classical one of [MATH], which is a generalization of a result of Yuan [CITATION].', '0803.2583-3-9-6': 'By the convergence result in the fourth section, we are able to prove that the volume of [MATH] coincides with the limit of normalized positive degrees, and therefore the sup-limit in [REF] is in fact a limit (Theorem [REF]).', '0803.2583-3-9-7': 'Here we also need the comparison result in the third section.', '0803.2583-3-9-8': 'Finally, we prove that the arithmetic bigness is equivalent to the positivity of asymptotic maximal slope (Theorem [REF]).', '0803.2583-3-9-9': 'In the sixth section, we establish the continuity of the limit of truncated polygons.', '0803.2583-3-9-10': 'Then we show in the seventh section how to compute the asymptotic polygon.'}",,
1412.7332,"{'1412.7332-1-0-0': 'Observations which are realizations from some continuous process are frequent in sciences, engineering, economics, and other fields.', '1412.7332-1-0-1': 'We consider a linear model where the responses are random functions in a suitable Sobolev space.', '1412.7332-1-0-2': ""The process can't be observed directly."", '1412.7332-1-0-3': 'With smoothing procedures from the original data, both the response curves and their derivatives can be reconstructed, even separately.', '1412.7332-1-0-4': 'From these functions we estimate the vector of functional parameters and we prove a functional version of the Gauss-Markov theorem.', '1412.7332-1-0-5': 'We also obtain experimental designs which are optimal for the estimation of the considered model.', '1412.7332-1-0-6': 'The advantages of this theory are finally showed in a real data-set.', '1412.7332-1-1-0': 'Keywords: functional data analysis; Sobolev spaces; linear models; repeated measurements; Gauss-Markov theorem; Riesz representation theorem; best linear unbiased estimator; experimental designs; optimal design.', '1412.7332-1-2-0': '# Introduction', '1412.7332-1-3-0': 'Observations which are realizations from some continuous process are ubiquitous in many fields like sciences, engineering, economics and other fields.', '1412.7332-1-3-1': 'For this reason, the interest for statistical modeling of functional data is increasing, with applications in many areas.', '1412.7332-1-3-2': 'Reference monographs on functional data analysis are, for instance, the books of [CITATION] and [CITATION], and the book of [CITATION] for the non-parametric approach.', '1412.7332-1-3-3': 'They cover topics like data representation, smoothing and registration; regression models; classification, discrimination and principal component analysis; derivatives and principal differential analysis; and more other.', '1412.7332-1-4-0': 'Regression models with functional variables can cover different situations: it may be the case of functional responses, or functional predictors, or both.', '1412.7332-1-4-1': 'In the present paper a of linear model with functional response and multivariate (or univariate) regressor is considered.', '1412.7332-1-4-2': 'In the model we consider the case of repeated measurements, which may be particularly useful in the context of experimental designs, but all the theoretical results proved remain valid in the standard case.', '1412.7332-1-4-3': 'Focus of the work is the best estimation of the functional coefficients of the regressors.', '1412.7332-1-5-0': 'The use of derivatives is very important for exploratory analysis of functional data as well as for inference and prediction methodologies.', '1412.7332-1-5-1': 'High quality derivative information can be provided, for instance, by reconstructing the functions with spline smoothing procedures.', '1412.7332-1-5-2': 'Recent developments on estimation of derivatives are contained in the works of [CITATION] and in [CITATION].', '1412.7332-1-5-3': 'See also [CITATION], who have obtained derivatives in the context of survival analysis, and [CITATION] who have estimated derivatives in a non-parametric model.', '1412.7332-1-6-0': 'In the literature the usual space for functional data is [MATH], and the data are used to reconstruct curve functions or derivatives.', '1412.7332-1-6-1': 'The particularity of the present work is that the curves are random elements of a suitable Sobolev space.', '1412.7332-1-6-2': 'The heuristic justification for this choice is that the data may provide information on both curve functions and their derivatives.', '1412.7332-1-6-3': 'If we take into consideration the whole information, about curves and derivatives, we may improve our estimates.', '1412.7332-1-6-4': 'The response processes cannot be observed directly; thus, we reconstruct the functions from a set of observed values.', '1412.7332-1-7-0': 'Two situations may occur: the sample of functions are reconstructed by a smoothing procedure of the data, and derivatives are then obtained by differentiation.', '1412.7332-1-7-1': 'At our knowledge, this is the most common method adopted in functional data analysis.', '1412.7332-1-7-2': 'Differently, the sample of functions and the sample of derivatives may be obtained separately.', '1412.7332-1-7-3': 'For instance, different smoothing techniques may be used to obtain the functions and the derivatives.', '1412.7332-1-7-4': 'Another possibility is when two sets of data are available, which are suitable to estimate functions and derivatives, respectively.', '1412.7332-1-7-5': 'For example, in the case of a motion process, data concerning positions and data about velocities may be observed.', '1412.7332-1-8-0': 'In this paper we propose a new method which incorporates the information provided by both the sample of functions and the sample of derivatives.', '1412.7332-1-8-1': 'The fundamental tool is the Riesz representative (in the Sobolev space) of a reconstructed function and a reconstructed derivative.', '1412.7332-1-8-2': 'From the two samples of reconstructed functions and derivatives just one sample of the Riesz representatives is obtained.', '1412.7332-1-8-3': 'We use this sample to estimate the functional parameters.', '1412.7332-1-8-4': 'In the simple situation when the derivatives are obtained by differentiating the reconstructed functions, the derivatives do not provide additional information to estimate the model and the Riesz representatives coincide with the functions.', '1412.7332-1-9-0': 'Using the Riesz representatives we provide the ordinary least square estimator of the functional parameter vector.', '1412.7332-1-9-1': 'In addition, the Gauss-Markov theorem is proved in the proper infinite-dimensional space.', '1412.7332-1-9-2': 'In the practice, however, the functions are represented is a basis expansion which is truncated at some order.', '1412.7332-1-9-3': 'Thus, we work in a finite dimensional space.', '1412.7332-1-9-4': 'We prove that the ordinary least square estimator in this finite-dimensional subset, is the projection of the ordinary least square estimator in the infinite-dimensional space.', '1412.7332-1-9-5': 'Moreover it is showed that, whenever a truncation of the basis expansions of functions is applied, the same results still holds in the finite-dimensional projection.', '1412.7332-1-10-0': 'As a consequence of the results proved, a rigorous generalization of the theory of optimal design of experiments in infinite-dimensional spaces is presented (see also [CITATION]).', '1412.7332-1-10-1': 'An application to an ergonomic data-set shows the advantages of this theory in the real world.', '1412.7332-1-11-0': 'The paper is organized as follows.', '1412.7332-1-11-1': 'Section [REF] contains the model and the theoretical results.', '1412.7332-1-11-2': 'Section [REF] presents some interesting considerations for the practice.', '1412.7332-1-11-3': 'Section [REF] is focused on the experimental context, and optimal designs for the model estimation are obtained.', '1412.7332-1-11-4': 'Section [REF] present a list of final remarks.', '1412.7332-1-11-5': 'Some additional results and proofs of theorems are postponed in Appendix.', '1412.7332-1-12-0': '# [MATH]-BLUE in functional linear models', '1412.7332-1-13-0': 'Let us consider a regression model where the response [MATH] is a random function which depends linearly on a vectorial (or scalar) known variable [MATH] through a functional coefficient, which needs to be estimated.', '1412.7332-1-13-1': 'We assume the following random effect model, for [MATH], [EQUATION] where [MATH] denote the response curve of the [MATH]-th observation for the [MATH]-th value of the regressor [MATH]; [MATH] is a [MATH]-dimensional vector of known functions; [MATH] is an unknown [MATH]-dimensional functional vector; [MATH] is a zero-mean process and denotes the random effect produced by the [MATH]-th experiment; [MATH] is a zero-mean error process.', '1412.7332-1-14-0': 'An example for the model [REF] is provided in Section [REF]; if [MATH] this model reduces to the functional response model described, for instance, in [CITATION].', '1412.7332-1-15-0': 'In a real world setting, the functions [MATH] are not directly observed.', '1412.7332-1-15-1': 'By a smoothing procedure from the original data, the investigator can reconstruct both the functions and their derivatives, obtaining [MATH] and [MATH], respectively.', '1412.7332-1-15-2': 'Hence we can assume that the model for the reconstructed functional data is [EQUATION] where', '1412.7332-1-16-0': 'the [MATH] bivariate vectors [MATH] are zero-mean independent and identically distributed couples of processes such that [MATH];', '1412.7332-1-17-0': 'all the [MATH] couples [MATH] are zero-mean identically distributed processes, each process being independent of all the other processes, with [MATH].', '1412.7332-1-18-0': 'Not that the investigator might reconstruct each function [MATH] and its derivative [MATH] separately.', '1412.7332-1-18-1': 'In this case, the terms of the second equation of [REF] are not the derivative of the terms of the first equation.', '1412.7332-1-18-2': 'The particular case when [MATH] is obtained deriving [MATH] is the most simple situation in model [REF].', '1412.7332-1-19-0': 'Let us consider an estimator [MATH] of [MATH], formed by [MATH] random functions in the Sobolev space [MATH].', '1412.7332-1-19-1': 'Recall that a function [MATH] is in [MATH] if [MATH] and its derivative function [MATH] belong to [MATH].', '1412.7332-1-19-2': 'Moreover, [MATH] is an Hilbert space with inner product [EQUATION]', '1412.7332-1-19-3': 'We define the [MATH]-generalized covariance matrix [MATH] of [MATH] as the [MATH] matrix whose [MATH]-th element is [EQUATION]', '1412.7332-1-19-4': 'In analogy with classical settings, we define the [MATH]-functional best linear unbiased estimator ([MATH]-BLUE) as the estimator with minimal (in the sense of Loewner Partial Order) [MATH]-generalized covariance matrix [REF], in the class of the linear unbiased estimators of [MATH].', '1412.7332-1-20-0': 'Given a couple [MATH], it may be defined a linear continuous operator on [MATH] as follows [EQUATION]', '1412.7332-1-20-1': 'From the Riesz representation theorem, there exists a unique [MATH] such that [EQUATION]', '1412.7332-1-20-2': 'We call [MATH] in [REF] the Riesz representative of the couple [MATH].', '1412.7332-1-21-0': 'This definition will be useful to provide a nice expression for the functional OLS estimator [MATH].', '1412.7332-1-21-1': 'Actually the Riesz representative synthesizes, in some sense, in [MATH] the information of both [MATH] and [MATH].', '1412.7332-1-22-0': 'The functional OLS estimator for the model [REF] is [EQUATION]', '1412.7332-1-22-1': 'The quantity [EQUATION] resembles [EQUATION] because [MATH] and [MATH] reconstruct [MATH] and its derivative function, respectively.', '1412.7332-1-22-2': 'The functional OLS estimator [MATH] minimizes, in this sense, the sum of the [MATH]-norm of the unobservable residuals [MATH].', '1412.7332-1-23-0': 'Given a model as in [REF],', '1412.7332-1-24-0': '[Proof of Theorem [REF]] Part a).', '1412.7332-1-24-1': 'We consider the sum of square residuals: [EQUATION]', '1412.7332-1-24-2': 'The Gateaux derivative', '1412.7332-1-25-0': 'of [MATH] at [MATH] in the direction of [MATH] is [EQUATION] where [EQUATION].', '1412.7332-1-26-0': 'The Gateaux derivatives [REF] is equal to [MATH] for any [MATH] for [MATH] such that [EQUATION].', '1412.7332-1-26-1': 'This proves the first statement of the theorem.', '1412.7332-1-27-0': 'Part b) The model [REF] implies that [EQUATION] where [MATH] and [MATH] denote the Riesz representatives of [MATH] and [MATH], respectively.', '1412.7332-1-27-1': 'From the hypothesis ([REF]) and ([REF]) in the model [REF], the left-hand side quantities in [REF] are zero-mean i.i.d. processes, for [MATH].', '1412.7332-1-27-2': 'Therefore, the generalized covariance matrix of [MATH] is [MATH], where [MATH].', '1412.7332-1-27-3': 'Hence, the generalized covariance matrix of [MATH] is [MATH].', '1412.7332-1-28-0': 'The functional OLS estimator obtained in Theorem [REF] by means of the Riesz representatives is also the best linear unbiased estimator in the Sobolev space, as stated in the next theorem.', '1412.7332-1-28-1': 'The proof is postponed in Appendix.', '1412.7332-1-29-0': 'The functional OLS estimator [MATH] for the model [REF] is a [MATH]-functional BLUE.', '1412.7332-1-30-0': '# Practical considerations', '1412.7332-1-31-0': 'In a real world context, we work with a finite dimensional subspace [MATH] of [MATH].', '1412.7332-1-31-1': 'Let [MATH] be a base of [MATH].', '1412.7332-1-31-2': 'Without loss of generality, we may assume that [MATH], where [EQUATION] is the Kronecker delta symbol, since a Gram-Schmidt orthonormalization procedure may be always applied.', '1412.7332-1-31-3': 'More precisely, given any base [MATH] in [MATH], the corresponding orthonormal base is given by:', '1412.7332-1-32-0': 'for [MATH], define [MATH],', '1412.7332-1-33-0': 'for [MATH], let [MATH] and [MATH].', '1412.7332-1-34-0': 'With this orthonormalized base, the projection [MATH] on [MATH] of the Riesz representative [MATH] of the couple [MATH] is given by [EQUATION] where the last equality comes from the definition [REF] of the Riesz representative.', '1412.7332-1-34-1': 'Now, if [MATH] is the [MATH]-th row of [MATH], then [EQUATION] hence [MATH].', '1412.7332-1-35-0': 'Let us note that, even if the Riesz representative [REF] is implicitly defined, its projection on [MATH] can be easily computed by [REF].', '1412.7332-1-35-1': 'From a practical point of view, the statistician can work with the data [MATH] projected on a finite linear subspace [MATH] and the corresponding OLS estimator [MATH] is the projection on [MATH] of the [MATH]-BLUE estimator [MATH] obtained in the Section [REF].', '1412.7332-1-35-2': 'For the projection, it is crucial to take a base of [MATH] which is orthonormal in [MATH].', '1412.7332-1-36-0': 'It is straightforward to prove that the estimator [REF] becomes [EQUATION] in two cases: when we do not take into consideration [MATH], or when [MATH].', '1412.7332-1-36-1': 'In both the cases, from the results obtained in the Section [REF], [MATH] is an [MATH]-BLUE.', '1412.7332-1-36-2': 'Up to our knowledge, this is the most common situation considered in the literature.', '1412.7332-1-37-0': '# Optimal designs of experiments in functional models', '1412.7332-1-38-0': 'In this section, we assume to work in an experimental setup.', '1412.7332-1-38-1': 'Therefore, [MATH], with [MATH], are not observed auxiliary variables; they can be freely chosen by an experimenter on the design space [MATH].', '1412.7332-1-38-2': 'The set of experimental conditions [MATH] is called an exact design.', '1412.7332-1-38-3': 'A more general definition is that of continuous design, as a probability measure [MATH] with support on [MATH] (see, for instance, [CITATION]).', '1412.7332-1-38-4': 'The choice of [MATH] may be done with the aim of obtaining accurate estimates of the model parameters.', '1412.7332-1-39-0': 'From Theorem [REF], [MATH] given in [REF] is the [MATH]-BLUE for the model [REF].', '1412.7332-1-39-1': 'This optimal estimator can be further improved by a ""clever"" choice of the design.', '1412.7332-1-39-2': 'In analogy with the criteria proposed in the finite-dimensional theory (see for instance, [CITATION]) we define a functional optimal design as a design which minimizes an appropriate convex function of the generalized covariance matrix [MATH] (see Definition [REF]).', '1412.7332-1-39-3': 'For instance, a functional D-optimum design is a design [MATH] which minimizes [MATH].', '1412.7332-1-39-4': 'Part b) of Theorem [REF] proves that [MATH].', '1412.7332-1-39-5': 'From the definition of continuous design [EQUATION] and hence [MATH] maximizes [MATH].', '1412.7332-1-40-0': '## An example: the ergonomic data', '1412.7332-1-41-0': 'Herein, we study the performance of the design proposed in [CITATION].', '1412.7332-1-42-0': 'In detail, to forecast the motion of drivers, some data are to be collected on the motion of a single subject to different locations within a test car.', '1412.7332-1-42-1': 'An experimental design is given by the choice of these locations and by the number of observations to be taken at each location.', '1412.7332-1-42-2': 'The response curve [MATH] is the angle formed at the right elbow between the upper and the lower arm, which is measured by a motion capture equipment.', '1412.7332-1-42-3': 'In the design used by [CITATION], 3 motion curves were observed at 20 different locations, spread around the glove compartment, gear shift, the central instrument panel and an overhead panel.', '1412.7332-1-42-4': ""These data are available from Faraway's website."", '1412.7332-1-42-5': 'In [CITATION], 3 different models are compared to predict the motion given the coordinates [MATH].', '1412.7332-1-42-6': 'From this comparison, the following quadratic model seems to be adequate: [EQUATION] where [EQUATION].', '1412.7332-1-42-7': 'Let us denote by [MATH] the exact design used by [CITATION] to collect observations [MATH].', '1412.7332-1-42-8': 'This design has 20 different locations at which the experiment is repeated three times, therefore the total number of observation is [MATH].', '1412.7332-1-43-0': 'We standardize support points (coordinates) of [MATH] so that they belong to [MATH].', '1412.7332-1-43-1': 'For this experimental domain, [CITATION] provides both the continuous D-optimum design, say [MATH], and an exact D-optimum design.', '1412.7332-1-43-2': 'Since [MATH] has support on the points of the [MATH] factorial (with [MATH]), [CITATION] proposes to search an exact D-optimum design for a small [MATH] over the points of the [MATH] factorial.', '1412.7332-1-43-3': 'To be consistent with the design used by [CITATION], we consider the exact design with [MATH] locations (some of which can be repeated) and we take [MATH] observations at each location.', '1412.7332-1-43-4': 'Thus again the total number of observation is [MATH].', '1412.7332-1-43-5': 'Let us call this design [MATH].', '1412.7332-1-43-6': 'In addition, we search an exact D-optimum design over the whole experimental domain [MATH].', '1412.7332-1-43-7': 'This exact design has 9 different locations which are not points of the [MATH] factorial, 5 different points of the [MATH] factorial which are not replicated and 3 different points of the [MATH] factorial which are replicated twice.', '1412.7332-1-43-8': 'Three observations are taken at each location.', '1412.7332-1-43-9': 'Let this exact design be denoted by [MATH].', '1412.7332-1-43-10': 'Finally, an exact D-optimum design over the [MATH] factorial has been find with Matlab 2014b routine rowexch, finding [MATH].', '1412.7332-1-44-0': 'In practice all designs are exact.', '1412.7332-1-44-1': 'Therefore, the continuous D-optimum design [MATH] is used only as a benchmark, in order to measure the goodness of a design [MATH] with respect to it.', '1412.7332-1-44-2': 'As a measure of goodness the D-efficiency is used: [EQUATION] which is proportional to design size, so that a design [MATH] with D-efficiency is equal to 0,5 needs the double of observations of [MATH] to get the same precision in the estimates.', '1412.7332-1-44-3': 'The number [MATH] is the number of unknown functions, [MATH] in this case.', '1412.7332-1-45-0': 'The following table lists the D-efficiencies of [MATH], [MATH], [MATH] and [MATH].', '1412.7332-1-46-0': 'From Table [REF] we have that the exact D-optimum designs [MATH] and [MATH] are almost equivalent.', '1412.7332-1-47-0': 'The D-efficiency of [MATH] is instead very low, only equal to [MATH].', '1412.7332-1-47-1': 'This example clearly shows the importance of choosing the experimental design according to an optimality criterion.', '1412.7332-1-48-0': '# Final discussion', '1412.7332-1-49-0': 'In this paper we propose an original method to incorporate both functions and derivatives information to estimate a functional linear regression model.', '1412.7332-1-49-1': 'Functional data are suitably modeled in Hilbert spaces (see [CITATION] and Bosq (2000)) and the space [MATH] is usually sufficient to handle the most of techniques presented in functional data analysis literature.', '1412.7332-1-49-2': 'We here consider proper Sobolev spaces and we extend the well known Gauss-Markov theorem in these infinite-dimensional spaces.', '1412.7332-1-49-3': 'The classical theory of estimation is extended by means of the Riesz representatives, which generates a new sample of functions from the data.', '1412.7332-1-50-0': 'The theory presented in this work may be generalized in other Sobolev spaces if a different weight is used for curves estimation and derivatives estimation, respectively, in the Riesz representatives.', '1412.7332-1-50-1': 'Another interesting generalization may be to add a weight function on [MATH].', '1412.7332-1-50-2': 'Both these generalizations, which have different impacts on applications, will be object of future work.', '1412.7332-1-51-0': 'As a consequence of the results obtained, in this paper the classical theory of optimal designs for model estimation has been extended to the case of linear models with functional responses.', '1412.7332-1-51-1': 'Interesting studies have been presented in the recent literature (see, for instance, [CITATION], [CITATION], [CITATION], [CITATION] and [CITATION]).', '1412.7332-1-51-2': 'We believe to provide a rigorous theoretical support in the proper infinite-dimensional spaces.', '1412.7332-1-52-0': 'A different generalization of optimal design theory to functional data is considered in [CITATION]: the design region is a subset of some functional space, while the response is a scalar quantity.', '1412.7332-1-52-1': 'In the following, we will study also different optimality criteria in experimental designs with functional data.', '1412.7332-1-53-0': '# Proof of Theorem [REF]', '1412.7332-1-54-0': 'The OLS [MATH] is a linear map which associates an element [MATH] in [MATH] to any [MATH]-tuple [MATH] in [MATH].', '1412.7332-1-54-1': 'In what follows, we show that it is the ""best"" among all the linear unbiased operators [MATH].', '1412.7332-1-54-2': 'This means that the difference of the corresponding generalized covariance matrices, [MATH] is a non-negative definite matrix.', '1412.7332-1-55-0': 'The model [REF] may be written in the following vectorial form: [EQUATION] where [MATH] is the column vector of length [MATH] with components equal to [MATH]: [MATH].', '1412.7332-1-56-0': 'Let us introduce the linear operator [EQUATION] where [MATH] is a vector, and [EQUATION] where [MATH] is the Riesz representative of [MATH] as in [REF].', '1412.7332-1-56-1': 'Therefore [EQUATION]', '1412.7332-1-56-2': 'Since [EQUATION] the thesis follows from the uncorrelation of [MATH] and [MATH].', '1412.7332-1-56-3': 'Since [MATH] is unbiased, [MATH], hence we have to prove that, for any couple of components [MATH], [EQUATION]', '1412.7332-1-57-0': 'First step: we prove that [MATH] applied to [MATH] is identically null.', '1412.7332-1-57-1': 'By [REF] this fact implies that [MATH] applied to the observations is the same of [MATH] applied to the random parts, i.e. [EQUATION]', '1412.7332-1-58-0': 'As a consequence of the linearity of [MATH] and of the zero-mean hypothesis ([REF]) and ([REF]), [EQUATION] where [REF] is due to the unbiasedness of [MATH].', '1412.7332-1-59-0': 'Now, [MATH] is the derivative of [MATH], and therefore the Riesz representation of each couple [MATH] is [MATH].', '1412.7332-1-59-1': 'Then [EQUATION]', '1412.7332-1-59-2': 'Combining [REF], [REF] and [REF] gives [EQUATION] hence [REF].', '1412.7332-1-60-0': 'Second step: representation of the linear operator [MATH].', '1412.7332-1-61-0': 'The linearity of the [MATH]-th component of [MATH] with respect to the bivariate observations [MATH] [EQUATION] where, for any [MATH] and [MATH], [MATH] is linear.', '1412.7332-1-61-1': 'The domain of [MATH] is contained in [MATH].', '1412.7332-1-61-2': 'Let [MATH] and [MATH] be suitable bases of [MATH] and of [MATH], that will be given in the fourth step.', '1412.7332-1-61-3': 'With this notation [EQUATION] where [EQUATION]', '1412.7332-1-62-0': 'Third step: Proof of [EQUATION] where [MATH] is the [MATH]-th row of [MATH].', '1412.7332-1-63-0': 'Let [MATH] be the null vector, except for the [MATH]-th component which is given by [MATH], and let [MATH].', '1412.7332-1-63-1': 'By [REF], [EQUATION]', '1412.7332-1-63-2': 'The arbitrary choice of [MATH] implies [REF].', '1412.7332-1-64-0': 'Fourth step Karhunen-Loeve representation of the noise process; definition of its ""Riesz representation"" [MATH], and definition of of the basis [MATH] and [MATH].', '1412.7332-1-65-0': 'For a given [MATH], the couple [MATH] is a process in [MATH].', '1412.7332-1-65-1': 'Let [MATH] be the spectral representation of the covariance matrix, which implies [MATH], and we assume, without loss of generalities, that the sequence [MATH] forms a orthonormal base (by completing it, defining [MATH] when needed).', '1412.7332-1-65-2': 'Note that [MATH] does not depend on [MATH] and [MATH], since, from the hypothesis ([REF]) and ([REF]) in the model [REF] [EQUATION] are identically distributed.', '1412.7332-1-65-3': 'By Karhunen-Loeve Theorem (see, e.g., [CITATION]), there exists an array of zero-mean unit variance random variables [MATH] such that [EQUATION] and hence [EQUATION]', '1412.7332-1-65-4': 'The independence assumptions in the hypothesis ([REF]) and ([REF]) ensure that [MATH] and [MATH] are independent if [MATH].', '1412.7332-1-65-5': 'Moreover, on the same unit statistics (when [MATH]), the Karhunen-Loeve representation gives [MATH].', '1412.7332-1-65-6': 'Finally, on different replication of the same experiment (when [MATH]) the common identically distributed bivariate process [MATH] yield a correlation not depending on the experiment [MATH] and on the replications [MATH]: [MATH].', '1412.7332-1-65-7': 'Summing up, the independence assumptions in the hypothesis ([REF]) and ([REF]) read [EQUATION] and hence [EQUATION]', '1412.7332-1-65-8': 'By the Riesz representation theorem, associated to any [MATH], there exists [MATH] such that [EQUATION] and then we define [EQUATION]', '1412.7332-1-65-9': 'Finally, since [MATH] is a bases for [MATH] and [MATH] for any [MATH], then [MATH] is a system of generators of [MATH].', '1412.7332-1-65-10': 'The base [MATH] may be chosen as a subsequence [MATH] of linearly independent functions which generates [MATH].', '1412.7332-1-66-0': 'To prove this, for any [MATH], we have [EQUATION].', '1412.7332-1-66-1': 'As a consequence, [MATH], and hence', '1412.7332-1-67-0': 'where the last equation is a consequence of [REF].', '1412.7332-1-67-1': 'The last result, together with [REF], [REF] and [REF] rewrites the thesis [REF] as', '1412.7332-1-68-0': 'for any choice of [MATH], where, again, [MATH] is the [MATH]-th row of [MATH].'}","{'1412.7332-2-0-0': 'Observations which are realizations from some continuous process are frequent in sciences, engineering, economics, and other fields.', '1412.7332-2-0-1': 'We consider linear models, with possible random effects, where the responses are random functions in a suitable Sobolev space.', '1412.7332-2-0-2': 'The processes can not be observed directly.', '1412.7332-2-0-3': 'With smoothing procedures from the original data, both the response curves and their derivatives can be reconstructed, even separately.', '1412.7332-2-0-4': 'From both these samples of functions, we estimate the vector of functional parameters and we provide a new and stronger functional version of the Gauss-Markov theorem.', '1412.7332-2-0-5': 'We also obtain experimental designs which are optimal for the estimation of the considered model.', '1412.7332-2-0-6': 'The advantages of this theory are finally showed in a real data-set.', '1412.7332-2-1-0': 'Keywords: functional data analysis; Sobolev spaces; linear models; repeated measurements; Gauss-Markov theorem; Riesz representation theorem; best linear unbiased estimator; experimental designs; optimal design.', '1412.7332-2-2-0': '# Introduction', '1412.7332-2-3-0': 'Observations which are realizations from some continuous process are ubiquitous in many fields like sciences, engineering, economics and other fields.', '1412.7332-2-3-1': 'For this reason, the interest for statistical modeling of functional data is increasing, with applications in many areas.', '1412.7332-2-3-2': 'Reference monographs on functional data analysis are, for instance, the books of [CITATION] and [CITATION], and the book of [CITATION] for the non-parametric approach.', '1412.7332-2-3-3': 'They cover topics like data representation, smoothing and registration; regression models; classification, discrimination and principal component analysis; derivatives and principal differential analysis; and more other.', '1412.7332-2-4-0': 'Regression models with functional variables can cover different situations: it may be the case of functional responses, or functional predictors, or both.', '1412.7332-2-4-1': 'In the present paper linear models with functional response and multivariate (or univariate) regressor are considered.', '1412.7332-2-4-2': 'We consider the case of repeated measurements, which may be particularly useful in the context of experimental designs, but all the theoretical results proved remain valid in the standard case.', '1412.7332-2-4-3': 'Focus of the work is the best estimation of the functional coefficients of the regressors.', '1412.7332-2-5-0': 'The use of derivatives is very important for exploratory analysis of functional data as well as for inference and prediction methodologies.', '1412.7332-2-5-1': 'High quality derivative information can be provided, for instance, by reconstructing the functions with spline smoothing procedures.', '1412.7332-2-5-2': 'Recent developments on estimation of derivatives are contained in the works of [CITATION] and in [CITATION].', '1412.7332-2-5-3': 'See also [CITATION], who have obtained derivatives in the context of survival analysis, and [CITATION] who have estimated derivatives in a non-parametric model.', '1412.7332-2-6-0': 'In the literature the usual space for functional data is [MATH], and the data are used to reconstruct curve functions or derivatives.', '1412.7332-2-6-1': 'The novelty of the present work is that the curves are random elements of a suitable Sobolev space.', '1412.7332-2-6-2': 'The heuristic justification for this choice is that the data may provide information on both curve functions and their derivatives.', '1412.7332-2-6-3': 'Of course, if we take into consideration the whole information, about curves and derivatives, we may improve our estimates.', '1412.7332-2-6-4': 'Curves and derivatives are actually reconstructed from a set of observed values, because the response processes cannot be observed directly.', '1412.7332-2-6-5': 'Two situations may occur: the sample of functions are reconstructed by a smoothing procedure of the data, and derivatives are then obtained by differentiation.', '1412.7332-2-6-6': 'At our knowledge, this is the most common method adopted in functional data analysis.', '1412.7332-2-6-7': 'However, the sample of functions and the sample of derivatives may be obtained separately.', '1412.7332-2-6-8': 'For instance, different smoothing techniques may be used to obtain the functions and the derivatives.', '1412.7332-2-6-9': 'Another possibility is when two sets of data are available, which are suitable to estimate functions and derivatives, respectively.', '1412.7332-2-6-10': 'For example, in the case of a motion process, data concerning positions and data about velocities may be observed.', '1412.7332-2-7-0': 'In this paper we propose a new method which incorporates the information provided by both the sample of functions and the sample of derivatives.', '1412.7332-2-7-1': 'A fundamental tool to prove all the theoretical results is the Riesz representative (in the Sobolev space) of a reconstructed function and a reconstructed derivative.', '1412.7332-2-7-2': 'From the two samples of reconstructed functions and derivatives just one sample of Riesz representatives is obtained.', '1412.7332-2-7-3': 'We use this sample to estimate the functional parameters.', '1412.7332-2-7-4': 'In the simple situation when the derivatives are obtained by differentiating the reconstructed functions, the derivatives do not provide additional information to estimate the model and the Riesz representatives coincide with the functions.', '1412.7332-2-8-0': 'Using the Riesz representatives we provide the ordinary least square estimator (OLS) of the functional parameter vector.', '1412.7332-2-8-1': 'In addition, a new version of the Gauss-Markov theorem is proved in the proper infinite-dimensional space ([MATH]).', '1412.7332-2-8-2': 'More precisely, we prove that any linear operator on [MATH] applied to the OLS estimator has minimum variance.', '1412.7332-2-8-3': 'In this sense, the OLS estimator is [MATH]-strong BLUE.', '1412.7332-2-8-4': 'As a particular case, the OLS estimator is also the best linear unbiased estimator in [MATH].', '1412.7332-2-8-5': 'In practice this [MATH]-BLUE cannot be computed explicetely, since the Riesz rapresentative is only implicetely defined.', '1412.7332-2-8-6': 'Neverthless, practitioners are used to represent functions in a basis expansion which is truncated at some order.', '1412.7332-2-8-7': 'This means to work in a finite-dimensional subspace [MATH] of [MATH].', '1412.7332-2-8-8': 'We prove that the OLS estimator in [MATH] is the projection of the [MATH]-BLUE and it is itself the [MATH]-BLUE in [MATH].', '1412.7332-2-8-9': 'Moreover, the OLS estimator in [MATH] can be practically computed, since the projection of the Riesz representative depends explicitly on the sample of functions and derivatives.', '1412.7332-2-9-0': 'As a consequence of the results proved, a rigorous generalization of the theory of optimal design of experiments in infinite-dimensional spaces is presented (see also [CITATION]).', '1412.7332-2-9-1': 'An application to an ergonomic data-set shows the advantages of this theory in the real world.', '1412.7332-2-10-0': 'The paper is organized as follows.', '1412.7332-2-10-1': 'Section [REF] contains the model and the theoretical results.', '1412.7332-2-10-2': 'Section [REF] presents some considerations which are useful from a practical point of view.', '1412.7332-2-10-3': 'Section [REF] is focused on the experimental context, and optimal designs for the model estimation are obtained.', '1412.7332-2-10-4': 'Section [REF] present a list of final remarks.', '1412.7332-2-10-5': 'Some additional results and proofs of theorems are deferred to Appendix A.', '1412.7332-2-11-0': '# Strong [MATH]-BLUE in functional linear models', '1412.7332-2-12-0': 'Let us consider a regression model where the response [MATH] is a random function which depends linearly on a vectorial (or scalar) known variable [MATH] through a functional coefficient, which needs to be estimated.', '1412.7332-2-12-1': 'In particular, we assume that [MATH] is an experimental condition and that [MATH] experiments are performed in batches at [MATH] (not necessarily different) experimental conditions [MATH].', '1412.7332-2-12-2': 'The [MATH]-th experiment is formed by [MATH] trials (repetitions) at the same [MATH].', '1412.7332-2-12-3': 'Therefore, the following random effect model is considered: [EQUATION] where [MATH] denotes the response curve of the [MATH]-th observation at the [MATH]-th experiment; [MATH] is a [MATH]-dimensional vector of known functions; [MATH] is an unknown [MATH]-dimensional functional vector; [MATH] is a zero-mean process which denotes the random effect due to the [MATH]-th experiment and takes into account the correlation among the [MATH] repetitions; [MATH] is a zero-mean error process.', '1412.7332-2-13-0': 'An example for the model [REF] is provided in Section [REF]; if [MATH] this model reduces to the functional response model described, for instance, in [CITATION].', '1412.7332-2-14-0': 'In a real world setting, the functions [MATH] are not directly observed.', '1412.7332-2-14-1': 'By a smoothing procedure from the original data, the investigator can reconstruct both the functions and their derivatives, obtaining [MATH] and [MATH], respectively.', '1412.7332-2-14-2': 'Hence we can assume that the model for the reconstructed functional data is [EQUATION] where', '1412.7332-2-15-0': 'the [MATH] bivariate vectors [MATH] are zero-mean independent and identically distributed couples of processes such that [MATH];', '1412.7332-2-16-0': 'all the [MATH] couples [MATH] are zero-mean identically distributed processes, each process being independent of all the other processes, with [MATH].', '1412.7332-2-17-0': 'Not that the investigator might reconstruct each function [MATH] and its derivative [MATH] separately.', '1412.7332-2-17-1': 'In this case, the right-hand term of the second equation in [REF] is not the derivative of the right-hand term of the first equation.', '1412.7332-2-17-2': 'The particular case when [MATH] is obtained by differentiation [MATH] is the most simple situation in model [REF].', '1412.7332-2-18-0': 'Let us consider an estimator [MATH] of [MATH], formed by [MATH] random functions in the Sobolev space [MATH].', '1412.7332-2-18-1': 'Recall that a function [MATH] is in [MATH] if [MATH] and its derivative [MATH] belong to [MATH].', '1412.7332-2-18-2': 'Moreover, [MATH] is an Hilbert space with inner product [EQUATION]', '1412.7332-2-18-3': 'We define the [MATH]-generalized covariance matrix [MATH] of an unbiased estimator [MATH] as the [MATH] matrix whose [MATH]-th element is [EQUATION]', '1412.7332-2-18-4': 'This global notion of covariance has been used also in [CITATION], in the context of predicting georeferenced functional data.', '1412.7332-2-18-5': 'These authors have found a BLUE estimator for the drift of their underlying process, which can be seen as an example of the results given in this paper.', '1412.7332-2-19-0': 'In analogy with classical settings, we define the [MATH]-functional best linear unbiased estimator ([MATH]-BLUE) as the estimator with minimal (in the sense of Loewner Partial Order) [MATH]-generalized covariance matrix [REF], in the class of the linear unbiased estimators of [MATH].', '1412.7332-2-20-0': 'It is simple to show that an equivalent definition of [MATH]-BLUE.', '1412.7332-2-20-1': 'In fact a [MATH]-BLUE minimizes the quantity [EQUATION] for any choice of [MATH], in the class of the linear unbiased estimators [MATH] of [MATH].', '1412.7332-2-20-2': 'In other words, the [MATH]-BLUE minimizes the [MATH]-variance of any linear combination of its components.', '1412.7332-2-20-3': 'A stronger request is the following.', '1412.7332-2-21-0': 'We define the [MATH]-strong functional best linear unbiased estimator ([MATH]-SBLUE) as the estimator with minimal variance, [EQUATION] for any choice of linear operator [MATH], in the class of the linear unbiased estimators [MATH] of [MATH].', '1412.7332-2-22-0': 'Given a couple [MATH], it may be defined a linear continuous operator on [MATH] as follows [EQUATION]', '1412.7332-2-22-1': 'From the Riesz representation theorem, there exists a unique [MATH] such that [EQUATION]', '1412.7332-2-22-2': 'The unique element [MATH] defined in [REF] is called the Riesz representative of the couple [MATH].', '1412.7332-2-23-0': 'This definition will be useful to provide a nice expression for the functional OLS estimator [MATH].', '1412.7332-2-23-1': 'Actually the Riesz representative synthesizes, in some sense, in [MATH] the information of both [MATH] and [MATH].', '1412.7332-2-24-0': 'The functional OLS estimator for the model [REF] is [EQUATION]', '1412.7332-2-24-1': 'The quantity [EQUATION] resembles [EQUATION] because [MATH] and [MATH] reconstruct [MATH] and its derivative function, respectively.', '1412.7332-2-24-2': 'The functional OLS estimator [MATH] minimizes, in this sense, the sum of the [MATH]-norm of the unobservable residuals [MATH].', '1412.7332-2-25-0': 'Given a model as in [REF],', '1412.7332-2-26-0': '[Proof of Theorem [REF]] Part a).', '1412.7332-2-26-1': 'We consider the sum of square residuals: [EQUATION]', '1412.7332-2-26-2': 'The Gateaux derivative of [MATH] at [MATH] in the direction of [MATH] is [EQUATION] where [MATH] and [MATH] are two [MATH] vectors whose [MATH]-th elements are [EQUATION].', '1412.7332-2-26-3': 'Developing the right-hand side of ([REF]), we have that the Gateaux derivative is [EQUATION] where [MATH] is a [MATH] vector whose [MATH]-th element is the Riesz representative of [MATH].', '1412.7332-2-27-0': 'The Gateaux derivative [REF] is equal to [MATH] for any [MATH] if and only if [MATH] is given by the following equation: [EQUATION] which proves the first statement of the theorem.', '1412.7332-2-28-0': 'Part b) Definition [REF] and model [REF] imply that, for any [MATH], [EQUATION] then [MATH], and hence [MATH] is unbiased.', '1412.7332-2-28-1': 'Moreover, [EQUATION] where [MATH] and [MATH] denote the Riesz representatives of [MATH] and [MATH], respectively.', '1412.7332-2-28-2': 'From the hypothesis ([REF]) and ([REF]) in the model [REF], the left-hand side quantities in [REF] are zero-mean i.i.d. processes, for [MATH].', '1412.7332-2-28-3': 'Therefore, the generalized covariance matrix of [MATH] is [MATH], where [MATH].', '1412.7332-2-28-4': 'Hence, the generalized covariance matrix of [MATH] is [MATH].', '1412.7332-2-29-0': 'The functional OLS estimator obtained in Theorem [REF] by means of the Riesz representatives is also the best linear unbiased estimator in the Sobolev space, as stated in the next theorem.', '1412.7332-2-29-1': 'The proof is deferred to Appendix A.', '1412.7332-2-30-0': 'The functional OLS estimator [MATH] for the model [REF] is a [MATH]-functional S BLUE.', '1412.7332-2-31-0': 'The theory and the results presented in this work may be generalized to other Sobolev spaces.', '1412.7332-2-31-1': 'The extension to [MATH], [MATH], is straightforward.', '1412.7332-2-31-2': 'Moreover, as in Bayesian context, the investigator might have a different a priori consideration of [MATH] and [MATH].', '1412.7332-2-31-3': 'Thus, different weights [MATH] and [MATH] may be used for curve functions and derivatives, respectively.', '1412.7332-2-31-4': 'Another interesting generalization might be to add positive weight functions [MATH] and [MATH] on [MATH], when, for instance, distinct zones of [MATH] are considered to have different relevance.', '1412.7332-2-31-5': 'Therefore, the inner product given in [REF] may be extended to [EQUATION].', '1412.7332-2-31-6': 'When [MATH] and [MATH], the Hilbert space is called weighted Sobolev space, see [CITATION].', '1412.7332-2-31-7': 'These generalizations, which have different impacts on applications, will be object of future works.', '1412.7332-2-32-0': '# Practical considerations', '1412.7332-2-33-0': 'In a real world context, we work with a finite dimensional subspace [MATH] of [MATH].', '1412.7332-2-33-1': 'Let [MATH] be a base of [MATH].', '1412.7332-2-33-2': 'Without loss of generality, we may assume that [MATH], where [EQUATION] is the Kronecker delta symbol, since a Gram-Schmidt orthonormalization procedure may be always applied.', '1412.7332-2-33-3': 'More precisely, given any base [MATH] in [MATH], the corresponding orthonormal base is given by:', '1412.7332-2-34-0': 'for [MATH], define [MATH],', '1412.7332-2-35-0': 'for [MATH], let [MATH] and [MATH].', '1412.7332-2-36-0': 'With this orthonormalized base, the projection [MATH] on [MATH] of the Riesz representative [MATH] of the couple [MATH] is given by [EQUATION] where the last equality comes from the definition [REF] of the Riesz representative.', '1412.7332-2-36-1': 'Now, if [MATH] is the [MATH]-th row of [MATH], then [EQUATION] hence [MATH].', '1412.7332-2-37-0': 'Let us note that, even if the Riesz representative [REF] is implicitly defined, its projection on [MATH] can be easily computed by [REF].', '1412.7332-2-37-1': 'From a practical point of view, the statistician can work with the data [MATH] projected on a finite linear subspace [MATH] and the corresponding OLS estimator [MATH] is the projection on [MATH] of the [MATH]-redS BLUE estimator [MATH] obtained in the Section [REF].', '1412.7332-2-37-2': 'As a consequence of the Theorem [REF], [MATH] is [MATH]-redS BLUE in [MATH], since it is unbiased and the projection is linear.', '1412.7332-2-37-3': 'For the projection, it is crucial to take a base of [MATH] which is orthonormal in [MATH].', '1412.7332-2-38-0': 'It is straightforward to prove that the estimator [REF] becomes [EQUATION] in two cases: when we do not take into consideration [MATH], or when [MATH].', '1412.7332-2-38-1': 'In both the cases, from the results obtained in the Section [REF], [MATH] is an [MATH]-BLUE.', '1412.7332-2-38-2': 'Up to our knowledge, this is the most common situation considered in the literature.', '1412.7332-2-39-0': '# Optimal designs of experiments in functional models', '1412.7332-2-40-0': 'In this section, we assume to work in an experimental setup.', '1412.7332-2-40-1': 'Therefore, [MATH], with [MATH], are not observed auxiliary variables; they can be freely chosen by an experimenter on the design space [MATH].', '1412.7332-2-40-2': 'The set of experimental conditions [MATH] is called an exact design.', '1412.7332-2-40-3': 'A more general definition is that of continuous design, as a probability measure [MATH] with support on [MATH] (see, for instance, [CITATION]).', '1412.7332-2-40-4': 'The choice of [MATH] may be done with the aim of obtaining accurate estimates of the model parameters.', '1412.7332-2-41-0': 'From Theorem [REF], [MATH] given in [REF] is the [MATH]-BLUE for the model [REF].', '1412.7332-2-41-1': 'This optimal estimator can be further improved by a ""clever"" choice of the design.', '1412.7332-2-41-2': 'In analogy with the criteria proposed in the finite-dimensional theory (see for instance, [CITATION]) we define a functional optimal design as a design which minimizes an appropriate convex function of the generalized covariance matrix [MATH] (see Definition [REF]).', '1412.7332-2-41-3': 'For instance, a functional D-optimum design is a design [MATH] which minimizes [MATH].', '1412.7332-2-41-4': 'Part b) of Theorem [REF] proves that [MATH].', '1412.7332-2-41-5': 'From the definition of continuous design [EQUATION] and hence [MATH] maximizes [MATH].', '1412.7332-2-42-0': '## An example: the ergonomic data', '1412.7332-2-43-0': 'Herein, we study the performance of the design proposed in [CITATION].', '1412.7332-2-43-1': 'In detail, to forecast the motion of drivers, some data are to be collected on the motion of a single subject to different locations within a test car.', '1412.7332-2-43-2': 'An experimental design is given by the choice of these locations (the different experimental conditions) and by the number of times that the experiment has to be replicated at each location.', '1412.7332-2-43-3': 'The response curve [MATH] is the angle formed at the right elbow between the upper and the lower arm, which is measured by a motion capture equipment.', '1412.7332-2-43-4': 'In the design used by [CITATION], 3 motion curves were observed at 20 different locations, spread around the glove compartment, gear shift, the central instrument panel and an overhead panel.', '1412.7332-2-43-5': ""These data are available from Faraway's website."", '1412.7332-2-43-6': 'In [CITATION], 3 different models are compared to predict the motion given the coordinates [MATH].', '1412.7332-2-43-7': 'From this comparison, the following quadratic model seems to be adequate: [EQUATION] where [EQUATION].', '1412.7332-2-43-8': 'Let us denote by [MATH] the exact design used in [CITATION] to collect observations [MATH].', '1412.7332-2-43-9': 'This design has 20 different locations at which three trials are repeated, therefore the total number of observations is [MATH].', '1412.7332-2-44-0': 'We standardize support points (coordinates) of [MATH] so that they belong to [MATH].', '1412.7332-2-44-1': 'For this experimental domain, [CITATION] provides both the continuous D-optimum design, say [MATH], and exact D-optimum designs.', '1412.7332-2-44-2': 'Since [MATH] has support on the points of the [MATH] factorial (with [MATH]), [CITATION] proposes to search an exact D-optimum design (for a small [MATH]) over the points of the [MATH] factorial.', '1412.7332-2-44-3': 'To be consistent with the design used in [CITATION], we consider the exact design with [MATH] locations, some of which may be repeated.', '1412.7332-2-44-4': 'At each location, an experiment is performed in batches of three trials.', '1412.7332-2-44-5': 'When a location is replicated more times, several experiments (each one formed by three trials) is performed at this location.', '1412.7332-2-44-6': 'Thus, again the total number of observation is [MATH].', '1412.7332-2-44-7': 'Let us call this design [MATH].', '1412.7332-2-45-0': 'In addition, we search an exact D-optimum design over the whole experimental domain [MATH].', '1412.7332-2-45-1': 'This exact design has 9 different locations which are not points of the [MATH] factorial, 5 different points of the [MATH] factorial which are not replicated and 3 different points of the [MATH] factorial which are replicated twice.', '1412.7332-2-45-2': 'Three observations are taken at each location.', '1412.7332-2-45-3': 'Let this exact design be denoted by [MATH].', '1412.7332-2-46-0': 'Finally, in a grid of [MATH] equally spaced points in [MATH], an exact D-optimum design over the [MATH] factorial has been find with Matlab 2014b routine rowexch, finding [MATH].', '1412.7332-2-47-0': 'In practice all designs are exact.', '1412.7332-2-47-1': 'Therefore, the continuous D-optimum design [MATH] is used only as a benchmark, in order to measure the goodness of a design [MATH] with respect to it.', '1412.7332-2-47-2': 'As a measure of goodness the D-efficiency is used: [EQUATION] where [MATH] is the dimension of the functional vector [MATH].', '1412.7332-2-47-3': 'The D-efficiency [MATH] is proportional to the design size, thus a design [MATH] with D-efficiency equal to 0,5 needs the double of observations of [MATH] to get the same precision in the estimates.', '1412.7332-2-48-0': 'The following table lists the D-efficiencies of [MATH], [MATH], [MATH] and [MATH].', '1412.7332-2-49-0': 'From Table [REF] we have that the exact D-optimum designs [MATH] and [MATH] are almost equivalent.', '1412.7332-2-49-1': 'The D-efficiency of [MATH] is instead very low, only equal to [MATH].', '1412.7332-2-49-2': 'This example clearly shows the importance of choosing the experimental design according to an optimality criterion.', '1412.7332-2-50-0': '# Summary', '1412.7332-2-51-0': 'Functional data are suitably modeled in separable Hilbert spaces (see [CITATION] and [CITATION]) and [MATH] is usually sufficient to handle the majority of the techniques proposed in the literature of functional data analysis.', '1412.7332-2-52-0': 'Differently, we consider proper Sobolev spaces, since we guess that the data may provide information on both curve functions and their derivatives.', '1412.7332-2-52-1': 'The classical theory for linear regression models is extended to this context by means of the Riesz representative in a Sobolev space.', '1412.7332-2-52-2': 'Roughly speaking, the Riesz representative is a quantity"" which incorporates both function and derivative information.', '1412.7332-2-53-0': 'Through the Riesz representative we provide a generalization of the well known Gauss-Markov theorem for a functional linear regression model.', '1412.7332-2-53-1': 'From this theoretical result, the classical D-optimality criterion is extended to the linear models with functional responses (similar models can be found, for instance, in [CITATION], [CITATION], [CITATION]).', '1412.7332-2-54-0': 'Interesting studies on experimental design with functional observations are presented in the recent literature (see, for instance, [CITATION], [CITATION] and [CITATION]).', '1412.7332-2-54-1': 'In the present work, however, a rigorous theoretical support is provided to apply optimal design theory in the proper infinite-dimensional spaces.', '1412.7332-2-55-0': 'As a matter of future work, we will study also different optimality criteria in the context of functional data models.', '1412.7332-2-55-1': 'In addition, we intend to develop our theory also in the functional setting considered by [CITATION], where the design region is a subset of some functional space and the response is a scalar quantity.', '1412.7332-2-56-0': '# Proof of Theorem [REF]', '1412.7332-2-57-0': 'The OLS [MATH] is a linear map which associates an element [MATH] in [MATH] to any [MATH]-tuple [MATH] in [MATH].', '1412.7332-2-57-1': 'In what follows, we show that it is the ""best"" among all the linear unbiased operators [MATH].', '1412.7332-2-58-0': 'The model [REF] may be written in the following vectorial form: [EQUATION] where [MATH] is the column vector of length [MATH] with all components equal to [MATH].', '1412.7332-2-59-0': 'Let [EQUATION] be a [MATH] block vector, with [MATH].', '1412.7332-2-59-1': 'Given any couple of [MATH] block vectors [MATH], we may define the following [MATH] dimensional vector [EQUATION] where [EQUATION] and [MATH] is the Riesz representative of [MATH] as in [REF].', '1412.7332-2-60-0': 'Now we can introduce the following linear operator [EQUATION]', '1412.7332-2-60-1': 'Hence, [EQUATION] and [EQUATION].', '1412.7332-2-60-2': 'The thesis follows immediately if we prove that [MATH] and [MATH] are uncorrelated.', '1412.7332-2-61-0': 'Since both [MATH] and [MATH] are unbiased, [MATH], and thus we have to prove that [EQUATION] for any choice of linear operator [MATH].', '1412.7332-2-62-0': 'The proof of equality ([REF]) is developed in five steps.', '1412.7332-2-63-0': 'First step.', '1412.7332-2-63-1': 'The goal of this step is to prove that [MATH] applied to the deterministic part of the model [MATH] is identically null.', '1412.7332-2-63-2': 'As a consequence, [EQUATION]', '1412.7332-2-64-0': 'Proof', '1412.7332-2-65-0': 'From the linearity of [MATH] and the zero-mean hypothesis ([REF]) and ([REF]), we have that [EQUATION]', '1412.7332-2-65-1': 'Since [MATH] we have that [EQUATION]', '1412.7332-2-65-2': 'In addition, from the definition ([REF]) if [EQUATION] then [EQUATION]', '1412.7332-2-65-3': 'Combining [REF], [REF] and [REF] gives [EQUATION] and hence [REF].', '1412.7332-2-66-0': 'Second step.', '1412.7332-2-66-1': 'Representation of the linear operator [MATH].', '1412.7332-2-67-0': 'For the linearity of the [MATH]-th component of [MATH] with respect to the bivariate observations [MATH]: [EQUATION] where, for any [MATH] and [MATH], [MATH] is linear.', '1412.7332-2-67-1': 'The domain of [MATH] is contained in [MATH].', '1412.7332-2-67-2': 'Let [MATH] a suitable base of [MATH] that will be specified in the fourth step, and [MATH] be an orthonormal base of [MATH].', '1412.7332-2-67-3': 'With this notation [EQUATION] where [EQUATION]', '1412.7332-2-68-0': 'Third step.', '1412.7332-2-68-1': 'Proof of [EQUATION] where [MATH] is the [MATH]-th row of [MATH].', '1412.7332-2-69-0': 'Let [MATH] be the null vector except for the [MATH]-th component which is [MATH], and let [MATH].', '1412.7332-2-69-1': 'Setting [MATH] in [REF], [EQUATION] where the last equality is due to ([REF]).', '1412.7332-2-70-0': 'From the Riesz representation theorem we have that, for any [MATH], there exists [MATH] such that [EQUATION]', '1412.7332-2-70-1': 'From ([REF]), equality ([REF]) becomes [EQUATION]', '1412.7332-2-70-2': 'The arbitrary choice of [MATH] implies [REF].', '1412.7332-2-71-0': 'Fourth step.', '1412.7332-2-71-1': 'Karhunen-Loeve representation of the noise process and definition of of the base [MATH].', '1412.7332-2-72-0': 'For a given [MATH], the couple [MATH] is a process in [MATH].', '1412.7332-2-72-1': 'Let [MATH] be the spectral representation of the covariance matrix, which implies [MATH], and we assume, without loss of generality, that the sequence [MATH] forms a orthonormal base (by completing it, defining [MATH] when needed).', '1412.7332-2-72-2': 'Note that [MATH] does not depend on [MATH] and [MATH], since, from the hypothesis ([REF]) and ([REF]) in the model [REF] [EQUATION] are identically distributed.', '1412.7332-2-72-3': 'From Karhunen-Loeve Theorem (see, e.g., [CITATION]), there exists an array of zero-mean unit variance random variables [MATH] such that [EQUATION]', '1412.7332-2-72-4': 'From ([REF]) the Riesz representative of the noise process ([REF]) is [MATH] and we can define the following means of replications, [EQUATION] which will be useful in the next fifth step.', '1412.7332-2-73-0': 'Finally, from Karhunen-Loeve Theorem, we may obtain the following relations which will be useful in the fifth step of the proof.', '1412.7332-2-73-1': 'From ([REF]) we have that for any [MATH], [MATH] and [MATH], [EQUATION]', '1412.7332-2-73-2': 'In addition, the independence assumptions in the hypothesis ([REF]) and ([REF]) ensure that [MATH] and [MATH] are independent if [MATH].', '1412.7332-2-73-3': 'For the same observation (i.e. [MATH]), the Karhunen-Loeve representation gives [MATH].', '1412.7332-2-73-4': 'Finally, for different replications of the same experiment (i.e. [MATH] but [MATH]) the identically distributed bivariate process [MATH] yields to a correlation which does not depend on the experiment [MATH] neither on the replications [MATH]: [MATH].', '1412.7332-2-73-5': 'Summing up, the independence assumptions given in the hypothesis ([REF]) and ([REF]) imply [EQUATION] and hence [EQUATION]', '1412.7332-2-74-0': 'Fifth step.', '1412.7332-2-74-1': 'Proof of [REF]: [EQUATION] for any choice of linear operator [MATH].', '1412.7332-2-75-0': 'From the definitions given in the part a of Theorem [REF] and from Equations [REF], [REF] and [REF] we have that, for any [MATH] [EQUATION] which proves that [EQUATION] where [MATH].', '1412.7332-2-76-0': 'From this last result and from ([REF]), [MATH], and hence [EQUATION] where the last equality is a consequence of [REF].', '1412.7332-2-77-0': 'From the linearity of the operator [MATH], we have that [EQUATION].', '1412.7332-2-77-1': 'Since [MATH], where [MATH], we have [EQUATION].', '1412.7332-2-77-2': 'Setting [EQUATION] then [EQUATION].', '1412.7332-2-77-3': 'Hence, from Equations ([REF]), [REF] and [REF], the thesis [REF] becomes [EQUATION]', '1412.7332-2-77-4': 'From [REF], [REF] and [REF], the left-hand side of the last equation becomes [EQUATION] the last equality being a consequence of [REF].'}","[['1412.7332-1-27-1', '1412.7332-2-28-2'], ['1412.7332-1-27-2', '1412.7332-2-28-3'], ['1412.7332-1-27-3', '1412.7332-2-28-4'], ['1412.7332-1-11-0', '1412.7332-2-10-0'], ['1412.7332-1-11-1', '1412.7332-2-10-1'], ['1412.7332-1-11-3', '1412.7332-2-10-3'], ['1412.7332-1-11-4', '1412.7332-2-10-4'], ['1412.7332-1-17-0', '1412.7332-2-16-0'], ['1412.7332-1-44-0', '1412.7332-2-47-0'], ['1412.7332-1-44-1', '1412.7332-2-47-1'], ['1412.7332-1-8-0', '1412.7332-2-7-0'], ['1412.7332-1-8-3', '1412.7332-2-7-3'], ['1412.7332-1-8-4', '1412.7332-2-7-4'], ['1412.7332-1-54-0', '1412.7332-2-57-0'], ['1412.7332-1-54-1', '1412.7332-2-57-1'], ['1412.7332-1-5-0', '1412.7332-2-5-0'], ['1412.7332-1-5-1', '1412.7332-2-5-1'], ['1412.7332-1-5-2', '1412.7332-2-5-2'], ['1412.7332-1-5-3', '1412.7332-2-5-3'], ['1412.7332-1-1-0', '1412.7332-2-1-0'], ['1412.7332-1-31-0', '1412.7332-2-33-0'], ['1412.7332-1-31-1', '1412.7332-2-33-1'], ['1412.7332-1-31-2', '1412.7332-2-33-2'], ['1412.7332-1-35-0', '1412.7332-2-37-0'], ['1412.7332-1-35-2', '1412.7332-2-37-3'], ['1412.7332-1-39-0', '1412.7332-2-41-0'], ['1412.7332-1-39-1', '1412.7332-2-41-1'], ['1412.7332-1-39-2', '1412.7332-2-41-2'], ['1412.7332-1-39-3', '1412.7332-2-41-3'], ['1412.7332-1-39-4', '1412.7332-2-41-4'], ['1412.7332-1-39-5', '1412.7332-2-41-5'], ['1412.7332-1-28-0', '1412.7332-2-29-0'], ['1412.7332-1-3-0', '1412.7332-2-3-0'], ['1412.7332-1-3-1', '1412.7332-2-3-1'], ['1412.7332-1-3-2', '1412.7332-2-3-2'], ['1412.7332-1-3-3', '1412.7332-2-3-3'], ['1412.7332-1-61-1', '1412.7332-2-67-1'], ['1412.7332-1-18-0', '1412.7332-2-17-0'], ['1412.7332-1-36-0', '1412.7332-2-38-0'], ['1412.7332-1-36-1', '1412.7332-2-38-1'], ['1412.7332-1-36-2', '1412.7332-2-38-2'], ['1412.7332-1-16-0', '1412.7332-2-15-0'], ['1412.7332-1-21-0', '1412.7332-2-23-0'], ['1412.7332-1-21-1', '1412.7332-2-23-1'], ['1412.7332-1-14-0', '1412.7332-2-13-0'], ['1412.7332-1-15-0', '1412.7332-2-14-0'], ['1412.7332-1-15-1', '1412.7332-2-14-1'], ['1412.7332-1-15-2', '1412.7332-2-14-2'], ['1412.7332-1-22-0', '1412.7332-2-24-0'], ['1412.7332-1-22-1', '1412.7332-2-24-1'], ['1412.7332-1-22-2', '1412.7332-2-24-2'], ['1412.7332-1-4-0', '1412.7332-2-4-0'], ['1412.7332-1-4-3', '1412.7332-2-4-3'], ['1412.7332-1-34-0', '1412.7332-2-36-0'], ['1412.7332-1-34-1', '1412.7332-2-36-1'], ['1412.7332-1-20-0', '1412.7332-2-22-0'], ['1412.7332-1-20-1', '1412.7332-2-22-1'], ['1412.7332-1-10-0', '1412.7332-2-9-0'], ['1412.7332-1-10-1', '1412.7332-2-9-1'], ['1412.7332-1-13-0', '1412.7332-2-12-0'], ['1412.7332-1-38-0', '1412.7332-2-40-0'], ['1412.7332-1-38-1', '1412.7332-2-40-1'], ['1412.7332-1-38-2', '1412.7332-2-40-2'], ['1412.7332-1-38-3', '1412.7332-2-40-3'], ['1412.7332-1-38-4', '1412.7332-2-40-4'], ['1412.7332-1-0-0', '1412.7332-2-0-0'], ['1412.7332-1-0-3', '1412.7332-2-0-3'], ['1412.7332-1-0-5', '1412.7332-2-0-5'], ['1412.7332-1-0-6', '1412.7332-2-0-6'], ['1412.7332-2-61-0', '1412.7332-3-61-0'], ['1412.7332-2-29-0', '1412.7332-3-30-0'], ['1412.7332-2-29-1', '1412.7332-3-30-1'], ['1412.7332-2-63-1', '1412.7332-3-63-1'], ['1412.7332-2-63-2', '1412.7332-3-63-2'], ['1412.7332-2-27-0', '1412.7332-3-28-0'], ['1412.7332-2-77-4', '1412.7332-3-77-4'], ['1412.7332-2-36-0', '1412.7332-3-37-0'], ['1412.7332-2-36-1', '1412.7332-3-37-1'], ['1412.7332-2-51-0', '1412.7332-3-52-0'], ['1412.7332-2-23-0', '1412.7332-3-24-0'], ['1412.7332-2-23-1', '1412.7332-3-24-1'], ['1412.7332-2-47-0', '1412.7332-3-48-0'], ['1412.7332-2-47-1', '1412.7332-3-48-1'], ['1412.7332-2-47-2', '1412.7332-3-48-2'], ['1412.7332-2-47-3', '1412.7332-3-48-3'], ['1412.7332-2-31-0', '1412.7332-3-32-0'], ['1412.7332-2-31-1', '1412.7332-3-32-1'], ['1412.7332-2-31-2', '1412.7332-3-32-2'], ['1412.7332-2-31-3', '1412.7332-3-32-3'], ['1412.7332-2-31-4', '1412.7332-3-32-4'], ['1412.7332-2-31-5', '1412.7332-3-32-5'], ['1412.7332-2-31-6', '1412.7332-3-32-6'], ['1412.7332-2-31-7', '1412.7332-3-32-7'], ['1412.7332-2-1-0', '1412.7332-3-1-0'], ['1412.7332-2-49-0', '1412.7332-3-50-0'], ['1412.7332-2-49-1', '1412.7332-3-50-1'], ['1412.7332-2-49-2', '1412.7332-3-50-2'], ['1412.7332-2-70-0', '1412.7332-3-70-0'], ['1412.7332-2-70-1', '1412.7332-3-70-1'], ['1412.7332-2-70-2', '1412.7332-3-70-2'], ['1412.7332-2-14-0', '1412.7332-3-15-0'], ['1412.7332-2-14-1', '1412.7332-3-15-1'], ['1412.7332-2-14-2', '1412.7332-3-15-2'], ['1412.7332-2-21-0', '1412.7332-3-22-0'], ['1412.7332-2-19-0', '1412.7332-3-20-0'], ['1412.7332-2-75-0', '1412.7332-3-75-0'], ['1412.7332-2-12-0', '1412.7332-3-13-0'], ['1412.7332-2-12-1', '1412.7332-3-13-1'], ['1412.7332-2-12-2', '1412.7332-3-13-2'], ['1412.7332-2-12-3', '1412.7332-3-13-3'], ['1412.7332-2-3-0', '1412.7332-3-3-0'], ['1412.7332-2-3-1', '1412.7332-3-3-1'], ['1412.7332-2-3-2', '1412.7332-3-3-2'], ['1412.7332-2-3-3', '1412.7332-3-3-3'], ['1412.7332-2-58-0', '1412.7332-3-58-0'], ['1412.7332-2-69-0', '1412.7332-3-69-0'], ['1412.7332-2-69-1', '1412.7332-3-69-1'], ['1412.7332-2-13-0', '1412.7332-3-14-0'], ['1412.7332-2-26-2', '1412.7332-3-27-2'], ['1412.7332-2-26-3', '1412.7332-3-27-3'], ['1412.7332-2-52-0', '1412.7332-3-53-0'], ['1412.7332-2-74-1', '1412.7332-3-74-1'], ['1412.7332-2-24-0', '1412.7332-3-25-0'], ['1412.7332-2-24-1', '1412.7332-3-25-1'], ['1412.7332-2-24-2', '1412.7332-3-25-2'], ['1412.7332-2-38-0', '1412.7332-3-39-0'], ['1412.7332-2-38-1', '1412.7332-3-39-1'], ['1412.7332-2-38-2', '1412.7332-3-39-2'], ['1412.7332-2-43-0', '1412.7332-3-44-0'], ['1412.7332-2-43-1', '1412.7332-3-44-1'], ['1412.7332-2-43-2', '1412.7332-3-44-2'], ['1412.7332-2-43-3', '1412.7332-3-44-3'], ['1412.7332-2-43-4', '1412.7332-3-44-4'], ['1412.7332-2-43-5', '1412.7332-3-44-5'], ['1412.7332-2-43-6', '1412.7332-3-44-6'], ['1412.7332-2-43-7', '1412.7332-3-44-7'], ['1412.7332-2-43-8', '1412.7332-3-44-8'], ['1412.7332-2-43-9', '1412.7332-3-44-9'], ['1412.7332-2-8-3', '1412.7332-3-9-2'], ['1412.7332-2-8-4', '1412.7332-3-9-3'], ['1412.7332-2-8-7', '1412.7332-3-9-6'], ['1412.7332-2-8-8', '1412.7332-3-9-7'], ['1412.7332-2-5-0', '1412.7332-3-5-0'], ['1412.7332-2-5-1', '1412.7332-3-5-1'], ['1412.7332-2-5-2', '1412.7332-3-5-2'], ['1412.7332-2-5-3', '1412.7332-3-5-3'], ['1412.7332-2-41-0', '1412.7332-3-42-0'], ['1412.7332-2-41-1', '1412.7332-3-42-1'], ['1412.7332-2-41-2', '1412.7332-3-42-2'], ['1412.7332-2-41-3', '1412.7332-3-42-3'], ['1412.7332-2-41-4', '1412.7332-3-42-4'], ['1412.7332-2-41-5', '1412.7332-3-42-5'], ['1412.7332-2-46-0', '1412.7332-3-47-0'], ['1412.7332-2-18-0', '1412.7332-3-19-0'], ['1412.7332-2-18-1', '1412.7332-3-19-1'], ['1412.7332-2-18-2', '1412.7332-3-19-2'], ['1412.7332-2-18-3', '1412.7332-3-19-3'], ['1412.7332-2-18-4', '1412.7332-3-19-4'], ['1412.7332-2-18-5', '1412.7332-3-19-5'], ['1412.7332-2-33-0', '1412.7332-3-34-0'], ['1412.7332-2-33-1', '1412.7332-3-34-1'], ['1412.7332-2-33-2', '1412.7332-3-34-2'], ['1412.7332-2-10-0', '1412.7332-3-11-0'], ['1412.7332-2-10-5', '1412.7332-3-11-5'], ['1412.7332-2-7-0', '1412.7332-3-8-0'], ['1412.7332-2-7-3', '1412.7332-3-8-4'], ['1412.7332-2-57-0', '1412.7332-3-57-0'], ['1412.7332-2-57-1', '1412.7332-3-57-1'], ['1412.7332-2-60-0', '1412.7332-3-60-0'], ['1412.7332-2-60-2', '1412.7332-3-60-2'], ['1412.7332-2-15-0', '1412.7332-3-16-0'], ['1412.7332-2-16-0', '1412.7332-3-17-0'], ['1412.7332-2-28-0', '1412.7332-3-29-0'], ['1412.7332-2-28-1', '1412.7332-3-29-1'], ['1412.7332-2-28-2', '1412.7332-3-29-2'], ['1412.7332-2-28-3', '1412.7332-3-29-3'], ['1412.7332-2-28-4', '1412.7332-3-29-4'], ['1412.7332-2-45-0', '1412.7332-3-46-0'], ['1412.7332-2-45-1', '1412.7332-3-46-1'], ['1412.7332-2-45-2', '1412.7332-3-46-2'], ['1412.7332-2-45-3', '1412.7332-3-46-3'], ['1412.7332-2-73-0', '1412.7332-3-73-0'], ['1412.7332-2-73-1', '1412.7332-3-73-1'], ['1412.7332-2-73-2', '1412.7332-3-73-2'], ['1412.7332-2-73-3', '1412.7332-3-73-3'], ['1412.7332-2-73-4', '1412.7332-3-73-4'], ['1412.7332-2-73-5', '1412.7332-3-73-5'], ['1412.7332-2-9-0', '1412.7332-3-10-0'], ['1412.7332-2-9-1', '1412.7332-3-10-1'], ['1412.7332-2-59-0', '1412.7332-3-59-0'], ['1412.7332-2-59-1', '1412.7332-3-59-1'], ['1412.7332-2-37-0', '1412.7332-3-38-0'], ['1412.7332-2-37-1', '1412.7332-3-38-1'], ['1412.7332-2-37-2', '1412.7332-3-38-2'], ['1412.7332-2-37-3', '1412.7332-3-38-3'], ['1412.7332-2-67-0', '1412.7332-3-67-0'], ['1412.7332-2-67-1', '1412.7332-3-67-1'], ['1412.7332-2-67-2', '1412.7332-3-67-2'], ['1412.7332-2-40-0', '1412.7332-3-41-0'], ['1412.7332-2-40-1', '1412.7332-3-41-1'], ['1412.7332-2-40-2', '1412.7332-3-41-2'], ['1412.7332-2-40-3', '1412.7332-3-41-3'], ['1412.7332-2-40-4', '1412.7332-3-41-4'], ['1412.7332-2-71-1', '1412.7332-3-71-1'], ['1412.7332-2-0-0', '1412.7332-3-0-0'], ['1412.7332-2-0-1', '1412.7332-3-0-1'], ['1412.7332-2-0-2', '1412.7332-3-0-2'], ['1412.7332-2-0-3', '1412.7332-3-0-3'], ['1412.7332-2-30-0', '1412.7332-3-31-0'], ['1412.7332-2-72-0', '1412.7332-3-72-0'], ['1412.7332-2-72-1', '1412.7332-3-72-1'], ['1412.7332-2-72-2', '1412.7332-3-72-2'], ['1412.7332-2-72-3', '1412.7332-3-72-3'], ['1412.7332-2-72-4', '1412.7332-3-72-4'], ['1412.7332-2-17-0', '1412.7332-3-18-0'], ['1412.7332-2-17-1', '1412.7332-3-18-1'], ['1412.7332-2-17-2', '1412.7332-3-18-2'], ['1412.7332-2-4-0', '1412.7332-3-4-0'], ['1412.7332-2-4-1', '1412.7332-3-4-1'], ['1412.7332-2-4-2', '1412.7332-3-4-2'], ['1412.7332-2-4-3', '1412.7332-3-4-3'], ['1412.7332-2-20-0', '1412.7332-3-21-0'], ['1412.7332-2-20-1', '1412.7332-3-21-1'], ['1412.7332-2-20-2', '1412.7332-3-21-2'], ['1412.7332-2-20-3', '1412.7332-3-21-3'], ['1412.7332-2-22-0', '1412.7332-3-23-0'], ['1412.7332-2-22-1', '1412.7332-3-23-1'], ['1412.7332-2-22-2', '1412.7332-3-23-2'], ['1412.7332-2-44-0', '1412.7332-3-45-0'], ['1412.7332-2-44-1', '1412.7332-3-45-1'], ['1412.7332-2-44-2', '1412.7332-3-45-2'], ['1412.7332-2-44-3', '1412.7332-3-45-3'], ['1412.7332-2-44-4', '1412.7332-3-45-4'], 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'1412.7332-3-74-0', '1412.7332-3-76-0', '1412.7332-3-77-0', '1412.7332-3-77-1', '1412.7332-3-77-2', '1412.7332-3-77-3', '1412.7332-4-29-0', '1412.7332-4-32-3', '1412.7332-4-33-0', '1412.7332-4-34-0', '1412.7332-4-45-1', '1412.7332-4-48-0', '1412.7332-4-59-0', '1412.7332-4-75-0', '1412.7332-4-75-1', '1412.7332-4-82-2', '1412.7332-4-84-0', '1412.7332-4-85-0', '1412.7332-4-86-0', '1412.7332-4-86-1', '1412.7332-4-86-2', '1412.7332-4-86-3', '1412.7332-4-87-0', '1412.7332-4-87-1', '1412.7332-4-88-4', '1412.7332-4-89-0', '1412.7332-4-89-1', '1412.7332-4-90-0', '1412.7332-4-90-1', '1412.7332-4-92-0', '1412.7332-4-93-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1412.7332,"{'1412.7332-3-0-0': 'Observations which are realizations from some continuous process are frequent in sciences, engineering, economics, and other fields.', '1412.7332-3-0-1': 'We consider linear models, with possible random effects, where the responses are random functions in a suitable Sobolev space.', '1412.7332-3-0-2': 'The processes can not be observed directly.', '1412.7332-3-0-3': 'With smoothing procedures from the original data, both the response curves and their derivatives can be reconstructed, even separately.', '1412.7332-3-0-4': 'From both these samples of functions, just one sample of representatives is obtained to estimate the vector of functional parameters.', '1412.7332-3-0-5': 'We hence get a strong functional version of the Gauss-Markov theorem.', '1412.7332-3-1-0': 'Keywords: functional data analysis; Sobolev spaces; linear models; repeated measurements; Gauss-Markov theorem; Riesz representation theorem; best linear unbiased estimator; experimental designs; optimal design.', '1412.7332-3-2-0': '# Introduction', '1412.7332-3-3-0': 'Observations which are realizations from some continuous process are ubiquitous in many fields like sciences, engineering, economics and other fields.', '1412.7332-3-3-1': 'For this reason, the interest for statistical modeling of functional data is increasing, with applications in many areas.', '1412.7332-3-3-2': 'Reference monographs on functional data analysis are, for instance, the books of [CITATION] and [CITATION], and the book of [CITATION] for the non-parametric approach.', '1412.7332-3-3-3': 'They cover topics like data representation, smoothing and registration; regression models; classification, discrimination and principal component analysis; derivatives and principal differential analysis; and more other.', '1412.7332-3-4-0': 'Regression models with functional variables can cover different situations: it may be the case of functional responses, or functional predictors, or both.', '1412.7332-3-4-1': 'In the present paper linear models with functional response and multivariate (or univariate) regressor are considered.', '1412.7332-3-4-2': 'We consider the case of repeated measurements, which may be particularly useful in the context of experimental designs, but all the theoretical results proved remain valid in the standard case.', '1412.7332-3-4-3': 'Focus of the work is the best estimation of the functional coefficients of the regressors.', '1412.7332-3-5-0': 'The use of derivatives is very important for exploratory analysis of functional data as well as for inference and prediction methodologies.', '1412.7332-3-5-1': 'High quality derivative information can be provided, for instance, by reconstructing the functions with spline smoothing procedures.', '1412.7332-3-5-2': 'Recent developments on estimation of derivatives are contained in the works of [CITATION] and in [CITATION].', '1412.7332-3-5-3': 'See also [CITATION], who have obtained derivatives in the context of survival analysis, and [CITATION] who have estimated derivatives in a non-parametric model.', '1412.7332-3-6-0': 'In the literature the usual space for functional data is [MATH], and the data are used to reconstruct curve functions or derivatives.', '1412.7332-3-6-1': 'The novelty of the present work is that the curves are random elements of a suitable Sobolev space.', '1412.7332-3-6-2': 'The heuristic justification for this choice is that the data may provide information on both curve functions and their derivatives.', '1412.7332-3-6-3': 'Of course, if we take into consideration the whole information, about curves and derivatives, we may improve our estimates.', '1412.7332-3-6-4': 'Curves and derivatives are actually reconstructed from a set of observed values, because the response processes cannot be observed directly.', '1412.7332-3-6-5': 'Two situations may occur: the sample of functions are reconstructed by a smoothing procedure of the data, and derivatives are then obtained by differentiation.', '1412.7332-3-6-6': 'At our knowledge, this is the most common method adopted in functional data analysis.', '1412.7332-3-7-0': 'However, the sample of functions and the sample of derivatives may be obtained separately.', '1412.7332-3-7-1': 'For instance, different smoothing techniques may be used to obtain the functions and the derivatives.', '1412.7332-3-7-2': 'Another possibility is when two sets of data are available, which are suitable to estimate functions and derivatives, respectively.', '1412.7332-3-7-3': 'For example, in the case of a motion process, data concerning positions and data about velocities may be observed.', '1412.7332-3-8-0': 'In this paper we propose a new method which incorporates the information provided by both the sample of functions and the sample of derivatives.', '1412.7332-3-8-1': 'We show that the full information of the sample of functions and the sample of derivatives is not a weighted mean between, e.g., the functions and the integrated derivatives.', '1412.7332-3-8-2': 'From the two samples of reconstructed functions and derivatives just one sample of representatives is obtained.', '1412.7332-3-8-3': 'In addition, even if the sample of functions and the sample of derivatives belong to [MATH], the information carried by them on the regression parameters is a sample of function in [MATH], which implies that this new sample cannot be obtained from a linear combination involving the original sample of functions.', '1412.7332-3-8-4': 'We use this sample to estimate the functional parameters.', '1412.7332-3-8-5': 'Once this method is found, the optimal results may appear as a straightforward extension of the well-known classical case, althought the proof requires much technical effort.', '1412.7332-3-9-0': 'A new version of the Gauss-Markov theorem is hence proved in the proper infinite-dimensional space ([MATH]), showing that our new sample carries all the relevant information on the parameters.', '1412.7332-3-9-1': 'More precisely, we prove that any linear operator on [MATH] applied to the OLS estimator on the reconstructed sample has minimum variance.', '1412.7332-3-9-2': 'In this sense, the OLS estimator is [MATH]-strong BLUE.', '1412.7332-3-9-3': 'As a particular case, the OLS estimator is also the best linear unbiased estimator in [MATH].', '1412.7332-3-9-4': 'In practice this [MATH]-BLUE cannot be computed explicitly, since the representative is only implicitly defined in infinite dimensions.', '1412.7332-3-9-5': 'Nevertheless, practitioners are used to represent functions in a basis expansion which is truncated at some order.', '1412.7332-3-9-6': 'This means to work in a finite-dimensional subspace [MATH] of [MATH].', '1412.7332-3-9-7': 'We prove that the OLS estimator in [MATH] is the projection of the [MATH]-BLUE and it is itself the [MATH]-BLUE in [MATH].', '1412.7332-3-9-8': 'Moreover, the OLS estimator in [MATH] can be practically computed, since the projection of the representative depends explicitly on the sample of functions and derivatives.', '1412.7332-3-10-0': 'As a consequence of the results proved, a rigorous generalization of the theory of optimal design of experiments in infinite-dimensional spaces is presented (see also [CITATION]).', '1412.7332-3-10-1': 'An application to an ergonomic data-set shows the advantages of this theory in the real world.', '1412.7332-3-11-0': 'The paper is organized as follows.', '1412.7332-3-11-1': 'Section [REF] contains the model description and the new theoretical results offered in this work.', '1412.7332-3-11-2': 'Section [REF] presents some considerations which are fundamental from a practical point of view.', '1412.7332-3-11-3': 'Section [REF] is focused on the experimental context, and optimal designs for the model estimation are derived.', '1412.7332-3-11-4': 'Section [REF] is a summary together with some final remarks.', '1412.7332-3-11-5': 'Some additional results and proofs of theorems are deferred to Appendix A.', '1412.7332-3-12-0': '# Strong [MATH]-BLUE in functional linear models', '1412.7332-3-13-0': 'Let us consider a regression model where the response [MATH] is a random function which depends linearly on a vectorial (or scalar) known variable [MATH] through a functional coefficient, which needs to be estimated.', '1412.7332-3-13-1': 'In particular, we assume that [MATH] is an experimental condition and that [MATH] experiments are performed in batches at [MATH] (not necessarily different) experimental conditions [MATH].', '1412.7332-3-13-2': 'The [MATH]-th experiment is formed by [MATH] trials (repetitions) at the same [MATH].', '1412.7332-3-13-3': 'Therefore, the following random effect model is considered: [EQUATION] where [MATH] denotes the response curve of the [MATH]-th observation at the [MATH]-th experiment; [MATH] is a [MATH]-dimensional vector of known functions; [MATH] is an unknown [MATH]-dimensional functional vector; [MATH] is a zero-mean process which denotes the random effect due to the [MATH]-th experiment and takes into account the correlation among the [MATH] repetitions; [MATH] is a zero-mean error process.', '1412.7332-3-14-0': 'An example for the model [REF] is provided in Section [REF]; if [MATH] this model reduces to the functional response model described, for instance, in [CITATION].', '1412.7332-3-15-0': 'In a real world setting, the functions [MATH] are not directly observed.', '1412.7332-3-15-1': 'By a smoothing procedure from the original data, the investigator can reconstruct both the functions and their derivatives, obtaining [MATH] and [MATH], respectively.', '1412.7332-3-15-2': 'Hence we can assume that the model for the reconstructed functional data is [EQUATION] where', '1412.7332-3-16-0': 'the [MATH] bivariate vectors [MATH] are zero-mean independent and identically distributed couples of processes such that [MATH];', '1412.7332-3-17-0': 'all the [MATH] couples [MATH] are zero-mean identically distributed processes, each process being independent of all the other processes, with [MATH].', '1412.7332-3-18-0': 'Not that the investigator might reconstruct each function [MATH] and its derivative [MATH] separately.', '1412.7332-3-18-1': 'In this case, the right-hand term of the second equation in [REF] is not the derivative of the right-hand term of the first equation.', '1412.7332-3-18-2': 'The particular case when [MATH] is obtained by differentiation [MATH] is the most simple situation in model [REF].', '1412.7332-3-19-0': 'Let us consider an estimator [MATH] of [MATH], formed by [MATH] random functions in the Sobolev space [MATH].', '1412.7332-3-19-1': 'Recall that a function [MATH] is in [MATH] if [MATH] and its derivative [MATH] belong to [MATH].', '1412.7332-3-19-2': 'Moreover, [MATH] is an Hilbert space with inner product [EQUATION]', '1412.7332-3-19-3': 'We define the [MATH]-generalized covariance matrix [MATH] of an unbiased estimator [MATH] as the [MATH] matrix whose [MATH]-th element is [EQUATION]', '1412.7332-3-19-4': 'This global notion of covariance has been used also in [CITATION], in the context of predicting georeferenced functional data.', '1412.7332-3-19-5': 'These authors have found a BLUE estimator for the drift of their underlying process, which can be seen as an example of the results given in this paper.', '1412.7332-3-20-0': 'In analogy with classical settings, we define the [MATH]-functional best linear unbiased estimator ([MATH]-BLUE) as the estimator with minimal (in the sense of Loewner Partial Order) [MATH]-generalized covariance matrix [REF], in the class of the linear unbiased estimators of [MATH].', '1412.7332-3-21-0': 'It is simple to show that an equivalent definition of [MATH]-BLUE.', '1412.7332-3-21-1': 'In fact a [MATH]-BLUE minimizes the quantity [EQUATION] for any choice of [MATH], in the class of the linear unbiased estimators [MATH] of [MATH].', '1412.7332-3-21-2': 'In other words, the [MATH]-BLUE minimizes the [MATH]-variance of any linear combination of its components.', '1412.7332-3-21-3': 'A stronger request is the following.', '1412.7332-3-22-0': 'We define the [MATH]-strong functional best linear unbiased estimator ([MATH]-SBLUE) as the estimator with minimal variance, [EQUATION] for any choice of linear operator [MATH], in the class of the linear unbiased estimators [MATH] of [MATH].', '1412.7332-3-23-0': 'Given a couple [MATH], it may be defined a linear continuous operator on [MATH] as follows [EQUATION]', '1412.7332-3-23-1': 'From the Riesz representation theorem, there exists a unique [MATH] such that [EQUATION]', '1412.7332-3-23-2': 'The unique element [MATH] defined in [REF] is called the Riesz representative of the couple [MATH].', '1412.7332-3-24-0': 'This definition will be useful to provide a nice expression for the functional OLS estimator [MATH].', '1412.7332-3-24-1': 'Actually the Riesz representative synthesizes, in some sense, in [MATH] the information of both [MATH] and [MATH].', '1412.7332-3-25-0': 'The functional OLS estimator for the model [REF] is [EQUATION]', '1412.7332-3-25-1': 'The quantity [EQUATION] resembles [EQUATION] because [MATH] and [MATH] reconstruct [MATH] and its derivative function, respectively.', '1412.7332-3-25-2': 'The functional OLS estimator [MATH] minimizes, in this sense, the sum of the [MATH]-norm of the unobservable residuals [MATH].', '1412.7332-3-26-0': 'Given a model as in [REF],', '1412.7332-3-27-0': '[Proof of Theorem [REF]] Part a).', '1412.7332-3-27-1': 'We consider the sum of square residuals: [EQUATION]', '1412.7332-3-27-2': 'The Gateaux derivative of [MATH] at [MATH] in the direction of [MATH] is [EQUATION] where [MATH] and [MATH] are two [MATH] vectors whose [MATH]-th elements are [EQUATION].', '1412.7332-3-27-3': 'Developing the right-hand side of ([REF]), we have that the Gateaux derivative is [EQUATION] where [MATH] is a [MATH] vector whose [MATH]-th element is the Riesz representative of [MATH].', '1412.7332-3-28-0': 'The Gateaux derivative [REF] is equal to [MATH] for any [MATH] if and only if [MATH] is given by the following equation: [EQUATION] which proves the first statement of the theorem.', '1412.7332-3-29-0': 'Part b) Definition [REF] and model [REF] imply that, for any [MATH], [EQUATION] then [MATH], and hence [MATH] is unbiased.', '1412.7332-3-29-1': 'Moreover, [EQUATION] where [MATH] and [MATH] denote the Riesz representatives of [MATH] and [MATH], respectively.', '1412.7332-3-29-2': 'From the hypothesis ([REF]) and ([REF]) in the model [REF], the left-hand side quantities in [REF] are zero-mean i.i.d. processes, for [MATH].', '1412.7332-3-29-3': 'Therefore, the generalized covariance matrix of [MATH] is [MATH], where [MATH].', '1412.7332-3-29-4': 'Hence, the generalized covariance matrix of [MATH] is [MATH].', '1412.7332-3-30-0': 'The functional OLS estimator obtained in Theorem [REF] by means of the Riesz representatives is also the best linear unbiased estimator in the Sobolev space, as stated in the next theorem.', '1412.7332-3-30-1': 'The proof is deferred to Appendix A.', '1412.7332-3-31-0': 'The functional OLS estimator [MATH] for the model [REF] is a [MATH]-functional S BLUE.', '1412.7332-3-32-0': 'The theory and the results presented in this work may be generalized to other Sobolev spaces.', '1412.7332-3-32-1': 'The extension to [MATH], [MATH], is straightforward.', '1412.7332-3-32-2': 'Moreover, as in Bayesian context, the investigator might have a different a priori consideration of [MATH] and [MATH].', '1412.7332-3-32-3': 'Thus, different weights [MATH] and [MATH] may be used for curve functions and derivatives, respectively.', '1412.7332-3-32-4': 'Another interesting generalization might be to add positive weight functions [MATH] and [MATH] on [MATH], when, for instance, distinct zones of [MATH] are considered to have different relevance.', '1412.7332-3-32-5': 'Therefore, the inner product given in [REF] may be extended to [EQUATION].', '1412.7332-3-32-6': 'When [MATH] and [MATH], the Hilbert space is called weighted Sobolev space, see [CITATION].', '1412.7332-3-32-7': 'These generalizations, which have different impacts on applications, will be object of future works.', '1412.7332-3-33-0': '# Practical considerations', '1412.7332-3-34-0': 'In a real world context, we work with a finite dimensional subspace [MATH] of [MATH].', '1412.7332-3-34-1': 'Let [MATH] be a base of [MATH].', '1412.7332-3-34-2': 'Without loss of generality, we may assume that [MATH], where [EQUATION] is the Kronecker delta symbol, since a Gram-Schmidt orthonormalization procedure may be always applied.', '1412.7332-3-34-3': 'More precisely, given any base [MATH] in [MATH], the corresponding orthonormal base is given by:', '1412.7332-3-35-0': 'for [MATH], define [MATH],', '1412.7332-3-36-0': 'for [MATH], let [MATH] and [MATH].', '1412.7332-3-37-0': 'With this orthonormalized base, the projection [MATH] on [MATH] of the Riesz representative [MATH] of the couple [MATH] is given by [EQUATION] where the last equality comes from the definition [REF] of the Riesz representative.', '1412.7332-3-37-1': 'Now, if [MATH] is the [MATH]-th row of [MATH], then [EQUATION] hence [MATH].', '1412.7332-3-38-0': 'Let us note that, even if the Riesz representative [REF] is implicitly defined, its projection on [MATH] can be easily computed by [REF].', '1412.7332-3-38-1': 'From a practical point of view, the statistician can work with the data [MATH] projected on a finite linear subspace [MATH] and the corresponding OLS estimator [MATH] is the projection on [MATH] of the [MATH]-redS BLUE estimator [MATH] obtained in the Section [REF].', '1412.7332-3-38-2': 'As a consequence of the Theorem [REF], [MATH] is [MATH]-redS BLUE in [MATH], since it is unbiased and the projection is linear.', '1412.7332-3-38-3': 'For the projection, it is crucial to take a base of [MATH] which is orthonormal in [MATH].', '1412.7332-3-39-0': 'It is straightforward to prove that the estimator [REF] becomes [EQUATION] in two cases: when we do not take into consideration [MATH], or when [MATH].', '1412.7332-3-39-1': 'In both the cases, from the results obtained in the Section [REF], [MATH] is an [MATH]-BLUE.', '1412.7332-3-39-2': 'Up to our knowledge, this is the most common situation considered in the literature.', '1412.7332-3-40-0': '# Optimal designs of experiments in functional models', '1412.7332-3-41-0': 'In this section, we assume to work in an experimental setup.', '1412.7332-3-41-1': 'Therefore, [MATH], with [MATH], are not observed auxiliary variables; they can be freely chosen by an experimenter on the design space [MATH].', '1412.7332-3-41-2': 'The set of experimental conditions [MATH] is called an exact design.', '1412.7332-3-41-3': 'A more general definition is that of continuous design, as a probability measure [MATH] with support on [MATH] (see, for instance, [CITATION]).', '1412.7332-3-41-4': 'The choice of [MATH] may be done with the aim of obtaining accurate estimates of the model parameters.', '1412.7332-3-42-0': 'From Theorem [REF], [MATH] given in [REF] is the [MATH]-BLUE for the model [REF].', '1412.7332-3-42-1': 'This optimal estimator can be further improved by a ""clever"" choice of the design.', '1412.7332-3-42-2': 'In analogy with the criteria proposed in the finite-dimensional theory (see for instance, [CITATION]) we define a functional optimal design as a design which minimizes an appropriate convex function of the generalized covariance matrix [MATH] (see Definition [REF]).', '1412.7332-3-42-3': 'For instance, a functional D-optimum design is a design [MATH] which minimizes [MATH].', '1412.7332-3-42-4': 'Part b) of Theorem [REF] proves that [MATH].', '1412.7332-3-42-5': 'From the definition of continuous design [EQUATION] and hence [MATH] maximizes [MATH].', '1412.7332-3-43-0': '## An example: the ergonomic data', '1412.7332-3-44-0': 'Herein, we study the performance of the design proposed in [CITATION].', '1412.7332-3-44-1': 'In detail, to forecast the motion of drivers, some data are to be collected on the motion of a single subject to different locations within a test car.', '1412.7332-3-44-2': 'An experimental design is given by the choice of these locations (the different experimental conditions) and by the number of times that the experiment has to be replicated at each location.', '1412.7332-3-44-3': 'The response curve [MATH] is the angle formed at the right elbow between the upper and the lower arm, which is measured by a motion capture equipment.', '1412.7332-3-44-4': 'In the design used by [CITATION], 3 motion curves were observed at 20 different locations, spread around the glove compartment, gear shift, the central instrument panel and an overhead panel.', '1412.7332-3-44-5': ""These data are available from Faraway's website."", '1412.7332-3-44-6': 'In [CITATION], 3 different models are compared to predict the motion given the coordinates [MATH].', '1412.7332-3-44-7': 'From this comparison, the following quadratic model seems to be adequate: [EQUATION] where [EQUATION].', '1412.7332-3-44-8': 'Let us denote by [MATH] the exact design used in [CITATION] to collect observations [MATH].', '1412.7332-3-44-9': 'This design has 20 different locations at which three trials are repeated, therefore the total number of observations is [MATH].', '1412.7332-3-45-0': 'We standardize support points (coordinates) of [MATH] so that they belong to [MATH].', '1412.7332-3-45-1': 'For this experimental domain, [CITATION] provides both the continuous D-optimum design, say [MATH], and exact D-optimum designs.', '1412.7332-3-45-2': 'Since [MATH] has support on the points of the [MATH] factorial (with [MATH]), [CITATION] proposes to search an exact D-optimum design (for a small [MATH]) over the points of the [MATH] factorial.', '1412.7332-3-45-3': 'To be consistent with the design used in [CITATION], we consider the exact design with [MATH] locations, some of which may be repeated.', '1412.7332-3-45-4': 'At each location, an experiment is performed in batches of three trials.', '1412.7332-3-45-5': 'When a location is replicated more times, several experiments (each one formed by three trials) is performed at this location.', '1412.7332-3-45-6': 'Thus, again the total number of observation is [MATH].', '1412.7332-3-45-7': 'Let us call this design [MATH].', '1412.7332-3-46-0': 'In addition, we search an exact D-optimum design over the whole experimental domain [MATH].', '1412.7332-3-46-1': 'This exact design has 9 different locations which are not points of the [MATH] factorial, 5 different points of the [MATH] factorial which are not replicated and 3 different points of the [MATH] factorial which are replicated twice.', '1412.7332-3-46-2': 'Three observations are taken at each location.', '1412.7332-3-46-3': 'Let this exact design be denoted by [MATH].', '1412.7332-3-47-0': 'Finally, in a grid of [MATH] equally spaced points in [MATH], an exact D-optimum design over the [MATH] factorial has been find with Matlab 2014b routine rowexch, finding [MATH].', '1412.7332-3-48-0': 'In practice all designs are exact.', '1412.7332-3-48-1': 'Therefore, the continuous D-optimum design [MATH] is used only as a benchmark, in order to measure the goodness of a design [MATH] with respect to it.', '1412.7332-3-48-2': 'As a measure of goodness the D-efficiency is used: [EQUATION] where [MATH] is the dimension of the functional vector [MATH].', '1412.7332-3-48-3': 'The D-efficiency [MATH] is proportional to the design size, thus a design [MATH] with D-efficiency equal to 0,5 needs the double of observations of [MATH] to get the same precision in the estimates.', '1412.7332-3-49-0': 'The following table lists the D-efficiencies of [MATH], [MATH], [MATH] and [MATH].', '1412.7332-3-50-0': 'From Table [REF] we have that the exact D-optimum designs [MATH] and [MATH] are almost equivalent.', '1412.7332-3-50-1': 'The D-efficiency of [MATH] is instead very low, only equal to [MATH].', '1412.7332-3-50-2': 'This example clearly shows the importance of choosing the experimental design according to an optimality criterion.', '1412.7332-3-51-0': '# Summary', '1412.7332-3-52-0': 'Functional data are suitably modeled in separable Hilbert spaces (see [CITATION] and [CITATION]) and [MATH] is usually sufficient to handle the majority of the techniques proposed in the literature of functional data analysis.', '1412.7332-3-53-0': 'Differently, we consider proper Sobolev spaces, since we guess that the data may provide information on both curve functions and their derivatives.', '1412.7332-3-53-1': 'The classical theory for linear regression models is extended to this context by means of the the sample of Riesz representatives.', '1412.7332-3-53-2': 'Roughly speaking, the Riesz representatives are ""quantities"" which incorporates both functions and derivatives information in a non trivial way.', '1412.7332-3-54-0': 'Using the sample of Riesz representatives, we provide a strong, generalized version of the well known Gauss-Markov theorem for the functional linear regression models considered.', '1412.7332-3-54-1': 'Despite the complexity of the problem we obtain an elegant and simple solution, through the use of the Riesz representatives which always belong to a Sobolev space.', '1412.7332-3-55-0': 'Interesting studies on experimental design with functional observations are presented in the recent literature (see, for instance, [CITATION], [CITATION] and [CITATION]).', '1412.7332-3-55-1': 'In the present work a rigorous theoretical support is provided to apply D-optimal designs to linear models with functional responses (similar models are considered, for instance, in [CITATION], [CITATION], [CITATION]).', '1412.7332-3-55-2': 'As a matter of future work, we will study also different optimality criteria in the context of functional data models.', '1412.7332-3-55-3': 'In addition, we intend to develop our theory also in the functional setting considered by [CITATION], where the design region is a subset of some functional space and the response is a scalar quantity.', '1412.7332-3-56-0': '# Proof of Theorem [REF]', '1412.7332-3-57-0': 'The OLS [MATH] is a linear map which associates an element [MATH] in [MATH] to any [MATH]-tuple [MATH] in [MATH].', '1412.7332-3-57-1': 'In what follows, we show that it is the ""best"" among all the linear unbiased operators [MATH].', '1412.7332-3-58-0': 'The model [REF] may be written in the following vectorial form: [EQUATION] where [MATH] is the column vector of length [MATH] with all components equal to [MATH].', '1412.7332-3-59-0': 'Let [EQUATION] be a [MATH] block vector, with [MATH].', '1412.7332-3-59-1': 'Given any couple of [MATH] block vectors [MATH], we may define the following [MATH] dimensional vector [EQUATION] where [EQUATION] and [MATH] is the Riesz representative of [MATH] as in [REF].', '1412.7332-3-60-0': 'Now we can introduce the following linear operator [EQUATION]', '1412.7332-3-60-1': 'Hence, [EQUATION] and [EQUATION].', '1412.7332-3-60-2': 'The thesis follows immediately if we prove that [MATH] and [MATH] are uncorrelated.', '1412.7332-3-61-0': 'Since both [MATH] and [MATH] are unbiased, [MATH], and thus we have to prove that [EQUATION] for any choice of linear operator [MATH].', '1412.7332-3-62-0': 'The proof of equality ([REF]) is developed in five steps.', '1412.7332-3-63-0': 'First step.', '1412.7332-3-63-1': 'The goal of this step is to prove that [MATH] applied to the deterministic part of the model [MATH] is identically null.', '1412.7332-3-63-2': 'As a consequence, [EQUATION]', '1412.7332-3-64-0': 'Proof', '1412.7332-3-65-0': 'From the linearity of [MATH] and the zero-mean hypothesis ([REF]) and ([REF]), we have that [EQUATION]', '1412.7332-3-65-1': 'Since [MATH] we have that [EQUATION]', '1412.7332-3-65-2': 'In addition, from the definition ([REF]) if [EQUATION] then [EQUATION]', '1412.7332-3-65-3': 'Combining [REF], [REF] and [REF] gives [EQUATION] and hence [REF].', '1412.7332-3-66-0': 'Second step.', '1412.7332-3-66-1': 'Representation of the linear operator [MATH].', '1412.7332-3-67-0': 'For the linearity of the [MATH]-th component of [MATH] with respect to the bivariate observations [MATH]: [EQUATION] where, for any [MATH] and [MATH], [MATH] is linear.', '1412.7332-3-67-1': 'The domain of [MATH] is contained in [MATH].', '1412.7332-3-67-2': 'Let [MATH] a suitable base of [MATH] that will be specified in the fourth step, and [MATH] be an orthonormal base of [MATH].', '1412.7332-3-67-3': 'With this notation [EQUATION] where [EQUATION]', '1412.7332-3-68-0': 'Third step.', '1412.7332-3-68-1': 'Proof of [EQUATION] where [MATH] is the [MATH]-th row of [MATH].', '1412.7332-3-69-0': 'Let [MATH] be the null vector except for the [MATH]-th component which is [MATH], and let [MATH].', '1412.7332-3-69-1': 'Setting [MATH] in [REF], [EQUATION] where the last equality is due to ([REF]).', '1412.7332-3-70-0': 'From the Riesz representation theorem we have that, for any [MATH], there exists [MATH] such that [EQUATION]', '1412.7332-3-70-1': 'From ([REF]), equality ([REF]) becomes [EQUATION]', '1412.7332-3-70-2': 'The arbitrary choice of [MATH] implies [REF].', '1412.7332-3-71-0': 'Fourth step.', '1412.7332-3-71-1': 'Karhunen-Loeve representation of the noise process and definition of of the base [MATH].', '1412.7332-3-72-0': 'For a given [MATH], the couple [MATH] is a process in [MATH].', '1412.7332-3-72-1': 'Let [MATH] be the spectral representation of the covariance matrix, which implies [MATH], and we assume, without loss of generality, that the sequence [MATH] forms a orthonormal base (by completing it, defining [MATH] when needed).', '1412.7332-3-72-2': 'Note that [MATH] does not depend on [MATH] and [MATH], since, from the hypothesis ([REF]) and ([REF]) in the model [REF] [EQUATION] are identically distributed.', '1412.7332-3-72-3': 'From Karhunen-Loeve Theorem (see, e.g., [CITATION]), there exists an array of zero-mean unit variance random variables [MATH] such that [EQUATION]', '1412.7332-3-72-4': 'From ([REF]) the Riesz representative of the noise process ([REF]) is [MATH] and we can define the following means of replications, [EQUATION] which will be useful in the next fifth step.', '1412.7332-3-73-0': 'Finally, from Karhunen-Loeve Theorem, we may obtain the following relations which will be useful in the fifth step of the proof.', '1412.7332-3-73-1': 'From ([REF]) we have that for any [MATH], [MATH] and [MATH], [EQUATION]', '1412.7332-3-73-2': 'In addition, the independence assumptions in the hypothesis ([REF]) and ([REF]) ensure that [MATH] and [MATH] are independent if [MATH].', '1412.7332-3-73-3': 'For the same observation (i.e. [MATH]), the Karhunen-Loeve representation gives [MATH].', '1412.7332-3-73-4': 'Finally, for different replications of the same experiment (i.e. [MATH] but [MATH]) the identically distributed bivariate process [MATH] yields to a correlation which does not depend on the experiment [MATH] neither on the replications [MATH]: [MATH].', '1412.7332-3-73-5': 'Summing up, the independence assumptions given in the hypothesis ([REF]) and ([REF]) imply [EQUATION] and hence [EQUATION]', '1412.7332-3-74-0': 'Fifth step.', '1412.7332-3-74-1': 'Proof of [REF]: [EQUATION] for any choice of linear operator [MATH].', '1412.7332-3-75-0': 'From the definitions given in the part a of Theorem [REF] and from Equations [REF], [REF] and [REF] we have that, for any [MATH] [EQUATION] which proves that [EQUATION] where [MATH].', '1412.7332-3-76-0': 'From this last result and from ([REF]), [MATH], and hence [EQUATION] where the last equality is a consequence of [REF].', '1412.7332-3-77-0': 'From the linearity of the operator [MATH], we have that [EQUATION].', '1412.7332-3-77-1': 'Since [MATH], where [MATH], we have [EQUATION].', '1412.7332-3-77-2': 'Setting [EQUATION] then [EQUATION].', '1412.7332-3-77-3': 'Hence, from Equations ([REF]), [REF] and [REF], the thesis [REF] becomes [EQUATION]', '1412.7332-3-77-4': 'From [REF], [REF] and [REF], the left-hand side of the last equation becomes [EQUATION] the last equality being a consequence of [REF].'}","{'1412.7332-4-0-0': 'Observations which are realizations from some continuous process are frequent in sciences, engineering, economics, and other fields.', '1412.7332-4-0-1': 'We consider linear models, with possible random effects, where the responses are random functions in a suitable Sobolev space.', '1412.7332-4-0-2': 'The processes cannot be observed directly.', '1412.7332-4-0-3': 'With smoothing procedures from the original data, both the response curves and their derivatives can be reconstructed, even separately.', '1412.7332-4-0-4': 'From both these samples of functions, just one sample of representatives is obtained to estimate the vector of functional parameters.', '1412.7332-4-0-5': 'A simulation study shows the benefits of this approach over the common method of using information either on curves or derivatives.', '1412.7332-4-0-6': 'The main theoretical result is a strong functional version of the Gauss-Markov theorem.', '1412.7332-4-0-7': 'This ensures that the proposed functional estimator is more efficient than the best linear unbiased estimator based only on curves or derivatives.', '1412.7332-4-1-0': 'Keywords: functional data analysis; Sobolev spaces; linear models; repeated measurements; Gauss-Markov theorem; Riesz representation theorem; best linear unbiased estimator.', '1412.7332-4-2-0': '# Introduction', '1412.7332-4-3-0': 'Observations which are realizations from some continuous process are ubiquitous in many fields like sciences, engineering, economics and other fields.', '1412.7332-4-3-1': 'For this reason, the interest for statistical modeling of functional data is increasing, with applications in many areas.', '1412.7332-4-3-2': 'Reference monographs on functional data analysis are, for instance, the books of [CITATION] and [CITATION], and the book of [CITATION] for the non-parametric approach.', '1412.7332-4-3-3': 'They cover topics like data representation, smoothing and registration; regression models; classification, discrimination and principal component analysis; derivatives and principal differential analysis; and many other.', '1412.7332-4-4-0': 'Regression models with functional variables can cover different situations: it may be the case of functional responses, or functional predictors, or both.', '1412.7332-4-4-1': 'In the present paper linear models with functional response and multivariate (or univariate) regressor are considered.', '1412.7332-4-4-2': 'We consider the case of repeated measurements but all the theoretical results remain valid in the standard case.', '1412.7332-4-4-3': 'Focus of the work is the best estimation of the functional coefficients of the regressors.', '1412.7332-4-5-0': 'The use of derivatives is very important for exploratory analysis of functional data as well as for inference and prediction methodologies.', '1412.7332-4-5-1': 'High quality derivative information can be provided, for instance, by reconstructing the functions with spline smoothing procedures.', '1412.7332-4-5-2': 'Recent developments on estimation of derivatives are contained in [CITATION] and in [CITATION].', '1412.7332-4-5-3': 'See also [CITATION], who have obtained derivatives in the context of survival analysis, and [CITATION] who have estimated derivatives in a non-parametric model.', '1412.7332-4-6-0': 'Curves and derivatives are actually reconstructed from a set of observed values, because the response processes cannot be observed directly.', '1412.7332-4-6-1': 'In the literature the usual space for functional data is [MATH], and the observed values are used to reconstruct either curve functions or derivatives.', '1412.7332-4-7-0': 'To our knowledge, the most common method to reconstruct derivatives is to build the sample of functions by a smoothing procedure of the data, and then to differentiate these curve functions.', '1412.7332-4-7-1': 'However, the sample of functions and the sample of derivatives may be obtained separately.', '1412.7332-4-7-2': 'For instance, different smoothing techniques may be used to obtain the functions and the derivatives.', '1412.7332-4-7-3': 'Another possibility is when two sets of data are available, which are suitable to estimate functions and derivatives, respectively.', '1412.7332-4-8-0': 'Some possible examples of data concerning curves and derivatives are: in studying how the velocity of a car on a particular street is influenced by some covariates, the velocity is measured by a police radar; in addition we could benefit of more information since its position is tracked by a GPS.', '1412.7332-4-8-1': 'In chemical experiments, data on reaction velocity and concentration may be collected separately.', '1412.7332-4-9-0': 'The novelty of the present work is that both information on curves and derivatives (that are not obtained by differentiation of the curves themselves) are used to estimate the functional coefficients.', '1412.7332-4-10-0': 'The heuristic justification for this choice is that the data may provide different information on curve functions and their derivatives and it is always recommended to use the whole available information.', '1412.7332-4-10-1': 'Actually, we prove that if we take into consideration both information about curves and derivatives, we obtain the best linear unbiased estimates for the functional coefficients.', '1412.7332-4-10-2': 'Therefore, the common method of using information on either curve functions or derivatives provides always a less efficient estimate (see Theorem [REF] and Remark [REF]).', '1412.7332-4-10-3': 'For this reason, our theoretical results may have a relevant impact in practice.', '1412.7332-4-11-0': 'More in detail, in analogy with the Riesz representation theorem we can find a representative function in [MATH] which incorporates the information provided by a curve function and a derivative (which belong to [MATH]).', '1412.7332-4-11-1': 'Hence, from the two samples of reconstructed functions and derivatives just one sample of representatives is obtained and we use this sample of representatives to estimate the functional parameters.', '1412.7332-4-11-2': 'Once this method is given, the consequent theoretical results may appear as a straightforward extension of the well-known classical ones; their proof, however, requires much more technical effort and it is not a straightforward extension at all.', '1412.7332-4-12-0': 'The OLS estimator (based on both curves and derivatives through their Riesz representatives in [MATH]) is provided and some practical considerations are drawn.', '1412.7332-4-12-1': 'In general, the OLS estimator is not a BLUE, because of the possible correlation between curves and derivatives.', '1412.7332-4-12-2': 'Therefore, a different representation of the data is provided (which takes into into account this correlation) and then a new version of the Gauss-Markov theorem is proved in the proper infinite-dimensional space ([MATH]), showing that our sample of representatives carries all the relevant information on the parameters.', '1412.7332-4-12-3': 'More in detail, we propose an unbiased estimator which is linear with respect to the new sample of representatives and which minimizes a suitable covariance matrix (called global variance).', '1412.7332-4-12-4': 'This estimator is denoted [MATH]-functional SBLUE.', '1412.7332-4-13-0': 'A simulation study shows numerically the superiority of the [MATH]-functional SBLUE with respect to both the OLS estimators based only on curves or derivatives.', '1412.7332-4-13-1': 'This suggests that both sources of information should be used jointly, when available.', '1412.7332-4-13-2': 'A rough way of considering information on both curves and derivatives is to make a convex combination of the two OLS estimators.', '1412.7332-4-13-3': 'However, simulations show that the [MATH]-functional SBLUE is more efficient, as expected.', '1412.7332-4-14-0': 'The paper is organized as follows.', '1412.7332-4-14-1': 'Section [REF] describes the model and proposes the OLS estimator obtained from the Riesz representation of the data.', '1412.7332-4-14-2': 'Section [REF] explains some considerations which are fundamental from a practical point of view.', '1412.7332-4-14-3': 'Section [REF] presents the construction of the functional strong BLUE.', '1412.7332-4-14-4': 'Finally, Section [REF] is devoted to the simulation study.', '1412.7332-4-14-5': 'Section [REF] is a summary together with some final remarks.', '1412.7332-4-14-6': 'Some additional results and the proofs of theorems are deferred to [REF].', '1412.7332-4-15-0': '# Model description and Riesz representation', '1412.7332-4-16-0': 'Let us consider a regression model where the response [MATH] is a random function which depends linearly on a vectorial (or scalar) known variable [MATH] through a functional coefficient, which needs to be estimated.', '1412.7332-4-16-1': 'In particular, we assume that there are [MATH] units (subjects or clusters), and [MATH] observations per unit at a condition [MATH]).', '1412.7332-4-16-2': 'Note that [MATH] are not necessarily different.', '1412.7332-4-16-3': 'In this context of repeated measurements, we consider the following random effect model: [EQUATION] where: [MATH] belongs to a compact set [MATH]; [MATH] denotes the response curve of the [MATH]-th observation at the [MATH]-th experiment; [MATH] is a [MATH]-dimensional vector of known functions; [MATH] is an unknown [MATH]-dimensional functional vector; [MATH] is a zero-mean process which denotes the random effect due to the [MATH]-th experiment and takes into account the correlation among the [MATH] repetitions; [MATH] is a zero-mean error process.', '1412.7332-4-17-0': 'Let us note that we are interested in precise estimation of the fixed effects [MATH]; herein the random effects are nuisance parameters.', '1412.7332-4-18-0': 'An example for the model [REF] can be found in [CITATION], where an ergonomic problem is considered (in this case there are [MATH] clusters of observations for the same individual); if [MATH] this model reduces to the functional response model described, for instance, in [CITATION].', '1412.7332-4-19-0': 'In a real world setting, the functions [MATH] are not directly observed.', '1412.7332-4-19-1': 'By a smoothing procedure from the original data, the investigator can reconstruct both the functions and their first derivatives, obtaining [MATH] and [MATH], respectively.', '1412.7332-4-19-2': 'Hence we can assume that the model for the reconstructed functional data is [EQUATION] where', '1412.7332-4-20-0': 'the [MATH] couples [MATH] are independent and identically distributed bivariate vectors of zero-mean processes such that [MATH], that is, [MATH], where [MATH];', '1412.7332-4-21-0': 'the [MATH] couples [MATH] are independent and identically distributed bivariate vectors of zero mean processes processes, with [MATH].', '1412.7332-4-22-0': 'As a consequence of the above assumptions: the data [MATH] and [MATH] can be correlated; the couples [MATH] and [MATH] are independent whenever [MATH].', '1412.7332-4-22-1': 'The possible correlation between [MATH] and [MATH] is due to the common random effect [MATH].', '1412.7332-4-23-0': 'Note that the investigator might reconstruct each function [MATH] and its derivative [MATH] separately.', '1412.7332-4-23-1': 'In this case, the right-hand term of the second equation in [REF] is not the derivative of the right-hand term of the first equation.', '1412.7332-4-23-2': 'The particular case when [MATH] is obtained by differentiation [MATH] is the most simple situation in model [REF].', '1412.7332-4-24-0': 'Let [MATH] be an estimator of [MATH], formed by [MATH] random functions in the Sobolev space [MATH].', '1412.7332-4-24-1': 'Recall that a function [MATH] is in [MATH] if [MATH] and its derivative [MATH] belong to [MATH].', '1412.7332-4-24-2': 'Moreover, [MATH] is a Hilbert space with inner product [EQUATION]', '1412.7332-4-24-3': 'We define the [MATH]-global covariance matrix [MATH] of an unbiased estimator [MATH] as the [MATH] matrix whose [MATH]-th element is [EQUATION]', '1412.7332-4-24-4': 'This global notion of covariance has been used also in [CITATION], in the context of predicting georeferenced functional data.', '1412.7332-4-24-5': 'These authors have found a BLUE estimator for the drift of their underlying process, which can be seen as an example of the results given in this paper.', '1412.7332-4-25-0': 'Given a couple [MATH], it may be defined a linear continuous operator on [MATH] as follows [EQUATION]', '1412.7332-4-25-1': 'From the Riesz representation theorem, there exists a unique [MATH] such that [EQUATION]', '1412.7332-4-25-2': 'The unique element [MATH] defined in [REF] is called the Riesz representative of the couple [MATH].', '1412.7332-4-26-0': 'This definition will be useful to provide a nice expression for the functional OLS estimator [MATH].', '1412.7332-4-26-1': 'Actually the Riesz representative synthesizes, in some sense, in [MATH] the information of both [MATH] and [MATH].', '1412.7332-4-27-0': 'Note that, since [EQUATION] the Riesz representative [MATH] may be seen as the projection of [MATH] onto the immersion of [MATH] in [MATH], a linear closed subspace.', '1412.7332-4-28-0': 'The functional OLS estimator for the model [REF] is [EQUATION]', '1412.7332-4-28-1': 'The quantity [EQUATION] resembles [EQUATION] because [MATH] and [MATH] reconstruct [MATH] and its derivative function, respectively.', '1412.7332-4-28-2': 'The functional OLS estimator [MATH] minimizes, in this sense, the sum of the [MATH]-norm of the unobservable residuals [MATH].', '1412.7332-4-29-0': 'Given model in [REF],', '1412.7332-4-30-0': 'The previous results may be generalized to other Sobolev spaces.', '1412.7332-4-30-1': 'The extension to [MATH], [MATH], is straightforward.', '1412.7332-4-30-2': 'Moreover, in Bayesian context, the investigator might have a different a priori consideration of [MATH] and [MATH].', '1412.7332-4-30-3': 'Thus, different weights may be used for curves and derivatives, and the inner product given in [REF] may be extended to [EQUATION].', '1412.7332-4-30-4': 'Let [MATH] be the OLS estimator obtained by using this last inner product.', '1412.7332-4-30-5': 'Note that, for [MATH], we obtain [MATH] defined in Theorem [REF].', '1412.7332-4-30-6': 'The behavior of the [MATH] is explored in Section [REF] for different choices of [MATH].', '1412.7332-4-31-0': '# Practical considerations', '1412.7332-4-32-0': 'In a real world context, we work with a finite dimensional subspace [MATH] of [MATH].', '1412.7332-4-32-1': 'Let [MATH] be a base of [MATH].', '1412.7332-4-32-2': 'Without loss of generality, we may assume that [MATH], where [EQUATION] is the Kronecker delta symbol, since a Gram-Schmidt orthonormalization procedure may be always applied.', '1412.7332-4-32-3': 'More precisely, given any base [MATH] in [MATH], the corresponding orthonormal base is given by:', '1412.7332-4-33-0': 'for [MATH], define [MATH],', '1412.7332-4-34-0': 'for [MATH], let [MATH] and [MATH].', '1412.7332-4-35-0': 'With this orthonormalized base, the projection [MATH] on [MATH] of the Riesz representative [MATH] of the couple [MATH] is given by [EQUATION] where the last equality comes from the definition [REF] of the Riesz representative.', '1412.7332-4-35-1': 'Now, if [MATH] is the [MATH]-th row of [MATH], then [EQUATION] hence [MATH].', '1412.7332-4-36-0': 'Let us note that, even if the Riesz representative [REF] is implicitly defined, its projection on [MATH] can be easily computed by [REF].', '1412.7332-4-36-1': 'From a practical point of view, the statistician can work with the data [MATH] projected on a finite linear subspace [MATH] and the corresponding OLS estimator [MATH] is the projection on [MATH] of the OLS estimator [MATH] given in Section [REF].', '1412.7332-4-37-0': 'It is straightforward to prove that the estimator [REF] becomes [EQUATION] in two cases: when we do not take into consideration [MATH], or when [MATH].', '1412.7332-4-37-1': 'Up to our knowledge, this is the most common situation considered in the literature (see [CITATION]).', '1412.7332-4-37-2': 'However, from the simulation study of Section [REF], the OLS estimator [MATH] is less efficient when it is based only on [MATH].', '1412.7332-4-38-0': '# Strong [MATH]-BLUE in functional linear models', '1412.7332-4-39-0': 'Let [MATH], where [MATH] is a linear closed operator; in this case [MATH] is called a linear estimator.', '1412.7332-4-39-1': 'The domain of [MATH], denoted by [MATH], will be defined in [REF].', '1412.7332-4-39-2': 'Theorem [REF] will ensure that the dataset [MATH] is contained in [MATH].', '1412.7332-4-40-0': 'In analogy with classical settings, we define the [MATH]-functional best linear unbiased estimator ([MATH]-BLUE) as the estimator with minimal (in the sense of Loewner Partial Order) [MATH]-global covariance matrix [REF], in the class of the linear unbiased estimators [MATH] of [MATH].', '1412.7332-4-41-0': 'From the definition of Loewner Partial Order, a [MATH]-BLUE minimizes the quantity [EQUATION] for any choice of [MATH], in the class of the linear unbiased estimators [MATH] of [MATH].', '1412.7332-4-41-1': 'In other words, the [MATH]-BLUE minimizes the [MATH]-global variance of any linear combination of its components.', '1412.7332-4-41-2': 'A stronger request is the following.', '1412.7332-4-42-0': 'We define the [MATH]-strong functional best linear unbiased estimator ([MATH]-SBLUE) as the estimator with minimal global variance, [EQUATION] for any choice of a (sufficiently regular) continuous linear operator [MATH], in the class of the linear unbiased estimators [MATH] of [MATH].', '1412.7332-4-43-0': '## [MATH]-representation on the Hilbert space [MATH]', '1412.7332-4-44-0': 'Recall that, for any given [MATH], the couple [MATH] is a process with values in [MATH].', '1412.7332-4-44-1': 'Let [MATH] be the spectral representation of the covariance matrix of the process [EQUATION] which means [MATH], [MATH] and the sequence [MATH] are orthonormal bivariate vectors in [MATH].', '1412.7332-4-44-2': 'Without loss of generality assume that the [MATH]-closure of the linear span of [MATH] includes [MATH] (see Remark [REF]): [MATH].', '1412.7332-4-44-3': 'Note that [MATH], the covariance matrix of the process [MATH], does not depend on [MATH].', '1412.7332-4-44-4': 'From Karhunen-Loeve Theorem (see, e.g., [CITATION]), there exists an array of zero-mean unit variance random variables [MATH] such that [EQUATION]', '1412.7332-4-44-5': 'The linearity of the covariance operator with respect to the first process, together with the symmetry in [MATH] given in the hypothesis ([REF]) and ([REF]), ensures that [EQUATION]', '1412.7332-4-44-6': 'Now, for [MATH], let [EQUATION] and hence [EQUATION].', '1412.7332-4-44-7': 'The independence assumptions in the hypothesis ([REF]) and ([REF]) ensures that the joint law of the processes [MATH] and [MATH] does not depend on [MATH], hence [EQUATION].', '1412.7332-4-44-8': 'From [REF], the linearity of the expectation ensures that [EQUATION]', '1412.7332-4-44-9': 'Let us observe that the elements of [MATH] are the functions [MATH] such that [MATH] and [MATH].', '1412.7332-4-44-10': 'In the following definition a stronger condition is required.', '1412.7332-4-45-0': 'Given the spectral representation of [MATH], let [EQUATION] be a new Hilbert space, with inner product [EQUATION]', '1412.7332-4-45-1': 'Note that [MATH] .', '1412.7332-4-45-2': 'An orthonormal base for [MATH] is given by [MATH], where [MATH] for any [MATH].', '1412.7332-4-46-0': 'Consider now the following linear closed dense subset of [MATH]: [EQUATION].', '1412.7332-4-46-1': 'Observe that [MATH] for all [MATH].', '1412.7332-4-46-2': 'If [MATH] is the [MATH]-dual space of [MATH], the Gelfand triple [MATH] implies that [MATH].', '1412.7332-4-47-0': 'In analogy with the geometric interpretation of the Riesz representation, we construct the [MATH]-representation in the following way.', '1412.7332-4-47-1': 'For any element [MATH], we call [MATH]-representative its [MATH]-projection on [MATH], and we denote it with the symbol [MATH].', '1412.7332-4-47-2': 'In particular, for any [MATH], let [MATH] be the [MATH]-representative of [MATH], that is, the unique element in [MATH] such that [EQUATION].', '1412.7332-4-47-3': 'Note that the [MATH]-representatives of the orthonormal system [MATH] of [MATH] are given by [MATH], where, by definition of projection, [EQUATION]', '1412.7332-4-47-4': 'Moreover, [EQUATION] and the [MATH]-representation of any [MATH] can be written as [EQUATION]', '1412.7332-4-47-5': 'When [MATH], it is again possible to define formally its [MATH]-representation in the following way: [EQUATION]', '1412.7332-4-47-6': 'In this case, if [MATH], an analogous of the standard projection can be obtained: [MATH] it is the unique element in [MATH] of the form [MATH] with [MATH] such that [EQUATION].', '1412.7332-4-47-7': 'It will be useful to observe that, as a consequence, when [MATH], then its [MATH]-representative is [MATH].', '1412.7332-4-48-0': 'Given [MATH] as in [REF], its [MATH]-representative [EQUATION] belongs to [MATH], for any [MATH].', '1412.7332-4-49-0': 'The following theorem is a direct consequence of the previous results.', '1412.7332-4-50-0': 'The following equation holds in [MATH]: [EQUATION] where each [MATH] is the [MATH]-representation of the mean [MATH] of the observations given in [REF].', '1412.7332-4-50-1': 'As a consequence, [MATH] belongs to [MATH], and hence [MATH] a.s.', '1412.7332-4-51-0': 'The vector [MATH] plays the role of the Riesz representative of Theorem [REF] in the following redSBLUE theorem.', '1412.7332-4-52-0': 'The functional estimator [EQUATION] for the model [REF] is a [MATH]-functional redS BLUE.', '1412.7332-4-53-0': 'From the proof of Theorem [REF] (see [REF]) we have that [MATH] is the best estimator among all the estimators [MATH] where [MATH] is any linear closed unbiased operator.', '1412.7332-4-53-1': 'Therefore, [MATH] is also better than the best linear unbiased estimators based only on [MATH] or [MATH], since they are defined by some linear unbiased operator.', '1412.7332-4-54-0': 'The assumption [MATH] ensures that the each component of the unknown [MATH] is in [MATH].', '1412.7332-4-54-1': 'As a consequence, we have noted that the [MATH]-representative of [MATH], is [MATH].', '1412.7332-4-54-2': 'If this assumption is not true, it may happen that [MATH] for some [MATH], and then [MATH] would have a nonzero projection on the orthogonal complement of [MATH].', '1412.7332-4-54-3': 'Since on the orthogonal complement we do not observe any noise, this means that we would have a deterministic subproblem, that, without loss of generality, we can ignore.', '1412.7332-4-55-0': '# Simulations', '1412.7332-4-56-0': 'In this section, it is explored, throughout a simulation study, when it is more convenient to use the whole information on both reconstructed functions and derivatives with respect to the partial use of [MATH] (or [MATH]).', '1412.7332-4-56-1': 'The idea is that using the whole information on curves and derivatives is much more convenient as the dependence between [MATH] and [MATH] is smaller and their spread is more comparable.', '1412.7332-4-57-0': 'In this study, for each scenario listed below, [MATH] datasets are simulated from model [REF] by a Montecarlo method, with [MATH], [MATH], [MATH], [EQUATION] and [EQUATION].', '1412.7332-4-57-1': 'In what follows, we compare the following different estimators: the S BLUE [MATH] (see Section [REF]), the OLS estimators [MATH] (see Remark [REF]), and [MATH], where [MATH] is the OLS estimator based on [MATH] and [MATH] is the OLS estimator based on [MATH], with [MATH].', '1412.7332-4-58-0': 'Let us note that [MATH] is a compound OLS estimator; it is a rough way of taking into account both the sources of information on [MATH] and [MATH].', '1412.7332-4-58-1': 'Of course, setting [MATH] we ignore completely the information on the functions and [MATH], viceversa setting [MATH] means to ignore the information on the derivatives and thus [MATH].', '1412.7332-4-59-0': 'All the computations are developed using R package.', '1412.7332-4-60-0': 'In Figure [REF] it is plotted: one dataset of curves and derivatives (black lines); the regression functions [MATH] and [MATH] (green lines); the S BLUE predictions [MATH] and [MATH] (blue lines); the OLS predictions [MATH] and [MATH] (red lines).', '1412.7332-4-61-0': '## Dependence between functions and derivatives', '1412.7332-4-62-0': 'We consider three different scenarios; we generate functional data [MATH] and [MATH] such that', '1412.7332-4-63-0': '[MATH] is independent on [MATH]; [MATH] and [MATH] are mildly dependent (the degree of dependence is randomly obtained); [MATH] and [MATH] are fully dependent: [MATH], and hence [MATH].', '1412.7332-4-64-0': 'The performance of the different estimators is evaluated by comparing the [MATH]-norm of the [MATH]-components of the estimation errors.', '1412.7332-4-64-1': 'Figures [REF] depicts the Montecarlo distribution of the [MATH]-norm of the first component: [MATH] for different values of [MATH] (red box-plot, [REF]), [MATH] for different values of [MATH] (yellow box-plots) and [MATH] (blue box-plot).', '1412.7332-4-65-0': 'From the comparison of the box-plots corresponding to [MATH] and [MATH] with the other cases, we can observe that it is always more convenient to use the whole information on [MATH] and [MATH] (this behaviour is more evident in scenario 1).', '1412.7332-4-65-1': 'Among the three estimators [MATH], [MATH] and [MATH], the SBLUE is the most precise, as expected.', '1412.7332-4-65-2': 'When there is a one-to-one dependence between [MATH] and [MATH], one source of information is redundant and all the functional estimators coincide (bottom panel of Figure [REF]).', '1412.7332-4-66-0': '## Spread of functions and derivatives', '1412.7332-4-67-0': 'Also in this case, we consider three different scenarios.', '1412.7332-4-67-1': 'Let [EQUATION] where [MATH] denotes the the [MATH]-global covariance matrix defined in [REF].', '1412.7332-4-67-2': 'We generate functional data [MATH] and [MATH] with a different spread, such that', '1412.7332-4-68-0': '[MATH] (in this sense, [MATH] is ""more concentrate"" than [MATH]); [MATH] and [MATH] have more or less the same spread); [MATH] is ""more concentrate"" than [MATH]).', '1412.7332-4-69-0': 'As before, the performance of the different estimators is evaluated by comparing the [MATH]-norm of the [MATH]-components of the estimation errors.', '1412.7332-4-69-1': 'Figures [REF] depicts the Montecarlo distribution of the [MATH]-norm of the first component: [MATH] for different values of [MATH] (red box-plot, [REF]), [MATH] for different values of [MATH] (yellow box-plots) and [MATH] (blue box-plot).', '1412.7332-4-70-0': 'From the comparison of the box-plots of [MATH] and [MATH] corresponding to [MATH] and [MATH] with the other cases, it seems more convenient to use just the less ""less spread"" information: [MATH] in Scenario 1 and [MATH] in Scenario 2.', '1412.7332-4-70-1': 'Comparing the precision of [MATH] and [MATH] with the one of the [MATH], however, the redSBLUE is the most precise, as expected.', '1412.7332-4-70-2': 'Hence, we suggest the use of the whole available information through the use of the redSBLUE.', '1412.7332-4-70-3': 'Of course, when one of the sources of information has a spread near to zero then the more precise estimator is the one that uses just that piece of information and [MATH] reflects this behaviour.', '1412.7332-4-71-0': '# Summary', '1412.7332-4-72-0': 'Functional data are suitably modelled in separable Hilbert spaces (see [CITATION] and [CITATION]) and [MATH] is usually sufficient to handle the majority of the techniques proposed in the literature of functional data analysis.', '1412.7332-4-73-0': 'Differently, we consider proper Sobolev spaces, since we guess that the data may provide information on both curve functions and their derivatives.', '1412.7332-4-73-1': 'The classical theory for linear regression models is extended to this context by means of the sample of Riesz representatives.', '1412.7332-4-73-2': 'Roughly speaking, the Riesz representatives are ""quantities"" which incorporate both functions and derivatives information in a non trivial way.', '1412.7332-4-73-3': 'More in detail, a generalization of the Riesz representatives is proposed to take into account the possible correlation between curves and derivatives.', '1412.7332-4-73-4': 'These generalized Riesz representatives are called just representatives"".', '1412.7332-4-73-5': 'Using a sample of representatives, we prove a strong, generalized version of the well known Gauss-Markov theorem for functional linear regression models.', '1412.7332-4-73-6': 'Despite the complexity of the problem we obtain an elegant and simple solution, through the use of the representatives which belong to a Sobolev space.', '1412.7332-4-73-7': 'This result states that the proposed estimator, which takes into account both information about curves and derivatives (throughout the representatives), is much more efficient than the usual OLS estimator based only on one sample of functions (curves or derivatives).', '1412.7332-4-73-8': 'The superiority of the proposed estimator is also showed in the simulation study described in Section [REF].', '1412.7332-4-74-0': '# Proofs', '1412.7332-4-75-0': '[Proof of Theorem [REF]] Part a).', '1412.7332-4-75-1': 'We consider the sum of square residuals: [EQUATION]', '1412.7332-4-75-2': 'The Gateaux derivative of [MATH] at [MATH] in the direction of [MATH] is [EQUATION] where [MATH] and [MATH] are two [MATH] vectors whose [MATH]-th elements are [EQUATION]', '1412.7332-4-75-3': 'Developing the right-hand side of ([REF]), we have that the Gateaux derivative is [EQUATION] where [MATH] is a [MATH] vector whose [MATH]-th element is the Riesz representative of [MATH].', '1412.7332-4-76-0': 'The Gateaux derivative [REF] is equal to [MATH] for any [MATH] if and only if [MATH] is given by the following equation: [EQUATION] which proves the first statement of the theorem.', '1412.7332-4-77-0': 'Part b) Definition [REF] and model [REF] imply that, for any [MATH], [EQUATION] then [MATH], and hence [MATH] is unbiased.', '1412.7332-4-77-1': 'Moreover, [EQUATION] where [MATH] and [MATH] denote the Riesz representatives of [MATH] and [MATH], respectively.', '1412.7332-4-77-2': 'From the hypothesis ([REF]) and ([REF]) in the model [REF], the left-hand side quantities in [REF] are zero-mean i.i.d. processes, for [MATH].', '1412.7332-4-77-3': 'Therefore, the global covariance matrix of [MATH] is [MATH], where [MATH].', '1412.7332-4-77-4': 'Hence, the global covariance matrix of [MATH] is [MATH].', '1412.7332-4-78-0': '[Proof of Lemma [REF]] We have that [EQUATION] where the last inequality follows from [REF].', '1412.7332-4-78-1': 'Since [MATH], we get the thesis.', '1412.7332-4-79-0': '## Proof of Theorem [REF]', '1412.7332-4-80-0': 'The estimator [MATH] is a linear map which associates an element [MATH] in [MATH] to any [MATH]-tuple [MATH].', '1412.7332-4-80-1': 'In what follows, we show that it is the ""best"" among all the linear unbiased closed operators [MATH].', '1412.7332-4-81-0': 'The model [REF] may be written in the following vectorial form: [EQUATION] where [MATH] is the column vector of length [MATH] with all components equal to [MATH].', '1412.7332-4-82-0': 'In general, if [EQUATION] and [EQUATION] are two [MATH] block vectors in [MATH], we may define the following [MATH] dimensional vector [EQUATION] where [MATH] is the [MATH] representation of [EQUATION].', '1412.7332-4-82-1': 'Now we can introduce the following linear operator [EQUATION]', '1412.7332-4-82-2': 'Hence, [EQUATION] and [EQUATION].', '1412.7332-4-82-3': 'The thesis follows immediately if we prove that [MATH] and [MATH] are uncorrelated.', '1412.7332-4-83-0': 'Since both [MATH] and [MATH] are unbiased, [MATH], and thus we have to prove that [EQUATION] for any choice of linear operator [MATH].', '1412.7332-4-84-0': 'The proof of equality ([REF]) is developed in four steps.', '1412.7332-4-85-0': 'First step.', '1412.7332-4-85-1': 'The goal of this step is to prove that [MATH] applied to the deterministic part of the model [MATH] is identically null.', '1412.7332-4-85-2': 'As a consequence, [EQUATION]', '1412.7332-4-86-0': 'From the linearity of the closed operator [MATH], and the zero-mean hypothesis ([REF]) and ([REF]), we have that [EQUATION]', '1412.7332-4-86-1': 'Since [MATH] we have that [EQUATION]', '1412.7332-4-86-2': 'In addition, from the definition ([REF]) if [EQUATION] then [EQUATION]', '1412.7332-4-86-3': 'Combining [REF], [REF] and [REF] gives [EQUATION] and hence [REF].', '1412.7332-4-87-0': 'Second step.', '1412.7332-4-87-1': 'Representation of the linear operator [MATH].', '1412.7332-4-88-0': 'For the linearity of the [MATH]-th component of [MATH] with respect to the bivariate observations [MATH]: [EQUATION] where, for any [MATH] and [MATH], [MATH] is linear.', '1412.7332-4-88-1': 'The domain of [MATH] is contained in [MATH].', '1412.7332-4-88-2': 'Let [MATH] be an orthonormal base of [MATH].', '1412.7332-4-88-3': 'We express the linear operator [MATH] in terms of the base [MATH] for [MATH] and [MATH] for [MATH].', '1412.7332-4-88-4': 'In fact, [MATH] and [MATH] (see [REF]).', '1412.7332-4-88-5': 'Accordingly, [EQUATION] where [EQUATION]', '1412.7332-4-89-0': 'Third step.', '1412.7332-4-89-1': 'Proof of', '1412.7332-4-90-0': '[EQUATION] where [MATH] is the [MATH]-th row of [MATH].', '1412.7332-4-90-1': 'In particular, since [MATH], [EQUATION]', '1412.7332-4-91-0': 'Let [MATH] be the null vector except for the [MATH]-th component which is [MATH], and let [MATH].', '1412.7332-4-91-1': 'Setting [MATH] in [REF], [EQUATION] where the last equality is due to ([REF]).', '1412.7332-4-92-0': 'Fourth step.', '1412.7332-4-92-1': 'Proof of [REF]: [EQUATION] for any choice of linear operator [MATH].', '1412.7332-4-93-0': 'From Theorem [REF] and from [REF], [MATH], and hence [EQUATION] where the last equality is a consequence of [REF].', '1412.7332-4-94-0': 'Since [MATH] (see [REF]), we express the linear operator [MATH] in terms of the base [MATH] for [MATH] and [MATH] for [MATH], where [MATH] is an orthonormal base of [MATH].', '1412.7332-4-94-1': 'To begin with, from the linearity of the operator [MATH], we have that [EQUATION].', '1412.7332-4-94-2': 'Since [MATH], where [MATH], we have [EQUATION].', '1412.7332-4-94-3': 'Now, setting [EQUATION] then we have the representation of [MATH] in terms of the required bases: [EQUATION].', '1412.7332-4-94-4': 'Hence, from Equations ([REF]), [REF] and [REF], the thesis [REF] becomes [EQUATION]', '1412.7332-4-95-0': 'From [REF] and [REF], since [MATH] the left-hand side of the last equation becomes [EQUATION] the last equality being a consequence of [REF].'}",
cond-mat-0006123,"{'cond-mat-0006123-1-0-0': 'We study a 1-D granular gas of point-like particles not subject to gravity between two walls at temperatures [MATH] and [MATH].', 'cond-mat-0006123-1-0-1': 'The system exhibits two distinct regimes, depending on the normalized temperature difference [MATH]: one completely fluidized and one in which a cluster coexists with the fluidized gas.', 'cond-mat-0006123-1-0-2': 'When [MATH] is above a certain threshold, cluster formation is fully inhibited, obtaining a completely fluidized state.', 'cond-mat-0006123-1-0-3': 'The mechanism that produces these two phases is explained.', 'cond-mat-0006123-1-0-4': 'In the fluidized state the distribution function exhibits peculiar non-Gaussian features.', 'cond-mat-0006123-1-0-5': 'For this state, comparison between integration of the Boltzmann equation using direct-simulation Monte Carlo and results stemming from microscopic Newtonian molecular dynamics gives good coincidence, establishing that the non-Gaussian features observed do not stem from the onset of correlations.', 'cond-mat-0006123-1-1-0': '2', 'cond-mat-0006123-1-2-0': '# Introduction', 'cond-mat-0006123-1-3-0': 'Granular systems have been extensively studied due both to the theoretical challenges they present[CITATION] and to the applications of industrial importance that stem from the rich phenomena they exhibit (see [CITATION] and references therein).', 'cond-mat-0006123-1-3-1': 'These systems are characterized by an energy loss in collisions.', 'cond-mat-0006123-1-3-2': 'This loss is at the base of many interesting phenomena, such as inelastic collapse[CITATION], where the particles collide infinitely often in finite time, and clustering (for theoretical, simulational, and experimental approaches see [CITATION]).', 'cond-mat-0006123-1-3-3': 'Different methods for keeping the system from collapsing have been devised, such as subjecting the particles to Brownian forces[CITATION], and forcing through the boundaries by putting the system in a box with one or more thermal-like walls (see for example [CITATION]).', 'cond-mat-0006123-1-4-0': 'In this paper we study a one-dimensional system of [MATH] point-like particles interacting via collisions that conserve momentum but dissipate kinetic energy.', 'cond-mat-0006123-1-4-1': 'To fix notation, the particle velocities after a collision are given by [EQUATION]', 'cond-mat-0006123-1-4-2': 'Here [MATH], with [MATH] the restitution coefficient.', 'cond-mat-0006123-1-4-3': 'For the elastic case ([MATH]) the particles simply exchange velocities.', 'cond-mat-0006123-1-4-4': 'Since the particles are point-like, the system is then indistinguishable from a system in which the particles do not interact.', 'cond-mat-0006123-1-5-0': 'This one-dimensional system is interesting because dissipation is the first order correction to a free gas.', 'cond-mat-0006123-1-5-1': 'Besides, results for the one-dimensional system have been found to have unexpected relevance for higher-dimensional problems.', 'cond-mat-0006123-1-5-2': 'For example, in two-dimensions the particles involved in inelastic collapse lie roughly on a line[CITATION].', 'cond-mat-0006123-1-5-3': 'Also, the dissipation-induced temperature gradients calculated in [CITATION] inspired the authors to look for dissipation-induced Rayleigh-Benard-like convection for a two-dimensional system without an externally imposed temperature gradient[CITATION].', 'cond-mat-0006123-1-6-0': 'For the nearly elastic system, forcing though thermal walls has been shown to avoid inelastic collapse, but though the system is kept from collapsing, a precursor of this state, clustering, may develop in the system.', 'cond-mat-0006123-1-6-1': 'This has also been observed for a granular gas where one wall is kept at a fixed temperature and the other(s) is(are) elastic.', 'cond-mat-0006123-1-6-2': 'Here the system clumps against the wall opposite the thermal wall, while a few fast particles (sometimes only one) go back and forth between the thermal wall and the cluster[CITATION].', 'cond-mat-0006123-1-7-0': 'For a system with one thermal wall and open on the other side, under the influence of gravity, the quasi-elastic system may be kept fluidized[CITATION]: any cluster that starts to form is forced against the thermal wall, where it evaporates.', 'cond-mat-0006123-1-7-1': 'Here the test-particle equation[CITATION], which is the 1-D Boltzmann equation where the limit [MATH] is taken, but keeping [MATH] fixed, is successfully applied[CITATION], with close matching of theory and simulations even at the level of the distribution function.', 'cond-mat-0006123-1-8-0': 'The one-dimensional system under study is left to evolve between two thermal walls at temperatures [MATH] and [MATH], with [MATH].', 'cond-mat-0006123-1-8-1': 'We define the parameter [EQUATION] to quantify how far from the symmetrical case is the system.', 'cond-mat-0006123-1-8-2': 'Under the same conditions, an elastic system has a perfectly bimodal velocity distribution with global homogeneous temperature equal to [MATH].', 'cond-mat-0006123-1-8-3': 'For the sake of comparison, we simulate systems with [MATH].', 'cond-mat-0006123-1-8-4': 'Here [MATH] represents a symmetrical setting, while [MATH] represents an infinitely strong temperature gradient.', 'cond-mat-0006123-1-9-0': 'For [MATH] a cluster unavoidably forms away from the thermal walls.', 'cond-mat-0006123-1-9-1': 'After forming, the cluster performs a random walk about the center, growing in size (and therefore mass) while the rest of the system grows more rarefied.', 'cond-mat-0006123-1-9-2': 'With the decrease in density of the surrounding gas and the increased inertia of the cluster, an eventual collision with one of the walls is inevitable.', 'cond-mat-0006123-1-9-3': 'When this happens part of the cluster evaporates, and what is left of it is expelled from the wall (see Fig. [REF]), thus restarting the growth process.', 'cond-mat-0006123-1-9-4': 'Thus not only is the system highly clumped, but also in a non-steady state.', 'cond-mat-0006123-1-9-5': 'Nevertheless, the gas that is far from the random-walk zone has a well-defined time average for the distribution function, as is seen in Fig. [REF].', 'cond-mat-0006123-1-9-6': 'A noteworthy feature of this distribution is that it exhibits apparently singular behavior for slow velocities.', 'cond-mat-0006123-1-10-0': 'Setting [MATH] the symmetry of the system is broken.', 'cond-mat-0006123-1-10-1': 'For [MATH] the cluster performs a slightly asymmetric random walk, spending more time near the colder wall, and therefore colliding more often with it.', 'cond-mat-0006123-1-10-2': 'Thus the cluster cannot grow as much as it did in the symmetric case before colliding with a wall.', 'cond-mat-0006123-1-10-3': 'By increasing [MATH] the cluster-wall collision frequency grows, even obtaining short ""windows"" in which the cluster completely evaporates.', 'cond-mat-0006123-1-10-4': 'By further increasing [MATH] these windows grow larger until a point is reached where no cluster forms.', 'cond-mat-0006123-1-10-5': 'In this fashion a totally fluidized state is achieved, which may be tractable with the dissipative Boltzmann equation.', 'cond-mat-0006123-1-10-6': 'However, the distribution function obtained from the molecular dynamics (MD) simulations exhibits a peculiar non-Gaussian feature for slow velocities.', 'cond-mat-0006123-1-10-7': 'This feature is a smoothed version of the apparent singularity of the symmetric case.', 'cond-mat-0006123-1-10-8': 'To discern whether this feature is due to correlations or is present before they settle in, we compared Newtonian molecular dynamics results with those obtained through direct-simulation Monte Carlo (DSMC), which neglects correlations.', 'cond-mat-0006123-1-10-9': 'The results agree very well, except when the system approaches the clustering regime.', 'cond-mat-0006123-1-11-0': '# Simulation method', 'cond-mat-0006123-1-12-0': 'We simulate the system through event-driven molecular dynamics[CITATION] and through direct-simulation Monte Carlo[CITATION].', 'cond-mat-0006123-1-12-1': 'The direct-simulation Monte Carlo procedures use the null-collision technique[CITATION] where, overestimating the collision frequency (using the maximum relative velocity within a cell), the number of collisions to be attempted is calculated through a Poisson process.', 'cond-mat-0006123-1-12-2': 'In the next step the collisions are attempted, choosing at random two particles within the cell, and making them collide with a probability proportional to their relative velocity.', 'cond-mat-0006123-1-12-3': 'Most of the molecular dynamics and DSMC simulations were done with [MATH] particles.', 'cond-mat-0006123-1-13-0': 'In the MD simulations we detect clusters using a geometric criterion: we consider chains of particles that are nearer than a critical distance (in our case [MATH] and [MATH], to be certain that the conclusions are independent of the choice).', 'cond-mat-0006123-1-13-1': 'The system size is one, and with a thousand particles the mean distance between neighbors for a homogeneous system is [MATH].', 'cond-mat-0006123-1-13-2': 'Thus we detect particles that are uncommonly near by three orders of magnitude.', 'cond-mat-0006123-1-13-3': 'We discard chains of length three or less, since they may be random encounters.', 'cond-mat-0006123-1-13-4': 'Measuring the total size of the cluster we have found that on average it is of the order of [MATH]; thus the choice of [MATH] as link-link distance is much larger than the true distance between them.', 'cond-mat-0006123-1-14-0': 'As already stated, the boundary conditions are such that the (homogeneous) temperature of the corresponding elastic system (equal to [MATH]) is one.', 'cond-mat-0006123-1-15-0': '# Results', 'cond-mat-0006123-1-16-0': '## Clustering regime', 'cond-mat-0006123-1-17-0': 'Figure [REF] shows the non-steady state of a granular system between two walls at the same temperature for [MATH].', 'cond-mat-0006123-1-17-1': 'A cluster forms away from the walls, performing a random walk of varying amplitude.', 'cond-mat-0006123-1-17-2': 'When the cluster reaches a wall, part of it evaporates, and the growth process begins anew.', 'cond-mat-0006123-1-18-0': 'As is usual for the quasi-elastic case, we relabel the particles when they collide.', 'cond-mat-0006123-1-18-1': 'This enables us to visualize this system as a group of barely interacting particles passing through each other.', 'cond-mat-0006123-1-19-0': 'The picture for cluster evolution, as explained before, is the following: the cluster grows because the slowest particles, due to the asymmetry of the distribution function, drift towards the cluster[CITATION].', 'cond-mat-0006123-1-19-1': 'As it grows, the density of the gas surrounding it decreases, with the consequent saturation in growth.', 'cond-mat-0006123-1-19-2': 'Thus we have a ""Brownian particle"" of increasing mass moving in an increasingly rarefied medium.', 'cond-mat-0006123-1-19-3': 'This ""particle"" will be increasingly less affected by the surrounding medium, until it can no longer be kept away from the walls.', 'cond-mat-0006123-1-20-0': 'The cluster moves several orders of magnitude slower than the thermal speed (three orders of magnitude in Fig. [REF]).', 'cond-mat-0006123-1-20-1': 'Upon reaching a wall, the front liners strike the wall and are expelled by it much faster than the other cluster members.', 'cond-mat-0006123-1-20-2': 'These particles pass through the cluster, transferring momentum to it, as described in [CITATION].', 'cond-mat-0006123-1-20-3': 'Thus these particles push the cluster away from the wall, where it can absorb particles again.', 'cond-mat-0006123-1-20-4': 'The fast particles, however, no longer belong to the cluster.', 'cond-mat-0006123-1-21-0': ""Since the slowest particles in the gas are the ones that will be absorbed by the cluster[CITATION], it is the number of slow particles in the gas that will determine the cluster's growth rate."", 'cond-mat-0006123-1-21-1': 'After the cluster strikes a wall, the expelled particles will be fast particles, and they will not contribute to the growth of the cluster during the time it takes for the gas to cool down again: the only particles available for absorption are the ones that were available before the cluster-wall collision.', 'cond-mat-0006123-1-21-2': 'This explains why the cluster keeps growing at approximately the same rate it did before the collision.', 'cond-mat-0006123-1-21-3': 'After the gas has cooled down, the growth rate returns to its normal value.', 'cond-mat-0006123-1-21-4': 'This is the end of the plateau shown in Fig. [REF].', 'cond-mat-0006123-1-22-0': 'To quantify the evaporation process we proceed as in [CITATION]: when the cluster reaches a wall, the first particle to reach it is expelled by the wall much faster than the cluster velocity, thus we may consider the ideal situation of a cluster of [MATH] particles at rest being stricken by a fast particle (in this case [MATH]).', 'cond-mat-0006123-1-22-1': ""After colliding with the first particle in the cluster, the new fast particle's speed will be [MATH]."", 'cond-mat-0006123-1-22-2': ""Thus, after traversing the cluster, the fast particle's velocity will be [MATH]."", 'cond-mat-0006123-1-22-3': 'Since momentum is conserved in collisions, the center of mass of the cluster will have acquired a speed of [EQUATION]', 'cond-mat-0006123-1-22-4': 'Considering the case [MATH] with fixed [MATH], as in [CITATION], we may simplify this expression to [EQUATION]', 'cond-mat-0006123-1-22-5': 'By further considering the case [MATH] we get [MATH].', 'cond-mat-0006123-1-22-6': 'In this limit, if the cluster reaches the wall with velocity [MATH], it will evaporate particles until it stops approaching it.', 'cond-mat-0006123-1-22-7': 'Thus the number [MATH] of particles evaporated will satisfy [MATH].', 'cond-mat-0006123-1-22-8': 'For the situation shown in Fig. [REF], [MATH] is typically [MATH] and [MATH], hence the number of particles evaporated will be of the order of [MATH].', 'cond-mat-0006123-1-23-0': 'Even if the system is in a non-steady state, the gas at the walls (far from the random-walk zone) has a well-defined time average for the distribution function.', 'cond-mat-0006123-1-23-1': 'The distribution function at the left wall is shown in Fig. [REF].', 'cond-mat-0006123-1-23-2': 'There is an apparent singularity for slow velocities.', 'cond-mat-0006123-1-23-3': 'The distribution is asymmetric as it should, since the particles leaving the wall ([MATH]) follow a Gaussian distribution.', 'cond-mat-0006123-1-23-4': 'Figure [REF] also shows the distribution multiplied by [MATH].', 'cond-mat-0006123-1-23-5': 'Since the limit of [MATH] for [MATH] is finite and nonzero, we conclude that the distribution function exhibits a singularity that behaves like [MATH] for slow velocities.', 'cond-mat-0006123-1-24-0': 'When [MATH] (the symmetric case) the distribution shown in Fig. [REF] is not a solution of the steady-state test-particle equation: [EQUATION] where [EQUATION]', 'cond-mat-0006123-1-24-1': 'To establish this, let us study the behavior for small [MATH] of a solution of this equation.', 'cond-mat-0006123-1-24-2': 'Assume that, for small [MATH], [MATH], with [MATH] in order to have a finite density in the vicinity of [MATH].', 'cond-mat-0006123-1-24-3': 'Furthermore, assume that [MATH].', 'cond-mat-0006123-1-24-4': 'Inserting this behavior in Eq. ([REF]) we obtain [EQUATION]', 'cond-mat-0006123-1-24-5': 'Thus, in order to keep [MATH] (the amplitude of the singularity) finite, we must have either [MATH] or [MATH].', 'cond-mat-0006123-1-24-6': 'Integrating the distribution of Fig. [REF] we obtain [MATH].', 'cond-mat-0006123-1-24-7': 'Since [MATH], we must have [MATH].', 'cond-mat-0006123-1-24-8': 'But this corresponds to a nonintegrable distribution, and therefore the distribution cannot be steady.', 'cond-mat-0006123-1-25-0': '## Inhibition of cluster formation', 'cond-mat-0006123-1-26-0': 'Profiles obtained from DSMC are compared with MD results in Figs. [REF] and [REF] for [MATH].', 'cond-mat-0006123-1-26-1': 'The temperature of the right wall [MATH] is chosen so that the global temperature for the elastic case (equal to [MATH]) is one.', 'cond-mat-0006123-1-26-2': 'The curves match almost exactly.', 'cond-mat-0006123-1-27-0': 'At the level of the distribution function, the results also match closely.', 'cond-mat-0006123-1-27-1': 'The peculiar non-Gaussian feature of the distribution function is clearly seen in Fig. [REF].', 'cond-mat-0006123-1-27-2': 'There is some slight mismatch near the peak.', 'cond-mat-0006123-1-28-0': 'For [MATH], there is cluster formation for [MATH].', 'cond-mat-0006123-1-28-1': 'Figures [REF] and [REF] show a comparison between MD and DSMC for [MATH].', 'cond-mat-0006123-1-28-2': 'The non-Gaussian feature of the distribution function shows a systematic deviation for DSMC simulations: there is overpopulation for slow velocities.', 'cond-mat-0006123-1-28-3': 'This is easily explained by considering that the DSMC method, like the Boltzmann equation, neglects correlations.', 'cond-mat-0006123-1-28-4': 'When the system approaches the clustering regime, increased dissipation induces correlations which tend to make the particles collide less[CITATION].', 'cond-mat-0006123-1-28-5': 'In DSMC these correlations are neglected, with the corresponding systematic overestimation in the collision frequency.', 'cond-mat-0006123-1-28-6': 'This overestimation results in a lower temperature of the system about the density peak.', 'cond-mat-0006123-1-29-0': 'We have shown that an external temperature gradient can inhibit cluster formation in a quasi-elastic 1-D granular system.', 'cond-mat-0006123-1-29-1': 'When [MATH] is very small, perturbation schemes may be applicable in some cases[CITATION].', 'cond-mat-0006123-1-29-2': 'Since this method is simulational, the question arose of whether an external temperature gradient may inhibit cluster formation in systems where [MATH] is not small.', 'cond-mat-0006123-1-29-3': 'To pursue this issue we simulated systems with [MATH] and [MATH] with increasing temperature differences until cluster formation was inhibited.', 'cond-mat-0006123-1-29-4': 'The results of the simulations with [MATH] are shown in Figs. [REF] and [REF].', 'cond-mat-0006123-1-29-5': 'Recall that, if [MATH], [MATH], so this is a system under a huge temperature gradient.', 'cond-mat-0006123-1-29-6': 'The agreement between MD and DSMC is still reasonable.', 'cond-mat-0006123-1-29-7': 'Note that the overpopulation of slow velocities is also present in this case.', 'cond-mat-0006123-1-30-0': 'For [MATH] the system exhibited cluster formation up to [MATH].', 'cond-mat-0006123-1-31-0': '# Conclusions', 'cond-mat-0006123-1-32-0': 'We have shown that a system not subject to gravity between thermal walls unavoidably reaches a non-steady state when the walls are at the same temperature.', 'cond-mat-0006123-1-32-1': 'A cluster forms in the bulk, slowly roaming about the system while absorbing particles.', 'cond-mat-0006123-1-32-2': 'As it grows, the amplitude of the random walk increases, until at last the surrounding gas cannot keep the cluster away from the walls.', 'cond-mat-0006123-1-32-3': 'When the cluster reaches a wall, a part of it is ejected by the wall through the cluster (relabeling the particles on collisions), effectively pushing the cluster away from the wall, and leaving it to grow again.', 'cond-mat-0006123-1-33-0': 'Most of the time the cluster is far from the walls.', 'cond-mat-0006123-1-33-1': 'Thus measuring the distribution function at a wall is measuring the distribution function of the gas that surrounds the cluster.', 'cond-mat-0006123-1-33-2': 'This distribution function has a well-defined time average, and exhibits apparently singular behavior for slow particles, diverging like [MATH].', 'cond-mat-0006123-1-34-0': 'Imposing an external temperature gradient forces the cluster against the colder wall, inhibiting its growth.', 'cond-mat-0006123-1-34-1': 'Increasing the temperature difference leads to a system in which the cluster never forms: the system is completely fluidized.', 'cond-mat-0006123-1-34-2': 'The distribution function of the gas exhibits peculiar non-Gaussian features: a smooth version of the aforementioned singularity.', 'cond-mat-0006123-1-34-3': 'Therefore, any attempt at solving the Boltzmann equation through moment methods must consider this feature in the initial ansatz, as is done for the problem of an infinitely strong shock wave in [CITATION] and [CITATION].', 'cond-mat-0006123-1-34-4': 'In fact, a solution for this problem was attempted using the four moment method of [CITATION].', 'cond-mat-0006123-1-34-5': 'As mentioned in [CITATION], the fourth balance equation could not be freely chosen when the boundary conditions were symmetric: some choices gave undefined results.', 'cond-mat-0006123-1-34-6': 'As is natural, from this calculation we obtained absurd results, such as higher temperature in the middle of the system than near the walls.', 'cond-mat-0006123-1-35-0': 'We compared molecular dynamics with direct-simulation Monte Carlo.', 'cond-mat-0006123-1-35-1': 'Agreement between these two methods shows that the non-Gaussian feature of the distribution function may be predicted by the dissipative Boltzmann equation.', 'cond-mat-0006123-1-35-2': 'As the system approaches cluster formation, correlations settle in.', 'cond-mat-0006123-1-35-3': 'These correlations reduce the collision frequency among particles.', 'cond-mat-0006123-1-35-4': 'DSMC neglects these correlations, and thus overestimates the number of collisions.', 'cond-mat-0006123-1-35-5': 'This exaggerates the effects of dissipation, producing steeper profiles.', 'cond-mat-0006123-1-36-0': 'We thank Aldo Frezzotti, Rosa Ramirez, and Rodrigo Soto for helpful discussions.', 'cond-mat-0006123-1-36-1': 'This work has been partially funded by FONDAP under grant 11980002, Fondecyt under grants 1000884 and 2990108, and by Fundacion Andes through a doctoral scholarship.'}","{'cond-mat-0006123-2-0-0': 'We study a 1-D granular gas of point-like particles not subject to gravity between two walls at temperatures [MATH] and [MATH].', 'cond-mat-0006123-2-0-1': 'The system exhibits two distinct regimes, depending on the normalized temperature difference [MATH]: one completely fluidized and one in which a cluster coexists with the fluidized gas.', 'cond-mat-0006123-2-0-2': 'When [MATH] is above a certain threshold, cluster formation is fully inhibited, obtaining a completely fluidized state.', 'cond-mat-0006123-2-0-3': 'The mechanism that produces these two phases is explained.', 'cond-mat-0006123-2-0-4': 'In the fluidized state the velocity distribution function exhibits peculiar non-Gaussian features.', 'cond-mat-0006123-2-0-5': 'For this state, comparison between integration of the Boltzmann equation using the direct-simulation Monte Carlo method and results stemming from microscopic Newtonian molecular dynamics gives good coincidence, establishing that the non-Gaussian features observed do not arise from the onset of correlations.', 'cond-mat-0006123-2-1-0': '# Introduction', 'cond-mat-0006123-2-2-0': 'Granular systems have been extensively studied due both to the theoretical challenges they present (for a recent review see [CITATION]) and to the applications of industrial importance that spring from the rich phenomena they exhibit (see [CITATION] and references therein).', 'cond-mat-0006123-2-2-1': 'These systems are characterized by an energy loss in collisions.', 'cond-mat-0006123-2-2-2': 'This loss is at the base of many interesting phenomena, such as inelastic collapse[CITATION], where the particles collide infinitely often in finite time, and clustering (for a sample of theoretical, simulational, and experimental approaches see [CITATION]).', 'cond-mat-0006123-2-2-3': 'Different methods for keeping the system from collapsing have been devised, such as subjecting the particles to Brownian forces[CITATION], and forcing through the boundaries by putting the system in a box with one or more thermal-like walls (see for example [CITATION]).', 'cond-mat-0006123-2-2-4': 'This work focuses on the latter method.', 'cond-mat-0006123-2-3-0': 'Being one of the simplest types of forcing, several authors[CITATION] have studied a one-dimensional system in a box with one or two heated (stochastic) walls.', 'cond-mat-0006123-2-3-1': 'Of these, [CITATION] study cluster formation, although [CITATION] are not strictly one-dimensional.', 'cond-mat-0006123-2-4-0': 'This article studies a quasielastic one-dimensional system not subject to gravity between two thermalizing walls.', 'cond-mat-0006123-2-4-1': 'We focus on two control parameters: the total inelasticity parameter [MATH], where [MATH] is the number of particles and [MATH] is the restitution coefficient, and the externally imposed temperature gradient.', 'cond-mat-0006123-2-4-2': 'The parameter [MATH] has been shown to be relevant for the quasielastic system[CITATION].', 'cond-mat-0006123-2-4-3': 'By varying these parameters we determine the region in parameter space where clustering is fully inhibited, obtaining a fluidized state.', 'cond-mat-0006123-2-4-4': 'We present a singular feature of the distribution function for the clustering regime, and then study how this feature is modified for the fluidized state.', 'cond-mat-0006123-2-5-0': 'In [CITATION] the authors study a one-dimensional system of point-like particles between an elastic and a heated wall.', 'cond-mat-0006123-2-5-1': 'They emphasize that a cluster inevitably forms away from the heated wall, regardless of how elastic the system is (as long as it is not perfectly elastic).', 'cond-mat-0006123-2-5-2': 'They also study the same system, but with both walls expelling the particles with a fixed velocity.', 'cond-mat-0006123-2-5-3': 'In this case they find that the cluster forms away from the walls and roams slowly about the system, with two groups of fast particles connecting the cluster with the ""heated"" walls.', 'cond-mat-0006123-2-6-0': 'In [CITATION] the same system is studied for different types of boundary conditions at the heated wall.', 'cond-mat-0006123-2-6-1': 'The stochastic boundary condition studied has the form of a power of the velocity times the ""thermal"" condition (the one that produces a Maxwell-Boltzmann distribution in the elastic case).', 'cond-mat-0006123-2-6-2': 'The authors show that when the power that multiplies the thermal condition is positive the test-particle equation (derived from the Boltzmann equation) has a steady-state solution.', 'cond-mat-0006123-2-6-3': 'Thus the thermal case does not have a steady state and develops a cluster away from the heated wall.', 'cond-mat-0006123-2-6-4': 'The mechanism for the growth of the cluster is explained and verified numerically.', 'cond-mat-0006123-2-7-0': 'In [CITATION] a similar system is studied: a long thin pipe of inelastic hard disks with heated walls (at the same temperature) at the ends of the pipe and periodic side walls.', 'cond-mat-0006123-2-7-1': 'The pipe is thin enough for the particle order to be preserved.', 'cond-mat-0006123-2-7-2': 'The probability distribution for the distance between the central particles is studied.', 'cond-mat-0006123-2-7-3': 'This distribution gives a markedly denser system near the center than in the elastic case, although the limit to the elastic case is smooth, unlike the strictly one-dimensional case of [CITATION].', 'cond-mat-0006123-2-7-4': 'In [CITATION] the same author studies the velocity correlations that this system develops as inelasticity is increased, showing that a consistent description must take these correlations into account.', 'cond-mat-0006123-2-8-0': 'In this paper we revisit the one-dimensional system of [MATH] point-like particles interacting via collisions that conserve momentum but dissipate kinetic energy.', 'cond-mat-0006123-2-8-1': 'To fix notation, the particle velocities after a collision are given by [EQUATION] where [MATH] is the velocity of particle [MATH] before a collision, and [MATH], being [MATH] the restitution coefficient.', 'cond-mat-0006123-2-8-2': 'For the elastic case ([MATH]) the particles simply exchange velocities.', 'cond-mat-0006123-2-8-3': 'Since the particles are point-like, the system is then indistinguishable from a system in which the particles do not interact.', 'cond-mat-0006123-2-9-0': 'This one-dimensional system is interesting because dissipation is the first order correction to a free gas.', 'cond-mat-0006123-2-9-1': 'Besides, results for the one-dimensional system have been found to have unexpected relevance for higher-dimensional problems.', 'cond-mat-0006123-2-9-2': 'For example, in two-dimensions the particles involved in inelastic collapse lie roughly on a line[CITATION].', 'cond-mat-0006123-2-9-3': 'Also, the dissipation-induced temperature gradients calculated in [CITATION] for the one-dimensional case inspired the authors to look for dissipation-induced Rayleigh-Benard-like convection for a two-dimensional system without an externally imposed temperature gradient[CITATION].', 'cond-mat-0006123-2-10-0': 'For a system with one thermal wall and open on the other side, under the influence of gravity, the quasielastic system may be kept fluidized[CITATION]: any cluster that starts to form is forced against the thermal wall, where it evaporates.', 'cond-mat-0006123-2-10-1': 'In [CITATION] the test-particle equation[CITATION]-which is the 1-D Boltzmann equation where the limit [MATH] is taken, but keeping [MATH] fixed-is successfully applied, with close matching of theory and simulations even at the level of the distribution function.', 'cond-mat-0006123-2-11-0': 'The one-dimensional system under study is left to evolve between two thermal walls at temperatures [MATH] and [MATH], with [MATH].', 'cond-mat-0006123-2-11-1': 'We define the parameter [EQUATION] to quantify how far from the symmetrical case is the system.', 'cond-mat-0006123-2-11-2': 'Under the same conditions, an elastic system has a perfectly bimodal velocity distribution with global homogeneous temperature equal to [MATH].', 'cond-mat-0006123-2-11-3': 'For the sake of comparison, we simulate systems with [MATH].', 'cond-mat-0006123-2-11-4': 'Here [MATH] represents a symmetrical setting, while [MATH] represents an infinitely strong temperature gradient.', 'cond-mat-0006123-2-12-0': 'As in [CITATION], for [MATH] a cluster unavoidably forms away from the thermal walls.', 'cond-mat-0006123-2-12-1': 'After forming, the cluster performs an apparently random walk about the center, growing in size (and therefore mass) while the rest of the system grows more rarefied.', 'cond-mat-0006123-2-12-2': 'With the decrease in density of the surrounding gas and the increased inertia of the cluster, an eventual collision with one of the walls is to be expected.', 'cond-mat-0006123-2-12-3': 'When this happens part of the cluster evaporates, and what is left of it is expelled from the wall (see Fig. [REF]), thus restarting the growth process.', 'cond-mat-0006123-2-12-4': 'Thus not only is the system highly clumped, but also in a non-steady state.', 'cond-mat-0006123-2-12-5': 'Nevertheless, the gas that is far from the random-walk zone has a well-defined time average for the distribution function, as is seen in Fig. [REF].', 'cond-mat-0006123-2-12-6': 'A noteworthy feature of this distribution is that it exhibits apparently singular behavior for slow velocities.', 'cond-mat-0006123-2-13-0': 'Setting [MATH] the symmetry of the system is broken.', 'cond-mat-0006123-2-13-1': 'For [MATH] the cluster performs a slightly asymmetric random walk, spending more time near the colder wall, and therefore colliding more often with it.', 'cond-mat-0006123-2-13-2': 'Thus the cluster cannot grow as much as it did in the symmetric case before colliding with a wall.', 'cond-mat-0006123-2-13-3': 'By increasing [MATH] the cluster-wall collision frequency grows, even obtaining short ""windows"" in which the cluster completely evaporates.', 'cond-mat-0006123-2-13-4': 'By further increasing [MATH] these windows grow larger until a point is reached where no cluster forms.', 'cond-mat-0006123-2-13-5': 'In this fashion a totally fluidized state is achieved, which may be tractable with the dissipative Boltzmann equation.', 'cond-mat-0006123-2-13-6': 'However, the distribution function obtained from the molecular dynamics (MD) simulations exhibits a peculiar non-Gaussian feature for slow velocities.', 'cond-mat-0006123-2-13-7': 'This feature is a smoothed version of the apparent singularity of the symmetric case.', 'cond-mat-0006123-2-13-8': 'To discern whether this feature is due to correlations or is present before they settle in, we compared Newtonian molecular dynamics results with those obtained through direct-simulation Monte Carlo (DSMC), which neglects correlations.', 'cond-mat-0006123-2-13-9': 'The results agree very well, except when the system approaches the clustering regime.', 'cond-mat-0006123-2-14-0': '# Simulation method', 'cond-mat-0006123-2-15-0': 'We simulate the system through event-driven molecular dynamics[CITATION] and through direct-simulation Monte Carlo[CITATION].', 'cond-mat-0006123-2-15-1': 'The direct-simulation Monte Carlo procedures use the null-collision technique[CITATION] where, overestimating the collision frequency (using the maximum relative velocity within a cell), the number of collisions to be attempted is calculated through a Poisson process.', 'cond-mat-0006123-2-15-2': 'In the next step the collisions are attempted, choosing at random two particles within the cell, and making them collide with a probability proportional to their relative velocity.', 'cond-mat-0006123-2-15-3': 'Most of the molecular dynamics and DSMC simulations were done with [MATH] particles.', 'cond-mat-0006123-2-16-0': 'In the MD simulations we detect clusters using a geometric criterion: we consider chains of particles that are nearer than a critical distance (in our case [MATH] and [MATH], to be certain that the conclusions are independent of the choice).', 'cond-mat-0006123-2-16-1': 'The system length is one, and with a thousand particles the mean distance between neighbors for a homogeneous system is [MATH].', 'cond-mat-0006123-2-16-2': 'Thus we detect particles that are uncommonly near by three orders of magnitude.', 'cond-mat-0006123-2-16-3': 'We discard chains of length three or less, since they may be random encounters.', 'cond-mat-0006123-2-16-4': 'Measuring the total length of the cluster we have found that on average it is of the order of [MATH]; thus the choice of [MATH] as link-link distance is much larger than the true distance between them.', 'cond-mat-0006123-2-17-0': 'As already stated, the boundary conditions are such that the (homogeneous) temperature of the corresponding elastic system (equal to [MATH]) is one.', 'cond-mat-0006123-2-18-0': '# Results', 'cond-mat-0006123-2-19-0': '## Clustering regime', 'cond-mat-0006123-2-20-0': 'Figure [REF] shows the non-steady state of a granular system between two walls at the same temperature for [MATH].', 'cond-mat-0006123-2-20-1': 'A cluster forms away from the walls, performing a random walk of varying amplitude.', 'cond-mat-0006123-2-20-2': 'When the cluster reaches a wall, part of it evaporates, and the growth process begins anew.', 'cond-mat-0006123-2-21-0': 'As is usual for the quasielastic case, we relabel the particles when they collide.', 'cond-mat-0006123-2-21-1': 'This enables us to visualize this system as a group of barely interacting particles passing through each other.', 'cond-mat-0006123-2-22-0': 'The picture for cluster evolution, as explained before, is the following: the cluster grows because the slowest particles, due to the asymmetry of the distribution function, drift towards the cluster[CITATION].', 'cond-mat-0006123-2-22-1': 'As it grows, the density of the gas surrounding it decreases, with the consequent saturation in growth.', 'cond-mat-0006123-2-22-2': 'Thus we have a ""Brownian particle"" of increasing mass moving in an increasingly rarefied medium.', 'cond-mat-0006123-2-22-3': 'This ""particle"" will be increasingly less affected by the surrounding medium, until it can no longer be kept away from the walls.', 'cond-mat-0006123-2-23-0': 'The cluster moves several orders of magnitude slower than the thermal speed (three orders of magnitude in Fig. [REF]).', 'cond-mat-0006123-2-23-1': 'Upon reaching a wall, the front liners strike the wall and are expelled by it much faster than the other cluster members.', 'cond-mat-0006123-2-23-2': 'These particles pass through the cluster, transferring momentum to it, as described in [CITATION].', 'cond-mat-0006123-2-23-3': 'Thus these fast particles push the cluster away from the wall, where it can absorb particles again.', 'cond-mat-0006123-2-23-4': 'The fast particles, however, no longer belong to the cluster.', 'cond-mat-0006123-2-24-0': ""Since the slowest particles in the gas are the ones that will be absorbed by the cluster[CITATION], it is the number of slow particles in the gas that will determine the cluster's growth rate."", 'cond-mat-0006123-2-24-1': 'After the cluster strikes a wall, the expelled particles will be fast particles, and they will not contribute to the growth of the cluster during the time it takes for the gas to cool down again: the only particles available for absorption are the ones that were available before the cluster-wall collision.', 'cond-mat-0006123-2-24-2': 'This explains why the cluster keeps growing at approximately the same rate it did before the collision.', 'cond-mat-0006123-2-24-3': 'After the gas has cooled down, the growth rate returns to its normal value.', 'cond-mat-0006123-2-24-4': 'This is the end of the plateau seen in Fig. [REF] after each cluster-wall collision.', 'cond-mat-0006123-2-25-0': 'To quantify the evaporation process we proceed as in [CITATION]: as soon as the first particle belonging to the cluster reaches a wall, it is expelled with a speed much higher than the cluster velocity; thus we may consider the ideal situation of a cluster of [MATH] particles at rest being stricken by a fast particle with velocity [MATH] (in this case [MATH]).', 'cond-mat-0006123-2-25-1': ""After colliding with the first particle in the cluster, the new fast particle's speed will be [MATH]."", 'cond-mat-0006123-2-25-2': ""Thus, after traversing the cluster, the fast particle's velocity will be [MATH]."", 'cond-mat-0006123-2-25-3': 'Since momentum is conserved in collisions, the center of mass of the cluster will have acquired a speed of [EQUATION]', 'cond-mat-0006123-2-25-4': 'Considering the case [MATH] with fixed [MATH], as in [CITATION], we may simplify this expression to [EQUATION]', 'cond-mat-0006123-2-25-5': 'By further considering the case [MATH] we get [MATH].', 'cond-mat-0006123-2-25-6': 'In this limit, if the cluster reaches the wall with velocity [MATH] and is expelled from it with velocity [MATH], the number [MATH] of particles evaporated will satisfy [MATH].', 'cond-mat-0006123-2-25-7': 'For the situation shown in Fig. [REF], [MATH] and [MATH] are typically of the order of [MATH] and [MATH], hence the number of particles evaporated will be of the order of [MATH].', 'cond-mat-0006123-2-26-0': 'Even if the system is in a non-steady state, the gas at the walls (far from the random-walk zone) has a well-defined time average for the distribution function.', 'cond-mat-0006123-2-26-1': 'The distribution function at the left wall is shown in Fig. [REF].', 'cond-mat-0006123-2-26-2': 'There is an apparent singularity for slow velocities.', 'cond-mat-0006123-2-26-3': 'The distribution is asymmetric as it should, since the particles leaving the wall ([MATH]) follow a Gaussian distribution.', 'cond-mat-0006123-2-26-4': 'Figure [REF] also shows the distribution multiplied by [MATH].', 'cond-mat-0006123-2-26-5': 'Since the limit of [MATH] for [MATH] is finite and nonzero, we conclude that the distribution function exhibits a singularity that behaves like [MATH] for slow velocities.', 'cond-mat-0006123-2-27-0': 'As shown in [CITATION], when [MATH] (the symmetric case) the distribution shown in Fig. [REF] is not a solution of the steady-state test-particle equation: [EQUATION] where [EQUATION]', 'cond-mat-0006123-2-27-1': 'To establish this, let us study the behavior for small [MATH] of a solution of this equation.', 'cond-mat-0006123-2-27-2': 'Assume that, for small [MATH], [MATH], with [MATH] in order to have a finite density in the vicinity of [MATH].', 'cond-mat-0006123-2-27-3': 'Furthermore, assume that [MATH].', 'cond-mat-0006123-2-27-4': 'Inserting this behavior in Eq. ([REF]) we obtain [EQUATION]', 'cond-mat-0006123-2-27-5': 'Thus, in order to keep [MATH] (the amplitude of the singularity) finite, we must have either [MATH] or [MATH].', 'cond-mat-0006123-2-27-6': 'Integrating the distribution of Fig. [REF] we obtain [MATH].', 'cond-mat-0006123-2-27-7': 'Since [MATH], we must have [MATH].', 'cond-mat-0006123-2-27-8': 'But this corresponds to a nonintegrable distribution, and therefore the distribution cannot be steady.', 'cond-mat-0006123-2-28-0': '## Inhibition of cluster formation', 'cond-mat-0006123-2-29-0': 'Figure [REF] shows the regions in [MATH]-space where clustering is inhibited for [MATH].', 'cond-mat-0006123-2-29-1': 'As is to be expected, as the inelasticity increases, a stronger temperature gradient is necessary to inhibit cluster formation.', 'cond-mat-0006123-2-30-0': 'To discern whether the non-Gaussian features of the velocity distribution function are derived from correlations in the system we compared results from MD simulations (full Newtonian dynamics) with results from DSMC simulations (no velocity correlations assumed).', 'cond-mat-0006123-2-30-1': 'Figures [REF] and [REF] show this comparison for a case far from the clustering threshold ([MATH] and [MATH]).', 'cond-mat-0006123-2-30-2': 'The temperature of the left and right walls are chosen so that the global temperature for the elastic case (equal to [MATH]) is one.', 'cond-mat-0006123-2-30-3': 'The curves match almost exactly.', 'cond-mat-0006123-2-31-0': 'At the level of the distribution function, the results also match closely.', 'cond-mat-0006123-2-31-1': 'The peculiar non-Gaussian feature of the distribution function is clearly seen in Fig. [REF].', 'cond-mat-0006123-2-31-2': 'There is some slight mismatch near the peak.', 'cond-mat-0006123-2-32-0': 'For the fluidized case, since momentum is conserved and the system is stationary, the pressure is constant throughout the system.', 'cond-mat-0006123-2-32-1': 'The number density and the granular temperature calculated here are related by [MATH] in energy units).', 'cond-mat-0006123-2-32-2': 'Thus when the normalized density [MATH] varies little throughout the system ([MATH]), the normalized temperature is [EQUATION] thus obtaining the nearly symmetric profiles seen in Figs. [REF] and [REF].', 'cond-mat-0006123-2-33-0': 'Figures [REF], [REF], and [REF] compare the MD and DSMC results for cases near cluster formation.', 'cond-mat-0006123-2-33-1': 'The non-Gaussian feature of the distribution function shows a systematic deviation for DSMC simulations: there is overpopulation for slow velocities.', 'cond-mat-0006123-2-33-2': 'This is explained by considering that the DSMC method, like the Boltzmann equation, neglects correlations.', 'cond-mat-0006123-2-33-3': 'When the system approaches the clustering regime, increased dissipation induces correlations which tend to make the particles collide less[CITATION].', 'cond-mat-0006123-2-33-4': 'In DSMC these correlations are neglected, with the corresponding systematic overestimation in the collision frequency.', 'cond-mat-0006123-2-33-5': 'This overestimation results in a lower temperature of the system about the density peak.', 'cond-mat-0006123-2-34-0': '# Conclusions', 'cond-mat-0006123-2-35-0': 'We have shown that a system not subject to gravity between thermal walls unavoidably reaches a non-steady state when the walls are at the same temperature.', 'cond-mat-0006123-2-35-1': 'A cluster forms in the bulk, slowly roaming about the system while absorbing particles.', 'cond-mat-0006123-2-35-2': 'As it grows, the amplitude of the random walk increases, until at last the surrounding gas cannot keep the cluster away from the walls.', 'cond-mat-0006123-2-35-3': 'When the cluster reaches a wall, a part of it is ejected by the wall through the cluster (relabeling the particles on collisions), effectively pushing the cluster away from the wall, and leaving it to grow again.', 'cond-mat-0006123-2-36-0': 'Most of the time the cluster is far from the walls.', 'cond-mat-0006123-2-36-1': 'Thus measuring the distribution function at a wall is measuring the distribution function of the gas that surrounds the cluster.', 'cond-mat-0006123-2-36-2': 'This distribution function has a well-defined time average, and exhibits apparently singular behavior for slow particles, diverging like [MATH].', 'cond-mat-0006123-2-37-0': 'Imposing an external temperature gradient forces the cluster against the colder wall, inhibiting its growth.', 'cond-mat-0006123-2-37-1': 'Increasing the temperature difference leads to a system in which the cluster never forms: the system is completely fluidized.', 'cond-mat-0006123-2-37-2': 'The distribution function of the gas exhibits peculiar non-Gaussian features: a smooth version of the aforementioned singularity.', 'cond-mat-0006123-2-37-3': 'Therefore, any attempt at solving the Boltzmann equation through moment methods must consider this feature in the initial ansatz, as is done for the problem of an infinitely strong shock wave in [CITATION] and [CITATION].', 'cond-mat-0006123-2-37-4': 'In fact, a solution for this problem was attempted using the four moment method of [CITATION].', 'cond-mat-0006123-2-37-5': 'As mentioned in [CITATION], the fourth balance equation could not be freely chosen when the boundary conditions were symmetric: some choices gave undefined results.', 'cond-mat-0006123-2-37-6': 'As is natural, by not including the non-Gaussian feature in the ansatz for this calculation we obtained absurd results, such as higher temperature in the middle of the system than near the walls.', 'cond-mat-0006123-2-38-0': 'We compared molecular dynamics with direct-simulation Monte Carlo.', 'cond-mat-0006123-2-38-1': 'Agreement between these two methods shows that the non-Gaussian feature of the distribution function may be predicted by the dissipative Boltzmann equation.', 'cond-mat-0006123-2-38-2': 'As the system approaches cluster formation, correlations settle in.', 'cond-mat-0006123-2-38-3': 'These correlations reduce the collision frequency among particles.', 'cond-mat-0006123-2-38-4': 'DSMC neglects these correlations, and thus overestimates the number of collisions.', 'cond-mat-0006123-2-38-5': 'This exaggerates the effects of dissipation, producing steeper profiles.', 'cond-mat-0006123-2-39-0': 'We thank Rodrigo Soto, Aldo Frezzotti, and Rosa Ramirez for helpful discussions.', 'cond-mat-0006123-2-39-1': 'This work has been partially funded by Fundacion Andes through a doctoral scholarship, Fondecyt through grants 2990108 and 1000884, and by FONDAP through grant 11980002.'}","[['cond-mat-0006123-1-18-1', 'cond-mat-0006123-2-21-1'], ['cond-mat-0006123-1-32-0', 'cond-mat-0006123-2-35-0'], ['cond-mat-0006123-1-32-1', 'cond-mat-0006123-2-35-1'], ['cond-mat-0006123-1-32-2', 'cond-mat-0006123-2-35-2'], ['cond-mat-0006123-1-32-3', 'cond-mat-0006123-2-35-3'], ['cond-mat-0006123-1-21-0', 'cond-mat-0006123-2-24-0'], ['cond-mat-0006123-1-21-1', 'cond-mat-0006123-2-24-1'], ['cond-mat-0006123-1-21-2', 'cond-mat-0006123-2-24-2'], ['cond-mat-0006123-1-21-3', 'cond-mat-0006123-2-24-3'], ['cond-mat-0006123-1-27-0', 'cond-mat-0006123-2-31-0'], ['cond-mat-0006123-1-27-1', 'cond-mat-0006123-2-31-1'], ['cond-mat-0006123-1-27-2', 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['cond-mat-0006123-1-24-0', 'cond-mat-0006123-2-27-0'], ['cond-mat-0006123-1-3-0', 'cond-mat-0006123-2-2-0'], ['cond-mat-0006123-1-3-2', 'cond-mat-0006123-2-2-2'], ['cond-mat-0006123-1-7-0', 'cond-mat-0006123-2-10-0'], ['cond-mat-0006123-1-7-1', 'cond-mat-0006123-2-10-1'], ['cond-mat-0006123-1-4-0', 'cond-mat-0006123-2-8-0'], ['cond-mat-0006123-1-5-3', 'cond-mat-0006123-2-9-3'], ['cond-mat-0006123-1-26-1', 'cond-mat-0006123-2-30-2']]",[],"[['cond-mat-0006123-1-21-4', 'cond-mat-0006123-2-24-4'], ['cond-mat-0006123-1-34-6', 'cond-mat-0006123-2-37-6'], ['cond-mat-0006123-1-22-6', 'cond-mat-0006123-2-25-6'], ['cond-mat-0006123-1-22-7', 'cond-mat-0006123-2-25-6'], ['cond-mat-0006123-1-28-1', 'cond-mat-0006123-2-33-0'], ['cond-mat-0006123-1-36-1', 'cond-mat-0006123-2-39-1'], ['cond-mat-0006123-1-4-1', 'cond-mat-0006123-2-8-1'], ['cond-mat-0006123-1-4-2', 'cond-mat-0006123-2-8-1']]",[],"['cond-mat-0006123-1-1-0', 'cond-mat-0006123-1-30-0']","{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}",https://arxiv.org/abs/cond-mat/0006123,,,
hep-ph-0408037,"{'hep-ph-0408037-1-0-0': 'The structure of the nucleon wave function as a bound system of the constituent quarks was considered in framework of the quasipotential method of description of the bound states with a fixed number of particles.', 'hep-ph-0408037-1-0-1': 'In the impulse approximation the wave function is reduced to the three-quark component of the vector state in the Fock-momentum space.', 'hep-ph-0408037-1-0-2': 'Spin structure of the wave function was studied by the decomposition method of the irreducible representations product of inhomogeneous Lorentz group.', 'hep-ph-0408037-1-0-3': 'Physical distinction of SU(6) symmetric solution is determined by uniqueness of this solution in the nonrelativistic limit.', 'hep-ph-0408037-1-0-4': 'The effective mass approximation for the relativistic theory was numerically studied, in which dispersion of the average quark momentum is small as compare to a big average quark momentum.', 'hep-ph-0408037-1-0-5': 'Relativistic generalization of the nonrelativistic three-particle oscillator was proposed.', 'hep-ph-0408037-1-1-0': 'Introduction.', 'hep-ph-0408037-1-2-0': 'The two approaches have a leading position among others to solve a question about the construction of the relativistic many particle wave function (WF).', 'hep-ph-0408037-1-2-1': 'The first approach based on many-time 4-dimensional formalizm and its various 3-dimentional reductions (quasipotential (QP) formulations), in which WF is defined on equal-time surface in the center-of-mass system or in laboratory system.', 'hep-ph-0408037-1-2-2': 'The second one is formulated in light front formalism [CITATION].', 'hep-ph-0408037-1-2-3': 'Vacuum fluctuations are absent in the light front formalism and this allows to apply the fixed number particle approximation.', 'hep-ph-0408037-1-2-4': 'The light front formalism is more suitable for description at high energies [CITATION].', 'hep-ph-0408037-1-2-5': 'It may be expected that the first approach listed above is preferable at low and medium energies.', 'hep-ph-0408037-1-2-6': 'The aim of this work is to demonstrate this property by using example of description of nucleon in the quark model in the Logunov-Tavkhelidze QP approach [CITATION], in covariant formulation [CITATION].', 'hep-ph-0408037-1-2-7': 'The vacuum fluctuations, which are called the main defect of this method [CITATION], may be suppressed at low energies.', 'hep-ph-0408037-1-2-8': 'One task for developing approach consists of the answer to the question: in what region mechanizm of constituent quark may be applied for the nucleon, and that must be solved by using of the experimental data.', 'hep-ph-0408037-1-2-9': 'The fixed number particles assumption may be formally expressed as a zero-order interaction in quark operators in the Bethe-Salpeter wave function.', 'hep-ph-0408037-1-2-10': 'This allows one to reduce QP WF to the decomposition component of the vector state in the Fock-momentum space.', 'hep-ph-0408037-1-2-11': 'Then it is possible to apply an apparatus of decomposition of the product of irreducible representations of the Lorentz-group [CITATION], which are analogous of the Clebsh-Gordan decomposition in quantum mechanics.', 'hep-ph-0408037-1-2-12': 'Thus the QP method is the analogy of quantum mechanics description, which was fruitfully used by many authors.', 'hep-ph-0408037-1-2-13': 'In the third section of the article this property of the QP approach is used to obtain a solution of the three particle relativistic oscillator.', 'hep-ph-0408037-1-3-0': '1.', 'hep-ph-0408037-1-3-1': 'Three quark wave function of nucleon in quasipotential approach.', 'hep-ph-0408037-1-4-0': 'Let us suppose that the Fock-momentum space for nucleon has a 3-quark basis and the nucleon vector state in this space decomposes as: [EQUATION]', 'hep-ph-0408037-1-4-1': 'Orthonormalization conditions for the nucleon and quark vector states are following: [EQUATION]', 'hep-ph-0408037-1-4-2': 'Here and further [MATH] means a set of values with indices k=1,2,3; [MATH]; [MATH] are momentum and mass of quark; [MATH], [MATH] is the nucleon mass; [MATH] are spins and [MATH] are isospins of nucleon and constinuent quarks; [MATH] is the third projection of x.', 'hep-ph-0408037-1-4-3': 'In this section the isospin indices will be omitted.', 'hep-ph-0408037-1-4-4': ""We assume the sum over the repeated spin and isospin indices (if we don't indicate espessially)."", 'hep-ph-0408037-1-4-5': 'In the quasipotential approach let us consider a covariant projection of the Bethe-Salpeter wave function on equal-times surface in the center-of-mass system [CITATION]: [EQUATION] where [MATH] is the Bethe-Salpeter wave function, [MATH] is the unit 4-momentum.', 'hep-ph-0408037-1-4-6': '[MATH]-functions make the times to be equal covariantly in the c.m.s. Using a property of translational invariance, it is possible to separate motion of center-of-mass system and then the Fourier-representation of the wave function ([REF]) is given by [CITATION], notation [MATH] is introduced: [EQUATION] where [MATH] is a matrix of the Lorentz-boost representation [MATH], [MATH], [MATH].', 'hep-ph-0408037-1-4-7': 'In ([REF]) the wave function depends on two independent 3-momenta.', 'hep-ph-0408037-1-4-8': 'For the sake of symmetry we keep all the momenta assuming their relation [MATH].', 'hep-ph-0408037-1-4-9': 'The expression for [MATH] has the form: [EQUATION]', 'hep-ph-0408037-1-4-10': 'In the quantum-field interpretation the fixed number particles assumption is expressed in replacement of the Heizenberg field operators to the Dirac ones (zero-order interaction in field operators).', 'hep-ph-0408037-1-4-11': 'Bound effects are contained in the Heizenberg vector state of the bound system.', 'hep-ph-0408037-1-4-12': 'We have applied this assumption in ([REF]) and substituted it in ([REF]) and as a result the wave function ([REF]) is expressed through the decomposition component ([REF]) in the form: [EQUATION]', 'hep-ph-0408037-1-4-13': 'For obtaining ([REF]) we have used the relations: [EQUATION] where [MATH] is the Wigner spin rotation matrix, which is defined as a [MATH] representation of the 3-dimensional rotation [MATH] [CITATION].', 'hep-ph-0408037-1-4-14': 'It has an explicit form [CITATION]: [EQUATION]', 'hep-ph-0408037-1-4-15': 'We multiply ([REF]) by [MATH] from left-handed side and sum the result over the spinor index: [EQUATION]', 'hep-ph-0408037-1-4-16': 'The function ([REF]) is named a wave function projected onto positive frequency states [CITATION].', 'hep-ph-0408037-1-4-17': 'Taking into account ([REF]) and ([REF]) we obtain: [EQUATION] where [MATH] is related with [MATH], defined in ([REF])), via: [EQUATION]', 'hep-ph-0408037-1-4-18': 'Using the method of the two-time Green function [CITATION] it is possible to obtain quasipotential equation for function [MATH]: [EQUATION]', 'hep-ph-0408037-1-4-19': 'In the impulse approximation the quasipotential [MATH] may be independent of the energy, because of the nucleon mass in the equation iteration ([REF]) may be replaced in quasipotential by sum of the free quark energy in c.m.s. [MATH] and then the next order in the coupling constant may be neglected.', 'hep-ph-0408037-1-4-20': 'The wave function normalization condition obtained by means of the Green function [CITATION] for the quasipotential independent of energy has a symmetric form relatively the particle permutations: [EQUATION]', 'hep-ph-0408037-1-5-0': '2.', 'hep-ph-0408037-1-5-1': 'Nucleon spin wave function.', 'hep-ph-0408037-1-6-0': 'Now let us use the general method to construct the state with a definite momentum in the system of free particles to study spin dependence of QP WF by using connection of QP WF with a decomposition component of the nucleon vector state in the Fock-momentum space ([REF]).', 'hep-ph-0408037-1-6-1': 'In the center-of mass system formula ([REF]) has the form: [EQUATION]', 'hep-ph-0408037-1-6-2': 'Now we decompose [MATH] over free particle states with a definite momentum in the c.m.s. [MATH].', 'hep-ph-0408037-1-6-3': 'The dots mean other observables from the complete set which will be indicated below.', 'hep-ph-0408037-1-6-4': 'Its choise depends on the ways of summing spins and orbital angular momenta.', 'hep-ph-0408037-1-6-5': 'Further we use the method of work [CITATION] to decompose the direct product of irreducible representations of the inhomogeneous Lorentz group.', 'hep-ph-0408037-1-6-6': 'In case of the two free particle system in its c.m.s. with 3-momentum [MATH] and [MATH]) and with the invariant system [1+2] mass [MATH], this decomposition has the following form: [EQUATION]', 'hep-ph-0408037-1-6-7': 'Here [MATH] is angle variables of momentum [MATH], [MATH] is spherical harmonics, [MATH] is the relative orbital momentum of the system [1+2], [MATH] is spin, [MATH] is the total spin of system [1+2].', 'hep-ph-0408037-1-6-8': 'Going from the rest system [1+2] to c.m.s. of nucleon, we have from ([REF]): [EQUATION] where [MATH], [MATH].', 'hep-ph-0408037-1-6-9': 'If we introduce the notation [MATH], then applying ([REF]) we obtain: [EQUATION]', 'hep-ph-0408037-1-6-10': 'Multiplying ([REF]) on the vector state of the third quark, we have obtained: [EQUATION]', 'hep-ph-0408037-1-6-11': 'In analogy with ([REF]) we obtain (notation [MATH] is introduced): [EQUATION] where [MATH] is the spin of the system [[1+2]+3], obtained by summation of [MATH] and [MATH]; [MATH] is the orbital angular momentum of the third quark with respect to the [1+2] system, [MATH] is the total spin of the system [[1+2]+3], obtained by summmation of [MATH] and [MATH].', 'hep-ph-0408037-1-6-12': 'Thus, the desired decomposition has the form: [EQUATION]', 'hep-ph-0408037-1-6-13': 'We have separated motion of c.m.s. in the scalar product of the bound system vector state and the free particle vector state: [EQUATION]', 'hep-ph-0408037-1-6-14': 'Independence of [MATH] on [MATH] is the result the symmetry relatively the reflection.', 'hep-ph-0408037-1-6-15': 'The relation ([REF]) was obtained by using unit operator [MATH].', 'hep-ph-0408037-1-6-16': 'According to ([REF]), ([REF]) and ([REF]), for relative motion WF [MATH] in full record with isospin indices we have obtained: [EQUATION]', 'hep-ph-0408037-1-6-17': 'Since the contributions of higher orbital moments in nucleon are small [CITATION], we write only the S-wave part ([MATH]) omitting spin, isospin and symbols [MATH]: [EQUATION] where [MATH].', 'hep-ph-0408037-1-6-18': 'Let us decompose functions [MATH] over two isospin basic functions [MATH]; [MATH] is the [1+2] system isospin : [MATH] (sum over [MATH]) and substitute it in ([REF]): [EQUATION]', 'hep-ph-0408037-1-6-19': 'The function [MATH] ([REF]) is symmetrical relatively the particle permutations and with antisymmetric colour fuction realizes antisymmetrical representation of the permutations group.', 'hep-ph-0408037-1-6-20': 'S-wave function ([REF]) with zero-orbital momentum of the [1+2] system and zero angular orbital momentum of the third quark with respect to the [1+2] system, is antisymmetric relatively the quark permutations.', 'hep-ph-0408037-1-6-21': 'Let us suppose, that interquark interaction does not depend on the spin and isospin: [MATH] and we choose the normalization constants to agree with the nonrelativistic theory: [MATH], [MATH].', 'hep-ph-0408037-1-6-22': 'Introducing the notation for the symmetric spin-isospin wave function [MATH] , we obtain [EQUATION]', 'hep-ph-0408037-1-6-23': 'Two [MATH]-matrices in ([REF]) represent the minimal kinematic violation of the [MATH] symmetry.', 'hep-ph-0408037-1-6-24': 'The general expression for WF ([REF]) indicates three ways of [MATH] violations, related to relativization of the model: (1) including the admixture of the mixed [MATH] symmetry in WF, (2) taking into account the quark interaction dependence on spin and isospin, (3) and P- and D-waves.', 'hep-ph-0408037-1-7-0': '3.', 'hep-ph-0408037-1-7-1': 'Effective mass approximation and relativistic oscillator.', 'hep-ph-0408037-1-8-0': 'According to ([REF]), ([REF]) and ([REF]), function [MATH] satisfies the equation and normalization condition: [EQUATION] where [MATH] is a quasipotential determined by relation: [MATH], where [MATH] is a matrix record of factors in square brackets of expression ([REF]).', 'hep-ph-0408037-1-8-1': 'Variables [MATH], [MATH] and [MATH] are equivalent and the choise of variables [MATH], [MATH] as independent does not change the three quark equivalent to calculate the average values.', 'hep-ph-0408037-1-8-2': 'For example, the three effective quark masses, obtained by formula [MATH] are equal ([MATH]).', 'hep-ph-0408037-1-8-3': 'The approximate relativistic models, using the idea of the effective quark mass, were considered in work [CITATION].', 'hep-ph-0408037-1-8-4': 'In these works the effective quark mass is introduced as a parameter instead of the quark mass.', 'hep-ph-0408037-1-8-5': 'In our work the effective quark mass has been introduced as a suitable approximation.', 'hep-ph-0408037-1-8-6': 'Let us pass to the semi-momenta [CITATION] in ([REF]): [MATH].', 'hep-ph-0408037-1-8-7': 'Equation ([REF]) takes a nonrelativistic form in terms of [MATH].', 'hep-ph-0408037-1-8-8': 'In equation ([REF]) we rewrite the quasipotential as an analog of nonrelativistic oscillator: [EQUATION]', 'hep-ph-0408037-1-8-9': 'Using new variables [MATH] and parameters in ([REF]) [MATH], we have obtained: [EQUATION]', 'hep-ph-0408037-1-8-10': 'Decomposing the quark energies in semi-momenta in the [MATH] neighborhood and taking into account the condition [MATH], it is possible to show that term [MATH] has a small contribution into the energy, if value [MATH] is small (the effective mass aproximation).', 'hep-ph-0408037-1-8-11': 'In the zero-order approximation relatively the [MATH], we obtain a solution of equation ([REF]) for the ground state ([MATH]): [MATH].', 'hep-ph-0408037-1-8-12': 'Taking into account the effective mass approximation, this solution is reduced to the form: [EQUATION]', 'hep-ph-0408037-1-8-13': 'Numerical calculations with the relativistic oscillator WF ([REF]) gives [MATH] for [MATH].', 'hep-ph-0408037-1-8-14': 'In this case [MATH] is positive and is in limits [MATH].', 'hep-ph-0408037-1-8-15': 'Smallness of [MATH] means the smallness of the momentum module dispersion.', 'hep-ph-0408037-1-8-16': 'Indeed, in the ground state [MATH] and therefore [MATH], where [MATH] We note that, WF in form [MATH], applied in many works, is different from ([REF]) by the order of [MATH] and has different asymptotic behaviour for high momenta.', 'hep-ph-0408037-1-9-0': 'Conclusion.', 'hep-ph-0408037-1-10-0': 'In the impulse approximation, represented by the zero-order field operators in the Bethe-Salpeter WF, the quasipotential wave function reduces to the three-quark component of the vector state decomposition of the bound system in the Fock-momentum space.', 'hep-ph-0408037-1-10-1': 'This allows us to apply the standart method for the decomposition of the irreducible representations product of the inhomogeneous Lorentz-group over states with the definite momentum to analyse of the QP WF structure.', 'hep-ph-0408037-1-10-2': 'The physical preference of the [MATH]-symmetric solution is determined by its uniqueness in the nonreletivistic limit.', 'hep-ph-0408037-1-10-3': 'The model of the relativistic three-particle oscillator being the direct generalization of the nonrelativistic oscillator is proposed.', 'hep-ph-0408037-1-10-4': 'Numerically it was shown that the effective mass approximation may be applyed in the wide region of the oscillator parameters, in which the ratio of the momentum dispersion to the average value of the quark momentum module is small.', 'hep-ph-0408037-1-10-5': 'Authors express gratitude to N.V.Maksimenko for support and E.A.Dey for usefull remarks.'}","{'hep-ph-0408037-2-0-0': 'The structure of the nucleon wave function as a bound system of the constituent quarks was considered in framework of the quasipotential method of description of the bound states with a fixed number of particles.', 'hep-ph-0408037-2-0-1': 'In the impulse approximation the wave function is reduced to the three-quark component of the vector state in the Fock-momentum space.', 'hep-ph-0408037-2-0-2': 'Spin structure of the wave function was studied by the decomposition method of the irreducible representations product of inhomogeneous Lorentz group.', 'hep-ph-0408037-2-0-3': 'Physical distinction of SU(6) symmetric solution is determined by uniqueness of this solution in the nonrelativistic limit.', 'hep-ph-0408037-2-0-4': 'The effective mass approximation for the relativistic theory was numerically studied, in which dispersion of the average quark momentum is small as compare to a big average quark momentum.', 'hep-ph-0408037-2-0-5': 'Relativistic generalization of the nonrelativistic three-particle oscillator was proposed.', 'hep-ph-0408037-2-1-0': 'Introduction.', 'hep-ph-0408037-2-2-0': 'The two approaches have a leading position among others to solve a question about the construction of the relativistic many particle wave function (WF).', 'hep-ph-0408037-2-2-1': 'The first approach based on many-time 4-dimensional formalizm and its various 3-dimentional reductions (quasipotential (QP) formulations), in which WF is defined on equal-time surface in the center-of-mass system or in laboratory system.', 'hep-ph-0408037-2-2-2': 'The second one is formulated in light front formalism [CITATION].', 'hep-ph-0408037-2-2-3': 'Vacuum fluctuations are absent in the light front formalism and this allows to apply the fixed number particle approximation.', 'hep-ph-0408037-2-2-4': 'The light front formalism is more suitable for description at high energies [CITATION].', 'hep-ph-0408037-2-2-5': 'It may be expected that the first approach listed above is preferable at low and medium energies.', 'hep-ph-0408037-2-2-6': 'The aim of this work is to demonstrate this property by using example of description of nucleon in the quark model in the Logunov-Tavkhelidze QP approach [CITATION], in covariant formulation [CITATION].', 'hep-ph-0408037-2-2-7': 'The vacuum fluctuations, which are called the main defect of this method [CITATION], may be suppressed at low energies.', 'hep-ph-0408037-2-2-8': 'One task for developing approach consists of the answer to the question: in what region mechanizm of constituent quark may be applied for the nucleon, and that must be solved by using of the experimental data.', 'hep-ph-0408037-2-2-9': 'The fixed number particles assumption may be formally expressed as a zero-order interaction in quark operators in the Bethe-Salpeter wave function.', 'hep-ph-0408037-2-2-10': 'This allows one to reduce QP WF to the decomposition component of the vector state in the Fock-momentum space.', 'hep-ph-0408037-2-2-11': 'Then it is possible to apply an apparatus of decomposition of the product of irreducible representations of the Lorentz-group [CITATION], which are analogous of the Clebsh-Gordan decomposition in quantum mechanics.', 'hep-ph-0408037-2-2-12': 'Thus the QP method is the analogy of quantum mechanics description, which was fruitfully used by many authors.', 'hep-ph-0408037-2-2-13': 'In the third section of the article this property of the QP approach is used to obtain a solution of the three particle relativistic oscillator.', 'hep-ph-0408037-2-3-0': '1.', 'hep-ph-0408037-2-3-1': 'Three quark wave function of nucleon in quasipotential approach.', 'hep-ph-0408037-2-4-0': 'Let us suppose that the Fock-momentum space for nucleon has a 3-quark basis and the nucleon vector state in this space decomposes as: [EQUATION]', 'hep-ph-0408037-2-4-1': 'Orthonormalization conditions for the nucleon and quark vector states are following: [EQUATION]', 'hep-ph-0408037-2-4-2': 'Here and further [MATH] means a set of values with indices k=1,2,3; [MATH]; [MATH] are momentum and mass of quark; [MATH], [MATH] is the nucleon mass; [MATH] are spins and [MATH] are isospins of nucleon and constinuent quarks; [MATH] is the third projection of x.', 'hep-ph-0408037-2-4-3': 'In this section the isospin indices will be omitted.', 'hep-ph-0408037-2-4-4': ""We assume the sum over the repeated spin and isospin indices (if we don't indicate espessially)."", 'hep-ph-0408037-2-4-5': 'In the quasipotential approach let us consider a covariant projection of the Bethe-Salpeter wave function on equal-times surface in the center-of-mass system [CITATION]: [EQUATION] where [MATH] is the Bethe-Salpeter wave function, [MATH] is the unit 4-momentum.', 'hep-ph-0408037-2-4-6': '[MATH]-functions make the times to be equal covariantly in the c.m.s. Using a property of translational invariance, it is possible to separate motion of center-of-mass system and then the Fourier-representation of the wave function ([REF]) is given by [CITATION], notation [MATH] is introduced: [EQUATION] where [MATH] is a matrix of the Lorentz-boost representation [MATH], [MATH], [MATH].', 'hep-ph-0408037-2-4-7': 'In ([REF]) the wave function depends on two independent 3-momenta.', 'hep-ph-0408037-2-4-8': 'For the sake of symmetry we keep all the momenta assuming their relation [MATH].', 'hep-ph-0408037-2-4-9': 'The expression for [MATH] has the form: [EQUATION]', 'hep-ph-0408037-2-4-10': 'In the quantum-field interpretation the fixed number particles assumption is expressed in replacement of the Heizenberg field operators to the Dirac ones (zero-order interaction in field operators).', 'hep-ph-0408037-2-4-11': 'Bound effects are contained in the Heizenberg vector state of the bound system.', 'hep-ph-0408037-2-4-12': 'We have applied this assumption in ([REF]) and substituted it in ([REF]) and as a result the wave function ([REF]) is expressed through the decomposition component ([REF]) in the form: [EQUATION]', 'hep-ph-0408037-2-4-13': 'For obtaining ([REF]) we have used the relations: [EQUATION] where [MATH] is the Wigner spin rotation matrix, which is defined as a [MATH] representation of the 3-dimensional rotation [MATH] [CITATION].', 'hep-ph-0408037-2-4-14': 'It has an explicit form [CITATION]: [EQUATION]', 'hep-ph-0408037-2-4-15': 'We multiply ([REF]) by [MATH] from left-handed side and sum the result over the spinor index: [EQUATION]', 'hep-ph-0408037-2-4-16': 'The function ([REF]) is named a wave function projected onto positive frequency states [CITATION].', 'hep-ph-0408037-2-4-17': 'Taking into account ([REF]) and ([REF]) we obtain: [EQUATION] where [MATH] is related with [MATH], defined in ([REF])), via: [EQUATION]', 'hep-ph-0408037-2-4-18': 'Using the method of the two-time Green function [CITATION] it is possible to obtain quasipotential equation for function [MATH]: [EQUATION]', 'hep-ph-0408037-2-4-19': 'In the impulse approximation the quasipotential [MATH] may be independent of the energy, because of the nucleon mass in the equation iteration ([REF]) may be replaced in quasipotential by sum of the free quark energy in c.m.s. [MATH] and then the next order in the coupling constant may be neglected.', 'hep-ph-0408037-2-4-20': 'The wave function normalization condition obtained by means of the Green function [CITATION] for the quasipotential independent of energy has a symmetric form relatively the particle permutations: [EQUATION]', 'hep-ph-0408037-2-5-0': '2.', 'hep-ph-0408037-2-5-1': 'Nucleon spin wave function.', 'hep-ph-0408037-2-6-0': 'Now let us use the general method to construct the state with a definite momentum in the system of free particles to study spin dependence of QP WF by using connection of QP WF with a decomposition component of the nucleon vector state in the Fock-momentum space ([REF]).', 'hep-ph-0408037-2-6-1': 'In the center-of mass system formula ([REF]) has the form: [EQUATION]', 'hep-ph-0408037-2-6-2': 'Now we decompose [MATH] over free particle states with a definite momentum in the c.m.s. [MATH].', 'hep-ph-0408037-2-6-3': 'The dots mean other observables from the complete set which will be indicated below.', 'hep-ph-0408037-2-6-4': 'Its choise depends on the ways of summing spins and orbital angular momenta.', 'hep-ph-0408037-2-6-5': 'Further we use the method of work [CITATION] to decompose the direct product of irreducible representations of the inhomogeneous Lorentz group.', 'hep-ph-0408037-2-6-6': 'In case of the two free particle system in its c.m.s. with 3-momentum [MATH] and [MATH]) and with the invariant system [1+2] mass [MATH], this decomposition has the following form: [EQUATION]', 'hep-ph-0408037-2-6-7': 'Here [MATH] is angle variables of momentum [MATH], [MATH] is spherical harmonics, [MATH] is the relative orbital momentum of the system [1+2], [MATH] is spin, [MATH] is the total spin of system [1+2].', 'hep-ph-0408037-2-6-8': 'Going from the rest system [1+2] to c.m.s. of nucleon, we have from ([REF]): [EQUATION] where [MATH], [MATH].', 'hep-ph-0408037-2-6-9': 'If we introduce the notation [MATH], then applying ([REF]) we obtain: [EQUATION]', 'hep-ph-0408037-2-6-10': 'Multiplying ([REF]) on the vector state of the third quark, we have obtained: [EQUATION]', 'hep-ph-0408037-2-6-11': 'In analogy with ([REF]) we obtain (notation [MATH] is introduced): [EQUATION] where [MATH] is the spin of the system [[1+2]+3], obtained by summation of [MATH] and [MATH]; [MATH] is the orbital angular momentum of the third quark with respect to the [1+2] system, [MATH] is the total spin of the system [[1+2]+3], obtained by summmation of [MATH] and [MATH].', 'hep-ph-0408037-2-6-12': 'Thus, the desired decomposition has the form: [EQUATION]', 'hep-ph-0408037-2-6-13': 'We have separated motion of c.m.s. in the scalar product of the bound system vector state and the free particle vector state: [EQUATION]', 'hep-ph-0408037-2-6-14': 'Independence of [MATH] on [MATH] is the result the symmetry relatively the reflection.', 'hep-ph-0408037-2-6-15': 'The relation ([REF]) was obtained by using unit operator [MATH].', 'hep-ph-0408037-2-6-16': 'According to ([REF]), ([REF]) and ([REF]), for relative motion WF [MATH] in full record with isospin indices we have obtained: [EQUATION]', 'hep-ph-0408037-2-6-17': 'Since the contributions of higher orbital moments in nucleon are small [CITATION], we write only the S-wave part ([MATH]) omitting spin, isospin and symbols [MATH]: [EQUATION] where [MATH].', 'hep-ph-0408037-2-6-18': 'Let us decompose functions [MATH] over two isospin basic functions [MATH]; [MATH] is the [1+2] system isospin : [MATH] (sum over [MATH]) and substitute it in ([REF]): [EQUATION]', 'hep-ph-0408037-2-6-19': 'The function [MATH] ([REF]) is symmetrical relatively the particle permutations and with antisymmetric colour fuction realizes antisymmetrical representation of the permutations group.', 'hep-ph-0408037-2-6-20': 'S-wave function ([REF]) with zero-orbital momentum of the [1+2] system and zero angular orbital momentum of the third quark with respect to the [1+2] system, is antisymmetric relatively the quark permutations.', 'hep-ph-0408037-2-6-21': 'Let us suppose, that interquark interaction does not depend on the spin and isospin: [MATH] and we choose the normalization constants to agree with the nonrelativistic theory: [MATH], [MATH].', 'hep-ph-0408037-2-6-22': 'Introducing the notation for the symmetric spin-isospin wave function [MATH] , we obtain [EQUATION]', 'hep-ph-0408037-2-6-23': 'Two [MATH]-matrices in ([REF]) represent the minimal kinematic violation of the [MATH] symmetry.', 'hep-ph-0408037-2-6-24': 'The general expression for WF ([REF]) indicates three ways of [MATH] violations, related to relativization of the model: (1) including the admixture of the mixed [MATH] symmetry in WF, (2) taking into account the quark interaction dependence on spin and isospin, (3) and P- and D-waves.', 'hep-ph-0408037-2-7-0': '3.', 'hep-ph-0408037-2-7-1': 'Effective mass approximation and relativistic oscillator.', 'hep-ph-0408037-2-8-0': 'According to ([REF]), ([REF]) and ([REF]), function [MATH] satisfies the equation and normalization condition: [EQUATION] where [MATH] is a quasipotential determined by relation: [MATH], where [MATH] is a matrix record of factors in square brackets of expression ([REF]).', 'hep-ph-0408037-2-8-1': 'Variables [MATH], [MATH] and [MATH] are equivalent and the choise of variables [MATH], [MATH] as independent does not change the three quark equivalent to calculate the average values.', 'hep-ph-0408037-2-8-2': 'For example, the three effective quark masses, obtained by formula [MATH] are equal ([MATH]).', 'hep-ph-0408037-2-8-3': 'The approximate relativistic models, using the idea of the effective quark mass, were considered in work [CITATION].', 'hep-ph-0408037-2-8-4': 'In these works the effective quark mass is introduced as a parameter instead of the quark mass.', 'hep-ph-0408037-2-8-5': 'In our work the effective quark mass has been introduced as a suitable approximation.', 'hep-ph-0408037-2-8-6': 'Let us pass to the semi-momenta [CITATION] in ([REF]): [MATH].', 'hep-ph-0408037-2-8-7': 'Equation ([REF]) takes a nonrelativistic form in terms of [MATH].', 'hep-ph-0408037-2-8-8': 'In equation ([REF]) we rewrite the quasipotential as an analog of nonrelativistic oscillator: [EQUATION]', 'hep-ph-0408037-2-8-9': 'Using new variables [MATH] and parameters in ([REF]) [MATH], we have obtained: [EQUATION]', 'hep-ph-0408037-2-8-10': 'Decomposing the quark energies in semi-momenta in the [MATH] neighborhood and taking into account the condition [MATH], it is possible to show that term [MATH] has a small contribution into the energy, if value [MATH] is small (the effective mass aproximation).', 'hep-ph-0408037-2-8-11': 'In the zero-order approximation relatively the [MATH], we obtain a solution of equation ([REF]) for the ground state ([MATH]): [MATH].', 'hep-ph-0408037-2-8-12': 'Taking into account the effective mass approximation, this solution is reduced to the form: [EQUATION]', 'hep-ph-0408037-2-8-13': 'Numerical calculations with the relativistic oscillator WF ([REF]) gives [MATH] for [MATH].', 'hep-ph-0408037-2-8-14': 'In this case [MATH] is positive and is in limits [MATH].', 'hep-ph-0408037-2-8-15': 'Smallness of [MATH] means the smallness of the momentum module dispersion.', 'hep-ph-0408037-2-8-16': 'Indeed, in the ground state [MATH] and therefore [MATH], where [MATH] We note that, WF in form [MATH], applied in many works, is different from ([REF]) by the order of [MATH] and has different asymptotic behaviour for high momenta.', 'hep-ph-0408037-2-9-0': 'Conclusion.', 'hep-ph-0408037-2-10-0': 'In the impulse approximation, represented by the zero-order field operators in the Bethe-Salpeter WF, the quasipotential wave function reduces to the three-quark component of the vector state decomposition of the bound system in the Fock-momentum space.', 'hep-ph-0408037-2-10-1': 'This allows us to apply the standart method for the decomposition of the irreducible representations product of the inhomogeneous Lorentz-group over states with the definite momentum to analyse of the QP WF structure.', 'hep-ph-0408037-2-10-2': 'The physical preference of the [MATH]-symmetric solution is determined by its uniqueness in the nonreletivistic limit.', 'hep-ph-0408037-2-10-3': 'The model of the relativistic three-particle oscillator being the direct generalization of the nonrelativistic oscillator is proposed.', 'hep-ph-0408037-2-10-4': 'Numerically it was shown that the effective mass approximation may be applyed in the wide region of the oscillator parameters, in which the ratio of the momentum dispersion to the average value of the quark momentum module is small.', 'hep-ph-0408037-2-10-5': 'Authors express gratitude to N.V.Maksimenko for support and E.A.Dey for usefull remarks.'}","[['hep-ph-0408037-1-0-0', 'hep-ph-0408037-2-0-0'], ['hep-ph-0408037-1-0-1', 'hep-ph-0408037-2-0-1'], 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['hep-ph-0408037-1-2-13', 'hep-ph-0408037-2-2-13']]",[],[],[],[],"['hep-ph-0408037-1-1-0', 'hep-ph-0408037-1-3-0', 'hep-ph-0408037-1-3-1', 'hep-ph-0408037-1-5-0', 'hep-ph-0408037-1-5-1', 'hep-ph-0408037-1-7-0', 'hep-ph-0408037-1-7-1', 'hep-ph-0408037-1-9-0', 'hep-ph-0408037-2-1-0', 'hep-ph-0408037-2-3-0', 'hep-ph-0408037-2-3-1', 'hep-ph-0408037-2-5-0', 'hep-ph-0408037-2-5-1', 'hep-ph-0408037-2-7-0', 'hep-ph-0408037-2-7-1', 'hep-ph-0408037-2-9-0']","{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/'}",https://arxiv.org/abs/hep-ph/0408037,,,
hep-th-0308084,"{'hep-th-0308084-1-0-0': 'We extend a deformation prescription recently introduced and present some new soluble nonlinear problems for kinks and lumps.', 'hep-th-0308084-1-0-1': 'In particular, we show how to generate models which present the basic ingredients needed to give rise to ""dimension bubbles,"" having different macroscopic space dimensions on the interior and the exterior of the bubble surface.', 'hep-th-0308084-1-0-2': 'Also, we show how to deform a model possessing lumplike solutions, relevant to the discussion of tachyonic excitations, to get a new one having topological solutions.', 'hep-th-0308084-1-1-0': '# Introduction', 'hep-th-0308084-1-2-0': 'Nonlinear models in (1+1)-dimensional space-time play important hole both in field theory and in quantum mechanics.', 'hep-th-0308084-1-2-1': 'Some of such models possess defect solutions representing domain walls that appear, for example, in high energy physics [CITATION] and condensed matter [CITATION].', 'hep-th-0308084-1-3-0': 'In this work we extend the method introduced in Ref. [CITATION] to present new potentials bearing topological (kinklike) or non topological (lumplike) solutions.', 'hep-th-0308084-1-3-1': 'For example, we use the extended deformation procedure to build a semi-vacuumless model, and the corresponding domain wall which serves as seed for generation of ""dimension bubbles,"" as proposed in Refs. [CITATION].', 'hep-th-0308084-1-3-2': 'We also show how to deform models having lumplike solutions, relevant to the discussion of tachyonic excitations, to generate new ones presenting topological solutions.', 'hep-th-0308084-1-4-0': 'To begin, consider a theory of a single scalar real field in a (1+1)-dimensional space-time, described by the Lagrangian density [EQUATION]', 'hep-th-0308084-1-4-1': 'We use the metric [MATH], and we work with dimensionless fields and coordinates, fields being defined in the whole space.', 'hep-th-0308084-1-4-2': 'The equation of motion for static fields is [EQUATION] where the prime stands for the derivative with respect to the argument.', 'hep-th-0308084-1-4-3': 'Mathematically, this corresponds to consider two-point boundary value problems (with conditions imposed at [MATH] and [MATH]) for second-order ordinary differential equations [CITATION].', 'hep-th-0308084-1-5-0': 'Consider the broad class of potentials having at least one critical point [MATH] (that is, [MATH]), for which [MATH].', 'hep-th-0308084-1-5-1': 'In this case, solutions satisfying the conditions [EQUATION] obey the first order equation (a first integral of ([REF])) [EQUATION] thus, such solutions equally share their energy densities between gradient and potential parts.', 'hep-th-0308084-1-6-0': 'Many important examples can be presented: the [MATH]-model, with [MATH], is the prototype of theories having topological solitons (kinklike solutions) connecting two minima, [MATH] in this case; a situation where non topological (lumplike) solutions exist is the ""inverted [MATH]-model"", with potential given by [MATH], lumplike defects being [MATH].', 'hep-th-0308084-1-6-1': 'One notice that the potential need not be nonnegative for all values of [MATH] but the solution must be such that [MATH] for the whole range [MATH].', 'hep-th-0308084-1-7-0': '# The deformation procedure', 'hep-th-0308084-1-8-0': 'Both topological and non topological solutions can be deformed, according to the prescription introduced in [CITATION], to generate infinitely many new soluble problems.', 'hep-th-0308084-1-8-1': 'This method can be stated in general form as the following theorem:', 'hep-th-0308084-1-9-0': 'Let [MATH] be a bijective function having continuous non-vanishing derivative.', 'hep-th-0308084-1-9-1': 'For each potential [MATH], bearing solutions satisfying conditions ([REF]) (or equivalently Eq. ([REF])), the [MATH]-deformed model, defined by [EQUATION] possesses solution given by [EQUATION] where [MATH] is a solution of the static equation of motion for the original potential [MATH].', 'hep-th-0308084-1-10-0': 'To prove this assertion, notice that the static equation of motion of the new theory is written in terms of the old potential as [EQUATION]', 'hep-th-0308084-1-10-1': 'On the other hand, taking the second derivative with respect to [MATH] of Eq. ([REF]), one finds [EQUATION]', 'hep-th-0308084-1-10-2': 'It follows from ([REF]), ([REF]) and ([REF]) that [MATH] and [MATH] so that [MATH] satisfies ([REF]), as stated.', 'hep-th-0308084-1-10-3': 'The energy density of the solution ([REF]) of the [MATH]-deformed potential ([REF]) is given by [EQUATION]', 'hep-th-0308084-1-10-4': 'Naturally, the deformation procedure heavily depends on the deformation function [MATH].', 'hep-th-0308084-1-10-5': 'Assume that [MATH] is bijective and has no critical points.', 'hep-th-0308084-1-10-6': 'In this case, the [MATH]-deformation (and the deformation implemented by its inverse [MATH]) can be applied successively and one can define equivalence classes of potentials related to each other by repeated applications of the [MATH]- (or the [MATH]-) deformation.', 'hep-th-0308084-1-10-7': 'Each of such classes possesses an enumerable number of elements which correspond to smooth deformations of a representative one, all having the same topological characteristics.', 'hep-th-0308084-1-10-8': 'The generation sequence of new theories is depicted in the diagram below.', 'hep-th-0308084-1-10-9': '[EQUATION]', 'hep-th-0308084-1-10-10': 'As an example not considered before, take the [MATH]-model.', 'hep-th-0308084-1-10-11': 'This model, for which the potential [MATH] has three degenerated minima at [MATH] and [MATH], is important since it allows the discussion of first-order transitions.', 'hep-th-0308084-1-10-12': 'It possesses kinklike solutions, [MATH], [MATH], connecting the central vacuum with the lateral ones.', 'hep-th-0308084-1-10-13': 'Take [MATH] as the deforming function.', 'hep-th-0308084-1-10-14': 'The sinh-deformed [MATH]-potential is [EQUATION] and the sinh-deformed defects are [EQUATION]', 'hep-th-0308084-1-10-15': 'Notice that, since [MATH] for the sinh-deformation, the energy of the deformed solutions is diminished with respect to the undeformed kinks.', 'hep-th-0308084-1-10-16': 'The reverse situation emerges if one takes the inverse deformation implemented with [MATH].', 'hep-th-0308084-1-11-0': 'Interesting situations arise if one takes polynomial functions implementing the deformations.', 'hep-th-0308084-1-11-1': 'Consider [EQUATION] with [MATH] for all [MATH].', 'hep-th-0308084-1-11-2': 'These are bijective functions from [MATH] into [MATH] possessing positive derivatives.', 'hep-th-0308084-1-11-3': 'Fixing [MATH] corresponds to a trivial rescaling of the field.', 'hep-th-0308084-1-11-4': 'For [MATH], taking [MATH], one has [MATH] with inverse given by [MATH].', 'hep-th-0308084-1-11-5': 'Thus, the [MATH]-deformed [MATH] model, for which the potential has the form [EQUATION] supports topological solitons given by [EQUATION]', 'hep-th-0308084-1-11-6': 'Naturally, the inverse deformation can be implemented leading to another new soluble problem.', 'hep-th-0308084-1-11-7': 'But if one takes [MATH], the inverse of [MATH] can not be in general expressed analytically in terms of known functions.', 'hep-th-0308084-1-11-8': 'This leads to circumstances where one knows analytically solutions of potentials which can not be expressed in term of known functions and, conversely, one has well-established potentials for which solitonic solutions exist but are not expressible in terms of known functions.', 'hep-th-0308084-1-11-9': 'For example, take [MATH].', 'hep-th-0308084-1-11-10': 'The [MATH]-deformed [MATH] model has potential given by [EQUATION] but its solutions [MATH] are not known analytically.', 'hep-th-0308084-1-11-11': 'On the other hand, [EQUATION] are topological solutions of the potential [MATH] which can not be written in terms of known functions.', 'hep-th-0308084-1-12-0': 'The procedure can also be applied to potentials presenting non topological, lumplike, solutions which are of direct interest to tachyions in String Theory [CITATION].', 'hep-th-0308084-1-12-1': 'Take, for example, the Lorentzian lump [EQUATION] which solves Eq. ([REF]) for the potential [EQUATION] and satisfies conditions ([REF]).', 'hep-th-0308084-1-12-2': 'Distinctly of the topological solitons, this kind of solution is not stable.', 'hep-th-0308084-1-12-3': ""In fact, the 'secondary potential', that appears in the linearized Schrodiger-like equation satisfied by the small perturbations around [MATH] [CITATION] is given by [EQUATION]"", 'hep-th-0308084-1-12-4': 'This potential, a symmetric volcano-shaped potential, has zero mode given by [MATH], which does not correspond to the lowest energy state since it has a node.', 'hep-th-0308084-1-12-5': 'Deforming the potential ([REF]) with [MATH] leads to the potential [MATH] which possesses the lumplike solution [MATH].', 'hep-th-0308084-1-13-0': '# The extended deformation prescription', 'hep-th-0308084-1-14-0': 'The deformation prescription is powerful.', 'hep-th-0308084-1-14-1': 'The conditions under which our theorem holds are maintained if we consider a function for which the contra-domain is an interval of [MATH], that is, if we take [MATH].', 'hep-th-0308084-1-14-2': 'In this case, however, the inverse transformation (engendered by [MATH]) can only be applied for models where the values of [MATH] are restricted to [MATH].', 'hep-th-0308084-1-15-0': 'We illustrate this possibility by asking for a deformation that leads to a model of the form needed in Ref. [CITATION], described by a ""semi-vacuumless"" potential, in contrast with the vacuumless potential studied in Ref. [CITATION].', 'hep-th-0308084-1-15-1': 'Consider the new deformation function [MATH], acting on the potential [MATH].', 'hep-th-0308084-1-15-2': 'The deformed potential is [EQUATION] which is depicted in Fig. 1.', 'hep-th-0308084-1-15-3': 'The kinklike solution is [EQUATION]', 'hep-th-0308084-1-15-4': 'The deformed potential ([REF]) engenders the required profile: it has a minimum at [MATH] and another one at [MATH].', 'hep-th-0308084-1-15-5': 'It is similar to the potential required in Ref. [CITATION] for the existence of dimension bubbles.', 'hep-th-0308084-1-15-6': 'The bubble can be generated from the above (deformed) model, after removing the degeneracy between [MATH] and [MATH], in a way similar to the standard situation, which is usually implemented with the [MATH] potential, the undeformed potential that we have used to generate ([REF]).', 'hep-th-0308084-1-15-7': 'A key issue is that such bubble is unstable against collapse, unless a mechanism to balance the inward pressure due to the surface tension in the bubble is found.', 'hep-th-0308084-1-15-8': 'In Ref. [CITATION], the mechanism used to stabilize the bubble requires another scalar field, in a way similar to the case of non topological solitons previously proposed in Ref. [CITATION].', 'hep-th-0308084-1-15-9': 'This naturally leads to another scenario, which involves at least two real scalar fields.', 'hep-th-0308084-1-16-0': 'The deformation procedure can be extended even further, by relaxing the requirement of [MATH] being a bijective function, under certain conditions.', 'hep-th-0308084-1-16-1': 'Suppose that [MATH] is not bijective but it is such that its inverse [MATH] (which exists in the context of binary relations) is a multi-valued function with all branches defined in the same interval [MATH].', 'hep-th-0308084-1-16-2': 'If the domain of definition of [MATH] contains the interval where the values of the solutions [MATH] of the original potential vary, then [MATH] is a solution of the new model obtained by implementing the deformation with [MATH].', 'hep-th-0308084-1-16-3': 'However, one has to check out whether the deformed potential [MATH] is well defined on the critical points of [MATH].', 'hep-th-0308084-1-16-4': 'In fact, this does not happen in general but occurs for some interesting cases.', 'hep-th-0308084-1-17-0': 'Consider, for example, the function [MATH]; it is defined for all values of [MATH] and its inverse is the double valued real function [MATH], defined in the interval [MATH].', 'hep-th-0308084-1-17-1': 'If we deform the [MATH] model with this function we end up with the potential [MATH].', 'hep-th-0308084-1-17-2': 'The deformed kink solutions are given by [MATH] with [MATH] replaced by the solutions ([MATH]) of the [MATH] model, which reproduce the known solutions of the [MATH] theory.', 'hep-th-0308084-1-17-3': 'The important aspect, in the present case, is that the [MATH]-kink corresponds to field values restricted to the interval [MATH] which is contained within the domain of definition of the two branches of [MATH].', 'hep-th-0308084-1-17-4': 'The fact that the [MATH] model can be obtained from the [MATH] potential in this way is interesting, since these models have distinct characteristics.', 'hep-th-0308084-1-17-5': 'Notice that the critical point of [MATH] at [MATH] does not disturb the deformation in this case; this always occur for potentials having a factor [MATH], since the denominator of [MATH] (Eq. [REF]) is cancelled out.', 'hep-th-0308084-1-17-6': 'One can go on and apply this deformation to the [MATH] model; now, one finds the deformed potential [EQUATION] with solutions given by [EQUATION] corresponding to kinks connecting neighbouring minima (located at [MATH], [MATH], [MATH], [MATH] and [MATH]) of the potential ([REF]).', 'hep-th-0308084-1-17-7': 'This potential is illustrated in Fig. 1.', 'hep-th-0308084-1-17-8': 'Repeating the procedure for the potential ([REF]), the deformed potential is a polynomial function of degree [MATH], having sixteen kink solutions connecting adjacent minima of the set [MATH], and so on and so forth.', 'hep-th-0308084-1-18-0': 'The deformation implemented by the function [MATH] can also be applied to a potential possessing lumplike solutions.', 'hep-th-0308084-1-18-1': 'Consider the inverted [MATH] potential [MATH], which has the lump solutions [MATH].', 'hep-th-0308084-1-18-2': 'The deformed potential, in this case, is given by [EQUATION] which vanishes for [MATH], has a minimum at [MATH] and maxima for [MATH].', 'hep-th-0308084-1-18-3': 'Figure 3 shows a plot of this potential.', 'hep-th-0308084-1-18-4': 'The deformed solutions are given by [EQUATION] corresponding to lump solutions running between the zeros of the potential ([REF]).', 'hep-th-0308084-1-18-5': 'One notice that, again, the number of solutions duplicates using such a deformation, whose inverse is a double-valued function.', 'hep-th-0308084-1-19-0': 'Potentials which have a factor [MATH] can also be deformed using the function [MATH], producing many interesting situations.', 'hep-th-0308084-1-19-1': 'In fact, suppose the potential can be written in the form [EQUATION] this is always possible for all well-behaved potentials that vanish at both values [MATH], as shown by Taylor expansion.', 'hep-th-0308084-1-19-2': 'Then, the sin-deformation leads to the potential [EQUATION] which is a periodic potential, the critical points of [MATH] not causing any problem to the deformation process.', 'hep-th-0308084-1-19-3': 'The inverse of the sine function is the infinitely valued function [MATH], with [MATH] and [MATH] being the first determination of [MATH] (which varies from [MATH], for [MATH], to [MATH], when [MATH]), defined in the interval [MATH].', 'hep-th-0308084-1-19-4': 'So to each solution of the original potential, whose field values range in the interval [MATH], one finds infinitely many solutions of the deformed, periodic, potential.', 'hep-th-0308084-1-20-0': 'Consider firstly the [MATH] model.', 'hep-th-0308084-1-20-1': 'Applying the sin-deformation to it, one gets [MATH] which is one of the forms of the sine-Gordon potential.', 'hep-th-0308084-1-20-2': 'The deformed solutions thus obtained is given by [MATH], which correspond to all the kink solutions (connecting neighbouring minima) of this sine-Gordon model.', 'hep-th-0308084-1-20-3': 'For example, the kink solutions [MATH], which connect the minima [MATH] of the [MATH] model in both directions, are deformed into the kinks [MATH] (which runs between [MATH] and [MATH]) if one takes [MATH] while, for [MATH], the resulting solutions connect the minima [MATH] and [MATH] of the deformed potential.', 'hep-th-0308084-1-21-0': 'This example can be readily extended to other polynomial potentials, leading to a large class of sine-Gordon type of potentials.', 'hep-th-0308084-1-21-1': 'For instance, the [MATH] model, [MATH], deformed by the sine function, becomes the potential [EQUATION] having kink solutions given by [EQUATION]', 'hep-th-0308084-1-21-2': 'If, on the other hand, one considers the potential [MATH], which is unbounded below and supports kinklike solutions connecting the two inflection points at [MATH], one gets the potential [EQUATION] with solutions given by [EQUATION]', 'hep-th-0308084-1-21-3': 'A particularly interesting situation appears if one consider the inverted [MATH] model, which presents lumplike solutions.', 'hep-th-0308084-1-21-4': 'The sin-deformation of the potential [MATH] leads to the potential [MATH].', 'hep-th-0308084-1-21-5': 'In this case, the lump solutions of [MATH], namely [MATH], are deformed into [MATH].', 'hep-th-0308084-1-21-6': 'Consider the [MATH]-solution and take initially [MATH].', 'hep-th-0308084-1-21-7': 'As [MATH] varies from [MATH] to [MATH], [MATH] goes from [MATH] to [MATH], and [MATH] changes from [MATH] to [MATH].', 'hep-th-0308084-1-21-8': 'If one continuously makes [MATH] goes from [MATH] to [MATH], then the deformed solution passes to the [MATH] branch of [MATH], [MATH] for [MATH]), which varies from [MATH] to [MATH] as [MATH] goes from [MATH] to [MATH].', 'hep-th-0308084-1-21-9': 'Thus, in this case, the lump solution [MATH] of the inverted [MATH] model is deformed in the kink of the sine-Gordon model connecting the minima [MATH] and [MATH].', 'hep-th-0308084-1-21-10': 'Under reversed conditions (taking the [MATH] branch before the [MATH] one), the lump solution [MATH] leads to the anti-kink solution of the sine-Gordon model running from the minimum [MATH] to [MATH].', 'hep-th-0308084-1-21-11': 'The other topological solutions of the sine-Gordon model are obtained considering the other adjacent branches of [MATH].', 'hep-th-0308084-1-21-12': 'This example is remarkable since one has non topological solutions being deformed in topological ones.', 'hep-th-0308084-1-22-0': 'Finally, we mention that many other soluble models can be construct following the procedure presented in this paper.', 'hep-th-0308084-1-22-1': 'Also, the discussion raised here might be extended to consider situations where the fields are constrained to intervals of [MATH], thus representing two-point boundary value problems defined in finite or semi-infinite intervals of the [MATH]-axis.', 'hep-th-0308084-1-22-2': 'Such study is left for future work.'}","{'hep-th-0308084-2-0-0': 'We extend a deformation prescription recently introduced and present some new soluble nonlinear problems for kinks and lumps.', 'hep-th-0308084-2-0-1': 'In particular, we show how to generate models which present the basic ingredients needed to give rise to ""dimension bubbles,"" having different macroscopic space dimensions on the interior and the exterior of the bubble surface.', 'hep-th-0308084-2-0-2': 'Also, we show how to deform a model possessing lumplike solutions, relevant to the discussion of tachyonic excitations, to get a new one having topological solutions.', 'hep-th-0308084-2-1-0': '# Introduction', 'hep-th-0308084-2-2-0': 'Nonlinear models in (1+1)-dimensional space-time play important hole both in field theory and in quantum mechanics.', 'hep-th-0308084-2-2-1': 'Some of such models possess defect solutions representing domain walls that appear, for example, in high energy physics [CITATION] and condensed matter [CITATION].', 'hep-th-0308084-2-3-0': 'In this work we extend the method introduced in Ref. [CITATION] to present new potentials bearing topological (kinklike) or non topological (lumplike) solutions.', 'hep-th-0308084-2-3-1': 'For example, we use the extended deformation procedure to build a semi-vacuumless model, and the corresponding domain wall which serves as seed for generation of ""dimension bubbles,"" as proposed in Refs. [CITATION].', 'hep-th-0308084-2-3-2': 'We also show how to deform models having lumplike solutions, relevant to the discussion of tachyonic excitations, to generate new ones presenting topological solutions.', 'hep-th-0308084-2-4-0': 'To begin, consider a theory of a single scalar real field in a (1+1)-dimensional space-time, described by the Lagrangian density [EQUATION]', 'hep-th-0308084-2-4-1': 'We use the metric [MATH], and we work with dimensionless fields and coordinates, fields being defined in the whole space.', 'hep-th-0308084-2-4-2': 'The equation of motion for static fields is [EQUATION] where the prime stands for the derivative with respect to the argument.', 'hep-th-0308084-2-4-3': 'Mathematically, this corresponds to consider two-point boundary value problems (with conditions imposed at [MATH] and [MATH]) for second-order ordinary differential equations [CITATION].', 'hep-th-0308084-2-5-0': 'Consider the broad class of potentials having at least one critical point [MATH] (that is, [MATH]), for which [MATH].', 'hep-th-0308084-2-5-1': 'In this case, solutions satisfying the conditions [EQUATION] obey the first order equation (a first integral of ([REF])) [EQUATION] thus, such solutions equally share their energy densities between gradient and potential parts.', 'hep-th-0308084-2-6-0': 'Many important examples can be presented: the [MATH]-model, with [MATH], is the prototype of theories having topological solitons (kinklike solutions) connecting two minima, [MATH] in this case; a situation where non topological (lumplike) solutions exist is the ""inverted [MATH]-model"", with potential given by [MATH], lumplike defects being [MATH].', 'hep-th-0308084-2-6-1': 'One notice that the potential need not be nonnegative for all values of [MATH] but the solution must be such that [MATH] for the whole range [MATH].', 'hep-th-0308084-2-7-0': '# The deformation procedure', 'hep-th-0308084-2-8-0': 'Both topological and non topological solutions can be deformed, according to the prescription introduced in [CITATION], to generate infinitely many new soluble problems.', 'hep-th-0308084-2-8-1': 'This method can be stated in general form as the following theorem:', 'hep-th-0308084-2-9-0': 'Let [MATH] be a bijective function having continuous non-vanishing derivative.', 'hep-th-0308084-2-9-1': 'For each potential [MATH], bearing solutions satisfying conditions ([REF]) (or equivalently Eq. ([REF])), the [MATH]-deformed model, defined by [EQUATION] possesses solution given by [EQUATION] where [MATH] is a solution of the static equation of motion for the original potential [MATH].', 'hep-th-0308084-2-10-0': 'To prove this assertion, notice that the static equation of motion of the new theory is written in terms of the old potential as [EQUATION]', 'hep-th-0308084-2-10-1': 'On the other hand, taking the second derivative with respect to [MATH] of Eq. ([REF]), one finds [EQUATION]', 'hep-th-0308084-2-10-2': 'It follows from ([REF]), ([REF]) and ([REF]) that [MATH] and [MATH] so that [MATH] satisfies ([REF]), as stated.', 'hep-th-0308084-2-10-3': 'The energy density of the solution ([REF]) of the [MATH]-deformed potential ([REF]) is given by [EQUATION]', 'hep-th-0308084-2-10-4': 'Naturally, the deformation procedure heavily depends on the deformation function [MATH].', 'hep-th-0308084-2-10-5': 'Assume that [MATH] is bijective and has no critical points.', 'hep-th-0308084-2-10-6': 'In this case, the [MATH]-deformation (and the deformation implemented by its inverse [MATH]) can be applied successively and one can define equivalence classes of potentials related to each other by repeated applications of the [MATH]- (or the [MATH]-) deformation.', 'hep-th-0308084-2-10-7': 'Each of such classes possesses an enumerable number of elements which correspond to smooth deformations of a representative one, all having the same topological characteristics.', 'hep-th-0308084-2-10-8': 'The generation sequence of new theories is depicted in the diagram below.', 'hep-th-0308084-2-10-9': '[EQUATION]', 'hep-th-0308084-2-10-10': 'As an example not considered before, take the [MATH]-model.', 'hep-th-0308084-2-10-11': 'This model, for which the potential [MATH] has three degenerated minima at [MATH] and [MATH], is important since it allows the discussion of first-order transitions.', 'hep-th-0308084-2-10-12': 'It possesses kinklike solutions, [MATH], [MATH], connecting the central vacuum with the lateral ones.', 'hep-th-0308084-2-10-13': 'Take [MATH] as the deforming function.', 'hep-th-0308084-2-10-14': 'The sinh-deformed [MATH]-potential is [EQUATION] and the sinh-deformed defects are [EQUATION]', 'hep-th-0308084-2-10-15': 'Notice that, since [MATH] for the sinh-deformation, the energy of the deformed solutions is diminished with respect to the undeformed kinks.', 'hep-th-0308084-2-10-16': 'The reverse situation emerges if one takes the inverse deformation implemented with [MATH].', 'hep-th-0308084-2-11-0': 'Interesting situations arise if one takes polynomial functions implementing the deformations.', 'hep-th-0308084-2-11-1': 'Consider [EQUATION] with [MATH] for all [MATH].', 'hep-th-0308084-2-11-2': 'These are bijective functions from [MATH] into [MATH] possessing positive derivatives.', 'hep-th-0308084-2-11-3': 'Fixing [MATH] corresponds to a trivial rescaling of the field.', 'hep-th-0308084-2-11-4': 'For [MATH], taking [MATH], one has [MATH] with inverse given by [MATH].', 'hep-th-0308084-2-11-5': 'Thus, the [MATH]-deformed [MATH] model, for which the potential has the form [EQUATION] supports topological solitons given by [EQUATION]', 'hep-th-0308084-2-11-6': 'Naturally, the inverse deformation can be implemented leading to another new soluble problem.', 'hep-th-0308084-2-11-7': 'But if one takes [MATH], the inverse of [MATH] can not be in general expressed analytically in terms of known functions.', 'hep-th-0308084-2-11-8': 'This leads to circumstances where one knows analytically solutions of potentials which can not be expressed in term of known functions and, conversely, one has well-established potentials for which solitonic solutions exist but are not expressible in terms of known functions.', 'hep-th-0308084-2-11-9': 'For example, take [MATH].', 'hep-th-0308084-2-11-10': 'The [MATH]-deformed [MATH] model has potential given by [EQUATION] but its solutions [MATH] are not known analytically.', 'hep-th-0308084-2-11-11': 'On the other hand, [EQUATION] are topological solutions of the potential [MATH] which can not be written in terms of known functions.', 'hep-th-0308084-2-12-0': 'The procedure can also be applied to potentials presenting non topological, lumplike, solutions which are of direct interest to tachyions in String Theory [CITATION].', 'hep-th-0308084-2-12-1': 'Take, for example, the Lorentzian lump [EQUATION] which solves Eq. ([REF]) for the potential [EQUATION] and satisfies conditions ([REF]).', 'hep-th-0308084-2-12-2': 'Distinctly of the topological solitons, this kind of solution is not stable.', 'hep-th-0308084-2-12-3': ""In fact, the 'secondary potential', that appears in the linearized Schrodiger-like equation satisfied by the small perturbations around [MATH] [CITATION] is given by [EQUATION]"", 'hep-th-0308084-2-12-4': 'This potential, a symmetric volcano-shaped potential, has zero mode given by [MATH], which does not correspond to the lowest energy state since it has a node.', 'hep-th-0308084-2-12-5': 'Deforming the potential ([REF]) with [MATH] leads to the potential [MATH] which possesses the lumplike solution [MATH].', 'hep-th-0308084-2-13-0': '# The extended deformation prescription', 'hep-th-0308084-2-14-0': 'The deformation prescription is powerful.', 'hep-th-0308084-2-14-1': 'The conditions under which our theorem holds are maintained if we consider a function for which the contra-domain is an interval of [MATH], that is, if we take [MATH].', 'hep-th-0308084-2-14-2': 'In this case, however, the inverse transformation (engendered by [MATH]) can only be applied for models where the values of [MATH] are restricted to [MATH].', 'hep-th-0308084-2-15-0': 'We illustrate this possibility by asking for a deformation that leads to a model of the form needed in Ref. [CITATION], described by a ""semi-vacuumless"" potential, in contrast with the vacuumless potential studied in Ref. [CITATION].', 'hep-th-0308084-2-15-1': 'Consider the new deformation function [MATH], acting on the potential [MATH].', 'hep-th-0308084-2-15-2': 'The deformed potential is [EQUATION] which is depicted in Fig. 1.', 'hep-th-0308084-2-15-3': 'The kinklike solution is [EQUATION]', 'hep-th-0308084-2-15-4': 'The deformed potential ([REF]) engenders the required profile: it has a minimum at [MATH] and another one at [MATH].', 'hep-th-0308084-2-15-5': 'It is similar to the potential required in Ref. [CITATION] for the existence of dimension bubbles.', 'hep-th-0308084-2-15-6': 'The bubble can be generated from the above (deformed) model, after removing the degeneracy between [MATH] and [MATH], in a way similar to the standard situation, which is usually implemented with the [MATH] potential, the undeformed potential that we have used to generate ([REF]).', 'hep-th-0308084-2-15-7': 'A key issue is that such bubble is unstable against collapse, unless a mechanism to balance the inward pressure due to the surface tension in the bubble is found.', 'hep-th-0308084-2-15-8': 'In Ref. [CITATION], the mechanism used to stabilize the bubble requires another scalar field, in a way similar to the case of non topological solitons previously proposed in Ref. [CITATION].', 'hep-th-0308084-2-15-9': 'This naturally leads to another scenario, which involves at least two real scalar fields.', 'hep-th-0308084-2-16-0': 'The deformation procedure can be extended even further, by relaxing the requirement of [MATH] being a bijective function, under certain conditions.', 'hep-th-0308084-2-16-1': 'Suppose that [MATH] is not bijective but it is such that its inverse [MATH] (which exists in the context of binary relations) is a multi-valued function with all branches defined in the same interval [MATH].', 'hep-th-0308084-2-16-2': 'If the domain of definition of [MATH] contains the interval where the values of the solutions [MATH] of the original potential vary, then [MATH] are solutions of the new model obtained by implementing the deformation with [MATH].', 'hep-th-0308084-2-16-3': 'However, one has to check out whether the deformed potential [MATH] is well defined on the critical points of [MATH].', 'hep-th-0308084-2-16-4': 'In fact, this does not happen in general but occurs for some interesting cases.', 'hep-th-0308084-2-17-0': 'Consider, for example, the function [MATH]; it is defined for all values of [MATH] and its inverse is the double valued real function [MATH], defined in the interval [MATH].', 'hep-th-0308084-2-17-1': 'If we deform the [MATH] model with this function we end up with the potential [MATH].', 'hep-th-0308084-2-17-2': 'The deformed kink solutions are given by [MATH] with [MATH] replaced by the solutions ([MATH]) of the [MATH] model, which reproduce the known solutions of the [MATH] theory.', 'hep-th-0308084-2-17-3': 'The important aspect, in the present case, is that the [MATH]-kink corresponds to field values restricted to the interval [MATH] which is contained within the domain of definition of the two branches of [MATH].', 'hep-th-0308084-2-17-4': 'The fact that the [MATH] model can be obtained from the [MATH] potential in this way is interesting, since these models have distinct characteristics.', 'hep-th-0308084-2-17-5': 'Notice that the critical point of [MATH] at [MATH] does not disturb the deformation in this case; this always occur for potentials having a factor [MATH], since the denominator of [MATH] (Eq. [REF]) is cancelled out.', 'hep-th-0308084-2-17-6': 'One can go on and apply this deformation to the [MATH] model; now, one finds the deformed potential [EQUATION] with solutions given by [EQUATION] corresponding to kinks connecting neighbouring minima (located at [MATH], [MATH], [MATH], [MATH] and [MATH]) of the potential ([REF]).', 'hep-th-0308084-2-17-7': 'This potential is illustrated in Fig. 2.', 'hep-th-0308084-2-17-8': 'Repeating the procedure for the potential ([REF]), the deformed potential is a polynomial function of degree [MATH], having sixteen kink solutions connecting adjacent minima of the set [MATH], and so on and so forth.', 'hep-th-0308084-2-18-0': 'The deformation implemented by the function [MATH] can also be applied to a potential possessing lumplike solutions.', 'hep-th-0308084-2-18-1': 'Consider the inverted [MATH] potential [MATH], which has the lump solutions [MATH].', 'hep-th-0308084-2-18-2': 'The deformed potential, in this case, is given by [EQUATION]', 'hep-th-0308084-2-18-3': 'This potential, which is also unbounded from below, vanishes for [MATH], has an absolute maximum at [MATH] and local minima and maxima for [MATH] and [MATH], respectively.', 'hep-th-0308084-2-18-4': 'Figure 3 shows a plot of this potential.', 'hep-th-0308084-2-18-5': 'If we take [MATH] as the original lump, then the deformed solutions obtained are given by [EQUATION] corresponding to lumps which start in the local minima (for [MATH]), go to the lateral zeros (when [MATH]) and come back to the same minima (for [MATH]) of the potential ([REF]).', 'hep-th-0308084-2-18-6': 'On the other hand, if the undeformed lump is [MATH], taking naively [MATH] leads to functions that do not have derivative at [MATH] and, therefore, are not acceptable solutions of the equations of motion.', 'hep-th-0308084-2-18-7': 'In fact, the deformation of the [MATH] lump results in deformed kinks given by [EQUATION]', 'hep-th-0308084-2-18-8': 'Each of such solutions (e.g., the [MATH] one) starts at a minimum ([MATH]) when [MATH], goes to the absolute maximum ([MATH]) at [MATH] running along one of the branchs of [MATH] (the lower branch) and, passing continuously to the other branch (the upper one), reaches the other local minimum ([MATH]) when [MATH].', 'hep-th-0308084-2-18-9': 'One notice that, again, the number of solutions duplicates using such a deformation, whose inverse is a double-valued function.', 'hep-th-0308084-2-18-10': 'But, in this case, novel topological solutions emerge as deformations of a non topological one.', 'hep-th-0308084-2-19-0': 'Potentials which have a factor [MATH] can also be deformed using the function [MATH], producing many interesting situations.', 'hep-th-0308084-2-19-1': 'In fact, suppose the potential can be written in the form [EQUATION] this is always possible for all well-behaved potentials that vanish at both values [MATH], as shown by Taylor expansion.', 'hep-th-0308084-2-19-2': 'Then, the sin-deformation leads to the potential [EQUATION] which is a periodic potential, the critical points of [MATH] not causing any problem to the deformation process.', 'hep-th-0308084-2-19-3': 'The inverse of the sine function is the infinitely valued function [MATH], with [MATH] and [MATH] being the first determination of [MATH] (which varies from [MATH], for [MATH], to [MATH], when [MATH]), defined in the interval [MATH].', 'hep-th-0308084-2-19-4': 'So to each solution of the original potential, whose field values range in the interval [MATH], one finds infinitely many solutions of the deformed, periodic, potential.', 'hep-th-0308084-2-20-0': 'Consider firstly the [MATH] model.', 'hep-th-0308084-2-20-1': 'Applying the sin-deformation to it, one gets [MATH] which is one of the forms of the sine-Gordon potential.', 'hep-th-0308084-2-20-2': 'The deformed solutions thus obtained is given by [MATH], which correspond to all the kink solutions (connecting neighbouring minima) of this sine-Gordon model.', 'hep-th-0308084-2-20-3': 'For example, the kink solutions [MATH], which connect the minima [MATH] of the [MATH] model in both directions, are deformed into the kinks [MATH] (which runs between [MATH] and [MATH]) if one takes [MATH] while, for [MATH], the resulting solutions connect the minima [MATH] and [MATH] of the deformed potential.', 'hep-th-0308084-2-21-0': 'This example can be readily extended to other polynomial potentials, leading to a large class of sine-Gordon type of potentials.', 'hep-th-0308084-2-21-1': 'For instance, the [MATH] model, [MATH], deformed by the sine function, becomes the potential [EQUATION] having kink solutions given by [EQUATION]', 'hep-th-0308084-2-21-2': 'If, on the other hand, one considers the potential [MATH], which is unbounded below and supports kinklike solutions connecting the two inflection points at [MATH], one gets the potential [EQUATION] with solutions given by [EQUATION]', 'hep-th-0308084-2-21-3': 'A particularly interesting situation appears if one consider the inverted [MATH] model, which presents lumplike solutions.', 'hep-th-0308084-2-21-4': 'The sin-deformation of the potential [MATH] leads to the potential [MATH].', 'hep-th-0308084-2-21-5': 'In this case, the lump solutions of [MATH], namely [MATH], are deformed into [MATH].', 'hep-th-0308084-2-21-6': 'Consider the [MATH]-solution and take initially [MATH].', 'hep-th-0308084-2-21-7': 'As [MATH] varies from [MATH] to [MATH], [MATH] goes from [MATH] to [MATH], and [MATH] changes from [MATH] to [MATH].', 'hep-th-0308084-2-21-8': 'If one continuously makes [MATH] goes from [MATH] to [MATH], then the deformed solution passes to the [MATH] branch of [MATH], [MATH] for [MATH]), which varies from [MATH] to [MATH] as [MATH] goes from [MATH] to [MATH].', 'hep-th-0308084-2-21-9': 'Thus, in this case, the lump solution [MATH] of the inverted [MATH] model is deformed in the kink of the sine-Gordon model connecting the minima [MATH] and [MATH].', 'hep-th-0308084-2-21-10': 'Under reversed conditions (taking the [MATH] branch before the [MATH] one), the lump solution [MATH] leads to the anti-kink solution of the sine-Gordon model running from the minimum [MATH] to [MATH].', 'hep-th-0308084-2-21-11': 'The other topological solutions of the sine-Gordon model are obtained considering the other adjacent branches of [MATH].', 'hep-th-0308084-2-21-12': 'This is another remarkable example since one has non topological solutions being deformed in topological ones.', 'hep-th-0308084-2-22-0': 'Finally, we mention that many other soluble models can be construct following the procedure presented in this paper.', 'hep-th-0308084-2-22-1': 'Also, the discussion raised here might be extended to consider situations where the fields are constrained to intervals of [MATH], thus representing two-point boundary value problems defined in finite or semi-infinite intervals of the [MATH]-axis.', 'hep-th-0308084-2-22-2': 'Such study is left for future work.'}","[['hep-th-0308084-1-4-0', 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1210.1859,"{'1210.1859-1-0-0': 'We study a quantum ladder of interacting fermions with coupled [MATH] and [MATH] orbitals.', '1210.1859-1-0-1': 'Such a model describes dipolar molecules or atoms loaded into a double-well optical lattice, dipole moments being aligned by an external field.', '1210.1859-1-0-2': 'The two orbital components have distinct hoppings.', '1210.1859-1-0-3': 'The tunneling between them is equivalent to a partial Rashba spin-orbital coupling when the orbital space ([MATH], [MATH]) is identified as spanned by pseudo-spin 1/2 states.', '1210.1859-1-0-4': 'A rich phase diagram, including incommensurate orbital density wave, pair density wave and other exotic superconducting phases, is proposed with bosonization analysis.', '1210.1859-1-0-5': 'In particular, superconductivity is found in the repulsive regime.', '1210.1859-1-1-0': 'Introduction.', '1210.1859-1-1-1': 'Orbital degree of freedom [CITATION] plays a fundamental role in understanding the unconventional properties in solid state materials [CITATION].', '1210.1859-1-1-2': 'Recent experiments in optical lattices have demonstrated that orbitals can also be used to construct quantum emulators of exotic models beyond natural crystals.', '1210.1859-1-1-3': 'Orbital lattices are attracting growing interests due to their unique and fascinating properties resulting from the spatial nature of the degenerate states .', '1210.1859-1-1-4': 'For example, the bosonic [MATH] superfluid [CITATION] state has been prepared on a bipartite square lattice [CITATION], and later the other complex superfluid with [MATH] and [MATH] orbitals correlated was observed on a hexagonal lattice [CITATION].', '1210.1859-1-2-0': 'Previous study on multicomponent cold gases mainly focused on hyperfine states of alkali atoms [CITATION].', '1210.1859-1-2-1': 'In a cold gas of atoms with two approximately degenerate hyperfine states, the realized pseudo-spin SU(2) symmetry makes it possible to emulate Fermi Hubbard model in optical lattices [CITATION].', '1210.1859-1-2-2': 'To engineer spin-orbital couplings and the resulting topological phases, one has to induce Raman transitions between the hyperfine states to break the pseudo-spin symmetry [CITATION].', '1210.1859-1-2-3': 'In contrast, due to the spatial nature of the orbital degrees of freedom, the symmetry in orbital gases, such as that in [MATH] superfluid [CITATION], can be controlled by simply changing the lattice geometry [CITATION].', '1210.1859-1-2-4': 'With a certain lattice geometry, a spin-orbital like coupling can naturally appear in an orbital gas with [MATH] and [MATH]-orbitals without Raman transitions [CITATION].', '1210.1859-1-2-5': 'Theoretical studies of orbital physics largely focusing on two or three dimensions suggest exotic orbital phases [CITATION] beyond the scope of spin physics.', '1210.1859-1-3-0': 'In this article, we study a one dimensional orbital ladder with [MATH] and [MATH] orbitals coupled [CITATION].', '1210.1859-1-3-1': 'We shall derive such an effective model for dipolar molecules or atoms loaded in a double-well optical lattice.', '1210.1859-1-3-2': 'The tunneling rates (or effective mass) of each orbital component are highly tunable by changing the lattice strength.', '1210.1859-1-3-3': 'The coupling between [MATH] and [MATH] orbitals mimics the spin-orbital couplings [CITATION].', '1210.1859-1-3-4': 'A rich phase diagram, including incommensurate orbital density wave (ODW), pair density wave (PDW) [CITATION], and other exotic superconducting phases, is found with bosonization analysis.', '1210.1859-1-3-5': 'The PDW phase realized here is a superconducting phase, that features an oscillating Cooper pair field with a period of [MATH].', '1210.1859-1-3-6': 'The incommensurate ODW phase has an oscillating particle-hole pair, which tends to break the time-reversal symmetry.', '1210.1859-1-3-7': 'An exotic superconducting phase on the repulsive side is also discovered.', '1210.1859-1-4-0': 'Model.', '1210.1859-1-4-1': 'Consider a cold ensemble of polar molecules or atoms, e.g. [MATH] [CITATION], [MATH] [CITATION] or Dy [CITATION], whose dipole moments are controlled by an external field as demonstrated in experiments.', '1210.1859-1-4-2': 'Let the ensemble trapped by a ladder-like optical lattice of the type studied in [CITATION].', '1210.1859-1-4-3': 'That is, the lattice consists of two chains of potentials of unequal depth, with the molecules or atoms residing on the [MATH] and [MATH] orbital levels of the shallow and deep chains, respectively, in the tight binding regime.', '1210.1859-1-4-4': 'Recent experiments have reported various lattice configurations of double-well potentials formed by interfering laser beams, showing the unprecedented tunability of the relative depth between two sub-wells [CITATION].', '1210.1859-1-4-5': 'The single particle Hamiltonian of the [MATH]-orbital ladder is then given as [CITATION] [EQUATION] where [MATH], and [MATH]) is the creation operator for the [MATH]-orbital ([MATH]-orbital).', '1210.1859-1-4-6': 'The lattice constant is set as the length unit in this paper.', '1210.1859-1-4-7': 'The band structure is readily obtained by Fourier transform [MATH].', '1210.1859-1-4-8': 'The Hamiltonian in the momentum space reads as [MATH], with [MATH] where [MATH], [MATH], [MATH] and [MATH].', '1210.1859-1-4-9': 'Here [MATH] is the identity matrix and [MATH] are Pauli matrices.', '1210.1859-1-4-10': 'The two bands are given by [MATH] which are shown in FIG. [REF].', '1210.1859-1-4-11': 'The Hamiltonian is rewritten as [MATH].', '1210.1859-1-4-12': 'We define an angle variable [MATH] by [MATH] and [MATH] to save writing.', '1210.1859-1-4-13': 'Here, we only consider lower than half filling, i.e., less than one particle per unit cell.', '1210.1859-1-4-14': 'The lower band is thus partially filled and the upper band is empty.', '1210.1859-1-4-15': 'Since we are interested in the low-energy physics, the spectrum [MATH] is linearized around the Fermi momenta [MATH].', '1210.1859-1-4-16': 'Here, [MATH] or [MATH], and [MATH] are inner Fermi points and [MATH] are outer Fermi points (FIG. [REF]).', '1210.1859-1-4-17': 'The resulting Fermi velocities are [MATH].', '1210.1859-1-4-18': 'The operators capturing the low energy fluctuations are defined with right ([MATH]) and left ([MATH]) moving modes [MATH], [MATH], [MATH] and [MATH].', '1210.1859-1-4-19': 'The field operators are introduced by [MATH] and [MATH].', '1210.1859-1-4-20': 'These field operators are related to lattice operators by [EQUATION] where [EQUATION] with [MATH] and [MATH].', '1210.1859-1-5-0': 'With polar molecules or atoms loaded on the [MATH]-ladder, we include all momentum-independent interactions (momentum-dependent part is irrelevant in the Renormalization group flow [CITATION]) allowed by symmetry.', '1210.1859-1-5-1': 'The Hamiltonian density of the interactions is given by [EQUATION] where [MATH] and [MATH].', '1210.1859-1-5-2': 'For the symmetric case [MATH], an Umklapp process [EQUATION] becomes allowed for the reason that [MATH].', '1210.1859-1-5-3': 'Since dipolar interactions between polar molecules or atoms decay as [MATH], the leading interaction in the proposed double-well lattice setup [CITATION] is [EQUATION].', '1210.1859-1-5-4': 'In the weak interacting limit, the [MATH]-ology couplings are related to [MATH] by [MATH], [MATH], [MATH], [MATH], [EQUATION] and [EQUATION] at tree level [CITATION].', '1210.1859-1-5-5': 'Considering strong interactions or the finite ranged tail of dipolar interactions, the g-ology couplings will be renormalized due to neglected irrelevant couplings.', '1210.1859-1-5-6': 'By manipulating the direction of dipole moments with an external field, the interaction can be either repulsive or attractive [CITATION].', '1210.1859-1-6-0': 'We follow the notation convention of Ref. [CITATION], where the bosonization identity takes the form [EQUATION] where [MATH] is the Klein factor and [MATH] is the dual field of boson field [MATH].', '1210.1859-1-6-1': 'The charge and orbital boson fields are further introduced here by [MATH], with the matrix [MATH] given by [EQUATION] and their duals fields are [MATH].', '1210.1859-1-6-2': 'The Bosonized Hamiltonian density reads [EQUATION] with [MATH] , [MATH] and [MATH] where [MATH], [MATH] and the transformed coupling matrices [MATH] and [MATH] are given by [MATH].', '1210.1859-1-6-3': 'The mixing term [MATH] vanishes for the symmetric case with [MATH].', '1210.1859-1-7-0': 'Symmetric case.', '1210.1859-1-7-1': 'For the symmetric case with [MATH] (FIG. [REF]), the Hamiltonian has an accidental [MATH] symmetry, [MATH] and Fermi momenta are related by [MATH].', '1210.1859-1-7-2': 'This [MATH] symmetry implies that [MATH], [MATH] and [MATH].', '1210.1859-1-7-3': 'We find that the transformed coupling matrices [MATH] and [MATH] are diagonal and that the orbital-charge mixing term [MATH] vanishes.', '1210.1859-1-7-4': 'In other words, the [MATH] symmetry guarantees orbital-charge separation.', '1210.1859-1-7-5': 'The charge part [MATH] is quadratic and the orbital part [MATH] is a Sine-Gordon model [CITATION].', '1210.1859-1-8-0': 'With attraction, we have [MATH], [MATH], and the Sine-Gordon term [MATH] is relevant (flows to [MATH]) in the renormalization group (RG) flow [CITATION].', '1210.1859-1-8-1': 'This corresponds to an orbital gapped phase with [MATH] locked at [MATH].', '1210.1859-1-8-2': 'In this phase, quantum fluctuations of [MATH] become massive, and the divergent susceptibilities are the following: charge density wave (CDW) and PDW [CITATION] given by the operators: [EQUATION]', '1210.1859-1-8-3': 'Due to orbital-charge separation, the CDW and PDW correlation functions are readily given by [EQUATION]', '1210.1859-1-8-4': 'Since [MATH] for attraction, the algebraic PDW order is dominant.', '1210.1859-1-8-5': 'In this phase, the superconducting pairing [MATH] oscillates in space with a period of [MATH].', '1210.1859-1-9-0': 'With repulsion, we have [MATH], and thus [MATH] is relevant [CITATION].', '1210.1859-1-9-1': 'This gives an orbital gapped phase with [MATH] locked at [MATH], because [MATH].', '1210.1859-1-9-2': 'The fluctuations of [MATH] are massive, and the divergent susceptibilities are ODW and superconducting SC[MATH] given by the operators: [EQUATION]', '1210.1859-1-9-3': 'Since [MATH] for repulsion, the dominant algebraic order here is ODW, for which the correlation function is given by [EQUATION]', '1210.1859-1-9-4': 'In the ODW phase, the particle-hole pairing in terms of lattice operators reads [MATH].', '1210.1859-1-9-5': 'This ODW order is incommensurate with an oscillation period [MATH] in real space.', '1210.1859-1-9-6': 'If we go beyond the one-dimensional limit and consider small transverse tunnelings [CITATION], a true long-range ODW order [MATH] is expected.', '1210.1859-1-9-7': 'Such an order breaks time-reversal symmetry.', '1210.1859-1-10-0': 'The ODW and PDW phases predicted by Bosonization analysis are further verified in numerical simulations with matrix products states.', '1210.1859-1-10-1': 'The superconducting correlation [MATH] and the orbital density wave correlation [MATH] are calculated.', '1210.1859-1-10-2': 'FIG. [REF] shows the Fourier transform of these correlations, defined by [MATH].', '1210.1859-1-10-3': 'The sharp peaks of [MATH] at momenta [MATH] on the attractive side tell the quantum state has a PDW order shown in Eq. [REF].', '1210.1859-1-10-4': 'On the repulsive side sharp dips of [MATH] at finite momenta verify the incommensurate ODW order shown in Eq. [REF].', '1210.1859-1-10-5': 'With numerical calculations, we also find the existence of PDW phase in the strongly attractive regime if [MATH].', '1210.1859-1-11-0': 'Asymmetric case.', '1210.1859-1-11-1': 'For the asymmetric case-[MATH] (FIG. [REF]), the Fermi velocity [MATH] and the orbital-charge separation no longer holds.', '1210.1859-1-11-2': 'Thus, the orbital and charge degrees of freedom cannot be treated separately.', '1210.1859-1-11-3': 'The other difference with the symmetric case is that the Umklapp process [MATH] does not exist.', '1210.1859-1-11-4': 'Since the effects of [MATH] couplings are just to renormalize the Fermi velocities [CITATION].', '1210.1859-1-11-5': 'For simplicity, we do not consider such effects and set [MATH] here.', '1210.1859-1-11-6': 'The one-loop RG equations are given by [CITATION], [EQUATION] where [MATH] is the flow parameter ([MATH]) and [MATH]) for [MATH]).', '1210.1859-1-11-7': 'The RG flow of the Sine-Gordon term [MATH] is obtained as [EQUATION] with [EQUATION] and [EQUATION]', '1210.1859-1-11-8': 'The function [MATH] is the hyperbolic function ""[MATH]"" (the trigonometric function ""[MATH]"") if [MATH]).', '1210.1859-1-11-9': 'When [MATH], [MATH] always flows to [MATH] and the system is in some gapped phase.', '1210.1859-1-11-10': 'When [MATH], [MATH] flows to [MATH] only if [MATH].', '1210.1859-1-11-11': '[MATH] is irrelevant only if [MATH] and [MATH].', '1210.1859-1-11-12': 'In the weak interacting regime, [MATH].', '1210.1859-1-11-13': 'We will consider the regime [MATH] (this condition holds when [MATH] is weak compared with [MATH]) in the following.', '1210.1859-1-12-0': 'With repulsion ([MATH], [MATH]), [MATH] is relevant and flows to [MATH] in RG flow.', '1210.1859-1-12-1': 'Then the dual orbital field [MATH] is locked with [MATH] and its fluctuations [MATH] are massive.', '1210.1859-1-12-2': 'The key effect of orbital-charge mixing can be seen from its modification of the dynamics of the conjugate fields, given as [EQUATION] where [MATH] is the conjugate field of [MATH].', '1210.1859-1-12-3': 'The Lagrangian is constructed by [MATH].', '1210.1859-1-12-4': 'With massive fluctuations of [MATH] integrated out, the Lagrangian of the charge field [MATH] is given by [EQUATION] with the renormalized Luttinger parameter and sound velocity given by [EQUATION]', '1210.1859-1-12-5': 'To zeroth order in the interaction [MATH], the renormalized Luttinger parameter is [MATH].', '1210.1859-1-12-6': 'Our result reproduces the perturbative result [CITATION] when the orbital-charge mixing term is small.', '1210.1859-1-12-7': 'The diverging susceptibilities are ODW and SC[MATH], and the corresponding correlation functions are given as [EQUATION]', '1210.1859-1-12-8': 'With sufficiently weak repulsion [MATH], the dominant order is SC[MATH], of which the pairing in terms of lattice operators is [MATH].', '1210.1859-1-12-9': 'We emphasize here that this pairing does not oscillate in real space.', '1210.1859-1-12-10': 'Such a superconducting phase arises in the repulsive regime due to the orbital-charge mixing and the pinning effect of the dual orbital field [MATH].', '1210.1859-1-12-11': 'The Sine-Gordon term [MATH] causing this pinning effect is finite only when the coupling of [MATH]-orbitals [MATH] is finite, and [MATH] is monotonically increasing when [MATH] is increased.', '1210.1859-1-12-12': 'Thus the transition temperature of this repulsive superconducting phase can be increased by tuning [MATH], which makes this exotic superconducting phase potentially realizable in experiments.', '1210.1859-1-12-13': 'With stronger repulsion, the renormalized Luttinger parameter [MATH] decreases.', '1210.1859-1-12-14': 'Eventually with repulsion larger than some critical strength, we have [MATH], and the repulsive superconducting phase gives way to the ODW phase.', '1210.1859-1-12-15': 'Our numerical results cannot distinguish the ODW and SC phases in the weak interacting regime due to computational complexity [CITATION].', '1210.1859-1-13-0': 'With attractive interaction, the condition [MATH] gives [MATH].', '1210.1859-1-13-1': 'Thus [MATH] is relevant and flows to [MATH] when [MATH].', '1210.1859-1-13-2': 'The Sine-Gordon term [MATH] is locked at [MATH], and the dominant order is superconducting SC[MATH], given by [EQUATION]', '1210.1859-1-13-3': 'In numerical simulations we find the SC[MATH] phase competing with PDW in the strongly attractive regime.', '1210.1859-1-13-4': 'When [MATH] is irrelevant ([MATH], [MATH]), the orbital ladder is in a two component Luttinger liquid phase exhibiting two gapless normal modes and each mode is a mixture of orbital and charge.'}","{'1210.1859-2-0-0': 'We study a quantum ladder of interacting fermions with coupled [MATH] and [MATH] orbitals.', '1210.1859-2-0-1': 'Such a model describes dipolar molecules or atoms loaded into a double-well optical lattice, dipole moments being aligned by an external field.', '1210.1859-2-0-2': 'The two orbital components have distinct hoppings.', '1210.1859-2-0-3': 'The tunneling between them is equivalent to a partial Rashba spin-orbital coupling when the orbital space ([MATH], [MATH]) is identified as spanned by pseudo-spin 1/2 states.', '1210.1859-2-0-4': 'A rich phase diagram, including incommensurate orbital density wave, pair density wave and other exotic superconducting phases, is proposed with bosonization analysis.', '1210.1859-2-0-5': 'In particular, superconductivity is found in the repulsive regime.', '1210.1859-2-1-0': '# Introduction', '1210.1859-2-2-0': 'Orbital degree of freedom [CITATION] plays a fundamental role in understanding the unconventional properties in solid state materials [CITATION].', '1210.1859-2-2-1': 'Recent experiments in optical lattices have demonstrated that orbitals can also be used to construct quantum emulators of exotic models beyond natural crystals.', '1210.1859-2-2-2': 'Orbital lattices are attracting growing interests due to their unique and fascinating properties resulting from the spatial nature of the degenerate states.', '1210.1859-2-2-3': 'For example, the bosonic [MATH] superfluid [CITATION] state has been prepared on a bipartite square lattice [CITATION], and later the other complex superfluid with [MATH] and [MATH] orbitals correlated was observed on a hexagonal lattice [CITATION].', '1210.1859-2-3-0': 'Previous study on multicomponent cold gases mainly focused on hyperfine states of alkali atoms [CITATION].', '1210.1859-2-3-1': 'In a cold gas of atoms with two approximately degenerate hyperfine states, the realized pseudo-spin SU(2) symmetry makes it possible to emulate Fermi Hubbard model in optical lattices [CITATION].', '1210.1859-2-3-2': 'To engineer spin-orbital couplings and the resulting topological phases, one has to induce Raman transitions between the hyperfine states to break the pseudo-spin symmetry [CITATION].', '1210.1859-2-3-3': 'In contrast, due to the spatial nature of the orbital degrees of freedom, the symmetry in orbital gases, such as that in [MATH] superfluid [CITATION], can be controlled by simply changing the lattice geometry as shown in Ref. [CITATION], where unprecedented tunability of double-wells has been demonstrated.', '1210.1859-2-3-4': 'With a certain lattice geometry, a spin-orbital like coupling can naturally appear in an orbital gas with [MATH] and [MATH]-orbitals without Raman transitions [CITATION].', '1210.1859-2-3-5': 'Theoretical studies of orbital physics largely focusing on two or three dimensions suggest exotic orbital phases [CITATION] beyond the scope of spin physics.', '1210.1859-2-4-0': 'In this article, we study a one dimensional orbital ladder with [MATH] and [MATH] orbitals coupled [CITATION].', '1210.1859-2-4-1': 'We shall derive such an effective model for dipolar molecules or atoms [CITATION] loaded in a double-well optical lattice.', '1210.1859-2-4-2': 'The tunneling rates (or effective mass) of each orbital component are highly tunable by changing the lattice strength.', '1210.1859-2-4-3': 'The coupling between [MATH] and [MATH] orbitals mimics the spin-orbital couplings [CITATION].', '1210.1859-2-4-4': 'This orbital system suggests the possibility of exploring the equivalent of the exciting spin-orbital coupled physics in dipolar gases yet without requiring the use of synthetic gauge fields, and hence it provides an interesting, simple alternative route.', '1210.1859-2-4-5': 'A rich phase diagram, including incommensurate orbital density wave (ODW), pair density wave (PDW) [CITATION], and other exotic superconducting phases, is found with bosonization analysis.', '1210.1859-2-4-6': 'The PDW phase realized here is a superconducting phase, that features an oscillating Cooper pair field with a period of [MATH].', '1210.1859-2-4-7': 'The incommensurate ODW phase has an oscillating particle-hole pair, which tends to break the time-reversal symmetry.', '1210.1859-2-4-8': 'An exotic superconducting phase on the repulsive side is also discovered.', '1210.1859-2-5-0': '# Model', '1210.1859-2-6-0': 'Consider a cold ensemble of polar molecules or atoms, e.g. [MATH] [CITATION], [MATH] [CITATION], [MATH] [CITATION], or Dy [CITATION], whose dipole moments are controlled by an external field as demonstrated in experiments.', '1210.1859-2-6-1': 'Long lived polar molecules have been realized in optical lattices [CITATION].', '1210.1859-2-6-2': 'Let the ensemble trapped by a ladder-like optical lattice of the type studied in [CITATION].', '1210.1859-2-6-3': 'As shown in the schematic picture in [CITATION], the lattice consists of two chains of potentials of unequal depth.', '1210.1859-2-6-4': 'We consider a single species of fermionic atoms/molecules occupying the [MATH] and [MATH] orbitals of the shallow and deep chains, respectively, with the low-lying [MATH] orbitals on the deep chain completely filled (FIG. [REF]).', '1210.1859-2-6-5': 'Alternatively, fermions can be directly loaded into the higher orbitals while keeping the lower [MATH] nearly empty by the techniques developed in recent experiments [CITATION].', '1210.1859-2-6-6': 'Coherent meta-stable states in high orbitals with long life time up to several hundred milliseconds were demonstrated achievable [CITATION].', '1210.1859-2-6-7': 'To suppress chemical reactions of polar molecules, the latter approach is preferable.', '1210.1859-2-6-8': 'The single particle Hamiltonian of the [MATH]-orbital ladder is then given as [CITATION] [EQUATION] where [MATH], and [MATH]) is the creation operator for the [MATH]-orbital ([MATH]-orbital).', '1210.1859-2-6-9': 'The lattice constant is set as the length unit in this paper.', '1210.1859-2-6-10': 'In the proposed optical lattice setup [CITATION], the ratios [MATH] and [MATH] are small (typically [MATH]).', '1210.1859-2-7-0': 'We emphasize here that [MATH]-orbital is parity even and that [MATH]-orbital is parity odd.', '1210.1859-2-7-1': 'The relative signs of hoppings are dictated by the parity nature of the [MATH]- and [MATH]-orbitals [CITATION]', '1210.1859-2-8-0': 'The band structure is readily obtained by Fourier transform [MATH].', '1210.1859-2-8-1': 'The Hamiltonian in the momentum space reads as [MATH], with [MATH] where [MATH], [MATH], [MATH] and [MATH].', '1210.1859-2-8-2': 'Here [MATH] is the identity matrix and [MATH] are Pauli matrices.', '1210.1859-2-8-3': 'The two bands are given by [MATH] which are shown in FIG. [REF].', '1210.1859-2-8-4': 'The Hamiltonian is rewritten as [MATH].', '1210.1859-2-8-5': 'We define an angle variable [MATH] by [MATH] and [MATH] to save writing.', '1210.1859-2-8-6': 'Here, we only consider lower than half filling, i.e., less than one particle per unit cell.', '1210.1859-2-8-7': 'The lower band is thus partially filled and the upper band is empty.', '1210.1859-2-8-8': 'Since we are interested in the low-energy physics, the spectrum [MATH] is linearized around the Fermi momenta [MATH].', '1210.1859-2-8-9': 'Here, [MATH] or [MATH], and [MATH] are inner Fermi points and [MATH] are outer Fermi points (FIG. [REF]).', '1210.1859-2-8-10': 'The resulting Fermi velocities are [MATH].', '1210.1859-2-8-11': 'The operators capturing the low energy fluctuations are defined with right ([MATH]) and left ([MATH]) moving modes [MATH], [MATH], [MATH] and [MATH].', '1210.1859-2-8-12': 'The field operators are introduced by [MATH] and [MATH].', '1210.1859-2-8-13': 'These field operators are related to lattice operators by [EQUATION] where [EQUATION] with [MATH] and [MATH].', '1210.1859-2-8-14': 'The substitution in Eq. ([REF]) and the energy linearization are valid for weakly interacting fermions at low temperature.', '1210.1859-2-9-0': 'With polar molecules or atoms loaded on the [MATH]-ladder, we include all momentum-independent interactions (momentum-dependent part is irrelevant in the Renormalization group flow [CITATION]) allowed by symmetry.', '1210.1859-2-9-1': 'The Hamiltonian density of the interactions is given by [EQUATION] where [MATH] and [MATH].', '1210.1859-2-9-2': 'For the symmetric case [MATH], an Umklapp process [EQUATION] becomes allowed for the reason that [MATH].', '1210.1859-2-9-3': 'Since dipolar interactions between polar molecules or atoms decay as [MATH], the leading interaction in the proposed double-well lattice setup [CITATION] is [EQUATION].', '1210.1859-2-9-4': 'The strength of [MATH] is tunable by changing the dipole moment, or by varying the distance between the shallow and deep wells (FIG. [REF]).', '1210.1859-2-9-5': 'By controlling this distance the leading interaction can be made significantly larger than sub-leading interactions (dipolar tails), which are neglected here.', '1210.1859-2-10-0': 'In the weak interacting limit, the [MATH]-ology couplings are related to [MATH] by [MATH], [MATH], [MATH], [MATH], [EQUATION] and [EQUATION] at tree level [CITATION].', '1210.1859-2-10-1': 'Considering strong interactions or the finite ranged tail of dipolar interactions, the g-ology couplings will be renormalized due to neglected irrelevant couplings.', '1210.1859-2-10-2': 'By manipulating the direction of dipole moments with an external field, the interaction can be either repulsive or attractive (FIG. [REF]) [CITATION].', '1210.1859-2-11-0': 'We follow the notation convention of Ref. [CITATION], where the bosonization identity takes the form [EQUATION] where [MATH] is the Klein factor and [MATH] is the dual field of boson field [MATH].', '1210.1859-2-11-1': 'The charge and orbital boson fields are further introduced here by [MATH], with the matrix [MATH] given by [EQUATION] and their dual fields are [MATH].', '1210.1859-2-11-2': 'The Bosonized Hamiltonian density reads [EQUATION] with [EQUATION] and [EQUATION] where [MATH], [MATH] and the transformed coupling matrices [MATH] and [MATH] are given by [MATH].', '1210.1859-2-11-3': 'The mixing term [MATH] vanishes for the symmetric case with [MATH].', '1210.1859-2-12-0': '# Quantum phases and transitions of the Symmetric case', '1210.1859-2-13-0': 'For the symmetric case with [MATH] (FIG. [REF]), the Hamiltonian has an accidental [MATH] symmetry, [MATH] and Fermi momenta are related by [MATH].', '1210.1859-2-13-1': 'This [MATH] symmetry implies that [MATH], [MATH] and [MATH].', '1210.1859-2-13-2': 'We find that the transformed coupling matrices [MATH] and [MATH] are diagonal and that the orbital-charge mixing term [MATH] vanishes.', '1210.1859-2-13-3': 'In other words, the [MATH] symmetry guarantees orbital-charge separation.', '1210.1859-2-13-4': 'The charge part [MATH] is quadratic and the orbital part [MATH] is a sine-Gordon model [CITATION].', '1210.1859-2-14-0': 'With attraction, we have [MATH], [MATH], and the sine-Gordon term [MATH] is relevant (flows to [MATH]) in the renormalization group (RG) flow [CITATION].', '1210.1859-2-14-1': 'This corresponds to an orbital gapped phase with [MATH] locked at [MATH].', '1210.1859-2-14-2': 'In this phase, quantum fluctuations of [MATH] become massive, and the divergent susceptibilities are the following: charge density wave (CDW) and PDW [CITATION] given by the operators: [EQUATION]', '1210.1859-2-14-3': 'Due to orbital-charge separation, the CDW and PDW correlation functions are readily given by [EQUATION]', '1210.1859-2-14-4': 'Since [MATH] for attraction, the algebraic PDW order is dominant.', '1210.1859-2-14-5': 'In this phase, the superconducting pairing [MATH] oscillates in space with a period of [MATH].', '1210.1859-2-15-0': 'With repulsion, we have [MATH], and thus [MATH] is relevant [CITATION].', '1210.1859-2-15-1': 'This gives an orbital gapped phase with [MATH] locked at [MATH], because [MATH].', '1210.1859-2-15-2': 'The fluctuations of [MATH] are massive, and the divergent susceptibilities are ODW and superconducting SC[MATH] given by the operators: [EQUATION]', '1210.1859-2-15-3': 'Since [MATH] for repulsion, the dominant algebraic order here is ODW, for which the correlation function is given by [EQUATION]', '1210.1859-2-15-4': 'In the ODW phase, the particle-hole pairing in terms of lattice operators reads [MATH].', '1210.1859-2-15-5': 'This ODW order is incommensurate with an oscillation period [MATH] in real space.', '1210.1859-2-15-6': 'If we go beyond the one-dimensional limit and consider small transverse tunnelings [CITATION], a true long-range ODW order [MATH] is expected.', '1210.1859-2-15-7': 'Such an order breaks time-reversal symmetry.', '1210.1859-2-16-0': 'The ODW and PDW phases predicted by Bosonization analysis are further verified in numerical simulations with matrix products state, in which open boundary condition is adopted.', '1210.1859-2-16-1': 'The superconducting correlation [EQUATION] and the orbital density wave correlation [EQUATION] are calculated.', '1210.1859-2-16-2': 'In our calculation, the two points [MATH] and [MATH] are [MATH] sites away from the boundaries to minimize the boundary effects.', '1210.1859-2-16-3': 'The convergence of these correlations is checked in numerical simulations.', '1210.1859-2-16-4': 'FIG. [REF] shows the Fourier transform of these correlations, defined by [MATH], which approaches to its thermodynamic limit with increasing system size (FIG. [REF]).', '1210.1859-2-16-5': 'The sharp peaks of [MATH] at momenta [MATH] on the attractive side tell the quantum state has a PDW order shown in Eq. [REF].', '1210.1859-2-16-6': 'On the repulsive side sharp dips of [MATH] at finite momenta verify the incommensurate ODW order shown in Eq. [REF].', '1210.1859-2-16-7': 'With numerical calculations, we also find the existence of PDW phase in the strongly attractive regime if [MATH].', '1210.1859-2-17-0': '# Quantum phases and transitions of the Asymmetric case', '1210.1859-2-18-0': 'For the asymmetric case-[MATH] (FIG. [REF]), the Fermi velocity [MATH] and the orbital-charge separation no longer holds.', '1210.1859-2-18-1': 'Thus, the orbital and charge degrees of freedom cannot be treated separately.', '1210.1859-2-18-2': 'The other difference with the symmetric case is that the Umklapp process [MATH] does not exist.', '1210.1859-2-18-3': 'Since the effects of [MATH] couplings are just to renormalize the Fermi velocities [CITATION].', '1210.1859-2-18-4': 'For simplicity, we do not consider such effects and set [MATH] here.', '1210.1859-2-18-5': 'The one-loop RG equations are given by [CITATION], [EQUATION] where [MATH] is the flow parameter ([MATH]) and [MATH]) for [MATH]).', '1210.1859-2-18-6': 'The RG flow of the sine-Gordon term [MATH] is obtained as [EQUATION] with [EQUATION] and [EQUATION]', '1210.1859-2-18-7': 'The function [MATH] is the hyperbolic function ""[MATH]"" (the trigonometric function ""[MATH]"") if [MATH]).', '1210.1859-2-18-8': 'When [MATH], [MATH] always flows to [MATH] and the system is in some gapped phase.', '1210.1859-2-18-9': 'When [MATH], [MATH] flows to [MATH] only if [MATH].', '1210.1859-2-18-10': '[MATH] is irrelevant only if [MATH] and [MATH].', '1210.1859-2-18-11': 'In the weak interacting regime, we have [EQUATION]', '1210.1859-2-18-12': 'We will consider the regime [MATH] (this condition holds when [MATH] is weak compared with [MATH]) in the following.', '1210.1859-2-19-0': 'With repulsion ([MATH], [MATH]), [MATH] is relevant and flows to [MATH] in RG flow.', '1210.1859-2-19-1': 'Then the dual orbital field [MATH] is locked with [MATH] and its fluctuations [MATH] are massive.', '1210.1859-2-19-2': 'The key effect of orbital-charge mixing can be seen from its modification of the dynamics of the conjugate fields, given as [EQUATION] where [MATH] is the conjugate field of [MATH].', '1210.1859-2-19-3': 'The Lagrangian is constructed by [EQUATION].', '1210.1859-2-19-4': 'With massive fluctuations of [MATH] integrated out, the Lagrangian of the charge field [MATH] is given by [EQUATION] with the renormalized Luttinger parameter and sound velocity given by [EQUATION]', '1210.1859-2-19-5': 'To zeroth order in the interaction [MATH], the renormalized Luttinger parameter is [EQUATION]', '1210.1859-2-19-6': 'Our result reproduces the perturbative result [CITATION] when the orbital-charge mixing term is small.', '1210.1859-2-19-7': 'The diverging susceptibilities are ODW and SC[MATH], and the corresponding correlation functions are given as [EQUATION]', '1210.1859-2-19-8': 'With sufficiently weak repulsion [MATH], the dominant order is SC[MATH], of which the pairing in terms of lattice operators is [MATH].', '1210.1859-2-19-9': 'We emphasize here that this pairing does not oscillate in real space.', '1210.1859-2-19-10': 'Such a superconducting phase arising in the repulsive regime results from that charge mode [MATH] is coupled with the orbital mode [MATH], which is strongly fluctuating with its conjugate field [MATH] pinned.', '1210.1859-2-19-11': 'The sine-Gordon term [MATH] causing this pinning effect is finite only when the coupling of [MATH]-orbitals [MATH] is finite, and [MATH] is monotonically increasing when [MATH] is increased.', '1210.1859-2-19-12': 'Thus the transition temperature of this repulsive superconducting phase can be increased by tuning [MATH], which makes this exotic superconducting phase potentially realizable in experiments.', '1210.1859-2-19-13': 'With stronger repulsion, the renormalized Luttinger parameter [MATH] decreases.', '1210.1859-2-19-14': 'Eventually with repulsion larger than some critical strength, we have [MATH], and the repulsive superconducting phase gives way to the ODW phase.', '1210.1859-2-20-0': 'With attractive interaction, the condition [MATH] gives [MATH].', '1210.1859-2-20-1': 'Thus [MATH] is relevant and flows to [MATH] when [MATH].', '1210.1859-2-20-2': 'The sine-Gordon term [MATH] is locked at [MATH], and the dominant order is superconducting SC[MATH], given by [EQUATION]', '1210.1859-2-20-3': 'In numerical simulations we find the SC[MATH] phase competing with PDW in the strongly attractive regime.', '1210.1859-2-20-4': 'When [MATH] is irrelevant ([MATH], [MATH]), the orbital ladder is in a two component Luttinger liquid phase exhibiting two gapless normal modes and each mode is a mixture of orbital and charge.', '1210.1859-2-21-0': 'Discussion of dipolar tails.', '1210.1859-2-21-1': '- In Ref. [CITATION], it is shown that the tail of dipolar interaction makes correlations decay as [MATH] in the gapped phase, and that the tail does not change the critical properties at the critical point.', '1210.1859-2-21-2': 'The present study finds Luttinger liquid phases, which are critical.', '1210.1859-2-21-3': 'The power law correlations characterizing our predicted critical phases decay much slower than [MATH].', '1210.1859-2-21-4': 'For example, with weak repulsive interaction, the scaling exponent for the SC[MATH] phase, [MATH], is found less than [MATH] from Eq. [REF], well below the dipolar exponent [MATH].', '1210.1859-2-21-5': 'We thus conclude that the dipolar tail corrections should be negligible.', '1210.1859-2-22-0': '# Conclusion', '1210.1859-2-23-0': 'To conclude, we have studied quantum phases of a one-dimensional [MATH]-coupled interacting Fermi gas, with both numeric and analytic methods.', '1210.1859-2-23-1': 'A PDW phase, featuring oscillating Cooper pair field, shows up naturally in the attractive regime.', '1210.1859-2-23-2': 'An incommensurate ODW phase is found in the repulsive regime.', '1210.1859-2-23-3': 'A repulsive superconducting phase emergent from orbital-charge mixing is also discussed.', '1210.1859-2-23-4': 'In experiments, radio-frequency spectroscopy can be used to probe spectra functions [CITATION], which exhibit the signatures of pairings of the predicted phases.', '1210.1859-2-23-5': 'In orbital density wave phases, where there are diverging correlations [MATH], the quench dynamics of occupation numbers of [MATH] and [MATH] orbitals is a probe of such orders [CITATION].'}","[['1210.1859-1-8-1', '1210.1859-2-14-1'], ['1210.1859-1-8-2', '1210.1859-2-14-2'], ['1210.1859-1-8-3', '1210.1859-2-14-3'], ['1210.1859-1-8-4', '1210.1859-2-14-4'], ['1210.1859-1-8-5', '1210.1859-2-14-5'], ['1210.1859-1-9-0', '1210.1859-2-15-0'], ['1210.1859-1-9-1', '1210.1859-2-15-1'], ['1210.1859-1-9-2', '1210.1859-2-15-2'], ['1210.1859-1-9-3', '1210.1859-2-15-3'], ['1210.1859-1-9-4', '1210.1859-2-15-4'], ['1210.1859-1-9-5', '1210.1859-2-15-5'], ['1210.1859-1-9-6', '1210.1859-2-15-6'], ['1210.1859-1-9-7', '1210.1859-2-15-7'], ['1210.1859-1-12-0', '1210.1859-2-19-0'], ['1210.1859-1-12-1', '1210.1859-2-19-1'], ['1210.1859-1-12-2', '1210.1859-2-19-2'], ['1210.1859-1-12-4', '1210.1859-2-19-4'], ['1210.1859-1-12-6', '1210.1859-2-19-6'], ['1210.1859-1-12-7', '1210.1859-2-19-7'], ['1210.1859-1-12-8', '1210.1859-2-19-8'], ['1210.1859-1-12-9', '1210.1859-2-19-9'], ['1210.1859-1-12-12', '1210.1859-2-19-12'], 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['1210.1859-1-6-2', '1210.1859-2-11-2'], ['1210.1859-1-10-0', '1210.1859-2-16-0'], ['1210.1859-1-10-1', '1210.1859-2-16-1'], ['1210.1859-1-4-1', '1210.1859-2-6-0'], ['1210.1859-1-5-6', '1210.1859-2-10-2']]",[],"[['1210.1859-1-12-10', '1210.1859-2-19-10'], ['1210.1859-1-11-12', '1210.1859-2-18-11'], ['1210.1859-1-10-2', '1210.1859-2-16-4']]",[],"['1210.1859-1-1-0', '1210.1859-1-4-0', '1210.1859-1-7-0', '1210.1859-1-11-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1210.1859,,,
1503.02046,"{'1503.02046-1-0-0': 'Nanoscale defect structure within the magnetic penetration depth of [MATH]100 nm is key to the performance limitations of niobium superconducting radio frequency (SRF) cavities.', '1503.02046-1-0-1': 'Using a unique combination of advanced thermometry during cavity RF measurements, and TEM structural and compositional characterization of the samples extracted from cavity walls, we discover the existence of nanoscale hydrides in electropolished cavities limited by the high field [MATH] slope, and show the decreased hydride formation in the electropolished cavity after 120[MATH]C baking.', '1503.02046-1-0-2': 'Furthermore, we demonstrate that adding [MATH]C hydrogen degassing followed by light buffered chemical polishing restores the hydride formation to the pre-120[MATH]C bake level.', '1503.02046-1-0-3': 'We also show absence of niobium oxides along the grain boundaries and the modifications of the surface oxide upon 120[MATH]C bake.', '1503.02046-1-1-0': '# Introduction', '1503.02046-1-2-0': 'Superconducting radio frequency (SRF) cavities is the state-of-the-art technology for particle acceleration implemented in most modern and future planned accelerators [CITATION].', '1503.02046-1-2-1': 'SRF cavities are predominantly made of bulk niobium and are typically operated at temperatures of 2 K or below, deep in superconducting state of niobium, which has superconducting critical temperature [MATH] K.', '1503.02046-1-2-2': 'The performance of SRF cavities is characterized by the maximum accelerating field ([MATH]) they can sustain, and the cavity quality factor [MATH] determining their efficiency of operation.', '1503.02046-1-2-3': 'Lower [MATH] leads to the increased dynamic heat load for the cryogenic system, and, if severe, can even lead to the limitation in [MATH] as it causes an increase in the inner cavity wall temperature that can trigger the localized loss of superconductivity - quench.', '1503.02046-1-2-4': 'The magnitude of [MATH] is determined by the average microwave surface resistance [MATH], which consists of the strongly temperature dependent part [MATH] and a temperature independent (residual) component [MATH].', '1503.02046-1-3-0': 'Recent investigations showed that for standard cavity preparation techniques [CITATION] as well as for a newly discovered nitrogen doping [CITATION] both [MATH] and [MATH] depend on the surface rf magnetic field magnitude [MATH].', '1503.02046-1-3-1': 'Since these field dependencies are determined by the surface treatments and the magnetic field only penetrates [MATH]100 nm inside niobium in superconducting state at 2 K, the nanostructure within this thickness and its changes with treatments is key to understanding changes in surface resistance and [MATH].', '1503.02046-1-4-0': 'One of the long-standing puzzles is a strong increase in the surface resistance of electropolished cavities above [MATH]100 mT surface magnetic field - a so-called high field [MATH] slope (HFQS).', '1503.02046-1-4-1': 'The effect persists in the absence of other well-known parasitic losses such as multipacting and field emission.', '1503.02046-1-4-2': 'HFQS can be removed by the empirically found ""mild baking"" at [MATH]-[MATH] in ultra high vacuum (UHV) for 24-48 hours [CITATION].', '1503.02046-1-5-0': 'Several models for the HFQS were proposed in the past, but most were shown to contradict at least one of the experimental observations [CITATION].', '1503.02046-1-5-1': 'The most recent promising model is based on the formation of lossy niobium nanohydrides in the penetration depth [CITATION].', '1503.02046-1-5-2': 'Nanohydrides may remain superconducting due to the proximity effect up to the breakdown field, which is determined by their size.', '1503.02046-1-5-3': 'The model attributes HFQS onset field to such a loss of proximity-induced superconductivity, which manifests as a strong increase in residual resistance and causes HFQS.', '1503.02046-1-5-4': 'The rationale for this theory is the presence of high concentration of interstitial hydrogen in the penetration depth [CITATION], which, upon cooling to 2 K, may coalesce into lumps of niobium hydrides.', '1503.02046-1-5-5': 'A challenging part is that in order to search for such nanohydrides directly, cryo-investigations at [MATH]100 K are required as at room temperature no hydrides are present.', '1503.02046-1-6-0': 'The characteristic feature of the HFQS is the localization of strong additional dissipation in the areas of cavity surface corresponding to highest surface magnetic fields [CITATION], as found out by advanced temperature mapping studies of outside cavity walls.', '1503.02046-1-6-1': 'A very powerful approach is based on localizing the strongest dissipative spots, which then can be extracted from cavity walls and their inner surface studied by different analytical techniques.', '1503.02046-1-6-2': 'Comparing cutouts from the cavities with the HFQS to the ones from the mild baked cavities without HFQS allows drawing conclusions on the possible underlying causes of the HFQS, and the origin of the mild baking effect.', '1503.02046-1-7-0': 'Previous comparative studies on such cavity cutouts provided clues into possible mechanisms at play.', '1503.02046-1-7-1': 'Low energy muon spin rotation spectroscopy (LE-[MATH]SR) showed that mild baking leads to a strong decrease in the electron mean free path [CITATION].', '1503.02046-1-7-2': 'Variable energy positron annihilation spectroscopy (VEPAS) showed that inward diffusion of vacancies and the hydrogen-trapping effect of vacancies may be behind this [MATH] suppression [CITATION].', '1503.02046-1-7-3': 'Another alternative mechanism of the mean free path suppression may be based on the inward diffusion of oxygen [CITATION].', '1503.02046-1-7-4': 'Both oxygen and vacancies may then serve as trapping centers for hydrogen, preventing the formation of lossy nanohydrides [CITATION].', '1503.02046-1-8-0': 'In this particular work we present structural and analytic comparison of FIB-prepared cross-sectional samples from mild baked and unbaked cavity cutouts using various TEM techniques at room and cryogenic (94 K) temperatures.', '1503.02046-1-8-1': 'Temperature dependent nano-area electron diffraction (NED) and scanning electron nano-area diffraction (SEND) reveal the formation of stoichiometric non-superconducting small niobium hydride inclusions.', '1503.02046-1-8-2': 'Mild baking is shown to decrease nanohydride sizes and/or density, which directly correlates with the observed suppression of the high field [MATH] slope in SRF cavities.', '1503.02046-1-8-3': 'Importantly, we also find nanohydrides after additional 800[MATH]C degassing for 3 hours followed by buffered chemical polishing for 20 [MATH]m material removal.', '1503.02046-1-8-4': 'Furthermore, size and/or density of such hydrides are comparable to the pre-120[MATH]C bake electropolished cutout.', '1503.02046-1-8-5': 'Additionally, high resolution TEM (HRTEM) and bright field (BF) imaging show similar surface oxide thickness and lack of any oxidation along grain boundaries.', '1503.02046-1-8-6': 'Electron energy loss spectroscopy (EELS) chemical characterization of the surface oxides suggests slight oxygen enrichment just below the oxides after the mild bake, consistent with the previous X-ray investigations [CITATION].', '1503.02046-1-9-0': '# Methods', '1503.02046-1-10-0': '## Identification of the cutout regions in unbaked and 120[MATH]C baked cavities', '1503.02046-1-11-0': 'In order to directly correlate different dissipation characteristics with surface nanostructure and to determine the underlying mechanisms of HFQS in Nb SRF cavities, we base our studies on the comparison of cutouts from cavities with and without HFQS, similar to previous studies [CITATION].', '1503.02046-1-11-1': 'Two Nb fine grain ([MATH]m) TESLA shape cavities with resonant frequency [MATH]1.3 GHz were used.', '1503.02046-1-11-2': 'Both cavities were electropolished, and one of them was additionally baked at [MATH] for 48 hours.', '1503.02046-1-11-3': 'Dependence of the quality factor on peak surface magnetic field at 2 K was measured for both cavities, and both also had temperature maps acquired during rf measurements in order to identify the regions for cutout.', '1503.02046-1-11-4': 'As expected, the EP-only cavity that had no final mild bake showed prominent HFQS while the performance of the EP+[MATH] baked cavity was free from the HFQS.', '1503.02046-1-12-0': 'Temperature maps of both cavities recorded at [MATH] MV/m, which is above the HFQS onset field, are shown in Fig. [MATH].', '1503.02046-1-12-1': 'The unbaked cavity shows large regions of elevated temperature (up to 0.4 K) - a standard feature of the HFQS - as compared to the cavity that was baked at [MATH], which had no such extended dissipative regions.', '1503.02046-1-12-2': 'Based on the temperature maps, samples with characteristic rf field dependences for each of the treatments (shown in Fig. [MATH]) were cut out from both cavities.', '1503.02046-1-12-3': 'The selected characteristic sample from the unbaked cavity (labeled ""EP"" for the following) shows a drastic increase of the local temperature as the surface rf field amplitude reaches about 100 mT. This is in contrast with the characteristic cutout from the [MATH] baked cavity (labeled ""EP120C"" for the following), which shows no such feature.', '1503.02046-1-12-4': 'Cutouts were of circular shape with 11 mm diameter and were extracted from the cavities by an automated milling machine with pure water used as a lubricant.', '1503.02046-1-13-0': 'Additionally, two further samples from the unbaked cavity were subjected to 800[MATH]C vacuum degassing for 3 hours, and 20 [MATH]m buffered chemical polishing.', '1503.02046-1-14-0': '## Characterization of cavity cutouts', '1503.02046-1-15-0': 'Cross sectional TEM samples were prepared from cutout samples by Focused Ion Beam (FIB) using a Helios 600 FEI instrument.', '1503.02046-1-15-1': 'Conventional polishing methods are not acceptable for Nb cavity investigations because polishing loads the sample with hydrogen.', '1503.02046-1-15-2': 'FIB lift-out technique allows one to prepare and mount a small rectangular cross sectional sample onto a standard copper TEM half-grid using an Omniprobe micromanipulator.', '1503.02046-1-15-3': 'Before milling, the top surface of each cross sectional cut was covered by a protective layer of platinum, in order to preserve the native niobium surface from Ga ions (Fig. [MATH]).', '1503.02046-1-15-4': 'Most FIB samples were intentionally prepared as bi-crystals in order to overcome possible tilting limitations of the TEM holder.', '1503.02046-1-15-5': 'For the temperature dependent electron diffraction experiments, every sample was used only once.', '1503.02046-1-15-6': 'This precaution was made to avoid possible niobium hydride nucleation centers that could be artificially produced during the fast warm up inside the TEM.', '1503.02046-1-16-0': 'Two types of TEMs were used for this work: field-emission gun (FEG) TEM and thermionic LaB[MATH] gun TEM.', '1503.02046-1-16-1': 'The key difference between these two microscopes is the higher brightness of FEG relative to the thermionic LaB[MATH] gun.', '1503.02046-1-16-2': 'Brightness, which defines the electron dose, is a crucial parameter for the investigation of dose-sensitive niobium nanohydrides, as will be described below.', '1503.02046-1-17-0': 'JEM 2010F Schottky FEG TEM at Materials Research Laboratory (MRL) at the University of Illinois at Urbana-Champaign (UIUC) operated at 197kV and equipped with a Gatan imaging filter (GIF), was used for the temperature dependent NED, SAED, and room temperature EELS.', '1503.02046-1-17-1': 'An approximately 80 nm sized NED parallel beam probe was used to record diffraction patterns onto Fuji imaging plates.', '1503.02046-1-17-2': 'EELS was collected in Scanning Transmission Electron Microscopy (STEM) mode.', '1503.02046-1-17-3': 'Energy dispersion was set to 0.3 eV/pixel for core-loss EELS niobium M-edge spectra.', '1503.02046-1-17-4': 'EELS collection time was set to 10-12 s for each of the spectra.', '1503.02046-1-18-0': 'JEOL JEM 2100 LaB[MATH] thermionic gun TEM at MRL/UIUC was used for temperature dependent NED and scanning electron nano-area diffraction (SEND) [CITATION].', '1503.02046-1-18-1': 'In SEND, approximately 170 nm and 100 nm probe sizes were used to obtain diffraction pattern ""maps"" by automated rastering of a parallel beam probe across the area of interest on a FIB sample.', '1503.02046-1-18-2': 'A SEND experiment produces a map of the diffraction patterns which represent a phase in a particular area of the sample.', '1503.02046-1-18-3': 'Automated positioning of the probe was achieved by using a custom-made Digital Micrograph script.', '1503.02046-1-18-4': 'Diffraction patterns were taken in sequential manner from a specific area of the sample which was imaged every time prior to scanning.', '1503.02046-1-18-5': 'Diffraction patterns were recorded using Gatan Ultrascan CCD camera, which is optimized for low contrast biological applications.', '1503.02046-1-18-6': 'The step length of the scans was set equal to the probe diameter to avoid oversampling and gaps in diffraction maps.', '1503.02046-1-19-0': 'JEOL JEM-2100 FasTEM at Northwestern University operated at 200 keV and equipped with GIF was used for room temperature SAED and high resolution TEM (HRTEM) imaging.', '1503.02046-1-20-0': 'Gatan liquid nitrogen cooled double-tilt stage was used for the low temperature measurements.', '1503.02046-1-20-1': 'FIB-prepared cavity cutout samples were cooled to 94 K inside the TEM in approximately 30 min.', '1503.02046-1-20-2': 'Additional 30 min was allowed before the measurements for temperature stabilization.', '1503.02046-1-20-3': 'SRF cavities have operational temperatures in the range of 1.2-4.2 K while all hydride precipitation happens during the cool-down at much higher temperatures of [MATH]100-150 K as follows from NbH phase diagram [CITATION] and confirmed by recent direct optical investigations [CITATION].', '1503.02046-1-20-4': 'Thus observing at 94 K is fully representative of the hydride state inside niobium at lower temperatures.', '1503.02046-1-21-0': '# results and discussion', '1503.02046-1-22-0': '## Temperature-dependent structural investigations', '1503.02046-1-23-0': '### Room temperature measurements', '1503.02046-1-24-0': 'Room temperature NED and SAED patterns were first acquired on all of the samples in order to investigate the state of the Nb-H system in the warm state.', '1503.02046-1-24-1': 'NED patterns were taken with a probe size of approximately 80 nm in diameter and areas directly underneath the niobium oxides, as well as areas a few hundred nanometers deep were explored.', '1503.02046-1-24-2': 'Similar diffraction patterns produced by body centered cubic (BCC) niobium with no additional ordered stoichiometric phases as shown in Fig. [MATH] were found on all the samples we investigated.', '1503.02046-1-24-3': 'SAED (not shown), which represents structural information from sample areas of a few micrometers, shows only BCC Nb reflections as well.', '1503.02046-1-24-4': 'Thus TEM electron diffraction shows that hydrogen behaves like a lattice gas and occupies random tetrahedral interstitial sites in Nb at room temperature.', '1503.02046-1-24-5': 'This phase is called solid solution ([MATH]-phase).', '1503.02046-1-25-0': '### Cryogenic temperature measurements', '1503.02046-1-26-0': 'Before TEM measurements, in order to confirm the absence of large bulk concentrations of hydrogen, cutouts from the same cavities were investigated in the optical cryogenic stage of the confocal microscope using the same methodology as in our previous study of [MATH] disease-causing larger hydrides [CITATION].', '1503.02046-1-26-1': 'No hydride formation was seen on any of the cutouts with the spatial resolution down to [MATH]m.', '1503.02046-1-27-0': 'Cryogenic temperature phase characterization of EP and EP120C samples was accomplished with SEND in thermionic gun TEM and with NED, SAED in FEG TEM.', '1503.02046-1-27-1': 'Fig. [MATH] shows a SEND map taken from the EP sample along with a TEM image of the sample at 94K.', '1503.02046-1-27-2': 'NED patterns were taken automatically in a sequential manner from the Nb near-surface region which was imaged prior to scanning.', '1503.02046-1-27-3': 'Every square in Fig. [MATH]a represents a sample area of diameter equal to the diameter of the diffraction probe.', '1503.02046-1-28-0': 'Fig. [MATH]b-d show NED representative patterns taken from the ""EP"" sample.', '1503.02046-1-28-1': '""Half-order"" additional reflections are clearly visible along with reflections from Nb matrix.', '1503.02046-1-28-2': 'Orientation of Nb crystal is close to [110] zone axis.', '1503.02046-1-28-3': 'Two niobium hydride phases were found in EP sample at 94K.', '1503.02046-1-29-0': 'Fig. [MATH]b shows [MATH]-phase niobium hydride diffraction pattern overlapped with [110] Nb.', '1503.02046-1-29-1': '[MATH]-phase was recognized by ""half-order"" reflections along the [110][MATH] direction [CITATION].', '1503.02046-1-29-2': '[MATH]-phase (Nb[MATH]H[MATH]) has non-centrosymmetric orthorhombic structure with [MATH] space group [CITATION].', '1503.02046-1-29-3': 'The [MATH]-phase forms in [MATH] + [MATH] alloys, by ordering of [MATH]-phase, when H/Nb [MATH] 0.7 at 207K [CITATION].', '1503.02046-1-29-4': 'The orientation of observed [MATH]-phase domains is close to the [114] Nb[MATH]H[MATH] zone axis.', '1503.02046-1-30-0': 'Fig. [MATH]c shows a [MATH]-phase niobium hydride diffraction pattern overlapped with [110] Nb.', '1503.02046-1-30-1': '[MATH]-phase was recognized by reflections at [MATH] in terms of cubic BCC Nb reflections [CITATION].', '1503.02046-1-30-2': '[MATH]-phase forms by ordering of the hydrogen interstitials on tetrahedral sites which lie on alternate (1[MATH] planes upon cooling over the composition range of 0.75 [MATH] H/Nb [MATH] 1.0 [CITATION].', '1503.02046-1-30-3': '[MATH]-phase (NbH) has face centered orthorhombic crystal structure with [MATH] space group [CITATION].', '1503.02046-1-30-4': 'The orientation of [MATH]-phase domains is close to the [100] NbH zone axis.', '1503.02046-1-31-0': 'SEND mapping of EP120C sample at 94K demonstrated only BCC Nb diffraction pattern with no additional reflections.', '1503.02046-1-31-1': 'One EP sample and one EP120C sample were evaluated by SEND at 94K.', '1503.02046-1-32-0': 'One unbaked cavity sample which received 800[MATH]C vacuum degassing for 3 hours, and 20 [MATH]m buffered chemical polishing (BCP), was evaluated by SEND at 94K.', '1503.02046-1-32-1': 'SEND mapping (Fig. [MATH]) shows the presence of the same low temperature niobium hydride phases as for the unbaked cavity sample without 800[MATH]C degassing followed by BCP.', '1503.02046-1-32-2': 'The diagram in Fig. [MATH] summarizes SEND mapping at 94K for all three types of samples: EP, EP which received 800[MATH]C degassing and BCP, and EP120C.', '1503.02046-1-33-0': 'Additional NED structural characterization of the cavity cutouts was performed in FEG TEM at 94 K.', '1503.02046-1-33-1': 'In order to collect diffraction patterns from the near-surface area, the NED probe was positioned by the deflection coils onto the TEM sample for each exposure.', '1503.02046-1-33-2': 'The size of the NED probe was approximately 80 nm.', '1503.02046-1-33-3': 'NED diffraction patterns were collected by sequentially moving the probe along the length of the sample, which is schematically represented in Fig. [REF]a. Fig. [REF] (a)-(c) and (d)-(f) show typical cryogenic temperatures NED patterns taken from EP and EP120C samples, respectively.', '1503.02046-1-33-4': 'Additional second phase reflections are clearly observed along with Nb matrix reflections at 94 K in both types of samples.', '1503.02046-1-33-5': 'Additional low temperature reflections in EP samples are more intense and frequent than additional reflections in EP120C samples.', '1503.02046-1-33-6': 'Comparing EP and EP120C samples, the number of probed spots exhibiting reflections of an additional low-temperature phase differ as shown in Fig. [REF].', '1503.02046-1-33-7': 'For the EP samples, 68% of probed spots showed additional reflections, whereas for the EP120C samples, 27% of the probed spots showed additional reflections.', '1503.02046-1-33-8': 'Three FIB prepared EP and three EP120C samples were investigated with NED.', '1503.02046-1-34-0': 'Most of the detected second phase reflections were not in compliance with any reported phase of niobium hydride.', '1503.02046-1-34-1': 'Only a few diffraction patterns taken from the EP samples show clear ""half-order"" reflections which can be associated with [MATH]- and [MATH]-phases of niobium hydrides (Fig. [REF]b,c).', '1503.02046-1-34-2': 'This can be explained in terms of dissociation of the native low temperature niobium hydride phases under the electron beam exposure.', '1503.02046-1-34-3': 'It has been noticed that additional low-temperature reflections rapidly vanish under exposure to the electron beam.', '1503.02046-1-34-4': 'Heating of the exposed area by electron bombardment allows hydrogen to regain its mobility and move to different parts of the sample, which can lead to severe distortion and dissociation of niobium hydrides.', '1503.02046-1-34-5': 'We believe that by the time the exposure was taken, the structure of the second phase could be already altered.', '1503.02046-1-34-6': 'A similar effect was previously observed by several researchers [CITATION].', '1503.02046-1-34-7': 'Due to this effect, relevant zone axis tilts were determined prior to cooling, and set up with the minimal sample exposure at low temperatures.', '1503.02046-1-35-0': 'Formation of low temperature stoichiometric niobium hydrides in Nb samples was previously detected by SAED for various hydrogen concentrations [CITATION].', '1503.02046-1-35-1': 'However, low temperature SAED on the samples prepared from the cavity cutouts did not reveal any additional reflections.', '1503.02046-1-35-2': 'The absence of an additional reflections in SAED patterns can be explained either by negligibly small SAED signal from nano-scale niobium hydrides or by their fast dissociation under the broad electron beam.', '1503.02046-1-36-0': '## Grain boundaries and surface oxides', '1503.02046-1-37-0': 'HRTEM imaging and EELS were used for detailed imaging and comparison of the surface oxides and grain boundaries in EP120C and EP samples.', '1503.02046-1-37-1': 'The appearance of an approximately 5 nm-thick amorphous oxide layer (Nb[MATH]O[MATH]) in HRTEM images is similar for both types of samples (Fig. [REF]).', '1503.02046-1-38-0': 'EELS investigations of the surface oxides in the EP120C and EP samples were performed as a way to make detailed comparisons of niobium valence across the oxide layer.', '1503.02046-1-39-0': 'Fig. [REF]a shows EELS spectra for niobium M[MATH] edge.', '1503.02046-1-39-1': 'EELS spectra were taken for the four regions marked in the STEM image of the niobium near surface (Fig. [REF]b).', '1503.02046-1-39-2': 'The M[MATH] edge of niobium is a result of the transition of Nb 3p electrons to unoccupied Nb 4d and 5s states.', '1503.02046-1-39-3': 'Spin-orbit coupling of the 3p orbital causes the appearance of two peaks (M[MATH] and M[MATH]).', '1503.02046-1-39-4': 'All niobium core loss spectra were calibrated with respect to carbon K-edge onset at 286 eV using the second derivative method [CITATION].', '1503.02046-1-39-5': 'Three spectra for each region were added after the background subtraction with log-polynomial function [CITATION].', '1503.02046-1-39-6': 'Thickness of the sample in the region of interest was estimated to be 41 nm.', '1503.02046-1-40-0': 'The linear relationship between the chemical shift of M-edge onset and niobium valence can be used to determine the niobium oxidation state [CITATION].', '1503.02046-1-40-1': 'For both samples, the M[MATH] peak for each region shows a clear chemical shift toward higher energy as a function of distance from the Nb metallic surface.', '1503.02046-1-40-2': 'The EP120C sample shows a greater shift than the EP sample.', '1503.02046-1-40-3': 'The observed shifts for both the EP and EP120C samples agree well with previous results of Tao et al [CITATION].', '1503.02046-1-40-4': 'Table [REF] summarizes the positions of M[MATH] peaks for both samples for each region.', '1503.02046-1-40-5': 'The last column shows the difference in position from region 1 to region 4.', '1503.02046-1-40-6': 'The M[MATH] peaks follow the same trend as the M[MATH] peaks.', '1503.02046-1-40-7': 'The comparatively larger shifts of the M[MATH] peaks of the EP120C sample is an indication of higher niobium valence in each region relative to that in the EP sample.', '1503.02046-1-40-8': 'This suggests inward (toward the bulk of Nb) oxygen diffusion during the mild bake.', '1503.02046-1-41-0': 'According to X-ray investigations of Nb/Nb-oxide interfaces [CITATION], an increase of [MATH] for NbO[MATH] underneath Nb[MATH]O[MATH] in EP120C sample can be caused by the enrichment in interstitial oxygen.', '1503.02046-1-42-0': 'Several samples with uniformly thin grain boundary regions were prepared from the cavity cutouts.', '1503.02046-1-42-1': 'Images of the grain boundaries in the EP and EP120C spot samples are represented in Fig. [REF].', '1503.02046-1-42-2': 'HRTEM images of the cavity cutouts do not show an amorphous contrast from the isolating niobium pentoxide along the grain boundaries, in contradiction to some literature models [CITATION].', '1503.02046-1-43-0': '## Comparison of dislocation structure', '1503.02046-1-44-0': 'Appearance of dislocations produced by the precipitation of niobium hydrides in Nb samples after the first cool down was reported in a number of studies.', '1503.02046-1-44-1': 'Therefore, EP spot samples that suffer from more prominent niobium hydride precipitation at low temperatures can possess higher dislocation density in the near-surface layer at room temperature.', '1503.02046-1-44-2': 'To look for this secondary effect, we used HRTEM and Bright Field imaging to compare dislocation content in the EP and EP120C samples.', '1503.02046-1-44-3': 'Fig.[REF] shows HRTEM images of EP120C and EP samples under [111] Nb zone axis.', '1503.02046-1-44-4': 'Diffraction contrast in Bright Field (BF) images of the EP120C and EP samples confirms a large amount of dislocations in both, which appear as dark streaks and spots (Fig.[REF]).', '1503.02046-1-44-5': 'This large number of pre-existing dislocations is likely a result of extensive plastic deformation of niobium introduced during cavity manufacturing steps (i.e. deep drawing).', '1503.02046-1-44-6': 'Such high dislocation density leads to complicated bending effects which make atomic column projections go in and out of focus in HRTEM images, which made it impossible to discern any effect of NbH precipitation.', '1503.02046-1-45-0': '# Discussion', '1503.02046-1-46-0': 'Presence and possible involvement of nanoscale niobium hydrides in the HFQS mechanism was recently proposed [CITATION] but gaining direct evidence of their existence remained a challenging task.', '1503.02046-1-46-1': 'One of the primary findings of our work is the clear demonstration in TEM by NEG and SEND that such nanohydrides do in fact exist, and therefore may indeed be the possible cause of the HFQS.', '1503.02046-1-46-2': 'In general, such nanohydrides represent a yet unaccounted for extrinsic mechanism of additional rf dissipation in hydrogen [MATH]-disease free SRF cavities, and the full range of their effect on the whole [MATH] curve has to be further understood.', '1503.02046-1-47-0': 'The significantly lower area affected by nanohydride precipitation in EP120C cutouts found by the near-surface NED investigations is consistent with the hydride precipitation suppression by the 120[MATH]C baking, as proposed recently [CITATION].', '1503.02046-1-47-1': 'At this stage, it is not yet possible to definitively say if it is the volume density or the size of the nanohydrides, which is affected.', '1503.02046-1-47-2': 'However, the removal of the HFQS suggests that it is likely the size.', '1503.02046-1-48-0': 'Another key finding is the reapperance of high nanohydride population in the cutout sample after additional 800[MATH]C vacuum treatment for 3 hours followed by 20 [MATH]m buffered chemical polishing.', '1503.02046-1-48-1': 'This is a standard processing sequence to guarantee the absence of the [MATH] disease in SRF cavities, which works by drastically lowering the bulk hydrogen content.', '1503.02046-1-48-2': 'However, as our results confirm, there is still enough hydrogen near surface - likely due to hydrogen reabsorption in the furnace during cool down and BCP - to cause the formation of nanohydrides.', '1503.02046-1-49-0': 'It was discussed in the past that niobium oxide structure may also get modified during the 120[MATH]C bake in several different ways [CITATION].', '1503.02046-1-49-1': 'Our investigations show that amorphous Nb[MATH]O[MATH] of about 5 nm thick is very similar in both EP and EP120C cutouts.', '1503.02046-1-49-2': 'The slightly increased niobium oxidation state in EP120C is consistent with the increased oxygen concentration right underneath the oxide, as found before by Delheusy et al [CITATION].', '1503.02046-1-49-3': 'This increased oxygen concentration may be a reason of the [MATH]1-2 n[MATH] higher residual resistance in 120[MATH]C baked cavities, which can be restored to the pre-120[MATH]C bake level by the hydrofluoric acid rinse [CITATION] since the oxygen-reach layer gets converted to the newly grown oxide.', '1503.02046-1-50-0': 'Finally, a very important finding is lack of any oxidation along grain boundaries.', '1503.02046-1-50-1': 'This contradicts a model of niobium surface, frequently used up to now, which suggests the presence of oxidized grain boundaries, crack corrosion, and isolated niobium suboxides islands [CITATION].', '1503.02046-1-50-2': 'Our investigations show that none of these features are present in SRF cavities.', '1503.02046-1-51-0': '# Summary and Conclusions', '1503.02046-1-52-0': 'Extensive microscopic comparison of the original cutouts from SRF cavities with and without HFQS was performed in order to elucidate the underlying cause of the HFQS and the mechanism of its cure.', '1503.02046-1-52-1': 'TEM comparison using cryogenic NED and SEND of EP and EP120C cutouts revealed for the first time the formation of the near-surface low temperature nanoscale niobium hydride phases, which area density and/or size directly correlates to the presence or absence of the HFQS.', '1503.02046-1-52-2': 'Mild [MATH]bake was demonstrated to reduce the amount of and/or change the distribution of niobium hydrides in the near-surface layer.', '1503.02046-1-52-3': 'Phase identification in SEND demonstrated the presence of [MATH]- and [MATH]-niobium hydrides in the EP cutout at 94K.', '1503.02046-1-53-0': 'Additional HRTEM and BF imaging, as well as EELS characterization of the cavity cutouts, were conducted in order to investigate any possible differences in grain boundaries, surface oxides, and dislocation structure.', '1503.02046-1-53-1': 'HRTEM investigation of grain boundaries showed no niobium pentoxide along the grain boundaries and a similar structure of grain boundaries in EP and EP120C samples.', '1503.02046-1-54-0': 'Identical thickness of the surface niobium pentoxide was found from HRTEM images of the EP and EP120C cutouts.', '1503.02046-1-54-1': 'EELS chemical characterization of the niobium oxidation state as a function of distance from the surface revealed that EP120C samples have higher chemical shifts for all regions, suggesting inward oxygen diffusion from the oxide into the bulk, which may be related to the hydrofluoric acid rinse beneficial effect on 120[MATH]C baked cavities.'}","{'1503.02046-2-0-0': 'Nanoscale defect structure within the magnetic penetration depth of [MATH]100 nm is key to the performance limitations of niobium superconducting radio frequency (SRF) cavities.', '1503.02046-2-0-1': 'Using a unique combination of advanced thermometry during cavity RF measurements, and TEM structural and compositional characterization of the samples extracted from cavity walls, we discover the existence of nanoscale hydrides in electropolished cavities limited by the high field [MATH] slope, and show the decreased hydride formation in the electropolished cavity after 120[MATH]C baking.', '1503.02046-2-0-2': 'Furthermore, we demonstrate that adding [MATH]C hydrogen degassing followed by light buffered chemical polishing restores the hydride formation to the pre-120[MATH]C bake level.', '1503.02046-2-0-3': 'We also show absence of niobium oxides along the grain boundaries and the modifications of the surface oxide upon 120[MATH]C bake.', '1503.02046-2-1-0': '# Introduction', '1503.02046-2-2-0': 'Superconducting radio frequency (SRF) cavities is the state-of-the-art technology for particle acceleration implemented in most modern and future planned accelerators [CITATION].', '1503.02046-2-2-1': 'SRF cavities are predominantly made of bulk niobium and are typically operated at temperatures of 2 K or below, deep in superconducting state of niobium, which has superconducting critical temperature [MATH] K.', '1503.02046-2-2-2': 'The performance of SRF cavities is characterized by the maximum accelerating field ([MATH]) they can sustain, and the cavity quality factor [MATH] determining their efficiency of operation.', '1503.02046-2-2-3': 'Lower [MATH] leads to the increased dynamic heat load for the cryogenic system, and, if severe, can even lead to the limitation in [MATH] as it causes an increase in the inner cavity wall temperature that can trigger the localized loss of superconductivity - quench.', '1503.02046-2-2-4': 'The magnitude of [MATH] is determined by the average microwave surface resistance [MATH], which consists of the strongly temperature dependent part [MATH] and a temperature independent (residual) component [MATH].', '1503.02046-2-3-0': 'Recent investigations showed that for standard cavity preparation techniques [CITATION] as well as for a newly discovered nitrogen doping [CITATION] both [MATH] and [MATH] depend on the surface rf magnetic field magnitude [MATH].', '1503.02046-2-3-1': 'Since these field dependencies are determined by the surface treatments and the magnetic field only penetrates [MATH]100 nm inside niobium in superconducting state at 2 K, the nanostructure within this thickness and its changes with treatments is key to understanding changes in surface resistance and [MATH].', '1503.02046-2-4-0': 'One of the long-standing puzzles is a strong increase in the surface resistance of electropolished cavities above [MATH]100 mT surface magnetic field - a so-called high field [MATH] slope (HFQS).', '1503.02046-2-4-1': 'The effect persists in the absence of other well-known parasitic losses such as multipacting and field emission.', '1503.02046-2-4-2': 'HFQS can be removed by the empirically found ""mild baking"" at [MATH]-[MATH] in ultra high vacuum (UHV) for 24-48 hours [CITATION].', '1503.02046-2-5-0': 'Several models for the HFQS were proposed in the past, but most were shown to contradict at least one of the experimental observations [CITATION].', '1503.02046-2-5-1': 'The most recent promising model is based on the formation of lossy niobium nanohydrides in the penetration depth [CITATION].', '1503.02046-2-5-2': 'Nanohydrides may remain superconducting due to the proximity effect up to the breakdown field, which is determined by their size.', '1503.02046-2-5-3': 'The model attributes HFQS onset field to such a loss of proximity-induced superconductivity, which manifests as a strong increase in residual resistance and causes HFQS.', '1503.02046-2-5-4': 'The rationale for this theory is the presence of high concentration of interstitial hydrogen in the penetration depth [CITATION], which, upon cooling to 2 K, may coalesce into lumps of niobium hydrides.', '1503.02046-2-5-5': 'A challenging part is that in order to search for such nanohydrides directly, cryo-investigations at [MATH]100 K are required as at room temperature no hydrides are present.', '1503.02046-2-6-0': 'The characteristic feature of the HFQS is the localization of strong additional dissipation in the areas of cavity surface corresponding to highest surface magnetic fields [CITATION], as found out by advanced temperature mapping studies of outside cavity walls.', '1503.02046-2-6-1': 'A very powerful approach is based on localizing the strongest dissipative spots, which then can be extracted from cavity walls and their inner surface studied by different analytical techniques.', '1503.02046-2-6-2': 'Comparing cutouts from the cavities with the HFQS to the ones from the mild baked cavities without HFQS allows drawing conclusions on the possible underlying causes of the HFQS, and the origin of the mild baking effect.', '1503.02046-2-7-0': 'Previous comparative studies on such cavity cutouts provided clues into possible mechanisms at play.', '1503.02046-2-7-1': 'Low energy muon spin rotation spectroscopy (LE-[MATH]SR) showed that mild baking leads to a strong decrease in the electron mean free path [CITATION].', '1503.02046-2-7-2': 'Variable energy positron annihilation spectroscopy (VEPAS) showed that inward diffusion of vacancies and the hydrogen-trapping effect of vacancies may be behind this [MATH] suppression [CITATION].', '1503.02046-2-7-3': 'Another alternative mechanism of the mean free path suppression may be based on the inward diffusion of oxygen [CITATION].', '1503.02046-2-7-4': 'Both oxygen and vacancies may then serve as trapping centers for hydrogen, preventing the formation of lossy nanohydrides [CITATION].', '1503.02046-2-8-0': 'In this particular work we present structural and analytic comparison of FIB-prepared cross-sectional samples from mild baked and unbaked cavity cutouts using various TEM techniques at room and cryogenic (94 K) temperatures.', '1503.02046-2-8-1': 'Temperature dependent nano-area electron diffraction (NED) and scanning electron nano-area diffraction (SEND) reveal the formation of stoichiometric non-superconducting small niobium hydride inclusions.', '1503.02046-2-8-2': 'Mild baking is shown to decrease nanohydride sizes and/or density, which directly correlates with the observed suppression of the high field [MATH] slope in SRF cavities.', '1503.02046-2-8-3': 'Importantly, we also find nanohydrides after additional 800[MATH]C degassing for 3 hours followed by buffered chemical polishing for 20 [MATH]m material removal.', '1503.02046-2-8-4': 'Furthermore, size and/or density of such hydrides are comparable to the pre-120[MATH]C bake electropolished cutout.', '1503.02046-2-8-5': 'Additionally, high resolution TEM (HRTEM) and bright field (BF) imaging show similar surface oxide thickness and lack of any oxidation along grain boundaries.', '1503.02046-2-8-6': 'Electron energy loss spectroscopy (EELS) chemical characterization of the surface oxides suggests slight oxygen enrichment just below the oxides after the mild bake, consistent with the previous X-ray investigations [CITATION].', '1503.02046-2-9-0': '# Methods', '1503.02046-2-10-0': '## Identification of the cutout regions in unbaked and 120[MATH]C baked cavities', '1503.02046-2-11-0': 'In order to directly correlate different dissipation characteristics with surface nanostructure and to determine the underlying mechanisms of HFQS in Nb SRF cavities, we base our studies on the comparison of cutouts from cavities with and without HFQS, similar to previous studies [CITATION].', '1503.02046-2-11-1': 'Two Nb fine grain ([MATH]m) TESLA shape cavities with resonant frequency [MATH]1.3 GHz were used.', '1503.02046-2-11-2': 'Both cavities were electropolished, and one of them was additionally baked at [MATH] for 48 hours.', '1503.02046-2-11-3': 'Dependence of the quality factor on peak surface magnetic field at 2 K was measured for both cavities, and both also had temperature maps acquired during rf measurements in order to identify the regions for cutout.', '1503.02046-2-11-4': 'As expected, the EP-only cavity that had no final mild bake showed prominent HFQS while the performance of the EP+[MATH] baked cavity was free from the HFQS.', '1503.02046-2-12-0': 'Temperature maps of both cavities recorded at [MATH] MV/m, which is above the HFQS onset field, are shown in Fig. [MATH].', '1503.02046-2-12-1': 'The unbaked cavity shows large regions of elevated temperature (up to 0.4 K) - a standard feature of the HFQS - as compared to the cavity that was baked at [MATH], which had no such extended dissipative regions.', '1503.02046-2-12-2': 'Based on the temperature maps, samples with characteristic rf field dependences for each of the treatments (shown in Fig. [MATH]) were cut out from both cavities.', '1503.02046-2-12-3': 'The selected characteristic sample from the unbaked cavity (labeled ""EP"" for the following) shows a drastic increase of the local temperature as the surface rf field amplitude reaches about 100 mT. This is in contrast with the characteristic cutout from the [MATH] baked cavity (labeled ""EP120C"" for the following), which shows no such feature.', '1503.02046-2-12-4': 'Cutouts were of circular shape with 11 mm diameter and were extracted from the cavities by an automated milling machine with pure water used as a lubricant.', '1503.02046-2-13-0': 'Additionally, two further samples from the unbaked cavity were subjected to 800[MATH]C vacuum degassing for 3 hours, and 20 [MATH]m buffered chemical polishing.', '1503.02046-2-14-0': '## Characterization of cavity cutouts', '1503.02046-2-15-0': 'Cross sectional TEM samples were prepared from cutout samples by Focused Ion Beam (FIB) using a Helios 600 FEI instrument.', '1503.02046-2-15-1': 'Conventional electrochemocal polishing methods are not acceptable for Nb cavity investigations because polishing loads the sample with hydrogen.', '1503.02046-2-15-2': 'The FIB lift-out technique allows one to prepare and mount a small rectangular cross sectional sample onto a standard copper TEM half-grid using an Omniprobe micromanipulator.', '1503.02046-2-15-3': 'Before FIB milling, the top surface of each cross sectional cut was covered by a protective layer of platinum, in order to protect the native niobium surface from Ga ions (Fig. [MATH]).', '1503.02046-2-15-4': 'Most FIB samples were intentionally prepared as Nb bi-crystals in order to overcome possible tilting limitations of the TEM holder.', '1503.02046-2-15-5': 'For every temperature dependent electron diffraction experiment, a new sample was prepared.', '1503.02046-2-15-6': 'This precaution was made to avoid possible niobium hydride nucleation centers that could be artificially produced during the fast warm up inside the TEM.', '1503.02046-2-16-0': 'Two types of TEMs were used for this work: field-emission gun (FEG) TEM and thermionic LaB[MATH] gun TEM.', '1503.02046-2-16-1': 'The key difference between these two microscopes is the higher brightness of FEG relative to the thermionic LaB[MATH] gun.', '1503.02046-2-16-2': 'Brightness, which defines the electron dose in a small probe, is a crucial parameter for the investigation of dose-sensitive niobium nanohydrides, as will be described below.', '1503.02046-2-17-0': 'The JEM 2010F Schottky FEG TEM at Materials Research Laboratory (MRL) at the University of Illinois at Urbana-Champaign (UIUC) operated at 197kV and equipped with a Gatan imaging filter (GIF), was used for the temperature dependent NED, SAED, and room temperature EELS.', '1503.02046-2-17-1': 'An approximately 80 nm sized parallel beam was used to record NED patterns onto the Fuji imaging plates.', '1503.02046-2-17-2': 'EELS was collected in Scanning Transmission Electron Microscopy (STEM) mode.', '1503.02046-2-17-3': 'Energy dispersion was set to 0.3 eV/pixel for the niobium M-edge core-loss EELS spectra.', '1503.02046-2-17-4': 'The EELS collection time was set to 10-12 s for each of the spectra.', '1503.02046-2-18-0': 'JEOL JEM 2100 LaB[MATH] thermionic gun TEM at MRL/UIUC was used for temperature dependent NED and scanning electron nano-area diffraction (SEND) [CITATION].', '1503.02046-2-18-1': 'In SEND, approximately 170 nm and 100 nm beam sizes were used to obtain diffraction pattern ""maps"" by automated rastering of a parallel beam across the area of interest on a FIB sample.', '1503.02046-2-18-2': 'A SEND experiment produces a map of the diffraction patterns which represent a phase in a particular area of the sample.', '1503.02046-2-18-3': 'Automated positioning of the beam was achieved by using a custom-made Digital Micrograph script.', '1503.02046-2-18-4': 'Diffraction patterns were taken sequentially from a specific area of the sample which was imaged prior to scanning.', '1503.02046-2-18-5': 'Diffraction patterns were recorded using a Gatan Ultrascan CCD camera, which is optimized for low contrast biological applications.', '1503.02046-2-18-6': 'The step length of the scans was set equal to the probe diameter to avoid oversampling and gaps in diffraction maps.', '1503.02046-2-19-0': 'JEOL JEM-2100 FasTEM at Northwestern University operated at 200 keV and equipped with GIF was used for room temperature SAED and high resolution TEM (HRTEM) imaging.', '1503.02046-2-20-0': 'Gatan liquid nitrogen cooled double-tilt stage was used for the low temperature measurements.', '1503.02046-2-20-1': 'FIB-prepared cavity cutout samples were cooled to 94 K inside the TEM in approximately 30 min.', '1503.02046-2-20-2': 'Additional 30 min was allowed before the measurements for temperature stabilization.', '1503.02046-2-20-3': 'SRF cavities have operational temperatures in the range of 1.2-4.2 K while all hydride precipitation happens during the cool-down at much higher temperatures of [MATH]100-150 K as follows from NbH phase diagram [CITATION] and confirmed by recent direct optical investigations [CITATION].', '1503.02046-2-20-4': 'Thus observing at 94 K is fully representative of the hydride state inside niobium at lower temperatures.', '1503.02046-2-21-0': '# results and discussion', '1503.02046-2-22-0': '## Temperature-dependent structural investigations', '1503.02046-2-23-0': '### Room temperature measurements', '1503.02046-2-24-0': 'Room temperature NED and SAED patterns were first acquired on all of the samples in order to investigate the state of the Nb-H system in the warm state.', '1503.02046-2-24-1': 'NED patterns were taken with a probe size of approximately 80 nm in diameter and areas directly underneath the niobium oxides, as well as areas a few hundred nanometers deep were explored.', '1503.02046-2-24-2': 'Similar diffraction patterns produced by body centered cubic (BCC) niobium with no additional ordered stoichiometric phases as shown in Fig. [MATH] were found on all the samples we investigated.', '1503.02046-2-24-3': 'SAED (not shown), which represents structural information from sample areas of a few micrometers, shows only BCC Nb reflections as well.', '1503.02046-2-24-4': 'Thus TEM electron diffraction shows that hydrogen behaves like a lattice gas and occupies random tetrahedral interstitial sites in Nb at room temperature.', '1503.02046-2-24-5': 'This phase is called solid solution ([MATH]-phase).', '1503.02046-2-25-0': '### Cryogenic temperature measurements', '1503.02046-2-26-0': 'Before TEM measurements, in order to confirm the absence of large bulk concentrations of hydrogen, cutouts from the same cavities were investigated in the optical cryogenic stage of the confocal microscope using the same methodology as in our previous study of [MATH] disease-causing larger hydrides [CITATION].', '1503.02046-2-26-1': 'No hydride formation was seen on any of the cutouts with the spatial resolution down to [MATH]m.', '1503.02046-2-27-0': 'Cryogenic temperature phase characterization of EP and EP120C samples was accomplished with SEND in thermionic gun TEM and with NED, SAED in FEG TEM.', '1503.02046-2-27-1': 'Fig. [MATH] shows a SEND map taken from the EP sample along with a TEM image of the sample at 94K.', '1503.02046-2-27-2': 'NED patterns were taken automatically in a sequential manner from the Nb near-surface region which was imaged prior to scanning.', '1503.02046-2-27-3': 'Every square in Fig. [MATH]a represents a sample area of diameter equal to the diameter of the diffraction probe.', '1503.02046-2-28-0': 'Fig. [MATH]b-d show NED representative patterns taken from the ""EP"" sample.', '1503.02046-2-28-1': '""Half-order"" additional reflections are clearly visible along with reflections from Nb matrix.', '1503.02046-2-28-2': 'Orientation of Nb crystal is close to [110] zone axis.', '1503.02046-2-28-3': 'Two niobium hydride phases were found in EP sample at 94K.', '1503.02046-2-29-0': 'Fig. [MATH]b shows [MATH]-phase niobium hydride diffraction pattern overlapped with [110] Nb.', '1503.02046-2-29-1': '[MATH]-phase was recognized by ""half-order"" reflections along the [110][MATH] direction [CITATION].', '1503.02046-2-29-2': '[MATH]-phase (Nb[MATH]H[MATH]) has non-centrosymmetric orthorhombic structure with [MATH] space group [CITATION].', '1503.02046-2-29-3': 'The [MATH]-phase forms in [MATH] + [MATH] alloys, by ordering of [MATH]-phase, when H/Nb [MATH] 0.7 at 207K [CITATION].', '1503.02046-2-29-4': 'The orientation of observed [MATH]-phase domains is close to the [114] Nb[MATH]H[MATH] zone axis.', '1503.02046-2-30-0': 'Fig. [MATH]c shows a [MATH]-phase niobium hydride diffraction pattern overlapped with [110] Nb.', '1503.02046-2-30-1': '[MATH]-phase was recognized by reflections at [MATH] in terms of cubic BCC Nb reflections [CITATION].', '1503.02046-2-30-2': '[MATH]-phase forms by ordering of the hydrogen interstitials on tetrahedral sites which lie on alternate (1[MATH] planes upon cooling over the composition range of 0.75 [MATH] H/Nb [MATH] 1.0 [CITATION].', '1503.02046-2-30-3': '[MATH]-phase (NbH) has face centered orthorhombic crystal structure with [MATH] space group [CITATION].', '1503.02046-2-30-4': 'The orientation of [MATH]-phase domains is close to the [100] NbH zone axis.', '1503.02046-2-31-0': 'SEND mapping of EP120C sample at 94K demonstrated only BCC Nb diffraction pattern with no additional reflections.', '1503.02046-2-31-1': 'One EP sample and one EP120C sample were evaluated by SEND at 94K.', '1503.02046-2-32-0': 'One unbaked cavity sample which received 800[MATH]C vacuum degassing for 3 hours, and 20 [MATH]m buffered chemical polishing (BCP), was evaluated by SEND at 94K.', '1503.02046-2-32-1': 'SEND mapping (Fig. [MATH]) shows the presence of the same low temperature niobium hydride phases as for the unbaked cavity sample without 800[MATH]C degassing followed by BCP.', '1503.02046-2-32-2': 'The diagram in Fig. [MATH] summarizes SEND mapping at 94K for all three types of samples: EP, EP which received 800[MATH]C degassing and BCP, and EP120C.', '1503.02046-2-33-0': 'Additional NED structural characterization of the cavity cutouts was performed in FEG TEM at 94 K.', '1503.02046-2-33-1': 'In order to collect diffraction patterns from the near-surface area, the NED probe was positioned by the deflection coils onto the TEM sample for each exposure.', '1503.02046-2-33-2': 'The size of the NED probe was approximately 80 nm.', '1503.02046-2-33-3': 'NED diffraction patterns were collected by sequentially moving the probe along the length of the sample, which is schematically represented in Fig. [REF]a. Fig. [REF] (a)-(c) and (d)-(f) show typical cryogenic temperatures NED patterns taken from EP and EP120C samples, respectively.', '1503.02046-2-33-4': 'Additional second phase reflections are clearly observed along with Nb matrix reflections at 94 K in both types of samples.', '1503.02046-2-33-5': 'Additional low temperature reflections in EP samples are more intense and frequent than additional reflections in EP120C samples.', '1503.02046-2-33-6': 'Comparing EP and EP120C samples, the number of probed spots exhibiting reflections of an additional low-temperature phase differ as shown in Fig. [REF].', '1503.02046-2-33-7': 'For the EP samples, 68% of probed spots showed additional reflections, whereas for the EP120C samples, 27% of the probed spots showed additional reflections.', '1503.02046-2-33-8': 'Three FIB prepared EP and three EP120C samples were investigated with NED.', '1503.02046-2-34-0': 'Most of the detected second phase reflections were not in compliance with any reported phase of niobium hydride.', '1503.02046-2-34-1': 'Only a few diffraction patterns taken from the EP samples show clear ""half-order"" reflections which can be associated with [MATH]- and [MATH]-phases of niobium hydrides (Fig. [REF]b,c).', '1503.02046-2-34-2': 'This can be explained in terms of dissociation of the native low temperature niobium hydride phases under the electron beam exposure.', '1503.02046-2-34-3': 'It has been noticed that additional low-temperature reflections rapidly vanish under exposure to the electron beam.', '1503.02046-2-34-4': 'Heating of the exposed area by electron bombardment or direct electron energy transfer allow hydrogen to regain its mobility and move to different parts of the sample, which can lead to severe distortion and dissociation of niobium hydrides.', '1503.02046-2-34-5': 'We believe that by the time the exposure was taken, the structure of the second phase could be already altered.', '1503.02046-2-34-6': 'A similar effect was previously observed by several researchers [CITATION].', '1503.02046-2-34-7': 'Due to this effect, relevant zone axis tilts were determined prior to cooling, and set up with the minimal sample exposure at low temperatures.', '1503.02046-2-35-0': 'Formation of low temperature stoichiometric niobium hydrides in Nb samples was previously detected by SAED for various hydrogen concentrations [CITATION].', '1503.02046-2-35-1': 'However, low temperature SAED on the samples prepared from the cavity cutouts did not reveal any additional reflections.', '1503.02046-2-35-2': 'The absence of an additional reflections in SAED patterns can be explained either by negligibly small SAED signal from nano-scale niobium hydrides or by their fast dissociation under the broad electron beam.', '1503.02046-2-36-0': '## Grain boundaries and surface oxides', '1503.02046-2-37-0': 'HRTEM imaging and EELS were used for detailed imaging and comparison of the surface oxides and grain boundaries in EP120C and EP samples.', '1503.02046-2-37-1': 'The appearance of an approximately 5 nm-thick amorphous oxide layer (Nb[MATH]O[MATH]) in HRTEM images is similar for both types of samples (Fig. [REF]).', '1503.02046-2-38-0': 'EELS investigations of the surface oxides in the EP120C and EP samples were performed as a way to make detailed comparisons of niobium valence across the oxide layer.', '1503.02046-2-39-0': 'Fig. [REF]a shows EELS spectra for niobium M[MATH] edge.', '1503.02046-2-39-1': 'EELS spectra were taken for the four regions marked in the STEM image of the niobium near surface (Fig. [REF]b).', '1503.02046-2-39-2': 'The M[MATH] edge of niobium is a result of the transition of Nb 3p electrons to unoccupied Nb 4d and 5s states.', '1503.02046-2-39-3': 'Spin-orbit coupling of the 3p orbital causes the appearance of two peaks (M[MATH] and M[MATH]).', '1503.02046-2-39-4': 'All niobium core loss spectra were calibrated with respect to carbon K-edge onset at 286 eV using the second derivative method [CITATION].', '1503.02046-2-39-5': 'Three spectra for each region were added after the background subtraction with log-polynomial function [CITATION].', '1503.02046-2-39-6': 'Thickness of the sample in the region of interest was estimated to be 41 nm from the Log-Ratio Method [CITATION].', '1503.02046-2-40-0': 'The linear relationship between the chemical shift of M-edge onset and niobium valence can be used to determine the niobium oxidation state [CITATION].', '1503.02046-2-40-1': 'For both samples, the M[MATH] peak for each region shows a clear chemical shift toward higher energy as a function of distance from the Nb metallic surface.', '1503.02046-2-40-2': 'The EP120C sample shows a greater shift than the EP sample.', '1503.02046-2-40-3': 'The observed shifts for both the EP and EP120C samples agree well with previous results of Tao et al [CITATION].', '1503.02046-2-40-4': 'Table [REF] summarizes the positions of M[MATH] peaks for both samples for each region.', '1503.02046-2-40-5': 'The last column shows the difference in position from region 1 to region 4.', '1503.02046-2-40-6': 'The M[MATH] peaks follow the same trend as the M[MATH] peaks.', '1503.02046-2-40-7': 'The comparatively larger shifts of the M[MATH] peaks of the EP120C sample is an indication of higher niobium valence in each region relative to that in the EP sample.', '1503.02046-2-40-8': 'This suggests inward (toward the bulk of Nb) oxygen diffusion during the mild bake.', '1503.02046-2-41-0': 'According to X-ray investigations of Nb/Nb-oxide interfaces [CITATION], an increase of [MATH] for NbO[MATH] underneath Nb[MATH]O[MATH] in EP120C sample can be caused by the enrichment in interstitial oxygen.', '1503.02046-2-42-0': 'Several samples with uniformly thin grain boundary regions were prepared from the cavity cutouts.', '1503.02046-2-42-1': 'Images of the grain boundaries in the EP and EP120C spot samples are represented in Fig. [REF].', '1503.02046-2-42-2': 'HRTEM images of the cavity cutouts do not show an amorphous contrast from the isolating niobium pentoxide along the grain boundaries, in contradiction to some literature models [CITATION].', '1503.02046-2-43-0': '## Comparison of dislocation structure', '1503.02046-2-44-0': 'Appearance of dislocations produced by the precipitation of niobium hydrides in Nb samples after the first cool down was reported in a number of studies.', '1503.02046-2-44-1': 'Therefore, EP spot samples that suffer from more prominent niobium hydride precipitation at low temperatures can possess higher dislocation density in the near-surface layer at room temperature.', '1503.02046-2-44-2': 'To look for this secondary effect, we used HRTEM and Bright Field imaging to compare dislocation content in the EP and EP120C samples.', '1503.02046-2-44-3': 'Fig.[REF] shows HRTEM images of EP120C and EP samples under [111] Nb zone axis.', '1503.02046-2-44-4': 'Diffraction contrast in Bright Field (BF) images of the EP120C and EP samples confirms a large amount of dislocations in both, which appear as dark streaks and spots (Fig.[REF]).', '1503.02046-2-44-5': 'This large number of pre-existing dislocations is likely a result of extensive plastic deformation of niobium introduced during cavity manufacturing steps (i.e. deep drawing).', '1503.02046-2-44-6': 'Such high dislocation density leads to complicated bending effects which make atomic column projections go in and out of focus in HRTEM images, which made it impossible to discern any effect of NbH precipitation.', '1503.02046-2-45-0': '# Discussion', '1503.02046-2-46-0': 'Presence and possible involvement of nanoscale niobium hydrides in the HFQS mechanism was recently proposed [CITATION] but gaining direct evidence of their existence remained a challenging task.', '1503.02046-2-46-1': 'One of the primary findings of our work is the clear demonstration in TEM by NEG and SEND that such nanohydrides do in fact exist, and therefore may indeed be the possible cause of the HFQS.', '1503.02046-2-46-2': 'In general, such nanohydrides represent a yet unaccounted for extrinsic mechanism of additional rf dissipation in hydrogen [MATH]-disease free SRF cavities, and the full range of their effect on the whole [MATH] curve has to be further understood.', '1503.02046-2-47-0': 'The significantly lower area affected by nanohydride precipitation in EP120C cutouts found by the near-surface NED investigations is consistent with the hydride precipitation suppression by the 120[MATH]C baking, as proposed recently [CITATION].', '1503.02046-2-47-1': 'At this stage, it is not yet possible to definitively say if it is the volume density or the size of the nanohydrides, which is affected.', '1503.02046-2-47-2': 'However, the removal of the HFQS suggests that it is likely the size.', '1503.02046-2-48-0': 'Another key finding is the reapperance of high nanohydride population in the cutout sample after additional 800[MATH]C vacuum treatment for 3 hours followed by 20 [MATH]m buffered chemical polishing.', '1503.02046-2-48-1': 'This is a standard processing sequence to guarantee the absence of the [MATH] disease in SRF cavities, which works by drastically lowering the bulk hydrogen content.', '1503.02046-2-48-2': 'However, as our results confirm, there is still enough hydrogen near surface - likely due to hydrogen reabsorption in the furnace during cool down and BCP - to cause the formation of nanohydrides.', '1503.02046-2-49-0': 'It was discussed in the past that niobium oxide structure may also get modified during the 120[MATH]C bake in several different ways [CITATION].', '1503.02046-2-49-1': 'Our investigations show that amorphous Nb[MATH]O[MATH] of about 5 nm thick is very similar in both EP and EP120C cutouts.', '1503.02046-2-49-2': 'The slightly increased niobium oxidation state in EP120C is consistent with the increased oxygen concentration right underneath the oxide, as found before [CITATION].', '1503.02046-2-49-3': 'This increased oxygen concentration may be a reason of the [MATH]1-2 n[MATH] higher residual resistance in 120[MATH]C baked cavities, which can be restored to the pre-120[MATH]C bake level by the hydrofluoric acid rinse [CITATION] since the oxygen-reach layer gets converted to the newly grown oxide.', '1503.02046-2-50-0': 'Finally, a very important finding is lack of any oxidation along grain boundaries.', '1503.02046-2-50-1': 'This contradicts a model of niobium surface, frequently used up to now, which suggests the presence of oxidized grain boundaries, crack corrosion, and isolated niobium suboxides islands [CITATION].', '1503.02046-2-50-2': 'Our investigations show that none of these features are present in SRF cavities.', '1503.02046-2-51-0': '# Summary and Conclusions', '1503.02046-2-52-0': 'Extensive microscopic comparison of the original cutouts from SRF cavities with and without HFQS was performed in order to elucidate the underlying cause of the HFQS and the mechanism of its cure.', '1503.02046-2-52-1': 'TEM comparison using cryogenic NED and SEND of EP and EP120C cutouts revealed for the first time the formation of the near-surface low temperature nanoscale niobium hydride phases, which area density and/or size directly correlates to the presence or absence of the HFQS.', '1503.02046-2-52-2': 'Mild [MATH]bake was demonstrated to reduce the amount of and/or change the distribution of niobium hydrides in the near-surface layer.', '1503.02046-2-52-3': 'Phase identification in SEND demonstrated the presence of [MATH]- and [MATH]-niobium hydrides in the EP cutout at 94K.', '1503.02046-2-53-0': 'Additional HRTEM and BF imaging, as well as EELS characterization of the cavity cutouts, were conducted in order to investigate any possible differences in grain boundaries, surface oxides, and dislocation structure.', '1503.02046-2-53-1': 'HRTEM investigation of grain boundaries showed no niobium pentoxide along the grain boundaries and a similar structure of grain boundaries in EP and EP120C samples.', '1503.02046-2-54-0': 'Identical thickness of the surface niobium pentoxide was found from HRTEM images of the EP and EP120C cutouts.', '1503.02046-2-54-1': 'EELS chemical characterization of the niobium oxidation state as a function of distance from the surface revealed that EP120C samples have higher chemical shifts for all regions, suggesting inward oxygen diffusion from the oxide into the bulk, which may be related to the hydrofluoric acid rinse beneficial effect on 120[MATH]C baked cavities.'}","[['1503.02046-1-50-0', '1503.02046-2-50-0'], ['1503.02046-1-50-1', '1503.02046-2-50-1'], ['1503.02046-1-50-2', '1503.02046-2-50-2'], ['1503.02046-1-34-0', '1503.02046-2-34-0'], ['1503.02046-1-34-1', '1503.02046-2-34-1'], ['1503.02046-1-34-2', '1503.02046-2-34-2'], ['1503.02046-1-34-3', '1503.02046-2-34-3'], ['1503.02046-1-34-5', '1503.02046-2-34-5'], ['1503.02046-1-34-6', '1503.02046-2-34-6'], ['1503.02046-1-34-7', '1503.02046-2-34-7'], ['1503.02046-1-26-0', 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'1503.02046-2-15-2'], ['1503.02046-1-15-3', '1503.02046-2-15-3'], ['1503.02046-1-15-4', '1503.02046-2-15-4'], ['1503.02046-1-49-2', '1503.02046-2-49-2'], ['1503.02046-1-16-2', '1503.02046-2-16-2']]",[],"[['1503.02046-1-39-6', '1503.02046-2-39-6'], ['1503.02046-1-15-5', '1503.02046-2-15-5']]",[],[],"{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1503.02046,,,
0906.2617,"{'0906.2617-1-0-0': ""We investigate the asymptotic properties of a slight deformation of the Sen's Lagrangian for the scalar tachyon, for a wide class of self-interaction potentials."", '0906.2617-1-0-1': 'It is found that the present model of modified tachyon cosmology is equivalent to the standard one under an appropiate transformation of the tachyon.', '0906.2617-1-0-2': 'The above equivalence is used to extend the dynamical systems study of standard tachyon cosmology to a wider class of self-interaction potentials beyond the (inverse) square-law and power-law ones.', '0906.2617-1-0-3': 'It is revealled that independent of the functional form of the potential, the matter-dominated solution and the ultra-relativistic (also matter-dominated) solution, are always associated with equilibrium points in the phase space of the tachyon models.', '0906.2617-1-0-4': 'The latter is always the past attractor, while the former is a saddle critical point.', '0906.2617-1-0-5': 'For inverse power-law potentials [MATH] the late-time attractor is always the de Sitter solution, while for sinh-like potentials [MATH], depending on the region of parameter space, the late-time attractor can be either the inflationary tachyon-dominated solution or the matter-scaling (also inflationary) phase.', '0906.2617-1-1-0': '# Introduction', '0906.2617-1-2-0': 'Inflationary models of the universe have been studied from the string theory perspective because inflation provides an explanation for the homogeneity and isotropy in the early universe.', '0906.2617-1-2-1': 'Additionally, recent astrophysical observations indicate us that the universe is presently undergoing a phase of accelerated expansion that has been attributed to a peculiar kind of source of the Einstein\'s field equations aknowledged as ""dark energy"" [CITATION] The crucial feature of the dark energy which ensures an accelerated expansion of the universe is that it breaks the strong energy condition.', '0906.2617-1-2-2': 'The tachyon field arising in the context of string theory [CITATION] provides one example of matter which does the job.', '0906.2617-1-2-3': ""The tachyon has been intensively studied during the last few years also in application to cosmology [CITATION]-[CITATION]; in this case one usually takes Sen's effective Lagrangian density for granted [CITATION]: [EQUATION] and studies its cosmological consequences without worrying about the string-theoretical origin of the action itself."", '0906.2617-1-2-4': 'In the above equation [MATH] is the scalar tachyon field, [MATH]-its self-interaction potential, and [MATH].', '0906.2617-1-3-0': ""In the present paper we will be considering a slight modification of Sen's Lagrangian density ([REF]): [EQUATION] that is inspired in the recently proposed nonlinear scalar field theories of the Dirac-Born-Infeld (DBI) type [CITATION]-[CITATION]."", '0906.2617-1-3-1': ""As it is the case for Sen's tachyon field, these theories have attracted much attention in recent years due to their role in inflation based on string theory [CITATION]."", '0906.2617-1-3-2': 'In the above scenarios the inflaton is identified with the position of a mobile D-brane moving on a compact 6-dimensional submanifold of spacetime (for reviews and references see [CITATION]), which means that the inflaton is interpreted as an open string mode.', '0906.2617-1-3-3': 'The effects of such DBI-motivated fields in a Friedmann-Robertson-Walker (FRW) cosmology have been already studied by means of the dynamical systems tools, obtaining scaling solutions when the equation of state of the perfect fluid is negative and in the ultra-relativistic limit [CITATION].', '0906.2617-1-3-4': 'Non-linear Born-Infeld scalar fields with negative potential have been also investigated within FRW cosmology [CITATION].', '0906.2617-1-4-0': 'Here we aim at studying the cosmological dynamics of the above modification of standard tachyon field - Lagrangian ([REF]) - through using the dynamical systems tools.', '0906.2617-1-4-1': 'We shall apply an approach formerly used, for instance in Ref. [CITATION], that allows to study a vast variety of self-interaction potentials.', '0906.2617-1-4-2': ""It will be shown that there exists an equivalence between standard (Sen's) tachyon Lagrangian ([REF]) and the present modification, that enables to extend former dynamical systems studies of tachyon field cosmology (see [CITATION], for instance) to a wider class of self-interaction potentials beyond the power-law ones."", '0906.2617-1-4-3': 'Here we use natural units [MATH].', '0906.2617-1-5-0': '# Non-linear DBI-inspired Scalar Field Cosmology', '0906.2617-1-6-0': 'Consider the following effective action for a D3-brane [CITATION]: [EQUATION] where [MATH] is the Yang-Mills coupling and, as already noted in the introduction, [MATH] is the potential for the DBI-type field.', '0906.2617-1-6-1': 'For a spatially flat FRW metric - [MATH], where the dot accounts for derivative in respect to the cosmic time - the equation of motion for the DBI-type scalar field [MATH] can be written in the following way: [EQUATION] where [MATH] is the Hubble expansion parameter, and the ""Lorentz boost"" [MATH] is defined in the following way: [EQUATION]', '0906.2617-1-6-2': 'Alternatively the equation of motion of the DBI-type field can be written in the form of a continuity equation: [EQUATION] where we have defined the following energy density and pressure of the DBI scalar field: [EQUATION]', '0906.2617-1-6-3': ""The slight modification of Sen's Lagrangian we are proposing relies in a particular choice of the function [MATH] in the above equations."", '0906.2617-1-7-0': ""The Einstein's field equations in (flat) FRW metric, sourced by a mixture of a perfect barotropic fluid with energy density and pressure [MATH] and [MATH] respectively ([MATH] is the equation of state parameter of the perfect fluid which, for dust, vanishes), and of a non-linear DBI-inspired scalar field with Lagrangian ([REF]) - effective action ([REF]) with [MATH] -, that can be derived from an action principle, are the following: [EQUATION] where [EQUATION] and the modified Lorentz boost [MATH] is defined as: [EQUATION]"", '0906.2617-1-7-1': 'From now on we shall call the model given by equations ([REF]), ([REF]), as modified tachyon cosmology (MTC).', '0906.2617-1-8-0': 'It is remarkable that the Lagrangian densities ([REF]) and ([REF]) are equivalent under the change of field variables [EQUATION]', '0906.2617-1-8-1': 'I. e., under the transformation ([REF]) the equations ([REF]) above transform into the cosmological equations of standard tachyon cosmology (STC), in particular: [EQUATION]', '0906.2617-1-8-2': 'As we will show latter in section V, the above change of variable will allow us to extend the dynamical systems study of standard tachyon cosmology (see [CITATION] for instance) to a wider class of self-interaction potentials beyond the inverse square-law and power-law ones.', '0906.2617-1-9-0': '# Dynamical Systems Study', '0906.2617-1-10-0': 'The dynamical systems tools offer a very useful approach to the study of the asymptotic properties of the cosmological models [CITATION].', '0906.2617-1-10-1': 'In order to be able to apply these tools one has to (unavoidably) follow the steps enumerated below.', '0906.2617-1-11-0': 'After this one is ready to apply the standard tools of the (linear) dynamical systems analysis.', '0906.2617-1-12-0': '## Autonomous System of ODE', '0906.2617-1-13-0': 'Let us to introduce the following dimensionless phase space variables in order to build an autonomous system out of the system of cosmological equations ([REF],[REF]): [EQUATION]', '0906.2617-1-13-1': 'After this choice of phase space variables we can write the following autonomous system of ordinary differential equations: [EQUATION] where we have introduced the barotropic index for matter [MATH] (do not confound with either [MATH] or [MATH] which play the role of the standard and modified Lorentz boosts respectively), and the tilde denotes derivative with respect to the time variable [MATH] - properly the number of e-foldings.', '0906.2617-1-13-2': 'While deriving the ordinary differential equations ([REF]), and ([REF]), the following Friedmann constraint has also been considered: [EQUATION]', '0906.2617-1-13-3': 'It will be helpful to have other parameters of observational importance such as [MATH] - the scalar field dimensionless energy density parameter, and the equation of state (EOS) parameter [MATH], written in terms of the variables of phase space: [EQUATION]', '0906.2617-1-13-4': 'Additionally, the deceleration parameter [MATH]: [EQUATION]', '0906.2617-1-13-5': 'For an exponential self-interaction potential of the form: [EQUATION] since [MATH], then the equations ([REF]) and ([REF]) form a closed autonomous system of ODE: [EQUATION]', '0906.2617-1-13-6': 'The phase space where to look for equilibrium points of the system of ODE ([REF]) - corresponding to the MTC model described by ([REF]) - can be defined as follows: [EQUATION]', '0906.2617-1-13-7': 'The equations ([REF]) coincide with Eqs. (8,9) of [CITATION], for a standard tachyon field with inverse square-law self-interaction potential of the form: [EQUATION].', '0906.2617-1-13-8': 'Hence, the same critical points as in [CITATION] are found in the present case, this time for the exponential potential.', '0906.2617-1-13-9': 'This coincidence is easily explained through the equivalence between the present MTC model and standard tachyon cosmology (STC) model, under the change of variable ([REF]).', '0906.2617-1-13-10': 'Actually, as will be shown in section V, under ([REF]): [EQUATION].', '0906.2617-1-14-0': 'In the Figure [REF] we show the trajectories in phase space for different sets of initial conditions for the model driven by an exponential potential.', '0906.2617-1-14-1': 'The free parameters have been arbitrarily set to [MATH] and [MATH].', '0906.2617-1-14-2': 'Due to this choice of the parameters, the equilibrium points [MATH] represent inflationary critical points - past attractors - in the phase space, while the matter-scaling solution is the late-time attractor.', '0906.2617-1-14-3': 'Points [MATH] are associated with ultra-relativistic behaviour since [MATH].', '0906.2617-1-15-0': 'Due to our definition of the variable [MATH], the past attractor represents a scaling of the potential and of the kinetic energy of the tachyon scalar [EQUATION].', '0906.2617-1-15-1': 'This means that, in case the inflationary past attractor be identified with early-time inflation, it can not be associated with the slow-roll approximation which implies that the potential energy of the scalar field dominates over the kinetic one (the latter may be disregarded).', '0906.2617-1-16-0': 'In the next section we will focus on the asymptotic properties of MTC models driven by several others self-interaction potentials of cosmological relevance.', '0906.2617-1-17-0': '# Self-interaction Potentials beyond the Exponential one', '0906.2617-1-18-0': 'As long as one considers just constant and exponential self-interaction potentials ([MATH] and [MATH] respectively), the equations ([REF]) and ([REF]) form a closed autonomous system of ODE.', '0906.2617-1-18-1': 'However, if one wants to go further to consider a wider class of self-interaction potentials beyond the exponential one, the system of ODE ([REF],[REF]) is not a closed system of equations any more, since, in general, [MATH] is a function of the scalar field [MATH].', '0906.2617-1-18-2': 'A way out of this difficulty can be based on the method developed in [CITATION].', '0906.2617-1-18-3': 'In order to be able to study arbitrary self-interaction potentials one needs to consider one more variable [MATH], that is related with the derivative of the self-interaction potential through the following expression [EQUATION]', '0906.2617-1-18-4': 'Hence, an extra equation [EQUATION] has to be added to the above autonomous system of equations.', '0906.2617-1-18-5': 'The quantity [MATH] in equation ([REF]) is, in general, a function of [MATH].', '0906.2617-1-18-6': 'The idea behind the method in [CITATION] is that [MATH] can be written as a function of the variable [MATH], and, perhaps, of several constant parameters.', '0906.2617-1-18-7': 'Indeed, for a wide class of potentials the above requirement - [MATH] -, is fulfilled.', '0906.2617-1-18-8': 'Let us introduce a new function [MATH] so that equation ([REF]) can be written in the more compact form: [EQUATION]', '0906.2617-1-18-9': 'Equations ([REF]), ([REF]), and ([REF]) form a three-dimensional - closed - autonomous system of ODE: [EQUATION] that can be safely studied with the help of the standard dynamical systems tools [CITATION].', '0906.2617-1-18-10': 'As previously shown, since the cosmological equations of standard tachyon cosmology are equivalent to ([REF]) under the change of variable ([REF]), the present approach enables us to further extend former studies - with the application of the dynamical systems tools to tachyon cosmology - to a wider variety of self-interaction potentials.', '0906.2617-1-19-0': 'A drawback of the approach undertaken here can be associated with the fact that, for potentials that vanish at the minimum - usually correlated with late-time dynamics - the variable [MATH] is undefined so that the corresponding behaviour can not be properly investigated in the phase space spanned by [MATH].', '0906.2617-1-20-0': 'The phase space where to look for equilibrium points of the system of ODE ([REF]), corresponding to the MTC model, can be defined as follows (we take into account only expanding universes so that only [MATH] are being considered): [EQUATION] where it has to be pointed out that the range of the variable [MATH] depends on the specific kind of self-interaction potential considered.', '0906.2617-1-20-1': 'Recall that, for constant and exponential self-interaction potentials one does not need to consider the latter variable, so that, the corresponding system of ODE is a two-dimensional one.', '0906.2617-1-21-0': '## The (inverse) power-law potential [MATH]', '0906.2617-1-22-0': 'The inverse power-law potential have been extensively studied within standard tachyon field model [CITATION].', '0906.2617-1-22-1': 'According to the definition ([REF]) of the variable [MATH], for this potential one has: [EQUATION] so that the following asymptotics holds true: [EQUATION]', '0906.2617-1-22-2': 'In the upper panel of Figure [REF] a plot of [MATH] vs [MATH] is shown for the chosen values of free parameters: [MATH], [MATH].', '0906.2617-1-22-3': 'Notice that the range [MATH] is covered by negative values of the variable [MATH], while positive values of [MATH] cover the range [MATH].', '0906.2617-1-22-4': 'In what follows, for definiteness, we will restrict ourselves to [MATH], so that the tachyon field variable takes values in the interval [MATH].', '0906.2617-1-22-5': 'In this case the function [MATH] in ([REF],[REF]) can be written in the following way: [EQUATION]', '0906.2617-1-22-6': 'The cosmic dynamics driven by this potential can be associated with a 3-dimensional phase space ([REF]), spanned by the variables [MATH], [MATH], and [MATH], where [MATH].', '0906.2617-1-23-0': 'The equilibrium points of the autonomous system of ODE ([REF]) in the phase space [MATH] defined above, are listed in table [REF] while the eigenvalues of the corresponding linearization matrixes are shown in table [REF].', '0906.2617-1-23-1': 'These are non-hyperbolic critical points meaning that only limited information on their stability properties can be retrieved by means of the present linearized analysis.', '0906.2617-1-23-2': 'To help us understanding the stability properties of these points one has to rely on the phase portraits which show the structure of the phase space through phase path-probes originated by given initial data.', '0906.2617-1-24-0': 'Existence of the matter-dominated solution (equilibrium point [MATH] in table [REF]), is independent on the value of the variable [MATH], meaning that this phase of the cosmic evolution may arise at early-to-intermediate times ([MATH]), as well as at late time ([MATH]).', '0906.2617-1-24-1': 'As seen from table [REF], since in this case the two non vanishing eigenvalues of the linearization matrix are of opposite sign, the matter-dominated solution is always a saddle equilibrium point of ([REF]).', '0906.2617-1-24-2': 'This solution is inflationary whenever [MATH].', '0906.2617-1-25-0': 'Something similar can be said about the equilibrium point [MATH] in Tab. [REF].', '0906.2617-1-25-1': 'This point can be also associated either with early-time, as well as with intermediate-time dynamics, due to the fact that [MATH] can be any value.', '0906.2617-1-25-2': 'However, unlike the matter-dominated solution, the critical point [MATH] could be a past attractor (source critical point, usually associated with early-time dynamics) in the phase space, since the two non-vanishing eigenvalues of the corresponding linearization matrix are positive.', '0906.2617-1-25-3': 'Notwithstanding, since the point is a non-hyperbolic one, no final statement can be made on its stability properties until the corresponding phase portraits are drawn.', '0906.2617-1-25-4': 'The point [MATH] is also a matter-dominated solution, however, this phase corresponds to the ultra-relativistic case (the Lorentz boost [MATH]) and, unlike the standard situation, it represents scaling between the kinetic and the potential energy densities of the tachyon scalar.', '0906.2617-1-25-5': 'As before, the solution is inflationary whenever [MATH].', '0906.2617-1-26-0': 'The late-time dynamics driven by the potential [MATH] is correlated with infinitely large values of the variable [MATH] which, in the phase space ([REF]) ([MATH]), is depicted by the equilibrium point with [MATH] in table [REF] (equilibrium point [MATH]).', '0906.2617-1-26-1': 'This point corresponds to the inflationary de Sitter solution [MATH].', '0906.2617-1-26-2': 'From Tab. [REF] it is seen that this equilibrium point could be a late-time attractor since the two non-vanishing eigenvalues of the corresponding linearization matrix are both negative.', '0906.2617-1-26-3': 'However, since as already said, this is a non-hyperbolic point, only after drawing the corresponding phase portraits one is able to make conclusive statements about its stability properties.', '0906.2617-1-27-0': 'In Fig. [REF], in order to illustrate the stability properties of the asymptotic solutions [MATH], [MATH], and [MATH], probe paths in phase space - trajectories in the phase space originated by given initial data - are drawn.', '0906.2617-1-27-1': 'As clearly seen, these trajectories emerge from the ultra-relativistic (matter-dominated) point [MATH] and converge towards the inflationary de Sitter attractor (point [MATH]) at late times, thus confirming our suspects that [MATH] is the past attractor, while [MATH] is the future attractor.', '0906.2617-1-28-0': '## The potential [MATH].', '0906.2617-1-29-0': 'This potential has been formerly studied in Ref. [CITATION] as a new cosmological tracker solution for quintessence.', '0906.2617-1-29-1': 'According to the definition ([REF]), for this potential one gets: [EQUATION] from which it follows, in particular, that [EQUATION]', '0906.2617-1-29-2': 'In the right-hand panel of Fig. [REF] a plot of [MATH] vs [MATH] is shown for the chosen values of free parameters: [MATH], [MATH] and [MATH].', '0906.2617-1-29-3': 'Notice that the range of the variable [MATH] is covered by [MATH], while the range [MATH] is covered by [MATH].', '0906.2617-1-29-4': 'In what follows, for definiteness we will restrict ourselves to the interval [MATH].', '0906.2617-1-29-5': 'For the above potential the function [MATH] defined in ([REF],[REF]) can be written in the following way: [EQUATION]', '0906.2617-1-29-6': 'The cosmic dynamics driven by [MATH] can be associated with the 3-dimensional phase space ([REF]), where the variable [MATH] is constrained to the interval [MATH].', '0906.2617-1-29-7': 'The equilibrium points of the autonomous system of ODE ([REF]) in the phase space [MATH] defined above, are listed in table [REF], while the eigenvalues of the corresponding linearization matrixes are shown in Tab. [REF].', '0906.2617-1-30-0': 'As for the power-law potential, the existence of the matter-dominated solution (equilibrium point [MATH] in table [REF]), is independent of the value of the variable [MATH], meaning that this phase of the cosmic evolution may arise at early-to-intermediate times ([MATH]), as well as at late times ([MATH]).', '0906.2617-1-30-1': 'As seen from Tab. [REF], since in this case the two non vanishing eigenvalues of the linearization matrix are of opposite sign, the matter-dominated solution is always a saddle equilibrium point of ([REF]).', '0906.2617-1-30-2': 'The corresponding cosmological solution represents decelerating expansion whenever [MATH].', '0906.2617-1-30-3': 'Unlike this, the matter-dominated equilibrium point [MATH] in Tab. [REF] can be associated with ultra-relativistic behaviour (large Lorentz boost).', '0906.2617-1-30-4': 'As already said this point represents scaling between the potential and the kinetic energies of the tachyon field.', '0906.2617-1-30-5': 'As it can be seen from the phase portraits, it is always the past attractor for any path in the phase space of the model.', '0906.2617-1-31-0': 'Equilibrium points [MATH] (the tachyon-dominated solution) and [MATH] (the matter-scaling solution) are associated with late-time dynamics since, according to ([REF]), [MATH] is correlated with infinitely large values of the variable [MATH].', '0906.2617-1-31-1': 'The scalar field-dominated solution [MATH] always exits and whenever [MATH] it is a stable equilibrium point (the late-time attractor), otherwise it is a saddle critical point in phase space.', '0906.2617-1-31-2': 'Whenever the matter-scaling solution [MATH] exists, it is a stable equilibrium point (the late-time attractor).', '0906.2617-1-31-3': 'This solution is always associated with accelerated expansion.', '0906.2617-1-31-4': 'As in Ref. [CITATION], one has to take caution since the critical point [MATH] does not exist if either [MATH] (matter-dominated era), or [MATH] (radiation domination).', '0906.2617-1-31-5': 'This is due to the fact that the existence of the matter-scaling solution requires fulfillment of the condition [MATH].', '0906.2617-1-31-6': 'In this sense this solution can not be associated with a realistic model of dark energy.', '0906.2617-1-32-0': 'In the figure [REF] the phase portrait for this case is depicted.', '0906.2617-1-32-1': 'The above discussed behaviour is clearly illustrated by the figure.', '0906.2617-1-32-2': 'The free parameters were taken in such a way that the [MATH] solution exists ([MATH], [MATH], [MATH]).', '0906.2617-1-32-3': 'In correspondence, the ultra-relativistic phase [MATH] is the past attractor, while the matter-scaling solution [MATH] is the late-time (inflationary) attractor.', '0906.2617-1-33-0': '# Equivalence between Modified and Standard Tachyon Cosmologies', '0906.2617-1-34-0': 'As already shown in sections II and III, there is an equivalence between the MTC model given by the Lagrangian density ([REF]), and the STC model portrayed by the Lagrangian ([REF]), under the transformation ([REF]): [EQUATION].', '0906.2617-1-34-1': 'Equivalence under ([REF]) implies a residual equivalence between magnitudes of physical relevance, in particular: [EQUATION]', '0906.2617-1-34-2': 'It is evident that, once the functional form of the self-interaction potential [MATH] (or [MATH]) is known, the functional relationship [MATH] (or [MATH]) can be obtained through integration in quadratures, so that one is able to transform the potential [MATH] (or [MATH]).', '0906.2617-1-34-3': 'Actually, from ([REF]) it follows that [EQUATION]', '0906.2617-1-34-4': 'By using equation ([REF]) it can be found that, the transformation ([REF]) implies equivalence between the following self-interaction potentials ([MATH] is for the modified tachyon, while [MATH] is for the standard tachyon): [EQUATION] where [MATH], and [MATH] is an integration constant.', '0906.2617-1-34-5': 'For the inverse power-law potential one gets that [EQUATION] where [MATH], and [MATH].', '0906.2617-1-34-6': 'Additonally, for the sinh-like potential [MATH], one obtains the following equivalence [EQUATION] where [MATH] and [MATH].', '0906.2617-1-35-0': 'Notice that under the transformation ([REF]) the exponential potential of MTC is equivalent to the square-law potential [MATH] of STC studied in [CITATION], while the (inverse) power-law potential [MATH] is equivalent to a (inverse) power-law potential [MATH] of STC, not fully investigated in the same reference.', '0906.2617-1-35-1': 'I. e., the inverse power-law potential is not transformed under ([REF]).', '0906.2617-1-35-2': 'In the later case, the only difference of physical significance is in the power of the potential since [MATH].', '0906.2617-1-35-3': 'It is seen that, for positive [MATH].', '0906.2617-1-36-0': 'The above discussed MTC/STC equivalence opens up the possibility to apply the present approach to investigate the dynamics of standard tachyon cosmology for self-interaction potentials beyond the square-law potential which has been studied in detail in Ref. [CITATION].', '0906.2617-1-37-0': '# Discussion', '0906.2617-1-38-0': 'The approach undertaken in this paper enables applying the dynamical systems tools to investigate the cosmic dynamics of a wide variety of (scalar) tachyon self-interaction potentials, without resorting to such obscure concepts as ""instantaneous critical point"", whose physical relevance is suspicious.', '0906.2617-1-38-1': ""This is due to the equivalence of standard tachyon dynamics depicted by Sen's Lagrangian ([REF]), to the dynamics of a modified tachyon field given by the Lagrangian ([REF]), under the transformation ([REF]) of the tachyon field."", '0906.2617-1-38-2': 'Actually, if such an equivalence is taken into consideration, the results obtained in section IV - after applying the linear analysis to study the dynamics of the model of ([REF]) - can be safely assigned to the case of the standard tachyon model portrayed by the Lagrangian density ([REF]).', '0906.2617-1-38-3': 'As shown in the former section there is a full equivalence between inverse power-law potential of MTC [MATH] and that of STC [MATH], so that, for this kind of potential the results displayed in the tables [REF] and [REF] hold true for the standard tachyon cosmological model of [CITATION], which means in turn, that a detailed study of this potential - within STC - is possible without resorting to such obscure concepts as ""instantaneous critical point"" frequently used in the and [REF]), for the potential [MATH], whenever [MATH], one obtains that the de Sitter solution - point [MATH] in Tab. [REF] - is always the late-time attractor in the phase space, while the ultra-relativistic matter-dominated solution - point [MATH] in Tab. [REF] - is the past attractor from which the phase paths originate.', '0906.2617-1-38-4': 'The matter-dominated solution [MATH] is always a saddle in the phase space.', '0906.2617-1-38-5': 'Therefore, the standard tachyon cosmology model driven by the inverse power-law potential could be a nice scenario to address the late-time cosmic acceleration.', '0906.2617-1-39-0': 'A remarkable property of the tachyon model studied here - whether MTC or STC - is that, independent of the kind of self-interaction potential considered, the matter-dominated solutions [MATH] and [MATH] - the ultra-relativistic matter-dominated solution, are always equilibrium points of the corresponding autonomous system of ODE ([REF]) (see tables [REF],[REF]).', '0906.2617-1-39-1': 'A straightforward inspection of the equations ([REF]) reveals why this happens.', '0906.2617-1-39-2': 'Actually, a crude inspection of the equations in the system of ODE ([REF]) shows that, independent of the functional form of the function [MATH] and of the value of the variable [MATH], since for [MATH] the system ([REF]) reduces to the simplified system of equations: [EQUATION] then, for [MATH] and [MATH], the points [MATH] in phase space [MATH], and [MATH], both are equilibrium points of the system of ODE ([REF]).', '0906.2617-1-39-3': 'Since the existence of these points is independent of the value of the variable [MATH], both phases of the cosmic evolution may arise at early, intermediate, as well as at late times.', '0906.2617-1-39-4': 'In fact, the point [MATH] is always a saddle critical point, while [MATH] is the past attractor for any path in the phase space of the model, otherwise, [MATH] is the point in phase space from which all of the phase trajectories are repelled.', '0906.2617-1-40-0': 'From the analysis of the equations ([REF]) it also arises that, in general, for potentials for which the function [MATH] vanishes for a non-vanishing value [MATH], since in this case the system ([REF]) reduces to the autonomous system of ODE ([REF]) for an exponential potential [MATH]), the scalar field-dominated solution, as well as the matter-scaling phase, both arise in connection with the late-time dynamics in the tachyon model.', '0906.2617-1-40-1': 'This conclusion is quite robust and has been formerly stated in [CITATION] in a different context.', '0906.2617-1-41-0': '# Conclusions', '0906.2617-1-42-0': 'In the present paper we have performed an exhaustive study of the phase space for the so called modified tachyon cosmology (MTC) model, that is generated by the Lagrangian density [EQUATION].', '0906.2617-1-42-1': 'Thanks to an approach formerly used, for instance, in Ref. [CITATION], we were able to apply the standard tools of the (linear) dynamical systems analysis to uncover the relevant features of the cosmic dynamics in the model for a wide class of self-interaction potentials.', '0906.2617-1-43-0': 'It was demonstrated here, that the MTC model and the standard tachyon cosmology (STC) model associated with the Lagrangian density [EQUATION] are equivalent under the transformation [MATH].', '0906.2617-1-43-1': 'This fact enables us to straightforwardly assign the results of the linear analysis of [MATH] to the standard model of [MATH].', '0906.2617-1-43-2': 'Hence, a wide class of potentials, including the (inverse) power-law potential (see [CITATION]), can be investigated this way.', '0906.2617-1-43-3': 'It is revealled that independent of the functional form of the potential, the matter-dominated solution and the ultra-relativistic (also matter-dominated) solution, are always associated with equilibrium points in the phase space of the tachyon models.', '0906.2617-1-43-4': 'The latter is always the past attractor, while the former is a saddle critical point.', '0906.2617-1-43-5': 'For the power-law potential [MATH] the de Sitter solution is the late-time attractor, while for [MATH], depending on the region of the parameter space considered, the late-time attractor can be either the tachyon-dominated inflationary solution or the matter-scaling (also inflationary) solution.', '0906.2617-1-43-6': 'It can be demonstrated that, in general, for potentials for which the function [MATH] vanishes for [MATH], both late-time solutions: the scalar field-dominated, and the matter-scaling phase, always arise.', '0906.2617-1-44-0': 'This work was partly supported by CONACyT Mexico, under grant number I0101/131/07 C-234/07, Instituto Avanzado de Cosmologia (IAC) collaboration.', '0906.2617-1-44-1': 'The authors aknowledge also the MES of Cuba for partial support of the research.'}","{'0906.2617-2-0-0': ""We investigate in detail the asymptotic properties of a slight deformation of Sen's Lagrangian for the scalar tachyon for a wide class of self-interaction potentials."", '0906.2617-2-0-1': 'It is found that the model of modified tachyon cosmology is mathematically equivalent to the standard model under an appropiate transformation of the tachyon.', '0906.2617-2-0-2': 'The above mathematical equivalence is used to extend the dynamical systems study of standard tachyon cosmology to a wider class of self-interaction potentials beyond the (inverse) square-law and power-law ones.', '0906.2617-2-0-3': 'It is revealed that independent of the functional form of the potential, the matter-dominated solution and the ultra-relativistic (also matter-dominated) solution, are always associated with equilibrium points in the phase space of the tachyon models.', '0906.2617-2-0-4': 'The latter is always the past attractor, while the former is a saddle critical point.', '0906.2617-2-0-5': 'For inverse power-law potentials [MATH] the late-time attractor is always the de Sitter solution, while for sinh-like potentials [MATH], depending on the region of parameter space, the late-time attractor can be either the inflationary tachyon-dominated solution or the matter-scaling (also inflationary) phase.', '0906.2617-2-1-0': '# Introduction', '0906.2617-2-2-0': 'Inflationary models of the universe have been studied from the string theory perspective because inflation provides an explanation for the homogeneity and isotropy of the early universe.', '0906.2617-2-2-1': 'Additionally, recent astrophysical observations indicate us that the universe is presently undergoing a phase of accelerated expansion that has been attributed to a peculiar kind of source of the Einstein\'s field equations acknowledged as ""dark energy"" [CITATION].', '0906.2617-2-2-2': 'The crucial feature of the dark energy which ensures an accelerated expansion of the universe is that it breaks the strong energy condition.', '0906.2617-2-2-3': 'The tachyon field arising in the context of string theory [CITATION] provides one example of matter which does the job.', '0906.2617-2-2-4': ""The tachyon has been intensively studied during the last few years also in application to cosmology [CITATION]-[CITATION]; in this case one usually takes Sen's effective Lagrangian density for granted [CITATION]: [EQUATION] and studies its cosmological consequences without worrying about the string-theoretical origin of the action itself."", '0906.2617-2-2-5': 'In the above equation [MATH] is the scalar tachyon field, [MATH]-its self-interaction potential, and [MATH].', '0906.2617-2-3-0': 'Recently generalized nonlinear scalar field theories of the Dirac-Born-Infeld (DBI) type have been proposed [CITATION]-[CITATION].', '0906.2617-2-3-1': ""As it is the case for Sen's tachyon field, these theories have attracted much attention in recent years due to their role in inflation based on string theory [CITATION]."", '0906.2617-2-3-2': 'In the above scenarios the inflaton is identified with the position of a mobile D-brane moving on a compact 6-dimensional submanifold of spacetime (for reviews and references see [CITATION]), which means that the inflaton is interpreted as an open string mode.', '0906.2617-2-3-3': 'The effects of such DBI-motivated fields in a Friedmann-Robertson-Walker (FRW) cosmology have been already studied by means of the dynamical systems tools, yielding scaling solutions when the equation of state of the perfect fluid is negative and in the ultra-relativistic limit [CITATION].', '0906.2617-2-3-4': 'Non-linear Born-Infeld scalar fields with negative potential have been also investigated within FRW cosmology [CITATION].', '0906.2617-2-4-0': 'A dynamical systems study of the FRW cosmology within simpler theories based on ([REF]) can be found in Ref. [CITATION].', '0906.2617-2-4-1': 'However the authors of [CITATION] were able to study self-interaction potentials of the power-law type only.', '0906.2617-2-4-2': 'For more general potentials the corresponding system of ordinary differential equations in the phase space is not a closed system of equations any more, and one has to rely on the notion of ""instantaneous critical points"" whose physical relevance is unclear.', '0906.2617-2-5-0': ""In the present paper, through using the dynamical systems tools, we aim at studying the cosmological dynamics of a slight modification of Sen's Lagrangian ([REF]): [EQUATION] that is inspired in the generalized kind of DBI tachyon theories proposed in [CITATION]-[CITATION]."", '0906.2617-2-5-1': 'We shall apply an approach formerly used, for instance in Ref. [CITATION], that allows to study a vast variety of self-interaction potentials.', '0906.2617-2-5-2': ""It will be shown that there exists a formal (mathematical) equivalence between standard (Sen's) tachyon Lagrangian ([REF]) and the present modification ([REF]), that enables to extend former dynamical systems studies of tachyon field cosmology [CITATION] to a wider class of self-interaction potentials beyond the power-law ones."", '0906.2617-2-5-3': 'Here we use natural units [MATH].', '0906.2617-2-6-0': 'The paper has been organized in the following manner.', '0906.2617-2-6-1': 'The mathematical aspects of the model of modified tachyon cosmology are given in section II.', '0906.2617-2-6-2': 'The main section III is devoted to the study of the asymptotic properties of the above model for a wide variety of self-interaction potentials, through the application of the dynamical systems tools.', '0906.2617-2-6-3': 'In section IV we discuss the relevant aspects of the mathematical equivalence among the modified tachyon model and the standard one, so that the results of section III can be safely translated to the case of standard tachyon cosmology model.', '0906.2617-2-6-4': 'It is demonstrated, in particular, that former studies within the standard tachyon model, constrained to power-law potentials only [CITATION], can be extended to exponential type of potentials.', '0906.2617-2-6-5': 'A detailed discussion of the main results of the paper is presented in section V, while the conclusions are given in section VI.', '0906.2617-2-7-0': '# Generalized Non-linear DBI-inspired Scalar Field Cosmology', '0906.2617-2-8-0': 'Consider the following effective action for a D3-brane [CITATION]: [EQUATION] where [MATH] is the Yang-Mills coupling and, as already noted in the introduction, [MATH] is the potential for the DBI-type field.', '0906.2617-2-8-1': 'For a spatially flat FRW metric - [MATH] (the dot accounts for derivative in respect to the cosmic time) - the equation of motion for the DBI-type scalar field [MATH] can be written in the following way: [EQUATION] where [MATH] is the Hubble expansion parameter, and the ""Lorentz boost"" [MATH] is defined in the following way: [EQUATION]', '0906.2617-2-8-2': 'Alternatively the equation of motion of the DBI-type field can be written in the form of a continuity equation: [EQUATION] where we have defined the following energy density and pressure of the DBI scalar field: [EQUATION]', '0906.2617-2-8-3': ""The slight modification of Sen's Lagrangian ([REF]) we are proposing, relies in a particular choice of the function [MATH] in the above equations: [EQUATION]."", '0906.2617-2-9-0': ""The Einstein's field equations in (flat) FRW metric, sourced by a mixture of a perfect barotropic fluid with energy density and pressure [MATH] and [MATH] respectively ([MATH] is the equation of state parameter of the perfect fluid which, for dust, vanishes), and of a non-linear DBI-inspired scalar field with Lagrangian ([REF]) - effective action ([REF]) with [MATH] -, that can be derived from an action principle, are the following: [EQUATION] where [EQUATION] and the modified Lorentz boost [MATH] is defined as: [EQUATION]"", '0906.2617-2-9-1': 'From now on we shall call the model described by equations ([REF],[REF]) as ""modified tachyon cosmology"" (MTC).', '0906.2617-2-10-0': 'It is remarkable that the Lagrangian densities ([REF]) and ([REF]) are equivalent under the change of field variables [EQUATION]', '0906.2617-2-10-1': 'Actually, under the transformation ([REF]) the equations ([REF]) above transform into the cosmological equations of standard tachyon cosmology (STC), in particular: [EQUATION]', '0906.2617-2-10-2': 'As we will show latter in section IV, the above change of variable will allow us to extend the dynamical systems study of standard tachyon cosmology to a wider class of self-interaction potentials beyond the inverse square-law and power-law ones (see [CITATION] for a similar study but for power-law potentials only).', '0906.2617-2-11-0': '# Dynamical Systems Study', '0906.2617-2-12-0': 'The dynamical systems tools offer a very useful approach to the study of the asymptotic properties of the cosmological models [CITATION].', '0906.2617-2-12-1': 'In order to be able to apply these tools one has to (unavoidably) follow the steps enumerated below.', '0906.2617-2-13-0': 'After this one is ready to apply the standard tools of the (linear) dynamical systems analysis.', '0906.2617-2-13-1': 'For this purpose, let us to introduce the following dimensionless phase space variables in order to build an autonomous system out of the system of cosmological equations ([REF],[REF]): [EQUATION]', '0906.2617-2-13-2': 'After this choice of phase space variables we can write the following autonomous system of ordinary differential equations: [EQUATION] where we have introduced the barotropic index for matter [MATH] (do not confound with either [MATH] or [MATH] which play the role of the standard and modified Lorentz boosts respectively), and the tilde denotes derivative with respect to the time variable [MATH] - properly the number of e-foldings of expansion.', '0906.2617-2-13-3': 'While deriving the ordinary differential equations ([REF]), and ([REF]), the following Friedmann constraint has also been considered: [EQUATION]', '0906.2617-2-13-4': 'It will be helpful to have other parameters of observational importance such as [MATH] - the scalar field dimensionless energy density parameter, and the equation of state (EOS) parameter [MATH], written in terms of the variables of phase space: [EQUATION]', '0906.2617-2-13-5': 'Additionally, the deceleration parameter [MATH]: [EQUATION]', '0906.2617-2-14-0': '## Exponential Potential', '0906.2617-2-15-0': 'For an exponential self-interaction potential of the form: [EQUATION] since [MATH], then the equations ([REF]) and ([REF]) form a closed autonomous system of ODE: [EQUATION]', '0906.2617-2-15-1': 'The phase space where to look for equilibrium points of the system of ODE ([REF]) - corresponding to the MTC model described by ([REF]) - can be defined as follows: [EQUATION]', '0906.2617-2-15-2': 'The equations ([REF]) coincide with Eqs. (8,9) of [CITATION], for a standard tachyon field with inverse square-law self-interaction potential of the form: [EQUATION].', '0906.2617-2-15-3': 'Hence, the same critical points as in [CITATION] are found in the present case, this time for the exponential potential.', '0906.2617-2-15-4': 'This coincidence is easily explained through the equivalence between the present MTC model and standard tachyon cosmology (STC) model, under the change of variable ([REF]).', '0906.2617-2-15-5': 'Actually, as will be shown in section IV, under ([REF]): [EQUATION].', '0906.2617-2-16-0': 'In the Figure [REF] we show the trajectories in phase space for different sets of initial conditions for the model driven by an exponential potential.', '0906.2617-2-16-1': 'The free parameters have been arbitrarily set to [MATH] and [MATH].', '0906.2617-2-16-2': 'Due to this choice of the parameters, the equilibrium points [MATH] represent inflationary critical points - past attractors - in the phase space, while the matter-scaling solution is the late-time attractor.', '0906.2617-2-16-3': 'Points [MATH] are associated with ultra-relativistic behavior since [MATH].', '0906.2617-2-17-0': 'Due to our definition of the variable [MATH], the past attractor represents a scaling of the potential and of the kinetic energy of the tachyon scalar [EQUATION].', '0906.2617-2-17-1': 'This means that, in case the inflationary past attractor be identified with early-time inflation, it can not be associated with the slow-roll approximation which implies that the potential energy of the scalar field dominates over the kinetic one (the latter may be disregarded).', '0906.2617-2-18-0': 'In the next section we will focus on the asymptotic properties of MTC models driven by several others self-interaction potentials of cosmological relevance.', '0906.2617-2-19-0': '## Self-interaction Potentials beyond the Exponential one', '0906.2617-2-20-0': 'As long as one considers just constant and exponential self-interaction potentials ([MATH] and [MATH] respectively), the equations ([REF]) and ([REF]) form a closed autonomous system of ODE.', '0906.2617-2-20-1': 'However, if one wants to go further to consider a wider class of self-interaction potentials beyond the exponential one, the system of ODE ([REF],[REF]) is not a closed system of equations any more, since, in general, [MATH] is a function of the scalar field [MATH].', '0906.2617-2-20-2': 'A way out of this difficulty can be based on the method developed in [CITATION].', '0906.2617-2-20-3': 'In order to be able to study arbitrary self-interaction potentials one needs to consider one more variable [MATH], that is related with the derivative of the self-interaction potential through the following expression [EQUATION]', '0906.2617-2-20-4': 'Hence, an extra equation [EQUATION] has to be added to the above autonomous system of equations.', '0906.2617-2-20-5': 'The quantity [MATH] in equation ([REF]) is, in general, a function of [MATH].', '0906.2617-2-20-6': 'The idea behind the method in [CITATION] is that [MATH] can be written as a function of the variable [MATH], and, perhaps, of several constant parameters.', '0906.2617-2-20-7': 'Indeed, for a wide class of potentials the above requirement - [MATH] -, is fulfilled.', '0906.2617-2-20-8': 'Let us introduce a new function [MATH] so that equation ([REF]) can be written in the more compact form: [EQUATION]', '0906.2617-2-20-9': 'Equations ([REF]), ([REF]), and ([REF]) form a three-dimensional - closed - autonomous system of ODE: [EQUATION] that can be safely studied with the help of the standard dynamical systems tools [CITATION].', '0906.2617-2-20-10': 'As previously shown, since the cosmological equations of standard tachyon cosmology are equivalent to ([REF]) under the change of variable ([REF]), the present approach enables us to further extend former studies - with the application of the dynamical systems tools to tachyon cosmology - to a wider variety of self-interaction potentials.', '0906.2617-2-21-0': 'A drawback of the approach undertaken here can be associated with the fact that, for potentials that vanish at the minimum - usually correlated with late-time dynamics - the variable [MATH] is undefined so that the corresponding behavior can not be properly investigated in the phase space spanned by [MATH].', '0906.2617-2-22-0': 'The phase space where to look for equilibrium points of the system of ODE ([REF]), corresponding to the MTC model, can be defined as follows (we take into account only expanding universes so that only [MATH] are being considered): [EQUATION] where it has to be pointed out that the range of the variable [MATH] depends on the specific kind of self-interaction potential considered.', '0906.2617-2-22-1': 'Recall that, for constant and exponential self-interaction potentials one does not need to consider the latter variable, so that, the corresponding system of ODE is a two-dimensional one.', '0906.2617-2-23-0': '### The (inverse) power-law potential [MATH]', '0906.2617-2-24-0': 'The inverse power-law potential have been extensively studied within standard tachyon field model [CITATION].', '0906.2617-2-24-1': 'According to the definition ([REF]) of the variable [MATH], for this potential one has: [EQUATION] so that the following asymptotics hold true: [EQUATION]', '0906.2617-2-24-2': 'In the left hand panel of Figure [REF] a plot of [MATH] vs [MATH] is shown for the chosen values of free parameters: [MATH], [MATH].', '0906.2617-2-24-3': 'Notice that the range [MATH] is covered by negative values of the variable [MATH], while positive values of [MATH] cover the range [MATH].', '0906.2617-2-24-4': 'In what follows, for definiteness, we will restrict ourselves to [MATH], so that the tachyon field variable takes values in the interval [MATH].', '0906.2617-2-24-5': 'In this case the function [MATH] in ([REF],[REF]) can be written in the following way: [EQUATION]', '0906.2617-2-24-6': 'The cosmic dynamics driven by this potential can be associated with a 3-dimensional phase space ([REF]), spanned by the variables [MATH], [MATH], and [MATH], where [MATH].', '0906.2617-2-25-0': 'The equilibrium points of the autonomous system of ODE ([REF]) in the phase space [MATH] defined above, are listed in table [REF] while the eigenvalues of the corresponding linearization matrices are shown in table [REF].', '0906.2617-2-25-1': 'These are non-hyperbolic critical points meaning that only limited information on their stability properties can be retrieved by means of the present linearized analysis.', '0906.2617-2-25-2': 'To help us understanding the stability properties of these points one has to rely on the phase portraits which show the structure of the phase space through phase path-probes originated by given initial data.', '0906.2617-2-26-0': 'Existence of the matter-dominated solution (equilibrium point [MATH] in table [REF]), is independent on the value of the variable [MATH], meaning that this phase of the cosmic evolution may arise at early-to-intermediate times ([MATH]), as well as at late time ([MATH]).', '0906.2617-2-26-1': 'As seen from table [REF], since in this case the two non vanishing eigenvalues of the linearization matrix are of opposite sign, the matter-dominated solution is always a saddle equilibrium point of ([REF]).', '0906.2617-2-26-2': 'This solution is inflationary whenever [MATH].', '0906.2617-2-27-0': 'Something similar can be said about the equilibrium point [MATH] in Tab. [REF].', '0906.2617-2-27-1': 'This point can be also associated either with early-time, as well as with intermediate-time dynamics, due to the fact that [MATH] can be any value.', '0906.2617-2-27-2': 'However, unlike the matter-dominated solution, the critical point [MATH] could be a past attractor (source critical point, usually associated with early-time dynamics) in the phase space, since the two non-vanishing eigenvalues of the corresponding linearization matrix are positive.', '0906.2617-2-27-3': 'Notwithstanding, since the point is a non-hyperbolic one, no final statement can be made on its stability properties until the corresponding phase portraits are drawn.', '0906.2617-2-27-4': 'The point [MATH] is also a matter-dominated solution, however, this phase corresponds to the ultra-relativistic case (the Lorentz boost [MATH]) and, unlike the standard situation, it represents scaling between the kinetic and the potential energy densities of the tachyon scalar.', '0906.2617-2-27-5': 'As before, the solution is inflationary whenever [MATH].', '0906.2617-2-28-0': 'The late-time dynamics driven by the potential [MATH] is correlated with infinitely large values of the variable [MATH] which, in the phase space ([REF]) ([MATH]), is depicted by the equilibrium point with [MATH] in table [REF] (equilibrium point [MATH]).', '0906.2617-2-28-1': 'This point corresponds to the inflationary de Sitter solution [MATH].', '0906.2617-2-28-2': 'From Tab. [REF] it is seen that this equilibrium point could be a late-time attractor since the two non-vanishing eigenvalues of the corresponding linearization matrix are both negative.', '0906.2617-2-28-3': 'However, since as already said, this is a non-hyperbolic point, only after drawing the corresponding phase portraits one is able to make conclusive statements about its stability properties.', '0906.2617-2-29-0': 'We want to notice that the above results remain true if one considered power-law potentials with negative values of the constant parameter [MATH].', '0906.2617-2-29-1': 'In particular the above results can be safely extended to the quadratic potential [MATH].', '0906.2617-2-30-0': 'In Fig. [REF], in order to illustrate the stability properties of the asymptotic solutions [MATH], [MATH], and [MATH], probe paths in phase space - trajectories in the phase space originated by given initial data - are drawn.', '0906.2617-2-30-1': 'As clearly seen, these trajectories emerge from the ultra-relativistic (matter-dominated) point [MATH] and converge towards the inflationary de Sitter attractor (point [MATH]) at late times, thus confirming our suspects that [MATH] is the past attractor, while [MATH] is the future attractor.', '0906.2617-2-31-0': '### The potential [MATH].', '0906.2617-2-32-0': 'This potential has been formerly studied in Ref. [CITATION] as a new cosmological tracker solution for quintessence.', '0906.2617-2-32-1': 'According to the definition ([REF]), for this potential one gets: [EQUATION] from which it follows, in particular, that [EQUATION]', '0906.2617-2-32-2': 'In the right-hand panel of Fig. [REF] a plot of [MATH] vs [MATH] is shown for the chosen values of free parameters: [MATH], [MATH] and [MATH].', '0906.2617-2-32-3': 'Notice that the range of the variable [MATH] is covered by [MATH], while the range [MATH] is covered by [MATH].', '0906.2617-2-32-4': 'In what follows, for definiteness we will restrict ourselves to the interval [MATH].', '0906.2617-2-32-5': 'For the above potential the function [MATH] defined in ([REF],[REF]) can be written in the following way: [EQUATION]', '0906.2617-2-32-6': 'The cosmic dynamics driven by [MATH] can be associated with the 3-dimensional phase space ([REF]), where the variable [MATH] is constrained to the interval [MATH].', '0906.2617-2-32-7': 'The equilibrium points of the autonomous system of ODE ([REF]) in the phase space [MATH] defined above, are listed in table [REF], while the eigenvalues of the corresponding linearization matrices are shown in Tab. [REF].', '0906.2617-2-33-0': 'As for the power-law potential, the existence of the matter-dominated solution (equilibrium point [MATH] in table [REF]), is independent of the value of the variable [MATH], meaning that this phase of the cosmic evolution may arise at early-to-intermediate times ([MATH]), as well as at late times ([MATH]).', '0906.2617-2-33-1': 'As seen from Tab. [REF], since in this case the two non vanishing eigenvalues of the linearization matrix are of opposite sign, the matter-dominated solution is always a saddle equilibrium point of ([REF]).', '0906.2617-2-33-2': 'The corresponding cosmological solution represents decelerating expansion whenever [MATH].', '0906.2617-2-33-3': 'Unlike this, the matter-dominated equilibrium point [MATH] in Tab. [REF] can be associated with ultra-relativistic behavior (large Lorentz boost).', '0906.2617-2-33-4': 'As already said this point represents scaling between the potential and the kinetic energies of the tachyon field.', '0906.2617-2-33-5': 'As it can be seen from the phase portraits, it is always the past attractor for any path in the phase space of the model.', '0906.2617-2-34-0': 'Equilibrium points [MATH] (the tachyon-dominated solution) and [MATH] (the matter-scaling solution) are associated with late-time dynamics since, according to ([REF]), [MATH] is correlated with infinitely large values of the variable [MATH].', '0906.2617-2-34-1': 'The scalar field-dominated solution [MATH] always exits and whenever [MATH] it is a stable equilibrium point (the late-time attractor), otherwise it is a saddle critical point in phase space.', '0906.2617-2-34-2': 'Whenever the matter-scaling solution [MATH] exists, it is a stable equilibrium point (the late-time attractor).', '0906.2617-2-34-3': 'This solution is always associated with accelerated expansion.', '0906.2617-2-34-4': 'As in Ref. [CITATION], one has to take caution since the critical point [MATH] does not exist if either [MATH] (matter-dominated era), or [MATH] (radiation domination).', '0906.2617-2-34-5': 'This is due to the fact that the existence of the matter-scaling solution requires fulfillment of the condition [MATH].', '0906.2617-2-34-6': 'In this sense this solution can not be associated with a realistic model of dark energy.', '0906.2617-2-35-0': 'In the figure [REF] the phase portrait for this case is depicted.', '0906.2617-2-35-1': 'The above discussed behavior is clearly illustrated by the figure.', '0906.2617-2-35-2': 'The free parameters were taken in such a way that the [MATH] solution exists ([MATH], [MATH], [MATH]).', '0906.2617-2-35-3': 'In correspondence, the ultra-relativistic phase [MATH] is the past attractor, while the matter-scaling solution [MATH] is the late-time (inflationary) attractor.', '0906.2617-2-36-0': '# Mathematical Equivalence between Modified and Standard Tachyon Cosmologies', '0906.2617-2-37-0': 'As already shown in sections II and III, there is a mathematical equivalence between the MTC model given by the Lagrangian density ([REF]), and the STC model portrayed by the Lagrangian ([REF]), under the transformation ([REF]): [EQUATION].', '0906.2617-2-37-1': 'Equivalence under ([REF]) implies a residual equivalence between magnitudes of physical relevance, in particular: [EQUATION]', '0906.2617-2-37-2': 'It is evident that, once the functional form of the self-interaction potential [MATH] (or [MATH]) is known, the functional relationship [MATH] (or [MATH]) can be obtained through integration in quadratures, so that one is able to transform the potential [MATH] (or [MATH]).', '0906.2617-2-37-3': 'Actually, from ([REF]) it follows that [EQUATION]', '0906.2617-2-37-4': 'By using equation ([REF]) it can be found that the transformation ([REF]) implies the following transformations between modified tachyon potentials [MATH] and standard tachyon ones [MATH]: [EQUATION] where [MATH], and [MATH] is an integration constant.', '0906.2617-2-37-5': 'For the inverse power-law potential one gets that [EQUATION] where [MATH], and [MATH].', '0906.2617-2-37-6': 'Additionally, for the sinh-like potential [MATH], one obtains the following equivalence [EQUATION] where [MATH] and [MATH].', '0906.2617-2-38-0': 'Notice that under the transformation ([REF]) the exponential potential of MTC is equivalent to the square-law potential [MATH] of STC studied in [CITATION], while the (inverse) power-law potential [MATH] is equivalent to a (inverse) power-law potential [MATH] of STC, not fully investigated in the same reference.', '0906.2617-2-38-1': 'I. e., the inverse power-law potential is not transformed under ([REF]).', '0906.2617-2-38-2': 'In the later case, the only difference of physical significance is in the power of the potential since [MATH].', '0906.2617-2-38-3': 'It is seen that, for positive [MATH].', '0906.2617-2-39-0': 'The above discussed MTC/STC equivalence opens up the possibility to apply the present approach to investigate the dynamics of standard tachyon cosmology for self-interaction potentials beyond the square-law potential which has been studied in detail in Ref. [CITATION].', '0906.2617-2-39-1': 'Actually, consider, for instance, the exponential self-interaction potential for the standard tachyon: [EQUATION].', '0906.2617-2-39-2': 'By using the relationship ([REF]) it can be shown that: [EQUATION].', '0906.2617-2-39-3': 'Fortunately this case for the modified tachyon dynamics has been already studied in subsection III.B.1 (just replace [MATH] in equations ([REF]) and ([REF]) and bear in mind that the non-negative range of the variable [MATH] is now associated with negative values of the tachyon scalar [MATH]).', '0906.2617-2-39-4': 'It remains just to translate the corresponding results so that one could discuss their physical implications for the SMT cosmological dynamics.', '0906.2617-2-40-0': '# Discussion', '0906.2617-2-41-0': 'Thanks to the mathematical equivalence among standard tachyon dynamics depicted by Sen\'s Lagrangian ([REF]) and the dynamics of a modified tachyon field given by the Lagrangian ([REF]), under the transformation ([REF]), the approach undertaken in this paper enables applying the standard tools of the dynamical systems to investigate the cosmic dynamics driven by a wide variety of (scalar) tachyon self-interaction potentials, without resorting to such obscure concepts as ""instantaneous critical points"", whose physical relevance is suspicious.', '0906.2617-2-41-1': 'Actually, if such a mathematical (and dynamical) equivalence is taken into consideration, the results obtained in section III - after applying the linear analysis to study the dynamics of the model of ([REF]) - can be safely translated to the case of the standard tachyon model portrayed by the Lagrangian density ([REF]).', '0906.2617-2-42-0': 'As shown in the former section there is a full equivalence between inverse power-law potential of MTC [MATH] and that of STC [MATH], so that, for this kind of potential the results displayed in the tables [REF] and [REF] for the modified tachyon hold true for the standard tachyon cosmological model of [CITATION], which means in turn, that a detailed study of the dynamics driven by this potential within STC is indeed possible.', '0906.2617-2-42-1': 'According to the results of section III (see Tabs.', '0906.2617-2-42-2': '[REF] and [REF]), for the potential [MATH], whenever [MATH], one obtains that the de Sitter solution - point [MATH] in Tab. [REF] - is always the late-time attractor in the phase space, while the ultra-relativistic matter-dominated solution - point [MATH] in Tab. [REF] - is the past attractor from which the phase paths originate.', '0906.2617-2-42-3': 'The matter-dominated solution [MATH] is always a saddle in the phase space.', '0906.2617-2-42-4': 'Therefore, the standard tachyon cosmology model driven by the inverse power-law potential could be a nice scenario to address the late-time cosmic acceleration.', '0906.2617-2-43-0': 'The study of the asymptotic properties of the STC for the exponential potential is mathematically equivalent to the study of the asymptotic properties of the MTC for the quadratic potential [MATH], which is a particular case of the study presented in III.B.1 when the constant parameter [MATH] is replaced by the particular negative value [MATH].', '0906.2617-2-43-1': 'The only think to be changed is the phase space itself if one keeps [MATH] - in this case, in place of the half of the phase space ([REF]) corresponding to positive [MATH]-s, one has to consider the complementary half defined by negative [MATH]-s instead -, or one might keep intact the phase space at the cost that the tachyon field itself takes values in the interval [MATH].', '0906.2617-2-44-0': 'A remarkable property of the tachyon model studied here - whether MTC or STC - is that, independent of the kind of self-interaction potential considered, the matter-dominated solutions [MATH] and [MATH] - the ultra-relativistic matter-dominated solution, are always equilibrium points of the corresponding autonomous system of ODE ([REF]) (see tables [REF],[REF]).', '0906.2617-2-44-1': 'A straightforward inspection of the equations ([REF]) reveals why this happens.', '0906.2617-2-44-2': 'Actually, a crude inspection of the equations in the system of ODE ([REF]) shows that, independent of the functional form of the function [MATH] and of the value of the variable [MATH], since for [MATH] the system ([REF]) reduces to the simplified system of equations: [EQUATION] then, for [MATH] and [MATH], the corresponding points [MATH] in phase space: [MATH], and [MATH], both are equilibrium points of the system of ODE ([REF]).', '0906.2617-2-44-3': 'Since the existence of these points is independent of the value of the variable [MATH], both phases of the cosmic evolution may arise at early, intermediate, as well as at late times.', '0906.2617-2-44-4': 'In fact, the point [MATH] is always a saddle critical point, while [MATH] is the past attractor for any path in the phase space of the model, otherwise, [MATH] is the point in phase space from which all of the phase trajectories are repelled.', '0906.2617-2-45-0': 'From the analysis of the equations ([REF]) it also arises that, in general, for potentials for which the function [MATH] vanishes for a non-vanishing value [MATH], since in this case the system ([REF]) reduces to the autonomous system of ODE ([REF]) for an exponential potential [MATH]), the scalar field-dominated solution, as well as the matter-scaling phase, both arise in connection with the late-time dynamics in the tachyon model.', '0906.2617-2-45-1': 'This conclusion is quite robust and has been formerly stated in [CITATION] in a different context.', '0906.2617-2-46-0': '# Conclusions', '0906.2617-2-47-0': 'In the present paper we have performed an exhaustive study of the phase space for the so called modified tachyon cosmology (MTC) model, that is generated by the Lagrangian density [EQUATION].', '0906.2617-2-47-1': 'Thanks to an approach formerly used, for instance, in Ref. [CITATION], we were able to apply the standard tools of the (linear) dynamical systems analysis to uncover the relevant features of the cosmic dynamics in the model for a wide class of self-interaction potentials.', '0906.2617-2-48-0': 'It was demonstrated here, that the MTC model and the standard tachyon cosmology (STC) model associated with the Lagrangian density [EQUATION] are related by the transformation [MATH].', '0906.2617-2-48-1': 'This fact enables us to straightforwardly translate the results of the linear analysis of [MATH] to the standard model of [MATH].', '0906.2617-2-48-2': 'Hence, a wide class of potentials, including the (inverse) power-law potential (see [CITATION]) and the exponential potential, can be easily investigated this way.', '0906.2617-2-48-3': 'It is revealed that independent of the functional form of the potential, the matter-dominated solution and the ultra-relativistic (also matter-dominated) solution, are always associated with equilibrium points in the phase space of the tachyon models.', '0906.2617-2-48-4': 'The latter is always the past attractor, while the former is a saddle critical point.', '0906.2617-2-48-5': 'For the power-law potential [MATH] the de Sitter solution is the late-time attractor, while for [MATH], depending on the region of the parameter space considered, the late-time attractor can be either the tachyon-dominated inflationary solution or the matter-scaling (also inflationary) solution.', '0906.2617-2-48-6': 'It can be demonstrated that, in general, for potentials for which the function [MATH] vanishes for [MATH], both late-time solutions: the scalar field-dominated, and the matter-scaling phase, always arise.', '0906.2617-2-49-0': 'This work was partly supported by CONACyT Mexico, under grants 49865-F, 54576-F, 56159-F, 49924-J, 105079, 52327, and by grant number I0101/131/07 C-234/07, Instituto Avanzado de Cosmologia (IAC) collaboration.', '0906.2617-2-49-1': 'T G, D G and Y N acknowledge the MES of Cuba for partial support of the research.', '0906.2617-2-49-2': 'R G-S acknowledges partial support from COFAA-IPN and EDI-IPN grants, and SIP-IPN 20090440.'}","[['0906.2617-1-27-0', '0906.2617-2-30-0'], ['0906.2617-1-27-1', '0906.2617-2-30-1'], ['0906.2617-1-39-0', '0906.2617-2-44-0'], ['0906.2617-1-39-1', '0906.2617-2-44-1'], ['0906.2617-1-39-3', '0906.2617-2-44-3'], ['0906.2617-1-39-4', '0906.2617-2-44-4'], ['0906.2617-1-3-1', '0906.2617-2-3-1'], ['0906.2617-1-3-2', '0906.2617-2-3-2'], ['0906.2617-1-3-4', '0906.2617-2-3-4'], ['0906.2617-1-26-0', '0906.2617-2-28-0'], ['0906.2617-1-26-1', '0906.2617-2-28-1'], ['0906.2617-1-26-2', '0906.2617-2-28-2'], ['0906.2617-1-26-3', '0906.2617-2-28-3'], ['0906.2617-1-34-1', '0906.2617-2-37-1'], ['0906.2617-1-34-2', '0906.2617-2-37-2'], ['0906.2617-1-34-3', '0906.2617-2-37-3'], ['0906.2617-1-34-5', '0906.2617-2-37-5'], ['0906.2617-1-14-0', '0906.2617-2-16-0'], ['0906.2617-1-14-1', '0906.2617-2-16-1'], ['0906.2617-1-14-2', '0906.2617-2-16-2'], 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['0906.2617-2-3-2', '0906.2617-3-3-2'], ['0906.2617-2-4-0', '0906.2617-3-4-0'], ['0906.2617-3-0-0', '0906.2617-4-0-0'], ['0906.2617-1-38-2', '0906.2617-2-41-1'], ['0906.2617-1-13-10', '0906.2617-2-15-5'], ['0906.2617-1-13-0', '0906.2617-2-13-1'], ['0906.2617-1-13-1', '0906.2617-2-13-2']]",[],"[['0906.2617-1-38-3', '0906.2617-2-42-2'], ['0906.2617-1-3-0', '0906.2617-2-3-0'], ['0906.2617-1-34-4', '0906.2617-2-37-4'], ['0906.2617-1-7-1', '0906.2617-2-9-1'], ['0906.2617-1-4-0', '0906.2617-2-5-0'], ['0906.2617-1-2-1', '0906.2617-2-2-1'], ['0906.2617-1-2-1', '0906.2617-2-2-2'], ['0906.2617-2-0-0', '0906.2617-3-0-0'], ['0906.2617-1-38-0', '0906.2617-2-41-0'], ['0906.2617-1-38-1', '0906.2617-2-41-0'], ['0906.2617-1-38-3', '0906.2617-2-42-0']]","[['0906.2617-1-44-1', '0906.2617-2-49-1'], ['0906.2617-2-5-1', '0906.2617-3-5-1'], ['0906.2617-2-6-0', '0906.2617-3-6-0'], ['0906.2617-2-6-1', '0906.2617-3-6-1'], ['0906.2617-2-6-2', '0906.2617-3-6-2'], ['0906.2617-2-6-3', '0906.2617-3-6-3'], ['0906.2617-2-6-4', '0906.2617-3-6-4'], ['0906.2617-2-6-5', '0906.2617-3-6-5'], ['0906.2617-2-5-3', '0906.2617-3-6-7']]","['0906.2617-2-49-0', '0906.2617-3-8-1', '0906.2617-3-8-2', '0906.2617-4-0-1', '0906.2617-4-6-0', '0906.2617-4-8-0', '0906.2617-4-10-1']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/0906.2617,"{'0906.2617-3-0-0': 'We investigate in detail the asymptotic properties of tachyon cosmology for a generic class of self-interaction potentials.', '0906.2617-3-0-1': 'The present approach relies on an existing (formal) mathematical equivalence between a Dirac-Born-Infeld (DBI) model - with a particular choice of the warp factor and of the potential for the DBI field - and standard tachyon cosmology, under an appropriate transformation of the DBI field.', '0906.2617-3-0-2': 'The above mathematical equivalence is used to generalize the dynamical systems study of tachyon cosmology to a wider class of self-interaction potentials beyond the (inverse) square-law and power-law ones.', '0906.2617-3-0-3': 'It is revealed that independent of the functional form of the potential, the matter-dominated solution and the ultra-relativistic (also matter-dominated) solution, are always associated with equilibrium points in the phase space of the tachyon models.', '0906.2617-3-0-4': 'The latter is always the past attractor, while the former is a saddle critical point.', '0906.2617-3-0-5': 'For inverse power-law potentials [MATH] the late-time attractor is always the de Sitter solution, while for sinh-like potentials [MATH], depending on the region of parameter space, the late-time attractor can be either the inflationary tachyon-dominated solution or the matter-scaling (also inflationary) phase.', '0906.2617-3-0-6': 'In general, for most part of known quintessential potentials, the late-time dynamics will be associated either with de Sitter inflation, or with matter-scaling, or scalar field-dominated solutions.', '0906.2617-3-1-0': '# Introduction', '0906.2617-3-2-0': 'Inflationary models of the universe have been studied from the string theory perspective because inflation provides an explanation for the homogeneity and isotropy of the early universe.', '0906.2617-3-2-1': 'Additionally, recent astrophysical observations indicate us that the universe is presently undergoing a phase of accelerated expansion that has been attributed to a peculiar kind of source of the Einstein\'s field equations acknowledged as ""dark energy"" [CITATION].', '0906.2617-3-2-2': 'The crucial feature of the dark energy which ensures an accelerated expansion of the universe is that it breaks the strong energy condition.', '0906.2617-3-2-3': 'The tachyon field arising in the context of string theory [CITATION] provides one example of matter which does the job.', '0906.2617-3-2-4': ""The tachyon has been intensively studied during the last few years also in application to cosmology [CITATION]-[CITATION]; in this case one usually takes Sen's effective Lagrangian density for granted [CITATION]: [EQUATION] and studies its cosmological consequences without worrying about the string-theoretical origin of the action itself."", '0906.2617-3-2-5': 'In the above equation [MATH] is the scalar tachyon field, [MATH]-its self-interaction potential, and [MATH].', '0906.2617-3-3-0': 'Recently generalized nonlinear scalar field theories of the Dirac-Born-Infeld (DBI) type have been proposed [CITATION]-[CITATION].', '0906.2617-3-3-1': ""As it is the case for Sen's tachyon field, these theories have attracted much attention in recent years due to their role in inflation based on string theory [CITATION]."", '0906.2617-3-3-2': 'In the above scenarios the inflaton is identified with the position of a mobile D3-brane, moving on a compact 6-dimensional submanifold of spacetime (for reviews and references see [CITATION]), which means that the inflaton is interpreted as an open string mode.', '0906.2617-3-3-3': 'The effects of such DBI-motivated fields in a Friedmann-Robertson-Walker (FRW) cosmology have been already studied by means of the dynamical systems tools, yielding scaling solutions when the equation of state of the perfect fluid is negative and in the ultra-relativistic limit [CITATION].', '0906.2617-3-3-4': 'Non-linear Born-Infeld scalar fields with negative potential have been also investigated within FRW cosmology [CITATION].', '0906.2617-3-3-5': 'The cosmological dynamics of a DBI field when the potential and the brane tension are arbitrary power-law or exponential functions of the DBI field, has been studied very recently in [CITATION].', '0906.2617-3-4-0': 'A dynamical systems study of the FRW cosmology within the simpler theories based on the tachyon Lagrangian ([REF]) can be found in Ref. [CITATION].', '0906.2617-3-4-1': 'However the authors of [CITATION] were able to study self-interaction potentials of the power-law type only.', '0906.2617-3-4-2': 'For more general potentials the corresponding system of ordinary differential equations in the phase space is not a closed system of equations any more, and one has to rely on the notion of ""instantaneous critical points"" whose physical relevance is unclear.', '0906.2617-3-5-0': 'In the present paper we aim at studying the cosmological dynamics of the tachyon model given by ([REF]), for a wider variety of tachyon self-interactions potentials than in Ref.[CITATION].', '0906.2617-3-5-1': 'To this end we shall exploit a formal mathematical equivalence existing between a DBI field - holding a particular relationship between the DBI potential and the warp factor - and the tachyon field, in combination with the application of an approach formerly used, for instance in Ref.[CITATION], that allows to study a vast variety of self-interaction potentials.', '0906.2617-3-5-2': 'The above mathematical equivalence enables to extend the results of dynamical systems studies of DBI models - including results of studies already existing in the - to tachyon field cosmology.', '0906.2617-3-5-3': 'We will be able, in particular, to note an equivalence between the results of Ref. [CITATION] within the context of tachyon cosmology, and those in [CITATION] for a DBI field with exponential potential and brane tension, not reported previously.', '0906.2617-3-6-0': 'The paper has been organized in the following manner.', '0906.2617-3-6-1': 'The mathematical aspects of the particular DBI cosmological model considered, are given in section II.', '0906.2617-3-6-2': 'Section III is devoted to the study of the asymptotic properties of the above model for a generic variety of self-interaction potentials, through the application of the dynamical systems tools.', '0906.2617-3-6-3': 'In section IV we discuss the relevant aspects of the mathematical equivalence among the DBI model whose dynamics has been discussed in the previous section, and the corresponding tachyon model, so that the results of section III can be safely translated to the case of standard tachyon cosmology.', '0906.2617-3-6-4': 'It is demonstrated, in particular, that former studies within the standard tachyon model, constrained to power-law potentials only [CITATION], can be generalized to exponential type of potentials.', '0906.2617-3-6-5': 'A detailed discussion of the main results of the paper is presented in section V, while the conclusions are given in section VI.', '0906.2617-3-6-6': 'For self-consistency and completeness, an appendix with the basic recipes of the application of the dynamical systems tools to cosmology has been added.', '0906.2617-3-6-7': 'Here we use natural units [MATH].', '0906.2617-3-7-0': '# Basic Equations and Set-Up', '0906.2617-3-8-0': 'Consider the following effective action for a DBI field, [MATH] [CITATION]: [EQUATION] where [MATH] is the inverse of the brane tension (also acknowledged as the warp factor of the warped throat geometry in the internal space) and, [MATH] is the potential for the DBI field, arising from quantum interactions between the D3-brane associated with [MATH], and other D-branes.', '0906.2617-3-8-1': 'In the [MATH][MATH]f()=const.', '0906.2617-3-8-2': '[MATH]f()[MATH]f()V()=1[MATH]f()[MATH]V()[MATH][MATH]H=a/a[MATH]_L[MATH]f()[MATH]_m[MATH]p_m=_m_m[MATH]_m[MATH][MATH]f()V()=1[MATH]f()V()=1[MATH]_m=_m+1[MATH]_L[MATH][MATH]a[MATH]_V=-1,0)[MATH]U[MATH]x1[MATH]x/V[MATH]_V=0[MATH]_V=const[MATH]_V[MATH][MATH]v[MATH]V_^2 V/(_V)^2[MATH][MATH][MATH]v[MATH]0i[MATH]g(v)=0[MATH]v_0i[MATH]v=0[MATH]v=0[MATH]V()=V_0[MATH]v_0i0[MATH]V_i()(-v_0i)[MATH]g(v)=0[MATH][MATH]v_0i0[MATH]v=0[MATH]g(v)[MATH]y=0[MATH]x=1[MATH]x=0[MATH]V()[MATH]x=1^2=V()[MATH]x=0[MATH]y=0[MATH]3H^2=_m[MATH]x=0[MATH]y0[MATH]y=1[MATH]v=0[MATH]g(v)[MATH]v=0[MATH]g(v)=0[MATH]3H^2=V_0[MATH]1-^2[MATH]v[MATH]v[MATH]V_^2 V/(_V)^2[MATH][MATH]V()^-2[MATH]v[MATH]v[MATH]x,y,v[MATH]y0[MATH]v[MATH]V()=V_0^-[MATH]v[MATH]v()[MATH][MATH]V_0=1[MATH]]-,0[[MATH]v[MATH]v>0[MATH]]0,[[MATH]v>0[MATH]]0,[[MATH]g(v)[MATH]x[MATH]y[MATH]v[MATH]0v<[MATH][MATH]M[MATH]v[MATH]0<v<[MATH]v=0[MATH]_m<2/3[MATH]U[MATH]v[MATH]U[MATH]U[MATH][MATH]_m<2/3[MATH]V=V_0 ^-[MATH][MATH]0v<[MATH]v=0[MATH]dS[MATH](3H^2=V)[MATH][MATH]V()^2[MATH]M[MATH]dS[MATH]U[MATH]U[MATH]dS[MATH]U[MATH]dS[MATH]V()=V_0[()]^-[MATH]v()[MATH][MATH]V_0=1[MATH]]-,0[[MATH]-<v-[MATH]]0,[[MATH]v<[MATH]v<[MATH]g(v)[MATH]V=V_0 [()]^-[MATH]v[MATH]v<[MATH][MATH]M[MATH]v[MATH]<v<[MATH]v=[MATH]_m>2/3[MATH]U[MATH]T[MATH]MS[MATH]v=[MATH][MATH]T[MATH]_m>^2^2(36+^4^4-^2^2)/18[MATH]MS[MATH]MS[MATH]_m=1[MATH]_m=4/3[MATH]0<_m< 1[MATH]MS[MATH]_m=.2[MATH]V_04/^2[MATH]_0[MATH]V_0=[V_0/(+1)^2]^1/+1[MATH]n=/(+1)[MATH]^-2()[MATH]V_0=1/^2[MATH]_0^2=1/V_0^2[MATH]^-2[MATH]^-2[MATH]^-2n[MATH]n=/(+1)[MATH]>0n1[MATH]-2[MATH]v[MATH]]-,0[[MATH]^-2[MATH]^-2n[MATH]V()^-2n[MATH]0<n1[MATH]dS[MATH]U[MATH]M[MATH]V()^2[MATH][MATH]-2[MATH]]0,[[MATH]v[MATH]v[MATH]]-,0[[MATH]V()[MATH]M[MATH]U[MATH]g(v)[MATH]v[MATH]y=0[MATH]x=0[MATH]x=1[MATH](x,y,v)[MATH]M=(0,0,v)[MATH]U=(1,0,v)[MATH]v[MATH]M[MATH]U[MATH]U[MATH]g(v)[MATH]v[MATH]g(v)=0[MATH]v=v_0i0[MATH]Ve^[MATH]V=[MATH]0<<1[MATH]-1=1-x^2[MATH]x0[MATH]x1[MATH]fV=1[MATH]V=1[MATH][MATH]d/V()[MATH]fV=1[MATH]v[MATH]n=1[MATH]_0=0[MATH]^-2n[MATH]^-2[MATH]n=/(+1)[MATH]^-2[MATH]-[MATH](^2-_0^2)^-1[MATH]^-2()[MATH]^-2n[MATH]g(v)[MATH]v[MATH]V()[MATH][MATH][MATH](x_1,x_2,...x_n)[MATH](x_1,x_2,...x_n)[MATH]p_k=p_k(x_1,x_2,...x_n)[MATH]k=1,2,...m[MATH]x_ix_i+e_i[MATH]e_i[MATH]e_i[MATH]f_i=f_i(p)=0[MATH][M^j_i]=f_i/x_j[MATH]_i[MATH]p[MATH][U^j_i][MATH]p[MATH](e_0)_i^j[MATH][MATH][MATH]'}","{'0906.2617-4-0-0': 'We investigate in detail the asymptotic properties of tachyon cosmology for a broad class of self-interaction potentials.', '0906.2617-4-0-1': 'The present approach relies in an appropriate re-definition of the tachyon field, which, in conjunction with a method formerly applied in the [MATH]V^-()[MATH][MATH]V[MATH]()^2g^__[MATH]V()[MATH]-1[MATH]c=8G=1[MATH][MATH][MATH]V()[MATH]_m[MATH]p_m=_m_m[MATH]_m[MATH][MATH][MATH]_m=_m+1[MATH][MATH]a[MATH]_V=-1,0)[MATH]U[MATH]x1[MATH]x/V[MATH]_V=0[MATH]_V=const[MATH]_V[MATH][MATH]v[MATH]V_^2 V/(_V)^2[MATH][MATH][MATH]v[MATH]0i[MATH]g(v)=0[MATH]v_0i[MATH]v=0[MATH]v=0[MATH]V()=V_0[MATH]v_0i0[MATH]V_i()(-v_0i)[MATH]g(v)=0[MATH][MATH]v_0i0[MATH]v=0[MATH]g(v)[MATH]y=0[MATH]x=1[MATH]x=0[MATH]V()[MATH]x=1^2=V()[MATH]x=0[MATH]y=0[MATH]3H^2=_m[MATH]x=0[MATH]y0[MATH]y=1[MATH]v=0[MATH]g(v)[MATH]v=0[MATH]g(v)=0[MATH]3H^2=V_0[MATH]1-^2[MATH]v[MATH]v[MATH]V_^2 V/(_V)^2[MATH][MATH]V()^-2[MATH][MATH]v[MATH]x,y,v[MATH]y0[MATH]v[MATH]V()=V_0^-[MATH]v[MATH]v()[MATH][MATH]V_0=1[MATH]]-,0[[MATH]v[MATH]v>0[MATH]]0,[[MATH]v>0[MATH]]0,[[MATH]g(v)[MATH]x[MATH]y[MATH]v[MATH]0v<[MATH][MATH]M[MATH]v[MATH]0<v<[MATH]v=0[MATH]_m<2/3[MATH]U[MATH]v[MATH]U[MATH]U[MATH][MATH]_m<2/3[MATH]V=V_0 ^-[MATH][MATH]0v<[MATH]v=0[MATH]dS[MATH](3H^2=V)[MATH][MATH]V()^2[MATH]M[MATH]dS[MATH]U[MATH]U[MATH]dS[MATH]U[MATH]dS[MATH]V()=V_0[()]^-[MATH]v()[MATH][MATH]V_0=1[MATH]]-,0[[MATH]-<v-[MATH]]0,[[MATH]v<[MATH]v<[MATH]g(v)[MATH]V=V_0 [()]^-[MATH]v[MATH]v<[MATH][MATH]M[MATH]v[MATH]<v<[MATH]v=[MATH]_m>2/3[MATH]U[MATH]T[MATH]MS[MATH]v=[MATH][MATH]T[MATH]_m>^2^2(36+^4^4-^2^2)/18[MATH]MS[MATH]MS[MATH]_m=1[MATH]_m=4/3[MATH]0<_m< 1[MATH]MS[MATH]_m=.2[MATH]V_04/(V_0^2)[MATH]_0[MATH]V_0=[V_0/(+1)^2]^1/+1[MATH]n=/(+1)[MATH]^-2()[MATH]V_0=1/^2[MATH]_0^2=1/V_0^2[MATH]^-2[MATH]^-2[MATH]^-2n[MATH]n=/(+1)[MATH]>0n1[MATH]-2[MATH]v[MATH]]-,0[[MATH]^-2[MATH]^-2n[MATH][MATH]V()^-2n[MATH]0<n1[MATH]dS[MATH]U[MATH]M[MATH]V()()[MATH]V()^2[MATH][MATH]-2[MATH]]0,[[MATH]v[MATH]v[MATH]]-,0[[MATH]V()[MATH]M[MATH]U[MATH]g(v)[MATH]v[MATH]y=0[MATH]x=0[MATH]x=1[MATH](x,y,v)[MATH]M=(0,0,v)[MATH]U=(1,0,v)[MATH]v[MATH]M[MATH]U[MATH]U[MATH]g(v)[MATH]v[MATH]g(v)=0[MATH]v=v_0i0[MATH]Ve^[MATH][MATH]-constant), or, also [MATH].', '0906.2617-4-0-2': 'As one can immediately see, this action contains [MATH] in ([REF]) as a particular case, if one chooses [MATH] in ([REF]) to be related with the potential of the DBI-field in the following particular form: [MATH].', '0906.2617-4-1-0': 'In reference [CITATION], for instance, the authors study (among other cases) the late-time dynamics of a DBI model with exponential potential and brane tension (here we use the notation of [CITATION]): [EQUATION].', '0906.2617-4-1-1': 'In the particular case where [MATH], a new class of solutions arise, for which [MATH]), i. e., [MATH], and [MATH].', '0906.2617-4-1-2': 'These solutions are the matter-scaling solution and the scalar field-dominated (kinetic/potential energy-scaling) phase.', '0906.2617-4-1-3': 'It is interesting noting that, under the field replacement ([REF]), and [MATH], the exponential potential above transforms into the (inverse) square tachyon potential [EQUATION] so that, given that [MATH], the former results are equivalent to the results of the dynamical systems study of tachyon cosmology with an inverse square potential of reference [CITATION].', '0906.2617-4-1-4': 'Actually, the equilibrium points (c) and (d1,d2) in Tab. I of Ref. [CITATION] are the above mentioned matter-scaling, and the scalar (tachyon) field-dominated solutions respectively.', '0906.2617-4-1-5': 'As long as we know, no such equivalence between generalized-DBI and tachyon dynamics has been reported before.', '0906.2617-4-2-0': '# Conclusions', '0906.2617-4-3-0': 'In the quest for alternative models of inflation (here we include both primordial and late-time inflation), the effective tachyon models and their generalization: the scalar DBI-field models, have played an important role.', '0906.2617-4-3-1': 'Due to complexity of the analysis, in particular in connection with difficulties to obtain closed (autonomous) systems of ordinary differential equations out of the corresponding cosmological equations, the study of the tachyon dynamics has been performed for a limited number of particular tachyon potentials.', '0906.2617-4-4-0': 'In the present paper we have proposed an approach based on a re-definition of the tachyon field variable [MATH], followed by straightforward application of a method already used in the ) to study a wide class of self-interaction potentials in a different context.', '0906.2617-4-4-1': 'The above mentioned field-variable re-definition is necessary if one wants to introduce formerly used phase space variables that make the autonomous system of ODE - obtained out of the corresponding cosmological equations - a closed system of ODE.', '0906.2617-4-4-2': 'The present approach allows to apply the standard dynamical systems tools to the study of a broad class of self-interaction potentials, provided that the function [EQUATION] where [EQUATION] can be written as a polynomial in the new phase space variable [MATH].', '0906.2617-4-4-3': 'The usefulness of our combined approach has been illustrated with the detailed study of the following tachyon potentials [EQUATION] where the latter potential is just a particular case of the former one for [MATH], [MATH].', '0906.2617-4-4-4': 'It has been demonstrated that the [MATH] potential is equivalent to the inverse power-law potential [MATH]), while the inverse square tachyon potential [MATH] is equivalent to the exponential potential [MATH].', '0906.2617-4-4-5': 'The particular inverse square tachyon potential [MATH] is equivalent to the sinh-like potential [MATH], therefore, different kinds of potentials in the transformed model might be equivalent to a same tachyon potential.', '0906.2617-4-4-6': 'In spite of the fact that only the dynamics driven by the inverse power-law tachyon potential [MATH], and by the exponential one [MATH], have been studied in details in this paper, nonetheless, the combined method used here can be applied to other kinds of potentials, provided that the function [MATH] can be written as a polynomial in [MATH] for the potential [MATH] of the transformed tachyon field [MATH].', '0906.2617-4-5-0': 'An existing mathematical equivalence between the dynamics of a generalized DBI-field [MATH] with the following relationship between the warp factor [MATH] of the internal geometry, and the DBI potential [MATH]: [MATH], and the dynamics of a tachyon field [MATH] driven by the effective action [EQUATION] under the field redefinition [MATH], has been also revealed.', '0906.2617-4-5-1': 'The above equivalence allows to translate the results of dynamical systems studies of DBI models with the condition [MATH], for given DBI potentials, to the corresponding studies of the equivalent tachyon models.', '0906.2617-4-6-0': 'Amongst the most interesting results of the present study we can list the following:', '0906.2617-4-7-0': 'The present approach can be safely (and moderately easy) applied to consider new classes of tachyon potentials beyond the (inverse) power-law and exponential ones.', '0906.2617-4-7-1': 'While the present paper was under consideration for publication, an alternative method to investigate the cosmic dynamics of a tachyon field for wide classes of potential, without performing any field re-definition, has been published [CITATION].', '0906.2617-4-8-0': 'This work was partly supported by CONACyT Mexico, under grants 49865-F, 54576-F, 56159-F, 49924-J, 105079, 52327, and by grant number I0101/131/07 C-234/07, Instituto Avanzado de Cosmologia (IAC) collaboration.', '0906.2617-4-8-1': 'T G, D G and Y N acknowledge the MES of Cuba for partial support of the research.', '0906.2617-4-8-2': 'R G-S acknowledges partial support from COFAA-IPN and EDI-IPN grants, and SIP-IPN 201006010.', '0906.2617-4-9-0': '# Appendix: Dynamical Systems', '0906.2617-4-10-0': 'Here we include brief tips of how to apply the dynamical systems tools to situations of cosmological interest.', '0906.2617-4-10-1': 'In order to apply these tools one has to follow the steps enumerated below:', '0906.2617-4-11-0': 'To identify the phase space variables that allow writing the system of cosmological equations in the form of an autonomus system of ordinary differential equations (ODE), say: [EQUATION]', '0906.2617-4-11-1': 'With the help of the chosen phase space variables, to build an autonomous system of ODE out of the original system of cosmological equations ([MATH] is the time-ordering variable, not necessarily the cosmic time): [EQUATION].', '0906.2617-4-11-2': 'Notice that the RHS of these equations do not depend explicitly on [MATH] (that is the reason why the system is called autonomous).', '0906.2617-4-12-0': 'To identify the phase space spanned by the chosen variables [MATH], that is relevant to the cosmological model under study.', '0906.2617-4-12-1': 'This amounts, basically, to define the range of the phase space variables that is appropriate to the problem at hand: [EQUATION]', '0906.2617-4-12-2': 'Finding the equilibrium points of the autonomous system of ODE, amounts to solve the following system of algebraic equations on [MATH]: [EQUATION]', '0906.2617-4-12-3': 'Next one linearly expands the equations of the autonomous system of ODE in the neighborhood of the equilibrium points [MATH], [MATH]: I. e., one replaces [MATH], where [MATH] are the small (linear) perturbations around the equilibrium points.', '0906.2617-4-12-4': 'Hence the system of ODE becomes a system of linear equations to determine the evolution of the [MATH]-s: [EQUATION] otherwise, since [MATH], then [EQUATION] where we have introduced the linearization or Jacobian matrix [MATH].', '0906.2617-4-13-0': 'The next step is to solve the secular equation to determine the eigenvalues [MATH] of the linearization matrix at the given equilibrium point [MATH]: [EQUATION] where [MATH] is the unit matrix.', '0906.2617-4-14-0': 'Once the eigenvalues of the linearization around a given equilibrium point [MATH] have been computed, the evolution of the perturbations is given by [EQUATION] where the amplitudes [MATH] are constants of integration.', '0906.2617-4-15-0': 'If all of the eigenvalues have negative real parts, the perturbations decay with [MATH], i. e., the equilibrium point is stable against linear perturabtions.', '0906.2617-4-15-1': 'The corresponding equilibrium point is said to be a future attractor.', '0906.2617-4-15-2': 'If at least one of the eigenvalues has positive real part, the perturbations grow with [MATH] so that these are not stable in the direction spanned by the given eigenvalue.', '0906.2617-4-15-3': 'Hence the point is said to be a saddle.', '0906.2617-4-15-4': 'The perturbations around a given equilibrium point are unstable, in other words the point is a past attractor (a source point in the phase space), if all of the eigenvalues have positive real parts.', '0906.2617-4-15-5': 'Points whose linearization is characterized by complex eigenvalues are said to be spiral equilibrium points, and are commonly associated with oscillatory behavior of the corresponding solution.', '0906.2617-4-15-6': 'If at least one of the eigenvalues has a vanishing real part, the equilibrium point is said to be non-hyperbolic.', '0906.2617-4-15-7': 'In the latter case, in general, and unless some of the non-vanishing real parts of the eigenvalues are of opposite sign, one can not give conclusive arguments on the stability of the equilibrium point.', '0906.2617-4-15-8': 'Other techniques have to be applied.'}",
hep-ph-9812485,"{'hep-ph-9812485-1-0-0': 'The method of searching for signals of new physics beyond the Standard Model based on the renormalization group equation and the decoupling theorem is developed.', 'hep-ph-9812485-1-0-1': 'It is applied to find signals of new heavy abelian [MATH]-boson and realized in the form of specific relations between scattering amplitudes at low energies.', 'hep-ph-9812485-1-0-2': 'Considering four-fermion processes, a basic one-loop renormalization group relation is derived which gives possibility to reduce the problem to the scattering of Standard Model particles in the ""external field"" substituting heavy virtual [MATH] state.', 'hep-ph-9812485-1-0-3': 'From the relations it is determined that the absolute value of the [MATH] coupling to the axial currents has to be the same for all fermions of the Standard Model and is strongly related to the [MATH] coupling to the scalar fields.', 'hep-ph-9812485-1-0-4': 'The corresponding dependences between the parameters of the effective Lagrangians describing deviations from the Standard Model are derived.', 'hep-ph-9812485-1-1-0': '# Introduction', 'hep-ph-9812485-1-2-0': 'An important problem of nowadays high energy physics is searching for deviations from the Standard Model (SM) of elementary particles.', 'hep-ph-9812485-1-2-1': 'The deviations may appear due to virtual states of new heavy particles entering the extended models and having the masses much greater than the W-boson mass [MATH].', 'hep-ph-9812485-1-2-2': 'One of approaches for the description of these heavy states is the construction of effective Lagrangians (EL) [CITATION].', 'hep-ph-9812485-1-2-3': 'Owing to decoupling of heavy particles the EL can be expressed in the form of the expansion in inverse powers of heavy mass [MATH]: [EQUATION] where [MATH] stands for various dimension-six operators constructed from the SM fields and derivatives, and [MATH] are some numbers depended on the structure of physics beyond the SM.', 'hep-ph-9812485-1-2-4': 'Since the full higher-energy theory is unknown, the parameters [MATH] are supposed to be arbitrary.', 'hep-ph-9812485-1-3-0': 'In principle, it is possible to write down a lot of different EL parameterizing new physics effects.', 'hep-ph-9812485-1-3-1': 'In Ref. [CITATION] the EL generated at a tree level in a general renormalizable gauge theory have been derived.', 'hep-ph-9812485-1-3-2': 'These objects by construction contain a great number of arbitrary parameters responsible for specific processes.', 'hep-ph-9812485-1-3-3': 'However, it is well known that a renormalizable theory contains a few number of independent constants due to relations between them.', 'hep-ph-9812485-1-3-4': 'The renormalizability of a theory is implemented in the renormalization group (RG) invariance of scattering amplitudes [CITATION].', 'hep-ph-9812485-1-3-5': 'It has been proven in Ref. [CITATION] that the RG equation for the [MATH]-matrix element can be used in a non-standard way to obtain a set of relations between the parameters of the EL.', 'hep-ph-9812485-1-4-0': 'Two main observations were used.', 'hep-ph-9812485-1-4-1': 'First, it has been shown that a heavy virtual state may be considered as an external field scattering SM light particles.', 'hep-ph-9812485-1-4-2': 'Second, the renormalization of the vertices, describing scattering on the external field, can be determined by the [MATH]- and [MATH]- functions calculated with light particles only.', 'hep-ph-9812485-1-4-3': 'Hence, it is possible to incorporate all new physics effects into a set of external fields substituting heavy virtual states.', 'hep-ph-9812485-1-4-4': 'Then, the RG equation can be treated as the algebraic relation between the external fields and appropriate [MATH] and [MATH] functions.', 'hep-ph-9812485-1-4-5': 'This relation has to be the identity valid in each order of perturbation theory being the necessary condition of the renormalizability.', 'hep-ph-9812485-1-4-6': 'If due to a symmetry the number of [MATH] and [MATH] functions is less than the number of RG relations, one can obtain a non-trivial system of equations for the new physics parameters.', 'hep-ph-9812485-1-4-7': 'In Ref. [CITATION], as an example, the SM with the heavy Higgs scalar was investigated.', 'hep-ph-9812485-1-4-8': 'It has been shown that the above mentioned scenario is realized for gauge couplings.', 'hep-ph-9812485-1-5-0': 'In present paper the model based on the gauge group [MATH] is investigated.', 'hep-ph-9812485-1-5-1': 'It is supposed to describe the low-energy phenomenology of some unknown higher-energy theory.', 'hep-ph-9812485-1-5-2': 'The model includes the minimal SM and the additional heavy [MATH]- boson called [MATH].', 'hep-ph-9812485-1-5-3': 'At the current energies [MATH] the heavy boson is decoupled producing EL ([REF]).', 'hep-ph-9812485-1-5-4': 'Since the new physics couplings in the model are of the gauge type, we are able to use RG relations in searching for dependences between the EL parameters.', 'hep-ph-9812485-1-6-0': 'We will formulate low-energy RG relations for four-fermion amplitudes in the order [MATH].', 'hep-ph-9812485-1-6-1': 'As it will be shown, the relations are satisfied only if the absolute value of [MATH] coupling to the axial currents is identical for all the SM fermions.', 'hep-ph-9812485-1-6-2': 'Hence, some of the parameters [MATH] are not independent.', 'hep-ph-9812485-1-6-3': 'Moreover, the fermion sector of the new physics is appeared to be related to the scalar one.', 'hep-ph-9812485-1-6-4': 'So, the number of parameters describing deviations from the SM is decreased.', 'hep-ph-9812485-1-7-0': 'The paper is organized as follows: in section [REF] the spontaneous breakdown of the electroweak symmetry is investigated and the Lagrangian of the model is derived in terms of the physical fields.', 'hep-ph-9812485-1-7-1': 'In section [REF] the general RG relation for [MATH]-matrix element is obtained when the decoupling of heavy particles takes place.', 'hep-ph-9812485-1-7-2': 'The relation is applied to four-fermion scattering processes in section [REF].', 'hep-ph-9812485-1-7-3': 'It is reduced to the relations for vertices describing scattering of the SM fermions on the external field substituting the virtual heavy boson.', 'hep-ph-9812485-1-7-4': 'As a result, the dependences between the new physics parameters are found.', 'hep-ph-9812485-1-7-5': 'In section [REF] the appropriate relations between the parameters of the EL are derived.', 'hep-ph-9812485-1-7-6': 'The obtained results are summarized in section [REF].', 'hep-ph-9812485-1-8-0': '# The Model', 'hep-ph-9812485-1-9-0': 'The model under consideration is based on the gauge group [MATH].', 'hep-ph-9812485-1-9-1': 'It was investigated before in Ref. [CITATION].', 'hep-ph-9812485-1-9-2': 'The corresponding additional abelian vector boson [MATH] is supposed to be massive with the mass [MATH] much greater than the [MATH]-boson mass [MATH].', 'hep-ph-9812485-1-9-3': 'In present paper the mechanism of heavy mass generation is not considered since the theory describing interactions at energies [MATH] is unknown.', 'hep-ph-9812485-1-9-4': 'So, the Lagrangian includes the explicit mass term for the [MATH]-boson, and the mass [MATH] is treated as the parameter of the model.', 'hep-ph-9812485-1-10-0': 'In Ref. [CITATION] the interaction of scalar fields with the [MATH]-boson was not considered.', 'hep-ph-9812485-1-10-1': 'However, to write down the RG equation one needs to parameterize the explicit [MATH] coupling to the scalar fields of the theory.', 'hep-ph-9812485-1-10-2': 'Since the vector boson interactions with the SM fermions [MATH] and the SM scalar doublet [MATH] are specified by covariant derivatives, the relevant Lagrangian reads [EQUATION] where [MATH] is the fermion charge in the positron charge units, [MATH] are the Pauli matrices, and the summation over all the SM left doublets [MATH] and the SM right singlets [MATH] is understood.', 'hep-ph-9812485-1-10-3': 'The charges [MATH] and the fields [MATH], [MATH], [MATH] correspond to the gauge groups [MATH], [MATH], [MATH], respectively.', 'hep-ph-9812485-1-10-4': 'The values of the dimensionless constants [MATH], [MATH], [MATH] depend on the particular higher-energy theory including [MATH]-boson.', 'hep-ph-9812485-1-11-0': 'Up to this point the electroweak symmetry [MATH] was supposed to be unbroken, and all the SM particles were massless.', 'hep-ph-9812485-1-11-1': 'The masses of the SM particles are generated by the spontaneous breakdown of the symmetry [MATH] originated due to the non-zero vacuum expectation value of the scalar doublet.', 'hep-ph-9812485-1-11-2': 'However, the mass eigenstates are appeared to be shifted away from the original fields [MATH], since the corresponding mass terms become non-diagonal.', 'hep-ph-9812485-1-11-3': 'The physical fields [MATH] are obtained by the orthogonal transformation: [EQUATION] where [MATH] is the SM value of the Weinberg angle ([MATH]), and [MATH] denotes the mixing angle relating physical states [MATH] to massive neutral components of [MATH] gauge fields.', 'hep-ph-9812485-1-11-4': 'The value of the angle [MATH] can be determined from the relation [EQUATION] confirming Ref. [CITATION].', 'hep-ph-9812485-1-12-0': 'The masses of the physical fields are given by the expressions: [EQUATION]', 'hep-ph-9812485-1-12-1': 'As is seen, the mass of the physical [MATH]-boson differs from the SM value [MATH] by the small quantity of order [MATH].', 'hep-ph-9812485-1-12-2': 'So, the mixing [MATH] is also small [MATH].', 'hep-ph-9812485-1-12-3': 'The difference [MATH] is negative and completely determined by the [MATH] coupling to the scalar doublet.', 'hep-ph-9812485-1-12-4': 'Thus, constraints on the [MATH] interaction with scalar fields can be derived by means of experimental detecting this observable: [EQUATION]', 'hep-ph-9812485-1-12-5': 'Using Eqs.([REF]) the Lagrangian of the model can be expressed in terms of the physical fields.', 'hep-ph-9812485-1-12-6': 'The dependence on the mixing [MATH] in the Eqs.([REF]) causes new interactions [MATH] originally absent in the Lagrangian ([REF]).', 'hep-ph-9812485-1-13-0': 'In present paper we will deal with four-fermion amplitudes of order [MATH] produced by virtual [MATH]-boson in the [MATH]-channel.', 'hep-ph-9812485-1-13-1': 'The corresponding tree level diagram is shown in Fig. [REF].', 'hep-ph-9812485-1-13-2': 'At low energies the effective four-fermion vertex is generated due to the heavy mass in the [MATH]-boson propagator [MATH].', 'hep-ph-9812485-1-13-3': 'So, expressions involving such amplitudes can be computed neglecting the terms proportional to [MATH] in the Lagrangian, since these terms lead to the expressions of higher order in [MATH].', 'hep-ph-9812485-1-13-4': ""Only linear in [MATH] terms of the Lagrangian are needed, which have the following form in the 't Hooft-Feynman gauge: [EQUATION] where [MATH]; [MATH]; [MATH] is the positron charge; [MATH] denotes the vacuum expectation value of the scalar doublet; [MATH] is the Higgs scalar particle; [MATH] and [MATH] are the Goldstone fields."", 'hep-ph-9812485-1-14-0': '# RG relations in a theory with decoupling', 'hep-ph-9812485-1-15-0': 'The model under consideration is supposed to be the low-energy limit of some unknown renormalizable theory describing interactions at energies [MATH].', 'hep-ph-9812485-1-15-1': 'As well known, in a renormalizable theory the [MATH]-matrix elements are invariant under the renormalization group (RG) transformations meaning the independence of physical amplitudes of the renormalization point [MATH].', 'hep-ph-9812485-1-15-2': 'In present section we will derive RG relations for scattering amplitudes at energy scales much below the heavy mass [MATH], when the [MATH]-boson (as well as other heavy fields) is decoupled.', 'hep-ph-9812485-1-16-0': 'We will use the notations [MATH] (where [MATH] and [MATH] denote the QCD charge and the scalar self-coupling, respectively) and [MATH] (where [MATH] represents all the fields of the theory, and [MATH] are particle masses) to refer to the charges, the fields and the masses.', 'hep-ph-9812485-1-16-1': 'Then the RG equation reads: [EQUATION] where renormalized (running) parameters [MATH] are the solutions to the equations: [EQUATION]', 'hep-ph-9812485-1-16-2': 'Hats over quantities mark the parameters of the original theory.', 'hep-ph-9812485-1-16-3': 'They include contributions of loops containing both the SM particles and the heavy ones.', 'hep-ph-9812485-1-17-0': 'In present paper the dimensional regularization and the [MATH] renormalization scheme [CITATION] are used.', 'hep-ph-9812485-1-17-1': 'The [MATH]-scheme is mass-independent, so the RG operator [MATH] has the same form at high, [MATH], and low, [MATH], energies.', 'hep-ph-9812485-1-18-0': 'In the energy range [MATH] all the heavy fields are decoupled.', 'hep-ph-9812485-1-18-1': 'So, one can introduce the running parameters [MATH] completely defined by the low-energy effective theory [CITATION]: [EQUATION] where [MATH]- and [MATH]- functions contain no contributions from loops with heavy particles.', 'hep-ph-9812485-1-18-2': 'These parameters can be expressed in terms of the original ones: [EQUATION] where the matching between the both sets of parameters ([MATH] and [MATH]) is chosen to be done at the normalization point [MATH]: [EQUATION]', 'hep-ph-9812485-1-18-3': 'In fact, Eq.([REF]) is the readjustment of the parameters of the theory.', 'hep-ph-9812485-1-18-4': 'As it has been shown in Refs.[CITATION], all the heavy particle loop contributions proportional to [MATH] are completely removed from the RG equation ([REF]) by such a redefinition.', 'hep-ph-9812485-1-18-5': 'This fact is the consequence of the decoupling theorem [CITATION].', 'hep-ph-9812485-1-19-0': 'Then, the RG equation ([REF]) for the [MATH]-matrix expressed in terms of the low-energy quantities [MATH] becomes: [EQUATION] where [MATH] and [MATH] denote, respectively, the RG operator and the [MATH]-matrix of the low-energy effective theory, calculated in assumption that no heavy particles are excited in loops.', 'hep-ph-9812485-1-20-0': 'The usual treatment of Eq.([REF]) is to improve scattering amplitudes obtained in a perturbation theory.', 'hep-ph-9812485-1-20-1': 'For this purpose the solution to Eq.([REF]) [MATH] is found as the function of the renormalization point.', 'hep-ph-9812485-1-20-2': 'In contrast to this familiar approach we consider Eq.([REF]) as the algebraic relation between the constants [MATH], [MATH], [MATH].', 'hep-ph-9812485-1-20-3': 'This relation has to be the identity valid in every order of perturbation theory being the necessary condition of the renormalizability.', 'hep-ph-9812485-1-20-4': 'We will show that the relation ([REF]) for different four-fermion processes at the one-loop level holds only if the constants [MATH], [MATH], [MATH] satisfy some specific relations.', 'hep-ph-9812485-1-20-5': 'Then, the relations between the constants [MATH], [MATH], [MATH] are resulted in the dependences between appropriate parameters [MATH] of the EL.', 'hep-ph-9812485-1-21-0': 'Let [MATH] and [MATH] represent the one-loop level part and the tree level part of the [MATH]-matrix element, respectively.', 'hep-ph-9812485-1-21-1': 'Introducing the one-loop RG operator: [EQUATION] where [MATH] and [MATH] functions are computed at the one-loop level, one can derive from Eq.([REF]) the following identity: [EQUATION]', 'hep-ph-9812485-1-21-2': 'However, since the parameters [MATH] and [MATH] differ at the one-loop level, one is able to substitute one set of them by another in Eq.([REF]).', 'hep-ph-9812485-1-22-0': 'Eq.([REF]) is the starting point to analyse the RG relations for amplitudes in the low-energy region.', 'hep-ph-9812485-1-22-1': 'It is the consequence of the renormalizability of the original theory.', 'hep-ph-9812485-1-22-2': 'The relation insures the leading logarithmic terms of the one-loop [MATH]-matrix element to reproduce the appropriate tree-level structure.', 'hep-ph-9812485-1-22-3': 'Naturally, if the explicit expressions for coupling constants in terms of the basic set of parameters of the original theory are known, the relation is trivially fulfilled.', 'hep-ph-9812485-1-22-4': 'But the situation is changed when the couplings are represented by arbitrary unknown parameters as it takes place in the considered model and, in general, in the effective Lagrangian approach for describing deviations from the SM [CITATION].', 'hep-ph-9812485-1-22-5': 'In this case the relation ([REF]) is the equation for the parameters and corresponding [MATH]- and [MATH]- functions.', 'hep-ph-9812485-1-22-6': 'If due to a symmetry the number of [MATH]- and [MATH]- functions is less than the number of RG relations, the non-trivial system of equations relating the originally independent parameters may occur.', 'hep-ph-9812485-1-22-7': 'As it has been shown in Ref. [CITATION], such a situation is realized for gauge invariant interactions.', 'hep-ph-9812485-1-22-8': 'Since the considered model describes the interactions of this type, we will use the relation ([REF]) to derive the dependences between the parameters [MATH], [MATH], [MATH].', 'hep-ph-9812485-1-23-0': '# RG relations for four-fermion amplitudes', 'hep-ph-9812485-1-24-0': 'In present section the one-loop RG relation ([REF]) is applied to the four-fermion amplitudes [MATH] containing [MATH]-boson as the virtual state.', 'hep-ph-9812485-1-24-1': 'In the lower order in [MATH] such amplitudes produce effective contact interactions of the type current[MATH]current described by the effective Lagrangians ([REF]) [CITATION].', 'hep-ph-9812485-1-25-0': 'Let the renormalized fields, masses and charges are defined by the following relations: [EQUATION] where [MATH] is the vertex describing interaction between fermion [MATH] and neutral vector boson; [MATH] and [MATH] are [MATH] matrices ([MATH]).', 'hep-ph-9812485-1-25-1': 'The fermion renormalization constant [MATH] is the SM one.', 'hep-ph-9812485-1-25-2': 'Its value is known [CITATION].', 'hep-ph-9812485-1-25-3': 'Quantities [MATH] and [MATH] are represented at the one-loop level by the divergent ([MATH]) terms of the gauge-independent part of the polarization operator of neutral vector bosons [MATH] being evaluated from the diagrams in Fig. [REF] [EQUATION] with [MATH] for [MATH] and [MATH] otherwise.', 'hep-ph-9812485-1-25-4': 'To find the constant [MATH] in one-loop order one has to calculate the divergent ([MATH]) part of the vertex function [MATH] (the diagrams in Fig. [REF]): [EQUATION] where [MATH] is the vertex [MATH] at the tree level.', 'hep-ph-9812485-1-26-0': 'The [MATH]-matrix element expressed in terms of the renormalized quantities is finite.', 'hep-ph-9812485-1-26-1': 'It can be evaluated using the diagrams in Figs. [REF],[REF] where shaded blobs stand for the finite parts of the vertex function [MATH] and the polarization operator [MATH].', 'hep-ph-9812485-1-26-2': 'On the mass-shell of fermions the gauge-dependent part of the [MATH] gives no contribution to the physical amplitude due to the transversality of the vertex [MATH].', 'hep-ph-9812485-1-27-0': 'Eq. ([REF]) for the renormalized [MATH]-matrix element can not be reduced to relations for vertices, because of one-loop mixing between neutral bosons.', 'hep-ph-9812485-1-27-1': 'To derive these relations one has to avoid the mixing.', 'hep-ph-9812485-1-27-2': 'The corresponding procedure was described in Ref. [CITATION].', 'hep-ph-9812485-1-28-0': 'One has to introduce the quantities [MATH] and [MATH] instead of fields [MATH], [MATH], [MATH] and vertices [MATH] according to the following relations: [EQUATION]', 'hep-ph-9812485-1-28-1': 'After such a redefinition the mixing between neutral bosons in the relation ([REF]) vanishes.', 'hep-ph-9812485-1-28-2': 'In what follows, we will use the quantities [MATH] and [MATH], therefore only the diagonal terms of the polarization operator are needed in computing of the [MATH]-matrix element [MATH].', 'hep-ph-9812485-1-28-3': 'Since the difference between two sets of parameters is of order [MATH], it is possible to substitute the set of tilded parameters by the one without them in the lower order amplitudes.', 'hep-ph-9812485-1-29-0': 'After the diagonalization of the leading logarithmic terms of the vector boson polarization operator Eq.([REF]) can be reduced to relations for vertices describing scattering of fermions in the external field [MATH] substituting the heavy [MATH]-boson: [EQUATION]', 'hep-ph-9812485-1-29-1': ""The quantities [MATH] and [MATH] in Eq.([REF]) are calculated in the 't Hooft-Feynman gauge using diagrams in Figs. [REF], [REF]."", 'hep-ph-9812485-1-29-2': 'The computation gives the following expressions for the left and right fermions, respectively: [EQUATION] with [EQUATION]', 'hep-ph-9812485-1-29-3': 'The notation [MATH] stands for the iso-partner of [MATH], [MATH], [MATH], [MATH]), [MATH] equals [MATH] for quarks and zero for leptons, [EQUATION] where [MATH] denotes leptons, and [MATH] is the third component of the SM isospin.', 'hep-ph-9812485-1-30-0': 'The SM values of the one-loop fermion anomalous dimensions are known [CITATION]: [EQUATION]', 'hep-ph-9812485-1-30-1': 'Employing Eqs.([REF])-([REF]), it is easy to compute [MATH] and [MATH] in one-loop order: [EQUATION] where [MATH].', 'hep-ph-9812485-1-31-0': 'Since the [MATH]- function is the same for both Eqs.([REF]), the following relations can be derived [EQUATION]', 'hep-ph-9812485-1-31-1': 'Hence, it follows that originally independent parameters [MATH], [MATH], [MATH] are appeared to be connected.', 'hep-ph-9812485-1-31-2': 'Finally, one can check that the RG relation ([REF]) is fulfilled only if Eq. ([REF]) holds.', 'hep-ph-9812485-1-32-0': 'In fact, the relation ([REF]) means that the [MATH]-boson couplings to the SM axial currents have the universal absolute value, if a single light scalar doublet exists.', 'hep-ph-9812485-1-32-1': 'So, among the four parameters [MATH], [MATH], [MATH], [MATH] parameterizing interaction between the [MATH]-boson and the [MATH] fermionic isodoublet only one is really arbitrary.', 'hep-ph-9812485-1-32-2': 'The rest ones can be expressed through it and the hypercharge [MATH] of the [MATH] coupling to the SM scalar doublet.', 'hep-ph-9812485-1-32-3': 'Thus, the fermion and the scalar sectors of the new physics are connected.', 'hep-ph-9812485-1-33-0': '# Dependences between the parameters of the effective Lagrangian', 'hep-ph-9812485-1-34-0': 'The relations ([REF]) are resulted in the dependences between the parameters of the effective Lagrangians ([REF]).', 'hep-ph-9812485-1-34-1': 'In Ref.[CITATION] all the effective operators [MATH] describing deviations from the SM at the tree level and appearing due to heavy virtual states have been derived.', 'hep-ph-9812485-1-34-2': 'Additional heavy [MATH]-boson produces the following effective interactions: [EQUATION] where [MATH] and [MATH] stand for leptons and quarks; [MATH] is the electroweak covariant derivative, and the definitions of [MATH] correspond to Ref.[CITATION], respectively: [EQUATION]', 'hep-ph-9812485-1-34-3': 'Thus, the relations ([REF]) produce the following dependences between the parameters [MATH]: [EQUATION]', 'hep-ph-9812485-1-34-4': 'So, only the constants parameterizing the processes [MATH] and [MATH] are to be arbitrary.', 'hep-ph-9812485-1-35-0': 'Naturally, the relations ([REF]) hold if the only heavy particle is the abelian neutral boson.', 'hep-ph-9812485-1-35-1': 'Models including other types of heavy particles may produce other specific relations between the parameters [MATH].', 'hep-ph-9812485-1-35-2': 'However, the advantage of the relations ([REF]) is its independence of the mechanism of arising of abelian [MATH]-boson in some theory describing physics at energies [MATH].', 'hep-ph-9812485-1-36-0': '# Discussion', 'hep-ph-9812485-1-37-0': 'Let us discuss the results obtained.', 'hep-ph-9812485-1-37-1': 'The RG relation for the four-fermion scattering amplitude with the virtual [MATH]-boson in the [MATH]-channel is derived in the low-energy region [MATH].', 'hep-ph-9812485-1-37-2': 'It can be expressed in terms of the low-energy physics parameters only, since the decoupling theorem allows one to remove all the heavy particle loop contributions [MATH] to the RG relation by means of the redefinition of charges, fields and masses.', 'hep-ph-9812485-1-37-3': 'Since the difference between the original and the redefined parameters is of the one-loop order one can substitute one set of parameters by other in the one-loop RG relation.', 'hep-ph-9812485-1-38-0': 'It was shown also that the heavy virtual state can be considered as an external field scattering SM light particles.', 'hep-ph-9812485-1-38-1': 'In order to formulate the RG relations for corresponding vertices one has to avoid the one-loop mixing between heavy and light fields.', 'hep-ph-9812485-1-38-2': 'This can be done by the diagonalization of the leading logarithmic terms of the neutral vector boson polarization operator.', 'hep-ph-9812485-1-38-3': 'Since the difference between the diagonalized fields and charges and the original ones is of one-loop order, it is possible to omit the one-loop mixing terms in the RG relation at the lower level.', 'hep-ph-9812485-1-38-4': 'Hence, the RG relation for [MATH]-matrix element is reduced to the ones for vertices describing scattering of SM particles in the external field [MATH] substituting the heavy virtual [MATH]-boson.', 'hep-ph-9812485-1-39-0': 'The obtained RG relations contain the external field [MATH] and appropriate [MATH] and [MATH] functions.', 'hep-ph-9812485-1-39-1': 'These relations are the identities valid in each order of perturbation theory being the consequence of renormalizability of the original theory.', 'hep-ph-9812485-1-39-2': 'Due to [MATH] symmetry of the model the number of unknown [MATH]- and [MATH]- functions is less than the number of RG relations.', 'hep-ph-9812485-1-39-3': 'Therefore, the non-trivial dependences between the parameters describing new physics have been derived.', 'hep-ph-9812485-1-39-4': 'As the result, the number of arbitrary parameters in the tree-level EL describing deviations from the SM induced by the abelian [MATH]-boson is decreased.', 'hep-ph-9812485-1-39-5': 'The obtained relations are independent of the mechanism which leads to [MATH] in some theory describing physics at energies [MATH].', 'hep-ph-9812485-1-40-0': 'In Ref. [CITATION] a general renormalization framework for [MATH] theory was presented, and, in particular, the set of relevant counterterms has been derived.', 'hep-ph-9812485-1-40-1': 'Owing to [MATH] symmetry the same counterterms were postulated for different [MATH] couplings to fermions.', 'hep-ph-9812485-1-40-2': 'However, as it was shown above, this fact requires the specific relations between the hypercharges [MATH], [MATH], [MATH].', 'hep-ph-9812485-1-40-3': 'To determine these relations one has to parameterize the [MATH] coupling to the light scalar fields of the model.', 'hep-ph-9812485-1-40-4': 'Naturally, the relations between hypercharges hold automatically for a specific renormalizable higher-energy theory.', 'hep-ph-9812485-1-40-5': 'However, if the hypercharges are treated as unknown parameters these relations are to be taken into account in order to respect the gauge symmetry.', 'hep-ph-9812485-1-41-0': 'In fact, the obtained relation ([REF]) demonstrates that the fermion and the scalar sectors of the new physics are strongly connected.', 'hep-ph-9812485-1-41-1': 'The couplings of the [MATH]-boson to the SM axial currents are completely determined by its interaction with the scalar fields, if a single light scalar doublet is assumed.', 'hep-ph-9812485-1-41-2': 'When the [MATH]-boson coupling to the scalar doublet is switched off ([MATH]), the [MATH]-boson interacts at the tree level with the SM vector currents, only ([MATH]).', 'hep-ph-9812485-1-41-3': 'But even in this case the couplings to the axial currents are to be produced by loops being suppressed by the additional small factor [MATH].', 'hep-ph-9812485-1-42-0': 'As it was mentioned in section [REF], the [MATH] interaction with the scalar doublet is completely determined by the observable [MATH] (see Eq. ([REF])).', 'hep-ph-9812485-1-42-1': 'Therefore, one is able to predict the coupling of [MATH] to the SM axial currents.', 'hep-ph-9812485-1-42-2': 'When [MATH]-boson does not interact with the scalar doublet the physical mass of [MATH]-boson is to be identical to its SM value.', 'hep-ph-9812485-1-42-3': 'It is worth noticing that the experimental value of [MATH] in Eq. ([REF]) must be properly chosen in order to satisfy the definition [MATH].', 'hep-ph-9812485-1-42-4': 'The [MATH] existence has not to affect the quantity [MATH] in order [MATH].', 'hep-ph-9812485-1-42-5': 'So, the determination of the Weinberg angle in terms of the QED coupling constant [MATH] and the Fermi constant [MATH] measured from the muon decay is preferable, since the tree level values of these parameters [MATH], [MATH] include the contributions of [MATH]-bosons and the QED sector, only.', 'hep-ph-9812485-1-43-0': 'The proposed method can be applied to find relations of the derived type giving possibility to search for other heavy virtual states.', 'hep-ph-9812485-1-43-1': 'Investigations of the specific models will be the problem for future.', 'hep-ph-9812485-1-44-0': 'wudka J. Wudka, Int.', 'hep-ph-9812485-1-44-1': 'J. of Modern Phys.', 'hep-ph-9812485-1-44-2': 'arzt C. Arzt, M. Einhorn, and J. Wudka, Nucl.', 'hep-ph-9812485-1-44-3': 'bog-shirk N. N. Bogoliubov and D. V. Shirkov, Introduction to the Theory of Quantum Fields, Nauka, Moscow, 1984.', 'hep-ph-9812485-1-44-4': 'hhiggs A. V. Gulov and V. V. Skalozub, Yad.', 'hep-ph-9812485-1-44-5': 'sirlin G. Degrassi and A. Sirlin, Phys.', 'hep-ph-9812485-1-44-6': 'Li, Gauge Theory of Elementary Particle Physics, Clarendon Press, Oxford, 1984.', 'hep-ph-9812485-1-44-7': 'bando M. Bando et al., Progress of Theor.', 'hep-ph-9812485-1-44-8': '2sc A. V. Gulov and V. V. Skalozub, hep-ph/9811286 (submitted for publication in Yad.', 'hep-ph-9812485-1-44-9': 'decoupling T. Appelquist and J. Carazzone, Phys.', 'hep-ph-9812485-1-45-0': 'J. C. Collins et al., Phys.', 'hep-ph-9812485-1-45-1': 'smvalues M. B[MATH]hm, W. Hollik, and H. Spiesberger, Fortschr.'}","{'hep-ph-9812485-2-0-0': 'The method of searching for signals of new physics beyond the Standard Model based on the renormalization group equation and the decoupling theorem is developed.', 'hep-ph-9812485-2-0-1': 'It is applied to find signals of new heavy abelian [MATH]-boson and realized in the form of specific relations between scattering amplitudes at low energies.', 'hep-ph-9812485-2-0-2': 'Considering four-fermion processes, a basic one-loop renormalization group relation is derived which gives possibility to reduce the problem to the scattering of Standard Model particles in the ""external field"" substituting heavy virtual [MATH] state.', 'hep-ph-9812485-2-0-3': 'From the set of derived relations it is determined that the absolute value of the [MATH] coupling to the axial currents has to be the same for all fermions of the Standard Model and is strongly related to the [MATH] coupling to the scalar fields.', 'hep-ph-9812485-2-0-4': 'The corresponding dependences between the parameters of the effective Lagrangians describing at low energies deviations from the Standard Model are derived.', 'hep-ph-9812485-2-1-0': '# Introduction', 'hep-ph-9812485-2-2-0': 'An important problem of nowadays high energy physics is searching for deviations from the Standard Model (SM) of elementary particles.', 'hep-ph-9812485-2-2-1': 'At energies of the present day accelerators, the deviations may appear due to virtual states of new heavy particles entering the extended models and having the masses much greater than the W-boson mass [MATH].', 'hep-ph-9812485-2-2-2': 'One of approaches to describe contributions of these heavy states is the construction of the effective Lagrangians (EL) [CITATION].', 'hep-ph-9812485-2-2-3': 'Owing to decoupling of heavy particles the EL can be expressed in the form of the expansion in inverse powers of heavy mass [MATH]: [EQUATION] where [MATH] stand for various dimension-six operators constructed from the SM fields and their derivatives, and [MATH] are some numbers which are dependent on an underlying theory beyond the SM.', 'hep-ph-9812485-2-2-4': 'Since at present time such a theory is unknown, the parameters [MATH] are usually considered as arbitrary numbers to be determined from experiments.', 'hep-ph-9812485-2-3-0': 'In principle, it is possible to write down a lot of different EL parameterizing new physics effects.', 'hep-ph-9812485-2-3-1': 'In Ref. [CITATION] the EL generated at a tree level in a general renormalizable gauge theory have been derived.', 'hep-ph-9812485-2-3-2': 'These objects by construction contain a great number of arbitrary parameters responsible for specific processes.', 'hep-ph-9812485-2-3-3': 'However, as is known, a renormalizable theory is fixed by few independent constants.', 'hep-ph-9812485-2-3-4': 'That has to manifest itself at different energy scales and, in particular, at low energies when [MATH] is applicable.', 'hep-ph-9812485-2-3-5': 'Therefore, one could expect to find some correlations between the parameters [MATH] followed from the renormalizability.', 'hep-ph-9812485-2-3-6': 'In Ref. [CITATION] a set of the relations between [MATH], called the renormalization group (RG) relations, has been derived.', 'hep-ph-9812485-2-3-7': 'It is the consequence of the RG equations for the [MATH]-matrix elements and the decoupling theorem [CITATION] giving possibility to move from high energies where heavy particles are excited inside loops to low energies where they are decoupled and determine [MATH] explicitly.', 'hep-ph-9812485-2-3-8': 'Some remnants of these properties can be used even if the underlying theory (for example, GUT) is not specified.', 'hep-ph-9812485-2-4-0': 'Two main observations were used.', 'hep-ph-9812485-2-4-1': 'First, it has been shown that a heavy virtual state can be treated as an external field scattering SM light particles.', 'hep-ph-9812485-2-4-2': 'Second, the renormalizability of the theory was used in a way to incorporate information about the [MATH]- and [MATH]- functions entering into RG operator at low energies in the framework of the external field problem.', 'hep-ph-9812485-2-4-3': 'The key point of the renormalizability of the vertices was exploited: the algebraic structure of the divergent part of the scattering amplitude calculated in higher orders of perturbation theory coincides with that of the tree-vertex.', 'hep-ph-9812485-2-4-4': 'Hence, the relation between the vertex coefficient and the [MATH]- and [MATH]- functions follows.', 'hep-ph-9812485-2-4-5': 'If an underlying (GUT) theory is specified, the relation is just the identity.', 'hep-ph-9812485-2-4-6': 'But if the theory is not assumed and the number of [MATH] and [MATH] functions is less than the number of the RG relations, a non-trivial system of algebraic equations for vertex coefficients and, hence, for parameters [MATH] in Eq. ([REF]) can be obtained.', 'hep-ph-9812485-2-4-7': 'In Ref. [CITATION], as an instance, the SM with the heavy Higgs scalar was investigated.', 'hep-ph-9812485-2-4-8': 'It has been shown that the above noted scenario is realized for gauge couplings.', 'hep-ph-9812485-2-5-0': 'In the present paper the model based on the gauge group [MATH] is investigated.', 'hep-ph-9812485-2-5-1': 'It is supposed to describe the low-energy phenomenology of some unknown GUT theory.', 'hep-ph-9812485-2-5-2': 'The model includes the minimal SM and the additional heavy [MATH]- boson called [MATH].', 'hep-ph-9812485-2-5-3': 'At the current energies [MATH] the heavy boson is decoupled producing EL ([REF]).', 'hep-ph-9812485-2-5-4': 'Since the new physics couplings in the model are of the gauge type, we are able to use RG relations in searching for dependences between the parameters [MATH].', 'hep-ph-9812485-2-6-0': 'We will formulate low-energy RG relations for four-fermion amplitudes in the order [MATH].', 'hep-ph-9812485-2-6-1': 'As it will be shown, the relations hold only if the absolute value of [MATH] coupling to the axial currents is the same for all the SM fermions.', 'hep-ph-9812485-2-6-2': 'Hence, some of the parameters [MATH] are not independent.', 'hep-ph-9812485-2-6-3': 'Moreover, the fermion sector of the new physics is appeared to be correlated with the scalar one.', 'hep-ph-9812485-2-6-4': 'Therefore, the number of parameters describing deviations from the SM is decreased.', 'hep-ph-9812485-2-7-0': 'The paper is organized as follows: in section [REF] the model investigated is described.', 'hep-ph-9812485-2-7-1': 'In section [REF] the general RG relation for an [MATH]-matrix element at low energies is derived.', 'hep-ph-9812485-2-7-2': 'It is applied to four-fermion scattering processes in section [REF] to obtain dependences between the vertex parameters describing new physics at low energies.', 'hep-ph-9812485-2-7-3': 'In section [REF] the corresponding relations for parameters [MATH] of the EL are derived.', 'hep-ph-9812485-2-7-4': 'The results of the investigation are summarized in section [REF].', 'hep-ph-9812485-2-8-0': '# The Model', 'hep-ph-9812485-2-9-0': 'The model under consideration is based on the gauge group [MATH] which is assumed to be a low energy remnant of some underlying theory.', 'hep-ph-9812485-2-9-1': 'It was investigated before in Ref. [CITATION].', 'hep-ph-9812485-2-9-2': 'The corresponding additional abelian vector boson [MATH] is supposed to be massive with the mass [MATH] much greater than the [MATH]-boson mass [MATH].', 'hep-ph-9812485-2-9-3': 'In the present paper the mechanism of the heavy mass generation is not considered since the theory describing interactions at energies [MATH] is unknown.', 'hep-ph-9812485-2-9-4': 'So, the Lagrangian includes the explicit mass term for the [MATH]-boson, and the mass [MATH] is treated as the parameter of the model.', 'hep-ph-9812485-2-10-0': 'In Ref. [CITATION] the interaction of scalar fields with the [MATH]-boson was not specified.', 'hep-ph-9812485-2-10-1': 'However, to write down the RG equation one needs to parameterize explicitly the [MATH] coupling to the scalar fields of the theory.', 'hep-ph-9812485-2-10-2': 'Since the vector boson interactions with the SM fermions [MATH] and the SM scalar doublet [MATH] are specified by covariant derivatives, the relevant Lagrangian reads [EQUATION] where [MATH] is the fermion charge in the positron charge units, [MATH] are the Pauli matrices, [MATH] is a [MATH] unity matrix, [MATH] for leptons and [MATH] for quarks, and the summation over all the SM left-handed doublets [MATH] and the SM right-handed singlets [MATH] is understood.', 'hep-ph-9812485-2-10-3': 'The charges [MATH] and the fields [MATH], [MATH], [MATH] correspond to the gauge groups [MATH], [MATH], [MATH], respectively.', 'hep-ph-9812485-2-10-4': 'The values of the dimensionless constants [MATH], [MATH], [MATH] depend on the particular higher-energy theory including [MATH]-boson.', 'hep-ph-9812485-2-10-5': 'Here, they are to be considered as arbitrary parameters.', 'hep-ph-9812485-2-11-0': 'Up to this point the electroweak symmetry [MATH] was supposed to be unbroken, and all the SM particles were massless.', 'hep-ph-9812485-2-11-1': 'The masses of the SM particles are generated by the spontaneous breakdown of the symmetry [MATH] originated due to the non-zero vacuum expectation value of the scalar doublet.', 'hep-ph-9812485-2-11-2': 'However, the mass eigenstates are appeared to be shifted away from the original fields [MATH], since the corresponding mass terms become non-diagonal.', 'hep-ph-9812485-2-11-3': 'The physical fields [MATH] are obtained by the orthogonal transformation: [EQUATION] where [MATH] is the SM value of the Weinberg angle ([MATH]), and [MATH] denotes the mixing angle relating physical states [MATH] to massive neutral components of [MATH] gauge fields.', 'hep-ph-9812485-2-11-4': 'The value of the angle [MATH] can be determined from the relation [EQUATION] confirming Ref. [CITATION].', 'hep-ph-9812485-2-12-0': 'The masses of the physical fields are given by the expressions: [EQUATION]', 'hep-ph-9812485-2-12-1': 'As is seen, the mass of the physical [MATH]-boson differs from the SM value [MATH] by the small quantity of order [MATH].', 'hep-ph-9812485-2-12-2': 'So, the mixing angle [MATH] is also small [MATH].', 'hep-ph-9812485-2-12-3': 'The difference [MATH] is negative and completely determined by the [MATH] coupling to the scalar doublet.', 'hep-ph-9812485-2-12-4': 'Thus, constraints on the [MATH] interaction with scalar fields can be derived by means of experimental detecting this observable: [EQUATION]', 'hep-ph-9812485-2-12-5': 'Using Eqs.([REF]) the Lagrangian of the model can be expressed in terms of the physical fields.', 'hep-ph-9812485-2-12-6': 'The dependence on the mixing [MATH] in the Eqs.([REF]) causes new interactions [MATH] originally absent in the Lagrangian ([REF]).', 'hep-ph-9812485-2-13-0': 'In the present paper we will deal with four-fermion amplitudes of order [MATH] produced by virtual [MATH]-boson in the [MATH]-channel.', 'hep-ph-9812485-2-13-1': 'The corresponding tree level diagram is shown in Fig. [REF].', 'hep-ph-9812485-2-13-2': 'At low energies the effective four-fermion vertex is generated due to the heavy mass in the [MATH]-boson propagator [MATH].', 'hep-ph-9812485-2-13-3': 'So, expressions involving such amplitudes can be computed neglecting the terms proportional to [MATH] in the Lagrangian.', 'hep-ph-9812485-2-13-4': ""Only linear in [MATH] terms of the Lagrangian are needed, which have the following form in the 't Hooft-Feynman gauge: [EQUATION] where [MATH]; [MATH]; [MATH] is the positron charge; [MATH] denotes the vacuum expectation value of the scalar doublet; [MATH] is the Higgs scalar particle; [MATH] and [MATH] are the Goldstone fields."", 'hep-ph-9812485-2-14-0': '# RG relations in a theory with decoupling', 'hep-ph-9812485-2-15-0': 'As we mentioned before, the model under consideration is supposed to be the low-energy limit of some unspecified renormalizable theory describing interactions at energies [MATH].', 'hep-ph-9812485-2-15-1': 'As is known, due to renormalizability the [MATH]-matrix elements are invariant with respect to the RG transformations which express independence of the amplitudes of the normalization point [MATH].', 'hep-ph-9812485-2-15-2': 'In a theory with different mass scales the RG flow and the decoupling of heavy loop contributions at the thresholds of heavy masses give possibility to determine all the parameters of scattering amplitudes.', 'hep-ph-9812485-2-15-3': 'The decoupling is resulted in the important property of the low energy amplitudes: the running of the proper functions is regulated by the loops of light particles.', 'hep-ph-9812485-2-15-4': 'Therefore, the [MATH] and [MATH] functions at low energies are determined by the SM particles, only.', 'hep-ph-9812485-2-15-5': 'The latter fact allows to formulate a series of the RG relations for scattering amplitudes.', 'hep-ph-9812485-2-15-6': 'We will derive them for the case of heavy [MATH]-boson which as well as other heavy particles is decoupled, [MATH].', 'hep-ph-9812485-2-16-0': 'We will use the notations: [MATH] (where [MATH], [MATH] and [MATH] denote the QCD charge, the coupling of Yukawa interaction between [MATH] and [MATH] and the scalar self-coupling, respectively) and [MATH] (where [MATH] represents all the fields of the theory, and [MATH] are particle masses) to refer to the charges, the fields and the masses.', 'hep-ph-9812485-2-16-1': 'Then the RG equation reads: [EQUATION] where renormalized (running) parameters [MATH] are the solutions to the equations: [EQUATION]', 'hep-ph-9812485-2-16-2': 'Hats over the quantities mark the parameters of the underlying theory.', 'hep-ph-9812485-2-16-3': 'They include the contributions of loops containing both the SM particles and the heavy ones.', 'hep-ph-9812485-2-17-0': 'Below, in carrying out calculations the dimensional regularization and the [MATH] renormalization scheme [CITATION] will be used.', 'hep-ph-9812485-2-17-1': 'The [MATH]-scheme is mass-independent, so the RG operator [MATH] has the same form at high, [MATH], and low, [MATH], energies.', 'hep-ph-9812485-2-18-0': 'In the energy range [MATH] all the heavy fields are decoupled.', 'hep-ph-9812485-2-18-1': 'So, one can introduce the running parameters [MATH] completely defined by the low-energy effective theory [CITATION]: [EQUATION] where [MATH]- and [MATH]- functions contain no contributions from loops with heavy particles.', 'hep-ph-9812485-2-18-2': 'These parameters can be expressed in terms of the original ones: [EQUATION] where the matching between the both sets of parameters ([MATH] and [MATH]) is chosen to be done at the normalization point [MATH]: [EQUATION]', 'hep-ph-9812485-2-18-3': 'Actually, Eq.([REF]) is the redefinition of the parameters of the theory.', 'hep-ph-9812485-2-18-4': 'As it has been shown in Refs.[CITATION], all the heavy particle loop contributions proportional to [MATH] are completely removed from the RG equation ([REF]) by such a redefinition.', 'hep-ph-9812485-2-18-5': 'This fact is the consequence of the decoupling theorem [CITATION].', 'hep-ph-9812485-2-19-0': 'Then, the RG equation ([REF]) for the [MATH]-matrix expressed in terms of the low-energy quantities [MATH] becomes: [EQUATION] where [MATH] and [MATH] denote, respectively, the RG operator and the [MATH]-matrix element of the low-energy effective theory, calculated without the heavy particle loops.', 'hep-ph-9812485-2-20-0': 'The familiar usage of Eq.([REF]) is to improve scattering amplitudes calculated in perturbation theory.', 'hep-ph-9812485-2-20-1': 'In contrast, in what follows we will apply Eq.([REF]) to obtain algebraic relation between the parameters [MATH], [MATH], [MATH], which are to be considered as arbitrary numbers, if the underlying theory is not assumed.', 'hep-ph-9812485-2-20-2': 'The reasons for that are as follow.', 'hep-ph-9812485-2-20-3': 'If the underlying theory is specified ([MATH], [MATH], [MATH] are to be calculated), this relation is just the identity for the [MATH], [MATH] functions and the [MATH]-matrix element valid in every order of perturbation theory.', 'hep-ph-9812485-2-20-4': 'In a sense, it is a necessary condition of the renormalizability.', 'hep-ph-9812485-2-20-5': 'If the underlying theory is not specified, then, as we will show for four-fermion processes at one-loop level, the relation ([REF]) holds if the parameters [MATH], [MATH], [MATH] satisfy some specific relations.', 'hep-ph-9812485-2-20-6': 'The latter are resulted in the dependences between the appropriate parameters [MATH] in the EL ([REF]).', 'hep-ph-9812485-2-21-0': 'Let [MATH] and [MATH] represent the one-loop and the tree parts of the [MATH]-matrix element, respectively.', 'hep-ph-9812485-2-21-1': 'Introducing the one-loop RG operator: [EQUATION] where [MATH] and [MATH] functions are computed in one-loop order, one can derive from Eq.([REF]) the following identity: [EQUATION]', 'hep-ph-9812485-2-21-2': 'However, since the parameters [MATH] and [MATH] differ at the one-loop level, one could substitute one set of them by another in Eq.([REF]).', 'hep-ph-9812485-2-22-0': 'Eq.([REF]) is the starting point to analyze the RG relations for amplitudes at low energies.', 'hep-ph-9812485-2-22-1': 'The relation ([REF]) insures the leading logarithmic terms of the one-loop [MATH]-matrix element to reproduce the appropriate tree-level structure that is the simple consequence of the renormalizability.', 'hep-ph-9812485-2-22-2': 'When the couplings are represented by arbitrary unknown parameters, as it takes place in the considered model and, in general, in the discussed effective Lagrangian approach, the relation ([REF]) is the one-loop equation for the parameters which must be determined with the computed [MATH]- and [MATH]- functions.', 'hep-ph-9812485-2-22-3': 'If due to a symmetry the number of [MATH]- and [MATH]- functions is less than the number of RG relations, the non-trivial system of equations correlating the originally independent parameters may occur.', 'hep-ph-9812485-2-22-4': 'We will use Eq. ([REF]) to derive the dependences between the parameters [MATH], [MATH], [MATH].', 'hep-ph-9812485-2-23-0': '# RG relations for four-fermion amplitudes', 'hep-ph-9812485-2-24-0': 'In this section the one-loop RG relation ([REF]) is applied to the four-fermion amplitudes [MATH] containing [MATH]-boson as the virtual state.', 'hep-ph-9812485-2-24-1': 'In the lower order in [MATH] such amplitudes produce the effective contact interactions of the type current[MATH]current described by the effective Lagrangians ([REF]) [CITATION].', 'hep-ph-9812485-2-25-0': 'Let the renormalized fields, masses and charges are defined as follows: [EQUATION] where [MATH] is the vertex describing interaction between fermion [MATH] and neutral vector boson; [MATH] and [MATH] are [MATH] matrices ([MATH]).', 'hep-ph-9812485-2-25-1': 'The fermion renormalization constant [MATH] is the SM one.', 'hep-ph-9812485-2-25-2': 'Its value is known [CITATION].', 'hep-ph-9812485-2-25-3': 'Quantities [MATH] and [MATH] are represented at the one-loop level by the divergent ([MATH]) terms of the gauge-independent part of the polarization operator of neutral vector bosons [MATH] being evaluated from the diagrams in Fig. [REF] [EQUATION] with [MATH] for [MATH] and [MATH] otherwise.', 'hep-ph-9812485-2-25-4': 'To find the constant [MATH] in one-loop order one has to calculate the divergent ([MATH]) part of the vertex function [MATH] (the diagrams in Fig. [REF]): [EQUATION] where [MATH] is the vertex [MATH] at the tree level.', 'hep-ph-9812485-2-26-0': 'The [MATH]-matrix element expressed in terms of the renormalized quantities is finite.', 'hep-ph-9812485-2-26-1': 'It can be evaluated using the diagrams in Figs. [REF],[REF] where shaded blobs stand for the finite parts of the vertex function [MATH] and the polarization operator [MATH].', 'hep-ph-9812485-2-26-2': 'On the mass-shell of fermions the gauge-dependent part of [MATH] gives no contribution to the physical amplitude due to the transversality of the vertex [MATH].', 'hep-ph-9812485-2-27-0': 'Eq. ([REF]) for the renormalized [MATH]-matrix element can not be reduced to relations for vertices, because of the one-loop mixing between neutral bosons.', 'hep-ph-9812485-2-27-1': 'To derive these relations one has to avoid the mixing.', 'hep-ph-9812485-2-27-2': 'The corresponding procedure was described in Ref. [CITATION].', 'hep-ph-9812485-2-28-0': 'One has to introduce the quantities [MATH] and [MATH] instead of fields [MATH], [MATH], [MATH] and vertices [MATH] according to the following relations: [EQUATION]', 'hep-ph-9812485-2-28-1': 'After such a redefinition the mixing between neutral bosons in the equation ([REF]) vanishes.', 'hep-ph-9812485-2-28-2': 'In what follows, we will use the quantities [MATH] and [MATH], therefore only the diagonal terms of the polarization operator are needed in computing of the [MATH]-matrix element [MATH].', 'hep-ph-9812485-2-28-3': 'Since the difference between the two sets of parameters is of order [MATH], one can substitute the tilded parameters by the other set in the amplitudes of lower order.', 'hep-ph-9812485-2-29-0': 'When the diagonalization of the leading logarithmic terms of the vector boson polarization operator is fulfilled, Eq.([REF]) can be reduced to relations for vertices describing scattering of fermions in the external field [MATH] substituting the heavy [MATH]-boson: [EQUATION]', 'hep-ph-9812485-2-29-1': ""The quantities [MATH] and [MATH] in Eq.([REF]) have been calculated in the 't Hooft-Feynman gauge using diagrams in Figs. [REF], [REF]."", 'hep-ph-9812485-2-29-2': 'The computation gives the following relations for the left-handed and right-handed fermions, respectively: [EQUATION] where [EQUATION]', 'hep-ph-9812485-2-29-3': 'The notation [MATH] stands for the iso-partner of [MATH], [MATH], [MATH], [MATH]), [MATH] equals to [MATH] for quarks and zero for leptons, [EQUATION] and [MATH] denotes leptons, [MATH] is the third component of the weak isospin.', 'hep-ph-9812485-2-30-0': 'The SM values of the one-loop fermion anomalous dimensions are known [CITATION]: [EQUATION]', 'hep-ph-9812485-2-30-1': 'By employing Eqs.([REF])-([REF]), it is easy to compute [MATH] and [MATH] in one-loop order: [EQUATION] where [MATH].', 'hep-ph-9812485-2-31-0': 'Since the [MATH]- function is the same for both Eqs.([REF]), the following relations can be derived [EQUATION]', 'hep-ph-9812485-2-31-1': 'Hence one can see that originally independent parameters [MATH], [MATH], [MATH] are appeared to be the connected ones.', 'hep-ph-9812485-2-31-2': 'Finally, one can also check that the RG relations ([REF]) are fulfilled only if Eq. ([REF]) holds.', 'hep-ph-9812485-2-32-0': 'In a sense, the relations ([REF]) mean that the [MATH]-boson couplings to the SM axial currents have the universal absolute value, if a single light scalar doublet exists.', 'hep-ph-9812485-2-32-1': 'So, among the four constants [MATH], [MATH], [MATH], [MATH] parameterizing interaction between the [MATH]-boson and the [MATH] fermion isodoublet only one is really arbitrary.', 'hep-ph-9812485-2-32-2': 'The rest ones can be expressed through it and the hypercharge [MATH] of the [MATH] coupling to the SM scalar doublet.', 'hep-ph-9812485-2-32-3': 'Thus, the fermion and the scalar sectors of new physics are to be correlated.', 'hep-ph-9812485-2-33-0': 'Being derived from the renormalizability condition for a scattering amplitude in the external field of heavy particle, the relations ([REF]) respect gauge invariance and have a transparent interpretation.', 'hep-ph-9812485-2-33-1': 'First of them means that that the upper and lower components of a left-handed doublet transform in the same way under [MATH] gauge group.', 'hep-ph-9812485-2-33-2': 'If it does not hold, [MATH] gauge invariance would violate.', 'hep-ph-9812485-2-33-3': 'The second of relations guarantees, that the SM Yukawa interaction terms have to be invariant under [MATH] gauge transformations.', 'hep-ph-9812485-2-34-0': '# Dependences between the parameters of the effective Lagrangian', 'hep-ph-9812485-2-35-0': 'The relations ([REF]) are resulted in the dependences between the parameters of the EL ([REF]).', 'hep-ph-9812485-2-35-1': 'In Ref.[CITATION] all the effective operators [MATH] describing deviations from the SM and appearing at the tree level due to heavy virtual states have been derived.', 'hep-ph-9812485-2-35-2': 'Additional heavy [MATH]-boson produces the following effective vertices: [EQUATION] where [MATH] and [MATH] stand for leptons and quarks; [MATH] is the electroweak covariant derivative, and the definitions of [MATH] correspond to Ref.[CITATION]: [EQUATION]', 'hep-ph-9812485-2-35-3': 'Thus, the relations ([REF]) give rise to the following dependences between the parameters [MATH]: [EQUATION]', 'hep-ph-9812485-2-35-4': 'Hence it follows that the arbitrary constants are to be the ones parameterizing the processes [MATH] and [MATH].', 'hep-ph-9812485-2-36-0': 'The relations ([REF]) hold if the only additional heavy particle extending the SM is the abelian [MATH]-boson.', 'hep-ph-9812485-2-36-1': 'Other models may lead to other type of correlations between the parameters [MATH].', 'hep-ph-9812485-2-36-2': 'The advantage of the expressions ([REF]) is their independence of the mechanism of arising of the abelian [MATH]-boson in a specific theory describing physics at energy scale [MATH].', 'hep-ph-9812485-2-37-0': '# Discussion', 'hep-ph-9812485-2-38-0': 'Let us discuss the main points of our analysis and further perspectives.', 'hep-ph-9812485-2-38-1': 'Usually, the EL ([REF]) describing at low energies physics beyond the SM contains a set of the operators with parameters [MATH] assumed to be arbitrary numbers which must be fixed in experiments.', 'hep-ph-9812485-2-38-2': 'However, as we have demonstrated above considering the additional [MATH]-boson as extension of the SM, the renormalizability of the underlying theory may result in some correlations between [MATH] which have to be of interest in searching for new physics effects.', 'hep-ph-9812485-2-38-3': 'Most important is that the derived relations ([REF]), being the consequence of the renormalizability formulated in the framework of scattering in the external field, are independent of the specific underlying (GUT) model.', 'hep-ph-9812485-2-39-0': 'The latter statement requires to be discussed in more detail.', 'hep-ph-9812485-2-39-1': 'The Lagrangian ([REF]) respects [MATH] gauge invariance but includes only the interactions of the renormalizable type.', 'hep-ph-9812485-2-39-2': 'The terms of the form [MATH] are omitted because they, being generated at GUT (or some intermediate [MATH]) mass scale, are suppressed by the factors [MATH].', 'hep-ph-9812485-2-39-3': 'Hence one can conclude that at low energies, [MATH], the renormalizable interactions are to be dominant.', 'hep-ph-9812485-2-39-4': 'Therefore, the only model dependent information in Eq. ([REF]) is the specific values of the parameters [MATH], [MATH] and [MATH].', 'hep-ph-9812485-2-39-5': 'The relations ([REF]) hold for arbitrary values of them, as it was demonstrated in section [REF].', 'hep-ph-9812485-2-39-6': 'As a consequence, the relations ([REF]) also hold for arbitrary [MATH], [MATH], [MATH] and are independent of them explicitly.', 'hep-ph-9812485-2-40-0': 'Now, let us consider a more general parameterization in Eq. ([REF]).', 'hep-ph-9812485-2-40-1': 'In principle, one could employ the most general form of the generator [MATH] which could be just the part of the appropriate generator of the underlying theory.', 'hep-ph-9812485-2-40-2': 'Then, the RG relations can be used to obtain all possible solutions for the numbers [MATH], [MATH] and [MATH].', 'hep-ph-9812485-2-40-3': 'As one can check, the two sets of correlations are allowed.', 'hep-ph-9812485-2-40-4': 'The first one, [EQUATION] describes the vector boson analogous to the SM field [MATH].', 'hep-ph-9812485-2-40-5': 'Such a field may appear as a component of the non-abelian gauge field.', 'hep-ph-9812485-2-40-6': 'The second solution leads to the relations derived in this paper: [EQUATION]', 'hep-ph-9812485-2-40-7': 'They correspond to the abelian vector boson considered.', 'hep-ph-9812485-2-40-8': 'Thus, the proposed method to relate the parameters [MATH] could be applied to interactions of light SM particles with other heavy particles (leptoquarks, for example) treated as the external fields.', 'hep-ph-9812485-2-41-0': 'In Ref. [CITATION] a general renormalization framework for [MATH] theory was presented, and, in particular, the set of relevant counterterms has been derived.', 'hep-ph-9812485-2-41-1': 'Owing to [MATH] symmetry the same counterterms were postulated for different [MATH] couplings to fermions.', 'hep-ph-9812485-2-41-2': 'However, as it was shown above, this fact requires the specific relations between the hypercharges [MATH], [MATH], [MATH].', 'hep-ph-9812485-2-41-3': 'Naturally, the relations hold automatically for a specific renormalizable higher-energy theory.', 'hep-ph-9812485-2-41-4': 'However, if the hypercharges are treated as unknown parameters these relations are to be taken into account in order to respect the gauge symmetry.', 'hep-ph-9812485-2-42-0': 'In fact, the obtained relations ([REF]) demonstrate that the fermion and the scalar sectors of the new physics are strongly connected.', 'hep-ph-9812485-2-42-1': 'The couplings of the [MATH]-boson to the SM axial currents are completely determined by its interaction with the scalar fields, if a single light scalar doublet is assumed.', 'hep-ph-9812485-2-42-2': 'When the [MATH]-boson coupling to the scalar doublet is switched off ([MATH]), the [MATH]-boson interacts at the tree level with the SM vector currents, only ([MATH]).', 'hep-ph-9812485-2-42-3': 'But even in this case the couplings to the axial currents are to be produced by loops being suppressed by the additional small factor [MATH].', 'hep-ph-9812485-2-43-0': 'As it was mentioned in section [REF], the [MATH] interaction with the scalar doublet is completely determined by the observable [MATH] (see Eq. ([REF])).', 'hep-ph-9812485-2-43-1': 'Therefore, one is able to predict the coupling of [MATH] to the SM axial currents.', 'hep-ph-9812485-2-43-2': 'When [MATH]-boson does not interact with the scalar doublet the physical mass of [MATH]-boson is to be identical to its SM value.', 'hep-ph-9812485-2-43-3': 'It is worth noticing that the experimental value of [MATH] in Eq. ([REF]) must be properly chosen in order to satisfy the definition [MATH].', 'hep-ph-9812485-2-43-4': 'The [MATH] existence has not to affect the quantity [MATH] in order [MATH].', 'hep-ph-9812485-2-43-5': 'So, the determination of the Weinberg angle in terms of the QED coupling constant [MATH] and the Fermi constant [MATH] measured from the muon decay is preferable, since the tree level values of these parameters [MATH], [MATH] include the contributions of [MATH]-bosons and the QED sector, only.', 'hep-ph-9812485-2-44-0': 'The proposed method can be applied to find relations of the derived type giving possibility to search for other heavy virtual states.', 'hep-ph-9812485-2-44-1': 'Investigations of the specific models will be the problem for future.', 'hep-ph-9812485-2-45-0': 'wudka J. Wudka, Int.', 'hep-ph-9812485-2-45-1': 'J. of Modern Phys.', 'hep-ph-9812485-2-45-2': 'arzt C. Arzt, M. Einhorn, and J. Wudka, Nucl.', 'hep-ph-9812485-2-45-3': 'hhiggs A. V. Gulov and V. V. Skalozub, Yad.', 'hep-ph-9812485-2-45-4': 'decoupling T. Appelquist and J. Carazzone, Phys.', 'hep-ph-9812485-2-46-0': 'J. C. Collins, F. Wilczek, and A. Zee, Phys.', 'hep-ph-9812485-2-46-1': 'sirlin G. Degrassi and A. Sirlin, Phys.', 'hep-ph-9812485-2-46-2': 'Li, Gauge Theory of Elementary Particle Physics, Clarendon Press, Oxford, 1984.', 'hep-ph-9812485-2-46-3': 'bando M. Bando, T. Kugo, N. Maekawa, and H. Nakano, Progress of Theor.', 'hep-ph-9812485-2-46-4': '2sc A. V. Gulov and V. V. Skalozub, hep-ph/9811286 (be published in Yad.', 'hep-ph-9812485-2-46-5': 'smvalues M. B[MATH]hm, W. Hollik, and H. Spiesberger, Fortschr.'}","[['hep-ph-9812485-1-31-0', 'hep-ph-9812485-2-31-0'], ['hep-ph-9812485-1-10-3', 'hep-ph-9812485-2-10-3'], ['hep-ph-9812485-1-10-4', 'hep-ph-9812485-2-10-4'], ['hep-ph-9812485-1-3-0', 'hep-ph-9812485-2-3-0'], ['hep-ph-9812485-1-3-1', 'hep-ph-9812485-2-3-1'], ['hep-ph-9812485-1-3-2', 'hep-ph-9812485-2-3-2'], ['hep-ph-9812485-1-40-0', 'hep-ph-9812485-2-41-0'], ['hep-ph-9812485-1-40-1', 'hep-ph-9812485-2-41-1'], ['hep-ph-9812485-1-40-2', 'hep-ph-9812485-2-41-2'], ['hep-ph-9812485-1-40-5', 'hep-ph-9812485-2-41-4'], ['hep-ph-9812485-1-32-2', 'hep-ph-9812485-2-32-2'], ['hep-ph-9812485-1-0-0', 'hep-ph-9812485-2-0-0'], ['hep-ph-9812485-1-0-1', 'hep-ph-9812485-2-0-1'], ['hep-ph-9812485-1-0-2', 'hep-ph-9812485-2-0-2'], ['hep-ph-9812485-1-43-0', 'hep-ph-9812485-2-44-0'], ['hep-ph-9812485-1-43-1', 'hep-ph-9812485-2-44-1'], ['hep-ph-9812485-1-28-0', 'hep-ph-9812485-2-28-0'], ['hep-ph-9812485-1-28-2', 'hep-ph-9812485-2-28-2'], ['hep-ph-9812485-1-4-0', 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'hep-ph-9812485-2-7-1'], ['hep-ph-9812485-1-7-5', 'hep-ph-9812485-2-7-3'], ['hep-ph-9812485-1-7-6', 'hep-ph-9812485-2-7-4'], ['hep-ph-9812485-1-2-4', 'hep-ph-9812485-2-2-4'], ['hep-ph-9812485-1-15-1', 'hep-ph-9812485-2-15-1'], ['hep-ph-9812485-1-9-0', 'hep-ph-9812485-2-9-0'], ['hep-ph-9812485-1-18-3', 'hep-ph-9812485-2-18-3'], ['hep-ph-9812485-2-4-1', 'hep-ph-9812485-3-7-4'], ['hep-ph-9812485-2-18-3', 'hep-ph-9812485-3-21-3'], ['hep-ph-9812485-2-7-0', 'hep-ph-9812485-3-10-0'], ['hep-ph-9812485-2-7-0', 'hep-ph-9812485-3-10-1'], ['hep-ph-9812485-2-7-4', 'hep-ph-9812485-3-10-5'], ['hep-ph-9812485-2-10-3', 'hep-ph-9812485-3-13-2'], ['hep-ph-9812485-2-9-0', 'hep-ph-9812485-3-12-0'], ['hep-ph-9812485-2-9-2', 'hep-ph-9812485-3-12-1'], ['hep-ph-9812485-2-20-3', 'hep-ph-9812485-3-23-3'], ['hep-ph-9812485-2-20-3', 'hep-ph-9812485-3-23-5'], ['hep-ph-9812485-2-20-5', 'hep-ph-9812485-3-23-5'], ['hep-ph-9812485-2-0-0', 'hep-ph-9812485-3-0-0'], ['hep-ph-9812485-2-0-4', 'hep-ph-9812485-3-0-5'], ['hep-ph-9812485-2-15-6', 'hep-ph-9812485-3-18-6'], ['hep-ph-9812485-2-44-0', 'hep-ph-9812485-3-49-0'], ['hep-ph-9812485-2-44-1', 'hep-ph-9812485-3-49-1'], ['hep-ph-9812485-1-44-7', 'hep-ph-9812485-2-46-3'], ['hep-ph-9812485-1-44-8', 'hep-ph-9812485-2-46-4'], ['hep-ph-9812485-1-45-0', 'hep-ph-9812485-2-46-0']]","[['hep-ph-9812485-1-37-3', 'hep-ph-9812485-2-28-3'], ['hep-ph-9812485-2-3-6', 'hep-ph-9812485-3-5-4'], ['hep-ph-9812485-2-3-7', 'hep-ph-9812485-3-5-5'], ['hep-ph-9812485-2-5-1', 'hep-ph-9812485-3-12-0']]",[],"{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/'}",https://arxiv.org/abs/hep-ph/9812485,"{'hep-ph-9812485-3-0-0': 'The method of searching for signals of heavy virtual particles is developed.', 'hep-ph-9812485-3-0-1': 'It is based on a renormalization group equation for the low energy effective Lagrangian and the decoupling theorem.', 'hep-ph-9812485-3-0-2': 'As an application, the model independent search for Abelian [MATH] boson in four-fermion processes is analyzed.', 'hep-ph-9812485-3-0-3': 'The basic one-loop renormalization group relation for the parameters of the effective Lagrangian is derived which gives possibility to reduce the problem to the scattering of the Standard Model particles in the ""external field"" substituting heavy virtual [MATH] state.', 'hep-ph-9812485-3-0-4': 'From the set of derived relations it is determined that the absolute value of the [MATH] coupling to the axial currents has to be the same for all fermions and is strongly correlated to the [MATH] coupling to the scalar fields.', 'hep-ph-9812485-3-0-5': 'The corresponding dependences between the parameters of the effective Lagrangians are derived.', 'hep-ph-9812485-3-1-0': '# Introduction', 'hep-ph-9812485-3-2-0': 'An important problem of nowadays high energy physics is searching for deviations from the Standard Model (SM) of elementary particles.', 'hep-ph-9812485-3-2-1': 'At energies of the present day accelerators, the deviations may appear due to virtual states of new heavy particles entering the extended models and having the masses [MATH] much greater than the [MATH]-boson mass: [MATH].', 'hep-ph-9812485-3-2-2': 'One of approaches to describe contributions of these heavy states is construction of the effective Lagrangians (EL) [CITATION].', 'hep-ph-9812485-3-3-0': 'It is generally believed that the EL result from the decoupling of contributions from virtual states of heavy particles to the amplitudes of scattering processes featuring SM particles.', 'hep-ph-9812485-3-3-1': 'Below, we employ a traditional definition of the EL as the sum of scattering amplitudes of SM particles at external momenta [MATH].', 'hep-ph-9812485-3-3-2': 'In the case of this definition, the contributions of virtual heavy particles to the amplitudes decouple in the form of local operators involving SM fields, their derivatives, and masses.', 'hep-ph-9812485-3-3-3': 'These operators, [MATH], by construction respect gauge symmetries of the SM, have the dimension [MATH], and are suppressed by the appropriate powers of heavy masses: [EQUATION]', 'hep-ph-9812485-3-3-4': 'The dimensionless parameters [MATH] characterize physics beyond the SM.', 'hep-ph-9812485-3-3-5': 'They are model-dependent and expressed in terms of dimensionless coupling constants.', 'hep-ph-9812485-3-4-0': 'In this way, one can write down the EL of type ([REF]) for any model supposed as the basis low-energy theory.', 'hep-ph-9812485-3-4-1': 'For instance, the minimal super-symmetric SM can be considered.', 'hep-ph-9812485-3-4-2': 'In this case the content of operators [MATH] must be substituted [CITATION].', 'hep-ph-9812485-3-4-3': 'In what follows, we choose the minimal SM as the basis low-energy theory.', 'hep-ph-9812485-3-5-0': 'All the types of operators [MATH] that may arise in specific renormalizable gauge theories, containing the SM as a subgroup, are investigated in Refs. [CITATION].', 'hep-ph-9812485-3-5-1': 'Then, the EL is constructed as the sum of the type ([REF]) with the coefficients [MATH] treated as independent parameters to be determined from experimental data.', 'hep-ph-9812485-3-5-2': 'This meaning of [MATH] is peculiar to the approaches used in Refs. [CITATION].', 'hep-ph-9812485-3-5-3': 'In contrast, the parameters [MATH] entering Eq. ([REF]) and defined via the scattering amplitudes are not independent in general.', 'hep-ph-9812485-3-5-4': 'For such a definition, a set of relations between [MATH] has been derived and called the renormalization group (RG) relations [CITATION].', 'hep-ph-9812485-3-5-5': 'It is a consequence of the RG equations for [MATH]-matrix elements and the decoupling theorem [CITATION].', 'hep-ph-9812485-3-5-6': 'In what follows, we extend the analysis to the case of searching for heavy [MATH] boson in four-fermionic processes.', 'hep-ph-9812485-3-5-7': 'We suppose that the effective (or basis) low-energy theory (the minimal SM in our case) is specified.', 'hep-ph-9812485-3-5-8': 'That means we choose the appropriate set of operators [MATH] in Eq. ([REF]).', 'hep-ph-9812485-3-5-9': 'We discriminate between the Abelian and the non-Abelian [MATH] bosons, assuming the different types of their interactions with light particles.', 'hep-ph-9812485-3-6-0': 'The low-energy model is the remnant of some unknown high-energy theory predicting the Abelian [MATH] boson.', 'hep-ph-9812485-3-6-1': 'The constraints on the [MATH] parameters can be derived with or without knowledge of the underlying theory.', 'hep-ph-9812485-3-6-2': 'In this regard, we will call them the model dependent and the model independent ones [CITATION].', 'hep-ph-9812485-3-6-3': 'The RG relations being the consequence of renormalizability are the model independent constraints.', 'hep-ph-9812485-3-7-0': 'Let us summarize the main steps in deriving the RG relations [CITATION].', 'hep-ph-9812485-3-7-1': 'First, as the decoupling theorem guarantees, the operators originating from interactions of the renormalizable types have to be preserved in the EL in leading order in [MATH].', 'hep-ph-9812485-3-7-2': 'Since the underlying theory is not specified, the corresponding [MATH] must be considered as arbitrary parameters.', 'hep-ph-9812485-3-7-3': 'Hence, one is able at least to relate different parameters describing different scattering processes.', 'hep-ph-9812485-3-7-4': 'Second, a heavy virtual state can be treated as an external field scattering SM light particles.', 'hep-ph-9812485-3-7-5': 'The transition amplitude is splitted in two scattering vertices with arbitrary vertex coefficients.', 'hep-ph-9812485-3-7-6': 'The renormalizability of the theory can be used in a way to incorporate information about the [MATH] and [MATH] functions entering the RG operator at low energies in the framework of the external field problem.', 'hep-ph-9812485-3-7-7': 'The key point of the renormalizability of the vertices exploited is that the algebraic structure of the divergent part of a scattering amplitude calculated in higher orders of perturbation theory coincides with that of the tree-level vertex.', 'hep-ph-9812485-3-7-8': 'Hence, the relation between the vertex coefficient and the [MATH] and [MATH] functions follows.', 'hep-ph-9812485-3-7-9': 'If the underlying theory is specified, the relation is just an identity.', 'hep-ph-9812485-3-7-10': 'But if the theory is not assumed and the number of [MATH] and [MATH] functions is less than the number of RG relations, a non-trivial system of algebraic equations for vertex coefficients and, hence, for parameters [MATH] in Eq. ([REF]) can be obtained.', 'hep-ph-9812485-3-8-0': 'Although the underlying theory predicting the Abelian [MATH] boson is not specified, it is possible to parameterize the [MATH] interactions with the SM particles assuming the effective gauge group to be [MATH] as it is often discussed in literature [CITATION].', 'hep-ph-9812485-3-8-1': 'This parameterization is natural, in regard of the decoupling theorem, when the leading order operators [MATH] are investigated.', 'hep-ph-9812485-3-8-2': 'In general, there are not reasons to assume a priori that the complete Lagrangian of the model is the [MATH] invariant.', 'hep-ph-9812485-3-8-3': 'However, for our approach the low-energy parameterization is important.', 'hep-ph-9812485-3-8-4': 'As it will be seen, the derived RG relations distinguish the Abelian [MATH] and the most general parameterizations of the [MATH] interactions with the SM fields.', 'hep-ph-9812485-3-8-5': 'Moreover, in the latter case the subgroup of [MATH] gauge group is fixed (due to the renormalizability).', 'hep-ph-9812485-3-9-0': 'In what follows, we will formulate the low-energy RG relations for four-fermion amplitudes in order [MATH].', 'hep-ph-9812485-3-9-1': 'As it will be shown, the relations hold only in the case if the absolute value of the [MATH] coupling to the axial-vector currents is the same for all the SM fermions.', 'hep-ph-9812485-3-9-2': 'Hence, some of the parameters [MATH] are not independent.', 'hep-ph-9812485-3-9-3': 'Moreover, being a consequence of the renormalizability, the RG relations ensure that the Yukawa terms of the SM Lagrangian respect the [MATH] gauge symmetry associated with the [MATH] boson.', 'hep-ph-9812485-3-10-0': 'The content is as follows.', 'hep-ph-9812485-3-10-1': 'In Sec. [REF] the model investigated is described.', 'hep-ph-9812485-3-10-2': 'In Sec. [REF] the general RG relation for a [MATH]-matrix element at low energies is derived.', 'hep-ph-9812485-3-10-3': 'In Sec. [REF] it is applied to four-fermion scattering processes to obtain correlations between the vertex parameters describing new physics at low energies.', 'hep-ph-9812485-3-10-4': 'In Sec. [REF] the corresponding relations for the parameters [MATH] of the EL are derived.', 'hep-ph-9812485-3-10-5': 'The results of the investigation as well as the prospects are discussed in Sec. [REF].', 'hep-ph-9812485-3-11-0': '# The Model', 'hep-ph-9812485-3-12-0': 'The model under consideration is assumed to be the low energy remnant of some unknown underlying theory.', 'hep-ph-9812485-3-12-1': 'It involves the SM and the additional Abelian vector boson [MATH] which is supposed to be a massive particle with the mass [MATH] much heavier than the [MATH]-boson mass.', 'hep-ph-9812485-3-12-2': 'In the present paper the mechanism of heavy mass generation is not considered since the underlying theory describing interactions at energies [MATH] is unspecified.', 'hep-ph-9812485-3-12-3': 'So, the Lagrangian includes the explicit mass term for the [MATH] boson, and the mass [MATH] is treated as a parameter of the model.', 'hep-ph-9812485-3-13-0': 'The most general parameterization of the interactions between the Abelian [MATH] and the SM fields can be performed on the base of the effective gauge group [MATH] [CITATION].', 'hep-ph-9812485-3-13-1': 'Then, the corresponding Lagrangian is [EQUATION] where [MATH] is the SM scalar doublet, and the summation over all the SM left-handed doublets [MATH] and the SM right-handed singlets [MATH] is understood.', 'hep-ph-9812485-3-13-2': 'The charges [MATH] and the fields [MATH], [MATH], [MATH] correspond to the gauge groups [MATH], [MATH], [MATH], respectively, [MATH] is the fermion charge in the positron charge units, [MATH] are the Pauli matrices, [MATH] is a [MATH] unity matrix, and [MATH] for leptons and [MATH] for quarks.', 'hep-ph-9812485-3-13-3': 'The values of the dimensionless constants [MATH], [MATH], [MATH] depend on the particular higher-energy theory including [MATH] boson.', 'hep-ph-9812485-3-13-4': 'Here, they are to be considered as arbitrary parameters.', 'hep-ph-9812485-3-13-5': 'This parameterization, for instance, accounts for the most general [MATH] interactions generated in string theories [CITATION].', 'hep-ph-9812485-3-14-0': 'Up to this point the electroweak symmetry [MATH] was supposed to be unbroken, and all the SM particles were massless.', 'hep-ph-9812485-3-14-1': 'The masses of the SM particles are generated by the spontaneous breakdown of the symmetry [MATH] originated due to the non-zero vacuum expectation value of the scalar doublet.', 'hep-ph-9812485-3-14-2': 'However, the mass eigenstates are appeared to be shifted away from the original fields [MATH], since the corresponding mass terms become non-diagonal.', 'hep-ph-9812485-3-14-3': 'The physical fields [MATH] are obtained by the orthogonal transformation: [EQUATION] where [MATH] is the SM value of the Weinberg angle ([MATH]), and [MATH] denotes the mixing angle relating physical states [MATH], [MATH] to massive neutral components of the [MATH] gauge fields.', 'hep-ph-9812485-3-14-4': 'The value of the angle [MATH] can be determined from the relation [EQUATION] confirming Ref. [CITATION].', 'hep-ph-9812485-3-15-0': 'The masses of the physical fields are given by the expressions: [EQUATION]', 'hep-ph-9812485-3-15-1': 'As is seen, the mass of the physical [MATH]-boson differs from the SM value [MATH] by the small quantity of order [MATH].', 'hep-ph-9812485-3-15-2': 'So, the mixing angle [MATH] is also small [MATH].', 'hep-ph-9812485-3-15-3': 'The difference [MATH] is negative and completely determined by the [MATH] coupling to the scalar doublet.', 'hep-ph-9812485-3-15-4': 'Thus, constraints on the [MATH] interaction with scalar fields can be derived by means of experimental detecting this observable: [EQUATION]', 'hep-ph-9812485-3-15-5': 'Using Eqs. ([REF]) the Lagrangian of the model can be expressed in terms of the physical fields.', 'hep-ph-9812485-3-15-6': 'The dependence on the mixing [MATH] in Eqs. ([REF]) causes new interactions [MATH] originally absent in the Lagrangian ([REF]).', 'hep-ph-9812485-3-16-0': 'In the present paper we will deal with four-fermion amplitudes of order [MATH] produced by virtual [MATH] boson in the [MATH] channel.', 'hep-ph-9812485-3-16-1': 'The corresponding tree-level diagram is shown in Fig. [REF].', 'hep-ph-9812485-3-16-2': 'At low energies the effective four-fermion vertex is generated due to the heavy mass in the [MATH]-boson propagator: [MATH].', 'hep-ph-9812485-3-16-3': 'So, expressions involving such amplitudes can be computed neglecting the terms proportional to [MATH] in the Lagrangian.', 'hep-ph-9812485-3-16-4': ""Only linear in [MATH] terms of the Lagrangian are needed, which have the following form in the 't Hooft-Feynman gauge: [EQUATION] where [MATH], [MATH], [MATH] is the positron charge, [MATH] denotes the vacuum expectation value of the scalar doublet, [MATH] is the Higgs scalar particle, and [MATH], [MATH] are the Goldstone fields."", 'hep-ph-9812485-3-17-0': '# RG relations in a theory with decoupling', 'hep-ph-9812485-3-18-0': 'As we mentioned before, the model under consideration is supposed to be the low-energy limit of some unspecified renormalizable theory describing interactions at energies [MATH].', 'hep-ph-9812485-3-18-1': 'As is known, due to renormalizability the [MATH]-matrix elements are invariant with respect to the RG transformations which express independence of the amplitudes of the normalization point [MATH].', 'hep-ph-9812485-3-18-2': 'In a theory with different mass scales the RG flow and the decoupling of heavy loop contributions at the thresholds of heavy masses give possibility to determine all the parameters of scattering amplitudes.', 'hep-ph-9812485-3-18-3': 'The decoupling is resulted in an important property of the low energy amplitudes: the running of the proper functions is regulated by loops of light particles.', 'hep-ph-9812485-3-18-4': 'Therefore, the [MATH] and [MATH] functions at low energies, [MATH], are determined by the SM particles, only.', 'hep-ph-9812485-3-18-5': 'The latter fact allows to formulate a series of the RG relations for scattering amplitudes.', 'hep-ph-9812485-3-18-6': 'We will derive them for the case of heavy [MATH] boson which, as well as other heavy particles, is decoupled.', 'hep-ph-9812485-3-19-0': 'We will use the notations [MATH] (where [MATH], [MATH] and [MATH] denote the QCD charge, the coupling of the Yukawa interaction between [MATH] and [MATH], and the scalar self-coupling, respectively) and [MATH] (where [MATH] represents all the fields of the theory, and [MATH] are particle masses) to refer to the charges, the fields, and the masses.', 'hep-ph-9812485-3-19-1': 'Then the RG equation reads: [EQUATION] where renormalized (running) parameters [MATH], [MATH] are the solutions to the equations: [EQUATION]', 'hep-ph-9812485-3-19-2': 'Hats over quantities mark the parameters of the underlying theory.', 'hep-ph-9812485-3-19-3': 'They include the contributions of loops containing both the SM particles and the heavy ones.', 'hep-ph-9812485-3-20-0': 'Below, in carrying out calculations the dimensional regularization and the [MATH] renormalization scheme [CITATION] will be used.', 'hep-ph-9812485-3-20-1': 'The [MATH] scheme is mass independent one, so the RG operator [MATH] has the same form at high, [MATH], and low, [MATH], energies.', 'hep-ph-9812485-3-21-0': 'In the energy range [MATH] all the heavy fields are decoupled.', 'hep-ph-9812485-3-21-1': 'So, one can introduce the running parameters [MATH], [MATH] completely defined by the low-energy effective theory [CITATION]: [EQUATION] where [MATH] and [MATH] functions contain no contributions from loops with heavy particles.', 'hep-ph-9812485-3-21-2': 'These parameters can be expressed in terms of the original ones: [EQUATION] where the matching between the both sets of parameters ([MATH], [MATH] and [MATH], [MATH]) is chosen to be done at the normalization point [MATH]: [EQUATION]', 'hep-ph-9812485-3-21-3': 'Actually, Eq. ([REF]) is the redefinition of the parameters of the theory.', 'hep-ph-9812485-3-21-4': 'As it has been shown in Refs. [CITATION], all the heavy particle loop contributions proportional to [MATH] are completely removed from the RG equation ([REF]) by such a redefinition.', 'hep-ph-9812485-3-21-5': 'This fact is a consequence of the decoupling theorem [CITATION].', 'hep-ph-9812485-3-22-0': 'Then, the RG equation ([REF]) for the [MATH] matrix expressed in terms of the low-energy quantities [MATH], [MATH] becomes: [EQUATION] where [MATH] and [MATH] denote, respectively, the RG operator and the [MATH]-matrix element of the low-energy effective theory, calculated without the heavy particle loops.', 'hep-ph-9812485-3-23-0': 'The familiar usage of Eq. ([REF]) is to improve scattering amplitudes calculated in perturbation theory.', 'hep-ph-9812485-3-23-1': 'In contrast, in what follows we will apply Eq. ([REF]) to obtain algebraic relations between the parameters [MATH], [MATH], [MATH], which are to be considered as arbitrary numbers, if the underlying theory is not assumed.', 'hep-ph-9812485-3-23-2': 'The reasons for that are as follow.', 'hep-ph-9812485-3-23-3': 'In the case when the underlying theory is specified ([MATH], [MATH], [MATH] are to be computed) and the [MATH] and [MATH] functions as well as the [MATH]-matrix elements are calculated in a fixed order of perturbation theory, Eq. ([REF]) is just the identity.', 'hep-ph-9812485-3-23-4': 'In a sense, it is a necessary condition of renormalizability.', 'hep-ph-9812485-3-23-5': 'If the underlying theory is not specified, whereas the [MATH], [MATH] functions and the [MATH]-matrix elements are computed in a fixed order of perturbation theory, then, as we will show for four-fermion processes at one-loop level, the equality ([REF]) holds only if the parameters [MATH], [MATH], [MATH] satisfy some specific correlations.', 'hep-ph-9812485-3-23-6': 'The latter result in the dependences between the appropriate parameters [MATH] in the EL ([REF]).', 'hep-ph-9812485-3-24-0': 'Let [MATH] and [MATH] represent the one-loop and the tree parts of the [MATH]-matrix element, respectively.', 'hep-ph-9812485-3-24-1': 'Introducing the one-loop RG operator: [EQUATION] where [MATH] and [MATH] functions are computed in one-loop order, one can derive from Eq. ([REF]) the following identity: [EQUATION]', 'hep-ph-9812485-3-24-2': 'Since the parameters [MATH], [MATH] and [MATH], [MATH] differ at the one-loop level, one could freely substitute one set of them by another in Eq. ([REF]).', 'hep-ph-9812485-3-25-0': 'Equation ([REF]) is the starting point to analyze the RG relations for amplitudes at low energies.', 'hep-ph-9812485-3-25-1': 'The relation ([REF]) ensures the leading logarithmic terms of the one-loop [MATH]-matrix element to reproduce the appropriate tree-level structure that is a simple consequence of renormalizability.', 'hep-ph-9812485-3-25-2': 'When the couplings are represented by arbitrary unknown parameters, as it takes place in the considered model and, in general, in the discussed EL approach, the relation ([REF]) is the one-loop equation for the parameters which must be determined with the computed [MATH] and [MATH] functions.', 'hep-ph-9812485-3-25-3': 'If due to a symmetry the number of [MATH] and [MATH] functions is less than the number of RG relations, the non-trivial system of equations correlating the originally independent parameters may occur.', 'hep-ph-9812485-3-25-4': 'We will use Eq. ([REF]) to derive dependences between the parameters [MATH], [MATH], and [MATH].', 'hep-ph-9812485-3-26-0': '# RG relations for four-fermion amplitudes', 'hep-ph-9812485-3-27-0': 'In this section the one-loop RG relation ([REF]) is applied to the four-fermion processes [MATH] containing [MATH] boson as the virtual state.', 'hep-ph-9812485-3-27-1': 'In lower order in [MATH] such amplitudes produce the effective contact interactions of the type current[MATH]current described by the EL ([REF]) [CITATION].', 'hep-ph-9812485-3-28-0': 'Let the renormalized fields, masses, and charges are defined as follows: [EQUATION] where [MATH] is the vertex describing interaction between the fermion [MATH] and the neutral vector boson [MATH] and [MATH] and [MATH] are [MATH] matrices ([MATH]).', 'hep-ph-9812485-3-28-1': 'The fermion renormalization constant [MATH] is the SM one.', 'hep-ph-9812485-3-28-2': 'Its value is known [CITATION].', 'hep-ph-9812485-3-28-3': 'Quantities [MATH] and [MATH] are represented at the one-loop level by the divergent terms ([MATH]) of the gauge-independent part of the polarization operator of neutral vector bosons [MATH] being evaluated from the diagrams in Fig. [REF]: [EQUATION] where [MATH] is the Kronecker symbol.', 'hep-ph-9812485-3-28-4': 'In order to find the constant [MATH] in one-loop order one has to calculate the divergent ([MATH]) part of the vertex function [MATH] (the diagrams in Fig. [REF]): [EQUATION] where [MATH] is the vertex [MATH] at the tree level.', 'hep-ph-9812485-3-29-0': 'The [MATH]-matrix element expressed in terms of the renormalized quantities is finite.', 'hep-ph-9812485-3-29-1': 'It can be evaluated using the diagrams in Figs. [REF] and [REF], where shaded blobs stand for the finite parts of the vertex function [MATH] and the polarization operator [MATH].', 'hep-ph-9812485-3-29-2': 'On the mass-shell of fermions the gauge-dependent part of [MATH]) gives no contribution to the physical amplitude due to transversality of the vertex [MATH].', 'hep-ph-9812485-3-30-0': 'Because of the one-loop mixing between neutral bosons, Eq. ([REF]) for the renormalized [MATH]-matrix element cannot be reduced to relations for vertices.', 'hep-ph-9812485-3-30-1': 'To derive these relations one has to avoid the mixing.', 'hep-ph-9812485-3-30-2': 'The corresponding procedure was described in Ref. [CITATION].', 'hep-ph-9812485-3-31-0': 'One has to introduce the quantities [MATH] and [MATH] instead of the fields [MATH], [MATH], [MATH] and vertices [MATH] according to the following relations: [EQUATION]', 'hep-ph-9812485-3-31-1': 'After such a redefinition the mixing between neutral bosons in Eq. ([REF]) vanishes.', 'hep-ph-9812485-3-31-2': 'In what follows, we will use the quantities [MATH] and [MATH], therefore only the diagonal terms of the polarization operator are needed in computing of the [MATH]-matrix element [MATH].', 'hep-ph-9812485-3-31-3': 'Since the difference between the two sets of parameters is of order [MATH], one can substitute the tilded parameters by the other set in the amplitudes in lower order.', 'hep-ph-9812485-3-32-0': 'When the diagonalization of the leading logarithmic terms of the vector boson polarization operator is fulfilled, Eq. ([REF]) can be reduced to relations for vertices describing scattering of fermions in the external field [MATH] substituting the heavy [MATH] boson: [EQUATION]', 'hep-ph-9812485-3-32-1': ""The quantities [MATH] and [MATH] in Eq. ([REF]) have been calculated in the 't Hooft-Feynman gauge using diagrams in Figs. [REF], [REF]."", 'hep-ph-9812485-3-32-2': 'The computation gives the following relations for the left-handed and the right-handed fermions, respectively: [EQUATION] where [EQUATION]', 'hep-ph-9812485-3-32-3': 'The notation [MATH] stands for the isopartner of [MATH], [MATH], [MATH], [MATH]), [MATH] equals to [MATH] for quarks and zero for leptons, [EQUATION] where [MATH] denotes leptons, and [MATH] is the third component of the weak isospin.', 'hep-ph-9812485-3-33-0': 'The SM values of the one-loop fermion anomalous dimensions are known [CITATION]: [EQUATION]', 'hep-ph-9812485-3-33-1': 'By employing Eqs. ([REF])-([REF]), it is easy to compute [MATH] and [MATH] in one-loop order: [EQUATION] where [MATH].', 'hep-ph-9812485-3-34-0': 'Since the [MATH] function is the same for both Eqs. ([REF]), the following relations can be derived: [EQUATION]', 'hep-ph-9812485-3-34-1': 'Hence, one can see that originally independent parameters [MATH], [MATH], and [MATH] appear to be the connected ones.', 'hep-ph-9812485-3-34-2': 'Finally, one can also check that the RG relations ([REF]) are fulfilled only if Eq. ([REF]) holds.', 'hep-ph-9812485-3-35-0': 'In a sense, the relations ([REF]) mean that the [MATH] boson couplings to the SM axial-vector currents have the universal absolute value, if a single light scalar doublet exists.', 'hep-ph-9812485-3-35-1': 'So, among the four constants [MATH], [MATH], [MATH], and [MATH] parameterizing the interaction between the [MATH] boson and the [MATH] fermion isodoublet only one is really arbitrary.', 'hep-ph-9812485-3-35-2': 'The rest ones can be expressed through it and the hypercharge [MATH] of the [MATH] coupling to the SM scalar doublet.', 'hep-ph-9812485-3-35-3': 'Thus, the fermion and the scalar sectors of new physics are to be correlated.', 'hep-ph-9812485-3-36-0': 'Being derived from the condition of renormalizability of a scattering amplitude in the external field of heavy particle, the relations ([REF]) respect gauge invariance and have a transparent interpretation.', 'hep-ph-9812485-3-36-1': 'First of them means that that the upper and the lower components of a left-handed doublet transform in the same way under [MATH] gauge group.', 'hep-ph-9812485-3-36-2': 'If it does not hold, [MATH] gauge invariance would violate.', 'hep-ph-9812485-3-36-3': 'The second of relations guarantees, that the SM Yukawa interaction terms have to be invariant under [MATH] gauge transformations.', 'hep-ph-9812485-3-37-0': '# Dependences between the parameters of the effective Lagrangian', 'hep-ph-9812485-3-38-0': 'The relations ([REF]) result in dependences between the parameters of the EL ([REF]).', 'hep-ph-9812485-3-38-1': 'In Ref. [CITATION] all the effective operators [MATH] describing deviations from the SM and appearing at the tree level due to heavy virtual states have been derived.', 'hep-ph-9812485-3-38-2': 'The heavy [MATH] boson produces the following effective vertices: [EQUATION] where [MATH] and [MATH] stand for leptons and quarks; [MATH] is the electroweak covariant derivative, and the definitions of [MATH] correspond to Ref. [CITATION]: [EQUATION]', 'hep-ph-9812485-3-38-3': 'Thus, the relations ([REF]) give rise to the following dependences between the parameters [MATH]: [EQUATION]', 'hep-ph-9812485-3-38-4': 'Hence it follows that the arbitrary constants are to be the ones parameterizing the processes [MATH] and [MATH].', 'hep-ph-9812485-3-39-0': 'The relations ([REF]) hold if the only additional heavy particle extending the SM is the Abelian [MATH] boson.', 'hep-ph-9812485-3-39-1': 'Other models may lead to other type of correlations between the parameters [MATH].', 'hep-ph-9812485-3-39-2': 'The advantage of the expressions ([REF]) is their independence of the mechanism of arising of the Abelian [MATH] in a specific theory describing physics at energy scale [MATH].', 'hep-ph-9812485-3-40-0': '# Discussion', 'hep-ph-9812485-3-41-0': 'Let us discuss the main points of our analysis and further perspectives.', 'hep-ph-9812485-3-41-1': 'Usually, the EL ([REF]) describing at low energies physics beyond the SM contains a set of the operators with parameters [MATH] assumed to be arbitrary numbers which must be fixed in experiments.', 'hep-ph-9812485-3-41-2': 'However, as we have demonstrated above considering the additional [MATH] boson as extension of the SM, the renormalizability of the underlying theory may result in some correlations between [MATH] which have to be of interest in searching for new physics effects.', 'hep-ph-9812485-3-41-3': 'Most important is that the derived relations ([REF]), being a consequence of renormalizability formulated in the framework of scattering in the external field, are independent of the specific underlying (GUT) model.', 'hep-ph-9812485-3-42-0': 'The latter statement requires to be discussed in more detail.', 'hep-ph-9812485-3-42-1': 'The Lagrangian ([REF]) respects [MATH] gauge invariance but includes only the interactions of renormalizable type.', 'hep-ph-9812485-3-42-2': 'The terms of the form [MATH] are omitted because they, being generated at GUT (or some intermediate [MATH]) mass scale, are suppressed by the factors [MATH].', 'hep-ph-9812485-3-42-3': 'Hence, one can conclude that at low energies, [MATH], the renormalizable interactions are to be dominant.', 'hep-ph-9812485-3-42-4': 'Therefore, the only model dependent information in Eq. ([REF]) is the specific values of the parameters [MATH], [MATH] and [MATH].', 'hep-ph-9812485-3-42-5': 'The relations ([REF]) hold for arbitrary values of them, as it was demonstrated in Sec. [REF].', 'hep-ph-9812485-3-42-6': 'As a consequence, the relations ([REF]) also hold for arbitrary [MATH], [MATH], [MATH] and are independent of them explicitly.', 'hep-ph-9812485-3-43-0': 'Here, it will be useful to stress once again the main features of our approach as compared to the ones, where the [MATH] parameters are assumed to be arbitrary numbers.', 'hep-ph-9812485-3-43-1': 'In the latter case, the origin of the composed operators is completely hidden.', 'hep-ph-9812485-3-43-2': 'In the former one, we first came to the one step back in the analysis and have taken into account the most important consequences of the renormalizability: 1) at low energies, the interactions of renormalizable type are dominant whereas the nonrenormalizable interaction terms are suppressed by the additional degree of [MATH]; 2) for any renormalizable interactions at low energies one can interrelate scattering processes due to the renormalizability of scattering amplitudes in the external field of heavy particle.', 'hep-ph-9812485-3-44-0': 'Now, let us consider a more general parameterization in Eq. ([REF]).', 'hep-ph-9812485-3-44-1': 'In principle, one could employ the most general form of the generator [MATH] which could be just the part of the appropriate generator of the underlying theory.', 'hep-ph-9812485-3-44-2': 'Then, the RG relations can be used to obtain all possible solutions for the numbers [MATH], [MATH], and [MATH].', 'hep-ph-9812485-3-44-3': 'As one can check, the two sets of correlations are allowed.', 'hep-ph-9812485-3-44-4': 'The first one, [EQUATION] describes the vector boson analogous to the SM field [MATH].', 'hep-ph-9812485-3-44-5': 'Such a field may appear as a component of the non-Abelian gauge field.', 'hep-ph-9812485-3-44-6': 'The second solution leads to the relations derived in this paper: [EQUATION]', 'hep-ph-9812485-3-44-7': 'They correspond to the Abelian vector boson considered.', 'hep-ph-9812485-3-44-8': 'Thus, the proposed method to relate the parameters [MATH] could be applied to interactions of light SM particles with other heavy particles (leptoquarks, for example) treated as the external fields.', 'hep-ph-9812485-3-45-0': 'In Ref. [CITATION] a general renormalization framework for the [MATH] theory was presented, and, in particular, the set of relevant counterterms has been derived.', 'hep-ph-9812485-3-45-1': 'Owing to [MATH] symmetry the same counterterms were postulated for different [MATH] couplings to fermions.', 'hep-ph-9812485-3-45-2': 'However, as it was shown above, this fact requires the specific relations between the hypercharges [MATH], [MATH], [MATH].', 'hep-ph-9812485-3-45-3': 'Naturally, the relations hold automatically for a specific renormalizable higher-energy theory.', 'hep-ph-9812485-3-45-4': 'However, if the hypercharges are treated as unknown parameters these relations are to be taken into account in order to respect the gauge symmetry.', 'hep-ph-9812485-3-46-0': 'In fact, the obtained relations ([REF]) demonstrate that the fermion and the scalar sectors of new physics are strongly connected.', 'hep-ph-9812485-3-46-1': 'The couplings of the [MATH] boson to the SM axial-vector currents are completely determined by its interaction with the scalar fields, if a single light scalar doublet is assumed.', 'hep-ph-9812485-3-46-2': 'When the [MATH] coupling to the scalar doublet is switched off ([MATH]), the [MATH] boson interacts at the tree level with the SM vector currents, only ([MATH]).', 'hep-ph-9812485-3-46-3': 'But even in this case the couplings to the axial currents are to be produced by loops being suppressed by the additional small factor [MATH].', 'hep-ph-9812485-3-47-0': 'As it follows from Eq. ([REF]), the RG relations respect the [MATH] symmetry of the Yukawa terms responsible for generating the masses of the SM fermions.', 'hep-ph-9812485-3-47-1': 'Therefore, assuming the basis low-energy theory to be, for example, the two-Higgs-doublet SM, one could expect that the relations identical to Eq. ([REF]) can be derived for the Abelian [MATH] couplings to each of the scalar doublets.', 'hep-ph-9812485-3-47-2': 'However, in general, the RG relations are not determined by the specific gauge invariance of the Lagrangian.', 'hep-ph-9812485-3-47-3': 'For instance, consider the non-Abelian [MATH] couplings described by the solution ([REF]).', 'hep-ph-9812485-3-47-4': 'As one can check, the relations ([REF]) require no specific gauge invariance of the Yukawa terms.', 'hep-ph-9812485-3-48-0': 'As it was mentioned in Sec. [REF], the [MATH] interaction with the scalar doublet is completely determined by the observable [MATH] (see Eq. ([REF])).', 'hep-ph-9812485-3-48-1': 'Therefore, one is able to predict the coupling of the [MATH] to the SM axial-vector currents.', 'hep-ph-9812485-3-48-2': 'When [MATH] boson does not interact with the scalar doublet, the physical mass of [MATH] boson is to be identical to its SM value.', 'hep-ph-9812485-3-48-3': 'It is worth noticing that the experimental value of [MATH] in Eq. ([REF]) must be properly chosen in order to satisfy the definition [MATH].', 'hep-ph-9812485-3-48-4': 'The [MATH] existence has not to affect the quantity [MATH] in order [MATH].', 'hep-ph-9812485-3-48-5': 'So, the determination of the Weinberg angle in terms of the QED coupling constant [MATH] and the Fermi constant [MATH] measured from the muon decay is preferable, since the tree level values of these parameters [MATH], [MATH] include the contributions of [MATH] bosons and the QED sector, only.', 'hep-ph-9812485-3-49-0': 'The proposed method can be applied to find the relations of the derived type for different effective four-fermion operators generated by the [MATH] (for instance, discussed in Refs. [CITATION]).', 'hep-ph-9812485-3-49-1': 'In general, it gives possibility to search for other heavy virtual states that is the problem for future.', 'hep-ph-9812485-3-50-0': 'The authors are grateful to S. Dittmaier, D. Kazakov, A. Pankov, N. Paver, D. Schildknecht, and O. Teryaev for discussions.'}",,
1805.06408,"{'1805.06408-1-0-0': 'We use available baryon acoustic oscillation distance measurements and Hubble parameter data to constrain the cosmological constant [MATH], dynamical dark energy, and spatial curvature in simple cosmological models.', '1805.06408-1-0-1': 'We find that the consensus spatially flat [MATH]CDM model provides a reasonable fit to the data, but depending on the Hubble constant prior and cosmological model, it can be up to 2[MATH] away from the best-fit model, which can favor mild dark energy dynamics or mildly closed spatial hypersurfaces.', '1805.06408-1-1-0': 'defcitealias4Planck Collaboration (2016)', '1805.06408-1-2-0': '# Introduction', '1805.06408-1-3-0': 'It is widely accepted that the universe is undergoing accelerated expansion today.', '1805.06408-1-3-1': 'The consensus cosmological model, [MATH]CDM, posits that this acceleration is driven by the spatially homogeneous, constant dark energy density [MATH] of the cosmological constant [MATH] .', '1805.06408-1-3-2': ""The consensus [MATH]CDM model assumes flat spatial hypersurfaces, but observations don't rule out mildly curved spatial hypersurfaces; observations also do not rule out the possibility that the dark energy density varies slowly with time."", '1805.06408-1-3-3': 'In this paper we examine, in addition to the general (not necessarily spatially flat) [MATH]CDM model, the XCDM parametrization of dynamical dark energy, and the [MATH]CDM model in which a scalar field [MATH] is the dynamical dark energy.', '1805.06408-1-3-4': 'In the XCDM and [MATH]CDM cases we allow for both vanishing and non-vanishing spatial curvature.', '1805.06408-1-3-5': 'Details of the three models we study are summarized in Sec. [REF], and more information can be found in [CITATION].', '1805.06408-1-4-0': '[CITATION] have recently shown that, in the spatially flat case, the Planck 2015 CMB anisotropy data fromtalias4 (and some baryon acoustic oscillation distance measurements) weakly favor the XCDM parametrization and the [MATH]CDM model of dynamical dark energy over the [MATH]CDM consensus model.', '1805.06408-1-4-1': 'The XCDM case results have been confirmed by [CITATION] for a much bigger compilation of cosmological data, including most available Type Ia supernova apparent magnitude observations, BAO distance measurements, growth factor data, and Hubble parameter observations.', '1805.06408-1-4-2': 'Also, spatially flat XCDM and [MATH]CDM both reduce the tension between CMB anisotropy and weak gravitational lensing estimates of [MATH], the rms fractional energy density inhomogeneity averaged over [MATH]Mpc radius spheres .', '1805.06408-1-4-3': 'defcitealias4Planck Collaboration 2016', '1805.06408-1-5-0': 'In non-flat models nonzero spatial curvature provides an additional length scale which invalidates usage of the power-law power spectrum for energy density inhomogeneities in the non-flat case (as was assumed in the analysis of non-flat models intalias4).', '1805.06408-1-5-1': 'Non-flat inflation models provide the only known physically-consistent mechanism for generating energy density inhomogeneities in the non-flat case; the resulting open and closed model power spectra are not power laws .', '1805.06408-1-5-2': 'defcitealias4Planck Collaboration (2016)Using these power spectra, [CITATION] have found that the Planck 2015 CMB anisotropy data in combination with a few BAO distance measurements no longer rule out the non-flat [MATH]CDM case (unlike the earliertalias4 analyses based on the incorrect assumption of a power-law power spectrum in the non-flat model).', '1805.06408-1-5-3': '[CITATION] confirmed these results for a bigger compilation of cosmological data.', '1805.06408-1-6-0': 'Additionally, the non-flat models provide a better fit to the observed low multipole CMB anisotropy power spectrum, and do better at reconciling the CMB anisotropy and weak lensing constraints on [MATH], but do a worse job at fitting the observed large multipole CMB anisotropy power spectrum .', '1805.06408-1-6-1': 'Given the non-standard normalization of the Planck 2015 CMB anisotropy likelihood and that the flat and non-flat [MATH]CDM models are not nested, it is not possible to compute the relative goodness of fit between the flat and non-flat [MATH]CDM models quantitatively, although qualitatively the flat [MATH]CDM model provides a better fit to the current data .', '1805.06408-1-7-0': 'In the analyses discussed above, the Planck 2015 CMB anisotropy data played the major role.', '1805.06408-1-7-1': 'Those authors found consistency between cosmological constraints derived using the CMB anisotropy data in combination with various non-CMB data sets.', '1805.06408-1-7-2': 'CMB anisotropy data are sensitive to the behavior of cosmological spatial inhomogeneities.', '1805.06408-1-7-3': 'Here we derive constraints on similar models from a combination of all available Hubble parameter data as well as all available radial and transverse BAO data.', '1805.06408-1-7-4': 'Unlike the CMB anisotropy data, the [MATH] and these BAO data are not sensitive to the behavior of cosmological spatial inhomogeneities.', '1805.06408-1-8-0': 'The constraints we derive here are consistent with, but weaker than, those of the papers cited above; this provides a necessary and useful consistency test of those results.', '1805.06408-1-8-1': 'In particular, we find that the consensus flat [MATH]CDM model is a reasonable fit, in most cases, to the BAO and [MATH] data we study here.', '1805.06408-1-8-2': 'However, depending somewhat on the Hubble constant prior we use, consensus flat [MATH]CDM can be 1[MATH] or 2[MATH] away from the best-fit parameter values in some cases, which can favor mild dark energy dynamics or mildly closed spatial hypersurfaces.', '1805.06408-1-9-0': 'In Sec. [REF] we provide a short summary of the models we studied.', '1805.06408-1-9-1': 'Sec. [REF] presents the data that we used, and in Sec. [REF] we describe the methods by which we analyzed these data.', '1805.06408-1-9-2': 'Sec. [REF] describes the results of our analyses, and our conclusions are given in Sec. [REF].', '1805.06408-1-10-0': '# Models', '1805.06408-1-11-0': 'The models we examine in this paper are characterized by their expansion rate as a function of redshift [MATH], [EQUATION]', '1805.06408-1-11-1': 'Here [MATH] is the Hubble parameter and [MATH] is the Hubble constant.', '1805.06408-1-12-0': 'In the [MATH]CDM model dark energy is a constant vacuum energy density with negative pressure, equivalent to an ideal fluid with equation of state parameter [EQUATION]', '1805.06408-1-12-1': 'Here [MATH] and [MATH] are the homogeneous parts of the pressure and energy density, respectively.', '1805.06408-1-12-2': 'The expansion rate can be written in terms of the density parameters [EQUATION] where [MATH] is the current value of the non-relativistic matter density parameter, [MATH] is the cosmological constant density parameter, and [MATH] (which is nonzero in general) is the current value of the spatial curvature density parameter.', '1805.06408-1-12-3': 'The [MATH]CDM model is characterized by two parameters: [MATH].', '1805.06408-1-13-0': 'In the XCDM parametrization of dark energy, [MATH] where [MATH] is a negative constant (in general [MATH]).', '1805.06408-1-13-1': 'Hence [EQUATION] where [MATH] is the current value of the dark energy density.', '1805.06408-1-13-2': 'In contrast to [MATH]CDM, the dark energy density parameter [MATH] varies with time.', '1805.06408-1-13-3': 'If, however, [MATH], then XCDM reduces to [MATH]CDM, with [MATH].', '1805.06408-1-13-4': 'In general, the XCDM model has three free parameters: [MATH].', '1805.06408-1-13-5': 'We shall also consider spatially flat XCDM, with [MATH].', '1805.06408-1-14-0': 'The [MATH]CDM model provides a simple, physically consistent description of dynamical dark energy.', '1805.06408-1-14-1': 'In this model, the dark energy is a scalar field [MATH] with a potential energy density given by [EQUATION]', '1805.06408-1-14-2': 'Here [MATH], [MATH], [MATH] is the gravitational constant, and [EQUATION]', '1805.06408-1-14-3': 'The spatially homogeneous part of the scalar field obeys [EQUATION] where [MATH] is the scale factor, and an overdot denotes differentiation with respect to time.', '1805.06408-1-14-4': 'This, together with the first Friedmann equation, [EQUATION] and [EQUATION] determines the dynamics of the field.', '1805.06408-1-14-5': 'In eq. ([REF]), [MATH] is the non-relativistic matter density, [MATH] is the scalar field energy density, and [MATH] for flat, closed, and open spatial hypersurfaces, respectively.', '1805.06408-1-15-0': 'The dark energy equation of state parameter of [MATH]CDM is [EQUATION] which, unlike in the [MATH]CDM and XCDM models, changes with time.', '1805.06408-1-15-1': 'The expansion rate in the [MATH]CDM model is [EQUATION] where [EQUATION]', '1805.06408-1-15-2': 'In contrast to [MATH], [MATH] is not an explicit function of a power of [MATH]; it must be determined numerically.', '1805.06408-1-16-0': 'In general, the [MATH]CDM model has three free parameters: [MATH].', '1805.06408-1-16-1': 'We also consider spatially flat [MATH]CDM with [MATH].', '1805.06408-1-17-0': '# Data', '1805.06408-1-18-0': 'BAO provide observers with a ""standard ruler"" which can be used to measure cosmological distances.', '1805.06408-1-18-1': 'These distances can be computed in a given cosmological model, so measurements of them can be used to constrain the parameters of the model in question.', '1805.06408-1-18-2': 'The BAO distance measurements we use are listed in Table [REF].', '1805.06408-1-19-0': 'The transverse co-moving distance is [EQUATION] where [EQUATION] and the volume-averaged angular diameter distance is defcitealias4Planck Collaboration 2016 [EQUATION] .', '1805.06408-1-19-1': 'All of the measurements in Table [REF] are scaled by the size of the sound horizon at the drag epoch ([MATH]).', '1805.06408-1-19-2': 'This quantity istextsee [CITATION] for a derivation: [EQUATION] where [MATH] and [MATH] are the values of [MATH], the ratio of the baryon to photon momentum density, [EQUATION] at the drag epoch and matter-radiation equality epoch, respectively.', '1805.06408-1-19-3': 'Here [MATH] is the scale of the particle horizon at the matter-radiation equality epoch, and [MATH] and [MATH] are the baryon and photon mass densities.', '1805.06408-1-19-4': 'In our analyses, where appropriate, the original data listed in Table [REF] have been rescaled to a fiducial sound horizon [MATH] Mpctextfrom Table 4, column 2, oftalias4.', '1805.06408-1-20-0': 'In Table [REF] we list 31 [MATH] measurements determined using the cosmic chronometric techniquetextsee e.g. [CITATION] for a discussion of cosmic chronometers.', '1805.06408-1-20-1': 'This compilation is the same as the cosmic chronometric [MATH] data used in [CITATION].', '1805.06408-1-20-2': 'defcitealias4Planck Collaboration (2016)', '1805.06408-1-21-0': '# Methods', '1805.06408-1-22-0': 'defcitealias4Planck Collaboration 2016', '1805.06408-1-23-0': 'To determine the values of the best-fit parameters, we minimized [EQUATION] where [MATH] is the likelihood function and [MATH] is the set of parameters of the model under consideration.', '1805.06408-1-23-1': 'If the likelihood function [MATH] depends on an uninteresting nuisance parameter [MATH] with a probability distribution [MATH], we marginalize the likelihood function by integrating [MATH] over [MATH] [EQUATION]', '1805.06408-1-23-2': 'In our [MATH] analyses [MATH] is a nuisance parameter.', '1805.06408-1-23-3': 'We assumed a Gaussian distribution for [MATH] [EQUATION] and marginalized over it.', '1805.06408-1-23-4': 'We considered two cases: [MATH] km s[MATH] Mpc[MATH] and [MATH] km s[MATH] Mpc[MATH].', '1805.06408-1-24-0': 'Most of the data we analyzed are uncorrelated, however six of the data pointstextthose from [CITATION], are correlated.', '1805.06408-1-24-1': 'For uncorrelated data points, [EQUATION] where [MATH] are the model predictions at redshifts [MATH], and [MATH] and [MATH] are the central values and error bars of the measurements listed in Table [REF] and the last five lines of Table [REF].', '1805.06408-1-24-2': 'The correlated data (the first six entries in Table [REF]) require [EQUATION] where [MATH] is the inverse of the covariance matrix [EQUATION] .', '1805.06408-1-24-3': '[MATH] (in eq. [REF]) are the measurements in the first six lines of Table [REF].', '1805.06408-1-25-0': 'In addition to [MATH], we also used the Bayes Information Criterion [EQUATION] and the Akaike Information Criterion [EQUATION] .', '1805.06408-1-25-1': 'In these equations [MATH] is the minimum value of [MATH], [MATH] is the number of parameters of the given model, and [MATH] is the number of data points.', '1805.06408-1-25-2': ""BIC and AIC provide means to compare models with different numbers of parameters; they penalize models with a higher [MATH] in favor of those with a lower [MATH], in effect enforcing Occam's Razor in the model selection process."", '1805.06408-1-26-0': 'To determine the confidence intervals [MATH] on the 1d best-fit parameters, we computed one-sided limits [MATH] by using [EQUATION] where [MATH] is the point at which [MATH] has its maximimum value, such that [MATH] and [MATH], [MATH].', '1805.06408-1-26-1': 'Because the one-dimensional likelihood function is not guaranteed to be symmetric about [MATH], we compute the upper and lower confidence intervals separately.', '1805.06408-1-26-2': 'In the [MATH]CDM model, for example, the 1-sigma confidence intervals on [MATH] are computed by first integrating the likelihood function [MATH] over [MATH] to obtain a marginalized likelihood function that only depends on [MATH], [EQUATION] and then inserting this marginalized likelihood function into eq. ([REF]).', '1805.06408-1-27-0': 'The ranges over which we marginalized the parameters of the [MATH]CDM model were [MATH] and [MATH].', '1805.06408-1-27-1': 'For the spatially flat XCDM parametrization, we used [MATH] and [MATH], and for the spatially flat [MATH]CDM model we used [MATH] and [MATH].', '1805.06408-1-27-2': 'For 3-parameter XCDM, we used [MATH], [MATH], and [MATH].', '1805.06408-1-27-3': 'For the 3-parameter [MATH]CDM model we considered [MATH], [MATH], and [MATH].', '1805.06408-1-28-0': 'We analyzed the data with two independent Python codes, written by S.D. and J.R., that produced almost identical results in the 2-parameter cases and the 3-parameter XCDM model, and results that agreed to within 1% in the 3-parameter [MATH]CDM case.', '1805.06408-1-29-0': '# Results', '1805.06408-1-30-0': 'The confidence contours for the models we considered are shown in Figs. [REF], [REF], and [REF].', '1805.06408-1-30-1': 'The solid black contours indicate the [MATH] km s[MATH] Mpc[MATH] prior constraints, the dashed black contours indicate the [MATH] km s[MATH] Mpc[MATH] prior constraints, and the red dots indicate the best-fit point in each prior case.', '1805.06408-1-30-2': 'Our results for the parameter values of the unmarginalized and marginalized cases are collected in Tables [REF]-[REF], along with their [MATH], AIC, and BIC values.', '1805.06408-1-30-3': 'Wherever [MATH], [MATH]AIC, and [MATH]BIC are given, these are computed relative to the [MATH], AIC, and BIC of the corresponding [MATH]CDM model of each prior case.', '1805.06408-1-31-0': 'In the 2-parameter case, the model with the lowest value of [MATH] is [MATH]CDM if the prior on [MATH] is chosen to be [MATH] km s[MATH] Mpc[MATH].', '1805.06408-1-31-1': 'If, on the other hand, the [MATH] prior is chosen to be [MATH] km s[MATH] Mpc[MATH] then the spatially flat XCDM parametrization has the lowest value of [MATH].', '1805.06408-1-31-2': 'These models also have lower AIC and BIC values than the 3-parameter XCDM parametrization and the 3-parameter [MATH]CDM model (see Tables [REF] and [REF]).', '1805.06408-1-31-3': 'On the other hand, the 3-parameter models typically have a lower [MATH] than the 2-parameter [MATH]CDM case.', '1805.06408-1-31-4': 'These differences, however, are not statistically significant.', '1805.06408-1-31-5': 'Focusing on the [MATH] km s[MATH] Mpc[MATH] prior case, the [MATH] differences indicate that the non-flat [MATH]CDM model and non-flat XCDM parametrization provide a 0.97[MATH] and 0.85[MATH] better fit to the data, respectively, while from [MATH]AIC we find that these two models are 59% and 53% as probable as the 2-parameter [MATH]CDM model, respectively.', '1805.06408-1-32-0': 'In Table [REF] ([REF]), we list the 1[MATH] and 2[MATH] confidence intervals on the parameters of each of the 2-parameter (3-parameter) models.', '1805.06408-1-32-1': 'We obtained these by marginalizing the 2-parameter (3-parameter) likelihood function as described in Sec. [REF].', '1805.06408-1-32-2': 'The best-fit points in these tables correspond to the maximum value of the relevant one-dimensional marginalized likelihood function.', '1805.06408-1-32-3': 'Table [REF] ([REF]) lists the corresponding two-dimensional (three-dimensional) best-fit points.', '1805.06408-1-33-0': 'From the figures and tables, we see that the spatially flat [MATH]CDM model is a reasonable fit to the [MATH] and BAO data we use.', '1805.06408-1-33-1': 'In particular, from the figures, for the [MATH] km s[MATH] Mpc[MATH] prior, flat [MATH]CDM is always within 1[MATH] of the best-fit value (except in the right panel of Fig. [REF] for the non-flat [MATH]CDM case, where a closed model is favored over flat [MATH]CDM by a little more than 1[MATH]).', '1805.06408-1-33-2': 'However, the [MATH] km s[MATH] Mpc[MATH] case favors some larger deviations from flat [MATH]CDM.', '1805.06408-1-33-3': 'For example in the middle panel of Fig. [REF] for the flat XCDM model it favors a phantom model over flat [MATH]CDM at more than 2[MATH], while in the right panel of Fig. [REF] for the non-flat [MATH]CDM case it also favors a closed model at more than 2[MATH].', '1805.06408-1-33-4': 'Similar conclusions may be drawn from the parameter limits listed in Tables [REF] and [REF].', '1805.06408-1-34-0': 'When both dynamical dark energy and spatial curvature are present (as opposed to cases with only dynamical dark energy or only spatial curvature) it is not as easy to constrain both parameters simultaneously.', '1805.06408-1-34-1': 'This can be seen by comparing the center and right panels of Fig. [REF] to the left panels of Figs. [REF] and [REF], respectively.', '1805.06408-1-34-2': 'When spatial curvature is allowed to vary, the confidence contours in the 3-parameter XCDM parametrization and the [MATH]CDM model expand along the [MATH] and [MATH] axes (these are the parameters that govern the dynamics of the dark energy).', '1805.06408-1-35-0': 'The consensus model, spatially flat [MATH]CDM, is consistent with current [MATH] + BAO data, but these data allow some nonzero spatial curvature.', '1805.06408-1-35-1': 'In particular, we find that the best-fit values of the parameters in the [MATH]CDM model imply a curvature density parameter of [MATH] for the [MATH] km s[MATH] Mpc[MATH] prior case, and [MATH] for the [MATH] km s[MATH] Mpc[MATH] prior case.', '1805.06408-1-35-2': 'More precisely, using the [MATH] and [MATH] best-fit values and error bars for flat [MATH]CDM from Table [REF], and combining the errors in quadrature, an approximate estimate is [MATH] and [MATH] for the [MATH] km s[MATH] Mpc[MATH] and [MATH] km s[MATH] Mpc[MATH] priors, with the data favoring a closed model at 1[MATH] in the first case and a little over 2[MATH] in the second.', '1805.06408-1-35-3': 'This is also true of the the 3-parameter models, in which [MATH] is not a priori assumed to be zero (see the center and right panels of Figs. [REF] and [REF]).', '1805.06408-1-35-4': 'While not very statistically significant, we note that these results are not inconsistent with those of [CITATION] and [CITATION], who found that CMB anisotropy data, in conjunction with other cosmological data, were not inconsistent with mildly closed spatial hypersurfaces.', '1805.06408-1-36-0': 'The current data are also not inconsistent with some mild dark energy dynamics, although the size of the effect varies depending on the choice of [MATH] prior and whether or not [MATH] is allowed to vary as a free parameter.', '1805.06408-1-36-1': 'In the [MATH]CDM model, for instance, [MATH] can be different from zero in the [MATH] km s[MATH] Mpc[MATH] prior case, but only if spatial hypersurfaces are also closed (see line 2 of Table [REF]).', '1805.06408-1-37-0': '# Conclusions', '1805.06408-1-38-0': 'We analyzed a total of 42 measurements, 31 of which consisted of uncorrelated [MATH] data points, with the remainder coming from BAO observations (some correlated, some not), to constrain dark energy dynamics and spatial curvature, by determining how well these measurements can be described by three common models of dark energy: [MATH]CDM, the XCDM parametrization, and [MATH]CDM.', '1805.06408-1-39-0': 'The consensus flat [MATH]CDM model is in reasonable accord with these data, but depending on the model analyzed and the [MATH] prior used, it can be as far as 2[MATH] away from the best-fit model.', '1805.06408-1-39-1': 'These data are consistent with mild dark energy dynamics as well as mildy closed spatial hypersurfaces.', '1805.06408-1-39-2': 'While these results are interesting and encouraging, more and better data are needed before we can make definitive statements about the spatial curvature of the universe and about dark energy dynamics.', '1805.06408-1-40-0': '# Acknowledgements', '1805.06408-1-41-0': 'tocsection'}","{'1805.06408-2-0-0': 'We use all available baryon acoustic oscillation distance measurements and Hubble parameter data to constrain the cosmological constant [MATH], dynamical dark energy, and spatial curvature in simple cosmological models.', '1805.06408-2-0-1': 'We find that the consensus spatially flat [MATH]CDM model provides a reasonable fit to the data, but depending on the Hubble constant prior and cosmological model, it can be a little more than 1[MATH] away from the best-fit model, which can favor mild dark energy dynamics or non-flat spatial hypersurfaces.', '1805.06408-2-1-0': 'defcitealias4Planck Collaboration (2016)', '1805.06408-2-2-0': '# Introduction', '1805.06408-2-3-0': 'It is widely accepted that the universe is undergoing accelerated expansion today.', '1805.06408-2-3-1': 'The consensus cosmological model, [MATH]CDM, posits that this acceleration is driven by the spatially homogeneous, constant dark energy density [MATH] of the cosmological constant [MATH] .', '1805.06408-2-3-2': 'For reviews of the accelerated cosmological expansion and of the [MATH]CDM model, see [CITATION], [CITATION], [CITATION], and [CITATION].', '1805.06408-2-3-3': 'In this model, cold dark matter (CDM) is the second largest contributor to the current energy budget and, with non-relativistic baryonic matter, powered the decelerating cosmological expansion at earlier times.', '1805.06408-2-4-0': ""The consensus [MATH]CDM model assumes flat spatial hypersurfaces, but observations don't rule out mildly curved spatial hypersurfaces; observations also do not rule out the possibility that the dark energy density varies slowly with time."", '1805.06408-2-4-1': 'In this paper we examine, in addition to the general (not necessarily spatially flat) [MATH]CDM model, the XCDM parametrization of dynamical dark energy, and the [MATH]CDM model in which a scalar field [MATH] is the dynamical dark energy.', '1805.06408-2-4-2': 'In the XCDM and [MATH]CDM cases we allow for both vanishing and non-vanishing spatial curvature.', '1805.06408-2-4-3': 'Details of the three models we study are summarized in Sec. [REF], and more information can be found in [CITATION].', '1805.06408-2-5-0': '[CITATION] have recently shown that, in the spatially flat case, the Planck 2015 CMB anisotropy data fromtalias4 (and some baryon acoustic oscillation distance measurements) weakly favor the XCDM parametrization and the [MATH]CDM model of dynamical dark energy over the [MATH]CDM consensus model.', '1805.06408-2-5-1': 'The XCDM case results have been confirmed by [CITATION] for a much bigger compilation of cosmological data, including most available Type Ia supernova apparent magnitude observations, BAO distance measurements, growth factor data, and Hubble parameter observations.', '1805.06408-2-5-2': 'Also, spatially flat XCDM and [MATH]CDM both reduce the tension between CMB temperature anisotropy and weak gravitational lensing estimates of [MATH], the rms fractional energy density inhomogeneity averaged over [MATH]Mpc radius spheres, where [MATH] is the Hubble constant in units of 100 km s[MATH] Mpc[MATH] .', '1805.06408-2-5-3': 'defcitealias4Planck Collaboration 2016', '1805.06408-2-6-0': 'In non-flat models nonzero spatial curvature provides an additional length scale which invalidates usage of the power-law power spectrum for energy density inhomogeneities in the non-flat case (as was assumed in the analysis of non-flat models intalias4).', '1805.06408-2-6-1': 'Non-flat inflation models provide the only known physically-consistent mechanism for generating energy density inhomogeneities in the non-flat case; the resulting open and closed model power spectra are not power laws .', '1805.06408-2-6-2': 'defcitealias4Planck Collaboration (2016)Using these power spectra, [CITATION] have found that the Planck 2015 CMB anisotropy data in combination with a few BAO distance measurements no longer rule out the non-flat [MATH]CDM case (unlike the earliertalias4 analyses based on the incorrect assumption of a power-law power spectrum in the non-flat model).', '1805.06408-2-6-3': '[CITATION] confirmed these results for a bigger compilation of cosmological data, and similar conclusions hold in the non-flat dynamical dark energy XCDM and [MATH]CDM cases .', '1805.06408-2-7-0': 'Additionally, the non-flat models provide a better fit to the observed low multipole CMB temperature anisotropy power spectrum, and do better at reconciling the CMB anisotropy and weak lensing constraints on [MATH], but do a worse job at fitting the observed large multipole CMB anisotropy temperature power spectrum .', '1805.06408-2-7-1': 'Given the non-standard normalization of the Planck 2015 CMB anisotropy likelihood and that the flat and non-flat [MATH]CDM models are not nested, it is not possible to compute the relative goodness of fit between the flat and non-flat [MATH]CDM models quantitatively, although qualitatively the flat [MATH]CDM model provides a better fit to the current data .', '1805.06408-2-8-0': 'In the analyses discussed above, the Planck 2015 CMB anisotropy data played the major role.', '1805.06408-2-8-1': 'Those authors found consistency between cosmological constraints derived using the CMB anisotropy data in combination with various non-CMB data sets.', '1805.06408-2-8-2': 'CMB anisotropy data are sensitive to the behavior of cosmological spatial inhomogeneities.', '1805.06408-2-8-3': 'Here we derive constraints on similar models from a combination of all available Hubble parameter data as well as all available radial and transverse BAO data.', '1805.06408-2-8-4': 'Unlike the CMB anisotropy data, the [MATH] and these BAO data are not sensitive to the behavior of cosmological spatial inhomogeneities.', '1805.06408-2-9-0': 'The models that we study, and the methods we use to analyze these data, are the same as those presented in [CITATION], and we also use some of the same [MATH] and baryon acoustic oscillation measurements.', '1805.06408-2-9-1': 'We differ from those studies by now using all currently available [MATH] and baryon acoustic oscillation data.', '1805.06408-2-10-0': 'The constraints we derive here are consistent with, but weaker than, those of the papers cited above; this provides a necessary and useful consistency test of those results.', '1805.06408-2-10-1': 'In particular, we find that the consensus flat [MATH]CDM model is a reasonable fit, in most cases, to the BAO and [MATH] data we study here.', '1805.06408-2-10-2': 'However, depending somewhat on the Hubble constant prior we use, consensus flat [MATH]CDM can be 1[MATH] away from the best-fit parameter values in some cases, which can favor mild dark energy dynamics or non-flat spatial hypersurfaces.', '1805.06408-2-11-0': 'In Sec. [REF] we provide a short summary of the models we studied.', '1805.06408-2-11-1': 'Sec. [REF] presents the data that we used, and in Sec. [REF] we describe the methods by which we analyzed these data.', '1805.06408-2-11-2': 'Sec. [REF] describes the results of our analyses, and our conclusions are given in Sec. [REF].', '1805.06408-2-12-0': '# Models', '1805.06408-2-13-0': 'The models we examine in this paper are characterized by their expansion rate as a function of redshift [MATH], [EQUATION]', '1805.06408-2-13-1': 'Here [MATH] is the Hubble parameter and [MATH] is the Hubble constant.', '1805.06408-2-14-0': 'In the [MATH]CDM model dark energy is a constant vacuum energy density with negative pressure, equivalent to an ideal fluid with equation of state parameter [EQUATION]', '1805.06408-2-14-1': 'Here [MATH] and [MATH] are the homogeneous parts of the pressure and energy density, respectively.', '1805.06408-2-14-2': 'The expansion rate can be written in terms of the density parameters [EQUATION] where [MATH] is the current value of the non-relativistic matter density parameter, [MATH] is the cosmological constant energy density parameter, and [MATH] (which is nonzero in general) is the current value of the spatial curvature energy density parameter.', '1805.06408-2-14-3': 'Here, and in the other models we study, we ignore the contributions from CMB photons and neutrinos, which are very small at the redshifts of the data we use, so the [MATH]CDM model is characterized by two parameters: [MATH].', '1805.06408-2-15-0': 'In the XCDM parametrization of dark energy, [MATH] where [MATH] is a negative constant (in general [MATH]).', '1805.06408-2-15-1': 'Hence [EQUATION] where [MATH] is the current value of the dark energy density.', '1805.06408-2-15-2': 'In contrast to [MATH]CDM, the dark energy density parameter [MATH] varies with time.', '1805.06408-2-15-3': 'If, however, [MATH], then XCDM reduces to [MATH]CDM, with [MATH].', '1805.06408-2-15-4': 'In general, the XCDM parametrization has three free parameters: [MATH].', '1805.06408-2-15-5': 'We shall also consider spatially flat XCDM, with [MATH].', '1805.06408-2-16-0': 'The [MATH]CDM model provides a simple, physically consistent description of dynamical dark energy.', '1805.06408-2-16-1': 'In this model, the dark energy is a scalar field [MATH] with a potential energy density given by [EQUATION]', '1805.06408-2-16-2': 'Here [MATH], [MATH], [MATH] is the gravitational constant, and [EQUATION]', '1805.06408-2-16-3': 'The spatially homogeneous part of the scalar field obeys [EQUATION] where [MATH] is the scale factor, and an overdot denotes differentiation with respect to time.', '1805.06408-2-16-4': 'This, together with the first Friedmann equation, [EQUATION] and [EQUATION] determines the dynamics of the field.', '1805.06408-2-16-5': 'In eq. ([REF]), [MATH] is the non-relativistic matter density, [MATH] is the scalar field energy density, and [MATH] for flat, closed, and open spatial hypersurfaces, respectively.', '1805.06408-2-17-0': 'The dark energy equation of state parameter of [MATH]CDM is [EQUATION] which, unlike in the [MATH]CDM and XCDM parametrizations, changes with time.', '1805.06408-2-17-1': 'The expansion rate in the [MATH]CDM model is [EQUATION] where [EQUATION]', '1805.06408-2-17-2': 'In contrast to [MATH], [MATH] is not an explicit function of a power of [MATH]; it must be determined numerically.', '1805.06408-2-18-0': 'In general, the [MATH]CDM model has three free parameters: [MATH].', '1805.06408-2-18-1': 'We also consider spatially flat [MATH]CDM with [MATH].', '1805.06408-2-19-0': '# Data', '1805.06408-2-20-0': 'BAO provide observers with a ""standard ruler"" which can be used to measure cosmological distances (see [CITATION] for a review).', '1805.06408-2-20-1': 'These distances can be computed in a given cosmological model, so measurements of them can be used to constrain the parameters of the model in question.', '1805.06408-2-20-2': 'The BAO distance measurements we use are listed in Table [REF].', '1805.06408-2-21-0': 'The transverse co-moving distance is [EQUATION] where [EQUATION] and the volume-averaged angular diameter distance is defcitealias4Planck Collaboration 2016 [EQUATION] .', '1805.06408-2-21-1': 'All of the measurements in Table [REF] are scaled by the size of the sound horizon at the drag epoch ([MATH]).', '1805.06408-2-21-2': 'This quantity istextsee [CITATION] for a derivation: [EQUATION] where [MATH] and [MATH] are the values of [MATH], the ratio of the baryon to photon momentum density, [EQUATION] at the drag epoch and matter-radiation equality epoch, respectively.', '1805.06408-2-21-3': 'Here [MATH] is the scale of the particle horizon at the matter-radiation equality epoch, and [MATH] and [MATH] are the baryon and photon mass densities.', '1805.06408-2-21-4': 'In our analyses, where appropriate, the original data listed in Table [REF] have been rescaled to a fiducial sound horizon [MATH] Mpctextfrom Table 4, column 3, oftalias4.', '1805.06408-2-21-5': 'This fiducial sound horizon was determined by using the [MATH]CDM model, so its value is model dependent, though not to a significant degree (as can be seen by comparing the computed [MATH] of thetalias4 baseline model to that measured using the spatially open [MATH]CDM and flat XCDM parametrization oftalias4).', '1805.06408-2-22-0': 'In Table [REF] we list 31 [MATH] measurements determined using the cosmic chronometric technique, which are the same as the cosmic chronometric [MATH] data used in [CITATION]textsee e.g. [CITATION] for a discussion of cosmic chronometers.', '1805.06408-2-22-1': 'With this method, the Hubble rate as a function of redshift is determined by using [EQUATION]', '1805.06408-2-22-2': 'Although this determination of [MATH] does not depend on a cosmological model, it does depend on the quality of the measurement of [MATH], which requires an accurate determination of the age-redshift relation for a given chronometer.', '1805.06408-2-22-3': 'See [CITATION] and [CITATION] for discussions of the strengths and weaknesses of this method.', '1805.06408-2-22-4': 'While their approach requires accurate knowledge of the star formation history and metallicity of massive, passively evolving early galaxies, and although the two different techniques they use give slightly different values, they also point out that the measurement of [MATH] from this method is relatively insensitive to changes in the chosen stellar population synthesis model.', '1805.06408-2-23-0': 'defcitealias4Planck Collaboration (2016)', '1805.06408-2-24-0': '# Methods', '1805.06408-2-25-0': 'defcitealias4Planck Collaboration 2016', '1805.06408-2-26-0': 'To determine the values of the best-fit parameters, we minimized [EQUATION] where [MATH] is the likelihood function and [MATH] is the set of parameters of the model under consideration.', '1805.06408-2-26-1': 'If the likelihood function [MATH] depends on an uninteresting nuisance parameter [MATH] with a probability distribution [MATH], we marginalize the likelihood function by integrating [MATH] over [MATH] [EQUATION]', '1805.06408-2-26-2': 'In our [MATH] analyses [MATH] is a nuisance parameter.', '1805.06408-2-26-3': 'We assumed a Gaussian distribution for [MATH] [EQUATION] and marginalized over it.', '1805.06408-2-26-4': 'We considered two cases: [MATH] km s[MATH] Mpc[MATH] and [MATH] km s[MATH] Mpc[MATH].', '1805.06408-2-27-0': 'Most of the data we analyzed are uncorrelated, however six of the data pointstextthose from [CITATION], are correlated.', '1805.06408-2-27-1': 'For uncorrelated data points, [EQUATION] where [MATH] are the model predictions at redshifts [MATH], and [MATH] and [MATH] are the central values and error bars of the measurements listed in Table [REF] and the last five lines of Table [REF].', '1805.06408-2-27-2': 'The correlated data (the first six entries in Table [REF]) require [EQUATION] where [MATH] is the inverse of the covariance matrix [EQUATION] .', '1805.06408-2-27-3': '[MATH] (in eq. [REF]) are the measurements in the first six lines of Table [REF].', '1805.06408-2-28-0': 'In addition to [MATH], we also used the Bayes Information Criterion [EQUATION] and the Akaike Information Criterion [EQUATION] .', '1805.06408-2-28-1': 'In these equations [MATH] is the minimum value of [MATH], [MATH] is the number of parameters of the given model, and [MATH] is the number of data points.', '1805.06408-2-28-2': ""BIC and AIC provide means to compare models with different numbers of parameters; they penalize models with a higher [MATH] in favor of those with a lower [MATH], in effect enforcing Occam's Razor in the model selection process."", '1805.06408-2-29-0': 'To determine the confidence intervals [MATH] on the 1d best-fit parameters, we computed one-sided limits [MATH] by using [EQUATION] where [MATH] is the point at which [MATH] has its maximimum value, such that [MATH] and [MATH], [MATH].', '1805.06408-2-29-1': 'Because the one-dimensional likelihood function is not guaranteed to be symmetric about [MATH], we compute the upper and lower confidence intervals separately.', '1805.06408-2-29-2': 'In the [MATH]CDM model, for example, the 1-sigma confidence intervals on [MATH] are computed by first integrating the likelihood function [MATH] over [MATH] to obtain a marginalized likelihood function that only depends on [MATH], [EQUATION] and then inserting this marginalized likelihood function into eq. ([REF]).', '1805.06408-2-30-0': 'The ranges over which we marginalized the parameters of the [MATH]CDM model were [MATH] and [MATH].', '1805.06408-2-30-1': 'For the spatially flat XCDM parametrization, we used [MATH] and [MATH], and for the spatially flat [MATH]CDM model we used [MATH] and [MATH].', '1805.06408-2-30-2': 'For 3-parameter XCDM, we used [MATH], [MATH], and [MATH].', '1805.06408-2-30-3': 'For the 3-parameter [MATH]CDM model we considered [MATH], [MATH], and [MATH].', '1805.06408-2-31-0': 'We analyzed the data with two independent Python codes, written by S.D. and J.R., that produced almost identical results in the 2-parameter cases and the 3-parameter XCDM parametrization, and results that agreed to within 1% in the 3-parameter [MATH]CDM case.', '1805.06408-2-32-0': '# Results', '1805.06408-2-33-0': 'The confidence contours for the models we considered are shown in Figs. [REF], [REF], and [REF].', '1805.06408-2-33-1': 'The solid black contours indicate the [MATH] km s[MATH] Mpc[MATH] prior constraints, the dashed black contours indicate the [MATH] km s[MATH] Mpc[MATH] prior constraints, and the red dots indicate the best-fit point in each prior case.', '1805.06408-2-33-2': 'Our results for the parameter values of the unmarginalized and marginalized cases are collected in Tables [REF]-[REF], along with their [MATH], AIC, and BIC values.', '1805.06408-2-33-3': 'Wherever [MATH], [MATH]AIC, and [MATH]BIC are given, these are computed relative to the [MATH], AIC, and BIC of the corresponding [MATH]CDM model of each prior case.', '1805.06408-2-34-0': 'In the 2-parameter case, the spatially flat XCDM parametrization has the lowest value of [MATH] if the prior on [MATH] is chosen to be [MATH] km s[MATH] Mpc[MATH].', '1805.06408-2-34-1': 'If, on the other hand, the [MATH] prior is chosen to be [MATH] km s[MATH] Mpc[MATH] then the spatially flat [MATH]CDM model has the lowest value of [MATH].', '1805.06408-2-34-2': 'These models also have lower AIC and BIC values than the 3-parameter XCDM parametrization and the 3-parameter [MATH]CDM model (see Tables [REF] and [REF]).', '1805.06408-2-34-3': 'On the other hand, the 3-parameter models typically have a lower [MATH] than the 2-parameter [MATH]CDM case.', '1805.06408-2-34-4': 'These differences, however, are not statistically significant.', '1805.06408-2-34-5': 'Focusing on the [MATH] km s[MATH] Mpc[MATH] prior case, the [MATH] differences indicate that the non-flat [MATH]CDM model and non-flat XCDM parametrization provide a 1.2[MATH] and 1.3[MATH] better fit to the data, respectively, while from [MATH]AIC we find that these two models are 79% and 86% as probable as the 2-parameter [MATH]CDM model, respectively.', '1805.06408-2-35-0': 'In Table [REF] ([REF]), we list the 1[MATH] and 2[MATH] confidence intervals on the parameters of each of the 2-parameter (3-parameter) models.', '1805.06408-2-35-1': 'We obtained these by marginalizing the 2-parameter (3-parameter) likelihood function as described in Sec. [REF].', '1805.06408-2-35-2': 'The best-fit points in these tables correspond to the maximum value of the relevant one-dimensional marginalized likelihood function.', '1805.06408-2-35-3': 'Table [REF] ([REF]) lists the corresponding two-dimensional (three-dimensional) best-fit points.', '1805.06408-2-36-0': 'From the figures and tables, we see that the spatially flat [MATH]CDM model is a reasonable fit to the [MATH] and BAO data we use (although the flat XCDM parametrization and flat [MATH]CDM model provide slightly better fits in the [MATH] km s[MATH] Mpc[MATH] case).', '1805.06408-2-36-1': 'In particular, from the figures, for the [MATH] km s[MATH] Mpc[MATH] prior, flat [MATH]CDM is always within about 1[MATH] of the best-fit value.', '1805.06408-2-36-2': 'However, the [MATH] km s[MATH] Mpc[MATH] case favors some larger deviations from flat [MATH]CDM.', '1805.06408-2-36-3': 'For example in the middle panel of Fig. [REF] for the flat XCDM parametrization it favors a phantom model over flat [MATH]CDM at a little more than 1[MATH], while in the center and right panels of Fig. [REF] for the non-flat [MATH]CDM case it also favors a closed model at a little more than 2[MATH].', '1805.06408-2-36-4': 'Similar conclusions may be drawn from the parameter limits listed in Tables [REF] and [REF].', '1805.06408-2-37-0': 'When both dynamical dark energy and spatial curvature are present (as opposed to cases with only dynamical dark energy or only spatial curvature) it is not as easy to constrain both parameters simultaneously.', '1805.06408-2-37-1': 'This can be seen by comparing the center and right panels of Fig. [REF] to the left panels of Figs. [REF] and [REF], respectively.', '1805.06408-2-37-2': 'When spatial curvature is allowed to vary, the confidence contours in the 3-parameter XCDM parametrization and the [MATH]CDM model expand along the [MATH] and [MATH] axes (these are the parameters that govern the dynamics of the dark energy).', '1805.06408-2-38-0': 'The consensus model, spatially flat [MATH]CDM, is consistent with current [MATH] + BAO data, but these data allow some nonzero spatial curvature.', '1805.06408-2-38-1': 'In particular, we find that the best-fit values of the parameters in the [MATH]CDM model imply a curvature energy density parameter of [MATH] for the [MATH] km s[MATH] Mpc[MATH] prior case, and [MATH] for the [MATH] km s[MATH] Mpc[MATH] prior case.', '1805.06408-2-38-2': 'More precisely, using the [MATH] and [MATH] best-fit values and error bars for flat [MATH]CDM from Table [REF], and combining the errors in quadrature, an approximate estimate is [MATH] and [MATH] for the [MATH] km s[MATH] Mpc[MATH] and [MATH] km s[MATH] Mpc[MATH] priors, with the data favoring a closed model at a little over 1[MATH] in the second case.', '1805.06408-2-38-3': 'The 3-parameter models, in both prior cases, favor closed spatial hypersurfaces, but the error bars are so large that these results only stand out in the [MATH] km s[MATH] Mpc[MATH] prior case of the [MATH]CDM model (see the center and right panels of [REF]).', '1805.06408-2-38-4': 'While not very statistically significant, we note that these results are not inconsistent with those of [CITATION] and [CITATION], who found that CMB anisotropy data, in conjunction with other cosmological data, were not inconsistent with mildly closed spatial hypersurfaces.', '1805.06408-2-39-0': 'The current data are also not inconsistent with some mild dark energy dynamics, although the size of the effect varies depending on the choice of [MATH] prior and whether or not [MATH] is allowed to vary as a free parameter.', '1805.06408-2-39-1': 'In the flat [MATH]CDM model, for instance, [MATH] can be different from zero only in the [MATH] km s[MATH] Mpc[MATH] prior case, whereas [MATH] can be different from zero in both prior cases if [MATH] is allowed to vary (see the right panel of [REF] and the left panel of [REF]).', '1805.06408-2-40-0': '# Conclusions', '1805.06408-2-41-0': 'We analyzed a total of 42 measurements, 31 of which consisted of uncorrelated [MATH] data points, with the remainder coming from BAO observations (some correlated, some not), to constrain dark energy dynamics and spatial curvature, by determining how well these measurements can be described by three common models of dark energy: [MATH]CDM, the XCDM parametrization, and [MATH]CDM.', '1805.06408-2-42-0': 'The consensus flat [MATH]CDM model is in reasonable accord with these data, but depending on the model analyzed and the [MATH] prior used, it can be a little more than 1[MATH] away from the best-fit model.', '1805.06408-2-42-1': 'These data are consistent with mild dark energy dynamics as well as non-flat spatial hypersurfaces.', '1805.06408-2-42-2': 'While these results are interesting and encouraging, more and better data are needed before we can make definitive statements about the spatial curvature of the universe and about dark energy dynamics.', '1805.06408-2-43-0': '# Acknowledgements', '1805.06408-2-44-0': 'tocsection'}","[['1805.06408-1-32-0', '1805.06408-2-35-0'], ['1805.06408-1-32-1', '1805.06408-2-35-1'], ['1805.06408-1-32-2', '1805.06408-2-35-2'], ['1805.06408-1-32-3', '1805.06408-2-35-3'], ['1805.06408-1-9-0', '1805.06408-2-11-0'], ['1805.06408-1-9-1', '1805.06408-2-11-1'], ['1805.06408-1-9-2', '1805.06408-2-11-2'], ['1805.06408-1-19-0', '1805.06408-2-21-0'], ['1805.06408-1-19-1', '1805.06408-2-21-1'], ['1805.06408-1-19-2', '1805.06408-2-21-2'], ['1805.06408-1-19-3', '1805.06408-2-21-3'], ['1805.06408-1-24-0', '1805.06408-2-27-0'], ['1805.06408-1-24-1', '1805.06408-2-27-1'], ['1805.06408-1-24-2', '1805.06408-2-27-2'], 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1106.4223,"{'1106.4223-1-0-0': 'Predictive recursion (PR) is a fast stochastic algorithm for nonparametric estimation of mixing distributions in mixture models.', '1106.4223-1-0-1': 'It is known that the PR estimates of both the mixing and mixture densities are consistent under fairly mild conditions, but currently very little is known about the rate of convergence.', '1106.4223-1-0-2': 'In this note we investigate asymptotic convergence properties of the PR estimate under model mis-specification in the special case of finite mixtures with known support.', '1106.4223-1-0-3': 'Tools from stochastic approximation are used to prove that the PR estimates converge at a nearly root-[MATH] rate.', '1106.4223-1-0-4': 'This result provides some important clues about the choice of weight sequence in the PR algorithm in general.', '1106.4223-1-1-0': 'Keywords and phrases: Density estimation; Kullback-Leibler divergence; Lyapunov function; mixture model; stochastic approximation.', '1106.4223-1-2-0': '# Introduction', '1106.4223-1-3-0': 'Nonparametric estimation of mixing distributions is an important and challenging problem in statistics.', '1106.4223-1-3-1': 'Recent progress along these lines has been made with the fast stochastic predictive recursion (PR) algorithm due to [CITATION], [CITATION], and [CITATION].', '1106.4223-1-3-2': 'PR is fundamentally different from existing algorithms, such as EM, in a number of ways.', '1106.4223-1-3-3': 'Most importantly, PR is not a hill-climbing algorithm.', '1106.4223-1-3-4': 'Instead, it learns sequentially like stochastic approximation .', '1106.4223-1-3-5': 'In addition, PR is able to estimate a mixing density with respect to any user-defined dominating measure.', '1106.4223-1-3-6': ""That is, unlike the nonparmetric maximum likelihood estimate, which is almost surely discrete , the PR estimate can be discrete, continuous, or both, depending on the user's choice of dominating measure."", '1106.4223-1-4-0': 'Theoretically, it has been shown that the PR estimates of both the mixing and mixture densities are consistent under certain conditions; see Section [REF] for more details.', '1106.4223-1-4-1': 'The goal of this note is to investigate the rate of convergence, about which very little is known.', '1106.4223-1-4-2': 'For this, we shall explore further the connection between PR and stochastic approximation developed in [CITATION].', '1106.4223-1-4-3': ""To the author's knowledge, results on the rate of convergence for general stochastic approximations are only fully developed in the finite-dimensional context."", '1106.4223-1-4-4': 'Therefore, we shall confine ourselves here to an analysis of PR when the possibly mis-specified model assumes that the data-generating distribution is a finite mixture with known support.', '1106.4223-1-4-5': 'In this case, we prove that the PR estimate of the mixing distribution converges almost surely at a nearly parametric root-[MATH] rate, where the limit is characterized by the mixture model closest to the true data-generating distribution based on the Kullback-Leibler divergence.', '1106.4223-1-4-6': ""This result also sheds light on how one should choose PR's tuning parameter in practical applications."", '1106.4223-1-5-0': '# Predictive recursion', '1106.4223-1-6-0': 'Suppose independent data [MATH] are available from a distribution with unknown density [MATH], which we model as a nonparametric mixture: [EQUATION] where [MATH] is a known kernel on [MATH] and [MATH] is unknown and to be estimated.', '1106.4223-1-6-1': 'Here [MATH] is the set of all densities with respect to a given [MATH]-finite Borel measure [MATH] on [MATH].', '1106.4223-1-6-2': '[CITATION] presents the following algorithm for nonparametric estimation of [MATH] and [MATH] based on [MATH].', '1106.4223-1-7-0': 'Choose a density [MATH] and a sequence of weights [MATH].', '1106.4223-1-7-1': 'Set [MATH].', '1106.4223-1-7-2': 'Then, for [MATH], compute [EQUATION]', '1106.4223-1-7-3': 'Return [MATH] and [MATH] as estimates of [MATH] and [MATH], respectively.', '1106.4223-1-8-0': 'PR has some interesting connections to the nonparametric Bayes estimate in the case where the unknown mixing distribution is modeled as a random draw from the Dirichlet process distribution.', '1106.4223-1-8-1': '[CITATION] take advantage of this connection to motivate a PR-based semiparametric mixture model analysis where an additional unknown structural parameter is estimated by maximizing a PR-induced approximate marginal likelihood.', '1106.4223-1-8-2': 'This general strategy has been used to develop PR-based solutions for inference in mixed-effects models, as well as by [CITATION] for large-scale nonparametric empirical Bayes multiple testing.', '1106.4223-1-9-0': 'Asymptotic convergence properties of the PR estimates [MATH] and [MATH] have only recently become available.', '1106.4223-1-9-1': 'Let [MATH] denote the set of mixture densities [MATH] as [MATH] ranges over [MATH].', '1106.4223-1-9-2': '[CITATION] build on the work of [CITATION] to show that when the mixture model is correctly specified (i.e., [MATH]), then both [MATH] and [MATH] converge almost surely to [MATH] and [MATH] in their respective topologies.', '1106.4223-1-9-3': '[CITATION] go one step further, showing that if [MATH], then [MATH] converges to the closest mixture density [MATH] as measured by the Kullback-Leibler divergence.', '1106.4223-1-9-4': 'As a corollary, if the postulated mixture model is identifiable, then [MATH] converges almost surely to [MATH] in the weak topology.', '1106.4223-1-9-5': 'They also establish a bound on the rate of convergence for [MATH] in terms of the PR weight sequence [MATH].', '1106.4223-1-9-6': 'For weights of the form [MATH], for suitable [MATH], [CITATION] obtain a [MATH] bound on the Hellinger convergence rate of [MATH] to [MATH] for a wide class of kernels [MATH].', '1106.4223-1-9-7': 'While this rate is comparable to the rate obtained in [CITATION], it leaves a lot to be desired.', '1106.4223-1-9-8': 'In fact, simulations in [CITATION] suggest that the upper bound corresponds to a ""worst case scenario"" rate of convergence, i.e., when [MATH] sits on the boundary of [MATH].', '1106.4223-1-9-9': 'The author conjectures that a nearly parametric root-[MATH] rate for [MATH], like that obtained by [CITATION], can be acheived by PR quite generally.', '1106.4223-1-9-10': 'In Section [REF] we show that this conjecture holds in the special known finite support case.', '1106.4223-1-10-0': '# Main results', '1106.4223-1-11-0': 'In this section we shall assume that [MATH] is modeled as a finite mixture.', '1106.4223-1-11-1': 'That is, [MATH] is a finite set of size [MATH] and [MATH] is counting measure.', '1106.4223-1-11-2': 'In this case, [MATH] denotes the [MATH]-dimensional probability simplex, and we write [MATH].', '1106.4223-1-11-3': 'Therefore, [MATH].', '1106.4223-1-11-4': 'For simplicity, all [MATH]-dimensional vectors [MATH] will be indexed by [MATH], i.e., [MATH].', '1106.4223-1-11-5': 'Also, [MATH] denotes the usual inner-product and [MATH] the corresponding norm.', '1106.4223-1-12-0': 'We begin by listing two basic assumptions about the mixture model.', '1106.4223-1-13-0': '[MATH] is continuous for each [MATH].', '1106.4223-1-14-0': 'Model [REF] is identifiable; i.e., [MATH] is one-to-one.', '1106.4223-1-15-0': 'For any density [MATH] on [MATH], define the Kullback-Leibler divergence of [MATH] from [MATH] as [MATH].', '1106.4223-1-15-1': 'Henceforth, we shall silently assume that [MATH] for all [MATH].', '1106.4223-1-15-2': 'Then the infimum [EQUATION] where [MATH] is the closure of [MATH], is finite.', '1106.4223-1-15-3': 'It follows from Assumption [REF] that there exists an [MATH] in [MATH] such that [MATH]; see Lemma 3.1 of [CITATION].', '1106.4223-1-15-4': 'Assumption [REF] ensures that [MATH] is unique.', '1106.4223-1-15-5': 'Allowing the model to be mis-specified is particularly important here, given that the assumption of known finite support is rather strong.', '1106.4223-1-15-6': 'For example, even if the support [MATH] is unknown, the results that follow show that PR does as well asymptotically as could be hoped for if we simply guess what [MATH] should be.', '1106.4223-1-16-0': 'Following [CITATION], we may express the predictive recursion update [MATH] as follows: [EQUATION] where, for generic [MATH] and [MATH], the mapping [MATH] is defined as [EQUATION].', '1106.4223-1-16-1': 'Equation [REF] shows that PR is a special case of a general Robbins-Monro type of stochastic approximation algorithm designed to find roots of the mapping [EQUATION]', '1106.4223-1-16-2': 'This [MATH] is nothing but the conditional expectation of [MATH], under the true density [MATH], given [MATH].', '1106.4223-1-16-3': 'The following result is an immediate consequence of the definitions and construction above.', '1106.4223-1-17-0': 'The sequence [MATH], for [MATH], given by [EQUATION] is a martingale difference sequence with respect to the [MATH]-algebra [MATH] generated by [MATH].', '1106.4223-1-17-1': 'Moreover, [MATH] is bounded for all [MATH].', '1106.4223-1-18-0': 'According to the stochastic approximation theory , convergence properties of [MATH], as [MATH], can be found by investigating the asymptotic behavior of solutions [MATH] of the ordinary differential equation (ODE) [MATH], as [MATH].', '1106.4223-1-18-1': 'For this purpose, we will need some basic definitions and results from the theory of ODEs.', '1106.4223-1-19-0': 'The mixing distribution [MATH] is an equilibrium point of the ODE [MATH]; in other words, [MATH] for all [MATH].', '1106.4223-1-20-0': 'Plugging [MATH] into the expression in [REF] gives [EQUATION].', '1106.4223-1-20-1': 'By the fact that [MATH] minimizes [MATH], it follows from Lemma 3.3 of [CITATION] that [MATH] for each [MATH].', '1106.4223-1-20-2': 'But since [MATH] vanishes, it must be that [MATH] for each [MATH], proving the claim.', '1106.4223-1-21-0': 'The goal is to show that [MATH] is a stable equilibrium in the sense that any solution to the ODE converges to [MATH], regardless of the initial condition.', '1106.4223-1-21-1': 'For this, a Lyapunov function will be useful.', '1106.4223-1-22-0': 'A real-valued function [MATH] is a Lyapunov function at [MATH] for the differential equation [MATH] if (i) [MATH] is continuously differentiable in a neighborhood of [MATH], (ii) [MATH] with equality if and only if [MATH], and (iii) [MATH].', '1106.4223-1-23-0': ""Lyapunov's theory, described beautifully in [CITATION], states that if a Lyapunov function [MATH] exists at [MATH], then [MATH] is a stable equilibrium point."", '1106.4223-1-23-1': 'Next we show that a slight variation of the Kullback-Leibler divergence is a Lyapunov function in the present context.', '1106.4223-1-24-0': 'The mapping [MATH] given by [EQUATION] is a Lyapunov function for the ODE [MATH].', '1106.4223-1-25-0': 'Properties (i) and (ii) in Definition [REF] are obvious.', '1106.4223-1-25-1': 'For property (iii), simple calculus reveals that [MATH], from which it follows that [MATH].', '1106.4223-1-25-2': 'That equality is obtained if and only if [MATH] follows from the fact that [MATH] is the unique minimizer of [MATH] and, hence, the only point at which [MATH] vanishes.', '1106.4223-1-26-0': 'The function [MATH] in [REF] can be viewed as a Lagrange multiplier version of the Kullback-Leibler divergence with the trivial constraint [MATH].', '1106.4223-1-26-1': 'This is consistent with the interpretation of PR as an algorithm that asymptotically minimizes [MATH] over [MATH] .', '1106.4223-1-26-2': 'Another important observation is that [MATH] is convex in [MATH]; see Lemma [REF].', '1106.4223-1-27-0': 'Next we state the extension of the PR convergence theorem in [CITATION] for the case where the true data-generating density [MATH] need not belong to the class [MATH] of mixture models [REF].', '1106.4223-1-27-1': 'For this we need', '1106.4223-1-28-0': '[MATH] and [MATH] for some [MATH].', '1106.4223-1-29-0': 'In practice, it is common to take [MATH] for [MATH].', '1106.4223-1-29-1': 'Then Assumption [REF] holds with [MATH].', '1106.4223-1-30-0': 'In light of Lemmas [REF]-[REF], the claim follows from Theorem 5.2.3 of [CITATION] and the continuity of [MATH]; see [CITATION].', '1106.4223-1-31-0': 'Our main result on a rate of convergence for PR will make use of a general theorem on convergence rates of stochastic approximation .', '1106.4223-1-31-1': ""For the readers' convenience, this theorem is stated in the Appendix."", '1106.4223-1-31-2': 'We first establish an important preliminary result.', '1106.4223-1-32-0': 'The sequence [MATH] in [REF] satisfies [MATH] almost surely for [MATH], where [MATH] is as in Assumption [REF].', '1106.4223-1-33-0': 'Let [MATH].', '1106.4223-1-33-1': 'By Lemma [REF], [MATH] is a martingale sequence and, since [MATH] is bounded, [EQUATION].', '1106.4223-1-33-2': 'Taking [MATH], it follows from Assumption [REF] that [MATH] is uniformly bounded in [MATH].', '1106.4223-1-33-3': 'Then the martingale convergence theorem implies that [MATH] converges almost surely, completing the proof.', '1106.4223-1-34-0': 'An additional assumption about the weights is required.', '1106.4223-1-34-1': 'For weights given by [MATH], this assumption holds as long as [MATH].', '1106.4223-1-35-0': '[MATH] satisfies [MATH].', '1106.4223-1-36-0': 'Let [MATH] denote the derivative of [MATH] evaluated at [MATH].', '1106.4223-1-36-1': 'If [MATH] is in the interior of [MATH], then all eigenvalues of [MATH] are negative.', '1106.4223-1-37-0': 'Simple calculus reveals that [MATH] is of the form [EQUATION].', '1106.4223-1-37-1': 'In matrix notation, we have [EQUATION] where [MATH] is a diagonal matrix with the elements of [MATH] as its diagonal entries, and [MATH] is the second derivative matrix of [MATH] evaluated at [MATH].', '1106.4223-1-37-2': 'Since [MATH] is in the interior of [MATH], all entries are positive and, hence, [MATH] is positive definite.', '1106.4223-1-37-3': 'Since [MATH] is convex on [MATH], [MATH] is also positive definite.', '1106.4223-1-37-4': 'The claim follows from the fact that the product of these two positive definite matrices, which is [MATH], must have positive eigenvalues.', '1106.4223-1-38-0': 'An interesting observation is that the matrix [MATH], the negative transpose of the Jacobian [MATH] in Lemma [REF], is a transition probability matrix for an irreducible, aperiodic Markov chain on [MATH].', '1106.4223-1-38-1': 'This chain is also reversible and has [MATH] as its stationary distribution.', '1106.4223-1-38-2': 'But how this observation might be useful in studying the asymptotic convergence of PR remains unclear.', '1106.4223-1-39-0': ""In light of Assumptions [REF]-[REF], Lemmas [REF] and [REF], and the existence of a Lyapunov function proved in Lemma [REF], we have our main result on the convergence rate of PR, a consequence of Chen's theorem in the Appendix."", '1106.4223-1-40-0': 'When the weights are given by [MATH], for [MATH], it follows from Theorem [REF] and the previous discussion that [MATH] almost surely.', '1106.4223-1-40-1': 'Since [MATH] can be chosen arbitrarily close to 1, it follows that the convergence rate can be made arbirarily close to [MATH] almost surely.', '1106.4223-1-41-0': 'A slightly stronger version of Theorem [REF] could be obtained if weight sequences were allowed to satisfy [MATH], with [MATH].', '1106.4223-1-41-1': 'For example, if [MATH], then [MATH].', '1106.4223-1-41-2': 'This extension would make the root-[MATH] rate possible, but it would require all eigenvalues of [MATH] in Lemma [REF] to be less than [MATH].', '1106.4223-1-41-3': ""At this point it is unclear whether this claim is true; standard bounds for eigenvalues, such as those in Gershgorin's theorem or Proposition 2 in [CITATION], are not helpful in this case."", '1106.4223-1-42-0': 'A consequence of Theorem [REF] is an almost sure [MATH] rate of convergence for the mixture [MATH] to [MATH].', '1106.4223-1-42-1': 'Let [MATH] denote the usual [MATH] norm.', '1106.4223-1-43-0': 'Under the conditions on Theorem [REF], [MATH] almost surely for [MATH] in Lemma [REF].', '1106.4223-1-44-0': '[CITATION] derive a bound of [MATH] for [MATH] in the general compact [MATH] case, where [MATH].', '1106.4223-1-44-1': 'Corollary [REF] provides a drastic improvement over this bound.', '1106.4223-1-44-2': 'In fact, when [MATH], the bound for [MATH] in [CITATION] becomes [MATH], which can be no faster than [MATH] under their conditions.', '1106.4223-1-44-3': 'Compare this to the sharp rate of [MATH] obtained from Corollary [REF], which is considerably faster than [MATH] for [MATH], albeit for the special known finite support case.', '1106.4223-1-44-4': 'So, regarding the PR weights [MATH], the message here, contrary to that in [CITATION], is that the faster the weights vanish the faster the overall convergence.', '1106.4223-1-45-0': '1=1', '1106.4223-1-46-0': '# Concluding remarks', '1106.4223-1-47-0': 'In this paper we have considered convergence properties of the PR estimate of a mixing density for the special case of known finite support.', '1106.4223-1-47-1': 'We have proved convergence in the case where the true data-generating density may or may not be a member of the postulated model, and have established a sharp bound on the rate of convergence, independent of the choice of kernel or the size of the finite support.', '1106.4223-1-47-2': 'In certain cases, a nearly root-[MATH] rate can be achieved.', '1106.4223-1-48-0': 'The importance of the rate of convergence is that it sheds light on how one should choose the PR weights [MATH].', '1106.4223-1-48-1': 'Indeed, the message of Theorem [REF] is that smaller weights lead to faster convergence.', '1106.4223-1-48-2': 'In particular, when [MATH], the convergence of [MATH] is faster for larger [MATH].', '1106.4223-1-48-3': 'Compare this with the message in [CITATION] which indicates that [MATH] converges faster for smaller [MATH].', '1106.4223-1-48-4': 'But one must keep in mind the context of these two results.', '1106.4223-1-48-5': ""On the one hand, the rate for [MATH] in [CITATION] is for the general compact [MATH] case, but it's only a conservative upper bound."", '1106.4223-1-48-6': 'On the other hand, the rate in the present paper is much faster and applies to both [MATH] and [MATH], but it covers only the simple finite-support case.', '1106.4223-1-49-0': '# Appendix: Convergence rates for stochastic approximation', '1106.4223-1-50-0': 'Consider a stochastic approximation process [MATH] which, for fixed initial value [MATH], is defined recursively as follows: [EQUATION].', '1106.4223-1-50-1': 'The process is designed so that [MATH] almost surely, where [MATH] satisfies [MATH].', '1106.4223-1-50-2': 'We shall assume that [MATH] bounded; otherwise, some truncation or projection techniques are needed .', '1106.4223-1-50-3': 'The PR estimates [MATH] are constrained to the simplex, so they satisfy this boundedness condition trivially.', '1106.4223-1-50-4': 'Next are the main assumptions of the theorem.', '1106.4223-1-51-0': 'Under A1-A4, [MATH] almost surely.'}","{'1106.4223-2-0-0': 'Predictive recursion (PR) is a fast stochastic algorithm for nonparametric estimation of mixing distributions in mixture models.', '1106.4223-2-0-1': 'It is known that the PR estimates of both the mixing and mixture densities are consistent under fairly mild conditions, but currently very little is known about the rate of convergence.', '1106.4223-2-0-2': 'Here I first investigate asymptotic convergence properties of the PR estimate under model misspecification in the special case of finite mixtures with known support.', '1106.4223-2-0-3': 'Tools from stochastic approximation theory are used to prove that the PR estimates converge, to the best Kullback-Leibler approximation, at a nearly root-[MATH] rate.', '1106.4223-2-0-4': 'When the support is unknown, PR can be used to construct an objective function which, when optimized, yields an estimate the support.', '1106.4223-2-0-5': 'I apply the known-support results to derive a rate of convergence for this modified PR estimate in the unknown support case, which compares favorably to known optimal rates.', '1106.4223-2-1-0': 'Keywords and phrases: Density estimation; Kullback-Leibler divergence; Lyapunov function; mixture model; stochastic approximation.', '1106.4223-2-2-0': '# Introduction', '1106.4223-2-3-0': 'Nonparametric estimation of mixing distributions is an important and challenging problem in statistics.', '1106.4223-2-3-1': 'Recent progress along these lines has been made with the fast stochastic predictive recursion (PR) algorithm due to [CITATION] and [CITATION].', '1106.4223-2-3-2': 'PR is fundamentally different from existing algorithms, such as EM, in a number of ways.', '1106.4223-2-3-3': 'Most importantly, PR is not a hill-climbing algorithm.', '1106.4223-2-3-4': 'Instead, it learns sequentially like stochastic approximation .', '1106.4223-2-3-5': 'In addition, PR is able to estimate a mixing density with respect to any user-defined dominating measure.', '1106.4223-2-3-6': ""That is, unlike the nonparmetric maximum likelihood estimate, which is almost surely discrete , the PR estimate can be discrete, continuous, or both, depending on the user's choice of dominating measure."", '1106.4223-2-4-0': 'Theoretically, it has been shown that the PR estimates of both the mixing and mixture densities are consistent under certain conditions; see Section [REF] for more details.', '1106.4223-2-4-1': 'The goal of this note is to investigate the rate of convergence, about which very little is known.', '1106.4223-2-4-2': 'For this, we shall explore further the connection between PR and stochastic approximation developed in [CITATION].', '1106.4223-2-4-3': ""To the author's knowledge, results on the rate of convergence for general stochastic approximations are only fully developed in the finite-dimensional context."", '1106.4223-2-4-4': 'Therefore, we shall confine ourselves here to an analysis of PR when the possibly misspecified model assumes that the data-generating distribution is a finite mixture with known support.', '1106.4223-2-4-5': 'In this case, we prove that the PR estimate of the mixing distribution converges almost surely at a nearly parametric root-[MATH] rate, where the limit is characterized by the mixture model closest to the true data-generating distribution based on the Kullback-Leibler divergence.', '1106.4223-2-4-6': ""This result also sheds light on how one should choose PR's tuning parameter in practical applications."", '1106.4223-2-5-0': 'The PR algorithm itself is not naturally suited for the case when the support of the finite mixture model is unknown.', '1106.4223-2-5-1': 'But, by applying the general principle in [CITATION], I show that PR yields a sort of objective function which can be optimized to estimate the unknown support.', '1106.4223-2-5-2': ""I apply the paper's known-support results to establish rates of convergence for this new PR-based unknown-support procedure."", '1106.4223-2-5-3': 'Two numerical examples are given to illustrate the method; for more examples and the full computational details, the reader is referred to [CITATION].', '1106.4223-2-6-0': '# Predictive recursion', '1106.4223-2-7-0': 'Suppose independent data [MATH] are available from a distribution with unknown density [MATH], which we model as a nonparametric mixture: [EQUATION] where [MATH] is a known kernel on [MATH] and [MATH] is unknown and to be estimated.', '1106.4223-2-7-1': 'Here [MATH] is the set of all densities with respect to a given [MATH]-finite Borel measure [MATH] on [MATH].', '1106.4223-2-7-2': '[CITATION] presents the following algorithm for nonparametric estimation of [MATH] and [MATH] based on [MATH].', '1106.4223-2-8-0': 'Choose a density [MATH] and a sequence of weights [MATH].', '1106.4223-2-8-1': 'Then, for [MATH], compute [MATH] and [EQUATION]', '1106.4223-2-8-2': 'Return [MATH] and [MATH] as estimates of [MATH] and [MATH], respectively.', '1106.4223-2-9-0': 'PR has some interesting connections to the nonparametric Bayes estimate in the case where the unknown mixing distribution is modeled as a random draw from the Dirichlet process distribution.', '1106.4223-2-9-1': '[CITATION] take advantage of this connection to motivate a PR-based semiparametric mixture model analysis where an additional unknown structural parameter is estimated by maximizing a PR-induced approximate marginal likelihood.', '1106.4223-2-9-2': '[CITATION] use this general strategy to develop a PR-based methodology for large-scale nonparametric empirical Bayes multiple testing.', '1106.4223-2-9-3': 'In Section [REF] I apply this method to mixtures with unknown support.', '1106.4223-2-10-0': 'Asymptotic convergence properties of the PR estimates [MATH] and [MATH] have only recently become available.', '1106.4223-2-10-1': 'Let [MATH] denote the set of mixture densities [MATH] as [MATH] ranges over [MATH].', '1106.4223-2-10-2': '[CITATION] build on the work of [CITATION] to show that when the mixture model is correctly specified (i.e., [MATH]), then both [MATH] and [MATH] converge almost surely to [MATH] and [MATH] in their respective topologies.', '1106.4223-2-10-3': '[CITATION] go one step further, showing that if [MATH], then [MATH] converges to the closest mixture density [MATH] as measured by the Kullback-Leibler divergence.', '1106.4223-2-10-4': 'As a corollary, if [MATH] is identifiable in the postulated mixture model, then [MATH] converges almost surely to [MATH] in the weak topology.', '1106.4223-2-10-5': 'They also establish a bound on the rate of convergence for [MATH] in terms of the PR weight sequence [MATH].', '1106.4223-2-10-6': 'For weights of the form [MATH], for suitable [MATH], [CITATION] obtain a [MATH] bound on the Hellinger convergence rate of [MATH] to [MATH] for a wide class of kernels [MATH].', '1106.4223-2-10-7': 'While this rate is comparable to the rate obtained in [CITATION], it leaves a lot to be desired.', '1106.4223-2-10-8': 'In fact, simulations in [CITATION] suggest that the upper bound corresponds to a ""worst case scenario"" rate of convergence, i.e., when [MATH] sits on the boundary of [MATH].', '1106.4223-2-10-9': 'I expect that a nearly parametric root-[MATH] rate for [MATH], like that obtained by [CITATION], can be achieved by PR, at least in some cases.', '1106.4223-2-10-10': 'In Section [REF] we show that this conjecture holds in the special known finite support case.', '1106.4223-2-11-0': '# Asymptotics for PR with known support', '1106.4223-2-12-0': 'Assume that the true density [MATH] is modeled as a finite mixture.', '1106.4223-2-12-1': 'That is, [MATH] is a finite set of size [MATH] and [MATH] is counting measure.', '1106.4223-2-12-2': 'In this case, [MATH] denotes the [MATH]-dimensional probability simplex, and I write [MATH].', '1106.4223-2-12-3': 'Then [MATH].', '1106.4223-2-12-4': 'Throughout, all [MATH]-dimensional vectors [MATH] will be indexed by [MATH], i.e., [MATH].', '1106.4223-2-12-5': 'Also, [MATH] denotes the usual inner-product and [MATH] the corresponding norm.', '1106.4223-2-13-0': 'We begin by listing two basic assumptions about the mixture model.', '1106.4223-2-14-0': '[MATH] is continuous for each [MATH].', '1106.4223-2-15-0': '[MATH] is identifiable in model [REF], i.e., [MATH] is one-to-one.', '1106.4223-2-16-0': 'For any density [MATH] on [MATH], define the Kullback-Leibler divergence of [MATH] from [MATH] as [MATH].', '1106.4223-2-16-1': 'Henceforth, I shall silently assume that [MATH] for all [MATH].', '1106.4223-2-16-2': 'Then the infimum [EQUATION] is finite.', '1106.4223-2-16-3': 'It follows from Assumption [REF] that there exists an [MATH] in the closure of [MATH] such that [MATH]; see Lemma 3.1 of [CITATION].', '1106.4223-2-16-4': 'Assumption [REF] ensures that [MATH] is unique.', '1106.4223-2-16-5': 'Allowing the model to be misspecified is particularly important here, given that the assumption of known finite support is rather strong.', '1106.4223-2-16-6': 'For example, even if the support [MATH] is unknown, the results that follow show that PR does as well asymptotically as could be hoped for if we simply guess at what [MATH] should be.', '1106.4223-2-17-0': 'Following [CITATION], express the PR update [MATH], [MATH], as follows: [EQUATION] where, for generic [MATH] and [MATH], the mapping [MATH] is defined as [EQUATION].', '1106.4223-2-17-1': 'Equation [REF] shows that PR is a special case of a general Robbins-Monro type of stochastic approximation algorithm designed to find roots of the mapping [EQUATION]', '1106.4223-2-17-2': 'This [MATH] is nothing but the conditional expectation of [MATH], under the true density [MATH], given [MATH] equals [MATH].', '1106.4223-2-17-3': 'The following result is an immediate consequence of the definitions and construction above.', '1106.4223-2-18-0': 'The sequence [MATH], for [MATH], given by [EQUATION] is a martingale difference sequence with respect to the [MATH]-algebra [MATH] generated by [MATH].', '1106.4223-2-18-1': 'Moreover, [MATH] is bounded for all [MATH].', '1106.4223-2-19-0': 'According to stochastic approximation theory , convergence properties of [MATH], as [MATH], can be found by investigating the asymptotic behavior of solutions of an appropriate ordinary differential equation (ODE).', '1106.4223-2-19-1': 'Specifically, let [MATH] denote a generic trajectory in [MATH].', '1106.4223-2-19-2': 'Then the limiting behavior of solutions [MATH] of the ODE [MATH], as [MATH], can be used to study the limiting behavior of the PR sequence [MATH], as [MATH].', '1106.4223-2-19-3': 'For this purpose, I will need some basic definitions and results from the theory of ODEs.', '1106.4223-2-20-0': 'The mixing distribution [MATH] is an equilibrium point of the ODE [MATH]; in other words, [MATH] for all [MATH].', '1106.4223-2-21-0': 'Plugging [MATH] into the expression in [REF] gives [EQUATION].', '1106.4223-2-21-1': 'By the fact that [MATH] minimizes [MATH], it follows from Lemma 3.3 of [CITATION] that [MATH] for each [MATH].', '1106.4223-2-21-2': 'But since [MATH] vanishes, it must be that [MATH] for each [MATH], proving the claim.', '1106.4223-2-22-0': 'The goal is to show that [MATH] is a stable equilibrium in the sense that any solution to the ODE converges to [MATH], regardless of the initial condition.', '1106.4223-2-22-1': 'For this, a Lyapunov function will be useful.', '1106.4223-2-23-0': 'A function [MATH] is a Lyapunov function at [MATH] for the ODE [MATH] if (i) [MATH] is continuously differentiable in a neighborhood of [MATH], (ii) [MATH] with equality if and only if [MATH], and (iii) [MATH].', '1106.4223-2-24-0': ""Lyapunov's theory, described beautifully in [CITATION], states that if a Lyapunov function [MATH] exists at [MATH], then [MATH] is a stable equilibrium point."", '1106.4223-2-24-1': 'Next I show that a slight variation of the Kullback-Leibler divergence is a Lyapunov function in the present context.', '1106.4223-2-25-0': 'The mapping [MATH] given by [EQUATION] is a Lyapunov function for the ODE [MATH].', '1106.4223-2-26-0': 'Properties (i) and (ii) in Definition [REF] are obvious.', '1106.4223-2-26-1': 'For property (iii), simple calculus reveals that [MATH], from which it follows that [MATH].', '1106.4223-2-26-2': 'That equality is obtained if and only if [MATH] follows from the fact that [MATH] is the unique minimizer of [MATH] and, hence, the only point at which [MATH] vanishes.', '1106.4223-2-27-0': 'The function [MATH] in [REF] can be viewed as a Lagrange multiplier version of the Kullback-Leibler divergence with the trivial constraint [MATH].', '1106.4223-2-27-1': 'This is consistent with the interpretation of PR as an algorithm that asymptotically minimizes [MATH] over [MATH] .', '1106.4223-2-27-2': 'Another important observation, used in Lemma [REF] below, is that [MATH] is convex.', '1106.4223-2-28-0': 'Next I state an extension of the PR convergence theorem in [CITATION] for the case where the true data-generating density [MATH] need not belong to the class [MATH] of mixture models [REF].', '1106.4223-2-28-1': 'For this we need', '1106.4223-2-29-0': '[MATH] and [MATH] for some [MATH].', '1106.4223-2-30-0': 'In practice, it is common to take [MATH] for [MATH].', '1106.4223-2-30-1': 'Then Assumption [REF] holds with [MATH].', '1106.4223-2-31-0': 'In light of Lemmas [REF]-[REF], the claim follows from Theorem 5.2.3 of [CITATION] and the continuity of [MATH]; see [CITATION].', '1106.4223-2-32-0': 'The main result on a rate of convergence for PR will make use of a general theorem on convergence rates of stochastic approximation ; see Appendix [REF].', '1106.4223-2-32-1': 'But two preliminary result are needed first.', '1106.4223-2-33-0': 'The sequence [MATH] in [REF] satisfies [MATH] almost surely for [MATH], where [MATH] is as in Assumption [REF].', '1106.4223-2-34-0': 'Let [MATH].', '1106.4223-2-34-1': 'By Lemma [REF], [MATH] is a martingale sequence and, since [MATH] is bounded, [EQUATION].', '1106.4223-2-34-2': 'Taking [MATH], it follows from Assumption [REF] that [MATH] is uniformly bounded in [MATH].', '1106.4223-2-34-3': 'Then the martingale convergence theorem implies that [MATH] converges almost surely, completing the proof.', '1106.4223-2-35-0': 'An additional assumption about the weights is required.', '1106.4223-2-35-1': 'For weights given by [MATH], this assumption holds as long as [MATH].', '1106.4223-2-36-0': '[MATH] satisfies [MATH].', '1106.4223-2-37-0': 'Let [MATH] denote the derivative of [MATH] evaluated at [MATH].', '1106.4223-2-37-1': 'If [MATH] is in the interior of [MATH], then all eigenvalues of [MATH] are negative.', '1106.4223-2-38-0': 'Simple calculus reveals that [MATH] is of the form [EQUATION].', '1106.4223-2-38-1': 'In matrix notation, write [MATH], where [MATH] is a diagonal matrix with the elements of [MATH] as its diagonal entries, and [MATH] is the second derivative matrix of [MATH] evaluated at [MATH].', '1106.4223-2-38-2': 'Since [MATH] is in the interior of [MATH], all entries are positive and, hence, [MATH] is positive definite.', '1106.4223-2-38-3': 'Since [MATH] is convex on [MATH], [MATH] is also positive definite.', '1106.4223-2-38-4': 'The claim follows from the fact that the product of these two positive definite matrices, which is [MATH], must have positive eigenvalues.', '1106.4223-2-39-0': 'An interesting observation is that the matrix [MATH], the negative transpose of the Jacobian [MATH] in Lemma [REF], is a transition probability matrix for an irreducible, aperiodic Markov chain on [MATH].', '1106.4223-2-39-1': 'This chain is also reversible and has [MATH] as its stationary distribution.', '1106.4223-2-39-2': 'But how this observation might be useful in studying the asymptotic convergence of PR remains unclear.', '1106.4223-2-40-0': ""In light of Assumptions [REF]-[REF], Lemmas [REF] and [REF], and the existence of a Lyapunov function proved in Lemma [REF], the main result on the convergence rate of PR is a consequence of Chen's theorem in Appendix [REF]."", '1106.4223-2-41-0': 'When the weights are given by [MATH], for [MATH], it follows from Theorem [REF] and the previous discussion that [MATH] almost surely.', '1106.4223-2-41-1': 'Since [MATH] can be chosen arbitrarily close to 1, it follows that the convergence rate can be made arbitrarily close to [MATH] almost surely.', '1106.4223-2-42-0': 'A slightly stronger version of Theorem [REF] could be obtained if weight sequences were allowed to satisfy [MATH], with [MATH].', '1106.4223-2-42-1': 'For example, if [MATH], then [MATH].', '1106.4223-2-42-2': 'This extension would make the root-[MATH] rate possible, but it would require all eigenvalues of [MATH] in Lemma [REF] to be less than [MATH].', '1106.4223-2-42-3': ""At this point it is unclear whether this claim is true; standard bounds for eigenvalues, such as those in Gershgorin's theorem or Proposition 2 in [CITATION], are not helpful in this case."", '1106.4223-2-43-0': 'Almost sure rates of convergence for the mixture density [MATH] to [MATH] are available as consequences of Theorem [REF].', '1106.4223-2-43-1': 'The [MATH] rate follows immediately from its definition, while the rate for the Kullback-Leibler contrast, [MATH], requires a simple second-order Taylor approximation of [MATH] at [MATH].', '1106.4223-2-44-0': 'Under the conditions on Theorem [REF], [MATH] almost surely for [MATH] in Lemma [REF].', '1106.4223-2-44-1': 'Likewise, [MATH].', '1106.4223-2-45-0': '[CITATION] derive a bound of [MATH] for [MATH] in the general compact [MATH] case, where [MATH].', '1106.4223-2-45-1': 'When [MATH], the bound for [MATH] in [CITATION] becomes [MATH], which can be no faster than [MATH] under their conditions.', '1106.4223-2-45-2': 'Compare this to the rate of [MATH] obtained from Corollary [REF], which is considerably faster than [MATH] for [MATH], albeit for the special known finite support case.', '1106.4223-2-45-3': 'So, regarding the PR weights [MATH], the message here, contrary to that in [CITATION], is that the faster the weights vanish the faster the overall convergence.', '1106.4223-2-46-0': '# PR with unknown support', '1106.4223-2-47-0': 'The PR convergence theory in the previous section assumes the finite support is known and only the mixing distribution is unknown.', '1106.4223-2-47-1': 'In practice, however, both the support and mixing distribution are unknown and to be estimated.', '1106.4223-2-47-2': 'To close this gap, I propose here a new PR-based approach for handling the unknown support case.', '1106.4223-2-47-3': 'The asymptotic results in Section [REF] will be used to prove consistency of this new procedure.', '1106.4223-2-47-4': 'Two simple examples are also given for illustration, but the computational details, simulations, and extensions will be presented elsewhere .', '1106.4223-2-48-0': '## Setup', '1106.4223-2-49-0': 'Let [MATH] be a compact set, large enough that there is a finite mixture supported in [MATH] that gives a sufficiently accurate approximation to [MATH].', '1106.4223-2-49-1': 'Take [MATH] to be a generic finite subset of [MATH].', '1106.4223-2-49-2': 'By treating [MATH] as the fixed support, a run of PR will produce a sequence of estimates [MATH] of the mixing and mixture distributions, whose dependence on the chosen support set [MATH] are now made explicit.', '1106.4223-2-49-3': 'In the same vein, write [MATH] for the [MATH]-dimensional probability simplex and define [MATH], the smallest Kullback-Leibler number for mixtures supported on [MATH].', '1106.4223-2-50-0': 'The jumping off point is that the result [MATH] of Corollary [REF] holds ""pointwise"" for all [MATH]; that is, the particular support [MATH] plays no role in the analysis of Section [REF].', '1106.4223-2-50-1': 'Thus, in the present case where the support is unknown, a reasonable strategy is to estimate the support by minimizing, over [MATH], some estimate of [MATH].', '1106.4223-2-50-2': 'This is the approach advocated by [CITATION].', '1106.4223-2-50-3': 'Indeed, by making connections to PR and Dirichlet process mixture models, they argue that, in the present context, the appropriate estimate of [MATH] is [EQUATION]', '1106.4223-2-50-4': 'Then the goal is to minimize [MATH] over [MATH].', '1106.4223-2-50-5': 'But since it is not possible to perform this optimization over all finite [MATH], some adjustment must be made.', '1106.4223-2-50-6': 'Consider starting with a fixed finite subset [MATH] of [MATH] obtained by chopping up [MATH] into a sufficiently fine grid, so that [MATH] is large.', '1106.4223-2-50-7': 'Then the collection of all subsets [MATH] of [MATH] is huge-it has [MATH] elements-but finite so it is possible to minimize [MATH] over [MATH].', '1106.4223-2-50-8': '[CITATION] uses a simulated annealing strategy to perform this optimization.', '1106.4223-2-50-9': 'Once the minimizer [MATH] of [MATH] is obtained, PR is run once more to produce [MATH] and [MATH] as estimates of the mixing and mixture distributions, respectively.', '1106.4223-2-51-0': '## Large-sample theory', '1106.4223-2-52-0': 'For simplicity, I will assume that the true density [MATH] is indeed a mixture density of the postulated form with support contained in [MATH]; the more general case can be handled similarly, but with an additional technical assumption .', '1106.4223-2-52-1': 'Also, assume that [MATH] for some [MATH].', '1106.4223-2-52-2': 'To get convergence of the approximation [MATH] to [MATH], I will need one additional assumption, stated next, which holds for many common kernels, including normal and Poisson.', '1106.4223-2-53-0': 'There exists a finite constant [MATH] such that [EQUATION].', '1106.4223-2-54-0': 'Under Assumptions [REF]-[REF], one can follow the proof of Theorem 2 in [CITATION] to conclude that, for each fixed [MATH], [EQUATION] almost surely, for any sequence [MATH] that satisfies [MATH] for some [MATH].', '1106.4223-2-54-1': 'It follows from Corollary [REF] that the summation in [REF] is of the order [MATH].', '1106.4223-2-54-2': 'So, if [MATH], the right-most term in the modulus in [REF] vanishes and, therefore, so must the left-most term.', '1106.4223-2-54-3': 'This proves that, for [MATH], [MATH] pointwise in [MATH] at a rate just slower than [MATH].', '1106.4223-2-54-4': 'But since [MATH] is finite, the convergence is also uniform.', '1106.4223-2-54-5': 'The following theorem summarizes this result.', '1106.4223-2-55-0': 'Choose weights [MATH] with [MATH] and let [MATH].', '1106.4223-2-55-1': 'Then, under Assumptions [REF]-[REF], [MATH] almost surely as [MATH].', '1106.4223-2-55-2': 'Moreover, since [MATH] ranges only over a finite set, [MATH], where [MATH] is the support of the true mixture distribution.', '1106.4223-2-56-0': 'If I define a distance [MATH] between two sets as the cardinality of their symmetric difference, then Theorem [REF] states that [MATH].', '1106.4223-2-56-1': 'In other words, [MATH] is a nearly root-[MATH]-consistent estimate of [MATH].', '1106.4223-2-56-2': 'Furthermore, a nearly root-[MATH] rate of convergence for [MATH] can be obtained, which I now sketch.', '1106.4223-2-56-3': 'With a slight abuse of notation, I can bound the total variation distance between [MATH] and [MATH] as follows: [EQUATION]', '1106.4223-2-56-4': 'The two outer-most terms on the right-hand side vanish at a nearly root-[MATH] rate according to Theorems [REF] and [REF], respectively.', '1106.4223-2-56-5': 'The middle term is more difficult to analyze, but it is clear that the data-dependent PR mapping [MATH] is, in some sense, continuous in [MATH].', '1106.4223-2-56-6': 'So, the convergence of [MATH] is also driven by [MATH].', '1106.4223-2-56-7': 'Therefore, the rate for [MATH] must also be nearly [MATH].', '1106.4223-2-57-0': 'Recall that [CITATION] showed that, for finite mixtures, the optimal rate of convergence is [MATH].', '1106.4223-2-57-1': 'In that case, the unknown finite support is allowed to be anything, essentially nonparametric, so the rates are relatively slow.', '1106.4223-2-57-2': 'In contrast, by restricting the set of candidate supports to subsets of a large but ultimately finite set [MATH], I am able to achieve a nearly parametric root-[MATH] rate of convergence.', '1106.4223-2-58-0': '## Examples', '1106.4223-2-59-0': 'Here I give two relatively simple real-data examples-a Gaussian location mixture and a Poisson mixture-to illustrate the potential of the proposed method.', '1106.4223-2-60-0': 'Under the Big Bang model, galaxies should form clusters and the relative velocities of the galaxies should be similar within clusters.', '1106.4223-2-60-1': '[CITATION] considers velocity data for [MATH] galaxies.', '1106.4223-2-60-2': 'She models this data as a finite Gaussian mixture, with the number and location of the mixture components unknown.', '1106.4223-2-60-3': 'The assumption is that each galactic cluster is a single component of the Gaussian mixture.', '1106.4223-2-60-4': 'The presence of multiple mixture components is consistent with the hypothesis of galaxy clustering.', '1106.4223-2-61-0': 'We apply the methodology outlined above to estimate the mixing distribution [MATH].', '1106.4223-2-61-1': 'We will consider a simple Gaussian mixture model in which each component has variance [MATH], based on the a priori considerations of [CITATION].', '1106.4223-2-61-2': 'From the observed velocities, it is apparent that the mixture components should be centered somewhere in the interval [MATH], so we choose a grid of candidate support points [MATH].', '1106.4223-2-61-3': 'Figure [REF] shows the corresponding estimates of the mixing and mixture distribution.', '1106.4223-2-61-4': 'The PR method identifies six galaxy clusters, and the estimates of [MATH] and [MATH] closely match those of [CITATION] and others.', '1106.4223-2-62-0': '[CITATION] present data on the number of defaulted installments in a Spanish financial institution.', '1106.4223-2-62-1': 'This data has a high number of zero counts, as well as substantial overdispersion.', '1106.4223-2-62-2': 'This suggests a Poisson mixture model, and here we compare the PR-based estimates to others presented in the literature.', '1106.4223-2-62-3': 'The first three rows of Table [REF] show the estimates of [MATH] for three methods in an zero-inflated Poisson mixture model.', '1106.4223-2-62-4': 'These include an estimate based on the AIC penalty, the SCAD-based penalized likelihood approach of [CITATION], and a minimum Hellinger distance method for count data .', '1106.4223-2-62-5': 'I start by bounding the support by [MATH] and taking [MATH] to be a set of 100 equispaced points in [MATH].', '1106.4223-2-62-6': 'All but the Woo-Sriram estimates have five support points, including zero.', '1106.4223-2-62-7': 'Besides this, we find that the corresponding estimates are quite similar.', '1106.4223-2-62-8': 'An attractive feature of this method is that no special adjustments are needed for zero-inflation.', '1106.4223-2-62-9': 'That is, zero-inflation can be achieved by simply including zero in the grid [MATH] and letting the data decide if a mass at zero is appropriate.', '1106.4223-2-62-10': 'Fitted values were obtained for each of the four methods (not shown) and I find that, for small [MATH]-values, where the observed counts are relatively large, the PR-based estimate appears to provide a better overall fit compared to the others.', '1106.4223-2-63-0': '1=1'}","[['1106.4223-1-33-1', '1106.4223-2-34-1'], ['1106.4223-1-33-2', '1106.4223-2-34-2'], ['1106.4223-1-33-3', '1106.4223-2-34-3'], ['1106.4223-1-27-1', '1106.4223-2-28-1'], ['1106.4223-1-1-0', '1106.4223-2-1-0'], 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'1106.4223-2-14-0', '1106.4223-2-15-0', '1106.4223-2-29-0', '1106.4223-2-33-0', '1106.4223-2-34-0', '1106.4223-2-36-0', '1106.4223-2-44-1', '1106.4223-2-53-0', '1106.4223-2-63-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1106.4223,,,
1608.00709,"{'1608.00709-1-0-0': 'We give explicit bounds for Jordan constants of groups of birational automorphisms of rationally connected threefolds over fields of zero characteristic, in particular, for Cremona groups of ranks [MATH] and [MATH].', '1608.00709-1-1-0': '# Introduction', '1608.00709-1-2-0': '## Jordan property', '1608.00709-1-3-0': 'The Cremona group of rank [MATH] is the group [MATH] of birational transformations of the projective space [MATH] over a field [MATH].', '1608.00709-1-3-1': 'It has been actively studied from various points of view for many years (see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and references therein).', '1608.00709-1-3-2': 'One of the approaches to this huge group is to try to understand its finite subgroups.', '1608.00709-1-3-3': 'It appeared that it is possible to obtain a complete classification of finite subgroups of [MATH] over an algebraically closed field [MATH] of characteristic [MATH] (see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION]), and to obtain partial classification results for [MATH] (see [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION]).', '1608.00709-1-3-4': 'Some results are also known for non algebraically closed fields, see e.g. [CITATION], [CITATION], and [CITATION].', '1608.00709-1-3-5': 'In general, it is partially known and partially expected that the collection of finite subgroups of a Cremona group shares certain features with the collection of finite subgroups of a group [MATH].', '1608.00709-1-4-0': '[C.Jordan, see e.g. [CITATION]]', '1608.00709-1-5-0': 'There is a constant [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-1-6-0': 'This leads to the following definition (cf. [CITATION]).', '1608.00709-1-7-0': 'A group [MATH] is called Jordan (alternatively, we say that [MATH] has Jordan property) if there is a constant [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-1-8-0': 'Theorem [REF] implies that all linear algebraic groups over an arbitrary field [MATH] of characteristic [MATH] are Jordan.', '1608.00709-1-8-1': 'Jordan property was also studied recently for groups of birational automorphisms of algebraic varieties.', '1608.00709-1-8-2': 'The starting point here was the following result of J.-P.Serre.', '1608.00709-1-9-0': '[J.-P.Serre [CITATION], [CITATION]] The Cremona group [MATH] over a field [MATH] of characteristic [MATH] is Jordan.', '1608.00709-1-10-0': 'Note that the assumption about characteristic is indispensable.', '1608.00709-1-10-1': 'Indeed, the group [MATH] contains [MATH], so that if the characteristic of the field [MATH] equals [MATH] and [MATH] is algebraically closed, then [MATH] contains a series of simple subgroups [MATH] of increasing order.', '1608.00709-1-11-0': 'It also appeared that there are surfaces with non-Jordan groups of birational selfmaps (see [CITATION]).', '1608.00709-1-11-1': 'V.Popov managed to give a complete classification of surfaces with Jordan groups of birational automorphisms.', '1608.00709-1-12-0': '[V.Popov [CITATION]] Let [MATH] be a surface over a field [MATH] of characteristic [MATH].', '1608.00709-1-12-1': 'Then the group [MATH] of birational automorphisms of [MATH] is Jordan if and only if [MATH] is not birational to [MATH], where [MATH] is an elliptic curve.', '1608.00709-1-13-0': 'In dimension [MATH] Jordan property is known for groups of birational automorphisms of rationally connected varieties (see e.g. [CITATION] for definition and basic background).', '1608.00709-1-14-0': '[see [CITATION]]', '1608.00709-1-15-0': 'Fix a field [MATH] of characteristic [MATH].', '1608.00709-1-15-1': 'Then there is a constant [MATH] such that for any rationally connected variety [MATH] of dimension [MATH] defined over [MATH] and any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-1-15-2': 'In particular, for any rationally connected threefold [MATH] the group [MATH] is Jordan.', '1608.00709-1-16-0': 'Actually, by [CITATION] the assertion of Theorem [REF] holds in arbitrary dimension modulo conjectural boundedness of terminal Fano threefolds (see e.g. [CITATION] or [CITATION]).', '1608.00709-1-16-1': 'For other results (in particular, for birational automorphisms of non rationally connected varieties, and for automorphisms of varieties of different types) see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION].', '1608.00709-1-17-0': '## Jordan constants', '1608.00709-1-18-0': 'Given a Jordan group [MATH], one may get interested in the minimal value of the constant involved in Definition [REF], and in the values of other relevant constants.', '1608.00709-1-19-0': 'Let [MATH] be a Jordan group.', '1608.00709-1-19-1': 'The Jordan constant [MATH] of the group [MATH] is the minimal number [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-1-19-2': 'The weak Jordan constant [MATH] of the group [MATH] is the minimal number [MATH] such that for any finite subgroup [MATH] there exists a (not necessarily normal) abelian subgroup [MATH] of index at most [MATH].', '1608.00709-1-20-0': 'It is more traditional to study Jordan constants than weak Jordan constants of Jordan groups, although there is no big difference between them.', '1608.00709-1-20-1': 'Indeed, one has [MATH] for any Jordan group [MATH] for obvious reasons.', '1608.00709-1-20-2': 'Moreover, if [MATH] is a finite group and [MATH] is an abelian subgroup of [MATH], then by [CITATION] one can find a normal abelian subgroup [MATH] of [MATH] such that [EQUATION]', '1608.00709-1-20-3': 'Therefore, if [MATH] is a Jordan group, one always has [MATH].', '1608.00709-1-20-4': 'On the other hand, the advantage of the weak Jordan constant is that it allows easy estimates using subgroups of the initial group.', '1608.00709-1-20-5': 'Namely, if [MATH] is a subgroup of finite index in a group [MATH], and [MATH] is Jordan, then [MATH] is Jordan with [EQUATION]', '1608.00709-1-20-6': 'Also, if [MATH] and [MATH] are Jordan groups, the group [MATH] is Jordan with [EQUATION]', '1608.00709-1-20-7': 'In particular, if [MATH] is a subgroup of [MATH], where [MATH] is a Jordan group and [MATH] is an abelian group, then [MATH] is Jordan with [MATH].', '1608.00709-1-21-0': 'Jordan constants are known for example for the groups [MATH] (see [CITATION]).', '1608.00709-1-21-1': 'In [CITATION] J.-P.Serre gave an explicit bound for the Jordan constant of the Cremona group [MATH] (see Remark [REF] below).', '1608.00709-1-21-2': 'Our first result also concerns the group [MATH].', '1608.00709-1-22-0': 'Suppose that the field [MATH] has characteristic [MATH].', '1608.00709-1-22-1': 'Then one has [EQUATION]', '1608.00709-1-22-2': 'The first of these bounds becomes an equality if [MATH] is algebraically closed.', '1608.00709-1-23-0': 'The main goal of this paper is to present a bound for Jordan constants of the groups of birational automorphisms of rationally connected threefolds, in particular, for the group [MATH].', '1608.00709-1-24-0': 'Let [MATH] be a rationally connected threefold over a field [MATH] of characteristic [MATH].', '1608.00709-1-24-1': 'Then one has [EQUATION]', '1608.00709-1-24-2': 'If moreover [MATH] is rational and [MATH] is algebraically closed, then the first of these bounds becomes an equality.', '1608.00709-1-25-0': 'It is known (see [CITATION]) that if [MATH] is a rationally connected threefold over a field of characteristic [MATH], then there is a constant [MATH] such that for any prime [MATH] any finite [MATH]-group [MATH] is abelian.', '1608.00709-1-25-1': 'An immediate consequence of Theorem [REF] is an explicit bound for the latter constant [MATH].', '1608.00709-1-26-0': 'Let [MATH] be a rationally connected threefold over a field [MATH] of characteristic [MATH], and let [MATH] be a prime.', '1608.00709-1-26-1': 'Let [MATH] be a finite [MATH]-group.', '1608.00709-1-26-2': 'Then [MATH] is abelian.', '1608.00709-1-27-0': 'We believe that one can significantly improve the bound given by Corollary [REF].', '1608.00709-1-28-0': 'J.-P.Serre showed (see the remark made after Theorem 5.3 in [CITATION]) that any finite subgroup [MATH] of the Cremona group [MATH] over a field [MATH] of characteristic [MATH] has a normal abelian subgroup [MATH] such that the index [MATH] divides the number [MATH].', '1608.00709-1-28-1': 'The result of Theorem [REF] is not that precise: we cannot say much about the primes that divide the index [MATH] in our case.', '1608.00709-1-28-2': 'This is explained by the fact that to obtain the bound we have to deal with terminal singularities on threefolds as compared to smooth surfaces.', '1608.00709-1-28-3': 'See Remark [REF] for our expectations on possible improvements of the bounds given by Proposition [REF] and Theorem [REF], and Remark [REF] for a further disclaimer in higher dimensions.', '1608.00709-1-29-0': 'The plan of the paper is as follows.', '1608.00709-1-29-1': 'In [REF] we compute weak Jordan constants for some linear groups.', '1608.00709-1-29-2': 'In [REF] we compute certain relevant constants for rational surfaces, and in particular prove Proposition [REF].', '1608.00709-1-29-3': 'In [REF] we study groups of automorphisms of three-dimensional terminal singularities and estimate their weak Jordan constants; then we use these estimates to bound weak Jordan constants for groups of automorphisms of non-Gorenstein terminal Fano threefolds.', '1608.00709-1-29-4': 'In [REF] we estimate weak Jordan constants for groups acting on three-dimensional [MATH]-Mori fiber spaces.', '1608.00709-1-29-5': 'In [REF] and [REF] we bound weak Jordan constants for groups of automorphisms of Gorenstein terminal (and in particular smooth) Fano threefolds.', '1608.00709-1-29-6': 'Finally, in [REF] we summarize the above partial results and complete the proof of Theorem [REF], and also make concluding remarks.', '1608.00709-1-29-7': 'In Appendix [REF] we collect some information about automorphism groups of two particular classes of smooth Fano varieties: complete intersections of quadrics, and complete intersections in weighted projective spaces; these results are well known to experts, but we decided to include them because we do not know proper references.'}","{'1608.00709-2-0-0': 'We give explicit bounds for Jordan constants of groups of birational automorphisms of rationally connected threefolds over fields of zero characteristic, in particular, for Cremona groups of ranks [MATH] and [MATH].', '1608.00709-2-1-0': '2010 MATH.', '1608.00709-2-1-1': 'SUBJ.', '1608.00709-2-1-2': 'CLASS.', '1608.00709-2-1-3': '14E07, 14J30', '1608.00709-2-2-0': 'KEY WORDS.', '1608.00709-2-2-1': 'Birational automorphisms, Cremona group, Jordan property, Jordan constant.', '1608.00709-2-3-0': '# Introduction', '1608.00709-2-4-0': '## Jordan property', '1608.00709-2-5-0': 'The Cremona group of rank [MATH] is the group [MATH] of birational transformations of the projective space [MATH] over a field [MATH].', '1608.00709-2-5-1': 'It has been actively studied from various points of view for many years (see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and references therein).', '1608.00709-2-5-2': 'One of the approaches to this huge group is to try to understand its finite subgroups.', '1608.00709-2-5-3': 'It appeared that it is possible to obtain a complete classification of finite subgroups of [MATH] over an algebraically closed field [MATH] of characteristic [MATH] (see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION]), and to obtain partial classification results for [MATH] (see [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION]).', '1608.00709-2-5-4': 'Some results are also known for non algebraically closed fields, see e.g. [CITATION], [CITATION], and [CITATION].', '1608.00709-2-5-5': 'In general, it is partially known and partially expected that the collection of finite subgroups of a Cremona group shares certain features with the collection of finite subgroups of a group [MATH].', '1608.00709-2-6-0': '[C.Jordan, see e.g. [CITATION]]', '1608.00709-2-7-0': 'There is a constant [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-2-8-0': 'This leads to the following definition (cf. [CITATION]).', '1608.00709-2-9-0': 'A group [MATH] is called Jordan (alternatively, we say that [MATH] has Jordan property) if there is a constant [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-2-10-0': 'Theorem [REF] implies that all linear algebraic groups over an arbitrary field [MATH] of characteristic [MATH] are Jordan.', '1608.00709-2-10-1': 'Jordan property was also studied recently for groups of birational automorphisms of algebraic varieties.', '1608.00709-2-10-2': 'The starting point here was the following result of J.-P.Serre.', '1608.00709-2-11-0': '[J.-P.Serre [CITATION], [CITATION]] The Cremona group [MATH] over a field [MATH] of characteristic [MATH] is Jordan.', '1608.00709-2-12-0': 'Note that the assumption about characteristic is indispensable.', '1608.00709-2-12-1': 'Indeed, the group [MATH] contains [MATH], so that if the characteristic of the field [MATH] equals [MATH] and [MATH] is algebraically closed, then [MATH] contains a series of simple subgroups [MATH] of increasing order.', '1608.00709-2-13-0': 'It also appeared that there are surfaces with non-Jordan groups of birational selfmaps (see [CITATION]).', '1608.00709-2-13-1': 'V.Popov managed to give a complete classification of surfaces with Jordan groups of birational automorphisms.', '1608.00709-2-14-0': '[V.Popov [CITATION]] Let [MATH] be a surface over a field [MATH] of characteristic [MATH].', '1608.00709-2-14-1': 'Then the group [MATH] of birational automorphisms of [MATH] is Jordan if and only if [MATH] is not birational to [MATH], where [MATH] is an elliptic curve.', '1608.00709-2-15-0': 'In dimension [MATH] Jordan property is known for groups of birational automorphisms of rationally connected varieties (see e.g. [CITATION] for definition and basic background).', '1608.00709-2-16-0': '[see [CITATION]]', '1608.00709-2-17-0': 'Fix a field [MATH] of characteristic [MATH].', '1608.00709-2-17-1': 'Then there is a constant [MATH] such that for any rationally connected variety [MATH] of dimension [MATH] defined over [MATH] and any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-2-17-2': 'In particular, for any rationally connected threefold [MATH] the group [MATH] is Jordan.', '1608.00709-2-18-0': 'Actually, by [CITATION] the assertion of Theorem [REF] holds in arbitrary dimension modulo boundedness of terminal Fano threefolds (see e.g. [CITATION] or [CITATION]); the latter boundedness was recently proved in [CITATION].', '1608.00709-2-18-1': 'For other results (in particular, for birational automorphisms of non rationally connected varieties, and for automorphisms of varieties of different types) see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION].', '1608.00709-2-19-0': '## Jordan constants', '1608.00709-2-20-0': 'Given a Jordan group [MATH], one may get interested in the minimal value of the constant involved in Definition [REF], and in the values of other relevant constants.', '1608.00709-2-21-0': 'Let [MATH] be a Jordan group.', '1608.00709-2-21-1': 'The Jordan constant [MATH] of the group [MATH] is the minimal number [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-2-21-2': 'The weak Jordan constant [MATH] of the group [MATH] is the minimal number [MATH] such that for any finite subgroup [MATH] there exists a (not necessarily normal) abelian subgroup [MATH] of index at most [MATH].', '1608.00709-2-22-0': 'It is more traditional to study Jordan constants than weak Jordan constants of Jordan groups, although there is no big difference between them.', '1608.00709-2-22-1': 'Indeed, one has [MATH] for any Jordan group [MATH] for obvious reasons.', '1608.00709-2-22-2': 'Moreover, if [MATH] is a finite group and [MATH] is an abelian subgroup of [MATH], then by [CITATION] one can find a normal abelian subgroup [MATH] of [MATH] such that [EQUATION]', '1608.00709-2-22-3': 'Therefore, if [MATH] is a Jordan group, one always has [MATH].', '1608.00709-2-22-4': 'On the other hand, the advantage of the weak Jordan constant is that it allows easy estimates using subgroups of the initial group.', '1608.00709-2-22-5': 'Namely, if [MATH] is a subgroup of finite index in a group [MATH], and [MATH] is Jordan, then [MATH] is Jordan with [EQUATION]', '1608.00709-2-22-6': 'Also, if [MATH] and [MATH] are Jordan groups, the group [MATH] is Jordan with [EQUATION]', '1608.00709-2-22-7': 'In particular, if [MATH] is a subgroup of [MATH], where [MATH] is a Jordan group and [MATH] is an abelian group, then [MATH] is Jordan with [MATH].', '1608.00709-2-23-0': 'Jordan constants are known for example for the groups [MATH] (see [CITATION]).', '1608.00709-2-23-1': 'In [CITATION] J.-P.Serre gave an explicit bound for the Jordan constant of the Cremona group [MATH] (see Remark [REF] below).', '1608.00709-2-23-2': 'Our first result also concerns the group [MATH].', '1608.00709-2-24-0': 'Suppose that the field [MATH] has characteristic [MATH].', '1608.00709-2-24-1': 'Then one has [EQUATION]', '1608.00709-2-24-2': 'The first of these bounds becomes an equality if [MATH] is algebraically closed.', '1608.00709-2-25-0': 'The main goal of this paper is to present a bound for Jordan constants of the groups of birational automorphisms of rationally connected threefolds, in particular, for the group [MATH].', '1608.00709-2-26-0': 'Let [MATH] be a rationally connected threefold over a field [MATH] of characteristic [MATH].', '1608.00709-2-26-1': 'Then one has [EQUATION]', '1608.00709-2-26-2': 'If moreover [MATH] is rational and [MATH] is algebraically closed, then the first of these bounds becomes an equality.', '1608.00709-2-27-0': 'It is known (see [CITATION]) that if [MATH] is a rationally connected threefold over a field of characteristic [MATH], then there is a constant [MATH] such that for any prime [MATH] any finite [MATH]-group [MATH] is abelian.', '1608.00709-2-27-1': 'An immediate consequence of Theorem [REF] is an explicit bound for the latter constant [MATH].', '1608.00709-2-28-0': 'Let [MATH] be a rationally connected threefold over a field [MATH] of characteristic [MATH], and let [MATH] be a prime.', '1608.00709-2-28-1': 'Let [MATH] be a finite [MATH]-group.', '1608.00709-2-28-2': 'Then [MATH] is abelian.', '1608.00709-2-29-0': 'We believe that one can significantly improve the bound given by Corollary [REF].', '1608.00709-2-30-0': 'J.-P.Serre showed (see the remark made after Theorem 5.3 in [CITATION]) that any finite subgroup [MATH] of the Cremona group [MATH] over a field [MATH] of characteristic [MATH] has a normal abelian subgroup [MATH] such that the index [MATH] divides the number [MATH].', '1608.00709-2-30-1': 'The result of Theorem [REF] is not that precise: we cannot say much about the primes that divide the index [MATH] in our case.', '1608.00709-2-30-2': 'This is explained by the fact that to obtain the bound we have to deal with terminal singularities on threefolds as compared to smooth surfaces.', '1608.00709-2-30-3': 'See Remark [REF] for our expectations on possible improvements of the bounds given by Proposition [REF] and Theorem [REF], and Remark [REF] for a further disclaimer in higher dimensions.', '1608.00709-2-31-0': 'The plan of the paper is as follows.', '1608.00709-2-31-1': 'In [REF] we compute weak Jordan constants for some linear groups.', '1608.00709-2-31-2': 'In [REF] we compute certain relevant constants for rational surfaces, and in particular prove Proposition [REF].', '1608.00709-2-31-3': 'In [REF] we study groups of automorphisms of three-dimensional terminal singularities and estimate their weak Jordan constants; then we use these estimates to bound weak Jordan constants for groups of automorphisms of non-Gorenstein terminal Fano threefolds.', '1608.00709-2-31-4': 'In [REF] we estimate weak Jordan constants for groups acting on three-dimensional [MATH]-Mori fiber spaces.', '1608.00709-2-31-5': 'In [REF] and [REF] we bound weak Jordan constants for groups of automorphisms of Gorenstein terminal (and in particular smooth) Fano threefolds.', '1608.00709-2-31-6': 'Finally, in [REF] we summarize the above partial results and complete the proof of Theorem [REF], and also make concluding remarks.', '1608.00709-2-31-7': 'In Appendix [REF] we collect some information about automorphism groups of two particular classes of smooth Fano varieties: complete intersections of quadrics, and complete intersections in weighted projective spaces; these results are well known to experts, but we decided to include them because we do not know proper references.'}","[['1608.00709-1-18-0', '1608.00709-2-20-0'], ['1608.00709-1-29-0', '1608.00709-2-31-0'], ['1608.00709-1-29-1', '1608.00709-2-31-1'], ['1608.00709-1-29-2', '1608.00709-2-31-2'], ['1608.00709-1-29-3', '1608.00709-2-31-3'], ['1608.00709-1-29-4', '1608.00709-2-31-4'], ['1608.00709-1-29-5', '1608.00709-2-31-5'], ['1608.00709-1-29-6', '1608.00709-2-31-6'], ['1608.00709-1-29-7', '1608.00709-2-31-7'], ['1608.00709-1-23-0', '1608.00709-2-25-0'], ['1608.00709-1-3-0', '1608.00709-2-5-0'], ['1608.00709-1-3-1', '1608.00709-2-5-1'], ['1608.00709-1-3-2', '1608.00709-2-5-2'], ['1608.00709-1-3-3', '1608.00709-2-5-3'], ['1608.00709-1-3-4', '1608.00709-2-5-4'], ['1608.00709-1-3-5', '1608.00709-2-5-5'], ['1608.00709-1-0-0', '1608.00709-2-0-0'], ['1608.00709-1-26-0', '1608.00709-2-28-0'], ['1608.00709-1-26-1', '1608.00709-2-28-1'], ['1608.00709-1-26-2', '1608.00709-2-28-2'], ['1608.00709-1-28-0', '1608.00709-2-30-0'], ['1608.00709-1-28-1', '1608.00709-2-30-1'], ['1608.00709-1-28-2', '1608.00709-2-30-2'], ['1608.00709-1-28-3', '1608.00709-2-30-3'], ['1608.00709-1-8-0', '1608.00709-2-10-0'], ['1608.00709-1-8-1', '1608.00709-2-10-1'], ['1608.00709-1-8-2', '1608.00709-2-10-2'], ['1608.00709-1-7-0', '1608.00709-2-9-0'], ['1608.00709-1-20-0', '1608.00709-2-22-0'], ['1608.00709-1-20-1', '1608.00709-2-22-1'], ['1608.00709-1-20-2', '1608.00709-2-22-2'], ['1608.00709-1-20-3', '1608.00709-2-22-3'], ['1608.00709-1-20-4', '1608.00709-2-22-4'], ['1608.00709-1-20-5', '1608.00709-2-22-5'], ['1608.00709-1-20-6', '1608.00709-2-22-6'], ['1608.00709-1-20-7', '1608.00709-2-22-7'], ['1608.00709-1-10-0', '1608.00709-2-12-0'], ['1608.00709-1-10-1', '1608.00709-2-12-1'], ['1608.00709-1-27-0', '1608.00709-2-29-0'], ['1608.00709-1-25-0', '1608.00709-2-27-0'], ['1608.00709-1-25-1', '1608.00709-2-27-1'], ['1608.00709-1-12-0', '1608.00709-2-14-0'], ['1608.00709-1-12-1', '1608.00709-2-14-1'], ['1608.00709-1-24-0', '1608.00709-2-26-0'], ['1608.00709-1-24-1', '1608.00709-2-26-1'], ['1608.00709-1-24-2', '1608.00709-2-26-2'], ['1608.00709-1-16-1', '1608.00709-2-18-1'], ['1608.00709-1-19-0', '1608.00709-2-21-0'], ['1608.00709-1-19-1', '1608.00709-2-21-1'], ['1608.00709-1-19-2', '1608.00709-2-21-2'], ['1608.00709-1-13-0', '1608.00709-2-15-0'], ['1608.00709-1-21-0', '1608.00709-2-23-0'], ['1608.00709-1-21-1', '1608.00709-2-23-1'], ['1608.00709-1-21-2', '1608.00709-2-23-2'], ['1608.00709-1-15-0', '1608.00709-2-17-0'], ['1608.00709-1-15-1', '1608.00709-2-17-1'], ['1608.00709-1-15-2', '1608.00709-2-17-2'], ['1608.00709-1-22-0', '1608.00709-2-24-0'], ['1608.00709-1-22-1', '1608.00709-2-24-1'], ['1608.00709-1-22-2', '1608.00709-2-24-2'], ['1608.00709-1-11-0', '1608.00709-2-13-0'], ['1608.00709-1-11-1', '1608.00709-2-13-1'], ['1608.00709-1-5-0', '1608.00709-2-7-0'], ['1608.00709-2-30-0', '1608.00709-3-30-0'], ['1608.00709-2-30-1', '1608.00709-3-30-1'], ['1608.00709-2-30-2', '1608.00709-3-30-2'], ['1608.00709-2-30-3', '1608.00709-3-30-3'], ['1608.00709-2-18-0', '1608.00709-3-18-0'], ['1608.00709-2-18-1', '1608.00709-3-18-1'], ['1608.00709-2-21-0', '1608.00709-3-21-0'], ['1608.00709-2-21-1', '1608.00709-3-21-1'], ['1608.00709-2-21-2', '1608.00709-3-21-2'], ['1608.00709-2-17-0', '1608.00709-3-17-0'], ['1608.00709-2-17-1', '1608.00709-3-17-1'], ['1608.00709-2-17-2', '1608.00709-3-17-2'], ['1608.00709-2-27-0', '1608.00709-3-27-0'], ['1608.00709-2-27-1', '1608.00709-3-27-1'], ['1608.00709-2-13-0', '1608.00709-3-13-0'], ['1608.00709-2-13-1', '1608.00709-3-13-1'], ['1608.00709-2-7-0', '1608.00709-3-7-0'], ['1608.00709-2-26-0', '1608.00709-3-26-0'], ['1608.00709-2-26-1', '1608.00709-3-26-1'], ['1608.00709-2-26-2', '1608.00709-3-26-2'], ['1608.00709-2-12-0', '1608.00709-3-12-0'], ['1608.00709-2-12-1', '1608.00709-3-12-1'], ['1608.00709-2-0-0', '1608.00709-3-0-0'], ['1608.00709-2-5-0', '1608.00709-3-5-0'], ['1608.00709-2-5-1', '1608.00709-3-5-1'], ['1608.00709-2-5-2', '1608.00709-3-5-2'], ['1608.00709-2-5-3', '1608.00709-3-5-3'], ['1608.00709-2-5-4', '1608.00709-3-5-4'], ['1608.00709-2-5-5', '1608.00709-3-5-5'], ['1608.00709-2-10-0', '1608.00709-3-10-0'], ['1608.00709-2-10-1', '1608.00709-3-10-1'], ['1608.00709-2-10-2', '1608.00709-3-10-2'], ['1608.00709-2-9-0', 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'1608.00709-4-20-0'], ['1608.00709-3-31-0', '1608.00709-4-31-0'], ['1608.00709-3-31-1', '1608.00709-4-31-1'], ['1608.00709-3-31-2', '1608.00709-4-31-2'], ['1608.00709-3-31-3', '1608.00709-4-31-3'], ['1608.00709-3-31-4', '1608.00709-4-31-4'], ['1608.00709-3-31-5', '1608.00709-4-31-5'], ['1608.00709-3-31-6', '1608.00709-4-31-6'], ['1608.00709-3-31-7', '1608.00709-4-31-7'], ['1608.00709-3-17-0', '1608.00709-4-17-0'], ['1608.00709-3-17-1', '1608.00709-4-17-1'], ['1608.00709-3-17-2', '1608.00709-4-17-2'], ['1608.00709-3-0-0', '1608.00709-4-0-0'], ['1608.00709-3-25-0', '1608.00709-4-25-0'], ['1608.00709-3-22-0', '1608.00709-4-22-0'], ['1608.00709-3-22-1', '1608.00709-4-22-1'], ['1608.00709-3-22-3', '1608.00709-4-22-3'], ['1608.00709-3-22-4', '1608.00709-4-22-4'], ['1608.00709-3-22-5', '1608.00709-4-22-5'], ['1608.00709-3-22-6', '1608.00709-4-22-6'], ['1608.00709-3-22-7', '1608.00709-4-22-7'], ['1608.00709-3-30-0', '1608.00709-4-30-0'], ['1608.00709-3-30-1', '1608.00709-4-30-1'], ['1608.00709-3-30-2', '1608.00709-4-30-2'], ['1608.00709-3-30-3', '1608.00709-4-30-3'], ['1608.00709-3-13-0', '1608.00709-4-13-0'], ['1608.00709-3-13-1', '1608.00709-4-13-1'], ['1608.00709-3-15-0', '1608.00709-4-15-0'], ['1608.00709-3-10-0', '1608.00709-4-10-0'], ['1608.00709-3-10-1', '1608.00709-4-10-1'], ['1608.00709-3-10-2', '1608.00709-4-10-2'], ['1608.00709-3-14-0', '1608.00709-4-14-0'], ['1608.00709-3-14-1', '1608.00709-4-14-1'], ['1608.00709-3-23-0', '1608.00709-4-23-0'], ['1608.00709-3-23-1', '1608.00709-4-23-1'], ['1608.00709-3-23-2', '1608.00709-4-23-2']]","[['1608.00709-1-16-0', '1608.00709-2-18-0'], ['1608.00709-3-18-0', '1608.00709-4-18-0'], ['1608.00709-3-18-1', '1608.00709-4-18-1'], ['1608.00709-3-22-2', '1608.00709-4-22-2']]",[],[],[],"['1608.00709-1-4-0', '1608.00709-1-6-0', '1608.00709-1-9-0', '1608.00709-1-14-0', '1608.00709-2-1-0', '1608.00709-2-1-1', '1608.00709-2-1-2', '1608.00709-2-1-3', '1608.00709-2-2-0', '1608.00709-2-2-1', '1608.00709-2-6-0', '1608.00709-2-8-0', '1608.00709-2-11-0', '1608.00709-2-16-0', '1608.00709-3-1-0', '1608.00709-3-1-1', '1608.00709-3-1-2', '1608.00709-3-1-3', '1608.00709-3-2-0', '1608.00709-3-2-1', '1608.00709-3-6-0', '1608.00709-3-8-0', '1608.00709-3-11-0', '1608.00709-3-16-0', '1608.00709-4-1-0', '1608.00709-4-1-1', '1608.00709-4-1-2', '1608.00709-4-1-3', '1608.00709-4-2-0', '1608.00709-4-2-1', '1608.00709-4-6-0', '1608.00709-4-8-0', '1608.00709-4-11-0', '1608.00709-4-16-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1608.00709,"{'1608.00709-3-0-0': 'We give explicit bounds for Jordan constants of groups of birational automorphisms of rationally connected threefolds over fields of zero characteristic, in particular, for Cremona groups of ranks [MATH] and [MATH].', '1608.00709-3-1-0': '2010 MATH.', '1608.00709-3-1-1': 'SUBJ.', '1608.00709-3-1-2': 'CLASS.', '1608.00709-3-1-3': '14E07, 14J30', '1608.00709-3-2-0': 'KEY WORDS.', '1608.00709-3-2-1': 'Birational automorphisms, Cremona group, Jordan property, Jordan constant.', '1608.00709-3-3-0': '# Introduction', '1608.00709-3-4-0': '## Jordan property', '1608.00709-3-5-0': 'The Cremona group of rank [MATH] is the group [MATH] of birational transformations of the projective space [MATH] over a field [MATH].', '1608.00709-3-5-1': 'It has been actively studied from various points of view for many years (see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and references therein).', '1608.00709-3-5-2': 'One of the approaches to this huge group is to try to understand its finite subgroups.', '1608.00709-3-5-3': 'It appeared that it is possible to obtain a complete classification of finite subgroups of [MATH] over an algebraically closed field [MATH] of characteristic [MATH] (see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION]), and to obtain partial classification results for [MATH] (see [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION]).', '1608.00709-3-5-4': 'Some results are also known for non algebraically closed fields, see e.g. [CITATION], [CITATION], and [CITATION].', '1608.00709-3-5-5': 'In general, it is partially known and partially expected that the collection of finite subgroups of a Cremona group shares certain features with the collection of finite subgroups of a group [MATH].', '1608.00709-3-6-0': '[C.Jordan, see e.g. [CITATION]]', '1608.00709-3-7-0': 'There is a constant [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-3-8-0': 'This leads to the following definition (cf. [CITATION]).', '1608.00709-3-9-0': 'A group [MATH] is called Jordan (alternatively, we say that [MATH] has Jordan property) if there is a constant [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-3-10-0': 'Theorem [REF] implies that all linear algebraic groups over an arbitrary field [MATH] of characteristic [MATH] are Jordan.', '1608.00709-3-10-1': 'Jordan property was also studied recently for groups of birational automorphisms of algebraic varieties.', '1608.00709-3-10-2': 'The starting point here was the following result of J.-P.Serre.', '1608.00709-3-11-0': '[J.-P.Serre [CITATION], [CITATION]] The Cremona group [MATH] over a field [MATH] of characteristic [MATH] is Jordan.', '1608.00709-3-12-0': 'Note that the assumption about characteristic is indispensable.', '1608.00709-3-12-1': 'Indeed, the group [MATH] contains [MATH], so that if the characteristic of the field [MATH] equals [MATH] and [MATH] is algebraically closed, then [MATH] contains a series of simple subgroups [MATH] of increasing order.', '1608.00709-3-13-0': 'It also appeared that there are surfaces with non-Jordan groups of birational selfmaps (see [CITATION]).', '1608.00709-3-13-1': 'V.Popov managed to give a complete classification of surfaces with Jordan groups of birational automorphisms.', '1608.00709-3-14-0': '[V.Popov [CITATION]] Let [MATH] be a surface over a field [MATH] of characteristic [MATH].', '1608.00709-3-14-1': 'Then the group [MATH] of birational automorphisms of [MATH] is Jordan if and only if [MATH] is not birational to [MATH], where [MATH] is an elliptic curve.', '1608.00709-3-15-0': 'In dimension [MATH] Jordan property is known for groups of birational automorphisms of rationally connected varieties (see e.g. [CITATION] for definition and basic background).', '1608.00709-3-16-0': '[see [CITATION]]', '1608.00709-3-17-0': 'Fix a field [MATH] of characteristic [MATH].', '1608.00709-3-17-1': 'Then there is a constant [MATH] such that for any rationally connected variety [MATH] of dimension [MATH] defined over [MATH] and any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-3-17-2': 'In particular, for any rationally connected threefold [MATH] the group [MATH] is Jordan.', '1608.00709-3-18-0': 'Actually, by [CITATION] the assertion of Theorem [REF] holds in arbitrary dimension modulo boundedness of terminal Fano threefolds (see e.g. [CITATION] or [CITATION]); the latter boundedness was recently proved in [CITATION].', '1608.00709-3-18-1': 'For other results (in particular, for birational automorphisms of non rationally connected varieties, and for automorphisms of varieties of different types) see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION].', '1608.00709-3-19-0': '## Jordan constants', '1608.00709-3-20-0': 'Given a Jordan group [MATH], one may get interested in the minimal value of the constant involved in Definition [REF], and in the values of other relevant constants.', '1608.00709-3-21-0': 'Let [MATH] be a Jordan group.', '1608.00709-3-21-1': 'The Jordan constant [MATH] of the group [MATH] is the minimal number [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-3-21-2': 'The weak Jordan constant [MATH] of the group [MATH] is the minimal number [MATH] such that for any finite subgroup [MATH] there exists a (not necessarily normal) abelian subgroup [MATH] of index at most [MATH].', '1608.00709-3-22-0': 'It is more traditional to study Jordan constants than weak Jordan constants of Jordan groups, although there is no big difference between them.', '1608.00709-3-22-1': 'Indeed, one has [MATH] for any Jordan group [MATH] for obvious reasons.', '1608.00709-3-22-2': 'Moreover, if [MATH] is a finite group and [MATH] is an abelian subgroup of [MATH], then by [CITATION] one can find a normal abelian subgroup [MATH] of [MATH] such that [EQUATION]', '1608.00709-3-22-3': 'Therefore, if [MATH] is a Jordan group, one always has [MATH].', '1608.00709-3-22-4': 'On the other hand, the advantage of the weak Jordan constant is that it allows easy estimates using subgroups of the initial group.', '1608.00709-3-22-5': 'Namely, if [MATH] is a subgroup of finite index in a group [MATH], and [MATH] is Jordan, then [MATH] is Jordan with [EQUATION]', '1608.00709-3-22-6': 'Also, if [MATH] and [MATH] are Jordan groups, the group [MATH] is Jordan with [EQUATION]', '1608.00709-3-22-7': 'In particular, if [MATH] is a subgroup of [MATH], where [MATH] is a Jordan group and [MATH] is an abelian group, then [MATH] is Jordan with [MATH].', '1608.00709-3-23-0': 'Jordan constants are known for example for the groups [MATH] (see [CITATION]).', '1608.00709-3-23-1': 'In [CITATION] J.-P.Serre gave an explicit bound for the Jordan constant of the Cremona group [MATH] (see Remark [REF] below).', '1608.00709-3-23-2': 'Our first result also concerns the group [MATH].', '1608.00709-3-24-0': 'Suppose that the field [MATH] has characteristic [MATH].', '1608.00709-3-24-1': 'Then one has [EQUATION]', '1608.00709-3-24-2': 'The first of these bounds becomes an equality if [MATH] is algebraically closed.', '1608.00709-3-25-0': 'The main goal of this paper is to present a bound for Jordan constants of the groups of birational automorphisms of rationally connected threefolds, in particular, for the group [MATH].', '1608.00709-3-26-0': 'Let [MATH] be a rationally connected threefold over a field [MATH] of characteristic [MATH].', '1608.00709-3-26-1': 'Then one has [EQUATION]', '1608.00709-3-26-2': 'If moreover [MATH] is rational and [MATH] is algebraically closed, then the first of these bounds becomes an equality.', '1608.00709-3-27-0': 'It is known (see [CITATION]) that if [MATH] is a rationally connected threefold over a field of characteristic [MATH], then there is a constant [MATH] such that for any prime [MATH] any finite [MATH]-group [MATH] is abelian.', '1608.00709-3-27-1': 'An immediate consequence of Theorem [REF] is an explicit bound for the latter constant [MATH].', '1608.00709-3-28-0': 'Let [MATH] be a rationally connected threefold over a field [MATH] of characteristic [MATH], and let [MATH] be a prime.', '1608.00709-3-28-1': 'Let [MATH] be a finite [MATH]-group.', '1608.00709-3-28-2': 'Then [MATH] is abelian.', '1608.00709-3-29-0': 'We believe that one can significantly improve the bound given by Corollary [REF].', '1608.00709-3-30-0': 'J.-P.Serre showed (see the remark made after Theorem 5.3 in [CITATION]) that any finite subgroup [MATH] of the Cremona group [MATH] over a field [MATH] of characteristic [MATH] has a normal abelian subgroup [MATH] such that the index [MATH] divides the number [MATH].', '1608.00709-3-30-1': 'The result of Theorem [REF] is not that precise: we cannot say much about the primes that divide the index [MATH] in our case.', '1608.00709-3-30-2': 'This is explained by the fact that to obtain the bound we have to deal with terminal singularities on threefolds as compared to smooth surfaces.', '1608.00709-3-30-3': 'See Remark [REF] for our expectations on possible improvements of the bounds given by Proposition [REF] and Theorem [REF], and Remark [REF] for a further disclaimer in higher dimensions.', '1608.00709-3-31-0': 'The plan of the paper is as follows.', '1608.00709-3-31-1': 'In [REF] we compute weak Jordan constants for some linear groups.', '1608.00709-3-31-2': 'In [REF] we compute certain relevant constants for rational surfaces, and in particular prove Proposition [REF].', '1608.00709-3-31-3': 'In [REF] we study groups of automorphisms of three-dimensional terminal singularities and estimate their weak Jordan constants; then we use these estimates to bound weak Jordan constants for groups of automorphisms of non-Gorenstein terminal Fano threefolds.', '1608.00709-3-31-4': 'In [REF] we estimate weak Jordan constants for groups acting on three-dimensional [MATH]-Mori fiber spaces.', '1608.00709-3-31-5': 'In [REF] and [REF] we bound weak Jordan constants for groups of automorphisms of Gorenstein terminal (and in particular smooth) Fano threefolds.', '1608.00709-3-31-6': 'Finally, in [REF] we summarize the above partial results and complete the proof of Theorem [REF], and also make concluding remarks.', '1608.00709-3-31-7': 'In Appendix [REF] we collect some information about automorphism groups of two particular classes of smooth Fano varieties: complete intersections of quadrics, and complete intersections in weighted projective spaces; these results are well known to experts, but we decided to include them because we do not know proper references.'}","{'1608.00709-4-0-0': 'We give explicit bounds for Jordan constants of groups of birational automorphisms of rationally connected threefolds over fields of zero characteristic, in particular, for Cremona groups of ranks [MATH] and [MATH].', '1608.00709-4-1-0': '2010 MATH.', '1608.00709-4-1-1': 'SUBJ.', '1608.00709-4-1-2': 'CLASS.', '1608.00709-4-1-3': '14E07, 14J30', '1608.00709-4-2-0': 'KEY WORDS.', '1608.00709-4-2-1': 'Birational automorphisms, Cremona group, Jordan property, Jordan constant.', '1608.00709-4-3-0': '# Introduction', '1608.00709-4-4-0': '## Jordan property', '1608.00709-4-5-0': 'The Cremona group of rank [MATH] is the group [MATH] of birational transformations of the projective space [MATH] over a field [MATH].', '1608.00709-4-5-1': 'It has been actively studied from various points of view for many years (see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and references therein).', '1608.00709-4-5-2': 'One of the approaches to this huge group is to try to understand its finite subgroups.', '1608.00709-4-5-3': 'It appeared that it is possible to obtain a complete classification of finite subgroups of [MATH] over an algebraically closed field [MATH] of characteristic [MATH] (see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION]), and to obtain partial classification results for [MATH] (see [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION]).', '1608.00709-4-5-4': 'Some results are also known for non algebraically closed fields, see e.g. [CITATION], [CITATION], and [CITATION].', '1608.00709-4-5-5': 'In general, it is partially known and partially expected that the collection of finite subgroups of a Cremona group shares certain features with the collection of finite subgroups of a group [MATH].', '1608.00709-4-6-0': '[C.Jordan, see e.g. [CITATION]]', '1608.00709-4-7-0': 'There is a constant [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-4-8-0': 'This leads to the following definition (cf. [CITATION]).', '1608.00709-4-9-0': 'A group [MATH] is called Jordan (alternatively, we say that [MATH] has Jordan property) if there is a constant [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-4-10-0': 'Theorem [REF] implies that all linear algebraic groups over an arbitrary field [MATH] of characteristic [MATH] are Jordan.', '1608.00709-4-10-1': 'Jordan property was also studied recently for groups of birational automorphisms of algebraic varieties.', '1608.00709-4-10-2': 'The starting point here was the following result of J.-P.Serre.', '1608.00709-4-11-0': '[J.-P.Serre [CITATION], [CITATION]] The Cremona group [MATH] over a field [MATH] of characteristic [MATH] is Jordan.', '1608.00709-4-12-0': 'Note that the assumption about characteristic is indispensable.', '1608.00709-4-12-1': 'Indeed, the group [MATH] contains [MATH], so that if the characteristic of the field [MATH] equals [MATH] and [MATH] is algebraically closed, then [MATH] contains a series of simple subgroups [MATH] of increasing order.', '1608.00709-4-13-0': 'It also appeared that there are surfaces with non-Jordan groups of birational selfmaps (see [CITATION]).', '1608.00709-4-13-1': 'V.Popov managed to give a complete classification of surfaces with Jordan groups of birational automorphisms.', '1608.00709-4-14-0': '[V.Popov [CITATION]] Let [MATH] be a surface over a field [MATH] of characteristic [MATH].', '1608.00709-4-14-1': 'Then the group [MATH] of birational automorphisms of [MATH] is Jordan if and only if [MATH] is not birational to [MATH], where [MATH] is an elliptic curve.', '1608.00709-4-15-0': 'In dimension [MATH] Jordan property is known for groups of birational automorphisms of rationally connected varieties (see e.g. [CITATION] for definition and basic background).', '1608.00709-4-16-0': '[see [CITATION]]', '1608.00709-4-17-0': 'Fix a field [MATH] of characteristic [MATH].', '1608.00709-4-17-1': 'Then there is a constant [MATH] such that for any rationally connected variety [MATH] of dimension [MATH] defined over [MATH] and any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-4-17-2': 'In particular, for any rationally connected threefold [MATH] the group [MATH] is Jordan.', '1608.00709-4-18-0': 'Actually, by [CITATION] the assertion of Theorem [REF] holds in arbitrary dimension modulo boundedness of terminal Fano varieties (see e.g. [CITATION] or [CITATION]); the latter boundedness was recently proved in [CITATION].', '1608.00709-4-18-1': 'For other results (in particular, for birational automorphisms of non rationally connected varieties, and for automorphisms of varieties of different types) see [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], [CITATION], and [CITATION].', '1608.00709-4-19-0': '## Jordan constants', '1608.00709-4-20-0': 'Given a Jordan group [MATH], one may get interested in the minimal value of the constant involved in Definition [REF], and in the values of other relevant constants.', '1608.00709-4-21-0': 'Let [MATH] be a Jordan group.', '1608.00709-4-21-1': 'The Jordan constant [MATH] of the group [MATH] is the minimal number [MATH] such that for any finite subgroup [MATH] there exists a normal abelian subgroup [MATH] of index at most [MATH].', '1608.00709-4-21-2': 'The weak Jordan constant [MATH] of the group [MATH] is the minimal number [MATH] such that for any finite subgroup [MATH] there exists a (not necessarily normal) abelian subgroup [MATH] of index at most [MATH].', '1608.00709-4-22-0': 'It is more traditional to study Jordan constants than weak Jordan constants of Jordan groups, although there is no big difference between them.', '1608.00709-4-22-1': 'Indeed, one has [MATH] for any Jordan group [MATH] for obvious reasons.', '1608.00709-4-22-2': 'Moreover, if [MATH] is a finite group and [MATH] is an abelian subgroup of [MATH], then by [CITATION] (see also [CITATION]) one can find a normal abelian subgroup [MATH] of [MATH] such that [EQUATION]', '1608.00709-4-22-3': 'Therefore, if [MATH] is a Jordan group, one always has [MATH].', '1608.00709-4-22-4': 'On the other hand, the advantage of the weak Jordan constant is that it allows easy estimates using subgroups of the initial group.', '1608.00709-4-22-5': 'Namely, if [MATH] is a subgroup of finite index in a group [MATH], and [MATH] is Jordan, then [MATH] is Jordan with [EQUATION]', '1608.00709-4-22-6': 'Also, if [MATH] and [MATH] are Jordan groups, the group [MATH] is Jordan with [EQUATION]', '1608.00709-4-22-7': 'In particular, if [MATH] is a subgroup of [MATH], where [MATH] is a Jordan group and [MATH] is an abelian group, then [MATH] is Jordan with [MATH].', '1608.00709-4-23-0': 'Jordan constants are known for example for the groups [MATH] (see [CITATION]).', '1608.00709-4-23-1': 'In [CITATION] J.-P.Serre gave an explicit bound for the Jordan constant of the Cremona group [MATH] (see Remark [REF] below).', '1608.00709-4-23-2': 'Our first result also concerns the group [MATH].', '1608.00709-4-24-0': 'Suppose that the field [MATH] has characteristic [MATH].', '1608.00709-4-24-1': 'Then one has [EQUATION]', '1608.00709-4-24-2': 'The first of these bounds becomes an equality if [MATH] is algebraically closed.', '1608.00709-4-25-0': 'The main goal of this paper is to present a bound for Jordan constants of the groups of birational automorphisms of rationally connected threefolds, in particular, for the group [MATH].', '1608.00709-4-26-0': 'Let [MATH] be a rationally connected threefold over a field [MATH] of characteristic [MATH].', '1608.00709-4-26-1': 'Then one has [EQUATION]', '1608.00709-4-26-2': 'If moreover [MATH] is rational and [MATH] is algebraically closed, then the first of these bounds becomes an equality.', '1608.00709-4-27-0': 'It is known (see [CITATION]) that if [MATH] is a rationally connected threefold over a field of characteristic [MATH], then there is a constant [MATH] such that for any prime [MATH] any finite [MATH]-group [MATH] is abelian.', '1608.00709-4-27-1': 'An immediate consequence of Theorem [REF] is an explicit bound for the latter constant [MATH].', '1608.00709-4-28-0': 'Let [MATH] be a rationally connected threefold over a field [MATH] of characteristic [MATH], and let [MATH] be a prime.', '1608.00709-4-28-1': 'Let [MATH] be a finite [MATH]-group.', '1608.00709-4-28-2': 'Then [MATH] is abelian.', '1608.00709-4-29-0': 'We believe that one can significantly improve the bound given by Corollary [REF].', '1608.00709-4-30-0': 'J.-P.Serre showed (see the remark made after Theorem 5.3 in [CITATION]) that any finite subgroup [MATH] of the Cremona group [MATH] over a field [MATH] of characteristic [MATH] has a normal abelian subgroup [MATH] such that the index [MATH] divides the number [MATH].', '1608.00709-4-30-1': 'The result of Theorem [REF] is not that precise: we cannot say much about the primes that divide the index [MATH] in our case.', '1608.00709-4-30-2': 'This is explained by the fact that to obtain the bound we have to deal with terminal singularities on threefolds as compared to smooth surfaces.', '1608.00709-4-30-3': 'See Remark [REF] for our expectations on possible improvements of the bounds given by Proposition [REF] and Theorem [REF], and Remark [REF] for a further disclaimer in higher dimensions.', '1608.00709-4-31-0': 'The plan of the paper is as follows.', '1608.00709-4-31-1': 'In [REF] we compute weak Jordan constants for some linear groups.', '1608.00709-4-31-2': 'In [REF] we compute certain relevant constants for rational surfaces, and in particular prove Proposition [REF].', '1608.00709-4-31-3': 'In [REF] we study groups of automorphisms of three-dimensional terminal singularities and estimate their weak Jordan constants; then we use these estimates to bound weak Jordan constants for groups of automorphisms of non-Gorenstein terminal Fano threefolds.', '1608.00709-4-31-4': 'In [REF] we estimate weak Jordan constants for groups acting on three-dimensional [MATH]-Mori fiber spaces.', '1608.00709-4-31-5': 'In [REF] and [REF] we bound weak Jordan constants for groups of automorphisms of Gorenstein terminal (and in particular smooth) Fano threefolds.', '1608.00709-4-31-6': 'Finally, in [REF] we summarize the above partial results and complete the proof of Theorem [REF], and also make concluding remarks.', '1608.00709-4-31-7': 'In Appendix [REF] we collect some information about automorphism groups of two particular classes of smooth Fano varieties: complete intersections of quadrics, and complete intersections in weighted projective spaces; these results are well known to experts, but we decided to include them because we do not know proper references.'}",
1601.00019,"{'1601.00019-1-0-0': 'Multiple-input multiple-output (MIMO) systems with large number of basestation antennas, often called massive MIMO systems, have received much attention in academia and industry as a means to improve the spectral efficiency, energy efficiency, and also processing complexity.', '1601.00019-1-0-1': 'Mobile communication industry has initiated a feasibility study to meet the increasing demand of future wireless systems.', '1601.00019-1-0-2': 'Field trials of the proof-of-concept systems have demonstrated the potential gain of the FD-MIMO and 3rd generation partnership project (3GPP) standard body has initiated the standardization activity for the seamless integration of this technology into current 4G cellular carrier frequency.', '1601.00019-1-0-3': 'A study item, process done before a formal standardization process, has been completed in June 2015 and the follow up (work item) process will be initiated shortly for the formal standardization of Release 13.', '1601.00019-1-0-4': 'In this article, we provides an overview of the FD-MIMO system, with emphasis on the discussion and debate conducted on standardization process of Release 13.', '1601.00019-1-0-5': 'We present key features for FD-MIMO systems, summary of the major issues for the standardization and practical system design, and performance evaluation for typical FD-MIMO scenarios.', '1601.00019-1-1-0': '# Introduction', '1601.00019-1-2-0': 'Multiple-input multiple-output (MIMO) systems with large number of basestation antennas, often referred to as massive MIMO systems, have received much attention in academia and industry as a means to improve the spectral efficiency, energy efficiency, and also processing complexity [CITATION].', '1601.00019-1-2-1': 'The wisdom behind the massive MIMO systems is that when the number of basestation antennas goes to infinity, multiuser interference caused by the downlink user co-scheduling and uplink multiple access approaches to zero, resulting in a dramatic increase in the throughput with relative simple transmitter and receiver operations.', '1601.00019-1-2-2': 'While massive MIMO technology is key enabler for the next generation cellular systems, there are many practical challenges down the road to the successful commercialization.', '1601.00019-1-2-3': 'These include design of low-cost and low-power basestation with acceptable antenna space, increase in the backhaul capacity between radio devices and control unit, acquisition of high dimensional channel state information (CSI), and many others.', '1601.00019-1-2-4': 'Recently, 3rd generation partnership project (3GPP) standard body initiated the standardization activity for the massive MIMO systems with an aim to satisfy the spectral efficiency requirement of future cellular systems [CITATION].', '1601.00019-1-2-5': 'Considering the implementation cost and complexity, and also the timeline to the real deployment, the standard body decided to use tens of antennas with two dimensional (2D) array structure as a starting point for the standardization.', '1601.00019-1-2-6': 'Full-Dimension MIMO (FD-MIMO), an official name for massive MIMO for 3GPP, targets the systems utilizing up to 64 antennas at the transmitter side.', '1601.00019-1-2-7': 'Recently, field trials of the proof-of-concept FD-MIMO systems have been conducted successfully[CITATION].', '1601.00019-1-2-8': 'A study item, process done before a formal standardization process, has been completed in June 2015 and the follow up process (work item) will soon be initiated for the formal standardization of Release 13.', '1601.00019-1-3-0': 'The purpose of this article is to provide an overview of the FD-MIMO systems with emphasis on the discussion and debate conducted on standardization process of Release 13.', '1601.00019-1-3-1': 'Specifically, we describe key features for FD-MIMO systems, summarize the main issues for the standardization and practical system design, provide performance evaluation results for typical scenarios, and also make some comments for the future direction.', '1601.00019-1-4-0': '# Key Features of FD-MIMO Systems', '1601.00019-1-5-0': 'One of the main feature of FD-MIMO systems distinct from the MIMO systems of the current LTE standards (up to Rel.12) is that a large number of antennas in the form of 2D antenna array are employed at basestation.', '1601.00019-1-5-1': 'In theory, as the number of basestation antennas [MATH] increases, cross-correlation of two random channel realizations goes to zero [CITATION].', '1601.00019-1-5-2': 'As a result, the interuser interference in the downlink can be controlled via a simple linear precoder and multiuser interference in the uplink can be eliminated via a simple receive combiner.', '1601.00019-1-5-3': 'Such benefit, however, can be realized only when the perfect CSI is available at the basestation.', '1601.00019-1-5-4': 'While the CSI acquisition in TDD systems is relatively simple due to the channel reciprocity, such is not the case for FDD systems.', '1601.00019-1-5-5': 'Basically, channel is measured in FDD systems via the donwlink pilots and then sent back to the basestation after the quantization.', '1601.00019-1-5-6': 'Even in TDD mode, one cannot solely rely on the channel reciprocity because the measurement at the transmitter does not capture the downlink interference from neighboring cells or co-scheduled UEs.', '1601.00019-1-5-7': 'As such, pilots are still required to capture channel quality indicator (CQI) for the TDD mode and thus the downlink pilot transmission and CSI feedback are essential ingredients for both duplex scenarios.', '1601.00019-1-5-8': 'Identifying the potential issues of CSI acquisition and developing the proper solutions is, therefore, of great importance for the successful commercialization of FD-MIMO systems.', '1601.00019-1-5-9': 'Two major problems related to the CSI acquisition process are as follows.', '1601.00019-1-6-0': 'Another interesting feature of the FD-MIMO system is an introduction of the active antenna with 2D planar array.', '1601.00019-1-6-1': 'In the active antenna-based systems, gain and phase are controlled by the active components such as power amplifier (PA) and low noise amplifier (LNA) attached to each antenna element.', '1601.00019-1-6-2': 'Also, when the antenna array has 2D structure, one can control the radio wave on both vertical (elevation) and horizontal (azimuth) direction so that the control of the transmit beam in three dimension space is possible.', '1601.00019-1-6-3': 'This type of wave control mechanism is often referred to as the 3D beamforming.', '1601.00019-1-6-4': 'Another benefit of 2D planar array is that it can accommodate a large number of antennas without increasing the space.', '1601.00019-1-6-5': 'For example, when 64 linear antenna arrays are deployed in a horizontal direction, under the common assumption that the antenna spacing is half wavelength ([MATH]) and the system is using LTE carrier frequency (2 GHz), it requires a horizontal room of [MATH] m. Due to the limited space on rooftops or mast, this space would be burdensome for most of cell sites.', '1601.00019-1-6-6': 'In contrast, when antennas are arranged in a square array, relatively small space is required (e.g., 1.0 [MATH] 0.5 m with dual-polarized 8 [MATH] 8 antenna array).', '1601.00019-1-7-0': 'When basic features of the FD-MIMO systems are given, the next step is to design a system maximizing performance in terms of throughput, spectral efficiency, and peak data rate.', '1601.00019-1-7-1': 'There are various issues to consider in the design of practical systems.', '1601.00019-1-7-2': 'Notable issues include investigation and characterization of the 3D channel model, development of realistic deployment scenario for the performance evaluation, design of new transmitter architecture for supporting 2D active antenna array, and pilot transmission and feedback strategy in accordance with these changes.', '1601.00019-1-7-3': 'In particular, following factors need to be highlighted:', '1601.00019-1-8-0': '# System Design and Standardization of FD-MIMO Systems', '1601.00019-1-9-0': '## Status of MIMO Standardization', '1601.00019-1-10-0': 'The main purpose of Rel.', '1601.00019-1-10-1': '13 study item was to identify key issues to support up to 64 transmit antenna placed in form of 2D antenna array.', '1601.00019-1-10-2': 'Standardization of the systems supporting up to 16 antennas is an initial target for Rel.', '1601.00019-1-10-3': '13 and issues to support more than 16 antennas would be discussed in subsequent releases.', '1601.00019-1-10-4': 'In the study item phase, there has been extensive discussion to support 2D array antennas, elaborated transceiver units, enhanced channel measurement and feedback schemes, and also increased number of co-scheduled users (up to 8 users).', '1601.00019-1-10-5': 'Among these, an item tightly coupled to the standardization is the CSI feedback mechanism.', '1601.00019-1-10-6': 'In order to obtain better understanding of the CSI feedback, the relationship between the beamforming and channel needs to be discussed.', '1601.00019-1-11-0': 'First, under the rich scattering environment, dominant paths between eNB and UE depend on the direction and width of the transmit signal.', '1601.00019-1-11-1': 'For example, when the pilots are transmitted in a narrow beamwidth, radiation energy will be focused in one direction so that the channel can be well expressed as a few dominant taps.', '1601.00019-1-11-2': 'In fact, by controlling the weight applied to CSI-RS, effective dimension of the channel vector can be reduced and hence one can design an efficient feedback scheme.', '1601.00019-1-11-3': 'In view of this, the primary goal of the FD-MIMO feedback is to design a system such that UE can easily track the weight change.', '1601.00019-1-11-4': 'Clearly, this is in contrast to the role of conventional MIMO feedback adapting to the channel variation.', '1601.00019-1-11-5': 'To support this operation, the basestation needs to report the weight changes periodically or UE needs to identify the weight change by itself.', '1601.00019-1-12-0': 'Second, it was shown in the recent field measurement of urban environment that the angular spread in the vertical direction is much smaller than that of the horizontal direction [CITATION].', '1601.00019-1-12-1': 'One implication of this observation is that the interuser interference when two UEs with different heights are co-scheduled will be much less than the case where two UEs with the same height are co-scheduled.', '1601.00019-1-12-2': 'In the scenario where UEs are located in different floors and different buildings, therefore, 3D beamforming would be an effective means to improve the system performance.', '1601.00019-1-13-0': 'It is clear from the discussion above that following issues need to be addressed in the FD-MIMO standardization process.', '1601.00019-1-14-0': '## Issues in FD-MIMO Standardization', '1601.00019-1-15-0': '### Deployment scenarios', '1601.00019-1-16-0': 'For the performance evaluation of FD-MIMO systems, a scenario where antenna array and UEs are located in different height are considered.', '1601.00019-1-16-1': 'To this end, two typical deployment scenarios, viz., 3D urban macro scenario (3D-UMa) and 3D urban micro (3D-UMi), are introduced (see Fig. [REF]).', '1601.00019-1-16-2': 'In the former case, transmit antennas are placed over the rooftop and in the latter case, those are located below the rooftop.', '1601.00019-1-16-3': 'In case of 3D-UMa, diffraction over the rooftop is a dominant factor for the propagation so that down-tilted transmission in the vertical direction is desirable.', '1601.00019-1-16-4': 'In fact, by transmitting beams with different steering angles, eNB can separate channels corresponding to multiple UEs.', '1601.00019-1-16-5': 'In the 3D-UMi scenario, on the other hand, location of users is higher than the height of antenna so that direct signal path is dominant.', '1601.00019-1-16-6': 'In this scenario, both up and down-tilting are needed to schedule UEs in different floors.', '1601.00019-1-16-7': 'Since the cell radius of the 3D-UMi scenario is usually smaller than the that for 3D-UMa, line-of-sight (LOS) channel condition is predominant and hence more UEs can be co-scheduled in the vertical direction.', '1601.00019-1-17-0': '### Antenna configurations', '1601.00019-1-18-0': 'Beam pattern in 2D array antenna structure should be designed in a way to maximize the cell coverage and beamforming gain.', '1601.00019-1-18-1': 'As shown in Fig. [REF](a), there are three key parameters characterizing the antenna array structure [MATH]: the number of elements [MATH] in vertical direction, the number of elements [MATH] in horizontal direction, and the polarization degree [MATH] is for co-polarization and [MATH] is for dual-polarization).', '1601.00019-1-18-2': 'As a benchmark setting, 2D planar array using dual polarized antenna ([MATH]) configuration with [MATH] spacing in vertical direction) and [MATH] spacing in horizontal direction) is suggested in the study item.', '1601.00019-1-18-3': 'In this setting, null direction, an angle to make the magnitude of beam pattern to zero, for the elevation beam pattern is 11[MATH] and that for the horizontal beam pattern is 30[MATH] (see Fig. [REF](c)).', '1601.00019-1-18-4': 'Since the null direction in the vertical domain is much smaller than that of the horizontal domain, one can control the interuser interference effectively by scheduling UEs in the vertical domain.', '1601.00019-1-19-0': '### TXRU architectures', '1601.00019-1-20-0': 'As mentioned, one interesting feature of the active antenna systems is that each TXRU contains PA and LNA so that eNB can control the gain and phase of each antenna elements.', '1601.00019-1-20-1': 'In order to make this possible, a power feeding network between TXRUs and antenna elements referred to as TXRU architecture is required [CITATION].', '1601.00019-1-20-2': 'TXRU architecture consists of three components: TXRU array, antenna array, and radio distribution networks (RDN).', '1601.00019-1-20-3': 'A role of the RDN is to deliver the transmit signal from PA to antenna array elements and received signals from antenna array to LNA.', '1601.00019-1-20-4': 'Among various options, array partitioning and array connected architecture are well-known.', '1601.00019-1-21-0': 'In the array partitioning architecture, antenna elements are divided into multiple groups and each TXRU is connected to one of them (see Fig. [REF](a)).', '1601.00019-1-21-1': 'Whereas, in the array connected structure, RDN is designed such that RF signals of multiple TXRUs are delivered to the single antenna element.', '1601.00019-1-21-2': 'To mix RF signals from multiple TXRUs, additional RF combining circuitry is needed as shown in Fig. [REF](b).', '1601.00019-1-21-3': 'The difference between two can be better understood when we discuss the transmission of the CSI-RS.', '1601.00019-1-21-4': 'In the array partitioning architecture, each TXRU transmits its own CSI-RS so that the UE should measure the channel information of all TXRUs.', '1601.00019-1-21-5': 'Specifically, [MATH] antenna elements are partitioned into [MATH] groups of TXRU and orthogonal CSI-RS is assigned for each group.', '1601.00019-1-21-6': 'In the array connected architecture, each antenna element is connected to [MATH] (out of [MATH]) TXRUs and orthogonal CSI-RS is assigned for each TXRU with [MATH] dimension weight vector.', '1601.00019-1-21-7': 'Thus, UE measures the precoded channel [MATH] from the beamformed CSI-RS observation [MATH].', '1601.00019-1-21-8': 'Since the CSI-RS is transmitted with the narrow and directional beam, channel for the array connected architecture experiences less scattering.', '1601.00019-1-22-0': '### Beamformed CSI-RS transmission', '1601.00019-1-23-0': 'In the study item phase, two CSI-RS transmission strategies, i.e., extension of conventional non-precoded CSI-RS and beamformed CSI-RS, are suggested.', '1601.00019-1-23-1': 'In the first case, UE observes non-precoded CSI-RS transmitted from the passive antennas.', '1601.00019-1-23-2': 'By choosing the precoder maximizing the properly designed performance criterion, UE can adapt to channel variation.', '1601.00019-1-23-3': 'In the second case, eNB transmits multiple beamformed CSI-RS and among these UE selects the preferred CSI-RS and then feeds back the index of it.', '1601.00019-1-23-4': 'In doing so, eNB obtains the channel direction information.', '1601.00019-1-24-0': 'Overall downlink precoder for data transmission [MATH] can be expressed as [EQUATION] where [MATH] is the precoder between TXRU to antenna element, and [MATH] is the precoder between CSI-RS to TXRU ([MATH] is the number of antenna ports), and [MATH] is the precoder obtained from the feedback.', '1601.00019-1-24-1': 'Note that the weight applied to the CSI-RS is the product of [MATH] and [MATH] while the weight applied to the data transmission is the product of [MATH], [MATH], and [MATH].', '1601.00019-1-24-2': 'In the following, we summarize details of two strategies.', '1601.00019-1-25-0': 'In Table [REF], we summarize the CSI-RS transmission schemes discussed in the study item phase.', '1601.00019-1-26-0': '## CSI Feedback Mechanisms for FD-MIMO Systems', '1601.00019-1-27-0': 'In the study item phase, various RS transmission and feedback schemes have been proposed.', '1601.00019-1-27-1': 'In this subsection, we briefly describe key features of these schemes.', '1601.00019-1-28-0': 'Partial CSI-RS: When we use the conventional feedback scheme, under the same CSI-RS density requirement (1 RE/port/RB pair), CSI-RS overhead will increase with the number of TXRUs.', '1601.00019-1-28-1': 'One simple yet effective approach for reducing the overhead is to use only subset of antennas for the CSI measurement.', '1601.00019-1-28-2': 'For example, by partitioning the 2D antenna array into horizontal and vertical ports, says [MATH] ports in the row and [MATH] ports in the column, the total number of CSI-RS can be reduced from [MATH] to [MATH].', '1601.00019-1-28-3': 'This scheme, dubbed as partial CSI-RS transmission, can be easily mapped onto a single RB without changing the current CSI-RS density.', '1601.00019-1-28-4': 'Overall channel information can be reconstructed by exploiting spatial and temporal correlation among antenna elements [CITATION].', '1601.00019-1-29-0': 'Composite codebook: In this scheme, overall codebook is divided into two (vertical and horizontal codebooks) and thus the channel information is separately delivered to eNB.', '1601.00019-1-29-1': 'By combining two codebook (e.g., Kronecker product of two codebooks [MATH]), eNB reconstructs whole channel information.', '1601.00019-1-29-2': 'Considering that the angular spread of the vertical direction is smaller than that of the horizontal direction, one can reduce the feedback overhead by setting relatively long reporting period for the vertical codebook.', '1601.00019-1-29-3': 'The conventional LTE codebook can be reused for horizontal codebook, but it might be better to newly design the vertical codebook to achieve better tradeoff between performance and feedback overhead.', '1601.00019-1-30-0': 'Hybrid CSI-RS transmission: The purpose of hybrid CSI-RS transmission is to achieve the benefit of the beamformed CSI-RS transmission and conventional (non-precoded) transmission simultaneously.', '1601.00019-1-30-1': 'First, in order to acquire long-term channel information, the eNB transmits [MATH] non-precoded CSI-RSs.', '1601.00019-1-30-2': 'After receiving sufficient long-term channel statistics from UE, eNB generates and then transmits the beamformed CSI-RSs which are used for short-term and subband feedbacks at UE.', '1601.00019-1-30-3': 'For example, channel dimension of the beamformed CSI-RS can be reduced by projecting full dimensional channel information into the dominant eigen-directions of long-term channel information.', '1601.00019-1-30-4': 'In doing so, short-term feedback overhead can be reduced substantially.', '1601.00019-1-30-5': 'Also, by transmitting the conventional CSI-RS with a long duty cycle, long-term feedback overhead can be reduced.', '1601.00019-1-31-0': ""Beam index feedback: To obtain the UE's CDI from beamformed CSI-RSs, the eNB needs to transmit multiple beamformed CSI-RSs."", '1601.00019-1-31-1': 'When the channel rank is one, feedback of a beam index and corresponding CQI is enough.', '1601.00019-1-31-2': 'When the channel rank is two, however, co-phase information is additionally required for dual-polarized antennas.', '1601.00019-1-31-3': 'For example, once the eNB obtains the CDI, this can be used for the beamforming vector of two-port CSI-RS and each CSI-RS port is mapped to the different polarized antennas.', '1601.00019-1-31-4': 'UE then estimates and feeds back short-term co-phase information between two ports.', '1601.00019-1-31-5': 'A drawback of this scheme is that a large number of beams are needed to obtain an accurate CDI.', '1601.00019-1-31-6': 'One way to reduce pilot overhead is to transmit multiple spatially separated beams on the same CSI-RS resource.', '1601.00019-1-32-0': 'Flexible codebook: In order to support various 2D antenna layouts without increasing the number of codebooks, flexible codebook scheme can be employed.', '1601.00019-1-32-1': 'In this approach, one master codebook is designed for a large number of TXRUs, say 16 TXRUs, and the specific codebook (e.g., ([MATH]), ([MATH]), or ([MATH])) is derived based on this.', '1601.00019-1-32-2': 'To support this, the eNB needs to send the layout information via separate signaling and using this UE reconstructs the actual codebook.', '1601.00019-1-32-3': 'To achieve further improvement in performance, one can additionally control the codebook resolution based on the user distribution.', '1601.00019-1-32-4': 'For example, when antenna is placed in front of high-rise building, the eNB can configure the resolution such that the vertical direction has better resolution than the horizontal direction.', '1601.00019-1-33-0': '# Performance For FD-MIMO System', '1601.00019-1-34-0': 'In order to observe the potential gain of the FD-MIMO systems, we performed system-level simulations under the realistic multicell environment.', '1601.00019-1-34-1': 'In our simulations, we tested two typical deployment scenarios (3D-UMa and 3D-UMi) with 2-tier hexagonal layout.', '1601.00019-1-34-2': 'To observe the system performance in various loading scenario, we use full buffer and finite traffic model (e.g., FTP model).', '1601.00019-1-34-3': 'In case of the full buffer model, each user has unlimited amount of data to transmit.', '1601.00019-1-34-4': 'In the FTP traffic model, each UE with distinct arrival time receives a file with finite size.', '1601.00019-1-34-5': 'In each subband, eNB selects a pair of UEs maximizing sum-capacity.', '1601.00019-1-34-6': 'As a performance metric, we use 50%-tile (representing cell average performance) and 5%-tile user throughput (representing cell edge performance).', '1601.00019-1-34-7': 'In our simulations, several CSI feedback strategies are investigated.', '1601.00019-1-35-0': 'Detailed simulation parameters are provided in Table [REF].', '1601.00019-1-35-1': 'In Fig. [REF](a), we plot the throughput of the conventional LTE systems with 8Tx ([MATH]) and FD-MIMO systems with 16 and 32Tx ([MATH]) for the full buffer traffic.', '1601.00019-1-35-2': 'Since the co-scheduling UEs in different floors of the same building results in lower interuser interference than the case of co-scheduling UEs in different buildings, FD-MIMO systems achieve substantial gain over the conventional LTE MIMO systems, resulting in [MATH]% and [MATH]% gain for cell average and [MATH]% and [MATH]% gain for cell edge of [MATH]Tx and [MATH]Tx, respectively.', '1601.00019-1-35-3': 'In Fig. [REF](c), we plot the performance for the more realistic finite traffic model.', '1601.00019-1-35-4': 'Under various network loading scenarios, FD-MIMO systems outperform the conventional MIMO system with a large margin, achieving 1.5x and 3x improvement in performance in [MATH]%-tile and [MATH]%-tile user throughput, respectively.', '1601.00019-1-35-5': 'In the low interference scenario (low network loading), gain of the FD-MIMO systems is coming from the 3D beamforming.', '1601.00019-1-35-6': 'Whereas, for the high interference scenario (medium to high network loading), gain is mainly due to the multiuser precoding by the 2D active antenna array.', '1601.00019-1-35-7': 'Fig. [REF](d) summarizes the throughput of [MATH]Tx FD-MIMO systems for various CSI feedback frameworks.', '1601.00019-1-35-8': 'Since the number of beamformed CSI-RSs ([MATH]) is much smaller than the number of the precoder (number of codebooks is 128) and further beamforming weights are fixed, it is not easy to obtain an accurate CDI for the beamformed transmission.', '1601.00019-1-35-9': 'Due to this, the beamformed transmission performs worse than the non-precoded and hybrid transmission for 50%-tile user throughput.', '1601.00019-1-35-10': 'Since the hybrid transmission can control weights with reduced CSI-RS resources, performance of this scheme is comparable to non-precoded transmission.', '1601.00019-1-35-11': 'Nevertheless, as shown in Fig. [REF](d), performance of beamformed transmission for the 5%-tile user throughput is comparable to non-precoded, beamformed, and hybrid transmission.', '1601.00019-1-36-0': '# Concluding Remarks', '1601.00019-1-37-0': 'In this article, we have provided an overview of FD-MIMO systems in 3GPP LTE (recently named as LTE-Advanced Pro) with emphasis on the discussion and debate conducted on the study item phase of Release 13.', '1601.00019-1-37-1': 'We discussed key features of FD-MIMO systems, issues in standardization and system design such as channel model, transceiver architectures, pilot transmission and CSI feedback scheme, and also presented the system level simulation results to demonstrate the potential gain of FD-MIMO systems.', '1601.00019-1-37-2': 'To make the most of large number of basestation antennas in a cost and space effective manner, many new features, distinct from MIMO system in conventional LTE-A, should be introduced in both standard and system design.', '1601.00019-1-37-3': 'These include new transceiver architecture (TXRU architecture), new RS transmission scheme (beamformed CSI-RS transmissions), and channel direction information feedback, and many more.', '1601.00019-1-37-4': 'These issues is important for the success of commercial FD-MIMO systems and still many technical issues remain unsolved and further study and investigation will be conducted through the standardization process.'}","{'1601.00019-2-0-0': 'Multiple-input multiple-output (MIMO) systems with large number of basestation antennas, often called massive MIMO systems, have received much attention in academia and industry as a means to improve the spectral efficiency, energy efficiency, and processing complexity.', '1601.00019-2-0-1': 'Mobile communication industry has initiated a feasibility study to meet the increasing demand of future wireless systems.', '1601.00019-2-0-2': 'Field trials of the proof-of-concept systems have demonstrated the potential gain of the Full-Dimension MIMO (FD-MIMO) and 3rd generation partnership project (3GPP) standard body has initiated the standardization activity for the seamless integration of this technology into current 4G LTE systems.', '1601.00019-2-0-3': 'A study item, process done before a formal standardization process, has been completed in June 2015 and the follow up work item process will be finalized shortly for the formal standardization of Release 13.', '1601.00019-2-0-4': 'In this article, we provide an overview of the FD-MIMO system, with emphasis on the discussion and debate conducted on standardization process of Release 13.', '1601.00019-2-0-5': 'We present key features for FD-MIMO systems, summary of the major issues for the standardization and practical system design, and performance evaluation for typical FD-MIMO scenarios.', '1601.00019-2-1-0': '# Introduction', '1601.00019-2-2-0': 'Multiple-input multiple-output (MIMO) systems with large number of basestation antennas, often referred to as massive MIMO systems, have received much attention in academia and industry as a means to improve the spectral efficiency, energy efficiency, and also processing complexity [CITATION].', '1601.00019-2-2-1': 'The wisdom behind the massive MIMO systems is that when the number of basestation antennas goes to infinity, multiuser interference caused by the downlink user co-scheduling and uplink multiple access approaches to zero, resulting in a dramatic increase in the throughput with relative simple transmitter and receiver operations.', '1601.00019-2-2-2': 'While the massive MIMO technology is one of key enabler for the next generation cellular systems, there are many practical challenges down the road to the successful commercialization.', '1601.00019-2-2-3': 'These include design of low-cost and low-power basestation with acceptable antenna space, improvement in the fronthaul capacity between radio and control units, acquisition of high dimensional channel state information (CSI), and many others.', '1601.00019-2-2-4': 'Recently, 3rd generation partnership project (3GPP) standard body initiated the standardization activity for the massive MIMO systems with an aim to satisfy the spectral efficiency requirement of future cellular systems [CITATION].', '1601.00019-2-2-5': 'Considering the implementation cost and complexity, and also the timeline to the real deployment, standard body decided to use tens of antennas with two dimensional (2D) array structure as a starting point.', '1601.00019-2-2-6': 'Full-Dimension MIMO (FD-MIMO), an official name for massive MIMO for 3GPP, targets the system utilizing up to 64 antenna ports at the transmitter side.', '1601.00019-2-2-7': 'Recently, field trials of the proof-of-concept FD-MIMO systems have been conducted successfully[CITATION].', '1601.00019-2-2-8': 'A study item, process done before a formal standardization process, has been completed in June 2015 and the follow up work item process will be finalized soon for the formal standardization of Release 13.', '1601.00019-2-3-0': 'The purpose of this article is to provide an overview of the FD-MIMO systems with an emphasis on the discussion and debate conducted on standardization process of Release 13.', '1601.00019-2-3-1': 'We note that preliminary studies addressed feasibility of 2D array antenna structure and performance evaluation in ideal scenarios (non-precoded CSI-RS and ideal feedback) [CITATION].', '1601.00019-2-3-2': 'This work is distinct from previous efforts in the sense that we put out emphasis on key features for standardization such as (TXRU architectures, beamformed CSI-RS, 3D beamforming), details of CSI feedback issues, and performance evaluation in realistic FD-MIMO scenarios with new feedback schemes.', '1601.00019-2-4-0': '# Key Features of FD-MIMO Systems', '1601.00019-2-5-0': 'In this section, we discuss key features of FD-MIMO systems.', '1601.00019-2-5-1': 'These include large number of basestation antennas, two dimensional active antenna array, 3D channel propagation, and new pilot transmission with CSI feedback.', '1601.00019-2-5-2': 'In what follows, we will use LTE terminology exclusively: enhanced node-B (eNB) for basestation, user equipment (UE) for the mobile terminal, and reference signal (RS) for pilot signal.', '1601.00019-2-6-0': '## Increase the number of transmit antennas', '1601.00019-2-7-0': 'One of the main feature of FD-MIMO systems distinct from the MIMO systems of the current LTE and LTE-Advanced standards is to employ a large number of antennas at eNB.', '1601.00019-2-7-1': 'In theory, as the number of eNB antennas [MATH] increases, cross-correlation of two random channel realizations goes to zero [CITATION].', '1601.00019-2-7-2': 'As a result, the interuser interference in the downlink can be controlled via a simple linear precoder and multiuser interference in the uplink can be eliminated via a simple receive combiner.', '1601.00019-2-7-3': 'Such benefit, however, can be realized only when the perfect CSI is available at the eNB.', '1601.00019-2-7-4': 'While the CSI acquisition in time division duplex (TDD) systems is relatively simple due to the channel reciprocity, such is not the case for frequency division duplex (FDD) systems.', '1601.00019-2-7-5': 'Basically, time variation and frequency response of channel is measured in FDD systems via the donwlink RSs and then sent back to the eNB after the quantization.', '1601.00019-2-7-6': 'Even in TDD mode, one cannot solely rely on the channel reciprocity because the measurement at the transmitter does not capture the downlink interference from neighboring cells or co-scheduled UEs.', '1601.00019-2-7-7': 'As such, downlink RSs are still required to capture channel quality indicator (CQI) for the TDD mode and thus the downlink RSs transmission and the uplink CSI feedback are essential for both duplex modes.', '1601.00019-2-7-8': 'Identifying the potential issues of CSI acquisition and developing the proper solutions is, therefore, of great importance for the successful commercialization of FD-MIMO systems.', '1601.00019-2-7-9': 'Two major problems related to the CSI acquisition process are as follows.', '1601.00019-2-8-0': '## 2D active antenna system (AAS)', '1601.00019-2-9-0': 'Another interesting feature of the FD-MIMO system is an introduction of the active antenna with 2D planar array.', '1601.00019-2-9-1': 'In the active antenna-based systems, gain and phase are controlled by the active components such as power amplifier (PA) and low noise amplifier (LNA) attached to each antenna element.', '1601.00019-2-9-2': 'In the 2D structured antenna array, one can control the radio wave on both vertical (elevation) and horizontal (azimuth) direction so that the control of the transmit beam in three dimension space is possible.', '1601.00019-2-9-3': 'This type of wave control mechanism is referred to as the 3D beamforming.', '1601.00019-2-9-4': 'Another benefit of 2D planar array is that it can accommodate a large number of antennas without increasing the deployment space.', '1601.00019-2-9-5': 'For example, when 64 linear antenna arrays are deployed in a horizontal direction, under the common assumption that the antenna spacing is half wavelength ([MATH]) and the system is using LTE carrier frequency (2 GHz), it requires a horizontal room of [MATH]m. Due to the limited space on rooftop or mast, this space would be burdensome for most of cell sites.', '1601.00019-2-9-6': 'In contrast, when antennas are arranged in a square array, relatively small space is required (e.g., 1.0 [MATH] 0.5m with dual-polarized 8 [MATH] 8 antenna array).', '1601.00019-2-10-0': '## 3D channel environment', '1601.00019-2-11-0': 'When basic features of the FD-MIMO systems are provided, the next step is to design a system maximizing performance in terms of throughput, spectral efficiency, and peak data rate in the realistic channel environment.', '1601.00019-2-11-1': 'There are various issues to consider in the design of practical systems.', '1601.00019-2-11-2': 'Notable issues include investigation and characterization of the channel model of realistic deployment scenario for the performance evaluation.', '1601.00019-2-11-3': 'While the conventional MIMO systems consider the propagation in the horizontal direction only, FD-MIMO systems employing 2D planar array should consider the propagation in both vertical and horizontal direction.', '1601.00019-2-11-4': 'To do so, geometric structure of the transmitter antenna array and propagation effect of the three-dimensional positions between the eNB and UE should be reflected in the channel model.', '1601.00019-2-11-5': 'Main features of 3D channel propagation obtained from real measurement are as follows [CITATION]:', '1601.00019-2-12-0': '## RS transmission for CSI acquisition', '1601.00019-2-13-0': 'From the Rel.', '1601.00019-2-13-1': '8 LTE to Rel.', '1601.00019-2-13-2': '12 LTE-Advanced, there has been substantial improvement in the RS scheme (See Fig. [REF](a)).', '1601.00019-2-13-3': 'In the LTE system, common RS (CRS) has been exclusively used for channel training and data demodulation.', '1601.00019-2-13-4': 'In LTE-Advanced system, two new RSs, viz., channel state information RS (CSI-RS) and demodulation RS (DM-RS) have been introduced to perform the CSI acquisition and demodulation of data channel, respectively.', '1601.00019-2-13-5': 'While the CRS is transmitted with 1ms period, CSI-RS, used mainly for the CSI measurement in fixed or low-mobility scenarios, is transmitted with a multiple of 5ms period.', '1601.00019-2-13-6': 'While the CSI-RS is common to all users in a cell and thus un-precoded, the DM-RS is UE-specific (dedicated to each UE) so that it is precoded by the same weight applied for the data transmission.', '1601.00019-2-13-7': 'Since the DM-RS is present only on time/frequency resources where the UE is scheduled, this is not appropriate for CSI measurements [CITATION].', '1601.00019-2-14-0': 'One of the key features of the FD-MIMO systems distinct from previous LTE standards is to use a beamformed RS, called beamformed CSI-RS, for the CSI acquisition.', '1601.00019-2-14-1': 'Beamformed RS transmission is a technique that changes the radiation pattern of RS using the precoding weight.', '1601.00019-2-14-2': 'This scheme provides many benefits over non-precoded CSI-RS in the massive MIMO regime.', '1601.00019-2-14-3': 'Some of benefits are summarized as follows:', '1601.00019-2-15-0': 'In order to support the beamformed CSI-RS scheme, new transmitter architecture called transceiver unit (TXRU) architecture has been introduced.', '1601.00019-2-15-1': 'By TXRU architecture, we mean a hardware connection between the baseband signal path and antenna array elements.', '1601.00019-2-15-2': 'In the active antenna system, patch antennas and active devices are integrated on the printed circuit board (PCB) so that one can easily design paths between TXRUs and antenna elements.', '1601.00019-2-15-3': 'Since this architecture facilitates the control of phase and gain in both digital and analog domain, more accurate control of the beamforming direction is possible.', '1601.00019-2-15-4': 'One thing to note is that the conventional codebook cannot measure the CSI of the beamformed transmission so that new channel feedback mechanism supporting the beamformed transmission should be introduced (See Section [REF].', '1601.00019-2-15-5': 'D for details).', '1601.00019-2-16-0': '# System Design and Standardization of FD-MIMO Systems', '1601.00019-2-17-0': 'The main purpose of Rel.', '1601.00019-2-17-1': '13 study item is to identify key issues to support up to 64 transmit antennas placed in form of 2D antenna array.', '1601.00019-2-17-2': 'Standardization of the systems supporting up to 16 antennas is an initial target of Rel.', '1601.00019-2-17-3': '13 and issues to support more than 16 antennas will be discussed in subsequent releases.', '1601.00019-2-17-4': 'In the study item phase, there has been extensive discussion to support 2D array antennas, elaborated TXRUs, enhanced channel measurement and feedback schemes, and also increased number of co-scheduled users (up to 8 users).', '1601.00019-2-17-5': 'Among these, an item tightly coupled to the standardization is the CSI measurement and feedback mechanism.', '1601.00019-2-17-6': 'In this subsection, we discuss the deployment scenarios, TXRU structure, new RS strategy and corresponding feedback mechanism.', '1601.00019-2-18-0': '## Deployment scenarios', '1601.00019-2-19-0': 'For the design and evaluation of FD-MIMO systems, a realistic scenario where antenna array and UEs are located in different height are considered.', '1601.00019-2-19-1': 'To this end, two typical deployment scenarios, viz., 3D urban macro scenario (3D-UMa) and 3D urban micro (3D-UMi), are introduced (see Fig. [REF]).', '1601.00019-2-19-2': 'In the former case, transmit antennas are placed over the rooftop and in the latter case, those are located below the rooftop.', '1601.00019-2-19-3': 'In case of 3D-UMa, diffraction over the rooftop is a dominant factor for the propagation so that down-tilted transmission in the vertical direction is desirable.', '1601.00019-2-19-4': 'In fact, by transmitting beams with different steering angles, eNB can separate channels corresponding to multiple UEs.', '1601.00019-2-19-5': 'In the 3D-UMi scenario, on the other hand, location of users is higher than the height of antenna so that direct signal path is dominant.', '1601.00019-2-19-6': 'In this scenario, both up and down-tilting are needed to schedule UEs in different floors.', '1601.00019-2-19-7': 'Since the cell radius of the 3D-UMi scenario is typically smaller than that of 3D-UMa, LOS channel condition is predominant so that more UEs can be co-scheduled without increasing the interuser interference [CITATION].', '1601.00019-2-19-8': 'Although not as strong as 3D-UMi scenario, LOS probability in 3D-UMa scenario also increases when the distance between eNB and UE decreases.', '1601.00019-2-20-0': '## Antenna configurations', '1601.00019-2-21-0': 'Unlike the conventional MIMO systems relying on the passive antenna, systems based on the active antenna can dynamically control the gain of an antenna element by applying the weight to low-power amplifiers attached to each antenna element.', '1601.00019-2-21-1': 'Since the radiation pattern depends on the antenna arrangement such as the number of the antenna elements and antenna spacing, the antenna system should be modeled in an element-level.', '1601.00019-2-21-2': 'As shown in Fig. [REF](a), there are three key parameters characterizing the antenna array structure [MATH]: the number of elements [MATH] in vertical direction, the number of elements [MATH] in horizontal direction, and the polarization degree [MATH] is for co-polarization and [MATH] is for dual-polarization).', '1601.00019-2-21-3': 'As a benchmark setting, 2D planar array using dual polarized antenna ([MATH]) configuration with [MATH] spacing in vertical direction) and [MATH] spacing in horizontal direction) is suggested.', '1601.00019-2-21-4': 'In this setting, null direction, an angle to make the magnitude of beam pattern to zero, for the elevation beam pattern is 11[MATH] and that for the horizontal beam pattern is 30[MATH] (see Fig. [REF](c)).', '1601.00019-2-21-5': 'Since the null direction in the vertical domain is much smaller than that of the horizontal domain, scheduling UEs in the vertical domain is more effective in controlling the interuser interference.', '1601.00019-2-21-6': 'Note that tall or fat array structure ([MATH] or [MATH]) is favorable since it will generate sharp beam but it might be less flexible in the situation where surrounding environment is changed.', '1601.00019-2-21-7': 'Also, large antenna spacing might be a desirable option but it can also increase inter-cell interference due to the coverage enhancement (caused by the effective SNR enhancement).', '1601.00019-2-21-8': 'For this reason, in a real deployment scenario, the design parameters should be carefully chosen by considering various factors such as user location, cell radius, building height, and antenna height.', '1601.00019-2-22-0': '## TXRU architectures', '1601.00019-2-23-0': 'As mentioned, one interesting feature of the active antenna systems is that each TXRU contains PA and LNA so that eNB can control the gain and phase of an individual antenna element.', '1601.00019-2-23-1': 'In order to support this, a power feeding network between TXRUs and antenna elements called TXRU architecture is introduced [CITATION].', '1601.00019-2-23-2': 'TXRU architecture consists of three components: TXRU array, antenna array, and radio distribution networks (RDN).', '1601.00019-2-23-3': 'A role of the RDN is to deliver the transmit signal from PA to antenna array elements and received signals from antenna array to LNA.', '1601.00019-2-23-4': 'Depending on the CSI-RS transmission strategy, two representative options, array partitioning and array connected architecture, are suggested.', '1601.00019-2-23-5': 'In the array partitioning architecture, antenna elements are divided into multiple groups and each TXRU is connected to one of them (see Fig. [REF](d)).', '1601.00019-2-23-6': 'Whereas, in the array connected structure, RDN is designed such that RF signals of multiple TXRUs are delivered to the single antenna element.', '1601.00019-2-23-7': 'To mix RF signals from multiple TXRUs, additional RF combining circuitry is needed as shown in Fig. [REF](e).', '1601.00019-2-23-8': 'The difference between two can be better understood when we discuss the transmission of the CSI-RS.', '1601.00019-2-23-9': 'In the array partitioning architecture, [MATH] antenna elements are partitioned into [MATH] groups of TXRU and orthogonal CSI-RS is assigned for each group.', '1601.00019-2-23-10': 'Each TXRU transmits its own CSI-RS so that the UE should measure the channel [MATH] from the non-precoded CSI-RS observation [MATH] of all TXRUs.', '1601.00019-2-23-11': 'In the array connected architecture, each antenna element is connected to [MATH] (out of [MATH]) TXRUs and orthogonal CSI-RS is assigned for each TXRU with [MATH] dimension weight vector (see Fig. [REF](e) for [MATH] and [MATH]).', '1601.00019-2-23-12': 'Letting [MATH] be the channel vector and [MATH] be the precoding weight, UE measures the precoded channel [MATH] from the beamformed CSI-RS observation [MATH].', '1601.00019-2-23-13': 'Due to the narrow and directional CSI-RS beam transmission with linear array, SNR of precoded channel is maximized at the target direction.', '1601.00019-2-24-0': '## New CSI-RS transmission strategy', '1601.00019-2-25-0': 'In the FD-MIMO systems, two CSI-RS transmission strategies, i.e., extension of conventional non-precoded CSI-RS and beamformed CSI-RS, are suggested.', '1601.00019-2-25-1': 'In the first strategy, UE observes non-precoded CSI-RS transmitted from each of partitioned antenna arrays (see Fig. [REF](d)).', '1601.00019-2-25-2': 'By choosing the precoder maximizing the properly designed performance criterion, UE can adapt to the channel variation.', '1601.00019-2-25-3': 'In the second strategy, eNB transmits multiple beamformed CSI-RS (for simplicity, we call it beam) using connected arrays architecture.', '1601.00019-2-25-4': 'Among these, UE selects the preferred beam and then feeds back its index.', '1601.00019-2-25-5': 'When eNB receives the beam index, the weight corresponding to the selected beam is used for data transmission.', '1601.00019-2-25-6': 'In doing so, eNB obtains the channel direction information.', '1601.00019-2-26-0': 'Overall downlink precoder for data transmission [MATH] can be expressed as [EQUATION] where [MATH] is the precoder between TXRU to antenna element, and [MATH] is the precoder between CSI-RS port to TXRU ([MATH] is the number of antenna ports), and [MATH] is the precoder between data channel to CSI-RS port.', '1601.00019-2-26-1': 'Note that the weight applied to the CSI-RS is the product of [MATH] and [MATH] while the weight applied to the data transmission is the product of [MATH], [MATH], and [MATH].', '1601.00019-2-26-2': 'In the following, we summarize details of two strategies.', '1601.00019-2-27-0': 'In Table [REF], we summarize the CSI-RS transmission schemes discussed in the FD-MIMO.', '1601.00019-2-28-0': '## CSI Feedback Mechanisms for FD-MIMO Systems', '1601.00019-2-29-0': 'In the study item phase, various RS transmission and feedback schemes have been proposed.', '1601.00019-2-29-1': 'As shown in Fig. [REF], capacity and overhead of class A and B feedback schemes are more or less similar in the initial target range (10 and 20 antennas) so that Rel.13 has decided to support both classes.', '1601.00019-2-29-2': 'In this subsection, we briefly describe codebook, RS transmission, CSI feedback schemes discuss in the standardization.', '1601.00019-2-29-3': 'Among various schemes, composite codebook and beam index feedback have received much attention as a main ingredients for class A and B CSI feedback.', '1601.00019-2-29-4': 'Rest will be considered in the future release.', '1601.00019-2-30-0': 'Composite codebook: In this scheme, overall codebook is divided into two (vertical and horizontal codebooks) and thus the channel information is separately delivered to eNB.', '1601.00019-2-30-1': 'By combining two codebooks (e.g., Kronecker product of two codebooks [MATH]), eNB reconstructs whole channel information.', '1601.00019-2-30-2': 'Considering that the angular spread of the vertical direction is smaller than that of the horizontal direction, one can reduce the feedback overhead by setting relatively long reporting period to the vertical codebook.', '1601.00019-2-30-3': 'The conventional LTE codebook can be reused for horizontal codebook, but it might be better to newly design the vertical codebook to achieve better tradeoff between performance and feedback overhead.', '1601.00019-2-31-0': ""Beam index feedback: To obtain the UE's channel direction information (CDI) from beamformed CSI-RSs, eNB needs to transmit multiple beamformed CSI-RSs."", '1601.00019-2-31-1': 'When the channel rank is one, feedback of a beam index and corresponding CQI is enough.', '1601.00019-2-31-2': 'Whereas, when the channel rank is two, co-phase information is additionally required for dual-polarized antennas.', '1601.00019-2-31-3': 'For example, once eNB obtains the CDI, this can be used for the beamforming vector of two-port CSI-RS and each CSI-RS port is mapped to the different polarized antennas.', '1601.00019-2-31-4': 'UE then estimates and feeds back short-term co-phase information between two ports.', '1601.00019-2-31-5': 'A drawback of this scheme is that a large number of beams are needed to obtain an accurate CDI.', '1601.00019-2-32-0': 'Partial CSI-RS with dimensional feedbacks: When we use the conventional feedback scheme, under the same CSI-RS density requirement (1 RE/port/RB pair), CSI-RS overhead will increase with the number of TXRUs.', '1601.00019-2-32-1': 'One simple yet effective approach for reducing the overhead is to use only subset of antennas for the CSI measurement.', '1601.00019-2-32-2': 'For example, by partitioning the 2D antenna array into horizontal and vertical ports, says [MATH] ports in the row and [MATH] ports in the column, the total number of CSI-RS can be reduced from [MATH] to [MATH].', '1601.00019-2-32-3': 'This scheme, dubbed as partial CSI-RS transmission, can be easily mapped into a single RB without changing the current CSI-RS density.', '1601.00019-2-32-4': 'Overall channel information can be reconstructed by exploiting spatial and temporal correlation among antenna elements [CITATION].', '1601.00019-2-33-0': 'Adaptive CSI feedback with class A and B: The purpose of this feedback scheme is to collect benefits of the beamformed and conventional (non-precoded) CSI-RS transmission simultaneously.', '1601.00019-2-33-1': 'First, in order to acquire long-term channel information, eNB transmits [MATH] non-precoded CSI-RSs.', '1601.00019-2-33-2': 'After receiving sufficient long-term channel statistics from UE, eNB generates and then transmits the beamformed CSI-RSs which are used for short-term and subband feedbacks.', '1601.00019-2-33-3': 'For example, channel dimension of the beamformed CSI-RS can be reduced by projecting full dimensional channel information into the dominant eigen-directions of long-term channel information.', '1601.00019-2-33-4': 'In doing so, short-term feedback overhead can be reduced substantially.', '1601.00019-2-33-5': 'Also, long-term feedback overhead can be reduced by transmitting the conventional CSI-RS with a long duty cycle.', '1601.00019-2-34-0': 'Flexible/Configurable codebook: In order to support various 2D antenna layouts without increasing the number of codebooks, flexible codebook scheme can be employed.', '1601.00019-2-34-1': 'In this approach, one master codebook is designed for a large number of TXRUs, say 16 TXRUs, and the specific codebook (e.g., ([MATH]), ([MATH]), or ([MATH])) is derived based on this.', '1601.00019-2-34-2': 'To support this, the eNB needs to send the layout information via separate signaling and UE reconstructs the actual codebook using this information.', '1601.00019-2-34-3': 'To achieve further improvement in performance, one can additionally control the codebook resolution based on the user distribution.', '1601.00019-2-34-4': 'For example, when the antenna is placed in front of high-rise building, the eNB can assign more quantization bits to the vertical direction such that the vertical direction has better resolution than the horizontal direction.', '1601.00019-2-35-0': '# Performance of FD-MIMO System', '1601.00019-2-36-0': 'In order to observe the potential gain of the FD-MIMO systems, we perform system-level simulations under the realistic multicell environment.', '1601.00019-2-36-1': 'In our simulations, we test two typical deployment scenarios (3D-UMa and 3D-UMi) with 2-tier hexagonal layout.', '1601.00019-2-36-2': 'As a performance metric, we use spectral efficiency for cell average and cell edge.', '1601.00019-2-36-3': 'Detailed simulation parameters are provided in Table [REF].', '1601.00019-2-36-4': 'We first investigate the system performance of FD-MIMO systems with two types of antenna configuration.', '1601.00019-2-36-5': 'For type I and II configurations, [MATH] and [MATH] are used, respectively.', '1601.00019-2-36-6': 'In the type II configuration, antenna spacing is set to four times larger than the spacing of type I. To investigate the effect of antenna structure, ideal feedback under full buffer traffic model (each user has unlimited amount of data to transmit) is used.', '1601.00019-2-36-7': 'In Fig. [REF](a), we plot the throughput of the conventional LTE systems with 8Tx ([MATH]) and FD-MIMO systems with 16, 32, and 64Tx ([MATH] and [MATH]) where [MATH] and [MATH] are the number of CSI-RS in vertical and horizontal dimension, respectively.', '1601.00019-2-36-8': 'This results shows that both antenna configuration provide large gain over the conventional 8Tx in LTE-A, resulting in [MATH] (type I) and [MATH] (type II) gain at cell edge, respectively.', '1601.00019-2-36-9': 'Due to the sufficient antenna spacing, cross-correlation between channels becomes negligible, and thus the spectral efficiency of type II increases linearly with the TXRU, resulting in [MATH] (cell average) and [MATH] gain (cell edge) when the number of TXRUs is doubled [CITATION].', '1601.00019-2-36-10': 'However, due to the insufficient antenna spacing, the spectral efficiency of type I configuration does not scale linearly with the number of TXRUs.', '1601.00019-2-37-0': 'We next investigate the system performance of FD-MIMO system under finite traffic model (e.g., FTP model) where each UE with distinct arrival time receives a file with finite size.', '1601.00019-2-37-1': 'As a performance metric, we use a user packet throughput (number of successively received packet during the transmission period).', '1601.00019-2-37-2': 'In order to support backward compatibility and also perform fair comparison among schemes under test, we employ the conventional MMSE based channel estimation.', '1601.00019-2-37-3': 'In our simulations, following CSI feedback strategies are considered.', '1601.00019-2-38-0': 'In Fig. [REF](c), we plot the user throughput of the finite traffic model as a function of packet arrival rate.', '1601.00019-2-38-1': 'Note that when the packet arrival rate is high, co-scheduled users increase and thus intercell and multiuser interference will also increase.', '1601.00019-2-38-2': 'In this realistic scenario, FD-MIMO systems outperform the conventional MIMO systems with a large margin, achieving 1.5[MATH] and 3[MATH] improvement in performance in cell average and edge user packet throughput, respectively.', '1601.00019-2-38-3': 'Note that in the low network loading (low interference scenario), gain of the FD-MIMO systems is coming from the 3D beamforming and in the medium to high network loading (high interference scenario), gain is mainly due to the multiuser precoding of the 2D active antenna array.', '1601.00019-2-38-4': 'Fig. [REF](d) summarizes the throughput of various CSI feedback frameworks.', '1601.00019-2-38-5': 'With the same feedback overhead (2bit), beamformed CSI-RS scheme I outperforms non-precoded scheme with a large margin.', '1601.00019-2-38-6': 'This is because the number of the codebook for the channel feedback is only 4, so that channel state information at eNB is very poor.', '1601.00019-2-38-7': 'Since the beamformed CSI-RS scheme II can adapt weights of beamformed CSI-RS to generate an accurate CDI, it performs better than other schemes.', '1601.00019-2-38-8': 'It is worth mentioning that the non-precoded CSI-RS scheme requires a large amount of feedback overhead (approximately 128 quantization levels) to achieve comparable performance.', '1601.00019-2-38-9': 'From this observation, we clearly see that the beamformed CSI-RS transmission is effective in controlling the precoding weights (in time, frequency, and space), feedback overhead, and pilot resource overhead, and we expect that this scheme would be a key ingredient for future release of FD-MIMO.', '1601.00019-2-39-0': '# Concluding Remarks', '1601.00019-2-40-0': 'In this article, we have provided an overview of FD-MIMO systems in 3GPP LTE (recently named as LTE-Advanced Pro) with emphasis on the discussion and debate conducted on the Rel.', '1601.00019-2-40-1': '13 phase.', '1601.00019-2-40-2': 'We discussed key features of FD-MIMO systems, main issues in standardization of system design such as channel model, transceiver architectures, pilot transmission and CSI feedback scheme.', '1601.00019-2-40-3': 'We also presented the system level simulation results to demonstrate the potential gain of FD-MIMO systems.', '1601.00019-2-40-4': 'To make the most of large number of eNB antennas in a cost and space effective manner, new key features, distinct from MIMO system in conventional LTE-A, should be introduced in both standard, system design, and transceiver implementation.', '1601.00019-2-40-5': 'These include new transmitter architecture (array connected architecture), new RS transmission scheme (beamformed CSI-RS transmissions), and enhanced channel feedback (beam index feedback), and many more.', '1601.00019-2-40-6': 'Although our work focus on the standardization in Rel.', '1601.00019-2-40-7': '13, there are still many issues for the successful deployment of FD-MIMO systems in the future including pilot transmission and advanced channel estimation exploiting channel sparsity, beam adaptation and optimization, and RS overhead reduction for throughput enhancement.'}","[['1601.00019-1-2-0', '1601.00019-2-2-0'], ['1601.00019-1-2-1', '1601.00019-2-2-1'], ['1601.00019-1-2-4', '1601.00019-2-2-4'], ['1601.00019-1-2-7', '1601.00019-2-2-7'], ['1601.00019-1-10-0', '1601.00019-2-17-0'], ['1601.00019-1-30-3', '1601.00019-2-33-3'], ['1601.00019-1-30-4', '1601.00019-2-33-4'], ['1601.00019-1-28-1', '1601.00019-2-32-1'], ['1601.00019-1-28-2', '1601.00019-2-32-2'], ['1601.00019-1-28-4', '1601.00019-2-32-4'], ['1601.00019-1-23-4', '1601.00019-2-25-6'], ['1601.00019-1-29-0', '1601.00019-2-30-0'], ['1601.00019-1-29-3', '1601.00019-2-30-3'], ['1601.00019-1-27-0', '1601.00019-2-29-0'], ['1601.00019-1-0-1', '1601.00019-2-0-1'], ['1601.00019-1-0-5', '1601.00019-2-0-5'], ['1601.00019-1-16-1', '1601.00019-2-19-1'], ['1601.00019-1-16-2', '1601.00019-2-19-2'], 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'1601.00019-3-14-4', '1601.00019-3-28-0', '1601.00019-3-40-1', '1601.00019-4-2-8', '1601.00019-4-3-1', '1601.00019-4-11-4', '1601.00019-4-14-4', '1601.00019-4-28-0', '1601.00019-4-40-1']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1601.00019,"{'1601.00019-3-0-0': 'Multiple-input multiple-output (MIMO) systems with a large number of basestation antennas, have received much attention in academia and industry as a means to improve the spectral efficiency, energy efficiency, and processing complexity of next generation cellular system.', '1601.00019-3-0-1': 'Mobile communication industry has initiated a feasibility study of MIMO systems with tens of basestation antennas to meet the increasing demand of future wireless systems.', '1601.00019-3-0-2': 'Field trials of the proof-of-concept systems have demonstrated the potential gain of the Full-Dimension MIMO (FD-MIMO), and 3rd generation partnership project (3GPP) standard body initiated standardization activity for the seamless integration of this technology into current 4G LTE systems.', '1601.00019-3-0-3': 'A study item, a process done before a formal standardization process, has been completed in June 2015, and the follow-up work item process will be finalized shortly for the formal standardization of Release 13.', '1601.00019-3-0-4': 'In this article, we provide an overview of the FD-MIMO system, with emphasis on the discussion and debate conducted on the standardization process of Release 13.', '1601.00019-3-0-5': 'We present key features for FD-MIMO systems, a summary of the major issues for the standardization and practical system design, and performance evaluations for typical FD-MIMO scenarios.', '1601.00019-3-1-0': '# Introduction', '1601.00019-3-2-0': 'Multiple-input multiple-output (MIMO) systems with a large number of basestation antennas, often referred to as massive MIMO systems, have received much attention in academia and industry as a means to improve the spectral efficiency, energy efficiency, and processing complexity [CITATION].', '1601.00019-3-2-1': 'While the massive MIMO technology is a promising technology, there are many practical challenges and technical hurdles down the road to the successful commercialization.', '1601.00019-3-2-2': 'These include design of low-cost and low-power basestation with acceptable antenna space, improvement in the fronthaul capacity between radio and control units, acquisition of high dimensional channel state information (CSI), and many others.', '1601.00019-3-2-3': 'Recently, 3rd generation partnership project (3GPP) standard body initiated the standardization activity to employ tens of antennas at basestation with an aim to satisfy the spectral efficiency requirement of future cellular systems [CITATION].', '1601.00019-3-2-4': 'Considering the implementation cost and complexity, and also the timeline to the real deployment, 3GPP decided to use tens of antennas with a two dimensional (2D) array structure as a starting point.', '1601.00019-3-2-5': 'Full-Dimension MIMO (FD-MIMO), the official name for the MIMO enhancement in 3GPP, targets the system utilizing up to 64 antenna ports at the transmitter side.', '1601.00019-3-2-6': 'Recently, field trials of the proof-of-concept FD-MIMO systems have been conducted successfully[CITATION].', '1601.00019-3-2-7': 'A study item, a process done before a formal standardization process, has been completed in June 2015, and the follow-up work item process will be finalized soon for the formal standardization of Release 13 (Rel.', '1601.00019-3-2-8': '13).', '1601.00019-3-3-0': 'The purpose of this article is to provide an overview of the FD-MIMO systems with an emphasis on the discussion and debate conducted on the standardization process of Rel.', '1601.00019-3-3-1': '13.', '1601.00019-3-3-2': 'We note that preliminary studies addressed the feasibility of 2D array antenna structure and performance evaluation in ideal pilot transmission and feedback scenarios [CITATION].', '1601.00019-3-3-3': 'This work is distinct from these in the sense that we put our emphasis on describing realistic issues in the standardization process, including TXRU architectures, beamformed CSI-RS, 3D beamforming, details of CSI feedback, and performance evaluation in realistic FD-MIMO scenarios with new feedback schemes.', '1601.00019-3-4-0': '# Key Features of FD-MIMO Systems', '1601.00019-3-5-0': 'In this section, we discuss key features of FD-MIMO systems.', '1601.00019-3-5-1': 'These include a large number of basestation antennas, 2D active antenna array, 3D channel propagation, and new pilot transmission with CSI feedback.', '1601.00019-3-5-2': 'In what follows, we will use LTE terminology exclusively: enhanced node-B (eNB) for basestation, user equipment (UE) for the mobile terminal, and reference signal (RS) for pilot signal.', '1601.00019-3-6-0': '## Increase the number of transmit antennas', '1601.00019-3-7-0': 'One of the main features of FD-MIMO systems distinct from the MIMO systems of the current LTE and LTE-Advanced standards is to use a large number of antennas at eNB.', '1601.00019-3-7-1': 'In theory, as the number of eNB antennas [MATH] increases, cross-correlation of two random channel realizations goes to zero [CITATION] so that the inter-user interference in the downlink can be controlled via a simple linear precoder.', '1601.00019-3-7-2': 'Such benefit, however, can be realized only when the perfect CSI is available at the eNB.', '1601.00019-3-7-3': 'While the CSI acquisition in time division duplex (TDD) systems is relatively simple due to the channel reciprocity, such is not the case for frequency division duplex (FDD) systems.', '1601.00019-3-7-4': 'Note that in the FDD systems, time variation and frequency response of the channel are measured via the downlink RSs and then sent back to the eNB after the quantization.', '1601.00019-3-7-5': 'Even in TDD mode, one cannot solely rely on the channel reciprocity because the measurement at the transmitter does not capture the downlink interference from neighboring cells or co-scheduled UEs.', '1601.00019-3-7-6': 'As such, downlink RSs are still required to capture the channel quality indicator (CQI) for the TDD mode, and thus the downlink RS and the uplink CSI feedback are essential for both duplex modes.', '1601.00019-3-7-7': 'Identifying the potential issues of CSI acquisition and developing the proper solutions are, therefore, of great importance for the successful commercialization of FD-MIMO systems.', '1601.00019-3-7-8': 'Before we go into detail, we briefly summarize two major problems related to the CSI acquisition.', '1601.00019-3-8-0': '## 2D active antenna system (AAS)', '1601.00019-3-9-0': 'Another interesting feature of the FD-MIMO system is an introduction of the active antenna with 2D planar array.', '1601.00019-3-9-1': 'In the active antenna-based systems, gain and phase are controlled by the active components, such as power amplifier (PA) and low noise amplifier (LNA), attached to each antenna element.', '1601.00019-3-9-2': 'In the 2D structured antenna array, one can control the radio wave on both vertical (elevation) and horizontal (azimuth) direction so that the control of the transmit beam in 3D space is possible.', '1601.00019-3-9-3': 'This type of wave control mechanism is also referred to as the 3D beamforming.', '1601.00019-3-9-4': 'Another important benefit of 2D AAS is that it can accommodate a large number of antennas without increasing the deployment space.', '1601.00019-3-9-5': 'For example, when 64 linear antenna arrays are deployed in a horizontal direction, under the common assumption that the antenna spacing is half wavelength ([MATH]) and the system is using LTE carrier frequency (2 GHz), it requires a horizontal room of [MATH]m. Due to the limited space on a rooftop or mast, this space would be burdensome for most of the cell sites.', '1601.00019-3-9-6': 'In contrast, when antennas are arranged in a square array, relatively small space is required for 2D antenna array (e.g., 1.0 [MATH] 0.5m with dual-polarized 8 [MATH] 8 antenna array).', '1601.00019-3-10-0': '## 3D channel environment', '1601.00019-3-11-0': 'When basic features of the FD-MIMO systems are determined, the next step is to design a system maximizing performance in terms of throughput, spectral efficiency, and peak data rate in the realistic channel environment.', '1601.00019-3-11-1': 'There are various issues to consider in the design of practical systems, such as investigation and characterization of the realistic channel model for the performance evaluation.', '1601.00019-3-11-2': 'While the conventional MIMO systems consider the propagation in the horizontal direction only, FD-MIMO systems employing 2D planar array should consider the propagation in both vertical and horizontal direction.', '1601.00019-3-11-3': 'To do so, geometric structure of the transmitter antenna array and propagation effect of the 3D positions between the eNB and UE should be reflected in the channel model.', '1601.00019-3-11-4': 'Main features of 3D channel propagation obtained from real measurement are as follows [CITATION]:', '1601.00019-3-12-0': '## RS transmission for CSI acquisition', '1601.00019-3-13-0': 'From the LTE to LTE-Advanced, there has been substantial improvement in the RS scheme for MIMO systems (see Fig. 1(a)).', '1601.00019-3-13-1': 'From the common RS (CRS) to the channel state information RS (CSI-RS), various RSs to perform the CSI acquisition have been introduced.', '1601.00019-3-13-2': 'While these are common to all users in a cell and thus un-precoded, the demodulation RS (DM-RS) is UE-specific (dedicated to each UE) so that it is precoded by the same weight applied for the data transmission.', '1601.00019-3-13-3': 'Since the DM-RS is present only on time/frequency resources where the UE is scheduled, this cannot be used for CSI measurements [CITATION].', '1601.00019-3-14-0': 'One of the new features of the FD-MIMO systems is to use a beamformed RS, called beamformed CSI-RS, for the CSI acquisition.', '1601.00019-3-14-1': 'Beamformed RS transmission is a channel training technique that uses multiple precoding weights in spatial domain.', '1601.00019-3-14-2': 'In this scheme, UE selects the best weight among transmitted and then feeds back its index.', '1601.00019-3-14-3': 'This scheme provides many benefits over non-precoded CSI-RS, in particular when [MATH] is large.', '1601.00019-3-14-4': 'Some of the benefits are summarized as follows:', '1601.00019-3-15-0': 'In order to support the beamformed CSI-RS scheme, new transmitter architecture called transceiver unit (TXRU) architecture has been introduced.', '1601.00019-3-15-1': 'By TXRU architecture, we mean a hardware connection between the baseband signal path and antenna array elements.', '1601.00019-3-15-2': 'Since this architecture facilitates the control of phase and gain in both digital and analog domain, more accurate control of the beamforming direction is possible.', '1601.00019-3-15-3': 'One thing to note is that the conventional codebook cannot measure the CSI of the beamformed transmission so that a new channel feedback mechanism supporting the beamformed transmission is required (see Section III.D for details).', '1601.00019-3-16-0': '# System Design and Standardization of FD-MIMO Systems', '1601.00019-3-17-0': 'The main purpose of the Rel.', '1601.00019-3-17-1': '13 study item is to identify key issues to support up to 64 transmit antennas placed in the form of a 2D antenna array.', '1601.00019-3-17-2': 'Standardization of the systems supporting up to 16 antennas is an initial target of Rel.', '1601.00019-3-17-3': '13 and issues to support more than 16 antennas will be discussed in subsequent releases.', '1601.00019-3-17-4': 'In the study item phase, there has been extensive discussion to support 2D array antennas, elaborated TXRUs, enhanced channel measurement and feedback schemes, and also an increased number of co-scheduled users (up to eight users).', '1601.00019-3-17-5': 'Among these, an item tightly coupled to the standardization is the CSI measurement and feedback mechanism.', '1601.00019-3-17-6': 'In this subsection, we discuss the deployment scenarios, antenna configurations, TXRU structure, new RS strategy, and feedback mechanisms.', '1601.00019-3-18-0': '## Deployment scenarios', '1601.00019-3-19-0': 'For the design and evaluation of FD-MIMO systems, a realistic scenario in which antenna array and UEs are located in different height is considered.', '1601.00019-3-19-1': 'To this end, two typical deployment scenarios, viz., 3D urban macro scenario (3D-UMa) and 3D urban micro (3D-UMi), are introduced (see Fig. [REF]).', '1601.00019-3-19-2': 'In the former case, transmit antennas are placed over the rooftop, and in the latter case, they are located below the rooftop.', '1601.00019-3-19-3': 'In case of 3D-UMa, diffraction over the rooftop is a dominant factor for the propagation so that down-tilted transmission in the vertical direction is desirable (see Fig. [REF](b)).', '1601.00019-3-19-4': 'In fact, by transmitting beams with different steering angles, eNB can separate channels corresponding to multiple UEs.', '1601.00019-3-19-5': 'In the 3D-UMi scenario, on the other hand, the location of users is higher than the height of the antenna so that direct signal path is dominant (see Fig. [REF](c)).', '1601.00019-3-19-6': 'In this scenario, both up and down-tilting can be used to schedule UEs in different floors.', '1601.00019-3-19-7': 'Since the cell radius of the 3D-UMi scenario is typically smaller than that of 3D-UMa, LOS channel condition is predominant, and thus more UEs can be co-scheduled without increasing the inter-user interference [CITATION].', '1601.00019-3-19-8': 'Although not as strong as the 3D-UMi scenario, LOS probability in the 3D-UMa scenario also increases when the distance between eNB and UE decreases.', '1601.00019-3-20-0': '## Antenna configurations', '1601.00019-3-21-0': 'Unlike the conventional MIMO systems relying on the passive antenna, systems based on the active antenna can dynamically control the gain of an antenna element by applying the weight of low-power amplifiers attached to each antenna element.', '1601.00019-3-21-1': 'Since the radiation pattern depends on the antenna arrangement, such as the number of the antenna elements and antenna spacing, the antenna system should be modeled in an element-level.', '1601.00019-3-21-2': 'As shown in Fig. [REF](a), there are three key parameters characterizing the antenna array structure [MATH]: the number of elements [MATH] in vertical direction, the number of elements [MATH] in horizontal direction, and the polarization degree [MATH] is for co-polarization and [MATH] is for dual-polarization).', '1601.00019-3-21-3': 'As a benchmark setting, 2D planar array using dual polarized antenna ([MATH]) configuration with [MATH] spacing in vertical direction) and [MATH] spacing in horizontal direction) is suggested.', '1601.00019-3-21-4': 'In this setting, null direction, an angle to make the magnitude of beam pattern to zero, for the elevation beam pattern is 11[MATH] and that for the horizontal beam pattern is 30[MATH] (see Fig. [REF](c)).', '1601.00019-3-21-5': 'Since the null direction in the vertical domain is much smaller than that of the horizontal domain, scheduling UEs in the vertical domain is more effective in controlling the inter-user interference.', '1601.00019-3-21-6': 'Also, a tall or fat array structure ([MATH] or [MATH]) is favorable since it will generate a sharp beam but it might be less flexible in the situation where the surrounding environment is changed.', '1601.00019-3-21-7': 'Further, large antenna spacing is not always a desirable option since it can increase the inter-cell interference due to the narrow beamforming for cell edge UEs (this phenomenon is called flash-light effect).', '1601.00019-3-21-8': 'For this reason, in a real deployment scenario, the design parameters should be carefully chosen by considering various factors, such as user location, cell radius, building height, and antenna height.', '1601.00019-3-22-0': '## TXRU architectures', '1601.00019-3-23-0': 'As mentioned, one interesting feature of the active antenna systems is that each TXRU contains PA and LNA so that eNB can control the gain and phase of an individual antenna element.', '1601.00019-3-23-1': 'In order to support this, a power feeding network between TXRUs and antenna elements called TXRU architecture is introduced [CITATION].', '1601.00019-3-23-2': 'TXRU architecture consists of three components: TXRU array, antenna array, and radio distribution networks (RDN).', '1601.00019-3-23-3': 'A role of the RDN is to deliver the transmit signal from PA to antenna array elements and the received signal from antenna array to LNA.', '1601.00019-3-23-4': 'Depending on the CSI-RS transmission and feedback strategy, two representative options, array partitioning and array connected architecture, are suggested.', '1601.00019-3-23-5': 'The former is for the conventional codebook scheme and the latter is for the beamforming scheme.', '1601.00019-3-24-0': 'In the array partitioning architecture, antenna elements are divided into multiple groups and each TXRU is connected to one of them (see Fig. [REF](d)).', '1601.00019-3-24-1': 'Whereas, in the array connected structure, RDN is designed such that RF signals of multiple TXRUs are delivered to the single antenna element.', '1601.00019-3-24-2': 'To mix RF signals from multiple TXRUs, additional RF combining circuitry is needed as shown in Fig. [REF](e).', '1601.00019-3-24-3': 'The difference between the two can be better understood when we discuss the transmission of the CSI-RS.', '1601.00019-3-24-4': 'In the array partitioning architecture, [MATH] antenna elements are partitioned into [MATH] groups of TXRU and orthogonal CSI-RS is assigned for each group.', '1601.00019-3-24-5': 'Each TXRU transmits its own CSI-RS so that the UE measures the channel [MATH] from the CSI-RS observation [MATH].', '1601.00019-3-24-6': 'In the array connected architecture, each antenna element is connected to [MATH] (out of [MATH]) TXRUs and orthogonal CSI-RS is assigned for each TXRU.', '1601.00019-3-24-7': 'Denoting [MATH] as the channel vector and [MATH] as the precoding weight ([MATH]) for each beamformed CSI-RS, the beamformed CSI-RS observation is [MATH] and the UE measures the precoded channel [MATH] from this.', '1601.00019-3-24-8': 'Due to the narrow and directional CSI-RS beam transmission with a linear array, SNR of the precoded channel is maximized at the target direction.', '1601.00019-3-25-0': '## New CSI-RS transmission strategy', '1601.00019-3-26-0': 'In the standardization process, two CSI-RS transmission strategies, i.e., extension of the conventional non-precoded CSI-RS and the beamformed CSI-RS, are suggested.', '1601.00019-3-26-1': 'In the first strategy, UE observes the non-precoded CSI-RS transmitted from each of partitioned antenna arrays (see Fig. [REF](d)).', '1601.00019-3-26-2': 'By sending the precoder maximizing the properly designed performance criterion to the eNB, UE can adapt to the channel variation.', '1601.00019-3-26-3': 'In the second strategy, eNB transmits multiple beamformed CSI-RS (we call it beam for simplicity) using connected arrays architecture.', '1601.00019-3-26-4': 'Among these, UE selects the preferred beam and then feeds back its index.', '1601.00019-3-26-5': 'When the eNB receives the beam index, the weight corresponding to the selected beam is used for the data transmission.', '1601.00019-3-27-0': 'Overall downlink precoder for data transmission [MATH] and CSI-RS transmission [MATH] can be expressed as [EQUATION] where [MATH] is the precoder between TXRU and the antenna element, [MATH] is the precoder between the CSI-RS port and the TXRU ([MATH] is the number of antenna ports), and [MATH] is the precoder between data channel to CSI-RS port.', '1601.00019-3-28-0': 'In the following, we summarize details of two strategies.', '1601.00019-3-29-0': 'In Table [REF], we summarize two CSI-RS transmission schemes discussed in the FD-MIMO.', '1601.00019-3-30-0': '## CSI feedback mechanisms for FD-MIMO systems', '1601.00019-3-31-0': 'In the study item phase, various RS transmission and feedback schemes have been proposed.', '1601.00019-3-31-1': 'As shown in Fig. [REF], capacity and overhead of class-A and class-B feedback schemes are more or less similar in the initial target range ([MATH]) so that Rel.', '1601.00019-3-31-2': '13 has decided to support both classes.', '1601.00019-3-31-3': 'In this subsection, we briefly describe the CSI feedback schemes associated with TXRU architectures.', '1601.00019-3-31-4': 'Among various schemes, composite codebook and beam index feedback have received much attention as main ingredients for class-A and class-B CSI feedback.', '1601.00019-3-31-5': 'The rest will be considered in a future release.', '1601.00019-3-32-0': 'Composite codebook: In this scheme, overall codebook is divided into two (vertical and horizontal codebooks) and thus the channel information is separately delivered to the eNB.', '1601.00019-3-32-1': 'By combining two codebooks (e.g., Kronecker product of two codebooks [MATH]), eNB reconstructs whole channel information.', '1601.00019-3-32-2': 'Considering that the angular spread of the vertical direction is smaller than that of the horizontal direction, one can reduce the feedback overhead by setting a relatively long reporting period to the vertical codebook.', '1601.00019-3-33-0': ""Beam index feedback: To obtain the UE's channel direction information (CDI) from beamformed CSI-RSs, eNB needs to transmit multiple beamformed CSI-RSs."", '1601.00019-3-33-1': 'When the channel rank is one, feedback of a beam index and corresponding CQI is enough.', '1601.00019-3-33-2': 'Whereas, when the channel rank is two with dual-polarized antennas, co-phase information is additionally required for adapting channel orthogonalization between layers.', '1601.00019-3-33-3': 'For example, once eNB obtains the CDI, this can be used for the beamforming vector of two-port CSI-RS and each CSI-RS port is mapped to the different polarized antennas.', '1601.00019-3-33-4': 'UE then estimates and feeds back short-term co-phase information between two ports.', '1601.00019-3-34-0': 'Other CSI feedback schemes: In the partial CSI-RS transmission, CSI-RS overhead can be reduced by partitioning the 2D antenna array into horizontal and vertical ports, say [MATH] ports in the row and [MATH] ports in the column.', '1601.00019-3-34-1': 'In doing so, the total number of CSI-RS can be reduced from [MATH] to [MATH].', '1601.00019-3-34-2': 'In the adaptive CSI feedback scheme, benefits of the beamformed and non-precoded CSI-RS transmission can be combined.', '1601.00019-3-34-3': 'First, in order to acquire long-term channel information, eNB transmits [MATH] non-precoded CSI-RSs.', '1601.00019-3-34-4': 'After receiving sufficient long-term channel statistics from UE, eNB determines spatial direction roughly and then transmits the beamformed CSI-RSs used for short-term and subband feedbacks.', '1601.00019-3-34-5': 'The flexible codebook scheme can support various 2D antenna layouts without increasing the number of codebooks.', '1601.00019-3-34-6': 'In this approach, one master codebook is designed for a large number of TXRUs, say 16 TXRUs, and the specific codebook (e.g., ([MATH]), ([MATH]), or ([MATH])) is derived based on this.', '1601.00019-3-34-7': 'To support this, the eNB needs to send the layout information via separate signaling.', '1601.00019-3-35-0': '# Performance of FD-MIMO System', '1601.00019-3-36-0': 'In order to observe the potential gain of the FD-MIMO systems, we perform system-level simulations under the realistic multicell environment.', '1601.00019-3-36-1': 'In our simulations, we test two typical deployment scenarios (3D-UMa and 3D-UMi) with 2-tier hexagonal layout.', '1601.00019-3-36-2': 'As a performance metric, we use spectral efficiency for cell average and cell edge.', '1601.00019-3-36-3': 'Detailed simulation parameters are provided in Table [REF].', '1601.00019-3-36-4': 'We first investigate the system performance of FD-MIMO systems with two types of antenna configurations.', '1601.00019-3-36-5': 'For type I and II configurations, [MATH] and [MATH] are used, respectively.', '1601.00019-3-36-6': 'In the type II configuration, antenna spacing is set to four times larger than the spacing of type I. To investigate the effect of antenna structure, the ideal feedback under the full buffer traffic model (each user has an unlimited amount of data to transmit) is used.', '1601.00019-3-36-7': 'In Fig. [REF](a), we plot the throughput of the conventional LTE systems with 8Tx ([MATH]) and FD-MIMO systems with 16, 32, and 64Tx ([MATH] and [MATH]), where [MATH] and [MATH] are the number of CSI-RS in vertical and horizontal dimensions, respectively.', '1601.00019-3-36-8': 'This result shows that both antenna configurations provide a large gain over the conventional 8Tx in LTE-A, resulting in [MATH] (type I) and [MATH] (type II) gain at cell edge, respectively.', '1601.00019-3-36-9': 'Due to the sufficient antenna spacing, cross-correlation between channels becomes negligible, and thus the spectral efficiency of type II increases linearly with the TXRU, resulting in [MATH] (cell average) and [MATH] gain (cell edge) when the number of TXRUs is doubled [CITATION].', '1601.00019-3-36-10': 'However, due to the insufficient antenna spacing, the spectral efficiency of type I configuration does not scale linearly with the number of TXRUs.', '1601.00019-3-37-0': 'We next investigate the system performance under the finite traffic model (e.g., FTP model) where each UE with distinct arrival time receives a file with finite size.', '1601.00019-3-37-1': 'As a performance metric, we use a user packet throughput, the number of successively received packets during the transmission period.', '1601.00019-3-37-2': 'In order to support the backward compatibility and also perform fair comparison among schemes under test, we employ the conventional MMSE-based channel estimation.', '1601.00019-3-37-3': 'In our simulations, the following CSI feedback strategies are considered.', '1601.00019-3-38-0': 'In Fig. [REF](c), we plot the user throughput of the finite traffic model as a function of packet arrival rate.', '1601.00019-3-38-1': 'Note that when the packet arrival rate is high, co-scheduled users need to be increased and thus the intercell and multiuser interference will also increase.', '1601.00019-3-38-2': 'In this realistic scenario, FD-MIMO systems outperform the conventional MIMO systems with a large margin, achieving 1.5[MATH] and 3[MATH] improvement in cell average and edge user packet throughput, respectively.', '1601.00019-3-38-3': 'Note that in the low network loading (low interference scenario), gain of the FD-MIMO systems is coming from the 3D beamforming.', '1601.00019-3-38-4': 'In the medium to high network loading (high interference scenario), this gain is mainly due to the multiuser precoding of the 2D active antenna array.', '1601.00019-3-38-5': 'Fig. [REF](d) summarizes the throughput of various CSI feedback frameworks.', '1601.00019-3-38-6': 'With the same feedback overhead (2bit), beamformed CSI-RS scheme I outperforms the non-precoded scheme with a large margin.', '1601.00019-3-38-7': 'This is because the number of codewords for the channel feedback is only four so that channel state information at eNB is very coarse.', '1601.00019-3-38-8': 'Since the beamformed CSI-RS scheme II can adapt weights of the beamformed CSI-RS to generate an accurate CDI, it performs best among all under tests.', '1601.00019-3-38-9': 'It is worth mentioning that the non-precoded CSI-RS scheme requires a large amount of feedback overhead (approximately 128 quantization levels) to achieve comparable performance to the CSI-RS scheme I. From this observation, we clearly see that the beamformed CSI-RS transmission is effective in controlling the precoding weights (in time, frequency, and space), feedback overhead, and pilot resource overhead.', '1601.00019-3-39-0': '# Concluding Remarks', '1601.00019-3-40-0': 'In this article, we have provided an overview of FD-MIMO systems in 3GPP LTE (recently named as LTE-Advanced Pro) with emphasis on the discussion and debate conducted on the Rel.', '1601.00019-3-40-1': '13 phase.', '1601.00019-3-40-2': 'We discussed key features of FD-MIMO systems and main issues in standardization of system design, such as channel model, transceiver architectures, pilot transmission, and CSI feedback scheme.', '1601.00019-3-40-3': 'To make the most of a large number of eNB antennas in a cost and space effective manner, new key features, distinct from MIMO systems in conventional LTE-A, should be introduced in the standardization, system design, and transceiver implementation.', '1601.00019-3-40-4': 'These include new transmitter architecture (array connected architecture), new RS transmission scheme (beamformed CSI-RS transmissions), and enhanced channel feedback (beam index feedback).', '1601.00019-3-40-5': 'Although our work focused primarily on the standardization in Rel.', '1601.00019-3-40-6': '13, there are still many issues for the successful deployment of FD-MIMO systems in the future, including pilot transmission and advanced channel estimation exploiting channel sparsity, beam adaptation and optimization, and RS overhead reduction for throughput enhancement.'}","{'1601.00019-4-0-0': 'Multiple-input multiple-output (MIMO) systems with a large number of basestation antennas, often called massive MIMO, have received much attention in academia and industry as a means to improve the spectral efficiency, energy efficiency, and processing complexity of next generation cellular system.', '1601.00019-4-0-1': 'Mobile communication industry has initiated a feasibility study of massive MIMO systems to meet the increasing demand of future wireless systems.', '1601.00019-4-0-2': 'Field trials of the proof-of-concept systems have demonstrated the potential gain of the Full-Dimension MIMO (FD-MIMO), an official name for the MIMO enhancement in 3rd generation partnership project (3GPP).', '1601.00019-4-0-3': '3GPP initiated standardization activity for the seamless integration of this technology into current 4G LTE systems.', '1601.00019-4-0-4': 'In this article, we provide an overview of the FD-MIMO system, with emphasis on the discussion and debate conducted on the standardization process of Release 13.', '1601.00019-4-0-5': 'We present key features for FD-MIMO systems, a summary of the major issues for the standardization and practical system design, and performance evaluations for typical FD-MIMO scenarios.', '1601.00019-4-1-0': '# Introduction', '1601.00019-4-2-0': 'Multiple-input multiple-output (MIMO) systems with a large number of basestation antennas, often referred to as massive MIMO systems, have received much attention in academia and industry as a means to improve the spectral efficiency, energy efficiency, and processing complexity [CITATION].', '1601.00019-4-2-1': 'While the massive MIMO technology is a promising technology, there are many practical challenges and technical hurdles down the road to the successful commercialization.', '1601.00019-4-2-2': 'These include design of low-cost and low-power basestation with acceptable antenna space, improvement in the fronthaul capacity between radio and control units, acquisition of high dimensional channel state information (CSI), and many others.', '1601.00019-4-2-3': 'Recently, 3rd generation partnership project (3GPP) standard body initiated the standardization activity to employ tens of antennas at basestation with an aim to satisfy the spectral efficiency requirement of future cellular systems [CITATION].', '1601.00019-4-2-4': 'Considering the implementation cost and complexity, and also the timeline to the real deployment, 3GPP decided to use tens of antennas with a two dimensional (2D) array structure as a starting point.', '1601.00019-4-2-5': 'Full-Dimension MIMO (FD-MIMO), the official name for the MIMO enhancement in 3GPP, targets the system utilizing up to 64 antenna ports at the transmitter side.', '1601.00019-4-2-6': 'Recently, field trials of the proof-of-concept FD-MIMO systems have been conducted successfully[CITATION].', '1601.00019-4-2-7': 'A study item, a process done before a formal standardization process, has been completed in June 2015, and the follow-up work item process will be finalized soon for the formal standardization of Release 13 (Rel.', '1601.00019-4-2-8': '13).', '1601.00019-4-3-0': 'The purpose of this article is to provide an overview of the FD-MIMO systems with an emphasis on the discussion and debate conducted on the standardization process of Rel.', '1601.00019-4-3-1': '13.', '1601.00019-4-3-2': 'We note that preliminary studies addressed the feasibility of 2D array antenna structure and performance evaluation in ideal pilot transmission and feedback scenarios [CITATION].', '1601.00019-4-3-3': 'This work is distinct from these in the sense that we put our emphasis on describing realistic issues in the standardization process, including TXRU architectures, beamformed CSI-RS, 3D beamforming, details of CSI feedback, and performance evaluation in realistic FD-MIMO scenarios with new feedback schemes.', '1601.00019-4-4-0': '# Key Features of FD-MIMO Systems', '1601.00019-4-5-0': 'In this section, we discuss key features of FD-MIMO systems.', '1601.00019-4-5-1': 'These include a large number of basestation antennas, 2D active antenna array, 3D channel propagation, and new pilot transmission with CSI feedback.', '1601.00019-4-5-2': 'In what follows, we will use LTE terminology exclusively: enhanced node-B (eNB) for basestation, user equipment (UE) for the mobile terminal, and reference signal (RS) for pilot signal.', '1601.00019-4-6-0': '## Increase the number of transmit antennas', '1601.00019-4-7-0': 'One of the main features of FD-MIMO systems distinct from the MIMO systems of the current LTE and LTE-Advanced standards is to use a large number of antennas at eNB.', '1601.00019-4-7-1': 'In theory, as the number of eNB antennas [MATH] increases, cross-correlation of two random channel realizations goes to zero [CITATION] so that the inter-user interference in the downlink can be controlled via a simple linear precoder.', '1601.00019-4-7-2': 'Such benefit, however, can be realized only when the perfect CSI is available at the eNB.', '1601.00019-4-7-3': 'While the CSI acquisition in time division duplex (TDD) systems is relatively simple due to the channel reciprocity, such is not the case for frequency division duplex (FDD) systems.', '1601.00019-4-7-4': 'Note that in the FDD systems, time variation and frequency response of the channel are measured via the downlink RSs and then sent back to the eNB after the quantization.', '1601.00019-4-7-5': 'Even in TDD mode, one cannot solely rely on the channel reciprocity because the measurement at the transmitter does not capture the downlink interference from neighboring cells or co-scheduled UEs.', '1601.00019-4-7-6': 'As such, downlink RSs are still required to capture the channel quality indicator (CQI) for the TDD mode, and thus the downlink RS and the uplink CSI feedback are essential for both duplex modes.', '1601.00019-4-7-7': 'Identifying the potential issues of CSI acquisition and developing the proper solutions are, therefore, of great importance for the successful commercialization of FD-MIMO systems.', '1601.00019-4-7-8': 'Before we go into detail, we briefly summarize two major problems related to the CSI acquisition.', '1601.00019-4-8-0': '## 2D active antenna system (AAS)', '1601.00019-4-9-0': 'Another interesting feature of the FD-MIMO system is an introduction of the active antenna with 2D planar array.', '1601.00019-4-9-1': 'In the active antenna-based systems, gain and phase are controlled by the active components, such as power amplifier (PA) and low noise amplifier (LNA), attached to each antenna element.', '1601.00019-4-9-2': 'In the 2D structured antenna array, one can control the radio wave on both vertical (elevation) and horizontal (azimuth) direction so that the control of the transmit beam in 3D space is possible.', '1601.00019-4-9-3': 'This type of wave control mechanism is also referred to as the 3D beamforming.', '1601.00019-4-9-4': 'Another important benefit of 2D AAS is that it can accommodate a large number of antennas without increasing the deployment space.', '1601.00019-4-9-5': 'For example, when 64 linear antenna arrays are deployed in a horizontal direction, under the common assumption that the antenna spacing is half wavelength ([MATH]) and the system is using LTE carrier frequency (2 GHz), it requires a horizontal room of [MATH]m. Due to the limited space on a rooftop or mast, this space would be burdensome for most of the cell sites.', '1601.00019-4-9-6': 'In contrast, when antennas are arranged in a square array, relatively small space is required for 2D antenna array (e.g., 1.0 [MATH] 0.5m with dual-polarized 8 [MATH] 8 antenna array).', '1601.00019-4-10-0': '## 3D channel environment', '1601.00019-4-11-0': 'When basic features of the FD-MIMO systems are determined, the next step is to design a system maximizing performance in terms of throughput, spectral efficiency, and peak data rate in the realistic channel environment.', '1601.00019-4-11-1': 'There are various issues to consider in the design of practical systems, such as investigation and characterization of the realistic channel model for the performance evaluation.', '1601.00019-4-11-2': 'While the conventional MIMO systems consider the propagation in the horizontal direction only, FD-MIMO systems employing 2D planar array should consider the propagation in both vertical and horizontal direction.', '1601.00019-4-11-3': 'To do so, geometric structure of the transmitter antenna array and propagation effect of the 3D positions between the eNB and UE should be reflected in the channel model.', '1601.00019-4-11-4': 'Main features of 3D channel propagation obtained from real measurement are as follows [CITATION]:', '1601.00019-4-12-0': '## RS transmission for CSI acquisition', '1601.00019-4-13-0': 'From the LTE to LTE-Advanced, there has been substantial improvement in the RS scheme for MIMO systems (see Fig. 1(a)).', '1601.00019-4-13-1': 'From the common RS (CRS) to the channel state information RS (CSI-RS), various RSs to perform the CSI acquisition have been introduced.', '1601.00019-4-13-2': 'While these are common to all users in a cell and thus un-precoded, the demodulation RS (DM-RS) is UE-specific (i.e., dedicated to each UE) so that it is precoded by the same weight applied for the data transmission.', '1601.00019-4-13-3': 'Since the DM-RS is present only on time/frequency resources where the UE is scheduled, this cannot be used for CSI measurements [CITATION].', '1601.00019-4-14-0': 'One of the new features of the FD-MIMO systems is to use a beamformed RS, called beamformed CSI-RS, for the CSI acquisition.', '1601.00019-4-14-1': 'Beamformed RS transmission is a channel training technique that uses multiple precoding weights in spatial domain.', '1601.00019-4-14-2': 'In this scheme, UE picks the best weight among transmitted and then feeds back its index.', '1601.00019-4-14-3': 'This scheme provides many benefits over non-precoded CSI-RS, in particular when [MATH] is large.', '1601.00019-4-14-4': 'Some of the benefits are summarized as follows:', '1601.00019-4-15-0': 'In order to support the beamformed CSI-RS scheme, new transmitter architecture called transceiver unit (TXRU) architecture has been introduced.', '1601.00019-4-15-1': 'By TXRU architecture, we mean a hardware connection between the baseband signal path and antenna array elements.', '1601.00019-4-15-2': 'Since this architecture facilitates the control of phase and gain in both digital and analog domain, more accurate control of the beamforming direction is possible.', '1601.00019-4-15-3': 'One thing to note is that the conventional codebook cannot measure the CSI of the beamformed transmission so that a new channel feedback mechanism supporting the beamformed transmission is required (see Section III.D for details).', '1601.00019-4-16-0': '# System Design and Standardization of FD-MIMO Systems', '1601.00019-4-17-0': 'The main purpose of the Rel.', '1601.00019-4-17-1': '13 study item is to identify key issues to support up to 64 transmit antennas placed in the form of a 2D antenna array.', '1601.00019-4-17-2': 'Standardization of the systems supporting up to 16 antennas is an initial target of Rel.', '1601.00019-4-17-3': '13 and issues to support more than 16 antennas will be discussed in subsequent releases.', '1601.00019-4-17-4': 'In the study item phase, there has been extensive discussion to support 2D array antennas, elaborated TXRUs, enhanced channel measurement and feedback schemes, and also an increased number of co-scheduled users (up to eight users).', '1601.00019-4-17-5': 'Among these, an item tightly coupled to the standardization is the CSI measurement and feedback mechanism.', '1601.00019-4-17-6': 'In this subsection, we discuss the deployment scenarios, antenna configurations, TXRU structure, new RS strategy, and feedback mechanisms.', '1601.00019-4-18-0': '## Deployment scenarios', '1601.00019-4-19-0': 'For the design and evaluation of FD-MIMO systems, a realistic scenario in which antenna array and UEs are located in different height is considered.', '1601.00019-4-19-1': 'To this end, two typical deployment scenarios, viz., 3D urban macro scenario (3D-UMa) and 3D urban micro (3D-UMi), are introduced (see Fig. [REF]).', '1601.00019-4-19-2': 'In the former case, transmit antennas are placed over the rooftop, and in the latter case, they are located below the rooftop.', '1601.00019-4-19-3': 'In case of 3D-UMa, diffraction over the rooftop is a dominant factor for the propagation so that down-tilted transmission in the vertical direction is desirable (see Fig. [REF](b)).', '1601.00019-4-19-4': 'In fact, by transmitting beams with different steering angles, eNB can separate channels corresponding to multiple UEs.', '1601.00019-4-19-5': 'In the 3D-UMi scenario, on the other hand, the location of users is higher than the height of the antenna so that direct signal path is dominant (see Fig. [REF](c)).', '1601.00019-4-19-6': 'In this scenario, both up and down-tilting can be used to schedule UEs in different floors.', '1601.00019-4-19-7': 'Since the cell radius of the 3D-UMi scenario is typically smaller than that of 3D-UMa, LOS channel condition is predominant, and thus more UEs can be co-scheduled without increasing the inter-user interference [CITATION].', '1601.00019-4-19-8': 'Although not as strong as the 3D-UMi scenario, LOS probability in the 3D-UMa scenario also increases when the distance between eNB and UE decreases.', '1601.00019-4-20-0': '## Antenna configurations', '1601.00019-4-21-0': 'Unlike the conventional MIMO systems relying on the passive antenna, systems based on the active antenna can dynamically control the gain of an antenna element by applying the weight of low-power amplifiers attached to each antenna element.', '1601.00019-4-21-1': 'Since the radiation pattern depends on the antenna arrangement, such as the number of the antenna elements and antenna spacing, the antenna system should be modeled in an element-level.', '1601.00019-4-21-2': 'As shown in Fig. [REF](a), there are three key parameters characterizing the antenna array structure [MATH]: the number of elements [MATH] in vertical direction, the number of elements [MATH] in horizontal direction, and the polarization degree [MATH] is for co-polarization and [MATH] is for dual-polarization).', '1601.00019-4-21-3': 'As a benchmark setting, 2D planar array using dual polarized antenna ([MATH]) configuration with [MATH] spacing in vertical direction) and [MATH] spacing in horizontal direction) is suggested.', '1601.00019-4-21-4': 'In this setting, null direction, an angle to make the magnitude of beam pattern to zero, for the elevation beam pattern is 11[MATH] and that for the horizontal beam pattern is 30[MATH] (see Fig. [REF](c)).', '1601.00019-4-21-5': 'Since the null direction in the vertical domain is much smaller than that of the horizontal domain, scheduling UEs in the vertical domain is more effective in controlling the inter-user interference.', '1601.00019-4-21-6': 'Also, a tall or fat array structure ([MATH] or [MATH]) is favorable since it will generate a sharp beam but it might be less flexible in the situation where the surrounding environment is changed.', '1601.00019-4-21-7': 'Further, large antenna spacing is not always a desirable option since it can increase the inter-cell interference due to the narrow beamforming for cell edge UEs (this phenomenon is called flash-light effect).', '1601.00019-4-21-8': 'For this reason, in a real deployment scenario, the design parameters should be carefully chosen by considering various factors, such as user location, cell radius, building height, and antenna height.', '1601.00019-4-22-0': '## TXRU architectures', '1601.00019-4-23-0': 'As mentioned, one interesting feature of the active antenna systems is that each TXRU contains PA and LNA so that eNB can control the gain and phase of an individual antenna element.', '1601.00019-4-23-1': 'In order to support this, a power feeding network between TXRUs and antenna elements called TXRU architecture is introduced [CITATION].', '1601.00019-4-23-2': 'TXRU architecture consists of three components: TXRU array, antenna array, and radio distribution networks (RDN).', '1601.00019-4-23-3': 'A role of the RDN is to deliver the transmit signal from PA to antenna array elements and the received signal from antenna array to LNA.', '1601.00019-4-23-4': 'Depending on the CSI-RS transmission and feedback strategy, two representative options, array partitioning and array connected architecture, are suggested.', '1601.00019-4-23-5': 'The former is for the conventional codebook scheme and the latter is for the beamforming scheme.', '1601.00019-4-24-0': 'In the array partitioning architecture, antenna elements are divided into multiple groups and each TXRU is connected to one of them (see Fig. [REF](d)).', '1601.00019-4-24-1': 'Whereas, in the array connected structure, RDN is designed such that RF signals of multiple TXRUs are delivered to the single antenna element.', '1601.00019-4-24-2': 'To mix RF signals from multiple TXRUs, additional RF combining circuitry is needed as shown in Fig. [REF](e).', '1601.00019-4-24-3': 'The difference between the two can be better understood when we discuss the transmission of the CSI-RS.', '1601.00019-4-24-4': 'In the array partitioning architecture, [MATH] antenna elements are partitioned into [MATH] groups of TXRU and orthogonal CSI-RS is assigned for each group.', '1601.00019-4-24-5': 'Each TXRU transmits its own CSI-RS so that the UE measures the channel [MATH] from the CSI-RS observation [MATH].', '1601.00019-4-24-6': 'In the array connected architecture, each antenna element is connected to [MATH] (out of [MATH]) TXRUs and orthogonal CSI-RS is assigned for each TXRU.', '1601.00019-4-24-7': 'Denoting [MATH] as the channel vector and [MATH] as the precoding weight ([MATH]) for each beamformed CSI-RS, the beamformed CSI-RS observation is [MATH] and the UE measures the precoded channel [MATH] from this.', '1601.00019-4-24-8': 'Due to the narrow and directional CSI-RS beam transmission with a linear array, SNR of the precoded channel is maximized at the target direction.', '1601.00019-4-25-0': '## New CSI-RS transmission strategy', '1601.00019-4-26-0': 'In the standardization process, two CSI-RS transmission strategies, i.e., extension of the conventional non-precoded CSI-RS and the beamformed CSI-RS, are suggested.', '1601.00019-4-26-1': 'In the first strategy, UE observes the non-precoded CSI-RS transmitted from each of partitioned antenna arrays (see Fig. [REF](d)).', '1601.00019-4-26-2': 'By sending the precoder maximizing the properly designed performance criterion to the eNB, UE can adapt to the channel variation.', '1601.00019-4-26-3': 'In the second strategy, eNB transmits multiple beamformed CSI-RS (we call it beam for simplicity) using connected arrays architecture.', '1601.00019-4-26-4': 'Among these, UE selects the preferred beam and then feeds back its index.', '1601.00019-4-26-5': 'When the eNB receives the beam index, the weight corresponding to the selected beam is used for the data transmission.', '1601.00019-4-27-0': 'Overall downlink precoder for data transmission [MATH] and CSI-RS transmission [MATH] can be expressed as [EQUATION] where [MATH] is the precoder between TXRU and the antenna element, [MATH] is the precoder between the CSI-RS port and the TXRU ([MATH] is the number of antenna ports), and [MATH] is the precoder between data channel to CSI-RS port.', '1601.00019-4-28-0': 'In the following, we summarize details of two strategies.', '1601.00019-4-29-0': 'In Table [REF], we summarize two CSI-RS transmission schemes discussed in the FD-MIMO.', '1601.00019-4-30-0': '## CSI feedback mechanisms for FD-MIMO systems', '1601.00019-4-31-0': 'In the study item phase, various RS transmission and feedback schemes have been proposed.', '1601.00019-4-31-1': 'As shown in Fig. [REF], capacity and overhead of class-A and class-B feedback schemes are more or less similar in the initial target range ([MATH]) so that Rel.', '1601.00019-4-31-2': '13 has decided to support both classes.', '1601.00019-4-31-3': 'In this subsection, we briefly describe the CSI feedback schemes associated with TXRU architectures.', '1601.00019-4-31-4': 'Among various schemes, composite codebook and beam index feedback have received much attention as main ingredients for class-A and class-B CSI feedback.', '1601.00019-4-31-5': 'The rest will be considered in a future release.', '1601.00019-4-32-0': 'Composite codebook: In this scheme, overall codebook is divided into two (vertical and horizontal codebooks) and thus the channel information is separately delivered to the eNB.', '1601.00019-4-32-1': 'By combining two codebooks (e.g., Kronecker product of two codebooks [MATH]), eNB reconstructs whole channel information.', '1601.00019-4-32-2': 'Considering that the angular spread of the vertical direction is smaller than that of the horizontal direction, one can reduce the feedback overhead by setting a relatively long reporting period to the vertical codebook.', '1601.00019-4-33-0': ""Beam index feedback: To obtain the UE's channel direction information (CDI) from beamformed CSI-RSs, eNB needs to transmit multiple beamformed CSI-RSs."", '1601.00019-4-33-1': 'When the channel rank is one, feedback of a beam index and corresponding CQI is enough.', '1601.00019-4-33-2': 'Whereas, when the channel rank is two with dual-polarized antennas, co-phase information is additionally required for adapting channel orthogonalization between layers.', '1601.00019-4-33-3': 'For example, once eNB obtains the CDI, this can be used for the beamforming vector of two-port CSI-RS and each CSI-RS port is mapped to the different polarized antennas.', '1601.00019-4-33-4': 'UE then estimates and feeds back short-term co-phase information between two ports.', '1601.00019-4-34-0': 'Other CSI feedback schemes: In the partial CSI-RS transmission, CSI-RS overhead can be reduced by partitioning the 2D antenna array into horizontal and vertical ports, say [MATH] ports in the row and [MATH] ports in the column.', '1601.00019-4-34-1': 'In doing so, the total number of CSI-RS can be reduced from [MATH] to [MATH].', '1601.00019-4-34-2': 'Overall channel information can be reconstructed by exploiting spatial and temporal correlation among antenna elements [CITATION].', '1601.00019-4-34-3': 'In the adaptive CSI feedback scheme, benefits of the beamformed and non-precoded CSI-RS transmission can be combined.', '1601.00019-4-34-4': 'First, in order to acquire long-term channel information, eNB transmits [MATH] non-precoded CSI-RSs.', '1601.00019-4-34-5': 'After receiving sufficient long-term channel statistics from UE, eNB determines spatial direction roughly and then transmits the beamformed CSI-RSs used for short-term and subband feedbacks.', '1601.00019-4-34-6': 'The flexible codebook scheme can support various 2D antenna layouts without increasing the number of codebooks.', '1601.00019-4-34-7': 'In this approach, one master codebook is designed for a large number of TXRUs, say 16 TXRUs, and the specific codebook (e.g., ([MATH]), ([MATH]), or ([MATH])) is derived based on this.', '1601.00019-4-34-8': 'To support this, the eNB needs to send the layout information via separate signaling.', '1601.00019-4-35-0': '# Performance of FD-MIMO System', '1601.00019-4-36-0': 'In order to observe the potential gain of the FD-MIMO systems, we perform system-level simulations under the realistic multicell environment.', '1601.00019-4-36-1': 'In our simulations, we test two typical deployment scenarios (3D-UMa and 3D-UMi) with 2-tier hexagonal layout.', '1601.00019-4-36-2': 'As a performance metric, we use spectral efficiency for cell average and cell edge.', '1601.00019-4-36-3': 'Detailed simulation parameters are provided in Table [REF].', '1601.00019-4-36-4': 'We first investigate the system performance of FD-MIMO systems with two types of antenna configurations.', '1601.00019-4-36-5': 'For type I and II configurations, [MATH] and [MATH] are used, respectively.', '1601.00019-4-36-6': 'In the type II configuration, antenna spacing is set to four times larger than the spacing of type I. To investigate the effect of antenna structure, the ideal feedback under the full buffer traffic model (each user has an unlimited amount of data to transmit) is used.', '1601.00019-4-36-7': 'In Fig. [REF](a), we plot the throughput of the conventional LTE systems with 8Tx ([MATH]) and FD-MIMO systems with 16, 32, and 64Tx ([MATH] and [MATH]), where [MATH] and [MATH] are the number of CSI-RS in vertical and horizontal dimensions, respectively.', '1601.00019-4-36-8': 'This result shows that both antenna configurations provide a large gain over the conventional 8Tx in LTE-A, resulting in [MATH] (type I) and [MATH] (type II) gain at cell edge, respectively.', '1601.00019-4-36-9': 'Due to the sufficient antenna spacing, cross-correlation between channels becomes negligible, and thus the spectral efficiency of type II increases linearly with the TXRU, resulting in [MATH] (cell average) and [MATH] gain (cell edge) when the number of TXRUs is doubled [CITATION].', '1601.00019-4-36-10': 'However, due to the insufficient antenna spacing, the spectral efficiency of type I configuration does not scale linearly with the number of TXRUs.', '1601.00019-4-37-0': 'We next investigate the system performance under the finite traffic model (e.g., FTP model) where each UE with distinct arrival time receives a file with finite size.', '1601.00019-4-37-1': 'As a performance metric, we use a user packet throughput, the number of successively received packets during the transmission period.', '1601.00019-4-37-2': 'In order to support the backward compatibility and also perform fair comparison among schemes under test, we employ the conventional MMSE-based channel estimation.', '1601.00019-4-37-3': 'In our simulations, the following CSI feedback strategies are considered.', '1601.00019-4-38-0': 'In Fig. [REF](c), we plot the user throughput of the finite traffic model as a function of packet arrival rate.', '1601.00019-4-38-1': 'Note that when the packet arrival rate is high, co-scheduled users need to be increased and thus the intercell and multiuser interference will also increase.', '1601.00019-4-38-2': 'In this realistic scenario, FD-MIMO systems outperform the conventional MIMO systems with a large margin, achieving 1.5[MATH] and 3[MATH] improvement in cell average and edge user packet throughput, respectively.', '1601.00019-4-38-3': 'Note that in the low network loading (low interference scenario), gain of the FD-MIMO systems is coming from the 3D beamforming.', '1601.00019-4-38-4': 'In the medium to high network loading (high interference scenario), this gain is mainly due to the multiuser precoding of the 2D active antenna array.', '1601.00019-4-38-5': 'Fig. [REF](d) summarizes the throughput of various CSI feedback frameworks.', '1601.00019-4-38-6': 'With the same feedback overhead (2bit), beamformed CSI-RS scheme I outperforms the non-precoded scheme with a large margin.', '1601.00019-4-38-7': 'This is because the number of codewords for the channel feedback is only four so that channel state information at eNB is very coarse.', '1601.00019-4-38-8': 'Since the beamformed CSI-RS scheme II can adapt weights of the beamformed CSI-RS to generate an accurate CDI, it performs best among all under tests.', '1601.00019-4-38-9': 'It is worth mentioning that the non-precoded CSI-RS scheme requires a large amount of feedback overhead (approximately 128 quantization levels) to achieve comparable performance to the CSI-RS scheme I. From this observation, we clearly see that the beamformed CSI-RS transmission is effective in controlling the precoding weights (in time, frequency, and space), feedback overhead, and pilot resource overhead.', '1601.00019-4-39-0': '# Concluding Remarks', '1601.00019-4-40-0': 'In this article, we have provided an overview of FD-MIMO systems in 3GPP LTE (recently named as LTE-Advanced Pro) with emphasis on the discussion and debate conducted on the Rel.', '1601.00019-4-40-1': '13 phase.', '1601.00019-4-40-2': 'We discussed key features of FD-MIMO systems and main issues in standardization of system design, such as channel model, transceiver architectures, pilot transmission, and CSI feedback scheme.', '1601.00019-4-40-3': 'To make the most of a large number of eNB antennas in a cost and space effective manner, new key features, distinct from MIMO systems in conventional LTE-A, should be introduced in the standardization, system design, and transceiver implementation.', '1601.00019-4-40-4': 'These include new transmitter architecture (array connected architecture), new RS transmission scheme (beamformed CSI-RS transmissions), and enhanced channel feedback (beam index feedback).', '1601.00019-4-40-5': 'Although our work focused primarily on the standardization in Rel.', '1601.00019-4-40-6': '13, there are still many issues for the successful deployment of FD-MIMO systems in the future, including pilot overhead reduction, beam adaptation and optimization, and advanced channel estimation exploiting time and angular domain sparsity [CITATION].'}",
0907.1345,"{'0907.1345-1-0-0': 'We show using a ""Thomas-Fermi"" variational wave function with a distribution function in momentum space, that a homogeneous two-component Fermi gas with dipolar interactions has a ferro-nematic ground state for non-zero (arbitrarily small) values of the dipolar coupling constant.', '0907.1345-1-0-1': 'The ferro-nematic phase is characterized by having a non-zero magnetization and long range orientational uniaxial order.', '0907.1345-1-0-2': 'The Fermi surface of spin up (down) component is elongated (compressed) along the direction of the magnetization.', '0907.1345-1-0-3': 'We calculate several thermodynamic quantities and discuss the prospects of experimental realizations.', '0907.1345-1-1-0': 'Cold dipolar Fermi gases have attracted much attention due to the novel anisotropic and long-range character of its interaction as compared with the usual short-range isotropic Lenard-Jones- type interaction.', '0907.1345-1-1-1': 'Recent studies of many-body effects were predicted to lead to an elongated Fermi surface (FS) in a one-component fully polarized Fermi gas with dipolar interactions along the polarization direction established by an external field[CITATION].', '0907.1345-1-1-2': 'A biaxial state with a critical value of the effective coupling constant [MATH] was proposed in Ref. [CITATION] ([MATH] is the dipole moment of the fermion, [MATH] is the total density and [MATH] is the Fermi energy of the free Fermi gas at the same density).', '0907.1345-1-1-3': 'One important result of Ref. [CITATION] is the system will exhibit violations of the Landau theory of the Fermi liquid both at quantum criticality and in the biaxial phase.', '0907.1345-1-1-4': 'More generally, understanding anisotropic non-Fermi liquid phases of cold atomic systems may shade light into the quantum liquid crystal phases in strongly correlated systems and high [MATH] superconductors [CITATION].', '0907.1345-1-2-0': 'The important question we want to address here is whether a cold Fermi gas of interacting dipoles become spontaneously magnetized.', '0907.1345-1-2-1': 'The classical version of this problem has been considered in mixtures of ferromagnetic particles with nematic liquid crystals[CITATION], and in dipolar colloidal fluids and ferrofluids[CITATION].', '0907.1345-1-2-2': 'Classical dipolar fluids have complex phase diagrams, typically featuring inhomogeneous phases with complex spacial structures.', '0907.1345-1-2-3': 'Much less is known about their quantum counterparts.', '0907.1345-1-2-4': 'In the case of simple quantum fluids, such as [MATH]He, the dipolar interaction plays a very small role compared to the short-range exchange interaction[CITATION].', '0907.1345-1-2-5': 'In the context of ultracold gases, a number of atomic and molecular systems with strong dipolar interactions, such as dysprosium, have been the focus of recent experiments.', '0907.1345-1-3-0': 'For simplicity in this paper we will consider a dipolar Fermi system with two spin (or hyperfine) states per particle.', '0907.1345-1-3-1': 'We use a variational wave function approach to show that a spin-[MATH] cold and dilute dipolar Fermi gas has a ferromagnetic ground state.', '0907.1345-1-3-2': 'In addition we show that, due to the anisotropic nature of the dipolar interaction, the ferromagnetic state also breaks spontaneously the symmetry of rotations in real space.', '0907.1345-1-3-3': 'Thus, this state should be regarded as a ferro-nematic state.', '0907.1345-1-3-4': '[CITATION] In addition of computing the magnetization and the nematic order parameter, we compute the related distortion of the Fermi surface (FS) as well as the pressure, compressibility and chemical potential for this gas.', '0907.1345-1-3-5': 'We also discuss the experimental prospects of observing ferro-nematicity in cold atomic systems.', '0907.1345-1-4-0': 'Consider a restricted Hilbert space of two hyperfine states, called [MATH] (""spin up"") and [MATH] (""spin down""), of a point-like magnetic atom at zero temperature.', '0907.1345-1-4-1': 'The particles have mass [MATH] and magnetic moment [MATH], where [MATH] and where [MATH] are the usual spin-[MATH] matrices (the factor of [MATH] which is absorbed into the definition of [MATH]).', '0907.1345-1-4-2': 'The Hamiltonian is [EQUATION] where the fields [MATH] destroy fermions on spin state with [MATH]-component [MATH] at position [MATH].', '0907.1345-1-4-3': 'The dipolar interaction is [EQUATION] where [MATH] is the relative position vector of the interacting particles and [MATH] is a unit vector in the direction of [MATH].', '0907.1345-1-4-4': 'The Fourier transform of Eq.([REF]) is [MATH].', '0907.1345-1-4-5': 'The Hamiltonian of Eq.[REF] and Eq.[REF] is invariant under simultaneous [MATH] transformations in spin space and [MATH] rotations in real space.', '0907.1345-1-4-6': 'We are neglecting spin-orbit interactions as well as short range isotropic interactions.', '0907.1345-1-4-7': 'The dipolar interaction has the standard contact term.', '0907.1345-1-4-8': 'We have not included the effects of short range isotropic interactions.', '0907.1345-1-5-0': 'The ferro-nematic state breaks simultaneously the rotational invariance in spin space and in real space, and its order parameters reflect this pattern of symmetry breaking.', '0907.1345-1-5-1': 'The order parameters are: a) the local magnetization vector [MATH]) that measures the spin polarization, b) the nematic order parameter, a [MATH] symmetric traceless matrix, [MATH]) that measures the breaking of rotational invariance in space, and c) the generalized ""nematic-spin-nematic"" order parameter [MATH], a tensor symmetric and traceless on the spatial ([MATH]) and spin ([MATH]) components, that measures the breaking of both symmetries [CITATION].', '0907.1345-1-5-2': 'General order parameters of the latter (nematic-spin-nematic) type were considered by Wu et al[CITATION] who gave a detailed description of their phases in 2D systems and partially in 3D systems.', '0907.1345-1-5-3': 'The ferro-nematic state has an unbroken uniaxial symmetry in real space.', '0907.1345-1-5-4': 'It is is a 3D generalization of the quadrupolar [MATH] phase described in Ref. [CITATION].', '0907.1345-1-5-5': 'Phases with more general broken symmetries will be considered elsewhere[CITATION].', '0907.1345-1-6-0': 'An intuitive way to describe these phases in a Fermi system is in terms of spontaneously deformed Fermi surfaces (FS)[CITATION].', '0907.1345-1-6-1': 'A ferromagnetic state is isotropic in real space and has a spherical FS of unequal size (and same shape) for both spin polarizations.', '0907.1345-1-6-2': 'The nematic phase is isotropic in spin space, and its FS has a uniaxial distortion in real space, with the up and down spin FS being identical in shape and size.', '0907.1345-1-6-3': 'For small values of the order parameter, the distorted FS are ellipsoids with an eccentricity determined by the magnitude of the order parameter.', '0907.1345-1-6-4': 'Due to the anisotropy of the dipolar interaction, a ferromagnetic state causes the FS to distort, thus driving the system into an uniaxial nematic state.', '0907.1345-1-6-5': 'Since the state is ferromagnetic, the up (down) FS is a prolate (oblate) revolution ellipsoid.', '0907.1345-1-6-6': 'Both FS are collinear, and have unequal distortion and volume.', '0907.1345-1-6-7': 'Hence, the nematic-spin-nematic order parameter [MATH] has a finite value.', '0907.1345-1-6-8': 'We find that all three orders are present even for arbitrarily small values of the dipolar coupling, where the ferromagnetic order is the strongest, the nematic intermediate, and the nematic-spin-nematic the weakest.', '0907.1345-1-7-0': 'We make a variational ansatz of the wave function of the form of a Slater determinant describing a state in which the spin up and down Fermi surfaces are spontaneously deformed away from their non-interacting spherical shape, much as in the Hartree-Fock description of the nematic Fermi fluid[CITATION].', '0907.1345-1-7-1': 'Since we are interested in magnetism we allow for the volume of the up and down Fermi surfaces to change as well.', '0907.1345-1-7-2': 'This results in a (""Thomas-Fermi"" like) distribution function of fermions in momentum space with 3 independent variational parameters, [MATH], [MATH], [MATH] and [MATH].', '0907.1345-1-7-3': 'We keep the total particle density [MATH] fixed,[CITATION] [EQUATION] where [MATH] is the step function and [MATH].', '0907.1345-1-7-4': 'If [MATH] the FS is a sphere.', '0907.1345-1-7-5': 'It has the property that [MATH], i.e. the total density does not depend on the distortion parameters, [MATH].', '0907.1345-1-8-0': 'Computing the energy in Hartree-Fock we obtain an expression for the ground state energy density in terms of the distribution functions of spin up and down particles [MATH], and [MATH], [EQUATION]', '0907.1345-1-8-1': 'Computing the integrals in Eqn. [REF] we obtain the total energy density given by [MATH] where [EQUATION] [MATH].', '0907.1345-1-8-2': 'Here we defined the dimensionless integrals [EQUATION] and [EQUATION] [MATH] is symmetric under the exchange of [MATH] and [MATH], and its derivative is discontinuous for [MATH].', '0907.1345-1-8-3': 'For the minimization of the energy we reduced this integral to 4 dimensions using its cylindrical symmetry and computed it numerically.', '0907.1345-1-8-4': 'The energy can be written in terms of the magnetization [MATH] and total density, [MATH], as [EQUATION] where [MATH].', '0907.1345-1-8-5': 'Note that in Eq. [REF], the difference [MATH] acts as an effective ""magnetic field"" that couples linearly to the magnetization [MATH].', '0907.1345-1-8-6': 'Since [MATH] and [MATH] parametrize FS distortions for components up and down, we conclude that, for any unequal distortions [MATH], the [MATH] spin symmetry is broken down to the residual [MATH] invariance of this uniaxial state.', '0907.1345-1-8-7': 'The equilibrium FS of the up and down spin components are shown in Fig. [REF].', '0907.1345-1-8-8': 'As a consequence of the anisotropic dipolar interactions, the FSs are unequally distorted for [MATH]: the up (down) spin FS is prolate (oblate), [MATH]) (see Fig [REF]).', '0907.1345-1-8-9': 'Both FSs are invariant under [MATH] rigid rotations about an axis parallel to [MATH].', '0907.1345-1-8-10': 'This feature is a consequence of the linear term in [MATH] in the energy, Eq. [REF].', '0907.1345-1-9-0': 'Pressure, bulk modulus and chemical potential.', '0907.1345-1-9-1': 'The total energy [MATH] and the equilibrium values of the FS distortions [MATH], [MATH] and the magnetization [MATH] are functions of the particle density [MATH].', '0907.1345-1-9-2': 'The equilibrium pressure [MATH], the chemical potential [MATH], and the compressibility [MATH] can be computed straightforwardly.', '0907.1345-1-9-3': 'The density dependence of [MATH] and [MATH], [MATH] were computed numerically.', '0907.1345-1-9-4': 'The pressure, compressibility and chemical potential are illustrated in Fig. [REF].', '0907.1345-1-9-5': 'As in the case of the fully polarized dipolar Fermi gas[CITATION], the compressibility of becomes negative at a critical value [MATH] (for the fully polarized state [MATH]) [CITATION].', '0907.1345-1-10-0': 'In current experiments on cold atoms it is possible to prepare a two component Fermionic system out of the hyperfine manifold of the atom.', '0907.1345-1-10-1': 'Hence, it may be possible to, in future experiments in strongly magnetic atoms such as [MATH], prepare a strongly two component dipolar system out of the hyperfine manifold.', '0907.1345-1-10-2': 'With a density of [MATH]cm[MATH], we find [MATH] [CITATION].', '0907.1345-1-10-3': 'Recent experimental advances in the study of strongly magnetic fermionic cold atomic systems, such as [MATH]Dy and [MATH]Dy and others, and the current efforts to cool them below their degeneracy temperatures, opens the possibility to detect the ferro-nematic state in the laboratory.', '0907.1345-1-10-4': 'Depending on the value of the dipole moment the FS for the two components may differ considerably, and free expansion experiments should be able to provide signatures of the ferro-nematic phase.', '0907.1345-1-10-5': 'In cold dilute systems interacting by s-wave contact pseudo-potential the spin components are conserved separately because there is no spin flip processes on the time scales of the experiments.', '0907.1345-1-10-6': 'This is contrast to a dense electron fluid in solids.', '0907.1345-1-10-7': 'For this reasons, it has been hard to observe ferromagnetism in cold and dilute Fermi gases.', '0907.1345-1-10-8': '[CITATION].', '0907.1345-1-10-9': 'Dipolar interactions do provide suitable spin-flip processes by which to observe this effects.', '0907.1345-1-10-10': 'Currently we are studying these processes.', '0907.1345-1-11-0': 'In this work we have shown the existence a new phase of matter, the ferro-nematic Fermi fluid, a ground state of a dipolar Fermi gas with a spontaneous magnetization and long range orientational order.', '0907.1345-1-11-1': 'In this state the up and down FS manifolds have different shapes and volumes.', '0907.1345-1-11-2': 'Since rotational invariance in real space and in spin space is simultaneously spontaneously broken in this state, it supports a rich spectrum of Goldstone exctitations.', '0907.1345-1-11-3': 'As a result the fluid is an optically anisotropic medium whose effects that may be detected by light scattering experiments.', '0907.1345-1-11-4': 'A quantum phase transition to a biaxial state may occur at a critical value of the dipolar coupling[CITATION].', '0907.1345-1-11-5': 'The effects of trapping potentials, temperature, atoms with spin [MATH], and complex inhomogeneous phases, as well as topological textures of this state and light scattering experiments, are will be discussed elsewhere[CITATION].', '0907.1345-1-12-0': 'We thank B. L. Lev, K. Sun and B. Uchoa for many discussions.', '0907.1345-1-12-1': 'EF thanks the Kavli Institute for Theoretical Physics (KITP) of the University of California Santa Barbara for hospitality under the program ""The Physics of Higher Temperature Superconductivity"".', '0907.1345-1-12-2': 'This work was supported in part by the National Science Foundation under the grants DMR 0758462 (EF) at the University of Illinois, and PHY05-51164 (EF) at the KITP, and by the Office of Science, U.S. Department of Energy under contracts DE-FG02-91ER45439 (EF, BMF) through the University of Illinois Frederick Seitz Materials Research Laboratory.'}","{'0907.1345-2-0-0': 'It is shown that a homogeneous two-component Fermi gas with (long range) dipolar and short-range isotropic interactions has a ferro-nematic phase for suitable values of the dipolar and short-range coupling constants.', '0907.1345-2-0-1': 'The ferro-nematic phase is characterized by having a non-zero magnetization and long range orientational uniaxial order.', '0907.1345-2-0-2': 'The Fermi surface of spin up (down) component is elongated (compressed) along the direction of the magnetization.', '0907.1345-2-1-0': 'Cold dipolar Fermi gases have attracted much attention due to the novel anisotropic and long-range character of dipole-dipole interactions.', '0907.1345-2-1-1': 'Recent studies of many-body effects predicted an elongated Fermi surface (FS) in a one-component fully polarized Fermi gas with dipolar interactions along the polarization direction established by an external field[CITATION].', '0907.1345-2-1-2': 'A biaxial state with a critical value of the effective coupling constant [MATH] was proposed in Ref. [CITATION] ([MATH] is the dipole moment of the fermion, [MATH] is the total density and [MATH] is the Fermi energy of the free Fermi gas at the same density).', '0907.1345-2-1-3': 'It was found[CITATION] that the system will exhibit violations of the Landau theory of the Fermi liquid both at quantum criticality and in the biaxial phase.', '0907.1345-2-1-4': 'More generally, understanding anisotropic non-Fermi liquid phases of cold atomic systems may shed light into the quantum liquid crystal phases in strongly correlated systems and high [MATH] superconductors [CITATION].', '0907.1345-2-2-0': 'The question we want to address here is whether a cold spin-12 Fermi gas with long range dipolar interactions can become spontaneously polarized and what is the nature of the broken symmetry state.', '0907.1345-2-2-1': 'The theory we present here is a generalization of the theory of the Stoner (ferromagnetic) transition in metals[CITATION] to take into account the effects of the long range and anisotropic dipolar interaction[CITATION].', '0907.1345-2-2-2': 'As we will see below, much as in the theory of Stoner ferromagnetism, the polarized state can only occur for sufficiently large values of the magnetic dipole moment and/or of the spin-flip scattering rate.', '0907.1345-2-2-3': 'However, unlike what happens in Stoner ferromagnetism, as a result of the structure of the dipolar interactions, the resulting polarized state is also spatially anisotropic, a ferro-nematic state.', '0907.1345-2-3-0': 'The classical version of this problem has been considered in mixtures of ferromagnetic particles with nematic liquid crystals[CITATION], and in dipolar colloidal fluids and ferrofluids[CITATION].', '0907.1345-2-3-1': 'Classical dipolar fluids have complex phase diagrams, typically featuring inhomogeneous phases with complex spacial structures.', '0907.1345-2-3-2': 'Much less is known about their quantum counterparts.', '0907.1345-2-3-3': 'In the case of simple quantum fluids, such as [MATH]He, the dipolar interaction plays a very small role compared to the short-range exchange interaction[CITATION].', '0907.1345-2-3-4': 'In the context of ultracold gases, a number of atomic and molecular systems with strong dipolar interactions, such as Dysprosium, have been the focus of recent experiments.', '0907.1345-2-4-0': 'Consider a restricted Hilbert space of two hyperfine states, called [MATH] (""spin up"") and [MATH] (""spin down""), of a point-like magnetic atom of mass [MATH] and magnetic moment, [MATH], with components [MATH]) and [MATH] are the usual spin-[MATH] Pauli matrices (the factor of [MATH] is absorbed in the definition of [MATH]).', '0907.1345-2-4-1': 'The Hamiltonian is [EQUATION] where the fields [MATH] destroy fermions on spin state with [MATH]-component [MATH] at position [MATH].', '0907.1345-2-4-2': 'We consider the model interaction is of the form [EQUATION] where [MATH] and [MATH] is a unit vector in the direction of [MATH].', '0907.1345-2-4-3': 'The last (ultra-local) term in Eq. [REF] represents the short-range isotropic (contact) interactions.', '0907.1345-2-4-4': 'It only affects the spin-triplet channel and we denote by [MATH] the associated coupling constant.', '0907.1345-2-4-5': 'The Fourier transform of the bare two body interaction is [MATH].', '0907.1345-2-4-6': 'The Hamiltonian is invariant under simultaneous [MATH] transformations in spin space and [MATH] rotations in real space.', '0907.1345-2-5-0': 'The ferro-nematic state breaks simultaneously the rotational invariance in spin space and in real space of the Hamiltonian, and its order parameters reflect this pattern of symmetry breaking.', '0907.1345-2-5-1': 'The order parameters are: a) the local magnetization vector [MATH]) that measures the spin polarization, b) the nematic order parameter, a [MATH] symmetric traceless matrix, [MATH]) that measures the breaking of rotational invariance in space, and c) the generalized ""nematic-spin-nematic"" order parameter [MATH], a tensor symmetric and traceless on the spatial ([MATH]) and spin ([MATH]) components, that measures the breaking of both symmetries [CITATION].', '0907.1345-2-5-2': 'General order parameters of the latter (nematic-spin-nematic) type were considered by Wu et.', '0907.1345-2-5-3': 'al.', '0907.1345-2-5-4': '[CITATION] who gave a detailed description in 2D systems and partially in 3D systems.', '0907.1345-2-5-5': 'The ferro-nematic state has an unbroken uniaxial symmetry in real space.', '0907.1345-2-5-6': 'It is a 3D generalization of the quadrupolar [MATH] of Ref. [CITATION].', '0907.1345-2-5-7': 'Phases with more general broken symmetries will be considered elsewhere[CITATION].', '0907.1345-2-6-0': 'An intuitive way to describe these phases in a Fermi system is in terms of spontaneously deformed Fermi surfaces (FS)[CITATION].', '0907.1345-2-6-1': 'A ferromagnetic state is isotropic in real space and has a spherical FS of unequal size for both spin polarizations.', '0907.1345-2-6-2': 'The nematic phase is isotropic in spin space, and its FS has a uniaxial distortion in real space, with the up and down spin FS being identical in shape and size.', '0907.1345-2-6-3': 'For small values of the order parameter, the distorted FS are ellipsoids with an eccentricity determined by the magnitude of the order parameter.', '0907.1345-2-6-4': 'Due to the anisotropy of the dipolar interaction, a ferromagnetism causes the FS to distort, thus driving the system into an uniaxial nematic state.', '0907.1345-2-6-5': 'Since the state is ferromagnetic, the up (down) FS is a prolate(oblate) revolution ellipsoid.', '0907.1345-2-6-6': 'Both FS are collinear, and have unequal distortion and volume.', '0907.1345-2-6-7': 'Hence, the nematic-spin-nematic order parameter [MATH] has a finite value.', '0907.1345-2-6-8': 'We find that all three order orders are present even for arbitrary small values of the dipolar coupling, where the ferromagnetic is the strongest, the nematic intermediate, and the nematic-spin-nematic the weakest.', '0907.1345-2-7-0': 'We make a variational ansatz of the wave function of the form of a Slater determinant describing a state in which the spin up and down Fermi surfaces are spontaneously deformed away from their non-interacting spherical shape, much as in the Hartree-Fock description of the nematic Fermi fluid[CITATION].', '0907.1345-2-7-1': 'Since we are interested in magnetism we allow for the volume of the up and down Fermi surfaces to change as well.', '0907.1345-2-7-2': 'This results in a (""Thomas-Fermi"" like) distribution function of fermions in momentum space with 4 variational parameters, [MATH], [MATH] and [MATH].', '0907.1345-2-7-3': 'We keep the total particle density [MATH] fixed,[CITATION] [EQUATION] where [MATH] is the step function and [MATH].', '0907.1345-2-7-4': 'If [MATH] the FS of are spheres.', '0907.1345-2-7-5': 'It has the property that [MATH], i.e. the total density does not depend on the distortion parameters, [MATH].', '0907.1345-2-8-0': 'Computing the energy in Hartree-Fock we obtain an expression for the ground state energy density in terms of the distribution functions of spin up and down particles [MATH], and [MATH], [EQUATION] which is a function of the magnetization [MATH], the particle density [MATH], and that the ratio of the undistorted Fermi surface volumes is [MATH].', '0907.1345-2-9-0': 'Computing the integrals we obtain the total energy density given by [MATH] where [EQUATION] where [MATH].', '0907.1345-2-9-1': 'The integrals [MATH] and [MATH] are given by [EQUATION] and [EQUATION]', '0907.1345-2-9-2': 'The total energy [MATH] becomes [EQUATION] where [MATH].', '0907.1345-2-9-3': 'Numerically minimizing the energy we obtain the phase diagram in the parameters [MATH] and [MATH] of Fig. [REF].', '0907.1345-2-10-0': 'The phase diagram shown in Fig. [REF] exhibits a ferro-nematic phase and a paramagnetic phase, separated by a phase boundary consisting of a line of 1st order transitions that meets a line of continuous transitions at a tricritical point (labeled by [MATH].)', '0907.1345-2-10-1': 'As expected the ferro-nematic state becomes more accessible as the [MATH]-wave coupling increases.', '0907.1345-2-10-2': 'This phase is fully polarized ([MATH]) for much of the phase diagram, except for a small region where the polarization is partial, [MATH].', '0907.1345-2-10-3': 'In the ferro-nematic phase with partial polarization the up and down Fermi surfaces are unequally distorted, while in the fully polarized regime only one distorted up FS exists.', '0907.1345-2-10-4': 'The conventional Stoner transition occurs at [MATH] and [MATH], where the up and down FS become distorted even for arbitrarily small values of the dipolar coupling (see Fig. [REF].)', '0907.1345-2-10-5': 'From the structure of the the free energy for small values of [MATH], [MATH] and [MATH] (valid in the vicinity of the continuous transition) we see that the dipolar interaction leads to a leading term of the form [MATH], which is invariant under [MATH].', '0907.1345-2-10-6': 'Since the ferromagnetic state is already favored by the contact term, this term also favors [MATH] and [MATH].', '0907.1345-2-11-0': 'In the ferro-nematic phase the [MATH] spin symmetry of the Hamiltonian is broken down to the residual [MATH] invariance of this uniaxial state.', '0907.1345-2-11-1': 'The equilibrium FS of the up and down spin components are shown in Fig. [REF] for several values of of the coupling constants.', '0907.1345-2-11-2': ""Both FS's are invariant under [MATH] rigid rotations about [MATH]."", '0907.1345-2-12-0': 'The total energy, the equilibrium values of the FS distortions and the magnetization are functions of the particle density [MATH].', '0907.1345-2-12-1': 'The pressure [MATH], chemical potential [MATH] and the compressibility [MATH] can be computed straightforwardly, and decrease monotonically as the dipolar coupling increases, see Fig. [REF].', '0907.1345-2-12-2': 'The compressibility vanishes at [MATH] [CITATION] where the Fermi gas becomes (formally) unstable to collapse[CITATION].', '0907.1345-2-13-0': 'Discussion.', '0907.1345-2-13-1': 'In current experiments on cold atoms it is possible to prepare a two-component Fermi gas out of the hyperfine manifold of the atom.', '0907.1345-2-13-2': 'One can imagine, a two-component dipolar Fermi gas made out of, say, the hyperfine manifold of strongly magnetic atom such as Dysprosium.', '0907.1345-2-13-3': 'For [MATH]Dy with a density of [MATH]cm[MATH], we estimate [MATH] [CITATION].', '0907.1345-2-13-4': 'Recent experimental advances in the study of strongly magnetic fermionic cold atomic systems, such as [MATH]Dy and [MATH]Dy and others, and the current efforts to cool them below their degeneracy temperatures, opens the possibility to detect the ferro-nematic state in the laboratory possibly by tuning the [MATH]-wave scattering amplitude.', '0907.1345-2-13-5': 'Depending on the value of the dipole moment and of the [MATH]-wave coupling the FS for the two components may differ considerably from the spherical shape, and free expansion experiments may be able to provide signatures of this state.', '0907.1345-2-13-6': 'Ferromagnetism in cold dilute systems interacting only by s-wave contact pseudo-potential has been difficult to observe as the spin states of fermions are conserved separately on the time scales of the experiments[CITATION].', '0907.1345-2-13-7': 'However, a ferro-nematic state can be set up experimentally in systems with a population imbalance of hyperfine states, which may also exhibit other anisotropic and inhomogeneous phases.', '0907.1345-2-14-0': 'In this work we have shown the existence a new phase of matter, the ferro-nematic Fermi fluid, a ground state of a dipolar Fermi gas with short range interactions with a spontaneous magnetization and long range orientational order.', '0907.1345-2-14-1': 'In this state the up and down FS manifolds have unequal shapes and volumes.', '0907.1345-2-14-2': 'Since rotational invariance in real space and in spin space is simultaneously spontaneously broken in this state, it supports a rich spectrum of Goldstone excitations.', '0907.1345-2-14-3': 'As a result the fluid is an optically anisotropic medium whose effects that may be detected by light scattering experiments.', '0907.1345-2-15-0': 'We thank B. L. Lev, B. Hsu, K. Sun, and B. Uchoa for many discussions.', '0907.1345-2-15-1': 'EF thanks the Kavli Institute for Theoretical Physics (KITP) of the University of California Santa Barbara for hospitality.', '0907.1345-2-15-2': 'This work was supported in part by the National Science Foundation under the grants DMR 0758462 (EF) at the University of Illinois, and PHY05-51164 (EF) at the KITP, and by the Office of Science, U.S. Department of Energy under contracts DE-FG02-91ER45439 (EF, BMF) through the University of Illinois Frederick Seitz Materials Research Laboratory.'}","[['0907.1345-1-7-0', '0907.1345-2-7-0'], ['0907.1345-1-7-1', '0907.1345-2-7-1'], ['0907.1345-1-7-3', '0907.1345-2-7-3'], ['0907.1345-1-7-5', '0907.1345-2-7-5'], ['0907.1345-1-11-3', '0907.1345-2-14-3'], ['0907.1345-1-2-1', '0907.1345-2-3-0'], ['0907.1345-1-2-2', '0907.1345-2-3-1'], ['0907.1345-1-2-3', '0907.1345-2-3-2'], ['0907.1345-1-2-4', '0907.1345-2-3-3'], ['0907.1345-1-12-2', '0907.1345-2-15-2'], ['0907.1345-1-0-1', '0907.1345-2-0-1'], 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'0907.1345-2-5-3', '0907.1345-2-13-0', '0907.1345-3-5-3', '0907.1345-3-13-0', '0907.1345-4-5-3']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/0907.1345,"{'0907.1345-3-0-0': 'It is shown that a homogeneous two-component Fermi gas with (long range) dipolar and short-range isotropic interactions has a ferro-nematic phase for suitable values of the dipolar and short-range coupling constants.', '0907.1345-3-0-1': 'The ferro-nematic phase is characterized by having a non-zero magnetization and long range orientational uniaxial order.', '0907.1345-3-0-2': 'The Fermi surface of spin up (down) component is elongated (compressed) along the direction of the magnetization.', '0907.1345-3-1-0': 'Cold dipolar Fermi gases have attracted much attention due to the novel anisotropic and long-range character of dipole-dipole interactions.', '0907.1345-3-1-1': 'Recent studies of many-body effects predicted an elongated Fermi surface (FS) in a one-component fully polarized Fermi gas with dipolar interactions along the polarization direction established by an external field[CITATION].', '0907.1345-3-1-2': 'A biaxial state with a critical value of the effective coupling constant [MATH] was proposed in Ref. [CITATION] ([MATH] is the dipole moment of the fermion, [MATH] is the total density and [MATH] is the Fermi energy of the free Fermi gas at the same density).', '0907.1345-3-1-3': 'It was found[CITATION] that the system will exhibit violations of the Landau theory of the Fermi liquid both at quantum criticality and in the biaxial phase.', '0907.1345-3-1-4': 'More generally, understanding anisotropic non-Fermi liquid phases of cold atomic systems may shed light into the quantum liquid crystal phases in strongly correlated systems and high [MATH] superconductors [CITATION].', '0907.1345-3-2-0': 'The question we want to address here is whether a cold spin-12 Fermi gas with long range dipolar interactions can become spontaneously polarized and what is the nature of the broken symmetry state.', '0907.1345-3-2-1': 'The theory we present here is a generalization of the theory of the Stoner (ferromagnetic) transition in metals[CITATION] to take into account the effects of the long range and anisotropic dipolar interaction[CITATION].', '0907.1345-3-2-2': 'As we will see below, much as in the theory of Stoner ferromagnetism, the polarized state can occur only for sufficiently large values of the magnetic dipole moment and/or of the spin-flip scattering rate.', '0907.1345-3-2-3': 'However, unlike what happens in Stoner ferromagnetism, as a result of the structure of the dipolar interactions, the resulting polarized state is also spatially anisotropic, a ferro-nematic state.', '0907.1345-3-3-0': 'The classical version of this problem has been considered in mixtures of ferromagnetic particles with nematic liquid crystals[CITATION], and in dipolar colloidal fluids and ferrofluids[CITATION].', '0907.1345-3-3-1': 'Classical dipolar fluids have complex phase diagrams, typically featuring inhomogeneous phases with complex spatial structures.', '0907.1345-3-3-2': 'Much less is known about their quantum counterparts.', '0907.1345-3-3-3': 'In the case of simple quantum fluids, such as [MATH]He, the dipolar interaction plays a small role compared to the short-range exchange interaction[CITATION].', '0907.1345-3-3-4': 'In the context of ultracold gases, a number of atomic and molecular systems with strong dipolar interactions, such as Dy, have been the focus of recent experiments (see Ref. [CITATION]).', '0907.1345-3-4-0': 'Consider a restricted Hilbert space of two hyperfine states, called [MATH] (""spin up"") and [MATH] (""spin down""), of a point-like magnetic atom of mass [MATH] and magnetic moment, [MATH], with components [MATH]) and [MATH] are the usual spin-[MATH] Pauli matrices (the factor of [MATH] is absorbed in the definition of [MATH]).', '0907.1345-3-4-1': 'The Hamiltonian is [EQUATION] where the fields [MATH] destroy fermions on spin state with [MATH]-component [MATH] at position [MATH].', '0907.1345-3-4-2': 'We consider the model interaction is of the form [EQUATION] where [MATH] and [MATH] is a unit vector in the direction of [MATH].', '0907.1345-3-4-3': 'The last (ultra-local) term represents the short-range isotropic (contact) interactions.', '0907.1345-3-4-4': 'It only affects the spin-triplet channel and we denote by [MATH] the associated coupling constant.', '0907.1345-3-4-5': 'The Fourier transform of the bare two body interaction is [MATH].', '0907.1345-3-4-6': 'The Hamiltonian is invariant under simultaneous [MATH] transformations in spin space and [MATH] rotations in real space.', '0907.1345-3-4-7': 'This mixing of orbital and spin degrees of freedom is, in essence, what relates the distortions of the shape of FS and the spin polarization.', '0907.1345-3-5-0': 'The ferro-nematic state breaks simultaneously the rotational invariance in spin space and in real space of the Hamiltonian, and its order parameters reflect this pattern of symmetry breaking.', '0907.1345-3-5-1': 'The order parameters are: a) the local magnetization vector [MATH]) that measures the spin polarization, b) the nematic order parameter, a [MATH] symmetric traceless matrix, [MATH]) that measures the breaking of rotational invariance in space, and c) the generalized ""nematic-spin-nematic"" order parameter [MATH], a tensor symmetric and traceless on the spatial ([MATH]) and spin ([MATH]) components, that measures the breaking of both symmetries [CITATION].', '0907.1345-3-5-2': 'General order parameters of the latter (nematic-spin-nematic) type were considered by Wu et.', '0907.1345-3-5-3': 'al.', '0907.1345-3-5-4': '[CITATION] who gave a detailed description in 2D systems and partially in 3D systems.', '0907.1345-3-5-5': 'The ferro-nematic state has an unbroken uniaxial symmetry in real space.', '0907.1345-3-5-6': 'It is a 3D generalization of the quadrupolar [MATH] phase of Ref. [CITATION].', '0907.1345-3-5-7': 'More general broken symmetry states will be considered elsewhere[CITATION].', '0907.1345-3-6-0': 'An intuitive way to describe these phases (and their order parameters) in a Fermi system is in terms of spontaneously deformed Fermi surfaces [CITATION].', '0907.1345-3-6-1': 'A ferromagnetic state is isotropic in real space and has a spherical FS of unequal size for both spin polarizations.', '0907.1345-3-6-2': 'The nematic phase is isotropic in spin space, and its FS has a uniaxial distortion in real space, with the up and down spin FS being identical in shape and size.', '0907.1345-3-6-3': 'For small values of the order parameter, the distorted FS are ellipsoids with an eccentricity determined by the magnitude of the order parameter [MATH].', '0907.1345-3-6-4': 'Due to the mixing of orbital and spin degrees of freedom by the dipolar interaction, ferromagnetism causes the FS to distort, thus driving the system into an uniaxial nematic state.', '0907.1345-3-6-5': 'Since the state is ferromagnetic, the up (down) FS is a prolate (oblate) revolution ellipsoid.', '0907.1345-3-6-6': 'Both FS are collinear, and have unequal distortion and volume.', '0907.1345-3-6-7': 'Hence, the nematic-spin-nematic order parameter [MATH] has a finite value.', '0907.1345-3-6-8': 'We find that all three order orders are present even for arbitrary small values of the dipolar coupling, where the ferromagnetic is the strongest, the nematic intermediate, and the nematic-spin-nematic the weakest.', '0907.1345-3-7-0': 'In this paper we use a Hartree-Fock (HF) variational wave function to determine the phase diagram as a function of the density [MATH], the (dimensionless) dipolar coupling constant [MATH] and of the (dimensionless) local exchange coupling [MATH].', '0907.1345-3-7-1': 'We consider an infinite system and significant finite size effects, such as the trap potential and the associated inhomogeneity of the gas, are not considered but can be included using the Thomas-Fermi approximation.', '0907.1345-3-7-2': 'We find two phases: a) an isotropic unpolarized state and b) a ferro-nematic phase.', '0907.1345-3-7-3': 'As in the conventional theory of the Stoner transition, we find that the values of [MATH] and [MATH] on the phase boundary are of order unity.', '0907.1345-3-7-4': 'In this regime, a HF wave function can only yield qualitative results, such as the broad structure of the phase diagram, but its is not not expected to be quantitatively accurate.', '0907.1345-3-7-5': 'Even within mean field theory, HF may miss important physics; a recent study[CITATION] found that to second order in [MATH] the usual continuous Stoner phase transition can turn first order.', '0907.1345-3-8-0': 'We take a variational HF wave function of the form of a Slater determinant describing a state in which the spin up and down Fermi surfaces are spontaneously deformed away from their non-interacting spherical shape.', '0907.1345-3-8-1': 'Since we are interested in magnetism we allow for the volume of the up and down Fermi surfaces to change as well.', '0907.1345-3-8-2': 'This results in a (""Thomas-Fermi"" like) distribution function of fermions in momentum space with 4 variational parameters, [MATH], [MATH] and [MATH].', '0907.1345-3-8-3': 'We keep the total particle density [MATH] fixed,[CITATION] [EQUATION] where [MATH] is the step function and [MATH].', '0907.1345-3-8-4': ""If [MATH] both FS's are spheres."", '0907.1345-3-8-5': 'Eq. [REF] has the property that [MATH]; the total density does not depend on the FS distortion parameters [MATH], with [MATH](oblate), [MATH] (prolate).', '0907.1345-3-9-0': 'Computing the energy in HF we obtain an expression for the ground state energy density in terms of the distribution functions of spin up and down particles [MATH], and [MATH], [EQUATION] where [MATH] labels the direction of [MATH] with respect to the [MATH]-axis.', '0907.1345-3-9-1': '[MATH] is a function of the magnetization [MATH], the density [MATH], and the ratio of the undistorted Fermi surface volumes is [MATH].', '0907.1345-3-9-2': 'The total energy density [MATH] is [EQUATION] where [MATH].', '0907.1345-3-9-3': 'The integrals [MATH] and [MATH] are given by [EQUATION]', '0907.1345-3-9-4': 'The total energy [MATH] becomes [EQUATION] where [MATH].', '0907.1345-3-9-5': 'Numerically minimizing the energy we obtain the phase diagram in the couplings [MATH] and [MATH] of Fig. [REF].', '0907.1345-3-10-0': 'The phase diagram shown in Fig. [REF] exhibits a ferro-nematic phase and a paramagnetic phase, separated by a phase boundary consisting of a line of 1st order transitions that meets a line of continuous transitions at a tricritical point (labeled by [MATH].)', '0907.1345-3-10-1': 'As expected the ferro-nematic state becomes more accessible as the [MATH]-wave coupling increases.', '0907.1345-3-10-2': 'This phase is fully polarized ([MATH]) for most of the phase diagram, except for a small region where the polarization is partial, [MATH].', '0907.1345-3-10-3': 'In the ferro-nematic phase with partial polarization the up and down Fermi surfaces are unequally distorted, while in the fully polarized regime only one distorted up FS exists.', '0907.1345-3-10-4': 'Partial magnetization in the conventional Stoner transition occurs for [MATH] and [MATH], where the up and down FS become distorted even for arbitrarily small values of the dipolar coupling (see Fig. [REF].)', '0907.1345-3-10-5': 'From the structure of the free energy for small values of [MATH], [MATH] and [MATH] (valid in the vicinity of the continuous transition) we see that the dipolar interaction leads to a leading term of the form [MATH], which is invariant under [MATH].', '0907.1345-3-10-6': 'Since the ferromagnetic state is already favored by the contact term, this term also favors [MATH] and [MATH].', '0907.1345-3-11-0': 'In the ferro-nematic phase the [MATH] spin symmetry of the Hamiltonian is broken down to the residual [MATH] invariance of this uniaxial state.', '0907.1345-3-11-1': 'The equilibrium FS of the up and down spin components are shown in Fig. [REF] for several values of of the coupling constants.', '0907.1345-3-11-2': ""Both FS's are invariant under [MATH] rigid rotations about [MATH]."", '0907.1345-3-12-0': 'The total energy, the equilibrium values of the FS distortions and the magnetization are functions of the particle density [MATH].', '0907.1345-3-12-1': 'The pressure [MATH], chemical potential [MATH] and the compressibility [MATH] can be computed straightforwardly, and decrease monotonically as the dipolar coupling increases, see Fig. [REF].', '0907.1345-3-12-2': 'The compressibility vanishes at [MATH] [CITATION] where the Fermi gas becomes (formally) unstable to collapse[CITATION].', '0907.1345-3-13-0': 'Discussion.', '0907.1345-3-13-1': 'In current experiments on cold atoms it is possible to prepare a two-component Fermi gas out of the hyperfine manifold of the atom.', '0907.1345-3-13-2': 'One can imagine, a two-component dipolar Fermi gas made out of, say, the hyperfine manifold of strongly magnetic atom such as Dy.', '0907.1345-3-13-3': 'For [MATH]Dy with a density of [MATH]cm[MATH], we estimate [MATH] [CITATION].', '0907.1345-3-13-4': 'The recent experimental observation of itinerant (Stoner) ferromagnetism in ultra-cold gases of [MATH]Li atoms[CITATION] opens the possibility to detect the ferro-nematic state in the laboratory possibly by tuning the [MATH]-wave scattering amplitude.', '0907.1345-3-13-5': 'Depending on the strength of the dipole moment and of the [MATH]-wave coupling, the FS for the two components may differ considerably from the spherical shape, and free expansion experiments may be able to provide signatures of this state.', '0907.1345-3-13-6': 'Ferromagnetism in cold dilute systems interacting only by s-wave contact pseudo-potential has been difficult to observe as the spin states of fermions are conserved separately on the time scales of the experiments[CITATION].', '0907.1345-3-13-7': 'However, a ferro-nematic state can be set up experimentally in systems with a population imbalance of hyperfine states, which may also exhibit other anisotropic and inhomogeneous phases.', '0907.1345-3-14-0': 'In this work we have shown the existence a new phase of matter, the ferro-nematic Fermi fluid, a ground state of a dipolar Fermi gas with short range interactions with a spontaneous magnetization and long range orientational order.', '0907.1345-3-14-1': 'In this state, the up and down FS manifolds have unequal shapes and volumes.', '0907.1345-3-14-2': 'Since rotational invariance in real space and in spin space is simultaneously spontaneously broken in this state, it supports a rich spectrum of Goldstone excitations.', '0907.1345-3-14-3': 'As a result the fluid is an optically anisotropic medium whose properties that may be detected by light scattering experiments.', '0907.1345-3-15-0': 'We thank B. L. Lev, B. Hsu, K. Sun, and B. Uchoa for many discussions.', '0907.1345-3-15-1': 'EF thanks the Kavli Institute for Theoretical Physics (KITP) of the University of California Santa Barbara for hospitality.', '0907.1345-3-15-2': 'This work was supported in part by the National Science Foundation under the grants DMR 0758462 (EF) at the University of Illinois, and PHY05-51164 (EF) at KITP, and by the Office of Science, U.S. Department of Energy under contracts DE-FG02-91ER45439 (EF, BMF) through the University of Illinois Frederick Seitz Materials Research Laboratory.'}","{'0907.1345-4-0-0': 'It is shown that a homogeneous two-component Fermi gas with (long range) dipolar and short-range isotropic interactions has a ferro-nematic phase for suitable values of the dipolar and short-range coupling constants.', '0907.1345-4-0-1': 'The ferro-nematic phase is characterized by having a non-zero magnetization and long range orientational uniaxial order.', '0907.1345-4-0-2': 'The Fermi surface of spin up (down) component is elongated (compressed) along the direction of the magnetization.', '0907.1345-4-1-0': 'Cold dipolar Fermi gases have attracted much attention due to the novel anisotropic and long-range character of dipole-dipole interactions.', '0907.1345-4-1-1': 'Recent studies of many-body effects predicted an elongated Fermi surface (FS) in a one-component fully polarized Fermi gas with dipolar interactions along the polarization direction established by an external field[CITATION].', '0907.1345-4-1-2': 'A biaxial state with a critical value of the effective coupling constant [MATH] was proposed in Ref. [CITATION] ([MATH] is the dipole moment of the fermion, [MATH] is the total density and [MATH] is the Fermi energy of the free Fermi gas at the same density).', '0907.1345-4-1-3': 'It was found[CITATION] that the system will exhibit violations of the Landau theory of the Fermi liquid both at quantum criticality and in the biaxial phase.', '0907.1345-4-1-4': 'More generally, understanding anisotropic non-Fermi liquid phases of cold atomic systems may shed light into the quantum liquid crystal phases in strongly correlated systems and high [MATH] superconductors [CITATION].', '0907.1345-4-2-0': 'The question we want to address here is whether a cold spin-12 Fermi gas with long range dipolar interactions can become spontaneously polarized and what is the nature of the broken symmetry state.', '0907.1345-4-2-1': 'The theory we present here is a generalization of the theory of the Stoner (ferromagnetic) transition in metals[CITATION] to take into account the effects of the long range and anisotropic dipolar interaction[CITATION].', '0907.1345-4-2-2': 'As we will see below, much as in the theory of Stoner ferromagnetism, the polarized state can occur only for sufficiently large values of the magnetic dipole moment and/or of the spin-flip scattering rate.', '0907.1345-4-2-3': 'However, unlike what happens in Stoner ferromagnetism, as a result of the structure of the dipolar interactions, the resulting polarized state is also spatially anisotropic, a ferro-nematic state.', '0907.1345-4-3-0': 'The classical version of this problem has been considered in mixtures of ferromagnetic particles with nematic liquid crystals[CITATION], and in dipolar colloidal fluids and ferrofluids[CITATION].', '0907.1345-4-3-1': 'Classical dipolar fluids have complex phase diagrams, typically featuring inhomogeneous phases with complex spatial structures.', '0907.1345-4-3-2': 'Much less is known about their quantum counterparts.', '0907.1345-4-3-3': 'In the case of simple quantum fluids, such as [MATH]He, the dipolar interaction plays a small role compared to the short-range exchange interaction[CITATION].', '0907.1345-4-3-4': 'In the context of ultracold gases, a number of atomic and molecular systems with strong dipolar interactions, such as Dy, have been the focus of recent experiments (see Ref. [CITATION]).', '0907.1345-4-4-0': 'Consider a restricted Hilbert space of two hyperfine states, called [MATH] (""spin up"") and [MATH] (""spin down""), of a point-like magnetic atom of mass [MATH] and magnetic moment, [MATH], with components [MATH]) and [MATH] are the usual spin-[MATH] Pauli matrices (the factor of [MATH] is absorbed in the definition of [MATH]).', '0907.1345-4-4-1': 'The Hamiltonian is [EQUATION] where the fields [MATH] destroy fermions on spin state with [MATH]-component [MATH] at position [MATH].', '0907.1345-4-4-2': 'We consider the model interaction is of the form [EQUATION] where [MATH] and [MATH] is a unit vector in the direction of [MATH].', '0907.1345-4-4-3': 'The last (ultra-local) term represents the short-range isotropic (contact) interactions.', '0907.1345-4-4-4': 'It only affects the spin-triplet channel and we denote by [MATH] the associated coupling constant.', '0907.1345-4-4-5': 'The Fourier transform of the bare two body interaction is [MATH].', '0907.1345-4-4-6': 'The Hamiltonian is invariant under simultaneous [MATH] transformations in spin space and [MATH] rotations in real space.', '0907.1345-4-4-7': 'This mixing of orbital and spin degrees of freedom is, in essence, what relates the distortions of the shape of FS and the spin polarization.', '0907.1345-4-5-0': 'The ferro-nematic state breaks simultaneously the rotational invariance in spin space and in real space of the Hamiltonian, and its order parameters reflect this pattern of symmetry breaking.', '0907.1345-4-5-1': 'The order parameters are: a) the local magnetization vector [MATH]) that measures the spin polarization, b) the nematic order parameter, a [MATH] symmetric traceless matrix, [MATH]) that measures the breaking of rotational invariance in space, and c) the generalized ""nematic-spin-nematic"" order parameter [MATH], a tensor symmetric and traceless on the spatial ([MATH]) and spin ([MATH]) components, that measures the breaking of both symmetries [CITATION].', '0907.1345-4-5-2': 'General order parameters of the latter (nematic-spin-nematic) type were considered by Wu et.', '0907.1345-4-5-3': 'al.', '0907.1345-4-5-4': '[CITATION] who gave a detailed description in 2D systems and partially in 3D systems.', '0907.1345-4-5-5': 'The ferro-nematic state has an unbroken uniaxial symmetry in real space.', '0907.1345-4-5-6': 'It is a 3D generalization of the quadrupolar [MATH] phase of Ref. [CITATION].', '0907.1345-4-6-0': 'An intuitive way to describe these phases (and their order parameters) in a Fermi system is in terms of spontaneously deformed Fermi surfaces [CITATION].', '0907.1345-4-6-1': 'A ferromagnetic state is isotropic in real space and has a spherical FS of unequal size for both spin polarizations.', '0907.1345-4-6-2': 'The nematic phase is isotropic in spin space, and its FS has a uniaxial distortion in real space, with the up and down spin FS being identical in shape and size.', '0907.1345-4-6-3': 'For small values of the order parameter, the distorted FS are ellipsoids with an eccentricity determined by the magnitude of the order parameter [MATH].', '0907.1345-4-6-4': 'Due to the mixing of orbital and spin degrees of freedom by the dipolar interaction, ferromagnetism causes the FS to distort, thus driving the system into an uniaxial nematic state.', '0907.1345-4-6-5': 'Since the state is ferromagnetic, the up (down) FS is a prolate (oblate) revolution ellipsoid.', '0907.1345-4-6-6': 'Both FS are collinear, and have unequal distortion and volume.', '0907.1345-4-6-7': 'Hence, the nematic-spin-nematic order parameter [MATH] has a finite value.', '0907.1345-4-6-8': 'We find that all three order orders are present even for arbitrary small values of the dipolar coupling, where the ferromagnetic is the strongest, the nematic intermediate, and the nematic-spin-nematic the weakest.', '0907.1345-4-7-0': 'In this paper we use a Hartree-Fock (HF) variational wave function to determine the phase diagram as a function of the density [MATH], the (dimensionless) dipolar coupling constant [MATH] and of the (dimensionless) local exchange coupling [MATH].', '0907.1345-4-7-1': 'We consider an infinite system and significant finite size effects, such as the trap potential and the associated inhomogeneity of the gas, are not considered but can be included using the Thomas-Fermi approximation.', '0907.1345-4-7-2': 'We find two phases: a) an isotropic unpolarized state and b) a ferro-nematic phase.', '0907.1345-4-7-3': 'As in the conventional theory of the Stoner transition, we find that the values of [MATH] and [MATH] on the phase boundary are of order unity.', '0907.1345-4-7-4': 'In this regime, a HF wave function can only yield qualitative results, such as the broad structure of the phase diagram, but its is not not expected to be quantitatively accurate.', '0907.1345-4-7-5': 'Even within mean field theory, HF may miss important physics; a recent study[CITATION] found that to second order in [MATH] the usual continuous Stoner phase transition can turn first order.', '0907.1345-4-8-0': 'We take a variational HF wave function of the form of a Slater determinant describing a state in which the spin up and down Fermi surfaces are spontaneously deformed away from their non-interacting spherical shape.', '0907.1345-4-8-1': 'We will not consider other interesting states with more complex order, such as biaxial[CITATION] and its generalizations.', '0907.1345-4-8-2': 'Since we are interested in magnetism we allow for the volume of the up and down Fermi surfaces to change as well.', '0907.1345-4-8-3': 'This results in a (""Thomas-Fermi"" like) distribution function of fermions in momentum space with 4 variational parameters, [MATH], [MATH] and [MATH].', '0907.1345-4-8-4': 'We keep the total particle density [MATH] fixed,[CITATION] [EQUATION] where [MATH] is the step function and [MATH].', '0907.1345-4-8-5': ""If [MATH] both FS's are spheres."", '0907.1345-4-8-6': 'Eq. [REF] has the property that [MATH]; the total density does not depend on the FS distortion parameters [MATH], with [MATH](oblate), [MATH] (prolate).', '0907.1345-4-9-0': 'Computing the energy in HF we obtain an expression for the ground state energy density in terms of the distribution functions of spin up and down particles [MATH], and [MATH], [EQUATION] where [MATH] labels the direction of [MATH] with respect to the [MATH]-axis.', '0907.1345-4-9-1': '[MATH] is a function of the magnetization [MATH], the density [MATH], and the ratio of the undistorted Fermi surface volumes is [MATH].', '0907.1345-4-9-2': 'The total energy density [MATH] is [EQUATION] where [MATH].', '0907.1345-4-9-3': 'The integrals [MATH] and [MATH] are given by [EQUATION]', '0907.1345-4-9-4': 'The total energy [MATH] becomes [EQUATION] where [MATH].', '0907.1345-4-9-5': 'Numerically minimizing the energy we obtain the phase diagram in the couplings [MATH] and [MATH] of Fig. [REF].', '0907.1345-4-10-0': 'The phase diagram shown in Fig. [REF] exhibits a ferro-nematic phase and a paramagnetic phase, separated by a phase boundary consisting of a line of 1st order transitions that meets a line of continuous transitions at a tricritical point (labeled by [MATH].)', '0907.1345-4-10-1': 'As expected the ferro-nematic state becomes more accessible as the [MATH]-wave coupling increases.', '0907.1345-4-10-2': 'This phase is fully polarized ([MATH]) for most of the phase diagram, except for a small region where the polarization is partial, [MATH].', '0907.1345-4-10-3': 'In the ferro-nematic phase with partial polarization the up and down Fermi surfaces are unequally distorted, while in the fully polarized regime only one distorted up FS exists.', '0907.1345-4-10-4': 'Partial magnetization in the conventional Stoner transition occurs for [MATH] and [MATH], where the up and down FS become distorted even for arbitrarily small values of the dipolar coupling (see Fig. [REF].)', '0907.1345-4-10-5': 'From the structure of the free energy for small values of [MATH], [MATH] and [MATH] (valid in the vicinity of the continuous transition) we see that the dipolar interaction leads to a leading term of the form [MATH], which is invariant under [MATH].', '0907.1345-4-10-6': 'Since the ferromagnetic state is already favored by the contact term, this term also favors [MATH] and [MATH].', '0907.1345-4-11-0': 'In the ferro-nematic phase the [MATH] spin symmetry of the Hamiltonian is broken down to the residual [MATH] invariance of this uniaxial state.', '0907.1345-4-11-1': 'The equilibrium FS of the up and down spin components are shown in Fig. [REF] for several values of of the coupling constants.', '0907.1345-4-11-2': ""Both FS's are invariant under [MATH] rigid rotations about [MATH]."", '0907.1345-4-12-0': 'The total energy, the equilibrium values of the FS distortions and the magnetization are functions of the particle density [MATH].', '0907.1345-4-12-1': 'The pressure [MATH], chemical potential [MATH] and the compressibility [MATH] can be computed straightforwardly, and decrease monotonically as the dipolar coupling increases, see Fig. [REF].', '0907.1345-4-12-2': 'The compressibility vanishes at [MATH] [CITATION] where the Fermi gas becomes (formally) unstable to collapse[CITATION].', '0907.1345-4-13-0': 'In current experiments on cold atoms it is possible to prepare a two-component Fermi gas out of the hyperfine manifold of the atom.', '0907.1345-4-13-1': 'One can imagine, a two-component dipolar Fermi gas made out of, say, the hyperfine manifold of strongly magnetic atom such as Dy.', '0907.1345-4-13-2': 'For [MATH]Dy with a density of [MATH]cm[MATH], we estimate [MATH] [CITATION].', '0907.1345-4-13-3': 'The recent experimental observation of itinerant (Stoner) ferromagnetism in ultra-cold gases of [MATH]Li atoms[CITATION] opens the possibility to detect the ferro-nematic state in the laboratory possibly by tuning the [MATH]-wave scattering amplitude.', '0907.1345-4-13-4': 'Depending on the strength of the dipole moment and of the [MATH]-wave coupling, the FS for the two components may differ considerably from the spherical shape, and free expansion experiments may be able to provide signatures of this state.', '0907.1345-4-13-5': 'Ferromagnetism in cold dilute systems interacting only by s-wave contact pseudo-potential has been difficult to observe as the spin states of fermions are conserved separately on the time scales of the experiments[CITATION].', '0907.1345-4-13-6': 'However, a ferro-nematic state can be set up experimentally in systems with a population imbalance of hyperfine states, which may also exhibit other anisotropic and inhomogeneous phases.', '0907.1345-4-13-7': 'In actual experiments the trap potential is anisotropic and weakly inhomogeneous.', '0907.1345-4-13-8': 'We estimate that such effects do not affect our main results provided the trap potential aspect ratios depart from 1 by perhaps up to about 30%.', '0907.1345-4-13-9': 'Nevertheless, the anisotropy acts as a weak symmetry breaking field, that orients the ferro-nematic order.', '0907.1345-4-14-0': 'In this work we have shown the existence a new phase of matter, the ferro-nematic Fermi fluid, a ground state of a dipolar Fermi gas with short range interactions with a spontaneous magnetization and long range orientational order.', '0907.1345-4-14-1': 'In this state, the up and down FS manifolds have unequal shapes and volumes.', '0907.1345-4-14-2': 'Since rotational invariance in real space and in spin space is simultaneously spontaneously broken in this state, it supports a rich spectrum of Goldstone excitations.', '0907.1345-4-14-3': 'As a result the fluid is an optically anisotropic medium whose properties that may be detected by light scattering experiments.', '0907.1345-4-15-0': 'We thank B. L. Lev, B. Hsu, K. Sun, and B. Uchoa for many discussions.', '0907.1345-4-15-1': 'EF thanks the Kavli Institute for Theoretical Physics (KITP) of the University of California Santa Barbara for hospitality.', '0907.1345-4-15-2': 'This work was supported in part by the National Science Foundation under the grants DMR 0758462 (EF) at the University of Illinois, and PHY05-51164 (EF) at KITP, and by the Office of Science, U.S. Department of Energy under contracts DE-FG02-91ER45439 (EF, BMF) through the University of Illinois Frederick Seitz Materials Research Laboratory.'}",
quant-ph-0208015,"{'quant-ph-0208015-1-0-0': 'We study optimal teleportation based on the Bell measurements.', 'quant-ph-0208015-1-0-1': 'An explicit expression for the quantum channel associated with the optimal teleportation with an arbitrary mixed state resource is presented.', 'quant-ph-0208015-1-0-2': 'The optimal transmission fidelity of the corresponding quantum channel is calculated and shown to be related to the fully entangled fraction of the quantum resource, rather than the singlet fraction as in the standard teleportation protocol.', 'quant-ph-0208015-1-1-0': '1truecm PACS: 03.67.', 'quant-ph-0208015-1-1-1': 'Hk; 89.70.', 'quant-ph-0208015-1-1-2': '+c; 03.65.-w', 'quant-ph-0208015-1-2-0': 'AMS classification: 81P68; 68P30', 'quant-ph-0208015-1-3-0': 'def22U_q(A^(2)_2)', 'quant-ph-0208015-1-4-0': 'Quantum teleportation protocols play an important role in quantum information processing.', 'quant-ph-0208015-1-4-1': 'In terms of a classical communication channel and a quantum resource (a nonlocal entangled state like an EPR-pair of particles), the teleportation protocol gives ways to transmit an unknown quantum state from a sender traditionally named ""Alice"" to a receiver ""Bob"" who are spatially separated.', 'quant-ph-0208015-1-4-2': 'These teleportation processes can be viewed as quantum channels.', 'quant-ph-0208015-1-4-3': 'The nature of a quantum channel is determined by the particular protocol and the state used as a teleportation resource [CITATION].', 'quant-ph-0208015-1-4-4': 'The standard teleportation protocol [MATH] proposed in [CITATION] uses Bell measurements and Pauli rotations.', 'quant-ph-0208015-1-4-5': 'When the maximally entangled pure state [MATH] is used as the quantum resource, it provides an ideal noiseless quantum channel [MATH].', 'quant-ph-0208015-1-4-6': 'However in realistic situation, instead of the pure maximally entangled states, Alice and Bob usually share a mixed entangled state due to the decoherence.', 'quant-ph-0208015-1-4-7': 'Teleportation using mixed state as an entangled resource is, in general, equivalent to having a noisy quantum channel.', 'quant-ph-0208015-1-4-8': 'Recently, an explicit expression for the output state of the quantum channel associated with the standard teleportation protocol [MATH] with an arbitrary mixed state resource has been obtained [CITATION].', 'quant-ph-0208015-1-5-0': 'In this paper, we consider the following problem.', 'quant-ph-0208015-1-5-1': 'Alice and Bob previously only share a pair of particles in an arbitrary mixed entangled state [MATH].', 'quant-ph-0208015-1-5-2': 'In order to teleport an unknown state to Bob, Alice first performs a joint Bell measurement on her particles (particle 1 and particle 2) and tell her result to Bob by the classical communication channel.', 'quant-ph-0208015-1-5-3': 'Then Bob, instead of the Pauli rotation like in the standard teleportation protocol [CITATION], tries his best to choose an particular unitary transformation which depends on the quantum resource [MATH], so as to get the maximal transmission fidelity.', 'quant-ph-0208015-1-5-4': 'We call our teleportation protocol the optimal teleportation based on the Bell measurement.', 'quant-ph-0208015-1-5-5': 'We derive an explicit expression for the quantum channel associated with the optimal teleportation with an arbitrary mixed state resource.', 'quant-ph-0208015-1-5-6': 'The transmission fidelity of the corresponding quantum channel is given in term of the fully entangled fraction of the quantum resource.', 'quant-ph-0208015-1-6-0': 'Let [MATH], [MATH], be an orthogonal normalized basis of an [MATH]-dimensional Hilbert space [MATH].', 'quant-ph-0208015-1-6-1': 'Any linear operator [MATH] can be represented by an [MATH]-matrix as follow: [EQUATION]', 'quant-ph-0208015-1-6-2': 'We shall only consider the following three-tensor Hilbert space: [MATH] where Alice has the first and the second Hilbert space, and the third one belongs to Bob.', 'quant-ph-0208015-1-6-3': 'Let [MATH] and [MATH] be [MATH] matrices such that [MATH], with [MATH].', 'quant-ph-0208015-1-6-4': 'We can introduce [MATH] linear-independent [MATH]-matrices [MATH], which satisfy [EQUATION]', 'quant-ph-0208015-1-6-5': 'One can also check that [MATH]satisfy the condition of bases of the unitary operators in the sense of [CITATION], i.e. [EQUATION] where [MATH] is the [MATH] identity matrix.', 'quant-ph-0208015-1-6-6': '[MATH] form a complete basis of [MATH]-matrices, namely, for any [MATH] matrix [MATH], [MATH] can be expressed as [EQUATION]', 'quant-ph-0208015-1-6-7': 'From [MATH], we can introduce the generalized Bell-states, [EQUATION] [MATH] are all maximally entangled states and form a complete orthogonal normalized basis of [MATH] shared by Alice and Bob.', 'quant-ph-0208015-1-6-8': 'For any state [MATH] shared by Alice and Bob, let us introduce the singlet fraction [CITATION]: [MATH].', 'quant-ph-0208015-1-6-9': 'In general, all the maximally entangled pure states are equivalent to [MATH]: [MATH], where [MATH] is a unitary transformation.', 'quant-ph-0208015-1-6-10': 'One can define the fully entangled fraction [CITATION] of a state [MATH] by [EQUATION]', 'quant-ph-0208015-1-6-11': 'Since the group of unitary transformations in n-dimensions is compact, there exists an unitary matrix [MATH] such that [EQUATION]', 'quant-ph-0208015-1-6-12': 'Suppose now Alice and Bob previously share a pair of particles in an arbitrary mixed entangled state [MATH].', 'quant-ph-0208015-1-6-13': 'To transform an unknown state to Bob, Alice first performs a joint Bell measurement based on the generalized Bell-states Eq. ([REF]) on her parties.', 'quant-ph-0208015-1-6-14': 'According to the measurement results of Alice, Bob chooses particular unitary transformations [MATH] to act on his particle.', 'quant-ph-0208015-1-7-0': 'The teleportation protocol defined by [MATH], when used with an arbitrary mixed state with density matrix [MATH] as a resource, acts as a quantum channel [EQUATION]', 'quant-ph-0208015-1-7-1': 'Proof.', 'quant-ph-0208015-1-7-2': 'The proof can be given in two steps:', 'quant-ph-0208015-1-8-0': 'Step 1.', 'quant-ph-0208015-1-8-1': 'Pure entangled state as a resource.', 'quant-ph-0208015-1-8-2': 'Each entangled pure state [MATH] shared by Alice and Bob has the form [EQUATION]', 'quant-ph-0208015-1-8-3': 'Let [MATH] be the [MATH] matrix with elements [MATH], [MATH].', 'quant-ph-0208015-1-8-4': 'Suppose Alice wishes to teleport the unknown pure state [MATH].', 'quant-ph-0208015-1-8-5': 'The initial state Alice and Bob have is then given by [EQUATION]', 'quant-ph-0208015-1-8-6': 'To transform the state [MATH] to Bob, Alice first performs a joint Bell measurement based on the generalized Bell-states Eq. ([REF]) on her party.', 'quant-ph-0208015-1-8-7': ""After her measurement with outcoming in the state [MATH], Bob's particle gets into an (unnormalized) state [EQUATION]"", 'quant-ph-0208015-1-8-8': 'Once Bob learns from Alice that she has obtained the result [MATH], he performs on his previously entangled particle (particle 3) a unitary transformation [MATH].', 'quant-ph-0208015-1-8-9': 'Then the final state becomes [MATH].', 'quant-ph-0208015-1-8-10': 'In terms of the density matrix, the teleportation based on the unitary matrices [MATH] , with quantum resource being a pure state [MATH], is a quantum channel with the output [EQUATION]', 'quant-ph-0208015-1-8-11': 'Step 2.', 'quant-ph-0208015-1-8-12': 'An arbitrary mixed entangled state as a resource.', 'quant-ph-0208015-1-8-13': 'Let [MATH] be a mixed state, [EQUATION]', 'quant-ph-0208015-1-8-14': 'Applying the teleportation protocol [MATH] with a mixed state [MATH], the final state of Bob becomes [EQUATION]', 'quant-ph-0208015-1-8-15': 'Since each matrix [MATH] can be decomposed in the basis of [MATH] by [MATH], ([REF]) becomes [EQUATION]', 'quant-ph-0208015-1-8-16': 'Using the definition of generalized Bell-states [MATH] in Eq.([REF]), after a lenthy calculation, we arrive at [EQUATION]', 'quant-ph-0208015-1-8-17': 'Substituting the above results into Eq.([REF]), one obtains Eq.([REF]).', 'quant-ph-0208015-1-8-18': 'Using Eq.([REF]) and the identity [EQUATION] the trace-preserving property of the quantum channel can be proved by [EQUATION] [MATH]', 'quant-ph-0208015-1-9-0': 'The fidelity of the teleportation is given by [EQUATION] averaged over all pure input states [MATH].', 'quant-ph-0208015-1-10-0': ""In order to calculate the transmission fidelity Eq. ([REF]), we need an irreducible n-dimensional representation of the unitary group [MATH], denoted by G. Let [MATH] be the unitary matrix representation of the element [MATH] of G. Recalling Schur's Lemma, one has the identity [EQUATION] for any operator [MATH] acting on the tensor space, where [MATH] is the flip operator such that [MATH]."", 'quant-ph-0208015-1-10-1': 'The invariant (Haar) measure [MATH] on [MATH] is normalized by [MATH].', 'quant-ph-0208015-1-11-0': 'The transmission fidelity of the teleportation protocol defined by [MATH] with arbitrary mixed state [MATH] as a resource is given by [EQUATION] 0.4truecm Proof.', 'quant-ph-0208015-1-11-1': 'From Theorem 1 and Eq. ([REF]), one has [EQUATION] where the identity [MATH], Eq. ([REF]) and Eq. ([REF]) have been used.', 'quant-ph-0208015-1-11-2': '[MATH]', 'quant-ph-0208015-1-12-0': 'Obviously when the term [MATH] is maximized, i.e., [MATH], one gets the maximal fidelity.', 'quant-ph-0208015-1-12-1': 'Recalling the definition of the fully entangled fraction Eq. ([REF]) and Eq. ([REF]), we arrive at our main result:', 'quant-ph-0208015-1-13-0': 'The optimal teleportation based on the Bell measurements, when used with an arbitrary mixed state with density matrix [MATH] as a resource, acts as a general trace-preserving quantum channel [EQUATION]', 'quant-ph-0208015-1-13-1': 'The corresponding transmission fidelity is given by [EQUATION] where [MATH] is the fully entangled fraction Eq. ([REF])and [MATH] is the unitary matrix which fulfills such a fully entangled fraction Eq. ([REF]).', 'quant-ph-0208015-1-14-0': 'Our results show that the maximally transmission fidelity of the teleportation based on the Bell measurement depends on the fully entangled fraction only, whereas that of a standard teleportation depends on the singlet fraction [CITATION].', 'quant-ph-0208015-1-14-1': 'Our result also agrees with the fidelity formula of the general optimal teleportation given by the Horodecki family [CITATION].', 'quant-ph-0208015-1-15-0': 'Summarizing, we obtain the explicit expression of the output state of the optimal teleportation, with arbitrary mixed entangled state as resource, in terms of some noisy quantum channel.', 'quant-ph-0208015-1-15-1': 'This allow us to calculate the transmission fidelity of the quantum channel.', 'quant-ph-0208015-1-15-2': 'It is shown that the transmission fidelity depends only on the fully entangled fraction of the quantum resource shared by the sender and the receiver.', 'quant-ph-0208015-1-15-3': 'The fidelity in our optimal teleportation protocol is in general greater than the one in standard teleportation protocol [CITATION].', 'quant-ph-0208015-1-16-0': '1.0truecm'}","{'quant-ph-0208015-2-0-0': 'We study optimal teleportation based on the Bell measurements.', 'quant-ph-0208015-2-0-1': 'An explicit expression for the quantum channel associated with the optimal teleportation with an arbitrary mixed state resource is presented.', 'quant-ph-0208015-2-0-2': 'The optimal transmission fidelity of the corresponding quantum channel is calculated and shown to be related to the fully entangled fraction of the quantum resource, rather than the singlet fraction as in the standard teleportation protocol.', 'quant-ph-0208015-2-1-0': '1truecm PACS: 03.67.', 'quant-ph-0208015-2-1-1': 'Hk; 89.70.', 'quant-ph-0208015-2-1-2': '+c; 03.65.-w', 'quant-ph-0208015-2-2-0': 'AMS classification: 81P68; 68P30', 'quant-ph-0208015-2-3-0': 'def22U_q(A^(2)_2)', 'quant-ph-0208015-2-4-0': 'Quantum teleportation protocols play an important role in quantum information processing.', 'quant-ph-0208015-2-4-1': 'In terms of a classical communication channel and a quantum resource (a nonlocal entangled state like an EPR-pair of particles), the teleportation protocol gives ways to transmit an unknown quantum state from a sender traditionally named ""Alice"" to a receiver ""Bob"" who are spatially separated.', 'quant-ph-0208015-2-4-2': 'These teleportation processes can be viewed as quantum channels.', 'quant-ph-0208015-2-4-3': 'The nature of a quantum channel is determined by the particular protocol and the state used as a teleportation resource [CITATION].', 'quant-ph-0208015-2-4-4': 'The standard teleportation protocol [MATH] proposed in [CITATION] uses Bell measurements and Pauli rotations.', 'quant-ph-0208015-2-4-5': 'When the maximally entangled pure state [MATH] is used as the quantum resource, it provides an ideal noiseless quantum channel [MATH].', 'quant-ph-0208015-2-4-6': 'However in realistic situation, instead of the pure maximally entangled states, Alice and Bob usually share a mixed entangled state due to the decoherence.', 'quant-ph-0208015-2-4-7': 'Teleportation using mixed state as an entangled resource is, in general, equivalent to having a noisy quantum channel.', 'quant-ph-0208015-2-4-8': 'Recently, an explicit expression for the output state of the quantum channel associated with the standard teleportation protocol [MATH] with an arbitrary mixed state resource has been obtained [CITATION].', 'quant-ph-0208015-2-5-0': 'In this paper, we consider the following problem.', 'quant-ph-0208015-2-5-1': 'Alice and Bob previously only share a pair of particles in an arbitrary mixed entangled state [MATH].', 'quant-ph-0208015-2-5-2': 'In order to teleport an unknown state to Bob, Alice first performs a joint Bell measurement on her particles (particle 1 and particle 2) and tell her result to Bob by the classical communication channel.', 'quant-ph-0208015-2-5-3': 'Then Bob, instead of the Pauli rotation like in the standard teleportation protocol [CITATION], tries his best to choose an particular unitary transformation which depends on the quantum resource [MATH], so as to get the maximal transmission fidelity.', 'quant-ph-0208015-2-5-4': 'We call our teleportation protocol the optimal teleportation based on the Bell measurement.', 'quant-ph-0208015-2-5-5': 'We derive an explicit expression for the quantum channel associated with the optimal teleportation with an arbitrary mixed state resource.', 'quant-ph-0208015-2-5-6': 'The transmission fidelity of the corresponding quantum channel is given in term of the fully entangled fraction of the quantum resource.', 'quant-ph-0208015-2-6-0': 'Let [MATH], [MATH], be an orthogonal normalized basis of an [MATH]-dimensional Hilbert space [MATH].', 'quant-ph-0208015-2-6-1': 'Any linear operator [MATH] can be represented by an [MATH]-matrix as follow: [EQUATION]', 'quant-ph-0208015-2-6-2': 'We shall only consider the following three-tensor Hilbert space: [MATH] where Alice has the first and the second Hilbert space, and the third one belongs to Bob.', 'quant-ph-0208015-2-6-3': 'Let [MATH] and [MATH] be [MATH] matrices such that [MATH], with [MATH].', 'quant-ph-0208015-2-6-4': 'We can introduce [MATH] linear-independent [MATH]-matrices [MATH], which satisfy [EQUATION]', 'quant-ph-0208015-2-6-5': 'One can also check that [MATH]satisfy the condition of bases of the unitary operators in the sense of [CITATION], i.e. [EQUATION] where [MATH] is the [MATH] identity matrix.', 'quant-ph-0208015-2-6-6': '[MATH] form a complete basis of [MATH]-matrices, namely, for any [MATH] matrix [MATH], [MATH] can be expressed as [EQUATION]', 'quant-ph-0208015-2-6-7': 'From [MATH], we can introduce the generalized Bell-states, [EQUATION] [MATH] are all maximally entangled states and form a complete orthogonal normalized basis of [MATH] shared by Alice and Bob.', 'quant-ph-0208015-2-6-8': 'For any state [MATH] shared by Alice and Bob, let us introduce the singlet fraction [CITATION]: [MATH].', 'quant-ph-0208015-2-6-9': 'In general, all the maximally entangled pure states are equivalent to [MATH]: [MATH], where [MATH] is a unitary transformation.', 'quant-ph-0208015-2-6-10': 'One can define the fully entangled fraction [CITATION] of a state [MATH] by [EQUATION]', 'quant-ph-0208015-2-6-11': 'Since the group of unitary transformations in n-dimensions is compact, there exists an unitary matrix [MATH] such that [EQUATION]', 'quant-ph-0208015-2-6-12': 'Suppose now Alice and Bob previously share a pair of particles in an arbitrary mixed entangled state [MATH].', 'quant-ph-0208015-2-6-13': 'To transform an unknown state to Bob, Alice first performs a joint Bell measurement based on the generalized Bell-states Eq. ([REF]) on her parties.', 'quant-ph-0208015-2-6-14': 'According to the measurement results of Alice, Bob chooses particular unitary transformations [MATH] to act on his particle.', 'quant-ph-0208015-2-7-0': 'The teleportation protocol defined by [MATH], when used with an arbitrary mixed state with density matrix [MATH] as a resource, acts as a quantum channel [EQUATION]', 'quant-ph-0208015-2-7-1': 'Proof.', 'quant-ph-0208015-2-7-2': 'The proof can be given in two steps:', 'quant-ph-0208015-2-8-0': 'Step 1.', 'quant-ph-0208015-2-8-1': 'Pure entangled state as a resource.', 'quant-ph-0208015-2-8-2': 'Each entangled pure state [MATH] shared by Alice and Bob has the form [EQUATION]', 'quant-ph-0208015-2-8-3': 'Let [MATH] be the [MATH] matrix with elements [MATH], [MATH].', 'quant-ph-0208015-2-8-4': 'Suppose Alice wishes to teleport the unknown pure state [MATH].', 'quant-ph-0208015-2-8-5': 'The initial state Alice and Bob have is then given by [EQUATION]', 'quant-ph-0208015-2-8-6': 'To transform the state [MATH] to Bob, Alice first performs a joint Bell measurement based on the generalized Bell-states Eq. ([REF]) on her party.', 'quant-ph-0208015-2-8-7': ""After her measurement with outcoming in the state [MATH], Bob's particle gets into an (unnormalized) state [EQUATION]"", 'quant-ph-0208015-2-8-8': 'Once Bob learns from Alice that she has obtained the result [MATH], he performs on his previously entangled particle (particle 3) a unitary transformation [MATH].', 'quant-ph-0208015-2-8-9': 'Then the final state becomes [MATH].', 'quant-ph-0208015-2-8-10': 'In terms of the density matrix, the teleportation based on the unitary matrices [MATH] , with quantum resource being a pure state [MATH], is a quantum channel with the output [EQUATION]', 'quant-ph-0208015-2-8-11': 'Step 2.', 'quant-ph-0208015-2-8-12': 'An arbitrary mixed entangled state as a resource.', 'quant-ph-0208015-2-8-13': 'Let [MATH] be a mixed state, [EQUATION]', 'quant-ph-0208015-2-8-14': 'Applying the teleportation protocol [MATH] with a mixed state [MATH], the final state of Bob becomes [EQUATION]', 'quant-ph-0208015-2-8-15': 'Since each matrix [MATH] can be decomposed in the basis of [MATH] by [MATH], ([REF]) becomes [EQUATION]', 'quant-ph-0208015-2-8-16': 'Using the definition of generalized Bell-states [MATH] in Eq.([REF]), after a lenthy calculation, we arrive at [EQUATION]', 'quant-ph-0208015-2-8-17': 'Substituting the above results into Eq.([REF]), one obtains Eq.([REF]).', 'quant-ph-0208015-2-8-18': 'Using Eq.([REF]) and the identity [EQUATION] the trace-preserving property of the quantum channel can be proved by [EQUATION] [MATH]', 'quant-ph-0208015-2-9-0': 'The fidelity of the teleportation is given by [EQUATION] averaged over all pure input states [MATH].', 'quant-ph-0208015-2-10-0': ""In order to calculate the transmission fidelity Eq. ([REF]), we need an irreducible n-dimensional representation of the unitary group [MATH], denoted by G. Let [MATH] be the unitary matrix representation of the element [MATH] of G. Recalling Schur's Lemma, one has the identity [EQUATION] for any operator [MATH] acting on the tensor space, where [MATH] is the flip operator such that [MATH]."", 'quant-ph-0208015-2-10-1': 'The invariant (Haar) measure [MATH] on [MATH] is normalized by [MATH].', 'quant-ph-0208015-2-11-0': 'The transmission fidelity of the teleportation protocol defined by [MATH] with arbitrary mixed state [MATH] as a resource is given by [EQUATION] 0.4truecm Proof.', 'quant-ph-0208015-2-11-1': 'From Theorem 1 and Eq. ([REF]), one has [EQUATION] where the identity [MATH], Eq. ([REF]) and Eq. ([REF]) have been used.', 'quant-ph-0208015-2-11-2': '[MATH]', 'quant-ph-0208015-2-12-0': 'Obviously when the term [MATH] is maximized, i.e., [MATH], one gets the maximal fidelity.', 'quant-ph-0208015-2-12-1': 'Recalling the definition of the fully entangled fraction Eq. ([REF]) and Eq. ([REF]), we arrive at our main result:', 'quant-ph-0208015-2-13-0': 'The optimal teleportation based on the Bell measurements, when used with an arbitrary mixed state with density matrix [MATH] as a resource, acts as a general trace-preserving quantum channel [EQUATION]', 'quant-ph-0208015-2-13-1': 'The corresponding transmission fidelity is given by [EQUATION] where [MATH] is the fully entangled fraction Eq. ([REF])and [MATH] is the unitary matrix which fulfills such a fully entangled fraction Eq. ([REF]).', 'quant-ph-0208015-2-14-0': 'Our results show that the maximally transmission fidelity of the teleportation based on the Bell measurement depends on the fully entangled fraction only, whereas that of a standard teleportation depends on the singlet fraction [CITATION].', 'quant-ph-0208015-2-14-1': 'Our result also agrees with the fidelity formula of the general optimal teleportation given by the Horodecki family [CITATION].', 'quant-ph-0208015-2-15-0': 'Summarizing, we obtain the explicit expression of the output state of the optimal teleportation, with arbitrary mixed entangled state as resource, in terms of some noisy quantum channel.', 'quant-ph-0208015-2-15-1': 'This allow us to calculate the transmission fidelity of the quantum channel.', 'quant-ph-0208015-2-15-2': 'It is shown that the transmission fidelity depends only on the fully entangled fraction of the quantum resource shared by the sender and the receiver.', 'quant-ph-0208015-2-15-3': 'The fidelity in our optimal teleportation protocol is in general greater than the one in standard teleportation protocol [CITATION].', 'quant-ph-0208015-2-16-0': '1.0truecm'}","[['quant-ph-0208015-1-7-0', 'quant-ph-0208015-2-7-0'], ['quant-ph-0208015-1-12-0', 'quant-ph-0208015-2-12-0'], ['quant-ph-0208015-1-11-0', 'quant-ph-0208015-2-11-0'], ['quant-ph-0208015-1-11-1', 'quant-ph-0208015-2-11-1'], ['quant-ph-0208015-1-13-0', 'quant-ph-0208015-2-13-0'], ['quant-ph-0208015-1-13-1', 'quant-ph-0208015-2-13-1'], ['quant-ph-0208015-1-14-0', 'quant-ph-0208015-2-14-0'], ['quant-ph-0208015-1-14-1', 'quant-ph-0208015-2-14-1'], ['quant-ph-0208015-1-15-0', 'quant-ph-0208015-2-15-0'], ['quant-ph-0208015-1-15-1', 'quant-ph-0208015-2-15-1'], ['quant-ph-0208015-1-15-2', 'quant-ph-0208015-2-15-2'], ['quant-ph-0208015-1-15-3', 'quant-ph-0208015-2-15-3'], ['quant-ph-0208015-1-5-0', 'quant-ph-0208015-2-5-0'], ['quant-ph-0208015-1-5-1', 'quant-ph-0208015-2-5-1'], ['quant-ph-0208015-1-5-2', 'quant-ph-0208015-2-5-2'], ['quant-ph-0208015-1-5-3', 'quant-ph-0208015-2-5-3'], ['quant-ph-0208015-1-5-4', 'quant-ph-0208015-2-5-4'], ['quant-ph-0208015-1-5-5', 'quant-ph-0208015-2-5-5'], ['quant-ph-0208015-1-5-6', 'quant-ph-0208015-2-5-6'], ['quant-ph-0208015-1-10-0', 'quant-ph-0208015-2-10-0'], ['quant-ph-0208015-1-10-1', 'quant-ph-0208015-2-10-1'], ['quant-ph-0208015-1-0-0', 'quant-ph-0208015-2-0-0'], ['quant-ph-0208015-1-0-1', 'quant-ph-0208015-2-0-1'], ['quant-ph-0208015-1-0-2', 'quant-ph-0208015-2-0-2'], ['quant-ph-0208015-1-8-1', 'quant-ph-0208015-2-8-1'], ['quant-ph-0208015-1-8-2', 'quant-ph-0208015-2-8-2'], ['quant-ph-0208015-1-8-3', 'quant-ph-0208015-2-8-3'], ['quant-ph-0208015-1-8-4', 'quant-ph-0208015-2-8-4'], ['quant-ph-0208015-1-8-5', 'quant-ph-0208015-2-8-5'], ['quant-ph-0208015-1-8-6', 'quant-ph-0208015-2-8-6'], ['quant-ph-0208015-1-8-7', 'quant-ph-0208015-2-8-7'], ['quant-ph-0208015-1-8-8', 'quant-ph-0208015-2-8-8'], ['quant-ph-0208015-1-8-9', 'quant-ph-0208015-2-8-9'], 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quant-ph-0209166,"{'quant-ph-0209166-1-0-0': 'We present a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via local operations and classical communication (LOCC).', 'quant-ph-0209166-1-0-1': 'This framework unifies and greatly simplifies the results of Refs. [CITATION].', 'quant-ph-0209166-1-0-2': 'A new necessary condition for ""pure contraction"" transformation is given.', 'quant-ph-0209166-1-0-3': 'Finally, constructive protocols to achieve both probabilistic and deterministic entanglement transformations are presented.', 'quant-ph-0209166-1-1-0': '# Introduction', 'quant-ph-0209166-1-2-0': 'The transformation of entangled states by means of LOCC is a key issue in quantum information processing [CITATION], for quantum computation [CITATION], quantum teleportation [CITATION], and quantum cryptography [CITATION].', 'quant-ph-0209166-1-2-1': 'However, the detrimental effect of losses and decoherence poses a serious problem for establishing entangled resources at distance or in a long computing network, since in a realistic transmission/computation the entanglement can be considerably degraded, thus preventing, for example, successful teleportation or dense coding.', 'quant-ph-0209166-1-2-2': 'For this reason, the use of transformations which can increase the available entanglement by means of LOCC-although with some probability-is crucial for practical purposes.', 'quant-ph-0209166-1-2-3': 'More generally, understanding entanglement transformations that are allowed by LOCC provides more insight in the structure and property of nonlocality, the most prominent character of quantum mechanics.', 'quant-ph-0209166-1-2-4': 'On entanglement transformations via LOCC there are two main results.', 'quant-ph-0209166-1-2-5': 'The first is the seminal work by Nielsen [CITATION] on deterministic transformations, which introduces majorization theory in this context.', 'quant-ph-0209166-1-2-6': 'The second is the work by Vidal [CITATION], which addresses more general probabilistic transformations.This work is based on the approach of entanglement monotones and gives conditions equivalent to weak majorization relations.', 'quant-ph-0209166-1-2-7': 'The two approaches are completely disconnected, and for practical applications of the theory, a unified framework would be needed, especially in consideration that the more general treatment by Vidal is more abstract and less constructive than Nielsen approach, which, however, is limited only to deterministic transformations.', 'quant-ph-0209166-1-2-8': 'Furthermore, from Vidal approach it is very difficult to recover Nielsen treatment as a special case, and it is quite surprising that such approach ends with a weak-majorization condition, without essentially using majorization theory.', 'quant-ph-0209166-1-2-9': 'This motivates a derivation of the general nondeterministic LOCC transformations of pure states with an approach completely based on majorization theory, generalizing Nielsen work [CITATION].', 'quant-ph-0209166-1-3-0': 'In this paper we present a general framework for entanglement transformations of bipartite pure states by means of LOCC.', 'quant-ph-0209166-1-3-1': 'In Sec. II we give a short and very simple proof of the theorem of Lo and Popescu [CITATION], which is at the basis of the theory of all LOCC, and which states that given two separate parties, say Alice and Bob, all LOCC transformations on a pure bipartite entangled state can be reduced to a contraction by Alice and a unitary transformation by Bob.', 'quant-ph-0209166-1-3-2': 'We include our derivation of this theorem, since it is particularly simple and is based on a new useful technique for operator transposition.', 'quant-ph-0209166-1-3-3': 'In Sec. III we derive the main theorem, which gives a necessary and sufficient condition for all entanglement transformations of pure states in terms of super-majorization conditions, generalizing Nielsen approach, and recovering the more general result of Vidal.', 'quant-ph-0209166-1-3-4': 'Here we also provide a new necessary condition for ""pure contraction"" transformations, namely those LOCC transformations in which the target state is achieved just for a single outcome of Alice measurement.', 'quant-ph-0209166-1-3-5': 'Such condition writes in terms of sub-majorization relation.', 'quant-ph-0209166-1-3-6': 'In Sec. IV explicit protocols to achieve pure, deterministic, and probabilistic transformations are given, using a new method which gives Alice contraction of the LOCC in terms of the Moore-Penrose pseudoinverse of the matrix of the entangled state.', 'quant-ph-0209166-1-3-7': 'Section V concludes the paper summarizing the results.', 'quant-ph-0209166-1-4-0': '# The Lo-Popescu theorem', 'quant-ph-0209166-1-5-0': 'For later convenience we introduce here the main notation used in the paper.', 'quant-ph-0209166-1-5-1': 'Given a linear operator [MATH] we denote its hermitian conjugate by [MATH].', 'quant-ph-0209166-1-5-2': 'On a fixed basis, we write the complex conjugate and the transpose of [MATH] as [MATH] and [MATH], respectively, so that [MATH].', 'quant-ph-0209166-1-5-3': 'With the notation [MATH] we denote the Moore-Penrose inverse of [MATH].', 'quant-ph-0209166-1-5-4': 'We recall that the Moore-Penrose inverse [MATH] is the unique matrix that satisfies [EQUATION] and we notice that [MATH] is the orthogonal projector over [MATH], whereas [MATH] is the orthogonal projector over [MATH].', 'quant-ph-0209166-1-5-5': 'We write the singular value decomposition (SVD) of [MATH] as follows [EQUATION] where [MATH] denotes the diagonal matrix whose entries are the singular values [MATH] of [MATH] taken in decreasing order, and [MATH] are unitary.', 'quant-ph-0209166-1-5-6': 'We remember that a quantum measurement is generally described by a Positive Operator-Valued Measurement (POVM), namely by a resolution of the identity [MATH], where each [MATH] corresponds to the possible outcomes.', 'quant-ph-0209166-1-5-7': 'Each operator [MATH] acts on the input states and is necessarily a contraction, namely the usual operator norm satisfies [MATH].', 'quant-ph-0209166-1-5-8': 'For bipartite pure states on the Hilbert space [MATH] we use the following notation [EQUATION] where [MATH] and [MATH] are two chosen orthonormal bases for [MATH] and [MATH], respectively.', 'quant-ph-0209166-1-5-9': 'Eq. ([REF]) introduces an isomorphism between vectors in [MATH] and [MATH] matrices, where [MATH] and [MATH] are the dimensions of [MATH] and [MATH].', 'quant-ph-0209166-1-5-10': 'One can easily check the relation [EQUATION] where transposition is defined with respect to the orthonormal basis [MATH].', 'quant-ph-0209166-1-5-11': 'Finally, we use the notation [MATH] for Hermitian operators [MATH] and [MATH] to denote the vector majorization relation [CITATION] [MATH], and in the same fashion we will write [MATH] and [MATH] for super- and sub-majorization, respectively.', 'quant-ph-0209166-1-5-12': 'In the last part of this section we provide a very short proof of the following theorem [CITATION].', 'quant-ph-0209166-1-5-13': 'All LOCC on a pure bipartite entangled state [MATH] can be reduced to a contraction by Alice and a unitary transformation by Bob.', 'quant-ph-0209166-1-5-14': 'This resorts to the equivalence of any Bob contraction [MATH] with the Alice contraction [MATH] assisted by Bob unitary transformation [MATH] as follows [EQUATION] where [EQUATION] and [MATH] is the unitary operator achieving the transposition of the operator [MATH], namely [EQUATION]', 'quant-ph-0209166-1-5-15': 'Proof.', 'quant-ph-0209166-1-5-16': 'To prove that every LOCC can be reduced to an Alice contraction and a Bob unitary transformation it is sufficient to prove the equivalence ([REF]), since: a) all possible elementary LOCC in a sequence will be reduced to an Alice contraction and a Bob unitary; b) the product of two contractions is a contraction; c) unitary transformations are particular cases of contraction.', 'quant-ph-0209166-1-5-17': 'Notice that given the SVD of any linear operator [MATH] as in Eq. ([REF]) one has [EQUATION] with [MATH].', 'quant-ph-0209166-1-5-18': 'Using Eq. ([REF]) Eq. ([REF]) rewrites as follows [EQUATION]', 'quant-ph-0209166-1-5-19': 'Then, from Eq. ([REF]) one has [EQUATION] which is just Eq. ([REF]) with [MATH] and [MATH] given as in Eq. ([REF]).', 'quant-ph-0209166-1-5-20': '[MATH]', 'quant-ph-0209166-1-6-0': '# LOCC transformations for pure states', 'quant-ph-0209166-1-7-0': 'In this section we will use the following useful lemmas:', 'quant-ph-0209166-1-8-0': '[MATH] for some [MATH] and [MATH].', 'quant-ph-0209166-1-9-0': '[MATH] for some [MATH] and [MATH].', 'quant-ph-0209166-1-10-0': 'Proof.', 'quant-ph-0209166-1-11-0': 'If for some [MATH] and [MATH], then [MATH] [EQUATION] namely [MATH].', 'quant-ph-0209166-1-11-1': 'Viceversa, if [MATH] one has [EQUATION]', 'quant-ph-0209166-1-11-2': 'Upon defining [EQUATION] clearly [MATH] and [MATH] [EQUATION] namely [MATH].', 'quant-ph-0209166-1-11-3': '[MATH]The second lemma can be proved along similar lines.', 'quant-ph-0209166-1-11-4': 'Moreover, we will make extensive use of the following theorem: [Uhlmann] For Hermitian operators [MATH] and [MATH] one has [MATH] if and only if there are probabilities [MATH] and unitary operators [MATH] such that [EQUATION]', 'quant-ph-0209166-1-11-5': 'Proof.', 'quant-ph-0209166-1-11-6': 'See Ref. [CITATION], pag.', 'quant-ph-0209166-1-11-7': '575.', 'quant-ph-0209166-1-11-8': 'We are now ready to prove the main theorem:', 'quant-ph-0209166-1-12-0': 'The state transformation [MATH] is possible by LOCC iff [EQUATION] where [MATH] is the probability of achieving the transformation.', 'quant-ph-0209166-1-12-1': 'A necessary condition to be satisfied is [MATH].', 'quant-ph-0209166-1-12-2': 'In particular, the transformation is deterministic ([MATH]) iff [MATH].', 'quant-ph-0209166-1-12-3': 'Finally, if there is a pure LOCC that achieves the state transformation with probability [MATH], we must have [EQUATION]', 'quant-ph-0209166-1-12-4': 'Proof.', 'quant-ph-0209166-1-12-5': 'According to Theorem [REF] every LOCC is equivalent to a measurement performed by Alice followed by a conditional unitary by Bob.', 'quant-ph-0209166-1-12-6': 'Therefore, the joint Alice-Bob state [MATH] will evolve as follows [EQUATION] where [EQUATION]', 'quant-ph-0209166-1-12-7': 'If we want to transform the state [MATH] into [MATH] we must have for a subset [MATH] of the possible outcomes [MATH] [EQUATION] where [MATH] denotes the overall probability of the transformation [MATH].', 'quant-ph-0209166-1-12-8': 'From Eq. ([REF]) we need to have [EQUATION] and, therefore, each probability [MATH] is given by [EQUATION] where [MATH] denotes the usual Frobenius norm.', 'quant-ph-0209166-1-12-9': 'The condition ([REF]) can be satisfied only if [MATH], i.e. we can only decrease the Schmidt number of the entangled state.', 'quant-ph-0209166-1-12-10': 'From Eq. ([REF]) one has [MATH] [EQUATION]', 'quant-ph-0209166-1-12-11': 'Therefore we have [EQUATION] and using Eq. ([REF]) we obtain [EQUATION]', 'quant-ph-0209166-1-12-12': 'By Uhlmann Theorem [REF] we have [EQUATION] and from Lemma [REF] we get the statement, namely [MATH].', 'quant-ph-0209166-1-12-13': 'Reversely, suppose that [MATH].', 'quant-ph-0209166-1-12-14': 'From Lemma [REF] there is an operator [MATH] with [MATH] and [MATH].', 'quant-ph-0209166-1-12-15': 'Then, Uhlmann Theorem [REF] guarantees the existence of a set of unitaries [MATH] and probabilities [MATH] such that [EQUATION]', 'quant-ph-0209166-1-12-16': 'Now, define Alice measurement [MATH] such that [EQUATION] namely [EQUATION]', 'quant-ph-0209166-1-12-17': 'We can always choose [MATH] such that [MATH], and show that [MATH] is a contraction since [EQUATION] and so [MATH].', 'quant-ph-0209166-1-12-18': 'Then, there exists a set of unitary operators [MATH] such that [MATH], namely [EQUATION] so that the transformation from [MATH] can be achieved deterministically.', 'quant-ph-0209166-1-12-19': 'Now, since [MATH] one can define the contraction [EQUATION] so that [EQUATION]', 'quant-ph-0209166-1-12-20': 'By using the SVD of both [MATH] and [MATH], Eq. ([REF]) rewrites as follows [EQUATION] which means that there exists a unitary transformation [MATH] such that [EQUATION] where [MATH].', 'quant-ph-0209166-1-12-21': 'Eq. ([REF]) is equivalent to the entanglement transformation [EQUATION] namely there is a LOCC occurring with probability [MATH] that transforms the state [MATH] into [MATH].', 'quant-ph-0209166-1-12-22': 'But, as we have seen before, the transformation [MATH] can be achieved deterministically, whence we conclude that [MATH] can be achieved probabilistically, namely the statement of the first part of the theorem.', 'quant-ph-0209166-1-12-23': 'Notice that along similar lines, using Lemma [REF] one can equivalently prove the existence of a transformation [MATH] that can be achieved via a contraction that transforms [MATH] in [MATH] with probability [MATH], followed by a deterministic transformation [MATH].', 'quant-ph-0209166-1-12-24': 'We now prove the last part of the theorem, regarding the pure LOCC case.', 'quant-ph-0209166-1-12-25': 'In a pure contraction transformation the target state is achieved just for a single outcome of Alice measurement.', 'quant-ph-0209166-1-12-26': 'Such a transformation that occurs with probability [MATH] is given by [EQUATION] and we need to have [EQUATION]', 'quant-ph-0209166-1-12-27': 'Again, this is possible if [MATH].', 'quant-ph-0209166-1-12-28': 'Using the SVD of [MATH] and [MATH] as in Eq. ([REF]), Eq. ([REF]) rewrites in terms of the diagonal matrices [MATH] and [MATH] as follows [EQUATION] with [EQUATION]', 'quant-ph-0209166-1-12-29': 'Eq. ([REF]) leads to [EQUATION] namely [EQUATION] where [MATH] is a sub-stochastic matrix, since', 'quant-ph-0209166-1-13-0': 'This prove that Eq. ([REF]) is a necessary condition for the LOCC transformation ([REF]), namely the last statement of the theorem.', 'quant-ph-0209166-1-13-1': '[MATH]The above theorem unifies and simplifies the result of Refs. [CITATION] and [CITATION].', 'quant-ph-0209166-1-13-2': 'The sub-majorization necessary condition is new.', 'quant-ph-0209166-1-14-0': '# The explicit protocols', 'quant-ph-0209166-1-15-0': '## Pure transformation', 'quant-ph-0209166-1-16-0': 'Pure LOCC transformations are achieved by a single contraction.', 'quant-ph-0209166-1-16-1': 'In this case we have the following theorem', 'quant-ph-0209166-1-17-0': 'The transformation [MATH] can be achieved with probability [MATH] by a single contraction on Alice side iff one can find a unitary operator [MATH] and linear operator [MATH] such that [EQUATION] is a contraction.', 'quant-ph-0209166-1-18-0': 'Proof.', 'quant-ph-0209166-1-18-1': 'Notice that the equation [EQUATION] is equivalent to the identity [EQUATION]', 'quant-ph-0209166-1-18-2': 'Since both sides of the identity must have the same kernel it follows [EQUATION] and, multiplying both sides by [MATH], we have [EQUATION]', 'quant-ph-0209166-1-18-3': 'The general solution of the last equation is [EQUATION] with arbitrary [MATH], and in fact one can easily check that [EQUATION]', 'quant-ph-0209166-1-18-4': 'The unitary [MATH] and the operator [MATH] should be taken such that [MATH] is a contraction.', 'quant-ph-0209166-1-18-5': 'This is not always possible.', 'quant-ph-0209166-1-18-6': 'However, a sufficient condition is [EQUATION] [MATH] ## Deterministic transformation', 'quant-ph-0209166-1-19-0': 'In the entanglement transformations, in order to perform the first part of the protocol, namely the deterministic transformation from [MATH] to [MATH], one needs to find the contractions [MATH] and the unitary operators [MATH] versus the unitary operators [MATH] that appear in Eq. ([REF]), such that one has [EQUATION]', 'quant-ph-0209166-1-19-1': 'We have seen that the general form of the solution of Eq. ([REF]) is given by [EQUATION]', 'quant-ph-0209166-1-19-2': 'We have now the following theorem', 'quant-ph-0209166-1-20-0': 'In Eq. ([REF]) we can always take [EQUATION] where [MATH] are the operators defined in Eq. ([REF]), such that [MATH] is a contraction and Eq. ([REF]) is satisfied.', 'quant-ph-0209166-1-21-0': 'Proof.', 'quant-ph-0209166-1-21-1': 'Substituting Eq. ([REF]) in Eq. ([REF]) one has [EQUATION]', 'quant-ph-0209166-1-21-2': 'From Eq. ([REF]) it follows [EQUATION]', 'quant-ph-0209166-1-21-3': 'Hence, [MATH] are contractions.', 'quant-ph-0209166-1-21-4': 'The completeness of the measurement can be guaranteed by the further contraction [EQUATION] where [MATH] is an arbitrary unitary operator.', 'quant-ph-0209166-1-21-5': 'For any outcome [MATH] on Alice side, Bob performs the unitary [MATH].', 'quant-ph-0209166-1-21-6': 'Using Eq. ([REF]) and ([REF]) one has [EQUATION]', 'quant-ph-0209166-1-21-7': 'From Eq. ([REF]) it follows [EQUATION]', 'quant-ph-0209166-1-21-8': 'Hence, by multiplying both sides of Eq. ([REF]) on the left by [MATH] and on the right by [MATH] one obtains [EQUATION]', 'quant-ph-0209166-1-21-9': 'The projector on [MATH] coincides with the projector on [MATH].', 'quant-ph-0209166-1-21-10': 'So Eq. ([REF]) gives [EQUATION] which is equivalent to Eq. ([REF]).', 'quant-ph-0209166-1-21-11': '[MATH]According to the our derivation, given explicitly Eq. ([REF]) that is implied by majorization, one can perform the contractions [MATH] and the unitaries [MATH] to achieve the transformation [MATH].', 'quant-ph-0209166-1-21-12': 'The problem of looking for a POVM with minimum number of outcomes (thus minimizing the amount of classical information sent to Bob) is reduced to find the transformation ([REF]) with minimum number of unitaries.', 'quant-ph-0209166-1-21-13': 'One can resort to a constructive algorithm to find a bistochastic matrix [MATH] which relates the vectors [MATH] and [MATH] of the singular values of [MATH] and [MATH], namely [EQUATION]', 'quant-ph-0209166-1-21-14': 'Then Birkhoff theorem allows to write [MATH] as a convex combination of permutation matrices [EQUATION]', 'quant-ph-0209166-1-21-15': 'In terms of [MATH] and [MATH] one has [EQUATION] where [MATH].', 'quant-ph-0209166-1-21-16': 'In this way one obtains Eq. ([REF]), with [MATH].', 'quant-ph-0209166-1-21-17': 'Using the corresponding expressions for the contractions [MATH] and unitaries [MATH] one recovers the result of Ref. [CITATION].', 'quant-ph-0209166-1-21-18': ""Notice that Caratheodory's theorem always allows to reduce the number of permutations in Eq. ([REF]) to [MATH], for [MATH]-dimensional Alice Hilbert space."", 'quant-ph-0209166-1-22-0': '## Probabilistic transformation', 'quant-ph-0209166-1-23-0': 'The second part of the protocol, namely the contraction which provides the state [MATH] from [MATH], is needed only for probabilistic transformations.', 'quant-ph-0209166-1-23-1': 'It is a pure contraction given by [EQUATION] with [EQUATION] and [EQUATION]', 'quant-ph-0209166-1-23-2': 'In fact for lemma [REF] one has [MATH], and so [MATH].', 'quant-ph-0209166-1-23-3': 'Then [EQUATION]', 'quant-ph-0209166-1-24-0': '# Conclusions', 'quant-ph-0209166-1-25-0': 'In this paper we presented a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via LOCC transformations.', 'quant-ph-0209166-1-25-1': 'We have generalized Nielsen work based on majorization theory [CITATION] in order to include the more general results by Vidal [CITATION], which were based on the approach of the entanglement monotones.', 'quant-ph-0209166-1-25-2': 'The main theorem gives an if and only if condition for all entanglement transformations of pure states in terms of super-majorization conditions.', 'quant-ph-0209166-1-25-3': 'We also provided a new necessary sub-majorization condition for pure transformations, which allows to write each contraction in terms of the Moore-Penrose pseudoinverse of the matrix of the entangled state.', 'quant-ph-0209166-1-25-4': 'This allowed us to give explicit protocols to achieve pure, deterministic, and probabilistic LOCC.', 'quant-ph-0209166-1-25-5': 'At the end we notice that all theorems have been derived in finite dimensions, but can be easily extended to infinite dimensions for contractions that are compact operators and for normalized entangled states corresponding to Hilbert-Schmidt operators.', 'quant-ph-0209166-1-25-6': 'Thus our results also apply to continuous variables.'}","{'quant-ph-0209166-2-0-0': 'We present a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via local operations and classical communication (LOCC).', 'quant-ph-0209166-2-0-1': 'This framework unifies and greatly simplifies the results of Refs. [CITATION].', 'quant-ph-0209166-2-0-2': 'A new necessary condition for ""pure contraction"" transformations is given.', 'quant-ph-0209166-2-0-3': 'Finally, constructive protocols to achieve both probabilistic and deterministic entanglement transformations are presented.', 'quant-ph-0209166-2-1-0': '# Introduction', 'quant-ph-0209166-2-2-0': 'The transformation of entangled states by means of LOCC is a key issue in quantum information processing [CITATION], for quantum computation [CITATION], quantum teleportation [CITATION], and quantum cryptography [CITATION].', 'quant-ph-0209166-2-2-1': 'However, the detrimental effect of losses and decoherence poses a serious problem for establishing entangled resources at distance or in a long computing network, since in a realistic transmission/computation the entanglement can be considerably degraded, thus preventing, for example, successful teleportation or dense coding.', 'quant-ph-0209166-2-2-2': 'For this reason, the use of transformations which can increase the available entanglement by means of LOCC-although with some probability-is crucial for practical purposes.', 'quant-ph-0209166-2-2-3': 'More generally, understanding entanglement transformations that are allowed by LOCC provides more insight in the structure and property of nonlocality, the most prominent character of quantum mechanics.', 'quant-ph-0209166-2-3-0': 'On entanglement transformations via LOCC there are two main results.', 'quant-ph-0209166-2-3-1': 'The first is the seminal work by Nielsen [CITATION] on deterministic transformations, which introduces majorization theory in this context.', 'quant-ph-0209166-2-3-2': 'The second is the work by Vidal [CITATION], which addresses the more general problem of probabilistic transformations.', 'quant-ph-0209166-2-3-3': 'Such work is based on the approach of entanglement monotones, and gives conditions equivalent to weak majorization relations.', 'quant-ph-0209166-2-3-4': 'The two approaches are completely disconnected, and for practical applications of the theory, a unified framework would be needed, especially in consideration that the more general treatment by Vidal is more abstract and less constructive than the Nielsen approach, which, however, is limited only to deterministic transformations.', 'quant-ph-0209166-2-3-5': 'Furthermore, from the Vidal approach it is very difficult to recover the Nielsen treatment as a special case, and it is quite surprising that such approach ends up as a weak-majorization condition, without essentially using majorization theory.', 'quant-ph-0209166-2-3-6': 'This motivates a derivation of the general nondeterministic LOCC transformations of pure states with an approach completely based on majorization theory, generalizing the Nielsen work [CITATION].', 'quant-ph-0209166-2-4-0': 'In this paper we present a general framework for entanglement transformations of bipartite pure states by means of LOCC.', 'quant-ph-0209166-2-4-1': 'In Sec. II we give a short and very simple proof of the theorem of Lo and Popescu [CITATION], which is at the basis of the theory of all LOCC, and which states that given two separate parties, say Alice and Bob, all LOCC transformations on a pure bipartite state can be reduced to a contraction by Alice and a unitary transformation by Bob.', 'quant-ph-0209166-2-4-2': 'We include our derivation of this theorem, since it is particularly simple and is based on a new useful technique for operator transposition.', 'quant-ph-0209166-2-4-3': 'In Sec. III we derive the main theorem, which gives a necessary and sufficient condition for all entanglement transformations of pure states in terms of super-majorization conditions, generalizing the Nielsen approach, and recovering the result of Vidal.', 'quant-ph-0209166-2-4-4': 'Here we also provide a new necessary condition for ""pure contraction"" transformations, namely those LOCC transformations in which the target state is achieved just for a single outcome of Alice measurement and a corresponding unitary on Bob side.', 'quant-ph-0209166-2-4-5': 'Such condition writes in terms of sub-majorization relation.', 'quant-ph-0209166-2-4-6': 'In Sec. IV explicit protocols to achieve pure, deterministic, and probabilistic transformations are given, using a new method which gives the Alice contraction of the LOCC in terms of the Moore-Penrose pseudoinverse of the matrix of the entangled state.', 'quant-ph-0209166-2-4-7': 'Section V concludes the paper summarizing the results.', 'quant-ph-0209166-2-5-0': '# The Lo-Popescu theorem', 'quant-ph-0209166-2-6-0': 'For later convenience we introduce here the main notation used in the paper.', 'quant-ph-0209166-2-6-1': 'Given a linear operator [MATH] we denote its hermitian conjugate by [MATH].', 'quant-ph-0209166-2-6-2': 'On a fixed basis, we write the complex conjugate and the transpose of [MATH] as [MATH] and [MATH], respectively, so that [MATH].', 'quant-ph-0209166-2-6-3': 'With the notation [MATH] we denote the Moore-Penrose inverse of [MATH].', 'quant-ph-0209166-2-6-4': 'We recall that the Moore-Penrose inverse [MATH] is the unique matrix that satisfies [EQUATION]', 'quant-ph-0209166-2-6-5': 'From Eq. ([REF]) it immediately follows that both [MATH] and [MATH] are orthogonal projectors, in particular [MATH] is the orthogonal projector over the range of [MATH], whereas [MATH] is the orthogonal projector over the support of [MATH].', 'quant-ph-0209166-2-6-6': 'We write the singular value decomposition (SVD) of [MATH] as follows [EQUATION] where [MATH] denotes the diagonal matrix whose entries are the singular values [MATH] of [MATH] taken in decreasing order, and [MATH] are unitary.', 'quant-ph-0209166-2-6-7': 'The Moore-Penrose inverse [MATH] then writes [EQUATION] where [MATH] is diagonal with entries [MATH] for [MATH], and zero entries for [MATH].', 'quant-ph-0209166-2-7-0': 'We remember that a quantum measurement (with discrete spectrum) is generally described by a Positive Operator-Valued Measurement (POVM), namely by a resolution of the identity [MATH], where each [MATH] corresponds to a possible outcome.', 'quant-ph-0209166-2-7-1': 'Each operator [MATH] acts on the input states and is necessarily a contraction, namely it satisfies [MATH], where [MATH] denotes the usual operator norm.', 'quant-ph-0209166-2-7-2': 'For bipartite pure states on the Hilbert space [MATH] we use the following notation [CITATION] [EQUATION] where [MATH] and [MATH] are two chosen orthonormal bases for [MATH] and [MATH], respectively.', 'quant-ph-0209166-2-7-3': 'Eq. ([REF]) introduces an isomorphism between vectors in [MATH] and [MATH] matrices, where [MATH] and [MATH] are the dimensions of [MATH] and [MATH].', 'quant-ph-0209166-2-7-4': 'One can easily check the relation [EQUATION] where the transposition is defined with respect to the orthonormal basis [MATH].', 'quant-ph-0209166-2-7-5': 'Finally, we use the notation [MATH] for Hermitian operators [MATH] and [MATH] to denote the vector majorization relation [CITATION] [MATH], and in the same fashion we will write [MATH] and [MATH] for super- and sub-majorization, respectively.', 'quant-ph-0209166-2-8-0': 'In the last part of this section we provide a very short proof of the following theorem [CITATION].', 'quant-ph-0209166-2-8-1': 'All LOCC on a pure bipartite state [MATH] can be reduced to a contraction by Alice and a unitary transformation by Bob.', 'quant-ph-0209166-2-8-2': 'This resorts to the equivalence of any Bob contraction [MATH] with the Alice contraction [MATH] assisted by Bob unitary transformation [MATH] as follows [EQUATION] where [EQUATION] and [MATH] is the unitary operator achieving the transposition of the operator [MATH], namely [EQUATION]', 'quant-ph-0209166-2-8-3': 'Proof.', 'quant-ph-0209166-2-8-4': 'To prove that every LOCC can be reduced to an Alice contraction and a Bob unitary transformation it is sufficient to prove the equivalence ([REF]), since: a) all possible elementary LOCC in a sequence will be reduced to an Alice contraction and a Bob unitary; b) the product of two contractions is a contraction; c) unitary transformations are particular cases of contraction.', 'quant-ph-0209166-2-8-5': 'Notice that given the SVD of any linear operator [MATH] as in Eq. ([REF]) one has [EQUATION] with [MATH].', 'quant-ph-0209166-2-8-6': 'Using Eq. ([REF]), Eq. ([REF]) rewrites as follows [EQUATION]', 'quant-ph-0209166-2-8-7': 'Then, from Eq. ([REF]) one has [EQUATION] which is just Eq. ([REF]) with [MATH] and [MATH] given as in Eq. ([REF]).', 'quant-ph-0209166-2-8-8': '[MATH]', 'quant-ph-0209166-2-9-0': '# LOCC transformations for pure states', 'quant-ph-0209166-2-10-0': 'In this section we will use the following useful lemmas:', 'quant-ph-0209166-2-11-0': 'If [MATH] then for some [MATH] and [MATH].', 'quant-ph-0209166-2-12-0': 'Proof.', 'quant-ph-0209166-2-13-0': 'If [MATH] one has for [MATH] [EQUATION] and [EQUATION]', 'quant-ph-0209166-2-13-1': 'Upon defining [EQUATION] clearly one has [MATH], and [MATH] [EQUATION] namely [MATH].', 'quant-ph-0209166-2-13-2': '[MATH] If for some [MATH] and [MATH], then [MATH].', 'quant-ph-0209166-2-14-0': 'Proof.', 'quant-ph-0209166-2-14-1': 'One has [MATH] [EQUATION] namely [MATH].', 'quant-ph-0209166-2-14-2': '[MATH] We notice that both the above lemmas hold also in the reverse direction (for a proof see Ref. [CITATION], pp. 11 and 123).', 'quant-ph-0209166-2-14-3': 'Moreover, we will make extensive use of the following theorem: [Uhlmann] For Hermitian operators [MATH] and [MATH] one has [MATH] if and only if there are probabilities [MATH] and unitary operators [MATH] such that [EQUATION]', 'quant-ph-0209166-2-14-4': 'Proof.', 'quant-ph-0209166-2-14-5': 'See Ref. [CITATION], p. 575.', 'quant-ph-0209166-2-15-0': 'Theorem [REF] relates majorization between hermitian operators with a particular form of completely positive maps, namely those achievable through a random unitary evolution.', 'quant-ph-0209166-2-15-1': 'In the terminology of quantum information channels, such maps correspond to external-random-field channels, that are a subclass of bistochastic channels (which send the identity operator into itself).', 'quant-ph-0209166-2-15-2': 'For a qubit system ([MATH]) the set of bistochastic and external-random-field channels coincide [CITATION].', 'quant-ph-0209166-2-16-0': 'In the following we derive the necessary and sufficient condition for all entanglement transformations of pure states in terms of super-majorization conditions.', 'quant-ph-0209166-2-16-1': 'The theorem generalizes Nielsen approach [CITATION] and recovers the result of Vidal probabilistic transformations [CITATION].', 'quant-ph-0209166-2-16-2': 'Moreover, we provide a new necessary condition for ""pure contraction"" transformations, namely those LOCC transformations in which the target state is achieved just for a single outcome of Alice measurement and a corresponding unitary on Bob side.', 'quant-ph-0209166-2-16-3': 'In the proof a relevant role is played by the intermediate state (denoted in the following by [MATH]).', 'quant-ph-0209166-2-16-4': 'In a transformation from [MATH] to [MATH] the state [MATH] will be reached from [MATH] deterministically, whereas the final probabilistic transformation [MATH] will be obtained through a pure contraction (for deterministic transformations one has [MATH]).', 'quant-ph-0209166-2-16-5': 'We are now ready to prove the following', 'quant-ph-0209166-2-17-0': 'The state transformation [MATH] is possible by LOCC iff [EQUATION] where [MATH] is the probability of achieving the transformation.', 'quant-ph-0209166-2-17-1': 'A necessary condition to be satisfied is [MATH].', 'quant-ph-0209166-2-17-2': 'In particular, the transformation is deterministic ([MATH]) iff [MATH].', 'quant-ph-0209166-2-17-3': 'Finally, if there is a pure LOCC that achieves the state transformation with probability [MATH], we must have [EQUATION]', 'quant-ph-0209166-2-17-4': 'Proof.', 'quant-ph-0209166-2-17-5': 'Suppose that [MATH].', 'quant-ph-0209166-2-17-6': 'From Lemma [REF] there is an operator [MATH] with [MATH] and [MATH].', 'quant-ph-0209166-2-17-7': 'The state [MATH] represents the intermediate state that can be achieved deterministically from [MATH].', 'quant-ph-0209166-2-17-8': 'In fact, Uhlmann Theorem [REF] guarantees the existence of a set of unitaries [MATH] and probabilities [MATH] such that [EQUATION]', 'quant-ph-0209166-2-17-9': 'Now, define the Alice measurement [MATH] such that [EQUATION] namely [EQUATION]', 'quant-ph-0209166-2-17-10': 'We can always choose [MATH] such that [MATH], and show that [MATH] is a contraction since [EQUATION] and so [MATH].', 'quant-ph-0209166-2-17-11': 'Then, there exists a set of unitary operators [MATH] such that [MATH], namely [EQUATION] so that the transformation from [MATH] can be achieved deterministically.', 'quant-ph-0209166-2-17-12': 'Now, since [MATH] one can define the contraction [EQUATION] so that [EQUATION]', 'quant-ph-0209166-2-17-13': 'By using the SVD of both [MATH] and [MATH], Eq. ([REF]) rewrites as follows [EQUATION] which means that there exists a unitary transformation [MATH] such that [EQUATION] where [MATH].', 'quant-ph-0209166-2-17-14': 'Eq. ([REF]) is equivalent to the entanglement transformation [EQUATION] namely there is a pure LOCC occurring with probability [MATH] that transforms the state [MATH] into [MATH].', 'quant-ph-0209166-2-17-15': 'But, as we have seen before, the transformation [MATH] can be achieved deterministically, whence we conclude that [MATH] can be achieved with probability [MATH], namely the statement of the first part of the theorem.', 'quant-ph-0209166-2-17-16': 'Reversely, suppose that the transformation [MATH] can be achieved via LOCC with probability [MATH].', 'quant-ph-0209166-2-17-17': 'According to Theorem [REF] every LOCC is equivalent to a measurement performed by Alice followed by a conditional unitary by Bob.', 'quant-ph-0209166-2-17-18': 'Therefore, the joint Alice-Bob state [MATH] will evolve as follows [EQUATION] where [EQUATION]', 'quant-ph-0209166-2-17-19': 'If the state [MATH] goes to [MATH] we must have for a subset [MATH] of the possible outcomes [MATH] [EQUATION] where [MATH] denotes the overall probability of the transformation [MATH].', 'quant-ph-0209166-2-17-20': 'From Eq. ([REF]) we need to have [EQUATION] and, therefore, each probability [MATH] is given by [EQUATION] where [MATH] denotes the usual Frobenius norm.', 'quant-ph-0209166-2-17-21': 'The condition ([REF]) can be satisfied only if [MATH], i.e. we can only decrease the Schmidt number of the entangled state.', 'quant-ph-0209166-2-17-22': 'From Eq. ([REF]) one has [MATH] [EQUATION]', 'quant-ph-0209166-2-17-23': 'From Eq ([REF]) we have [EQUATION] and using Eq. ([REF]) we obtain [EQUATION]', 'quant-ph-0209166-2-17-24': 'By Uhlmann Theorem [REF] we have [EQUATION] and from Lemma [REF] we get the statement, namely [MATH].', 'quant-ph-0209166-2-18-0': 'We now prove the last part of the theorem, regarding the pure LOCC case.', 'quant-ph-0209166-2-18-1': 'In a pure contraction transformation the target state is achieved just for a single outcome of Alice measurement and a unitary performed by Bob.', 'quant-ph-0209166-2-18-2': 'Such a transformation that occurs with probability [MATH] is given by [EQUATION] and we need to have [EQUATION]', 'quant-ph-0209166-2-18-3': 'Again, this is possible if [MATH].', 'quant-ph-0209166-2-18-4': 'Using the SVD of [MATH] and [MATH] as in Eq. ([REF]), Eq. ([REF]) rewrites in terms of the diagonal matrices [MATH] and [MATH] as follows [EQUATION] with [EQUATION]', 'quant-ph-0209166-2-18-5': 'Eq. ([REF]) leads to [EQUATION] namely [EQUATION] where [MATH] is a sub-stochastic matrix [CITATION], since', 'quant-ph-0209166-2-19-0': 'Since Eq. ([REF]) with [MATH] sub-stochastic is equivalent [CITATION] to [MATH], namely [MATH], we have proved that Eq. ([REF]) is a necessary condition for the LOCC transformation ([REF]), namely the last statement of the theorem.', 'quant-ph-0209166-2-19-1': '[MATH]', 'quant-ph-0209166-2-20-0': '# The explicit protocols', 'quant-ph-0209166-2-21-0': '## Pure transformation', 'quant-ph-0209166-2-22-0': 'Pure LOCC transformations are achieved by a single contraction on Alice side, assisted by a unitary by Bob.', 'quant-ph-0209166-2-22-1': 'In this case we have the following theorem', 'quant-ph-0209166-2-23-0': 'The transformation [MATH] can be achieved with probability [MATH] by a pure LOCC transformation iff one can find a unitary operator [MATH] and linear operator [MATH] such that [EQUATION] is a contraction.', 'quant-ph-0209166-2-23-1': 'The transformation is then obtained as [EQUATION]', 'quant-ph-0209166-2-23-2': 'Proof.', 'quant-ph-0209166-2-23-3': 'Notice that Eq. ([REF]) is equivalent to the identity [EQUATION]', 'quant-ph-0209166-2-23-4': 'Since both sides of the identity must have the same kernel it follows that [EQUATION] and, multiplying both sides by [MATH], we have [EQUATION]', 'quant-ph-0209166-2-23-5': 'The general solution of the last equation is [EQUATION] with arbitrary [MATH], and in fact one can easily check that [EQUATION]', 'quant-ph-0209166-2-23-6': 'The unitary [MATH] and the operator [MATH] should be taken such that [MATH] is a contraction.', 'quant-ph-0209166-2-23-7': 'This is not always possible.', 'quant-ph-0209166-2-23-8': 'However, a sufficient condition is [EQUATION]', 'quant-ph-0209166-2-23-9': 'In fact, by taking [MATH] and [MATH] one has [MATH], and then, for Eq. ([REF]), [MATH].', 'quant-ph-0209166-2-23-10': '[MATH] ## Deterministic transformation', 'quant-ph-0209166-2-24-0': 'In the entanglement transformations, the first part of the protocol is a deterministic transformation from [MATH] to [MATH].', 'quant-ph-0209166-2-24-1': 'The majorization relation [MATH] implies Theorem [REF], namely the existence of a set of unitaries [MATH] and probabilities [MATH] such that [EQUATION]', 'quant-ph-0209166-2-24-2': 'In order to construct explicitly the protocol one needs to find contractions [MATH] and unitaries [MATH] versus the unitary operators [MATH] that appear in Eq. ([REF]), such that one has [EQUATION]', 'quant-ph-0209166-2-24-3': 'We have seen that the general form of the solution of Eq. ([REF]) is given by [EQUATION]', 'quant-ph-0209166-2-24-4': 'We have now the following theorem', 'quant-ph-0209166-2-25-0': 'In Eq. ([REF]) we can always take [EQUATION] where [MATH] are the operators defined in Eq. ([REF]), such that [MATH] is a contraction and Eq. ([REF]) is satisfied.', 'quant-ph-0209166-2-26-0': 'Proof.', 'quant-ph-0209166-2-26-1': 'Substituting Eq. ([REF]) in Eq. ([REF]) one has [EQUATION] and using Eq. ([REF]) one has [EQUATION]', 'quant-ph-0209166-2-26-2': 'Hence, [MATH] are contractions.', 'quant-ph-0209166-2-26-3': 'The completeness of the measurement can be guaranteed by the further contraction [EQUATION] where [MATH] is an arbitrary unitary operator.', 'quant-ph-0209166-2-26-4': 'For any outcome [MATH] on Alice side, Bob performs the unitary [MATH].', 'quant-ph-0209166-2-26-5': 'Using Eq. ([REF]) and ([REF]) one has [EQUATION]', 'quant-ph-0209166-2-26-6': 'From Eq. ([REF]), it follows that [MATH], namely [EQUATION]', 'quant-ph-0209166-2-26-7': 'Hence, by multiplying both sides of Eq. ([REF]) on the left by [MATH] and on the right by [MATH] one obtains [EQUATION]', 'quant-ph-0209166-2-26-8': 'The projector on [MATH] coincides with the projector on [MATH].', 'quant-ph-0209166-2-26-9': 'So Eq. ([REF]) gives [EQUATION] which is equivalent to Eq. ([REF]).', 'quant-ph-0209166-2-26-10': '[MATH]', 'quant-ph-0209166-2-27-0': 'According to our derivation, given explicitly Eq. ([REF]) one can perform the contractions [MATH] and the unitaries [MATH] to achieve the transformation [MATH].', 'quant-ph-0209166-2-27-1': 'The problem of looking for a POVM with minimum number of outcomes (thus minimizing the amount of classical information sent to Bob) is reduced to find the transformation ([REF]) with minimum number of unitaries.', 'quant-ph-0209166-2-27-2': 'One can resort to a constructive algorithm to find a bistochastic matrix [MATH] which relates the vectors [MATH] and [MATH] of the singular values of [MATH] and [MATH], namely [EQUATION]', 'quant-ph-0209166-2-27-3': 'Then Birkhoff theorem [CITATION] allows to write [MATH] as a convex combination of permutation matrices [EQUATION]', 'quant-ph-0209166-2-27-4': 'In terms of [MATH] and [MATH] one has [EQUATION] where [MATH].', 'quant-ph-0209166-2-27-5': 'In this way one obtains Eq. ([REF]), with [MATH].', 'quant-ph-0209166-2-27-6': 'Using the corresponding expressions for the contractions [MATH] and unitaries [MATH] one recovers the result of Ref. [CITATION].', 'quant-ph-0209166-2-27-7': ""Notice that Caratheodory's theorem always allows to reduce the number of permutations in Eq. ([REF]) to [MATH], for [MATH]-dimensional Alice Hilbert space."", 'quant-ph-0209166-2-28-0': '## Probabilistic transformation', 'quant-ph-0209166-2-29-0': 'The second part of the protocol, namely the contraction which provides the state [MATH] from [MATH], is needed only for probabilistic transformations.', 'quant-ph-0209166-2-29-1': 'It is a pure contraction given by [EQUATION] with [EQUATION] and [EQUATION]', 'quant-ph-0209166-2-29-2': 'In fact for lemma [REF] one has [MATH], which implies that [MATH].', 'quant-ph-0209166-2-29-3': 'Then [EQUATION]', 'quant-ph-0209166-2-30-0': '# Conclusions', 'quant-ph-0209166-2-31-0': 'In this paper we presented a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via LOCC transformations.', 'quant-ph-0209166-2-31-1': 'We have generalized Nielsen work based on majorization theory [CITATION] in order to include the more general results by Vidal [CITATION], which were based on the approach of the entanglement monotones.', 'quant-ph-0209166-2-31-2': 'The main theorem gives an if and only if condition for all entanglement transformations of pure states in terms of super-majorization conditions.', 'quant-ph-0209166-2-31-3': 'We also provided a new necessary sub-majorization condition for pure transformations, which allows to write each contraction in terms of the Moore-Penrose pseudoinverse of the matrix of the entangled state.', 'quant-ph-0209166-2-31-4': 'This lead to explicit protocols to achieve pure, deterministic, and probabilistic LOCC.', 'quant-ph-0209166-2-32-0': 'We notice that all theorems have been derived in finite dimensions, but they can be easily extended to infinite dimensions for contractions that are compact operators and for normalized entangled states corresponding to Hilbert-Schmidt operators.', 'quant-ph-0209166-2-32-1': 'Thus our results also apply to continuous variables.'}","[['quant-ph-0209166-1-3-0', 'quant-ph-0209166-2-4-0'], ['quant-ph-0209166-1-3-2', 'quant-ph-0209166-2-4-2'], ['quant-ph-0209166-1-3-5', 'quant-ph-0209166-2-4-5'], ['quant-ph-0209166-1-3-7', 'quant-ph-0209166-2-4-7'], ['quant-ph-0209166-1-23-0', 'quant-ph-0209166-2-29-0'], ['quant-ph-0209166-1-23-1', 'quant-ph-0209166-2-29-1'], ['quant-ph-0209166-1-20-0', 'quant-ph-0209166-2-25-0'], ['quant-ph-0209166-1-11-4', 'quant-ph-0209166-2-14-3'], ['quant-ph-0209166-1-16-1', 'quant-ph-0209166-2-22-1'], ['quant-ph-0209166-1-19-1', 'quant-ph-0209166-2-24-3'], ['quant-ph-0209166-1-19-2', 'quant-ph-0209166-2-24-4'], ['quant-ph-0209166-1-0-0', 'quant-ph-0209166-2-0-0'], ['quant-ph-0209166-1-0-1', 'quant-ph-0209166-2-0-1'], ['quant-ph-0209166-1-0-3', 'quant-ph-0209166-2-0-3'], ['quant-ph-0209166-2-7-0', 'quant-ph-0209166-3-7-0'], ['quant-ph-0209166-2-7-1', 'quant-ph-0209166-3-7-1'], ['quant-ph-0209166-2-7-2', 'quant-ph-0209166-3-7-2'], ['quant-ph-0209166-2-7-3', 'quant-ph-0209166-3-7-3'], ['quant-ph-0209166-2-7-4', 'quant-ph-0209166-3-7-4'], ['quant-ph-0209166-2-7-5', 'quant-ph-0209166-3-7-5'], ['quant-ph-0209166-2-23-0', 'quant-ph-0209166-3-23-0'], ['quant-ph-0209166-2-23-1', 'quant-ph-0209166-3-23-1'], ['quant-ph-0209166-2-23-3', 'quant-ph-0209166-3-23-3'], ['quant-ph-0209166-2-23-4', 'quant-ph-0209166-3-23-4'], ['quant-ph-0209166-2-23-5', 'quant-ph-0209166-3-23-5'], ['quant-ph-0209166-2-23-6', 'quant-ph-0209166-3-23-6'], ['quant-ph-0209166-2-23-7', 'quant-ph-0209166-3-23-7'], ['quant-ph-0209166-2-23-8', 'quant-ph-0209166-3-23-8'], ['quant-ph-0209166-2-23-9', 'quant-ph-0209166-3-23-9'], ['quant-ph-0209166-2-27-0', 'quant-ph-0209166-3-27-0'], ['quant-ph-0209166-2-27-1', 'quant-ph-0209166-3-27-1'], ['quant-ph-0209166-2-27-2', 'quant-ph-0209166-3-27-2'], ['quant-ph-0209166-2-27-3', 'quant-ph-0209166-3-27-3'], ['quant-ph-0209166-2-27-4', 'quant-ph-0209166-3-27-4'], ['quant-ph-0209166-2-27-5', 'quant-ph-0209166-3-27-5'], ['quant-ph-0209166-2-27-6', 'quant-ph-0209166-3-27-6'], ['quant-ph-0209166-2-27-7', 'quant-ph-0209166-3-27-7'], ['quant-ph-0209166-2-2-0', 'quant-ph-0209166-3-2-0'], ['quant-ph-0209166-2-2-1', 'quant-ph-0209166-3-2-1'], ['quant-ph-0209166-2-2-2', 'quant-ph-0209166-3-2-2'], ['quant-ph-0209166-2-2-3', 'quant-ph-0209166-3-2-3'], ['quant-ph-0209166-2-17-0', 'quant-ph-0209166-3-17-0'], ['quant-ph-0209166-2-17-1', 'quant-ph-0209166-3-17-1'], ['quant-ph-0209166-2-17-2', 'quant-ph-0209166-3-17-2'], ['quant-ph-0209166-2-17-3', 'quant-ph-0209166-3-17-3'], ['quant-ph-0209166-2-17-6', 'quant-ph-0209166-3-17-6'], ['quant-ph-0209166-2-17-7', 'quant-ph-0209166-3-17-7'], ['quant-ph-0209166-2-17-8', 'quant-ph-0209166-3-17-8'], ['quant-ph-0209166-2-17-9', 'quant-ph-0209166-3-17-9'], ['quant-ph-0209166-2-17-10', 'quant-ph-0209166-3-17-10'], ['quant-ph-0209166-2-17-11', 'quant-ph-0209166-3-17-11'], ['quant-ph-0209166-2-17-12', 'quant-ph-0209166-3-17-12'], ['quant-ph-0209166-2-17-13', 'quant-ph-0209166-3-17-13'], ['quant-ph-0209166-2-17-14', 'quant-ph-0209166-3-17-14'], ['quant-ph-0209166-2-17-15', 'quant-ph-0209166-3-17-15'], ['quant-ph-0209166-2-17-16', 'quant-ph-0209166-3-17-16'], ['quant-ph-0209166-2-17-17', 'quant-ph-0209166-3-17-17'], ['quant-ph-0209166-2-17-18', 'quant-ph-0209166-3-17-18'], ['quant-ph-0209166-2-17-19', 'quant-ph-0209166-3-17-19'], ['quant-ph-0209166-2-17-20', 'quant-ph-0209166-3-17-20'], ['quant-ph-0209166-2-17-21', 'quant-ph-0209166-3-17-21'], ['quant-ph-0209166-2-17-23', 'quant-ph-0209166-3-17-23'], ['quant-ph-0209166-2-17-24', 'quant-ph-0209166-3-17-24'], ['quant-ph-0209166-2-16-0', 'quant-ph-0209166-3-16-0'], ['quant-ph-0209166-2-16-1', 'quant-ph-0209166-3-16-1'], ['quant-ph-0209166-2-16-2', 'quant-ph-0209166-3-16-2'], ['quant-ph-0209166-2-16-3', 'quant-ph-0209166-3-16-3'], ['quant-ph-0209166-2-16-4', 'quant-ph-0209166-3-16-4'], ['quant-ph-0209166-2-16-5', 'quant-ph-0209166-3-16-5'], ['quant-ph-0209166-2-29-0', 'quant-ph-0209166-3-29-0'], ['quant-ph-0209166-2-29-1', 'quant-ph-0209166-3-29-1'], ['quant-ph-0209166-2-29-2', 'quant-ph-0209166-3-29-2'], ['quant-ph-0209166-2-32-0', 'quant-ph-0209166-3-32-0'], ['quant-ph-0209166-2-32-1', 'quant-ph-0209166-3-32-1'], ['quant-ph-0209166-2-3-0', 'quant-ph-0209166-3-3-0'], ['quant-ph-0209166-2-3-1', 'quant-ph-0209166-3-3-1'], ['quant-ph-0209166-2-3-2', 'quant-ph-0209166-3-3-2'], ['quant-ph-0209166-2-3-3', 'quant-ph-0209166-3-3-3'], ['quant-ph-0209166-2-3-4', 'quant-ph-0209166-3-3-4'], ['quant-ph-0209166-2-3-5', 'quant-ph-0209166-3-3-5'], ['quant-ph-0209166-2-3-6', 'quant-ph-0209166-3-3-6'], ['quant-ph-0209166-2-0-0', 'quant-ph-0209166-3-0-0'], ['quant-ph-0209166-2-0-1', 'quant-ph-0209166-3-0-1'], ['quant-ph-0209166-2-0-2', 'quant-ph-0209166-3-0-2'], ['quant-ph-0209166-2-0-3', 'quant-ph-0209166-3-0-3'], ['quant-ph-0209166-2-6-0', 'quant-ph-0209166-3-6-0'], ['quant-ph-0209166-2-6-1', 'quant-ph-0209166-3-6-1'], ['quant-ph-0209166-2-6-2', 'quant-ph-0209166-3-6-2'], ['quant-ph-0209166-2-6-3', 'quant-ph-0209166-3-6-3'], ['quant-ph-0209166-2-6-4', 'quant-ph-0209166-3-6-4'], ['quant-ph-0209166-2-6-5', 'quant-ph-0209166-3-6-5'], ['quant-ph-0209166-2-6-6', 'quant-ph-0209166-3-6-6'], ['quant-ph-0209166-2-6-7', 'quant-ph-0209166-3-6-7'], ['quant-ph-0209166-2-22-0', 'quant-ph-0209166-3-22-0'], ['quant-ph-0209166-2-22-1', 'quant-ph-0209166-3-22-2'], ['quant-ph-0209166-2-8-0', 'quant-ph-0209166-3-8-0'], ['quant-ph-0209166-2-8-1', 'quant-ph-0209166-3-8-1'], ['quant-ph-0209166-2-8-2', 'quant-ph-0209166-3-8-2'], ['quant-ph-0209166-2-8-4', 'quant-ph-0209166-3-8-4'], ['quant-ph-0209166-2-8-5', 'quant-ph-0209166-3-8-5'], ['quant-ph-0209166-2-8-6', 'quant-ph-0209166-3-8-6'], ['quant-ph-0209166-2-8-7', 'quant-ph-0209166-3-8-7'], ['quant-ph-0209166-2-18-0', 'quant-ph-0209166-3-18-0'], ['quant-ph-0209166-2-18-1', 'quant-ph-0209166-3-18-1'], ['quant-ph-0209166-2-18-2', 'quant-ph-0209166-3-18-2'], ['quant-ph-0209166-2-18-3', 'quant-ph-0209166-3-18-3'], ['quant-ph-0209166-2-18-4', 'quant-ph-0209166-3-18-4'], ['quant-ph-0209166-2-18-5', 'quant-ph-0209166-3-18-5'], ['quant-ph-0209166-2-25-0', 'quant-ph-0209166-3-25-0'], ['quant-ph-0209166-2-15-0', 'quant-ph-0209166-3-15-0'], ['quant-ph-0209166-2-15-1', 'quant-ph-0209166-3-15-1'], ['quant-ph-0209166-2-15-2', 'quant-ph-0209166-3-15-2'], ['quant-ph-0209166-2-14-1', 'quant-ph-0209166-3-14-1'], ['quant-ph-0209166-2-14-2', 'quant-ph-0209166-3-14-2'], ['quant-ph-0209166-2-14-3', 'quant-ph-0209166-3-14-3'], ['quant-ph-0209166-2-31-0', 'quant-ph-0209166-3-31-0'], ['quant-ph-0209166-2-31-1', 'quant-ph-0209166-3-31-1'], ['quant-ph-0209166-2-31-2', 'quant-ph-0209166-3-31-2'], ['quant-ph-0209166-2-31-3', 'quant-ph-0209166-3-31-3'], ['quant-ph-0209166-2-31-4', 'quant-ph-0209166-3-31-4'], ['quant-ph-0209166-2-4-0', 'quant-ph-0209166-3-4-0'], ['quant-ph-0209166-2-4-1', 'quant-ph-0209166-3-4-1'], ['quant-ph-0209166-2-4-2', 'quant-ph-0209166-3-4-2'], ['quant-ph-0209166-2-4-3', 'quant-ph-0209166-3-4-3'], ['quant-ph-0209166-2-4-4', 'quant-ph-0209166-3-4-4'], ['quant-ph-0209166-2-4-5', 'quant-ph-0209166-3-4-5'], ['quant-ph-0209166-2-4-6', 'quant-ph-0209166-3-4-6'], ['quant-ph-0209166-2-4-7', 'quant-ph-0209166-3-4-7'], ['quant-ph-0209166-2-26-1', 'quant-ph-0209166-3-26-1'], ['quant-ph-0209166-2-26-2', 'quant-ph-0209166-3-26-2'], ['quant-ph-0209166-2-26-3', 'quant-ph-0209166-3-26-3'], ['quant-ph-0209166-2-26-4', 'quant-ph-0209166-3-26-4'], ['quant-ph-0209166-2-26-6', 'quant-ph-0209166-3-26-6'], ['quant-ph-0209166-2-26-7', 'quant-ph-0209166-3-26-7'], ['quant-ph-0209166-2-26-8', 'quant-ph-0209166-3-26-8'], ['quant-ph-0209166-2-26-9', 'quant-ph-0209166-3-26-9'], ['quant-ph-0209166-2-19-0', 'quant-ph-0209166-3-19-0'], ['quant-ph-0209166-2-24-0', 'quant-ph-0209166-3-24-0'], ['quant-ph-0209166-2-24-1', 'quant-ph-0209166-3-24-1'], ['quant-ph-0209166-2-24-2', 'quant-ph-0209166-3-24-2'], ['quant-ph-0209166-2-24-3', 'quant-ph-0209166-3-24-3'], ['quant-ph-0209166-2-24-4', 'quant-ph-0209166-3-24-4'], ['quant-ph-0209166-3-2-2', 'quant-ph-0209166-4-2-2'], ['quant-ph-0209166-3-2-3', 'quant-ph-0209166-4-2-3'], ['quant-ph-0209166-3-3-0', 'quant-ph-0209166-4-3-0'], ['quant-ph-0209166-3-3-1', 'quant-ph-0209166-4-3-1'], ['quant-ph-0209166-3-3-2', 'quant-ph-0209166-4-3-2'], ['quant-ph-0209166-3-3-3', 'quant-ph-0209166-4-3-3'], ['quant-ph-0209166-3-3-4', 'quant-ph-0209166-4-3-4'], ['quant-ph-0209166-3-3-5', 'quant-ph-0209166-4-3-5'], ['quant-ph-0209166-3-3-6', 'quant-ph-0209166-4-3-6'], ['quant-ph-0209166-3-26-2', 'quant-ph-0209166-4-26-2'], ['quant-ph-0209166-3-26-3', 'quant-ph-0209166-4-26-3'], ['quant-ph-0209166-3-26-4', 'quant-ph-0209166-4-26-4'], ['quant-ph-0209166-3-26-8', 'quant-ph-0209166-4-26-8'], ['quant-ph-0209166-3-26-9', 'quant-ph-0209166-4-26-9'], ['quant-ph-0209166-3-23-0', 'quant-ph-0209166-4-23-0'], ['quant-ph-0209166-3-23-1', 'quant-ph-0209166-4-23-1'], ['quant-ph-0209166-3-23-3', 'quant-ph-0209166-4-23-3'], ['quant-ph-0209166-3-23-4', 'quant-ph-0209166-4-23-4'], ['quant-ph-0209166-3-23-6', 'quant-ph-0209166-4-23-6'], ['quant-ph-0209166-3-23-7', 'quant-ph-0209166-4-23-7'], ['quant-ph-0209166-3-23-8', 'quant-ph-0209166-4-23-8'], ['quant-ph-0209166-3-4-7', 'quant-ph-0209166-4-4-7'], ['quant-ph-0209166-3-16-1', 'quant-ph-0209166-4-16-1'], ['quant-ph-0209166-3-16-3', 'quant-ph-0209166-4-16-3'], ['quant-ph-0209166-3-16-5', 'quant-ph-0209166-4-16-5'], ['quant-ph-0209166-3-14-2', 'quant-ph-0209166-4-14-2'], ['quant-ph-0209166-3-24-0', 'quant-ph-0209166-4-24-0'], ['quant-ph-0209166-3-24-1', 'quant-ph-0209166-4-24-1'], ['quant-ph-0209166-3-24-3', 'quant-ph-0209166-4-24-3'], ['quant-ph-0209166-3-24-4', 'quant-ph-0209166-4-24-4'], ['quant-ph-0209166-3-31-1', 'quant-ph-0209166-4-31-1'], ['quant-ph-0209166-3-31-2', 'quant-ph-0209166-4-31-2'], ['quant-ph-0209166-3-22-0', 'quant-ph-0209166-4-22-0'], ['quant-ph-0209166-3-22-1', 'quant-ph-0209166-4-22-1'], ['quant-ph-0209166-3-7-1', 'quant-ph-0209166-4-7-1'], ['quant-ph-0209166-3-7-2', 'quant-ph-0209166-4-7-2'], ['quant-ph-0209166-3-7-4', 'quant-ph-0209166-4-7-4'], 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'quant-ph-0209166-1-9-0', 'quant-ph-0209166-1-10-0', 'quant-ph-0209166-1-11-5', 'quant-ph-0209166-1-11-6', 'quant-ph-0209166-1-11-7', 'quant-ph-0209166-1-11-8', 'quant-ph-0209166-1-12-4', 'quant-ph-0209166-1-12-10', 'quant-ph-0209166-1-18-0', 'quant-ph-0209166-1-21-0', 'quant-ph-0209166-1-21-6', 'quant-ph-0209166-1-23-3', 'quant-ph-0209166-2-8-3', 'quant-ph-0209166-2-8-8', 'quant-ph-0209166-2-10-0', 'quant-ph-0209166-2-11-0', 'quant-ph-0209166-2-12-0', 'quant-ph-0209166-2-13-0', 'quant-ph-0209166-2-13-1', 'quant-ph-0209166-2-13-2', 'quant-ph-0209166-2-14-0', 'quant-ph-0209166-2-14-4', 'quant-ph-0209166-2-14-5', 'quant-ph-0209166-2-17-4', 'quant-ph-0209166-2-17-5', 'quant-ph-0209166-2-17-22', 'quant-ph-0209166-2-19-1', 'quant-ph-0209166-2-23-2', 'quant-ph-0209166-2-23-10', 'quant-ph-0209166-2-26-0', 'quant-ph-0209166-2-26-5', 'quant-ph-0209166-2-26-10', 'quant-ph-0209166-2-29-3', 'quant-ph-0209166-3-8-3', 'quant-ph-0209166-3-8-8', 'quant-ph-0209166-3-10-0', 'quant-ph-0209166-3-11-0', 'quant-ph-0209166-3-12-0', 'quant-ph-0209166-3-13-0', 'quant-ph-0209166-3-13-1', 'quant-ph-0209166-3-13-2', 'quant-ph-0209166-3-14-0', 'quant-ph-0209166-3-14-4', 'quant-ph-0209166-3-14-5', 'quant-ph-0209166-3-17-4', 'quant-ph-0209166-3-17-5', 'quant-ph-0209166-3-17-22', 'quant-ph-0209166-3-19-1', 'quant-ph-0209166-3-23-2', 'quant-ph-0209166-3-23-10', 'quant-ph-0209166-3-26-0', 'quant-ph-0209166-3-26-5', 'quant-ph-0209166-3-26-10', 'quant-ph-0209166-3-29-3', 'quant-ph-0209166-4-8-3', 'quant-ph-0209166-4-8-8', 'quant-ph-0209166-4-10-0', 'quant-ph-0209166-4-11-0', 'quant-ph-0209166-4-12-0', 'quant-ph-0209166-4-13-0', 'quant-ph-0209166-4-13-1', 'quant-ph-0209166-4-13-2', 'quant-ph-0209166-4-14-0', 'quant-ph-0209166-4-14-5', 'quant-ph-0209166-4-14-6', 'quant-ph-0209166-4-17-4', 'quant-ph-0209166-4-17-5', 'quant-ph-0209166-4-17-22', 'quant-ph-0209166-4-19-1', 'quant-ph-0209166-4-23-2', 'quant-ph-0209166-4-23-10', 'quant-ph-0209166-4-26-0', 'quant-ph-0209166-4-26-5', 'quant-ph-0209166-4-26-10', 'quant-ph-0209166-4-29-3']","{'1': 'http://arxiv.org/licenses/assumed-1991-2003/', '2': 'http://arxiv.org/licenses/assumed-1991-2003/', '3': 'http://arxiv.org/licenses/assumed-1991-2003/', '4': 'http://arxiv.org/licenses/assumed-1991-2003/'}",https://arxiv.org/abs/quant-ph/0209166,"{'quant-ph-0209166-3-0-0': 'We present a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via local operations and classical communication (LOCC).', 'quant-ph-0209166-3-0-1': 'This framework unifies and greatly simplifies the results of Refs. [CITATION].', 'quant-ph-0209166-3-0-2': 'A new necessary condition for ""pure contraction"" transformations is given.', 'quant-ph-0209166-3-0-3': 'Finally, constructive protocols to achieve both probabilistic and deterministic entanglement transformations are presented.', 'quant-ph-0209166-3-1-0': '# Introduction', 'quant-ph-0209166-3-2-0': 'The transformation of entangled states by means of LOCC is a key issue in quantum information processing [CITATION], for quantum computation [CITATION], quantum teleportation [CITATION], and quantum cryptography [CITATION].', 'quant-ph-0209166-3-2-1': 'However, the detrimental effect of losses and decoherence poses a serious problem for establishing entangled resources at distance or in a long computing network, since in a realistic transmission/computation the entanglement can be considerably degraded, thus preventing, for example, successful teleportation or dense coding.', 'quant-ph-0209166-3-2-2': 'For this reason, the use of transformations which can increase the available entanglement by means of LOCC-although with some probability-is crucial for practical purposes.', 'quant-ph-0209166-3-2-3': 'More generally, understanding entanglement transformations that are allowed by LOCC provides more insight in the structure and property of nonlocality, the most prominent character of quantum mechanics.', 'quant-ph-0209166-3-3-0': 'On entanglement transformations via LOCC there are two main results.', 'quant-ph-0209166-3-3-1': 'The first is the seminal work by Nielsen [CITATION] on deterministic transformations, which introduces majorization theory in this context.', 'quant-ph-0209166-3-3-2': 'The second is the work by Vidal [CITATION], which addresses the more general problem of probabilistic transformations.', 'quant-ph-0209166-3-3-3': 'Such work is based on the approach of entanglement monotones, and gives conditions equivalent to weak majorization relations.', 'quant-ph-0209166-3-3-4': 'The two approaches are completely disconnected, and for practical applications of the theory, a unified framework would be needed, especially in consideration that the more general treatment by Vidal is more abstract and less constructive than the Nielsen approach, which, however, is limited only to deterministic transformations.', 'quant-ph-0209166-3-3-5': 'Furthermore, from the Vidal approach it is very difficult to recover the Nielsen treatment as a special case, and it is quite surprising that such approach ends up as a weak-majorization condition, without essentially using majorization theory.', 'quant-ph-0209166-3-3-6': 'This motivates a derivation of the general nondeterministic LOCC transformations of pure states with an approach completely based on majorization theory, generalizing the Nielsen work [CITATION].', 'quant-ph-0209166-3-4-0': 'In this paper we present a general framework for entanglement transformations of bipartite pure states by means of LOCC.', 'quant-ph-0209166-3-4-1': 'In Sec. II we give a short and very simple proof of the theorem of Lo and Popescu [CITATION], which is at the basis of the theory of all LOCC, and which states that given two separate parties, say Alice and Bob, all LOCC transformations on a pure bipartite state can be reduced to a contraction by Alice and a unitary transformation by Bob.', 'quant-ph-0209166-3-4-2': 'We include our derivation of this theorem, since it is particularly simple and is based on a new useful technique for operator transposition.', 'quant-ph-0209166-3-4-3': 'In Sec. III we derive the main theorem, which gives a necessary and sufficient condition for all entanglement transformations of pure states in terms of super-majorization conditions, generalizing the Nielsen approach, and recovering the result of Vidal.', 'quant-ph-0209166-3-4-4': 'Here we also provide a new necessary condition for ""pure contraction"" transformations, namely those LOCC transformations in which the target state is achieved just for a single outcome of Alice measurement and a corresponding unitary on Bob side.', 'quant-ph-0209166-3-4-5': 'Such condition writes in terms of sub-majorization relation.', 'quant-ph-0209166-3-4-6': 'In Sec. IV explicit protocols to achieve pure, deterministic, and probabilistic transformations are given, using a new method which gives the Alice contraction of the LOCC in terms of the Moore-Penrose pseudoinverse of the matrix of the entangled state.', 'quant-ph-0209166-3-4-7': 'Section V concludes the paper summarizing the results.', 'quant-ph-0209166-3-5-0': '# The Lo-Popescu theorem', 'quant-ph-0209166-3-6-0': 'For later convenience we introduce here the main notation used in the paper.', 'quant-ph-0209166-3-6-1': 'Given a linear operator [MATH] we denote its hermitian conjugate by [MATH].', 'quant-ph-0209166-3-6-2': 'On a fixed basis, we write the complex conjugate and the transpose of [MATH] as [MATH] and [MATH], respectively, so that [MATH].', 'quant-ph-0209166-3-6-3': 'With the notation [MATH] we denote the Moore-Penrose inverse of [MATH].', 'quant-ph-0209166-3-6-4': 'We recall that the Moore-Penrose inverse [MATH] is the unique matrix that satisfies [EQUATION]', 'quant-ph-0209166-3-6-5': 'From Eq. ([REF]) it immediately follows that both [MATH] and [MATH] are orthogonal projectors, in particular [MATH] is the orthogonal projector over the range of [MATH], whereas [MATH] is the orthogonal projector over the support of [MATH].', 'quant-ph-0209166-3-6-6': 'We write the singular value decomposition (SVD) of [MATH] as follows [EQUATION] where [MATH] denotes the diagonal matrix whose entries are the singular values [MATH] of [MATH] taken in decreasing order, and [MATH] are unitary.', 'quant-ph-0209166-3-6-7': 'The Moore-Penrose inverse [MATH] then writes [EQUATION] where [MATH] is diagonal with entries [MATH] for [MATH], and zero entries for [MATH].', 'quant-ph-0209166-3-7-0': 'We remember that a quantum measurement (with discrete spectrum) is generally described by a Positive Operator-Valued Measurement (POVM), namely by a resolution of the identity [MATH], where each [MATH] corresponds to a possible outcome.', 'quant-ph-0209166-3-7-1': 'Each operator [MATH] acts on the input states and is necessarily a contraction, namely it satisfies [MATH], where [MATH] denotes the usual operator norm.', 'quant-ph-0209166-3-7-2': 'For bipartite pure states on the Hilbert space [MATH] we use the following notation [CITATION] [EQUATION] where [MATH] and [MATH] are two chosen orthonormal bases for [MATH] and [MATH], respectively.', 'quant-ph-0209166-3-7-3': 'Eq. ([REF]) introduces an isomorphism between vectors in [MATH] and [MATH] matrices, where [MATH] and [MATH] are the dimensions of [MATH] and [MATH].', 'quant-ph-0209166-3-7-4': 'One can easily check the relation [EQUATION] where the transposition is defined with respect to the orthonormal basis [MATH].', 'quant-ph-0209166-3-7-5': 'Finally, we use the notation [MATH] for Hermitian operators [MATH] and [MATH] to denote the vector majorization relation [CITATION] [MATH], and in the same fashion we will write [MATH] and [MATH] for super- and sub-majorization, respectively.', 'quant-ph-0209166-3-8-0': 'In the last part of this section we provide a very short proof of the following theorem [CITATION].', 'quant-ph-0209166-3-8-1': 'All LOCC on a pure bipartite state [MATH] can be reduced to a contraction by Alice and a unitary transformation by Bob.', 'quant-ph-0209166-3-8-2': 'This resorts to the equivalence of any Bob contraction [MATH] with the Alice contraction [MATH] assisted by Bob unitary transformation [MATH] as follows [EQUATION] where [EQUATION] and [MATH] is the unitary operator achieving the transposition of the operator [MATH], namely [EQUATION]', 'quant-ph-0209166-3-8-3': 'Proof.', 'quant-ph-0209166-3-8-4': 'To prove that every LOCC can be reduced to an Alice contraction and a Bob unitary transformation it is sufficient to prove the equivalence ([REF]), since: a) all possible elementary LOCC in a sequence will be reduced to an Alice contraction and a Bob unitary; b) the product of two contractions is a contraction; c) unitary transformations are particular cases of contraction.', 'quant-ph-0209166-3-8-5': 'Notice that given the SVD of any linear operator [MATH] as in Eq. ([REF]) one has [EQUATION] with [MATH].', 'quant-ph-0209166-3-8-6': 'Using Eq. ([REF]), Eq. ([REF]) rewrites as follows [EQUATION]', 'quant-ph-0209166-3-8-7': 'Then, from Eq. ([REF]) one has [EQUATION] which is just Eq. ([REF]) with [MATH] and [MATH] given as in Eq. ([REF]).', 'quant-ph-0209166-3-8-8': '[MATH]', 'quant-ph-0209166-3-9-0': '# LOCC transformations for pure states', 'quant-ph-0209166-3-10-0': 'In this section we will use the following useful lemmas:', 'quant-ph-0209166-3-11-0': 'If [MATH] then for some [MATH] and [MATH].', 'quant-ph-0209166-3-12-0': 'Proof.', 'quant-ph-0209166-3-13-0': 'If [MATH] one has for [MATH] [EQUATION] and [EQUATION]', 'quant-ph-0209166-3-13-1': 'Upon defining [EQUATION] clearly one has [MATH], and [MATH] [EQUATION] namely [MATH].', 'quant-ph-0209166-3-13-2': '[MATH] If for some [MATH] and [MATH], then [MATH].', 'quant-ph-0209166-3-14-0': 'Proof.', 'quant-ph-0209166-3-14-1': 'One has [MATH] [EQUATION] namely [MATH].', 'quant-ph-0209166-3-14-2': '[MATH] We notice that both the above lemmas hold also in the reverse direction (for a proof see Ref. [CITATION], pp. 11 and 123).', 'quant-ph-0209166-3-14-3': 'Moreover, we will make extensive use of the following theorem: [Uhlmann] For Hermitian operators [MATH] and [MATH] one has [MATH] if and only if there are probabilities [MATH] and unitary operators [MATH] such that [EQUATION]', 'quant-ph-0209166-3-14-4': 'Proof.', 'quant-ph-0209166-3-14-5': 'See Ref. [CITATION], p. 575.', 'quant-ph-0209166-3-15-0': 'Theorem [REF] relates majorization between hermitian operators with a particular form of completely positive maps, namely those achievable through a random unitary evolution.', 'quant-ph-0209166-3-15-1': 'In the terminology of quantum information channels, such maps correspond to external-random-field channels, that are a subclass of bistochastic channels (which send the identity operator into itself).', 'quant-ph-0209166-3-15-2': 'For a qubit system ([MATH]) the set of bistochastic and external-random-field channels coincide [CITATION].', 'quant-ph-0209166-3-16-0': 'In the following we derive the necessary and sufficient condition for all entanglement transformations of pure states in terms of super-majorization conditions.', 'quant-ph-0209166-3-16-1': 'The theorem generalizes Nielsen approach [CITATION] and recovers the result of Vidal probabilistic transformations [CITATION].', 'quant-ph-0209166-3-16-2': 'Moreover, we provide a new necessary condition for ""pure contraction"" transformations, namely those LOCC transformations in which the target state is achieved just for a single outcome of Alice measurement and a corresponding unitary on Bob side.', 'quant-ph-0209166-3-16-3': 'In the proof a relevant role is played by the intermediate state (denoted in the following by [MATH]).', 'quant-ph-0209166-3-16-4': 'In a transformation from [MATH] to [MATH] the state [MATH] will be reached from [MATH] deterministically, whereas the final probabilistic transformation [MATH] will be obtained through a pure contraction (for deterministic transformations one has [MATH]).', 'quant-ph-0209166-3-16-5': 'We are now ready to prove the following', 'quant-ph-0209166-3-17-0': 'The state transformation [MATH] is possible by LOCC iff [EQUATION] where [MATH] is the probability of achieving the transformation.', 'quant-ph-0209166-3-17-1': 'A necessary condition to be satisfied is [MATH].', 'quant-ph-0209166-3-17-2': 'In particular, the transformation is deterministic ([MATH]) iff [MATH].', 'quant-ph-0209166-3-17-3': 'Finally, if there is a pure LOCC that achieves the state transformation with probability [MATH], we must have [EQUATION]', 'quant-ph-0209166-3-17-4': 'Proof.', 'quant-ph-0209166-3-17-5': 'Suppose that [MATH].', 'quant-ph-0209166-3-17-6': 'From Lemma [REF] there is an operator [MATH] with [MATH] and [MATH].', 'quant-ph-0209166-3-17-7': 'The state [MATH] represents the intermediate state that can be achieved deterministically from [MATH].', 'quant-ph-0209166-3-17-8': 'In fact, Uhlmann Theorem [REF] guarantees the existence of a set of unitaries [MATH] and probabilities [MATH] such that [EQUATION]', 'quant-ph-0209166-3-17-9': 'Now, define the Alice measurement [MATH] such that [EQUATION] namely [EQUATION]', 'quant-ph-0209166-3-17-10': 'We can always choose [MATH] such that [MATH], and show that [MATH] is a contraction since [EQUATION] and so [MATH].', 'quant-ph-0209166-3-17-11': 'Then, there exists a set of unitary operators [MATH] such that [MATH], namely [EQUATION] so that the transformation from [MATH] can be achieved deterministically.', 'quant-ph-0209166-3-17-12': 'Now, since [MATH] one can define the contraction [EQUATION] so that [EQUATION]', 'quant-ph-0209166-3-17-13': 'By using the SVD of both [MATH] and [MATH], Eq. ([REF]) rewrites as follows [EQUATION] which means that there exists a unitary transformation [MATH] such that [EQUATION] where [MATH].', 'quant-ph-0209166-3-17-14': 'Eq. ([REF]) is equivalent to the entanglement transformation [EQUATION] namely there is a pure LOCC occurring with probability [MATH] that transforms the state [MATH] into [MATH].', 'quant-ph-0209166-3-17-15': 'But, as we have seen before, the transformation [MATH] can be achieved deterministically, whence we conclude that [MATH] can be achieved with probability [MATH], namely the statement of the first part of the theorem.', 'quant-ph-0209166-3-17-16': 'Reversely, suppose that the transformation [MATH] can be achieved via LOCC with probability [MATH].', 'quant-ph-0209166-3-17-17': 'According to Theorem [REF] every LOCC is equivalent to a measurement performed by Alice followed by a conditional unitary by Bob.', 'quant-ph-0209166-3-17-18': 'Therefore, the joint Alice-Bob state [MATH] will evolve as follows [EQUATION] where [EQUATION]', 'quant-ph-0209166-3-17-19': 'If the state [MATH] goes to [MATH] we must have for a subset [MATH] of the possible outcomes [MATH] [EQUATION] where [MATH] denotes the overall probability of the transformation [MATH].', 'quant-ph-0209166-3-17-20': 'From Eq. ([REF]) we need to have [EQUATION] and, therefore, each probability [MATH] is given by [EQUATION] where [MATH] denotes the usual Frobenius norm.', 'quant-ph-0209166-3-17-21': 'The condition ([REF]) can be satisfied only if [MATH], i.e. we can only decrease the Schmidt number of the entangled state.', 'quant-ph-0209166-3-17-22': 'From Eq. ([REF]) one has [MATH] [EQUATION]', 'quant-ph-0209166-3-17-23': 'From Eq ([REF]) we have [EQUATION] and using Eq. ([REF]) we obtain [EQUATION]', 'quant-ph-0209166-3-17-24': 'By Uhlmann Theorem [REF] we have [EQUATION] and from Lemma [REF] we get the statement, namely [MATH].', 'quant-ph-0209166-3-18-0': 'We now prove the last part of the theorem, regarding the pure LOCC case.', 'quant-ph-0209166-3-18-1': 'In a pure contraction transformation the target state is achieved just for a single outcome of Alice measurement and a unitary performed by Bob.', 'quant-ph-0209166-3-18-2': 'Such a transformation that occurs with probability [MATH] is given by [EQUATION] and we need to have [EQUATION]', 'quant-ph-0209166-3-18-3': 'Again, this is possible if [MATH].', 'quant-ph-0209166-3-18-4': 'Using the SVD of [MATH] and [MATH] as in Eq. ([REF]), Eq. ([REF]) rewrites in terms of the diagonal matrices [MATH] and [MATH] as follows [EQUATION] with [EQUATION]', 'quant-ph-0209166-3-18-5': 'Eq. ([REF]) leads to [EQUATION] namely [EQUATION] where [MATH] is a sub-stochastic matrix [CITATION], since', 'quant-ph-0209166-3-19-0': 'Since Eq. ([REF]) with [MATH] sub-stochastic is equivalent [CITATION] to [MATH], namely [MATH], we have proved that Eq. ([REF]) is a necessary condition for the LOCC transformation ([REF]), namely the last statement of the theorem.', 'quant-ph-0209166-3-19-1': '[MATH]', 'quant-ph-0209166-3-20-0': '# The explicit protocols', 'quant-ph-0209166-3-21-0': '## Pure transformation', 'quant-ph-0209166-3-22-0': 'Pure LOCC transformations are achieved by a single contraction on Alice side, assisted by a unitary by Bob.', 'quant-ph-0209166-3-22-1': 'These are the most general one-way LOCC operations.', 'quant-ph-0209166-3-22-2': 'In this case we have the following theorem', 'quant-ph-0209166-3-23-0': 'The transformation [MATH] can be achieved with probability [MATH] by a pure LOCC transformation iff one can find a unitary operator [MATH] and linear operator [MATH] such that [EQUATION] is a contraction.', 'quant-ph-0209166-3-23-1': 'The transformation is then obtained as [EQUATION]', 'quant-ph-0209166-3-23-2': 'Proof.', 'quant-ph-0209166-3-23-3': 'Notice that Eq. ([REF]) is equivalent to the identity [EQUATION]', 'quant-ph-0209166-3-23-4': 'Since both sides of the identity must have the same kernel it follows that [EQUATION] and, multiplying both sides by [MATH], we have [EQUATION]', 'quant-ph-0209166-3-23-5': 'The general solution of the last equation is [EQUATION] with arbitrary [MATH], and in fact one can easily check that [EQUATION]', 'quant-ph-0209166-3-23-6': 'The unitary [MATH] and the operator [MATH] should be taken such that [MATH] is a contraction.', 'quant-ph-0209166-3-23-7': 'This is not always possible.', 'quant-ph-0209166-3-23-8': 'However, a sufficient condition is [EQUATION]', 'quant-ph-0209166-3-23-9': 'In fact, by taking [MATH] and [MATH] one has [MATH], and then, for Eq. ([REF]), [MATH].', 'quant-ph-0209166-3-23-10': '[MATH] ## Deterministic transformation', 'quant-ph-0209166-3-24-0': 'In the entanglement transformations, the first part of the protocol is a deterministic transformation from [MATH] to [MATH].', 'quant-ph-0209166-3-24-1': 'The majorization relation [MATH] implies Theorem [REF], namely the existence of a set of unitaries [MATH] and probabilities [MATH] such that [EQUATION]', 'quant-ph-0209166-3-24-2': 'In order to construct explicitly the protocol one needs to find contractions [MATH] and unitaries [MATH] versus the unitary operators [MATH] that appear in Eq. ([REF]), such that one has [EQUATION]', 'quant-ph-0209166-3-24-3': 'We have seen that the general form of the solution of Eq. ([REF]) is given by [EQUATION]', 'quant-ph-0209166-3-24-4': 'We have now the following theorem', 'quant-ph-0209166-3-25-0': 'In Eq. ([REF]) we can always take [EQUATION] where [MATH] are the operators defined in Eq. ([REF]), such that [MATH] is a contraction and Eq. ([REF]) is satisfied.', 'quant-ph-0209166-3-26-0': 'Proof.', 'quant-ph-0209166-3-26-1': 'Substituting Eq. ([REF]) in Eq. ([REF]) one has [EQUATION] and using Eq. ([REF]) one has [EQUATION]', 'quant-ph-0209166-3-26-2': 'Hence, [MATH] are contractions.', 'quant-ph-0209166-3-26-3': 'The completeness of the measurement can be guaranteed by the further contraction [EQUATION] where [MATH] is an arbitrary unitary operator.', 'quant-ph-0209166-3-26-4': 'For any outcome [MATH] on Alice side, Bob performs the unitary [MATH].', 'quant-ph-0209166-3-26-5': 'Using Eq. ([REF]) and ([REF]) one has [EQUATION]', 'quant-ph-0209166-3-26-6': 'From Eq. ([REF]), it follows that [MATH], namely [EQUATION]', 'quant-ph-0209166-3-26-7': 'Hence, by multiplying both sides of Eq. ([REF]) on the left by [MATH] and on the right by [MATH] one obtains [EQUATION]', 'quant-ph-0209166-3-26-8': 'The projector on [MATH] coincides with the projector on [MATH].', 'quant-ph-0209166-3-26-9': 'So Eq. ([REF]) gives [EQUATION] which is equivalent to Eq. ([REF]).', 'quant-ph-0209166-3-26-10': '[MATH]', 'quant-ph-0209166-3-27-0': 'According to our derivation, given explicitly Eq. ([REF]) one can perform the contractions [MATH] and the unitaries [MATH] to achieve the transformation [MATH].', 'quant-ph-0209166-3-27-1': 'The problem of looking for a POVM with minimum number of outcomes (thus minimizing the amount of classical information sent to Bob) is reduced to find the transformation ([REF]) with minimum number of unitaries.', 'quant-ph-0209166-3-27-2': 'One can resort to a constructive algorithm to find a bistochastic matrix [MATH] which relates the vectors [MATH] and [MATH] of the singular values of [MATH] and [MATH], namely [EQUATION]', 'quant-ph-0209166-3-27-3': 'Then Birkhoff theorem [CITATION] allows to write [MATH] as a convex combination of permutation matrices [EQUATION]', 'quant-ph-0209166-3-27-4': 'In terms of [MATH] and [MATH] one has [EQUATION] where [MATH].', 'quant-ph-0209166-3-27-5': 'In this way one obtains Eq. ([REF]), with [MATH].', 'quant-ph-0209166-3-27-6': 'Using the corresponding expressions for the contractions [MATH] and unitaries [MATH] one recovers the result of Ref. [CITATION].', 'quant-ph-0209166-3-27-7': ""Notice that Caratheodory's theorem always allows to reduce the number of permutations in Eq. ([REF]) to [MATH], for [MATH]-dimensional Alice Hilbert space."", 'quant-ph-0209166-3-28-0': '## Probabilistic transformation', 'quant-ph-0209166-3-29-0': 'The second part of the protocol, namely the contraction which provides the state [MATH] from [MATH], is needed only for probabilistic transformations.', 'quant-ph-0209166-3-29-1': 'It is a pure contraction given by [EQUATION] with [EQUATION] and [EQUATION]', 'quant-ph-0209166-3-29-2': 'In fact for lemma [REF] one has [MATH], which implies that [MATH].', 'quant-ph-0209166-3-29-3': 'Then [EQUATION]', 'quant-ph-0209166-3-30-0': '# Conclusions', 'quant-ph-0209166-3-31-0': 'In this paper we presented a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via LOCC transformations.', 'quant-ph-0209166-3-31-1': 'We have generalized Nielsen work based on majorization theory [CITATION] in order to include the more general results by Vidal [CITATION], which were based on the approach of the entanglement monotones.', 'quant-ph-0209166-3-31-2': 'The main theorem gives an if and only if condition for all entanglement transformations of pure states in terms of super-majorization conditions.', 'quant-ph-0209166-3-31-3': 'We also provided a new necessary sub-majorization condition for pure transformations, which allows to write each contraction in terms of the Moore-Penrose pseudoinverse of the matrix of the entangled state.', 'quant-ph-0209166-3-31-4': 'This lead to explicit protocols to achieve pure, deterministic, and probabilistic LOCC.', 'quant-ph-0209166-3-32-0': 'We notice that all theorems have been derived in finite dimensions, but they can be easily extended to infinite dimensions for contractions that are compact operators and for normalized entangled states corresponding to Hilbert-Schmidt operators.', 'quant-ph-0209166-3-32-1': 'Thus our results also apply to continuous variables.'}","{'quant-ph-0209166-4-0-0': 'We present a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via local operations and classical communication.', 'quant-ph-0209166-4-0-1': 'This framework unifies and greatly simplifies previous works.', 'quant-ph-0209166-4-0-2': 'A necessary condition for ""pure contraction"" transformations is given.', 'quant-ph-0209166-4-0-3': 'Finally, constructive protocols to achieve both probabilistic and deterministic entanglement transformations are presented.', 'quant-ph-0209166-4-1-0': '# Introduction', 'quant-ph-0209166-4-2-0': 'The transformation of entangled states by means of local operations and classical communication (LOCC) is a key issue in quantum information processing [CITATION], for quantum computation [CITATION], quantum teleportation [CITATION], and quantum cryptography [CITATION].', 'quant-ph-0209166-4-2-1': 'However, the detrimental effect of losses and decoherence poses a serious problem for establishing entangled resources at distance or in a long computing network, since in a realistic transmission or computation the entanglement can be considerably degraded, thus preventing, for example, successful teleportation or dense coding.', 'quant-ph-0209166-4-2-2': 'For this reason, the use of transformations which can increase the available entanglement by means of LOCC-although with some probability-is crucial for practical purposes.', 'quant-ph-0209166-4-2-3': 'More generally, understanding entanglement transformations that are allowed by LOCC provides more insight in the structure and property of nonlocality, the most prominent character of quantum mechanics.', 'quant-ph-0209166-4-3-0': 'On entanglement transformations via LOCC there are two main results.', 'quant-ph-0209166-4-3-1': 'The first is the seminal work by Nielsen [CITATION] on deterministic transformations, which introduces majorization theory in this context.', 'quant-ph-0209166-4-3-2': 'The second is the work by Vidal [CITATION], which addresses the more general problem of probabilistic transformations.', 'quant-ph-0209166-4-3-3': 'Such work is based on the approach of entanglement monotones, and gives conditions equivalent to weak majorization relations.', 'quant-ph-0209166-4-3-4': 'The two approaches are completely disconnected, and for practical applications of the theory, a unified framework would be needed, especially in consideration that the more general treatment by Vidal is more abstract and less constructive than the Nielsen approach, which, however, is limited only to deterministic transformations.', 'quant-ph-0209166-4-3-5': 'Furthermore, from the Vidal approach it is very difficult to recover the Nielsen treatment as a special case, and it is quite surprising that such approach ends up as a weak-majorization condition, without essentially using majorization theory.', 'quant-ph-0209166-4-3-6': 'This motivates a derivation of the general nondeterministic LOCC transformations of pure states with an approach completely based on majorization theory, generalizing the Nielsen work [CITATION].', 'quant-ph-0209166-4-4-0': 'In this paper, we present a general framework for entanglement transformations of bipartite pure states by means of LOCC.', 'quant-ph-0209166-4-4-1': 'In Sec. II, we give a short and very simple proof of the theorem of Lo and Popescu [CITATION], which is at the basis of the theory of all LOCC, and which states that given two separate parties, say Alice and Bob, all LOCC transformations on a pure bipartite state can be reduced to a contraction by Alice and a unitary transformation by Bob.', 'quant-ph-0209166-4-4-2': 'We include our derivation of this theorem, since it is particularly simple and is based on a useful technique for operator transposition.', 'quant-ph-0209166-4-4-3': 'In Sec. III, we derive the main theorem, which gives a necessary and sufficient condition for all entanglement transformations of pure states in terms of supermajorization conditions, generalizing the Nielsen approach, and recovering the result of Vidal.', 'quant-ph-0209166-4-4-4': 'Here, we also provide a necessary condition for ""pure contraction"" transformations, namely those LOCC transformations in which the target state is achieved just for a single outcome of Alice measurement and a corresponding unitary on Bob side.', 'quant-ph-0209166-4-4-5': 'Such a condition is written in terms of submajorization relation.', 'quant-ph-0209166-4-4-6': 'In Sec. IV, explicit protocols to achieve pure, deterministic, and probabilistic transformations are given, using a method that gives the Alice contraction of the LOCC in terms of the Moore-Penrose pseudoinverse of the matrix of the entangled state.', 'quant-ph-0209166-4-4-7': 'Section V concludes the paper summarizing the results.', 'quant-ph-0209166-4-5-0': '# The Lo-Popescu theorem', 'quant-ph-0209166-4-6-0': 'For later convenience, we introduce here the main notation used in the paper.', 'quant-ph-0209166-4-6-1': 'Given a linear operator [MATH], we denote its Hermitian conjugate by [MATH].', 'quant-ph-0209166-4-6-2': 'On a fixed basis, we write the complex conjugate and the transpose of [MATH] as [MATH] and [MATH], respectively, so that [MATH].', 'quant-ph-0209166-4-6-3': 'With the notation [MATH] we denote the Moore-Penrose inverse of [MATH].', 'quant-ph-0209166-4-6-4': 'We recall that the Moore-Penrose inverse [MATH] is the unique matrix that satisfies [EQUATION]', 'quant-ph-0209166-4-6-5': 'From Eq. ([REF]), it immediately follows that both [MATH] and [MATH] are orthogonal projectors, in particular, [MATH] is the orthogonal projector over the range of [MATH], whereas [MATH] is the orthogonal projector over the support of [MATH].', 'quant-ph-0209166-4-6-6': 'We write the singular value decomposition (SVD) of [MATH] as follows [EQUATION] where [MATH] denotes the diagonal matrix whose entries are the singular values [MATH] of [MATH] taken in a decreasing order, and [MATH] are unitary.', 'quant-ph-0209166-4-6-7': 'The Moore-Penrose inverse [MATH] then writes [EQUATION] where [MATH] is diagonal with entries [MATH] for [MATH], and zero entries for [MATH].', 'quant-ph-0209166-4-7-0': 'We remember that a quantum measurement (with discrete spectrum) is generally described by a positive operator-valued measurement (POVM), namely by a resolution of the identity [MATH], where each [MATH] corresponds to a possible outcome.', 'quant-ph-0209166-4-7-1': 'Each operator [MATH] acts on the input states and is necessarily a contraction, namely it satisfies [MATH], where [MATH] denotes the usual operator norm.', 'quant-ph-0209166-4-7-2': 'For bipartite pure states on the Hilbert space [MATH] we use the following notation [CITATION] [EQUATION] where [MATH] and [MATH] are two chosen orthonormal bases for [MATH] and [MATH], respectively.', 'quant-ph-0209166-4-7-3': 'Equation ([REF]) introduces an isomorphism between vectors in [MATH] and [MATH] matrices, where [MATH] and [MATH] are the dimensions of [MATH] and [MATH].', 'quant-ph-0209166-4-7-4': 'One can easily check the relation [EQUATION] where the transposition is defined with respect to the orthonormal basis [MATH].', 'quant-ph-0209166-4-7-5': 'Finally, we use the notation [MATH] for Hermitian operators [MATH] and [MATH] to denote the vector majorization relation [CITATION] [MATH], and in the same manner we will write [MATH] and [MATH] for super- and sub-majorization, respectively.', 'quant-ph-0209166-4-8-0': 'In the last part of this section, we provide a very short proof of the following theorem [CITATION].', 'quant-ph-0209166-4-8-1': 'All LOCC on a pure bipartite state [MATH] can be reduced to a contraction by Alice and a unitary transformation by Bob.', 'quant-ph-0209166-4-8-2': 'This resorts to the equivalence of any Bob contraction [MATH] with the Alice contraction [MATH] assisted by Bob unitary transformation [MATH] as follows [EQUATION] where [EQUATION] and [MATH] is the unitary operator achieving the transposition of the operator [MATH], namely, [EQUATION]', 'quant-ph-0209166-4-8-3': 'Proof.', 'quant-ph-0209166-4-8-4': 'To prove that every LOCC can be reduced to an Alice contraction and a Bob unitary transformation, it is sufficient to prove the equivalence ([REF]), since a) all possible elementary LOCC in a sequence will be reduced to an Alice contraction and a Bob unitary; b) the product of two contractions is a contraction; c) unitary transformations are particular cases of contraction.', 'quant-ph-0209166-4-8-5': 'Notice that given the SVD of any linear operator [MATH] as in Eq. ([REF]), one has [EQUATION] with [MATH].', 'quant-ph-0209166-4-8-6': 'Using Eq. ([REF]), Eq. ([REF]) rewrites as follows [EQUATION]', 'quant-ph-0209166-4-8-7': 'Then, from Eq. ([REF]) one has [EQUATION] which is just Eq. ([REF]) with [MATH] and [MATH] given as in Eq. ([REF]).', 'quant-ph-0209166-4-8-8': '[MATH]', 'quant-ph-0209166-4-9-0': '# LOCC transformations for pure states', 'quant-ph-0209166-4-10-0': 'In this section, we will use the following useful lemmas.', 'quant-ph-0209166-4-11-0': 'If [MATH], then for some [MATH] and [MATH].', 'quant-ph-0209166-4-12-0': 'Proof.', 'quant-ph-0209166-4-13-0': 'If [MATH] one has for [MATH] [EQUATION] and [EQUATION]', 'quant-ph-0209166-4-13-1': 'Upon defining [EQUATION] clearly one has [MATH], and [EQUATION] namely [MATH].', 'quant-ph-0209166-4-13-2': '[MATH] If for some [MATH] and [MATH], then [MATH].', 'quant-ph-0209166-4-14-0': 'Proof.', 'quant-ph-0209166-4-14-1': 'One has [EQUATION] namely, [MATH].', 'quant-ph-0209166-4-14-2': '[MATH] We notice that both the above lemmas hold also in the reverse direction (for a proof see Ref. [CITATION], pp. 11 and 123).', 'quant-ph-0209166-4-14-3': 'Moreover, we will make extensive use of the following theorem.', 'quant-ph-0209166-4-14-4': '[Uhlmann] For Hermitian operators [MATH] and [MATH], one has [MATH] if and only if there are probabilities [MATH] and unitary operators [MATH] such that [EQUATION]', 'quant-ph-0209166-4-14-5': 'Proof.', 'quant-ph-0209166-4-14-6': 'See Ref. [CITATION], p. 575.', 'quant-ph-0209166-4-15-0': 'Theorem [REF] relates majorization between Hermitian operators with a particular form of completely positive maps, namely those achievable through a random unitary evolution.', 'quant-ph-0209166-4-15-1': 'In the terminology of quantum-information channels, such maps correspond to external-random-field channels that are a subclass of bistochastic channels (which send the identity operator into itself).', 'quant-ph-0209166-4-15-2': 'For a qubit system ([MATH]), the set of bistochastic and external-random-field channels coincide [CITATION].', 'quant-ph-0209166-4-16-0': 'In the following, we derive the necessary and sufficient condition for all entanglement transformations of pure states in terms of supermajorization conditions.', 'quant-ph-0209166-4-16-1': 'The theorem generalizes Nielsen approach [CITATION] and recovers the result of Vidal probabilistic transformations [CITATION].', 'quant-ph-0209166-4-16-2': 'Moreover, we provide a necessary condition for ""pure contraction"" transformations, namely, those LOCC transformations in which the target state is achieved just for a single outcome of Alice measurement and a corresponding unitary on Bob side.', 'quant-ph-0209166-4-16-3': 'In the proof a relevant role is played by the intermediate state (denoted in the following by [MATH]).', 'quant-ph-0209166-4-16-4': 'In a transformation from [MATH] to [MATH], the state [MATH] will be reached from [MATH] deterministically, whereas the final probabilistic transformation [MATH] will be obtained through a pure contraction (for deterministic transformations one has [MATH]).', 'quant-ph-0209166-4-16-5': 'We are now ready to prove the following', 'quant-ph-0209166-4-17-0': 'The state transformation [MATH] is possible by LOCC iff [EQUATION] where [MATH] is the probability of achieving the transformation.', 'quant-ph-0209166-4-17-1': 'A necessary condition to be satisfied is [MATH].', 'quant-ph-0209166-4-17-2': 'In particular, the transformation is deterministic ([MATH]) iff [MATH].', 'quant-ph-0209166-4-17-3': 'Finally, if there is a pure LOCC that achieves the state transformation with probability [MATH], we must have [EQUATION]', 'quant-ph-0209166-4-17-4': 'Proof.', 'quant-ph-0209166-4-17-5': 'Assume that [MATH].', 'quant-ph-0209166-4-17-6': 'From Lemma [REF] there is an operator [MATH] with [MATH] and [MATH].', 'quant-ph-0209166-4-17-7': 'The state [MATH] represents the intermediate state that can be achieved deterministically from [MATH].', 'quant-ph-0209166-4-17-8': 'In fact, Uhlmann Theorem [REF] guarantees the existence of a set of unitaries [MATH] and probabilities [MATH] such that [EQUATION]', 'quant-ph-0209166-4-17-9': 'Now, define the Alice measurement [MATH] such that [EQUATION] namely, [EQUATION]', 'quant-ph-0209166-4-17-10': 'We can always choose [MATH] such that [MATH], and show that [MATH] is a contraction, since [EQUATION] and so [MATH].', 'quant-ph-0209166-4-17-11': 'Then, there exists a set of unitary operators [MATH] such that [MATH], namely, [EQUATION] so that the transformation from [MATH] can be achieved deterministically.', 'quant-ph-0209166-4-17-12': 'Now, since [MATH] one can define the contraction [EQUATION] so that [EQUATION]', 'quant-ph-0209166-4-17-13': 'By using the SVD of both [MATH] and [MATH], Eq. ([REF]) rewrites as follows [EQUATION] which means that there exists a unitary transformation [MATH] such that [EQUATION] where [MATH].', 'quant-ph-0209166-4-17-14': 'Equation ([REF]) is equivalent to the entanglement transformation [EQUATION] namely, there is a pure LOCC occurring with probability [MATH], which transforms the state [MATH] into [MATH].', 'quant-ph-0209166-4-17-15': 'As we have seen before, the transformation [MATH] can be achieved deterministically, whence we conclude that [MATH] can be achieved with probability [MATH], namely the statement of the first part of the theorem.', 'quant-ph-0209166-4-17-16': 'Reversely, assume that the transformation [MATH] can be achieved via LOCC with probability [MATH].', 'quant-ph-0209166-4-17-17': 'According to Theorem [REF] every LOCC is equivalent to a measurement performed by Alice followed by a conditional unitary by Bob.', 'quant-ph-0209166-4-17-18': 'Therefore, the joint Alice-Bob state [MATH] will evolve as follows: [EQUATION] where [EQUATION]', 'quant-ph-0209166-4-17-19': 'If the state [MATH] goes to [MATH], we must have for a subset [MATH] of the possible outcomes [MATH] [EQUATION] where [MATH] denotes the overall probability of the transformation [MATH].', 'quant-ph-0209166-4-17-20': 'From Eq. ([REF]), we need to have [EQUATION] and, therefore, each probability [MATH] is given by [EQUATION] where [MATH] denotes the usual Frobenius norm.', 'quant-ph-0209166-4-17-21': 'The condition ([REF]) can be satisfied only if [MATH], i.e. we can only decrease the Schmidt number of the entangled state.', 'quant-ph-0209166-4-17-22': 'From Eq. ([REF]), one has [MATH] [EQUATION]', 'quant-ph-0209166-4-17-23': 'From Eq. ([REF]), we have [EQUATION] and using Eq. ([REF]), we obtain [EQUATION]', 'quant-ph-0209166-4-17-24': 'By Uhlmann Theorem [REF], we have [EQUATION] and from Lemma [REF] we get the statement, namely, [MATH].', 'quant-ph-0209166-4-18-0': 'We now prove the last part of the theorem, regarding the pure LOCC case.', 'quant-ph-0209166-4-18-1': 'In a pure contraction transformation the target state is achieved just for a single outcome of Alice measurement and a unitary performed by Bob.', 'quant-ph-0209166-4-18-2': 'Such a transformation that occurs with probability [MATH] is given by [EQUATION] and we need to have [EQUATION]', 'quant-ph-0209166-4-18-3': 'Again, this is possible if [MATH].', 'quant-ph-0209166-4-18-4': 'Using the SVD of [MATH] and [MATH] as in Eq. ([REF]), Eq. ([REF]) rewrites in terms of the diagonal matrices [MATH] and [MATH] as follows [EQUATION] with [EQUATION]', 'quant-ph-0209166-4-18-5': 'Equation ([REF]) leads to [EQUATION] namely, [EQUATION] where [MATH] is a substochastic matrix [CITATION], since', 'quant-ph-0209166-4-19-0': 'Since Eq. ([REF]) with [MATH] substochastic is equivalent [CITATION] to [MATH], namely [MATH], we have proved that Eq. ([REF]) is a necessary condition for the LOCC transformation ([REF]), namely, the last statement of the theorem.', 'quant-ph-0209166-4-19-1': '[MATH]', 'quant-ph-0209166-4-20-0': '# Explicit protocols', 'quant-ph-0209166-4-21-0': '## Pure transformation', 'quant-ph-0209166-4-22-0': 'Pure LOCC transformations are achieved by a single contraction on Alice side, assisted by a unitary by Bob.', 'quant-ph-0209166-4-22-1': 'These are the most general one-way LOCC operations.', 'quant-ph-0209166-4-22-2': 'In this case, we have the following theorem.', 'quant-ph-0209166-4-23-0': 'The transformation [MATH] can be achieved with probability [MATH] by a pure LOCC transformation iff one can find a unitary operator [MATH] and linear operator [MATH] such that [EQUATION] is a contraction.', 'quant-ph-0209166-4-23-1': 'The transformation is then obtained as [EQUATION]', 'quant-ph-0209166-4-23-2': 'Proof.', 'quant-ph-0209166-4-23-3': 'Notice that Eq. ([REF]) is equivalent to the identity [EQUATION]', 'quant-ph-0209166-4-23-4': 'Since both sides of the identity must have the same kernel it follows that [EQUATION] and, multiplying both sides by [MATH], we have [EQUATION]', 'quant-ph-0209166-4-23-5': 'The general solution of the last equation is [EQUATION] with arbitrary [MATH] and, in fact, one can easily check that [EQUATION]', 'quant-ph-0209166-4-23-6': 'The unitary [MATH] and the operator [MATH] should be taken such that [MATH] is a contraction.', 'quant-ph-0209166-4-23-7': 'This is not always possible.', 'quant-ph-0209166-4-23-8': 'However, a sufficient condition is [EQUATION]', 'quant-ph-0209166-4-23-9': 'In fact, by taking [MATH] and [MATH], one has [MATH], and then, for Eq. ([REF]), [MATH].', 'quant-ph-0209166-4-23-10': '[MATH] ## Deterministic transformation', 'quant-ph-0209166-4-24-0': 'In the entanglement transformations, the first part of the protocol is a deterministic transformation from [MATH] to [MATH].', 'quant-ph-0209166-4-24-1': 'The majorization relation [MATH] implies Theorem [REF], namely the existence of a set of unitaries [MATH] and probabilities [MATH] such that [EQUATION]', 'quant-ph-0209166-4-24-2': 'In order to construct explicitly the protocol, one needs to find contractions [MATH] and unitaries [MATH] versus the unitary operators [MATH] that appear in Eq. ([REF]) such that one has [EQUATION]', 'quant-ph-0209166-4-24-3': 'We have seen that the general form of the solution of Eq. ([REF]) is given by [EQUATION]', 'quant-ph-0209166-4-24-4': 'We have now the following theorem', 'quant-ph-0209166-4-25-0': 'In Eq. ([REF]), we can always take [EQUATION] where [MATH] are the operators defined in Eq. ([REF]) such that [MATH] is a contraction and Eq. ([REF]) is satisfied.', 'quant-ph-0209166-4-26-0': 'Proof.', 'quant-ph-0209166-4-26-1': 'Substituting Eq. ([REF]) in Eq. ([REF]), one has [EQUATION] and using Eq. ([REF]), one has [EQUATION]', 'quant-ph-0209166-4-26-2': 'Hence, [MATH] are contractions.', 'quant-ph-0209166-4-26-3': 'The completeness of the measurement can be guaranteed by the further contraction [EQUATION] where [MATH] is an arbitrary unitary operator.', 'quant-ph-0209166-4-26-4': 'For any outcome [MATH] on Alice side, Bob performs the unitary [MATH].', 'quant-ph-0209166-4-26-5': 'Using Eqs. ([REF]) and ([REF]), one has [EQUATION]', 'quant-ph-0209166-4-26-6': 'From Eq. ([REF]), it follows that [MATH], namely, [EQUATION]', 'quant-ph-0209166-4-26-7': 'Hence, by multiplying both sides of Eq. ([REF]) on the left by [MATH] and on the right by [MATH], one obtains [EQUATION]', 'quant-ph-0209166-4-26-8': 'The projector on [MATH] coincides with the projector on [MATH].', 'quant-ph-0209166-4-26-9': 'So Eq. ([REF]) gives [EQUATION] which is equivalent to Eq. ([REF]).', 'quant-ph-0209166-4-26-10': '[MATH]', 'quant-ph-0209166-4-27-0': 'According to our derivation, given explicitly Eq. ([REF]), one can perform the contractions [MATH] and the unitaries [MATH] to achieve the transformation [MATH].', 'quant-ph-0209166-4-27-1': 'The problem of looking for a POVM with minimum number of outcomes (thus, minimizing the amount of classical information sent to Bob) is reduced to find the transformation ([REF]) with minimum number of unitaries.', 'quant-ph-0209166-4-27-2': 'One can resort to a constructive algorithm to find a bistochastic matrix [MATH] that relates the vectors [MATH] and [MATH] of the singular values of [MATH] and [MATH], namely [EQUATION]', 'quant-ph-0209166-4-27-3': 'Then Birkhoff theorem [CITATION] allows to write [MATH] as a convex combination of permutation matrices [EQUATION]', 'quant-ph-0209166-4-27-4': 'In terms of [MATH] and [MATH] one has [EQUATION] where [MATH].', 'quant-ph-0209166-4-27-5': 'In this way one obtains Eq. ([REF]), with [MATH].', 'quant-ph-0209166-4-27-6': 'Using the corresponding expressions for the contractions [MATH] and unitaries [MATH] one recovers the result of Ref. [CITATION].', 'quant-ph-0209166-4-27-7': ""Notice that Caratheodory's theorem always allows to reduce the number of permutations in Eq. ([REF]) to [MATH], for [MATH]-dimensional Alice Hilbert space."", 'quant-ph-0209166-4-28-0': '## Probabilistic transformation', 'quant-ph-0209166-4-29-0': 'The second part of the protocol, namely, the contraction that provides the state [MATH] from [MATH], is needed only for probabilistic transformations.', 'quant-ph-0209166-4-29-1': 'It is a pure contraction given by [EQUATION] with [EQUATION] and [EQUATION]', 'quant-ph-0209166-4-29-2': 'In fact for Lemma [REF], one has [MATH], which implies that [MATH].', 'quant-ph-0209166-4-29-3': 'Then [EQUATION]', 'quant-ph-0209166-4-30-0': '# Conclusions', 'quant-ph-0209166-4-31-0': 'In this paper, we presented a general theoretical framework for both deterministic and probabilistic entanglement transformations of bipartite pure states achieved via LOCC transformations.', 'quant-ph-0209166-4-31-1': 'We have generalized Nielsen work based on majorization theory [CITATION] in order to include the more general results by Vidal [CITATION], which were based on the approach of the entanglement monotones.', 'quant-ph-0209166-4-31-2': 'The main theorem gives an if and only if condition for all entanglement transformations of pure states in terms of super-majorization conditions.', 'quant-ph-0209166-4-31-3': 'We also provided a necessary submajorization condition for pure transformations, which allows to write each contraction in terms of the Moore-Penrose pseudoinverse of the matrix of the entangled state.', 'quant-ph-0209166-4-31-4': 'This led to explicit protocols to achieve pure, deterministic, and probabilistic LOCC.', 'quant-ph-0209166-4-32-0': 'We notice that all theorems have been derived in finite dimensions, but they can be easily extended to infinite dimensions for contractions that are compact operators and for normalized entangled states corresponding to Hilbert-Schmidt operators.', 'quant-ph-0209166-4-32-1': 'Thus, our results also apply to continuous variables.'}",
1809.06726,"{'1809.06726-1-0-0': 'The dynamics associated with invasion and metastasis, which hallmarks of cancer, and involve collective cell migration by blood or interstitial flow, is poorly understood.', '1809.06726-1-0-1': 'In the present contribution, we describe a theory to the essence of fluid shear stress (FSS) driven collective motion using the stochastic quantization method.', '1809.06726-1-0-2': 'Long range hydrodynamic interactions between cells exhibit complex collective properties whose understanding may be crucial for describing metastasis.', '1809.06726-1-0-3': 'Theory predicts a shear flow induced migration of colony of cells characterized by super-diffusive motion.', '1809.06726-1-0-4': 'Our findings, suggest a mechanism of shear flow induced directed motion of cancer cells, and provide the much needed insight into a recent experimental observation concerning the invasion of cancer cells.', '1809.06726-1-1-0': '# Introduction', '1809.06726-1-2-0': 'The collective migration of cells is relevant for many processes, like tissue remodeling events that underlie embryonic morphogenesis, wound repair, and cancer invasion[CITATION].', '1809.06726-1-2-1': 'Collective cancer cell invasion, followed by local or distant metastasis, is a hallmark of cancer[CITATION].', '1809.06726-1-2-2': 'Cancer cells can employ different cellular and molecular modes of invasion, depending on the specific cell-type.', '1809.06726-1-2-3': 'Prominent examples include the autonomous mechanism characterized by stochastic birth-death process, as well as reactive mechanisms induced by the local microenvironment.', '1809.06726-1-2-4': 'In such migration, cells move as sheets, strands, clusters or ducts rather than individually, and use similar actin and myosin mediated protrusions, and guidance by extrinsic chemotactic, and mechanical cues as used by single migratory cells[CITATION].', '1809.06726-1-2-5': 'Transition from epithelial to mesenchymal states plays an important role in cancer metastasis, is a subject of contemporary medical importance[CITATION].', '1809.06726-1-2-6': 'Tumor cells reside in an extracellular matrix (ECM) containing interstitial fluid that transports nutrients and signaling molecules.', '1809.06726-1-2-7': 'The interstitial flow has been shown to affect the morphology and migration of cells such as fibroblasts, cancer cells, endothelial cells, and mesenchymal stem cells[CITATION].', '1809.06726-1-2-8': 'Recent experiment has demonstrated that interstitial flows promote amoeboid over mesenchymal motility of brest cancer cells using a three dimensional microfluidic model[CITATION].', '1809.06726-1-3-0': 'How cancer cells respond to fluid shear stress (FSS) is less understood.', '1809.06726-1-3-1': 'Cancer cells escape from the primary tumor, and travel through blood stream to secondary organ sites[CITATION].', '1809.06726-1-3-2': 'During this metastatic spread, cancer cells are exposed to a microenvironment of dramatically varying physical forces.', '1809.06726-1-3-3': 'Cancer cells in the bloodstream experience hydrodynamic forces, which are alien to cancer cells[CITATION].', '1809.06726-1-3-4': 'FSS is mainly generated during blood flow in the vascular microenvironment, as well as interstitial flow in the tumor microenvironment[CITATION].', '1809.06726-1-3-5': 'FSS could contribute to the metastatic process by enhancing tumor cell invasion, and circulating tumor cell (CTC) adhesion to blood vessels.', '1809.06726-1-3-6': 'In this article, we develop a theory to describe how cancer cells respond to fluid shear stress.', '1809.06726-1-3-7': 'The cumulative effect arising from the non-equilibrium description of living cells using a stochastic birth-death process, and intercellular hydrodynamic interactions, lead to complex dynamics, which may have far reaching implications in our understanding of tumor growth.', '1809.06726-1-3-8': 'One of the major difficulties in the study of collective behavior of the cells far from equilibrium is the breakdown of a fluctuation-dissipation theorem (FDT); hence, independent diagrammatic expansions for the response function and the correlation function.', '1809.06726-1-3-9': 'The equilibrium distribution is not known and averages can be computed only for the statistical noise.', '1809.06726-1-3-10': 'We study the relevant continuum description of collective behavior of a colony of cells in the long time scale, using stochastic quantization technique, originally proposed by Parisi and Wu[CITATION].', '1809.06726-1-3-11': 'We show that non-linear terms that determine the long range hydrodynamic interactions among living cells dictate the complex collective behavior, when a continuum description of the cellular colony is invoked, in the long time limit.', '1809.06726-1-3-12': 'The hydrodynamic interactions lead to directed motions of cells.', '1809.06726-1-3-13': 'We find that cells exhibit super-diffusive motion in the intermediate time, where the mean-square displacement of the cells behaves as [MATH], with, [MATH].', '1809.06726-1-3-14': 'In the long time limit the motion becomes ballistic.', '1809.06726-1-4-0': '# Theory', '1809.06726-1-5-0': 'We begin by considering the dynamics of a colony of cells in a dissipative environment where inertial effects are negligible.', '1809.06726-1-5-1': 'Each cell experiences mechanical forces, such as adhesion, excluded volume interactions due to neighbors, and a random force characterized by Gaussian white noise.', '1809.06726-1-5-2': 'The equation of motion for single cell [MATH] is [CITATION] [EQUATION]', '1809.06726-1-5-3': 'The first term on the r.h.s. of Eq.([REF]) is the unperturbed velocity profile, where [MATH] is the velocity of fluid far from the cell surface, [MATH] the Darcy permeability of the porous medium, [MATH] the height, and [MATH] the unit vector parallel to the [MATH]-axis.', '1809.06726-1-6-0': 'The second term on the r.h.s. of Eq.([REF]) is the term associated with the motility of the cells with local fluid velocity.', '1809.06726-1-6-1': 'A recent experiment shows that a fluid force present in the tumor microenvironment modulates gene expression downstream of ROCK-LIMK-cofilin activated YAP1 signaling, resulting in developing fine cytoplasmic extensions or filopodia.', '1809.06726-1-6-2': 'Therefore, mortile cells exert force dipole [MATH] on the fluid.', '1809.06726-1-6-3': 'The fluid exerts equal and opposite force on the cell and the net force is zero at all times.', '1809.06726-1-6-4': 'The cell is assumed to be at neutral buoyancy.', '1809.06726-1-6-5': 'We assume that at low Reynolds number regime, in the absence of fluid fluctuations, the mortile cell is convected along with the fluid at the local fluid velocity, [MATH].', '1809.06726-1-6-6': 'The dynamics of the cell is then described by [MATH], where [MATH] is the position of the cell.', '1809.06726-1-6-7': 'The flow velocity of the fluid at [MATH] is determined by the solution of the Stokes equation, [MATH], where [MATH] is the force density by the cell on the fluid, i.e., [MATH].', '1809.06726-1-6-8': 'We assume that the fluid is incompressible, i.e., [MATH].', '1809.06726-1-6-9': 'From these set of equations, the Stokes equation is readily solved to obtain the flow field at the center of the cell as, u(r)=[MATH], where [MATH] is the mobility matrix.', '1809.06726-1-7-0': 'The potential [MATH], determining the configuration-dependent forces experinced by the cells consists of three terms: [MATH], [MATH] and [MATH].', '1809.06726-1-7-1': '[MATH] determines the excluded volume interaction with a range [MATH], [MATH] determines the favorable attractive interactions between cells with a range [MATH], and interaction strengths [MATH], and [MATH] respectively.', '1809.06726-1-7-2': 'The potential term [MATH] describes the cell-substrate or cell surface-ECM interactions as a function of [MATH], the average distance between cell and wall or ECM.', '1809.06726-1-7-3': 'Cells interact with the substrate through receptor-ligand interactions, described by short range potential.', '1809.06726-1-7-4': 'Negative sign associated with the adhesion term, [MATH], indicates a reduction of hydrodynamic interactions between cells [CITATION].', '1809.06726-1-7-5': 'To obtain an analytical solution, we use a Gaussian potential to describe adhesion.', '1809.06726-1-7-6': 'The third term in Eq. ([REF]) is the effect of force acting on cell [MATH] creates a hydrodynamic flow-field in the fluid, thereby entraining cell [MATH].', '1809.06726-1-7-7': ""Fourth term in the Langevin equation is due to the spacial variation of the cell's self motilities."", '1809.06726-1-7-8': 'It is introduced to compensate the flux caused by the position dependent random velocity contributions [MATH], which are assumed to be Gaussian random vectors exhibiting hydrodynamic correlations according to fluctuation-dissipation theorem (FDT), [MATH].', '1809.06726-1-7-9': ""Hydrodynamic effects are incorporated via the mobility matrix [MATH], which is obtained from the Green's function of Stokes equation that satisfies the no-slip condition on a planar boundary[CITATION], [EQUATION] with the relative coordinate between cell [MATH] and [MATH], [MATH] and the relative coordinate between cell [MATH] and the image of the cell [MATH], [MATH]."", '1809.06726-1-7-10': ""The wall Green's function [MATH] contains the bulk Oseen tensor, [MATH], the Stokes doublet [MATH], and the source doublet [MATH], with solvent viscosity [MATH]."", '1809.06726-1-7-11': 'The finite radii of the spheres are approximately taken into account via a multipole expansion to the second order, in terms of the cell radius [MATH] [CITATION], [MATH] for [MATH].', '1809.06726-1-7-12': 'The self motility tensor [MATH] is obtained in the limit [MATH] and regularized far away from the surface by the bulk sphere motility [MATH].', '1809.06726-1-8-0': 'We consider the evolution of the density function for a single cell [MATH].', '1809.06726-1-8-1': 'A closed form Langevin equation for the density, [MATH] can be obtained using standard approach [CITATION].', '1809.06726-1-8-2': 'The time evolution of [MATH] is given by, [EQUATION]', '1809.06726-1-8-3': 'Note that the density equation contains same the information as N-body stochastic Langevin equations.', '1809.06726-1-8-4': 'We extend the model phenomenologically by adding the source term that describes both cell division and death, and a noise term that breaks the cell number conservation.', '1809.06726-1-8-5': 'Our argument follows from the Doi-Peliti formalism [CITATION], which was introduced in the context of reaction-diffusion process.', '1809.06726-1-9-0': 'The final Langevin equation, for the time-dependent changes in the density, [MATH] is [EQUATION] with [MATH] satisfying [MATH].', '1809.06726-1-9-1': 'The source term [MATH], represents the cell death-birth term.', '1809.06726-1-9-2': 'The coefficient [MATH] denotes the strength of the noise corresponding to number fluctuations, and is a function of concentration, which has been derived using stochastic birth and death processes.', '1809.06726-1-9-3': 'This is an out of equilibrium problem characterized by the absence of fluctuation-dissipation theorem due to long range hydrodynamic term.', '1809.06726-1-9-4': 'Eq. ([REF]) can be studied analytically by treating the non-linear terms using a perturbative approach, based on the stochastic quantization scheme [CITATION].', '1809.06726-1-10-0': 'We assume that the density fluctuates around a constant value, which simplifies the multiplicative noise term (last term in Eq. ([REF])).', '1809.06726-1-10-1': 'Hence, we can define the density using [MATH], and write down the equation for density fluctuations in Fourier space as follows: [EQUATION] with [MATH].', '1809.06726-1-10-2': 'For [MATH], scaling exponents is determined by linear theory, where [MATH] an upper critical dimension.', '1809.06726-1-10-3': 'The non-trivial exponent is determined by non-linear terms for [MATH].', '1809.06726-1-10-4': 'To understand the role of non-linearity, we introduce a change of scale [MATH], [MATH] and [MATH] where [MATH] is the exponent corresponding to cell density fluctuation, and [MATH] is the dynamic exponent.', '1809.06726-1-10-5': 'The nonlinear term [MATH] involves hydrodynamic interaction, which scales as [MATH].', '1809.06726-1-10-6': 'The other non-linear term [MATH], which arise due to adhesion to the substrate, and cell-cell interaction, which scales as [MATH].', '1809.06726-1-10-7': 'The non-linear term, [MATH], arises due the force dipole on the fluid, determines the scaling solutions in the long time and long distance limit.', '1809.06726-1-10-8': 'The non-linear term [MATH], may dominate in the intermediate time, if interaction strengths are sufficiently strong.', '1809.06726-1-10-9': 'This term determines the scaling in the intermediate time regime.', '1809.06726-1-10-10': '[MATH] scales as [MATH] at large scale limit.', '1809.06726-1-10-11': 'Therefore in the hydrodynamic limit(i.e. [MATH] and [MATH]), scaling behavior is determined by the long range hydrodynamic term.', '1809.06726-1-11-0': '## Limiting cases', '1809.06726-1-12-0': 'In the absence of hydrodynamic interactions, the non-linear term [MATH] determines the scaling in the intermediate regime, and, the linear terms determine the scaling in the long time limit.', '1809.06726-1-12-1': 'We find the caging characterized by sub-diffusive motion of cells in the intermediate time and normal diffusion in the long time limit [CITATION].', '1809.06726-1-12-2': 'In the homeostatic limit, where the death and birth balance, we find normal diffusion.', '1809.06726-1-13-0': '## Stochastic quantization approach', '1809.06726-1-14-0': 'To understand the dynamics of collections of cells, we use the stochastic quantization method developed by Parisi-Wu in another context.', '1809.06726-1-14-1': 'The collective migration of cells described by Eq. ([REF]) is an out of equilibrium problem characterized by the absence of FDT, which relates the correlation and response function in momentum space as [MATH].', '1809.06726-1-14-2': 'The usual analytic route to get the scaling solution of this problem, one can introduce a response field [MATH].', '1809.06726-1-14-3': 'We need to calculate both the response function ([MATH]) and correlation function ([MATH]) because of the absence of a fluctuation dissipation relation.', '1809.06726-1-14-4': 'The key advantage of the present method is that we do not need to calculate both the correlation, and the response functions.', '1809.06726-1-14-5': 'The FDT is constructed in fictitious time introduced in the problem.', '1809.06726-1-14-6': 'The FDT relation enables us to obtain the scaling of the correlation function, once the scaling of the response function is known.', '1809.06726-1-14-7': 'By taking the infinite limit in fictitious time, one can obtain the correlation function in real time.', '1809.06726-1-14-8': 'The scaling solution of the problem can be obtained by power counting analysis instead of doing renormalization group calculation.', '1809.06726-1-15-0': ""We now exploit the Parisi-Wu stochastic quantization scheme [CITATION], and introduce a fictitious time [MATH]', and consider all variables to be functions of [MATH] in addition to k and [MATH]."", '1809.06726-1-15-1': ""A Langevin equation in [MATH]' space is, [EQUATION] with [MATH]."", '1809.06726-1-15-2': 'This ensures that as [MATH], the distribution function will be given by the action [MATH], because in the [MATH]-space a fluctuation dissipation theorem (FDT) is preserved.', '1809.06726-1-15-3': 'The action [MATH] can be obtained by writing down the probability distribution [MATH] corresponding to the noise term [MATH], and the action [MATH] in terms of [MATH] with the help of Eq. ([REF]).', '1809.06726-1-15-4': 'The expression for the [MATH] is in the appendix.', '1809.06726-1-16-0': 'The correlation functions, calculated from Eq. ([REF]), lead to the required correlation functions of the original theory, in the [MATH] limit.', '1809.06726-1-16-1': 'In order to obtain the scaling laws, it suffices to work at arbitrary [MATH].', '1809.06726-1-16-2': 'It is obvious from Eq. ([REF]) that in the absence of the non-linear terms, the Greens function [MATH] is given by, [EQUATION] where [MATH] is the frequency corresponding to the fictitious time [MATH].', '1809.06726-1-16-3': ""As is customary, the effect of non-linear terms, can be included perturbatively leading to the Dysons' equation [EQUATION]"", '1809.06726-1-16-4': 'Here, we are concerned with the behavior of [MATH], which becomes non-linear when expanded to second order.', '1809.06726-1-16-5': 'We note that the contribution comes from two different sources (1) a one-loop contribution from the second order term (containing three [MATH] fields) in Eq. ([REF]) (second term in Fig. [REF]) and (2) a two-loop contribution from the first order term ( containing two [MATH] fields) in Eq. ([REF]) (first term in Fig. [REF]).', '1809.06726-1-16-6': 'The contribution arising coming from the term containing three [MATH] fields, in Eq. ([REF]) can be readily obtained by contracting two of the [MATH] fields.', '1809.06726-1-16-7': 'The second order term coming from the one loop contribution in Eq. ([REF]) does not have any new momentum dependance.', '1809.06726-1-16-8': 'Hence it is the second-order contribution (first term in Fig. ([REF])), coming from the two-loop contribution in Eq. ([REF]), which is significant.', '1809.06726-1-16-9': 'The correlation function is given by the FDT as [MATH].', '1809.06726-1-16-10': 'With these observations, Eq. ([REF]) can be written as, [EQUATION] where [MATH], and [MATH] is defined by [EQUATION] with [MATH].', '1809.06726-1-16-11': 'Expanding [MATH], [MATH] about [MATH] and [MATH], respectively, and noting that the renormalization of [MATH] dominates, we get [EQUATION]', '1809.06726-1-17-0': '# Results', '1809.06726-1-18-0': 'Super-diffusive motion: In the intermediate time, hydrodynamic interactions arising due to adhesion determine the scaling behavior.', '1809.06726-1-18-1': 'In a self consistent mode coupling theory, we now replace [MATH] by [MATH] in the self energy term [MATH] in Fig. ([REF]), use [MATH] as given by Eq. ([REF]), and [MATH], which follows from the FDT.', '1809.06726-1-18-2': 'According to scale transformation, we know that [MATH], [MATH], [MATH], [MATH] and the vertex factor [MATH].', '1809.06726-1-18-3': 'The self energy term in Fig. ([REF]) can be written as [MATH].', '1809.06726-1-18-4': 'By carrying out the momentum count of [MATH], and using [MATH], we find that [MATH].', '1809.06726-1-18-5': 'Using Eq. ([REF]), we have [MATH], which leads to [MATH].', '1809.06726-1-18-6': 'In the context of dynamic renormalization group theoty, the usual approach to avoid infrared divergence in the loop integral, we integrate higher momentum and rescaling the momentum.', '1809.06726-1-18-7': 'One can end up with the flow equation for the coupling constant and by finding out the fixed-points; one can calculate the dynamic exponents.', '1809.06726-1-18-8': 'We do not consider that to be of any particular significance one way or another.', '1809.06726-1-18-9': 'By introducing fictitious time, we are able to bring back the fluctuation-dissipation theorem, albeit in fictitious time, and this allows a power counting analysis for the problem.', '1809.06726-1-18-10': 'In order to avoid the infrared divergence, we get the following condition, [MATH], implies [MATH], therefore, MSD exponent [MATH] in 3d.', '1809.06726-1-18-11': 'The dynamic exponent we obtain [MATH], consistent with the finiteness condition.', '1809.06726-1-19-0': 'Therefore the single cell mean-square displacement behaves as: [EQUATION]', '1809.06726-1-19-1': 'In this regime, [MATH], implying super-diffusive motion.', '1809.06726-1-19-2': 'In the absence of shear flow, the adhesion term determines the caging behavior in the intermediate time, and leads to sub-diffusive motion with MSD exponent, [MATH] [CITATION].', '1809.06726-1-19-3': 'Hence, we see the evidence of shear flow driven fluidization in the intermediate time regime.', '1809.06726-1-19-4': 'The MSD exponent agrees very well the recent experimental measurement of MSD using microfluidic model[CITATION].', '1809.06726-1-20-0': 'In the long time limit, the motility term that arises due to force dipoles on the fluid, determines the scaling behavior.', '1809.06726-1-20-1': 'In a self consistent mode coupling theory, similar arguments (as in the intermediate time) shows that [MATH].', '1809.06726-1-20-2': 'The single cell mean-square displacement behaves as [MATH], with [MATH].', '1809.06726-1-20-3': 'Hence, we see that the shear flow driven directed motion in a colony of cells arises due to long range hydrodynamic interactions.', '1809.06726-1-20-4': 'The scaling exponents depend on the form of interactions used in the theory.', '1809.06726-1-20-5': 'For the analytical solution of the present theory, gaussian potential has been used.', '1809.06726-1-20-6': 'Other forms of short range interactions may give different numerical values of the exponents, but the qualitative features of the underlying physics described by the theory, is likely to be preserved.', '1809.06726-1-21-0': '# Conclusion', '1809.06726-1-22-0': 'In the present contribution, using a new theoretical framework, we provide insight into the dynamics of a colony of cancer cells driven by shear flow, and birth-death of cells.', '1809.06726-1-22-1': 'The conventional practice in dealing with this out of equilibrium problems is to use a set of fictitious fields called response fields, which provide a field theoretic prescription for the response function.', '1809.06726-1-22-2': 'In contrast, we propose the introduction of a fictitious time in which a FDT is valid, and thereby only correlation functions need to be calculated.', '1809.06726-1-22-3': 'Our approach greatly simplifies the evaluation of scaling exponents.', '1809.06726-1-22-4': 'We see that the non-linear terms in the density evolution equation, arising from mechanical interactions, determine the scaling behavior in the long time limit.', '1809.06726-1-22-5': 'The super-diffusive motion of a colony of cells in 3D is induced by hydrodynamic interactions.', '1809.06726-1-22-6': 'A recent experiment shows that the fluid force present in the tumor microenvironment modulates gene expression downstream of ROCK-LIMK-cofilin-activated YAP1 signaling[CITATION].', '1809.06726-1-22-7': 'Hence, the fluid force can trigger cell motility by developing fine cytoplasmic extensions or filopodia, in the highly metastatic PC3 human prostate cancer cell line.', '1809.06726-1-22-8': 'The filopodia formation modifies the cell-cell interaction and cell-substrate interaction, and hence modifies the hydrodynamic interactions in the present theory.', '1809.06726-1-22-9': 'The new theoretical framework introduced here provides evidence of shear flow directed collective motion and could explain the invasion of cancer cells under shear flow, observed in a recent experiment[CITATION].', '1809.06726-1-22-10': 'The theory introduced here could help us understand how cancer cells spread by invading adjacent tissues involved in metastasis [CITATION].', '1809.06726-1-23-0': '# acknowledgements', '1809.06726-1-24-0': 'I wish to express my gratitude to Prof. D. Thirumalai for supporting me as a postdoc at the department of chemistry at UT, Austin.', '1809.06726-1-24-1': 'I am indebted to Dr. Debayan Chakrabarty for his valuable comments.', '1809.06726-1-24-2': 'Appendix:', '1809.06726-1-25-0': 'The expression for [MATH]: The equation for density fluctuations in Fourier space reads, [EQUATION] with [MATH].', '1809.06726-1-26-0': ""We introduce a fictitious time [MATH]', and consider all variables to be functions of [MATH] in addition to k and [MATH]."", '1809.06726-1-26-1': ""A Langevin equation in [MATH]' space is, [EQUATION] with [MATH]."", '1809.06726-1-26-2': 'This ensures that as [MATH], the distribution function will be given by the action [MATH], because in the [MATH]-space a fluctuation dissipation theorem (FDT) is preserved.', '1809.06726-1-26-3': 'The action [MATH] can be obtained by writing down the probability distribution [MATH], corresponding to the noise term [MATH], and the action [MATH] in term of [MATH] with the help of Eq. ([REF]).', '1809.06726-1-27-0': 'The expression for the [MATH] is,', '1809.06726-1-28-0': '[MATH]', '1809.06726-1-29-0': 'Diagrammatic representation of density equations : We write down the density equation (Eq. ([REF])) with the help of action [MATH], and, can be reprented diagrammatically as,', '1809.06726-1-30-0': 'Greens function for density equation:', '1809.06726-1-31-0': 'The Greens function [MATH] is given by [EQUATION]', '1809.06726-1-31-1': 'The contributing term for the power counting analysis in the self energy [MATH] is the two loop contribution from the first order term (contains two [MATH] fields, second term in fig. (3)) in the fictitious time equation.', '1809.06726-1-32-0': 'The expression for [MATH]: [EQUATION] where veterx term, [MATH]'}","{'1809.06726-2-0-0': 'The migration behaviors of cancer cells are known to be heterogeneous.', '1809.06726-2-0-1': 'However, the interplay between the adhesion interactions, dynamical shape changes and fluid flows in regulating cell migration heterogeneity and plasticity during cancer metastasis is still elusive.', '1809.06726-2-0-2': 'To further quantitative understanding of cell motility and morphology, we develop a theory using stochastic quantization method that describes the role of biophysical cues in regulating diverse cell motility.', '1809.06726-2-0-3': 'We show that the cumulative effect of time dependent adhesion interactions that determine the structural rearrangements and self-generated force due to actin remodeling, dictate the super-diffusive motion of mesenchymal phenotype in the absence of flow.', '1809.06726-2-0-4': 'Interstitial flows regulate cell motility phenotype and promote the amoeboid over mesenchymal motility through adhesion interactions.', '1809.06726-2-0-5': 'Cells exhibit a dynamical slowing down of collective migration, with a decreasing degree of super-diffusion.', '1809.06726-2-0-6': 'Our findings, suggest a mechanism of Interstitial flow induced directed motion of cancer cells through adhesion, and provide the much needed insight into a recent experimental observation concerning the diverse motility of breast cancer cells.', '1809.06726-2-1-0': 'Collective cancer cell invasion, followed by local and distant metastasis, is a hallmark of cancer[CITATION].', '1809.06726-2-1-1': 'Cancer metastasis is a multistep process, where tumor cells detach from primary tumor, invade through the interstitial extracellular matrix, intravasation of tumor cells into vascular vessels, extravasation of circulating tumor cells to peripheral tissues, and establish a secondary tumor at distant organ[CITATION].', '1809.06726-2-1-2': 'Dynamics associated with invasion and metastasis, involve the collective cell migration regulated by biomechanical (e.g. cytokines secreted by cells and nutrients) and biophysical cues (e.g. fluid flows and ECM)[CITATION].', '1809.06726-2-1-3': 'Tumor cells reside in an extracellular matrix (ECM) containing interstitial fluid that transports nutrients and signaling molecules.', '1809.06726-2-1-4': 'The interstitial flow has been shown to affect the morphology and migration of cells such as fibroblasts, cancer cells, endothelial cells, and mesenchymal stem cells[CITATION].', '1809.06726-2-1-5': 'Interstitial flows are particularly important for tumor cell invasion because it is elevated in tumor microenvironment due to the heightened interstitial fluid pressure as well as the abnormal angiogenic, lymphanogenic blood and lymphatic vessels[CITATION].', '1809.06726-2-1-6': 'The flow speed associated with interstitial flows are in the order of a few micrometers per second in normal tissue[CITATION].', '1809.06726-2-1-7': 'Two type of motility (e.g. amoeboid and mesenchymal) have been broadly categorized in cell migration in 3D architechture[CITATION].', '1809.06726-2-1-8': 'Cells with aspect ratio smaller than 2.0 are considered rounded or amoeboid and cells with aspect ratio greater than 2.0 are considered elongated or mesenchymal (see fig.1)[CITATION].', '1809.06726-2-2-0': 'Recent experiment has demonstrated that interstitial flows regulate the cancer cell migration heterogeneity within a three dimensional biomatrix.', '1809.06726-2-2-1': 'Using a microfluidic model, authors show that breast cancer cells (MDA-MB-231) embedded in collagen matrix, exhibit both amoeboid and mesenchymal motility phenotype and interstitial flows promote amoeboid over mesenchymal motility of breast cancer cells[CITATION].', '1809.06726-2-3-0': 'How interstitial flows promote cancer cell invasion is largely unknown.', '1809.06726-2-3-1': 'The understanding of this process could help to develop drugs that inhibit the process and prevent cancer from metastasizing.', '1809.06726-2-3-2': 'How the biophysical forces modulate tumor cell migration heterogeneity and plasticity, and creates a complex spatiotemporal dynamical property during cancer metastasis is still elusive.', '1809.06726-2-4-0': 'In this article, we develop a theory to describe how biophysical cues regulate the diverse collective cell motility.', '1809.06726-2-4-1': 'The cumulative effect arising from the non-equilibrium description of living cells using a time dependent mechanical interaction, and flows, lead to complex dynamics, which may have far reaching implications in our understanding of cancer metastasis.', '1809.06726-2-5-0': 'One of the major difficulties in the study of collective behavior of the cells far from equilibrium is the breakdown of a fluctuation-dissipation theorem (FDT); hence, independent diagrammatic expansions for the response function and the correlation function.', '1809.06726-2-5-1': 'The equilibrium distribution is not known and averages can be computed only for the statistical noise.', '1809.06726-2-6-0': 'We study the relevant continuum description of collective behavior of a colony of cells in the physical time scale, using stochastic quantization technique, originally proposed by Parisi and Wu[CITATION].', '1809.06726-2-6-1': 'We show that the time dependent adhesion interactions that determine the structural rearrangements, the long range hydrodynamic interactions among living cells, and self-generated force due to actin remodeling dictate the complex collective behavior, when a continuum description of the cellular colony is invoked, in the physical time scale.', '1809.06726-2-6-2': 'We find that cells exhibit both amoeboid and mesenchymal motility characterized by super-diffusive motion.', '1809.06726-2-6-3': 'The mean-square displacement (MSD) of the cells for the mesenchymal motility behaves as [MATH], with, [MATH].', '1809.06726-2-6-4': 'The interstitial flows impair the collective migration with a gradually decreasing degree of super-diffusion and promote the amoeboid motility phenotype over mesenchymal motility.', '1809.06726-2-6-5': 'In the case of flow, the MSD exponent [MATH], which reflects the dynamical slowing down of the spatiotemporal collective migration.', '1809.06726-2-7-0': 'Theory: In the absence of flow, cells exhibit mesenchymal motility phenotype in 3D collagen matrix.', '1809.06726-2-7-1': 'Cells secreted fibronectin molecules into a fibrillar form, and form long lived adhesions with the collagen fibers, which trigger the downstream signaling that activates actin-network expansion and thus exhibit cell migration.', '1809.06726-2-7-2': 'We consider the dynamics of a colony of cells in a dissipative environment where inertial effects are negligible.', '1809.06726-2-7-3': 'Each cell experiences systematic forces arising from mechanical interactions, and a Gaussian random force with white noise spectrum.', '1809.06726-2-7-4': 'The equation of motion for a single mesenchymal cell [MATH] is [MATH] where [MATH] contains repulsive interactions with range [MATH], adhesion interactions with collagen matrix with range [MATH], and favorable attractive interactions between cells with range [MATH], and with strengths [MATH], [MATH] and [MATH] respectively.', '1809.06726-2-7-5': 'We use Gaussian potentials (see the Supplementary Information (SI) for details) in order to obtain analytical solutions.', '1809.06726-2-7-6': 'Needless to say that the conclusions would be valid for any short-ranged [MATH].', '1809.06726-2-7-7': 'We assume that the adhesion strength is changing during the topological rearrangement via [MATH][CITATION].', '1809.06726-2-7-8': '[MATH] and [MATH] are initial and final interaction strengths ([MATH] stands for [MATH] and [MATH]) and the time scale for changing the receptor-ligand interaction is given by [MATH].', '1809.06726-2-7-9': 'The Gaussian white noise, satisfies [MATH].', '1809.06726-2-7-10': 'The mesenchymal cells are subject to a self-generated force of actin-network remodeling.', '1809.06726-2-7-11': 'The cells are, thus in addition subject to a random self-generated force with amplitude [MATH].', '1809.06726-2-7-12': 'The randomness is modeled by an athermal noise [MATH], which is exponentially correlated over a time scale [MATH].', '1809.06726-2-7-13': 'The statistics of the [MATH] is given by [MATH], [MATH] and [MATH].', '1809.06726-2-7-14': 'Where [MATH] is the dimensionless constant.', '1809.06726-2-7-15': 'The athermal noise in general does not obey the FDT.', '1809.06726-2-8-0': 'In the presence of interstitial flow, cells exhibit amoeboid motility phenotype.', '1809.06726-2-8-1': 'The flows carried away the cell-secreted fibronectin molecules before they were assembled into fibrils and attached to the collagen fibers.', '1809.06726-2-8-2': 'The round shaped amoeboid cells extend their protrusions in all directions and form a short lived adhesions with collagen matrix.', '1809.06726-2-8-3': 'The amoeboid cells migrate through squeezing the matrix when find a suitable path.', '1809.06726-2-8-4': 'We begin by considering the dynamics of a colony of cells in the presence of flow in a dissipative environment where inertial effects are negligible.', '1809.06726-2-8-5': 'Each cell experiences mechanical forces, such as adhesion, excluded volume interactions due to neighbors, and a random force characterized by Gaussian white noise.', '1809.06726-2-8-6': 'The equation of motion for single amoeboid cell [MATH] is [CITATION] [EQUATION]', '1809.06726-2-8-7': 'The first term on the r.h.s. of Eq. ([REF]) is the effect of force acting on cell [MATH] creates a hydrodynamic flow-field in the fluid, thereby entraining cell [MATH].', '1809.06726-2-8-8': 'Where [MATH] contains repulsive interactions with range [MATH], adhesion interactions with collagen matrix with range [MATH], and favorable attractive interactions between cells with range [MATH], and with strengths [MATH], [MATH], and [MATH] respectively (see the Supplementary Information (SI) for details).', '1809.06726-2-8-9': ""Second term in the Langevin equation is due to the spacial variation of the cell's self motilities."", '1809.06726-2-8-10': 'It is introduced to compensate the flux caused by the position dependent random velocity contributions [MATH], which are assumed to be Gaussian random vectors exhibiting hydrodynamic correlations according to fluctuation-dissipation theorem (FDT), [MATH].', '1809.06726-2-8-11': ""Hydrodynamic effects are incorporated via the mobility matrix [MATH], which is obtained from the Green's function of Stokes equation as [MATH]."", '1809.06726-2-8-12': 'with the relative coordinate between cell [MATH] and [MATH], [MATH].', '1809.06726-2-8-13': 'The protrusive flowing of the anterior actin network of the cell and squeezing actomyosin contractions of the trailing edge are modeled as the cells are subjected to a self-generated force with amplitude [MATH] during their path finding motion in collagen matrix.', '1809.06726-2-8-14': 'The statistics of the [MATH] is the same as for mesenchymal motility.', '1809.06726-2-9-0': 'We consider the evolution of the density function for a single cell [MATH].', '1809.06726-2-9-1': 'A closed form Langevin equation for the density, [MATH] can be obtained using standard approach [CITATION].', '1809.06726-2-9-2': 'The time evolution of [MATH] is given by, [EQUATION]', '1809.06726-2-9-3': 'Note that the density equation contains same information as N-body stochastic Langevin equations.', '1809.06726-2-9-4': 'This is an out of equilibrium problem characterized by the absence of fluctuation-dissipation theorem due to long range hydrodynamic term and self-generated force due to actin remodeling.', '1809.06726-2-9-5': 'Eq. ([REF]) can be studied analytically by treating the non-linear terms using a perturbative approach, based on the stochastic quantization scheme [CITATION].', '1809.06726-2-10-0': 'Stochastic quantization approach: To understand the dynamics of collections of cells, we use the stochastic quantization method developed by Parisi-Wu in another context.', '1809.06726-2-10-1': 'The collective migration of cells described by Eq. ([REF]) is an out of equilibrium problem characterized by the absence of FDT, which relates the correlation and response function in momentum space as [MATH].', '1809.06726-2-10-2': 'The usual analytic route to get the scaling solution of this problem, one can introduce a response field [MATH].', '1809.06726-2-10-3': 'We need to calculate both the response function ([MATH]) and correlation function ([MATH]) because of the absence of a fluctuation dissipation relation.', '1809.06726-2-10-4': 'The key advantage of the present method is that we do not need to calculate both the correlation, and the response functions.', '1809.06726-2-10-5': 'The FDT is constructed in fictitious time introduced in the problem.', '1809.06726-2-10-6': 'The FDT relation enables us to obtain the scaling of the correlation function, once the scaling of the response function is known.', '1809.06726-2-10-7': 'By taking the infinite limit in fictitious time, one can obtain the correlation function in real time.', '1809.06726-2-10-8': 'The scaling solution of the problem can be obtained by power counting analysis instead of doing renormalization group calculation.', '1809.06726-2-11-0': ""We now exploit the Parisi-Wu stochastic quantization scheme [CITATION], and introduce a fictitious time [MATH]', and consider all variables to be functions of [MATH] in addition to k and [MATH]."", '1809.06726-2-11-1': ""A Langevin equation in [MATH]' space is, [EQUATION] with [MATH]."", '1809.06726-2-11-2': 'This ensures that as [MATH], the distribution function will be given by the action [MATH], because in the [MATH]-space a fluctuation dissipation theorem (FDT) is preserved.', '1809.06726-2-11-3': 'The action [MATH] can be obtained by writing down the probability distribution [MATH] corresponding to the noise term [MATH], and the action [MATH] in terms of [MATH] with the help of Eq. ([REF]).', '1809.06726-2-11-4': 'The expression for the [MATH] is in the appendix.', '1809.06726-2-12-0': 'We follow the procedure of obtaining scaling laws of the problem, which has been demonstrated in earlier works [CITATION].', '1809.06726-2-12-1': 'The dynamics of Eq. ([REF]) requires only the calculation of response functions, the correlation functions in this dynamics are related to response function through FDT relation, i.e., in Fourier space, [MATH].', '1809.06726-2-12-2': 'We can obtain the scaling laws in real space and time in a straight forward fashion from the solution in the fictitious time [MATH] space.', '1809.06726-2-13-0': 'We obtain the following self-consistent equation for the self energy from the calculation of response function using Eq.([REF]): [EQUATION] where, [MATH], [MATH], and [MATH] is the self-energy term, a two-loop contribution from the first order term ( containing two [MATH] fields) in Eq. ([REF]) (first term in Fig. [REF]) will contribute in the scaling laws for the cell in the finite time.', '1809.06726-2-13-1': 'We use Eq.([REF]) for getting the scaling laws of both the amoeboid and mesenchymal cells phenotype.', '1809.06726-2-14-0': 'Results: Mesenchymal motility: The mesenchymal cell phenotype forms long-lived integrin-based adhesions with the collagen matrix and migrate via either remodeling of actin network or degrading the matrix.', '1809.06726-2-14-1': 'The non-linear term i.e. the adhesion interaction plays an important role in the complex dynamics of collective migration of mesenchymal cell phenotype.', '1809.06726-2-14-2': 'In a self consistent mode coupling theory, we now replace [MATH] by [MATH] in the self energy term [MATH] in the first term in Fig. ([REF]), use [MATH] as from Eq. ([REF]), and [MATH], which follows from the FDT.', '1809.06726-2-14-3': 'In the absence of flow, [MATH].', '1809.06726-2-14-4': 'According to scale transformation, we know that [MATH], [MATH], [MATH], [MATH] and the vertex factor [MATH].', '1809.06726-2-14-5': 'The self energy term in Fig. ([REF]) can be written as [MATH].', '1809.06726-2-14-6': 'By carrying out the momentum count of [MATH], and using [MATH], we find that [MATH].', '1809.06726-2-14-7': 'Using Eq. ([REF]) and [MATH], we have [MATH], which leads to [MATH].', '1809.06726-2-14-8': 'MSD exponent [MATH].', '1809.06726-2-14-9': 'In 3D, [MATH], i.e., the mesenchymal cells undergo super-diffusion.', '1809.06726-2-14-10': 'The non-linear term arising from cell-cell adhesion that determines the dynamical shape change during cell motion and self-generated force, produce super-diffusive motion.', '1809.06726-2-14-11': 'The theoretical result is in excellent agreement with the recent experimental result using microfluidic model[CITATION].', '1809.06726-2-15-0': 'Interstitial flow induced amoeboid motility: In the presence of interstitial flow, cells exhibit amoeboid motility.', '1809.06726-2-15-1': 'The amoeboid cells form short lived adhesion with the collagen matrix.', '1809.06726-2-15-2': 'The time scale [MATH] is small compared to the mesenchymal motility phenotype.', '1809.06726-2-15-3': 'In case of flow, the cells exhibit long range hydrodynamic interactions that determine the complex spatiotemporal dynamics of amoeboid cell phenotype.', '1809.06726-2-15-4': 'The self generating force of actin remodeling helps to propel in path finding fashion through collagen matrix.', '1809.06726-2-15-5': 'In a self consistent mode coupling theory, we now replace [MATH] by [MATH] in the self energy term [MATH] in the first term in Fig. ([REF]), use [MATH] as from Eq. ([REF]) and [MATH], which follows from the FDT.', '1809.06726-2-15-6': 'According to scale transformation, we know that [MATH], [MATH], [MATH], [MATH] and the vertex factor [MATH].', '1809.06726-2-15-7': 'The self energy term in Fig. ([REF]) can be written as [MATH].', '1809.06726-2-15-8': 'By carrying out the momentum count of [MATH], and using [MATH], we find that [MATH].', '1809.06726-2-15-9': 'Using Eq. ([REF]) and [MATH], we have [MATH], which leads to [MATH].', '1809.06726-2-15-10': 'MSD exponent [MATH].', '1809.06726-2-15-11': 'In 3D, [MATH], i.e., the amoeboid cells undergo superdiffusion.', '1809.06726-2-15-12': 'The non-linear term due to long-range hydrodynamic interactions among cells in addition with self-generated force, produce super-diffusive motion for amoeboid cell phenotype.', '1809.06726-2-15-13': 'The decrease of MSD exponent determines the dynamical slowing down of initial collective migration of mesenchymal cell phenotype.', '1809.06726-2-15-14': 'Therefore, interstitial flows impair the cells ability to spread by sweeping away the adhesion molecules with the flow and making the cells as amoeboid phenotype with short-lived adhesion with the collagen fiber.', '1809.06726-2-15-15': 'The cells migrate via squeezing through the pore of the collagen fiber when find a suitable path[CITATION].', '1809.06726-2-15-16': 'The theoretical result is in excellent agreement with the recent experimental result using microfluidic model[CITATION].', '1809.06726-2-16-0': 'Conclusion: In the present contribution, using a new theoretical framework, we provide insight into the dynamics of a colony of cancer cells driven by biophysical cues.', '1809.06726-2-16-1': 'The theory reveals that the interstitial flows regulate cancer cell morphology and motility phenotypes, emphasizing the role of fluid flows in regulating cancer cell migration heterogeneity.', '1809.06726-2-16-2': 'The conventional practice in dealing with this out of equilibrium problems is to use a set of fictitious fields called response fields, which provide a field theoretic prescription for the response function.', '1809.06726-2-16-3': 'In contrast, we propose the introduction of a fictitious time in which a FDT is valid, and thereby only correlation functions need to be calculated.', '1809.06726-2-16-4': 'Our approach greatly simplifies the evaluation of scaling exponents.', '1809.06726-2-16-5': 'We find that the non-linear term in the density evolution equation arising from mechanical interactions along with self generating force due to actin remodeling, determine the scaling behavior for the collective migration of cells.', '1809.06726-2-16-6': 'In the absence of flow, cells exhibit collective migration of mesenchymal motility phenotype induced by time dependent interaction potentials that determine the structural rearrangement during their path generating migration through collagen matrix.', '1809.06726-2-16-7': 'In contrast, the cells exhibit the amoeboid motility in the presence of flow and exhibit a dynamical slowing down of directed migration, with a gradually decreasing degree of super-diffusion.', '1809.06726-2-16-8': 'The long-range hydrodynamic interactions among cells in the presence of interstitial flow, determines the collective migration of cells through collagen matrix in a path finding fashion.', '1809.06726-2-16-9': 'The theoretical framework introduced here provides evidence of interstitial flow directed collective motion heterogeneity and could explain the invasion of cancer cells under interstitial flow, observed in a recent experiment[CITATION].', '1809.06726-2-16-10': 'The theory introduced here could help us understand how cancer cells spread by invading adjacent tissues involved in metastasis [CITATION].', '1809.06726-2-17-0': 'I wish to express my gratitude to Prof. D. Thirumalai for supporting me as a postdoc at the department of chemistry at UT, Austin.', '1809.06726-2-17-1': 'I am indebted to Dr. Debayan Chakrabarty for his valuable comments.', '1809.06726-2-18-0': '# Supplementary Information: Interstitial flows regulate collective cell migration heterogeneity through adhesion', '1809.06726-2-19-0': '## Short-range interaction', '1809.06726-2-20-0': 'To obtain the dynamics of an evolving collection of cells, we use the following simplified form for cell-cell interaction, [EQUATION] where [MATH] and [MATH] are the strengths of excluded volume and attractive interactions, respectively.', '1809.06726-2-21-0': 'In addition, the cell surface-ECM interactions [MATH], determining the configuration-dependent forces experienced by the cells: [MATH].', '1809.06726-2-21-1': 'The potential term [MATH] describes the cell surface-collagen interactions as a function of [MATH], the average distance between cell and collagen.', '1809.06726-2-21-2': 'Cells interact with the collagen through receptor-ligand interactions, described by short range potential.', '1809.06726-2-21-3': 'Mesenchymal cells form long-lived adhesion with collagen.', '1809.06726-2-21-4': 'We assume the adhesion strength is changing during the topological rearrangement via, [MATH].', '1809.06726-2-21-5': '[MATH] and [MATH] are initial and final interaction strengths and the time scale of changing the receptor-ligand interaction is given by [MATH].', '1809.06726-2-21-6': 'Where [MATH] stands for [MATH] and [MATH].', '1809.06726-2-21-7': 'In contrast, amoeboid cells form short lived adhesion with the collagen, i.e., the time scale for the adhesive interaction [MATH], is small compared to mesenchymal cells.', '1809.06726-2-22-0': '## Density equation', '1809.06726-2-23-0': 'To simplify the multiplicative noise term (last term in Eq. (2) in the main text), we assume that the density fluctuates around a constant value.', '1809.06726-2-23-1': 'Hence, we define the density using [MATH], and expand Eq. (1) in the main text in [MATH] up to the lowest order in non-linearity.', '1809.06726-2-23-2': 'In Fourier space, the equation for the density fluctuation becomes, [EQUATION] with [MATH].', '1809.06726-2-23-3': 'Where, [MATH].', '1809.06726-2-23-4': 'The Greens function [MATH] is given by, [EQUATION] where, [MATH], and [MATH] is the self energy term contributed from non-linear adhesion interactions.', '1809.06726-2-24-0': ""## Stochastic quantization technique We now exploit the Parisi-Wu stochastic quantization scheme [CITATION], and introduce a fictitious time [MATH]', and consider all the variables to be functions of [MATH]."", '1809.06726-2-24-1': ""A Langevin equation in [MATH]' space is, [EQUATION] with [MATH]."", '1809.06726-2-24-2': 'Because FDT is valid in the fictitious time it follows that as [MATH], the distribution function will be given by the action [MATH].', '1809.06726-2-24-3': 'The action [MATH] can be obtained by writing down the probability distribution [MATH] corresponding to the noise term [MATH] in Eq. ([REF]), and the action [MATH] in terms of [MATH] using Eq. ([REF]).', '1809.06726-2-24-4': 'The expression for the action [MATH] obtained using Eq. ([REF]) is,', '1809.06726-2-25-0': '[MATH]', '1809.06726-2-26-0': '## Greens function for density equation', '1809.06726-2-27-0': 'The correlation functions, calculated from Eq. ([REF]), lead to the required correlation functions of the original theory, in the [MATH] limit.', '1809.06726-2-27-1': 'In order to obtain the scaling laws, it suffices to work at arbitrary [MATH].', '1809.06726-2-27-2': 'It is obvious from Eq. ([REF]) that in the absence of the non-linear terms, the Greens function [MATH] is given by, [EQUATION] where [MATH] is the frequency corresponding to the fictitious time [MATH].', '1809.06726-2-27-3': ""As is customary, the effect of non-linear terms, can be included perturbatively leading to the Dyson's' equation [EQUATION]"", '1809.06726-2-27-4': 'Here, we are concerned with the behavior of [MATH], which becomes non-linear when expanded to second order.', '1809.06726-2-27-5': 'We note that the contribution comes from two different sources (1) a one-loop contribution from the second order term (containing three [MATH] fields) in Eq. ([REF]) (second term in Fig. [REF]) and (2) a two-loop contribution from the first order term ( containing two [MATH] fields) in Eq. ([REF]) (first term in Fig. [REF]).', '1809.06726-2-27-6': 'The contribution arising coming from the term containing three [MATH] fields, in Eq. ([REF]) can be readily obtained by contracting two of the [MATH] fields.', '1809.06726-2-27-7': 'The second order term coming from the one loop contribution in Eq. ([REF]) does not have any new momentum dependance.', '1809.06726-2-27-8': 'Hence it is the second-order contribution (first term in Fig. ([REF])), coming from the two-loop contribution in Eq. ([REF]), which is significant.', '1809.06726-2-27-9': 'The correlation function is given by the FDT as [MATH].', '1809.06726-2-27-10': 'With these observations, Eq. ([REF]) can be written as, [EQUATION] where [MATH], and [MATH] is defined by [EQUATION] with [MATH].', '1809.06726-2-27-11': 'Expanding [MATH], [MATH] about [MATH] and [MATH], respectively, and noting that the renormalization of [MATH] dominates, we get [EQUATION]', '1809.06726-2-28-0': '## Scaling exponents', '1809.06726-2-29-0': 'In a self consistent mode coupling theory, we now replace [MATH] by [MATH] in the self energy term [MATH] in the first term in Fig. ([REF]), use [MATH] as from Eq. ([REF]) and [MATH], which follows from the FDT.', '1809.06726-2-29-1': 'According to scale transformation, we know that [MATH], [MATH], [MATH], [MATH] and the vertex factor [MATH].', '1809.06726-2-29-2': 'The self energy term in Fig. ([REF]) can be written as [MATH].', '1809.06726-2-29-3': 'By carrying out the momentum count of [MATH], and using [MATH], we find that [MATH].', '1809.06726-2-29-4': 'Using Eq. ([REF]) and [MATH], we have [MATH], which leads to [MATH].', '1809.06726-2-29-5': 'Where, [MATH] and use [MATH] is small because in the case of flow, cells are amoeboid phenotype with short lived adhesion with collagen fiber.', '1809.06726-2-29-6': 'MSD exponent [MATH].', '1809.06726-2-29-7': 'In 3D, [MATH], i.e., the amoeboid cells undergo super-diffusion.', '1809.06726-2-30-0': '## The expression for [MATH]', '1809.06726-2-31-0': '[EQUATION] where veterx term, [MATH]'}","[['1809.06726-1-22-1', '1809.06726-2-16-2'], ['1809.06726-1-22-2', '1809.06726-2-16-3'], ['1809.06726-1-22-3', '1809.06726-2-16-4'], ['1809.06726-1-22-10', '1809.06726-2-16-10'], ['1809.06726-1-24-0', '1809.06726-2-17-0'], ['1809.06726-1-24-1', '1809.06726-2-17-1'], ['1809.06726-1-18-2', '1809.06726-2-14-4'], ['1809.06726-1-18-3', '1809.06726-2-14-5'], ['1809.06726-1-18-4', '1809.06726-2-14-6'], ['1809.06726-1-15-0', '1809.06726-2-11-0'], ['1809.06726-1-15-1', '1809.06726-2-11-1'], ['1809.06726-1-15-2', '1809.06726-2-11-2'], ['1809.06726-1-15-3', '1809.06726-2-11-3'], ['1809.06726-1-15-4', '1809.06726-2-11-4'], ['1809.06726-1-14-1', '1809.06726-2-10-1'], ['1809.06726-1-14-2', '1809.06726-2-10-2'], ['1809.06726-1-14-3', '1809.06726-2-10-3'], ['1809.06726-1-14-4', '1809.06726-2-10-4'], ['1809.06726-1-14-5', '1809.06726-2-10-5'], ['1809.06726-1-14-6', '1809.06726-2-10-6'], ['1809.06726-1-14-7', '1809.06726-2-10-7'], ['1809.06726-1-14-8', '1809.06726-2-10-8'], ['1809.06726-2-9-0', '1809.06726-3-12-0'], ['1809.06726-2-9-1', '1809.06726-3-12-1'], ['1809.06726-2-9-2', '1809.06726-3-12-2'], ['1809.06726-2-9-5', '1809.06726-3-12-5'], ['1809.06726-2-12-0', '1809.06726-3-15-0'], ['1809.06726-2-12-1', '1809.06726-3-15-1'], ['1809.06726-2-12-2', '1809.06726-3-15-2'], ['1809.06726-2-17-0', '1809.06726-3-23-0'], ['1809.06726-2-17-1', '1809.06726-3-23-1'], ['1809.06726-2-15-0', '1809.06726-3-18-0'], ['1809.06726-2-15-2', '1809.06726-3-18-2'], ['1809.06726-2-15-5', '1809.06726-3-18-5'], ['1809.06726-2-15-6', '1809.06726-3-18-6'], ['1809.06726-2-15-7', '1809.06726-3-18-7'], ['1809.06726-2-15-8', '1809.06726-3-18-8'], ['1809.06726-2-15-9', '1809.06726-3-18-9'], ['1809.06726-2-15-11', '1809.06726-3-18-11'], ['1809.06726-2-6-2', '1809.06726-3-7-2'], ['1809.06726-2-6-3', '1809.06726-3-7-3'], ['1809.06726-2-6-4', '1809.06726-3-7-4'], ['1809.06726-2-6-5', '1809.06726-3-7-5'], ['1809.06726-2-16-0', '1809.06726-3-22-0'], ['1809.06726-2-16-1', '1809.06726-3-22-1'], ['1809.06726-2-16-4', '1809.06726-3-22-4'], ['1809.06726-2-16-7', '1809.06726-3-22-7'], ['1809.06726-2-16-9', '1809.06726-3-22-10'], ['1809.06726-2-16-10', '1809.06726-3-22-11'], ['1809.06726-2-0-0', '1809.06726-3-0-0'], ['1809.06726-2-0-4', '1809.06726-3-0-4'], ['1809.06726-2-0-5', '1809.06726-3-0-5'], ['1809.06726-2-1-0', '1809.06726-3-1-0'], ['1809.06726-2-1-1', '1809.06726-3-1-1'], ['1809.06726-2-1-2', '1809.06726-3-1-2'], ['1809.06726-2-1-3', '1809.06726-3-1-3'], ['1809.06726-2-1-4', '1809.06726-3-1-4'], ['1809.06726-2-1-6', '1809.06726-3-1-6'], ['1809.06726-2-7-0', '1809.06726-3-8-0'], ['1809.06726-2-7-2', '1809.06726-3-8-2'], ['1809.06726-2-7-4', '1809.06726-3-8-4'], ['1809.06726-2-7-6', '1809.06726-3-8-6'], ['1809.06726-2-7-7', '1809.06726-3-8-10'], ['1809.06726-2-7-8', '1809.06726-3-8-11'], ['1809.06726-2-7-9', '1809.06726-3-8-18'], ['1809.06726-2-7-10', '1809.06726-3-8-19'], ['1809.06726-2-7-11', '1809.06726-3-8-20'], ['1809.06726-2-7-12', '1809.06726-3-8-21'], ['1809.06726-2-7-13', '1809.06726-3-8-22'], ['1809.06726-2-7-14', '1809.06726-3-8-23'], ['1809.06726-2-7-15', '1809.06726-3-8-24'], ['1809.06726-2-3-0', '1809.06726-3-5-0'], ['1809.06726-2-3-1', '1809.06726-3-5-1'], ['1809.06726-2-14-0', '1809.06726-3-17-0'], ['1809.06726-2-14-1', '1809.06726-3-17-1'], ['1809.06726-2-14-2', '1809.06726-3-17-2'], ['1809.06726-2-14-3', '1809.06726-3-17-3'], ['1809.06726-2-14-4', '1809.06726-3-17-4'], ['1809.06726-2-14-5', '1809.06726-3-17-5'], ['1809.06726-2-14-6', '1809.06726-3-17-6'], ['1809.06726-2-14-7', '1809.06726-3-17-7'], ['1809.06726-2-14-9', '1809.06726-3-17-9'], ['1809.06726-2-14-10', '1809.06726-3-17-10'], ['1809.06726-2-11-0', '1809.06726-3-14-0'], ['1809.06726-2-11-1', '1809.06726-3-14-1'], ['1809.06726-2-11-2', '1809.06726-3-14-2'], ['1809.06726-2-11-4', '1809.06726-3-14-4'], ['1809.06726-2-13-1', '1809.06726-3-16-1'], ['1809.06726-2-10-0', '1809.06726-3-13-0'], ['1809.06726-2-10-1', '1809.06726-3-13-1'], ['1809.06726-2-10-2', '1809.06726-3-13-2'], ['1809.06726-2-10-5', '1809.06726-3-13-5'], ['1809.06726-2-10-6', '1809.06726-3-13-6'], ['1809.06726-2-10-8', '1809.06726-3-13-8'], ['1809.06726-3-10-0', '1809.06726-4-13-0'], ['1809.06726-3-10-1', '1809.06726-4-13-1'], ['1809.06726-3-10-2', '1809.06726-4-13-2'], ['1809.06726-3-10-3', '1809.06726-4-13-3'], ['1809.06726-3-10-4', '1809.06726-4-13-14'], ['1809.06726-3-10-5', '1809.06726-4-13-15'], ['1809.06726-3-10-6', '1809.06726-4-13-16'], ['1809.06726-3-10-7', '1809.06726-4-13-17'], ['1809.06726-3-10-9', 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'1809.06726-4-36-4', '1809.06726-4-37-0', '1809.06726-4-41-6', '1809.06726-4-43-0']","{'1': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '2': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '3': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/', '4': 'http://arxiv.org/licenses/nonexclusive-distrib/1.0/'}",https://arxiv.org/abs/1809.06726,"{'1809.06726-3-0-0': 'The migration behaviors of cancer cells are known to be heterogeneous.', '1809.06726-3-0-1': 'However, the interplay between the adhesion interactions, dynamical shape changes and fluid flow in regulating cell migration heterogeneity and plasticity during cancer metastasis is still elusive.', '1809.06726-3-0-2': 'To further quantitative understanding of cell motility and morphology, we develop a theory using a stochastic quantization method that describes the role of biophysical cues in regulating diverse cell motility.', '1809.06726-3-0-3': 'We show that the cumulative effect of time-dependent adhesion interactions that determine the structural rearrangements and self-generated force due to actin remodeling dictate the super-diffusive motion of mesenchymal phenotype in the absence of flow.', '1809.06726-3-0-4': 'Interstitial flows regulate cell motility phenotype and promote the amoeboid over mesenchymal motility through adhesion interactions.', '1809.06726-3-0-5': 'Cells exhibit a dynamical slowing down of collective migration, with a decreasing degree of super-diffusion.', '1809.06726-3-0-6': 'Mesenchymal cells are more persistent and diffusive compared to amoeboid cells.', '1809.06726-3-0-7': 'Our findings suggest a mechanism of Interstitial flow-induced directed motion of cancer cells through adhesion and provide the much-needed insight into a recent experimental observation concerning the diverse motility of breast cancer cells.', '1809.06726-3-1-0': 'Collective cancer cell invasion, followed by local and distant metastasis, is a hallmark of cancer[CITATION].', '1809.06726-3-1-1': 'Cancer metastasis is a multistep process, where tumor cells detach from primary tumor, invade through the interstitial extracellular matrix, intravasation of tumor cells into vascular vessels, extravasation of circulating tumor cells to peripheral tissues, and establish a secondary tumor at distant organ[CITATION].', '1809.06726-3-1-2': 'Dynamics associated with invasion and metastasis, involve the collective cell migration regulated by biomechanical (e.g. cytokines secreted by cells and nutrients) and biophysical cues (e.g. fluid flows and ECM)[CITATION].', '1809.06726-3-1-3': 'Tumor cells reside in an extracellular matrix (ECM) containing interstitial fluid that transports nutrients and signaling molecules.', '1809.06726-3-1-4': 'The interstitial flow has been shown to affect the morphology and migration of cells such as fibroblasts, cancer cells, endothelial cells, and mesenchymal stem cells[CITATION].', '1809.06726-3-1-5': 'Interstitial flows are particularly important for tumor cell invasion because it is elevated in tumor microenvironment due to the heightened interstitial fluid pressure as well as the abnormal angiogenic, lymphangiogenic blood and lymphatic vessels[CITATION].', '1809.06726-3-1-6': 'The flow speed associated with interstitial flows are in the order of a few micrometers per second in normal tissue[CITATION].', '1809.06726-3-1-7': 'Two types of motility (e.g. amoeboid and mesenchymal) have been broadly categorized in cell migration in 3D architechture[CITATION].', '1809.06726-3-1-8': 'Cells with aspect ratio smaller than 2.0 are considered rounded or amoeboid and cells with an aspect ratio greater than 2.0 are considered elongated or mesenchymal (see fig.1)[CITATION].', '1809.06726-3-2-0': 'Recent experiment has demonstrated that interstitial flows regulate the cancer cell migration heterogeneity within a three-dimensional biomatrix.', '1809.06726-3-2-1': 'Using a microfluidic model, authors show that breast cancer cells (MDA-MB-231) embedded in a collagen matrix, exhibit both amoeboid and mesenchymal motility phenotype and interstitial flows promote amoeboid over mesenchymal motility of breast cancer cells[CITATION].', '1809.06726-3-3-0': 'Mesenchymal migration is characterized by an elongated fibroblast-like morphology, highly condensed cell-matrix adhesions, and formation of contractile actomyosin bundles[CITATION].', '1809.06726-3-3-1': 'The elongated cells demonstrated a mesenchymal phenotype where the actin filaments formed a highly polarized bundle.', '1809.06726-3-3-2': 'It is known that mesenchymal cells form long-lived adhesions with the ECM fiber bundles, which trigger the downstream signaling that activates actin remodeling and climbed along the fibers, and thus cell migration.', '1809.06726-3-3-3': 'Fibronectin (FN) is an important adhesion molecule in mediating mammalian cell migration.', '1809.06726-3-3-4': 'Cell-secreted FNs assemble into a fibrillar form and bind to collagen, which promotes a mesenchymal cell phenotype[CITATION].', '1809.06726-3-3-5': 'In the mesenchymal mode, cells wider than the matrix pore size extend protrusions at the leading edge to probe the surrounding ECM fibers, form stable adhesions at the poles of the elongated cell.', '1809.06726-3-3-6': 'Further polarization and strengthening of adhesions is accompanied by a rise in actomyosin contractility exerting traction force and proteolytic degradation of ECM fibers at cell-ECM junctions, and rupture the adhesions at the trailing edge of the cell, thus migrating in a ‘‘path-generating’’ manner[CITATION].', '1809.06726-3-4-0': 'Amoeboid migration is characterized by a rounded morphology, formation of bleb-like protrusions, restriction of actomyosin contractility to the cell cortex, and transient, short-lived adhesions with the ECM, and squeezing through the matrix pore when finding a suitable path[CITATION].', '1809.06726-3-4-1': 'The inhibition of proteolysis or integrin-dependent adhesion can be compensated by a weak or non-adhesive amoeboid mode of migration in which the cell adopts a rounded morphology and changes its shape by generating hydrostatic pressure at the cell cortex, thus forcibly extruding processes through available spaces in the porous matrix and eventually deforming the cell body.', '1809.06726-3-4-2': 'In the case of flow, the interstitial-flow-induced amoeboid cell motility was likely caused by the lack of assembled endogenous adhesion molecules such as FN.', '1809.06726-3-4-3': 'More specifically, the flows carried away the cell-secreted adhesion molecules before they were assembled into fibrils and anchored to the collagen fibers[CITATION].', '1809.06726-3-5-0': 'How interstitial flows promote cancer cell invasion is largely unknown.', '1809.06726-3-5-1': 'The understanding of this process could help to develop drugs that inhibit the process and prevent cancer from metastasizing.', '1809.06726-3-5-2': 'How the biophysical forces modulate tumor cell migration heterogeneity and plasticity and create a complex spatiotemporal dynamical property during cancer metastasis is still elusive.', '1809.06726-3-6-0': 'In this article, we develop a theory to describe how biophysical cues regulate the diverse collective cell motility.', '1809.06726-3-6-1': 'The cumulative effect arising from the non-equilibrium description of living cells using a time-dependent mechanical interaction, and flows, lead to complex dynamics, which may have far-reaching implications in our understanding of cancer metastasis.', '1809.06726-3-6-2': 'One of the major difficulties in the study of collective behavior of the cells far from equilibrium is the breakdown of a fluctuation-dissipation theorem (FDT); hence, independent diagrammatic expansions for the response function and the correlation function.', '1809.06726-3-6-3': 'The equilibrium distribution is not known and averages can be computed only for the statistical noise.', '1809.06726-3-7-0': 'We study the relevant continuum description of the collective behavior of a colony of cells in the physical time scale, using the stochastic quantization technique, originally proposed by Parisi and Wu[CITATION].', '1809.06726-3-7-1': 'We show that the time-dependent adhesion interactions that determine the structural rearrangements, the long-range hydrodynamic interactions among living cells, and self-generated force due to actin remodeling dictate the complex collective behavior, when a continuum description of the cellular colony is invoked, in the physical time scale.', '1809.06726-3-7-2': 'We find that cells exhibit both amoeboid and mesenchymal motility characterized by super-diffusive motion.', '1809.06726-3-7-3': 'The mean-square displacement (MSD) of the cells for the mesenchymal motility behaves as [MATH], with, [MATH].', '1809.06726-3-7-4': 'The interstitial flows impair the collective migration with a gradually decreasing degree of super-diffusion and promote the amoeboid motility phenotype over mesenchymal motility.', '1809.06726-3-7-5': 'In the case of flow, the MSD exponent [MATH], which reflects the dynamical slowing down of the spatiotemporal collective migration.', '1809.06726-3-7-6': 'The mesenchymal cell migration is more persistent than the amoeboid motility phenotype.', '1809.06726-3-8-0': 'Theory: In the absence of flow, cells exhibit mesenchymal motility phenotype in 3D collagen matrix.', '1809.06726-3-8-1': 'Cells secreted fibronectin molecules into a fibrillar form, and form long-lived adhesions with the collagen fibers, which trigger the downstream signaling that activates actin-network expansion and thus exhibit cell migration[CITATION].', '1809.06726-3-8-2': 'We consider the dynamics of a colony of cells in a dissipative environment where inertial effects are negligible.', '1809.06726-3-8-3': 'Each cell experiences systematic forces arising from mechanical interactions, and a Gaussian random force with a white noise spectrum.', '1809.06726-3-8-4': 'The equation of motion for a single mesenchymal cell [MATH] is [MATH] where [MATH] contains repulsive interactions with range [MATH], adhesion interactions with collagen matrix with range [MATH], and favorable attractive interactions between cells with range [MATH], and with strengths [MATH], [MATH] and [MATH] respectively.', '1809.06726-3-8-5': 'We use Gaussian potentials (see the Supplementary Information (SI) for details[CITATION]) in order to obtain analytical solutions.', '1809.06726-3-8-6': 'Needless to say that the conclusions would be valid for any short-ranged [MATH].', '1809.06726-3-8-7': 'The highly deformable cells with flexible cell boundaries connected through cell-cell junctions, which can be ruptured and reconnected.', '1809.06726-3-8-8': 'Cell structural rearrangement for changing nearest-neighbor cells has been seen during the migration of cells in movie S1 in [CITATION].', '1809.06726-3-8-9': 'Therefore we model the cell-cell interaction as time-dependent potential.', '1809.06726-3-8-10': 'We assume that the adhesion strength is changing during the topological rearrangement via [MATH][CITATION].', '1809.06726-3-8-11': '[MATH] and [MATH] are initial and final interaction strengths ([MATH] stands for [MATH] and [MATH]) and the time scale for changing the receptor-ligand interaction is given by [MATH].', '1809.06726-3-8-12': 'In addition to short-range interactions, the cells that are well separated, interact with each other via the matrix.', '1809.06726-3-8-13': 'The cells in the collagen matrix can be modeled as force concentration dipoles.', '1809.06726-3-8-14': 'The cells interact via long-ranged elastic interaction potentials ([MATH] with distance [MATH])[CITATION].', '1809.06726-3-8-15': 'The interaction potentials depend on elastic constants, geometry, and cellular orientations.', '1809.06726-3-8-16': 'For simplicity of the calculation, we consider the interaction is isotropic and repulsive.', '1809.06726-3-8-17': 'The potential [MATH], where [MATH] is the magnitude of the force dipole, [MATH], and [MATH], where [MATH] and [MATH] are the Lam[MATH] coefficients of the isotropic elastic medium[CITATION].', '1809.06726-3-8-18': 'The Gaussian white noise, satisfies [MATH].', '1809.06726-3-8-19': 'The mesenchymal cells are subject to a self-generated force of actin-network remodeling.', '1809.06726-3-8-20': 'The cells are, thus in addition subject to a random self-generated force with amplitude [MATH].', '1809.06726-3-8-21': 'The randomness is modeled by an athermal noise [MATH], which is exponentially correlated over a time scale [MATH].', '1809.06726-3-8-22': 'The statistics of the [MATH] is given by [MATH], [MATH] and [MATH].', '1809.06726-3-8-23': 'Where [MATH] is the dimensionless constant.', '1809.06726-3-8-24': 'The athermal noise in general does not obey the FDT.', '1809.06726-3-9-0': 'For the mesenchymal phenotype, the cells form long-lived adhesion with the matrix and exhibit matrix mediated long-range elastic interactions when they are well separated.', '1809.06726-3-9-1': 'On the other hand, the amoeboid cells form short-lived adhesion with the matrix and migrate squeezing through the pore.', '1809.06726-3-9-2': 'Therefore, we assume amoeboid cells do not feel long-range elastic interactions in the long-time limit.', '1809.06726-3-10-0': 'In the absence of flow, cell-secreted FNs assemble into a fibrillar form and bind to collagen, which promotes a mesenchymal cell phenotype.', '1809.06726-3-10-1': 'In the case of flow, the interstitial-flow carried away the cell-secreted adhesion molecules before they were assembled into fibrils and anchored to the collagen fibers and thus amoeboid cell motility was likely caused by the lack of assembled endogenous adhesion molecules such as FN.', '1809.06726-3-10-2': 'Therefore, in the case of flow cells form short-lived adhesion with matrix and form rounded morphology and thus amoeboid motility[CITATION].', '1809.06726-3-10-3': 'The experiment in [CITATION] investigates the role of exogenous FN in cell morphology and motility in the absence/presence of the flows.', '1809.06726-3-10-4': 'The amoeboid cells migrate through squeezing the matrix pore when finding a suitable path.', '1809.06726-3-10-5': 'Therefore we consider the long-range hydrodynamic interactions between amoeboid cells leading to collective behavior of cell motility.', '1809.06726-3-10-6': 'We begin by considering the dynamics of a colony of cells in the presence of flow in a dissipative environment where inertial effects are negligible.', '1809.06726-3-10-7': 'Each cell experiences mechanical forces, such as adhesion, excluded volume interactions due to neighbors, and a random force characterized by Gaussian white noise.', '1809.06726-3-10-8': 'The equation of motion for single amoeboid cell [MATH] is [CITATION] [EQUATION]', '1809.06726-3-10-9': 'The first term on the r.h.s. of Eq. ([REF]) is the effect of force acting on cell [MATH] creates a hydrodynamic flow-field in the fluid, thereby entraining cell [MATH].', '1809.06726-3-10-10': 'Where [MATH] contains repulsive interactions with range [MATH], adhesion interactions with collagen matrix with range [MATH], and favorable attractive interactions between cells with range [MATH], and with strengths [MATH], [MATH], and [MATH] respectively (see the Supplementary Information (SI) for details).', '1809.06726-3-11-0': '[MATH] is assumed to be Gaussian random vectors exhibiting hydrodynamic correlations according to fluctuation-dissipation theorem (FDT), [MATH].', '1809.06726-3-11-1': 'The dynamics of the cell is described by [MATH], where [MATH] is the position of the cell.', '1809.06726-3-11-2': 'The flow velocity of the fluid at [MATH] is determined by the solution of the Stokes equation, [MATH], where, [MATH].', '1809.06726-3-11-3': 'We assume that the fluid is incompressible, i.e., [MATH].', '1809.06726-3-11-4': 'From these set of equations, the Stokes equation is readily solved to obtain a fluid velocity at the center of the cell at [MATH] for a point force [MATH] as, u(r)=[MATH], where [MATH] is the mobility matrix with the relative coordinate between cell [MATH] and [MATH], [MATH].', '1809.06726-3-11-5': 'The long-ranged nature of hydrodynamic interaction apparent from [MATH] suggests that a collection of cells that dynamically exert forces on the fluid medium they are immersed in could influence each other very strongly, leading to the possibility of collective behaviors.', '1809.06726-3-11-6': 'The protrusive flowing of the anterior actin network of the cell and squeezing actomyosin contractions of the trailing edge are modeled as the cells are subjected to a self-generated force with amplitude [MATH] during their pathfinding motion in the collagen matrix.', '1809.06726-3-11-7': 'The statistics of the [MATH] is the same as for mesenchymal motility.', '1809.06726-3-12-0': 'We consider the evolution of the density function for a single cell [MATH].', '1809.06726-3-12-1': 'A closed form Langevin equation for the density, [MATH] can be obtained using standard approach [CITATION].', '1809.06726-3-12-2': 'The time evolution of [MATH] is given by, [EQUATION]', '1809.06726-3-12-3': 'Note that the density equation contains the same information as N-body stochastic Langevin equations.', '1809.06726-3-12-4': 'This is an out of equilibrium problem characterized by the absence of fluctuation-dissipation theorem due to long-range hydrodynamic term and self-generated force due to actin remodeling.', '1809.06726-3-12-5': 'Eq. ([REF]) can be studied analytically by treating the non-linear terms using a perturbative approach, based on the stochastic quantization scheme [CITATION].', '1809.06726-3-13-0': 'Stochastic quantization approach: To understand the dynamics of collections of cells, we use the stochastic quantization method developed by Parisi-Wu in another context.', '1809.06726-3-13-1': 'The collective migration of cells described by Eq. ([REF]) is an out of equilibrium problem characterized by the absence of FDT, which relates the correlation and response function in momentum space as [MATH].', '1809.06726-3-13-2': 'The usual analytic route to get the scaling solution of this problem, one can introduce a response field [MATH].', '1809.06726-3-13-3': 'We need to calculate both the response function ([MATH]) and correlation function ([MATH]) because of the absence of a fluctuation-dissipation relation.', '1809.06726-3-13-4': 'The key advantage of the present method is that we do not need to calculate both the correlation and response functions.', '1809.06726-3-13-5': 'The FDT is constructed in fictitious time introduced in the problem.', '1809.06726-3-13-6': 'The FDT relation enables us to obtain the scaling of the correlation function, once the scaling of the response function is known.', '1809.06726-3-13-7': 'By taking the infinite limit in fictitious time, one can obtain the correlation function in real-time.', '1809.06726-3-13-8': 'The scaling solution of the problem can be obtained by power counting analysis instead of doing renormalization group calculation.', '1809.06726-3-14-0': ""We now exploit the Parisi-Wu stochastic quantization scheme [CITATION], and introduce a fictitious time [MATH]', and consider all variables to be functions of [MATH] in addition to k and [MATH]."", '1809.06726-3-14-1': ""A Langevin equation in [MATH]' space is, [EQUATION] with [MATH]."", '1809.06726-3-14-2': 'This ensures that as [MATH], the distribution function will be given by the action [MATH], because in the [MATH]-space a fluctuation dissipation theorem (FDT) is preserved.', '1809.06726-3-14-3': 'The action [MATH] can be obtained by writing down the probability distribution [MATH] corresponding to the noise term [MATH], and the action [MATH] in terms of [MATH] with the help of Eq. ([REF]), where [MATH].', '1809.06726-3-14-4': 'The expression for the [MATH] is in the appendix.', '1809.06726-3-15-0': 'We follow the procedure of obtaining scaling laws of the problem, which has been demonstrated in earlier works [CITATION].', '1809.06726-3-15-1': 'The dynamics of Eq. ([REF]) requires only the calculation of response functions, the correlation functions in this dynamics are related to response function through FDT relation, i.e., in Fourier space, [MATH].', '1809.06726-3-15-2': 'We can obtain the scaling laws in real space and time in a straight forward fashion from the solution in the fictitious time [MATH] space.', '1809.06726-3-16-0': 'We obtain the following self-consistent equation for the self-energy from the calculation of response function using Eq.([REF]): [EQUATION] where, [MATH], [MATH], and [MATH] is the self-energy term, a two-loop contribution from the first order term ( containing two [MATH] fields) in Eq. ([REF]) (first term in Fig. [REF]) will contribute in the scaling laws for the cell in the finite time.', '1809.06726-3-16-1': 'We use Eq.([REF]) for getting the scaling laws of both the amoeboid and mesenchymal cells phenotype.', '1809.06726-3-17-0': 'Results: Mesenchymal motility: The mesenchymal cell phenotype forms long-lived integrin-based adhesions with the collagen matrix and migrate via either remodeling of actin network or degrading the matrix.', '1809.06726-3-17-1': 'The non-linear term i.e. the adhesion interaction plays an important role in the complex dynamics of collective migration of mesenchymal cell phenotype.', '1809.06726-3-17-2': 'In a self consistent mode coupling theory, we now replace [MATH] by [MATH] in the self energy term [MATH] in the first term in Fig. ([REF]), use [MATH] as from Eq. ([REF]), and [MATH], which follows from the FDT.', '1809.06726-3-17-3': 'In the absence of flow, [MATH].', '1809.06726-3-17-4': 'According to scale transformation, we know that [MATH], [MATH], [MATH], [MATH] and the vertex factor [MATH].', '1809.06726-3-17-5': 'The self energy term in Fig. ([REF]) can be written as [MATH].', '1809.06726-3-17-6': 'By carrying out the momentum count of [MATH], and using [MATH], we find that [MATH].', '1809.06726-3-17-7': 'Using Eq. ([REF]) and [MATH], we have [MATH], which leads to [MATH].', '1809.06726-3-17-8': 'MSD exponent [MATH].', '1809.06726-3-17-9': 'In 3D, [MATH], i.e., the mesenchymal cells undergo super-diffusion.', '1809.06726-3-17-10': 'The non-linear term arising from cell-cell adhesion that determines the dynamical shape change during cell motion and self-generated force, produce super-diffusive motion.', '1809.06726-3-17-11': 'The theoretical result is in good agreement with the recent experimental result ([MATH]) using the microfluidic model[CITATION].', '1809.06726-3-17-12': 'If we wait longer than experimental observation time, the cells become apart from each other.', '1809.06726-3-17-13': 'The short-range interactions do not play any role in cell motility.', '1809.06726-3-17-14': 'The long-range elastic interactions dictate the dynamics of the mesenchymal phenotype.', '1809.06726-3-17-15': 'The similar scaling analysis shows the super-diffusive behavior of cells characterized by the single-cell MSD exponent [MATH].', '1809.06726-3-17-16': 'The degree of superdiffusion decreases in the long-time where long-range elastic interactions play a role in cell motility.', '1809.06726-3-18-0': 'Interstitial flow induced amoeboid motility: In the presence of interstitial flow, cells exhibit amoeboid motility.', '1809.06726-3-18-1': 'The amoeboid cells form short-lived adhesion with the collagen matrix.', '1809.06726-3-18-2': 'The time scale [MATH] is small compared to the mesenchymal motility phenotype.', '1809.06726-3-18-3': 'In the case of flow, the cells exhibit long-range hydrodynamic interactions that determine the complex spatiotemporal dynamics of amoeboid cell phenotype.', '1809.06726-3-18-4': 'The self-generating force of actin remodeling helps to propel in pathfinding fashion through the collagen matrix.', '1809.06726-3-18-5': 'In a self consistent mode coupling theory, we now replace [MATH] by [MATH] in the self energy term [MATH] in the first term in Fig. ([REF]), use [MATH] as from Eq. ([REF]) and [MATH], which follows from the FDT.', '1809.06726-3-18-6': 'According to scale transformation, we know that [MATH], [MATH], [MATH], [MATH] and the vertex factor [MATH].', '1809.06726-3-18-7': 'The self energy term in Fig. ([REF]) can be written as [MATH].', '1809.06726-3-18-8': 'By carrying out the momentum count of [MATH], and using [MATH], we find that [MATH].', '1809.06726-3-18-9': 'Using Eq. ([REF]) and [MATH], we have [MATH], which leads to [MATH].', '1809.06726-3-18-10': 'MSD exponent [MATH].', '1809.06726-3-18-11': 'In 3D, [MATH], i.e., the amoeboid cells undergo superdiffusion.', '1809.06726-3-18-12': 'The non-linear term due to long-range hydrodynamic interactions among cells besides self-generated force, produce super-diffusive motion for amoeboid cell phenotype.', '1809.06726-3-18-13': 'The decrease of MSD exponent determines the dynamical slowing down of initial collective migration of the mesenchymal cell phenotype.', '1809.06726-3-18-14': ""Therefore, interstitial flows impair the cell's ability to spread by sweeping away the adhesion molecules with the flow and making the cells as amoeboid phenotype with short-lived adhesion with the collagen fiber[CITATION]."", '1809.06726-3-18-15': 'The cells migrate via squeezing through the pore of the collagen fiber when finding a suitable path[CITATION].', '1809.06726-3-18-16': 'The theoretical result is in good agreement with the recent experimental result ([MATH]) using the microfluidic model[CITATION].', '1809.06726-3-18-17': 'If we wait longer than experimental observation time, the cells become apart from each other.', '1809.06726-3-18-18': 'The short-range interactions do not play any role in cell motility.', '1809.06726-3-18-19': 'The cells undergo normal diffusion for the amoeboid motility phenotype characterized by the single-cell MSD exponent [MATH].', '1809.06726-3-19-0': 'We calculate velocity auto-correlation function [MATH], and time-dependent diffusion coefficient, [MATH], where [MATH] is the single cell MSD.', '1809.06726-3-19-1': 'The blue line dictates the mesenchymal motility and red line shows the amoeboid motility.', '1809.06726-3-19-2': 'The slower decay of velocity correlation function shows that the mesenchymal cells are more persistent than amoeboid cells.', '1809.06726-3-19-3': 'Similarly, the time-dependent diffusion coefficient for a mesenchymal cell is higher than the amoeboid cells, i.e, the mesenchymal cell is more diffusive than amoeboid cells.', '1809.06726-3-19-4': 'The theoretical result is in agreement with the experiment[CITATION].', '1809.06726-3-20-0': 'The MSD for a single cell in the experiment in [CITATION] shows the super-diffusion with the exponent [MATH] for both with and without the flow.', '1809.06726-3-20-1': 'Average for MSD calculation has been taken over all cell trajectories (approximate over 60 motile cells).', '1809.06726-3-20-2': 'The movies (S1 and S2) in [CITATION] show that cells are interacting through adhesion interactions that influence each other very strongly leading to the possibility of collective behaviors.', '1809.06726-3-20-3': 'That implies that the super-diffusion arises due to the collective motion of interacting cells for both with and without flow.', '1809.06726-3-20-4': 'Although it is a single-cell measurement, the super-diffusive motion of a cell arises due to the collective migration of interacting cells.', '1809.06726-3-21-0': 'In theory, the flow effect that we model through long-range hydrodynamic interactions among cells.', '1809.06726-3-21-1': 'That induces the super-diffusion for motile amoeboid cells.', '1809.06726-3-21-2': 'On the other hand, the time-dependent short-range interactions with the coupling with self-motility produce super-diffusive motion in mesenchymal cells.', '1809.06726-3-21-3': 'The single-cell MSD shows super-diffusion.', '1809.06726-3-21-4': 'The time-dependent short-range interactions and long-range hydrodynamic interactions between cells influence each other very strongly and produce non-linear terms in the density equations, leading to the possibility of novel collective behaviors.', '1809.06726-3-21-5': 'Therefore the super-diffusion effect is a collective phenomenon.', '1809.06726-3-22-0': 'Conclusion: In the present contribution, using a new theoretical framework, we provide insight into the dynamics of a colony of cancer cells driven by biophysical cues.', '1809.06726-3-22-1': 'The theory reveals that the interstitial flows regulate cancer cell morphology and motility phenotypes, emphasizing the role of fluid flows in regulating cancer cell migration heterogeneity.', '1809.06726-3-22-2': 'The conventional practice in dealing with this out of equilibrium problems is to use a set of fictitious fields called response fields, which provide a field-theoretic prescription for the response function.', '1809.06726-3-22-3': 'In contrast, we propose the introduction of a fictitious time in which an FDT is valid, and thereby only correlation functions need to be calculated.', '1809.06726-3-22-4': 'Our approach greatly simplifies the evaluation of scaling exponents.', '1809.06726-3-22-5': 'We find that the non-linear term in the density evolution equation arising from mechanical interactions along with self-generating force due to actin remodeling determines the scaling behavior for the collective migration of cells.', '1809.06726-3-22-6': 'In the absence of flow, cells exhibit collective migration of mesenchymal motility phenotype induced by time-dependent interaction potentials that determine the structural rearrangement during their path generating migration through the collagen matrix.', '1809.06726-3-22-7': 'In contrast, the cells exhibit the amoeboid motility in the presence of flow and exhibit a dynamical slowing down of directed migration, with a gradually decreasing degree of super-diffusion.', '1809.06726-3-22-8': 'The long-range hydrodynamic interactions among cells in the presence of interstitial flow determine the collective migration of cells through the matrix pore in a pathfinding fashion.', '1809.06726-3-22-9': 'The mesenchymal cells are more persistent and diffusive compared to amoeboid cells.', '1809.06726-3-22-10': 'The theoretical framework introduced here provides evidence of interstitial flow directed collective motion heterogeneity and could explain the invasion of cancer cells under interstitial flow, observed in a recent experiment[CITATION].', '1809.06726-3-22-11': 'The theory introduced here could help us understand how cancer cells spread by invading adjacent tissues involved in metastasis [CITATION].', '1809.06726-3-23-0': 'I wish to express my gratitude to Prof. D. Thirumalai for supporting me as a postdoc at the department of chemistry at UT, Austin.', '1809.06726-3-23-1': 'I am indebted to Dr. Debayan Chakrabarty for his valuable comments.'}","{'1809.06726-4-0-0': 'The migration behaviors of cancer cells are known to be heterogeneous.', '1809.06726-4-0-1': 'However, the interplay between the adhesion interactions, dynamical shape changes and fluid flow in regulating cell migration heterogeneity and plasticity during cancer metastasis is still elusive.', '1809.06726-4-0-2': 'To further quantitative understanding of cell motility and morphology, we develop a theory using a stochastic quantization method that describes the role of biophysical cues in regulating diverse cell motility.', '1809.06726-4-0-3': 'We show that the cumulative effect of time-dependent adhesion interactions that determine the structural rearrangements and self-generated force due to actin remodeling dictate the super-diffusive motion of mesenchymal phenotype in the absence of flow.', '1809.06726-4-0-4': 'Interstitial flows regulate cell motility phenotype and promote the amoeboid over mesenchymal motility through adhesion interactions.', '1809.06726-4-0-5': 'Cells exhibit a dynamical slowing down of collective migration, with a decreasing degree of super-diffusion.', '1809.06726-4-0-6': 'Mesenchymal cells are more persistent and diffusive compared to amoeboid cells.', '1809.06726-4-0-7': 'Our findings suggest a mechanism of Interstitial flow-induced directed motion of cancer cells through adhesion and provide the much-needed insight into a recent experimental observation concerning the diverse motility of breast cancer cells.', '1809.06726-4-1-0': '# Introduction', '1809.06726-4-2-0': 'Collective cancer cell invasion, followed by local and distant metastasis, is a hallmark of cancer[CITATION].', '1809.06726-4-2-1': 'Cancer metastasis is a multistep process, where tumor cells detach from primary tumor, invade through the interstitial extracellular matrix, intravasation of tumor cells into vascular vessels, extravasation of circulating tumor cells to peripheral tissues, and establish a secondary tumor at distant organ[CITATION].', '1809.06726-4-2-2': 'Dynamics associated with invasion and metastasis, involve the collective cell migration regulated by biomechanical (e.g. cytokines secreted by cells and nutrients) and biophysical cues (e.g. fluid flows and ECM)[CITATION].', '1809.06726-4-2-3': 'Tumor cells reside in an extracellular matrix (ECM) containing interstitial fluid that transports nutrients and signaling molecules.', '1809.06726-4-2-4': 'The interstitial flow has been shown to affect the morphology and migration of cells such as fibroblasts, cancer cells, endothelial cells, and mesenchymal stem cells[CITATION].', '1809.06726-4-2-5': 'Interstitial flows are particularly important for tumor cell invasion because it is elevated in tumor microenvironment due to the heightened interstitial fluid pressure as well as the abnormal angiogenic, lymphangiogenic blood and lymphatic vessels[CITATION].', '1809.06726-4-2-6': 'The flow speed associated with interstitial flows are in the order of a few micrometers per second in normal tissue[CITATION].', '1809.06726-4-2-7': 'Two types of motility (e.g. amoeboid and mesenchymal) have been broadly categorized in cell migration in 3D architechture[CITATION].', '1809.06726-4-2-8': 'Cells with aspect ratio smaller than 2.0 are considered rounded or amoeboid and cells with an aspect ratio greater than 2.0 are considered elongated or mesenchymal (see fig.1)[CITATION].', '1809.06726-4-3-0': 'Recent experiment has demonstrated that interstitial flows regulate the cancer cell migration heterogeneity within a three-dimensional biomatrix.', '1809.06726-4-3-1': 'Using a microfluidic model, authors show that breast cancer cells (MDA-MB-231) embedded in a collagen matrix, exhibit both amoeboid and mesenchymal motility phenotype and interstitial flows promote amoeboid over mesenchymal motility of breast cancer cells[CITATION].', '1809.06726-4-4-0': 'How interstitial flows promote cancer cell invasion is largely unknown.', '1809.06726-4-4-1': 'The understanding of this process could help to develop drugs that inhibit the process and prevent cancer from metastasizing.', '1809.06726-4-4-2': 'How the biophysical forces modulate tumor cell migration heterogeneity and plasticity and create a complex spatiotemporal dynamical property during cancer metastasis is still elusive.', '1809.06726-4-5-0': 'In this article, we develop a theory to describe how biophysical cues regulate the diverse collective cell motility.', '1809.06726-4-5-1': 'The cumulative effect arising from the non-equilibrium description of living cells using a time-dependent mechanical interaction, and flows, lead to complex dynamics, which may have far-reaching implications in our understanding of cancer metastasis.', '1809.06726-4-5-2': 'One of the major difficulties in the study of collective behavior of the cells far from equilibrium is the breakdown of a fluctuation-dissipation theorem (FDT); hence, independent diagrammatic expansions for the response function and the correlation function.', '1809.06726-4-5-3': 'The equilibrium distribution is not known and averages can be computed only for the statistical noise.', '1809.06726-4-6-0': 'We study the relevant continuum description of the collective behavior of a colony of cells in the physical time scale, using the stochastic quantization technique, originally proposed by Parisi and Wu[CITATION].', '1809.06726-4-6-1': 'We show that the time-dependent adhesion interactions that determine the structural rearrangements, the long-range hydrodynamic interactions among living cells, and self-generated force due to actin remodeling dictate the complex collective behavior, when a continuum description of the cellular colony is invoked, in the physical time scale.', '1809.06726-4-6-2': 'We find that cells exhibit both amoeboid and mesenchymal motility characterized by super-diffusive motion.', '1809.06726-4-6-3': 'The mean-square displacement (MSD) of the cells for the mesenchymal motility behaves as [MATH], with, [MATH].', '1809.06726-4-6-4': 'The interstitial flows impair the collective migration with a gradually decreasing degree of super-diffusion and promote the amoeboid motility phenotype over mesenchymal motility.', '1809.06726-4-6-5': 'In the case of flow, the MSD exponent [MATH], which reflects the dynamical slowing down of the spatiotemporal collective migration.', '1809.06726-4-6-6': 'The mesenchymal cell migration is more persistent than the amoeboid motility phenotype.', '1809.06726-4-7-0': 'The anomalous diffusion has been observed in many contexts from single to collective motions of particles[CITATION].', '1809.06726-4-7-1': 'Sancho et.al.', '1809.06726-4-7-2': '[CITATION] shows the anomalous diffusion of particles on a solid surface controlled by the friction coefficient.', '1809.06726-4-7-3': 'In the present model, we focus on the dynamics of cells in the low Reynolds number regime, where mechanical interactions exhibit collective phenomena characterized by the super-diffusive behavior of cells.', '1809.06726-4-7-4': 'We identify the interactions based on the experiment[CITATION] that exhibit collective behavior for the different motility phenotype.', '1809.06726-4-7-5': 'According to the experiment, the mesenchymal cells develop long-lived adhesion, we model the adhesion interaction time-dependent with a time scale of attachment [MATH].', '1809.06726-4-7-6': 'We show that how the time scale of attachment is reduced by flow-induced sweeping away the FN molecules and cells exhibit amoeboid motility phenotype.', '1809.06726-4-7-7': 'We write down the density field equation for the cells.', '1809.06726-4-7-8': 'We are interested in the dynamics of cells, for example, the mean square displacement (MSD) of a cell which is given by the relation MSD[MATH], with the dynamic exponent [MATH].', '1809.06726-4-7-9': 'The dynamics are out of equilibrium due to the self motility of cells and time-dependent interactions.', '1809.06726-4-7-10': 'We are interested in finite-time behavior.', '1809.06726-4-7-11': 'We use the Stochastic quantization technique to obtain the dynamic exponent, in which the correlation function is obtained in fictitious time [MATH].', '1809.06726-4-7-12': 'The real-time correlation is obtained by taking the asymptotic limit in the fictitious time direction.', '1809.06726-4-7-13': 'The scaling exponent is obtained by simple power counting analysis.', '1809.06726-4-8-0': 'Mesenchymal migration is characterized by an elongated fibroblast-like morphology, highly condensed cell-matrix adhesions, and formation of contractile actomyosin bundles[CITATION].', '1809.06726-4-8-1': 'The elongated cells demonstrated a mesenchymal phenotype where the actin filaments formed a highly polarized bundle.', '1809.06726-4-8-2': 'It is known that mesenchymal cells form long-lived adhesions with the ECM fiber bundles, which trigger the downstream signaling that activates actin remodeling and climbed along the fibers, and thus cell migration.', '1809.06726-4-8-3': 'Fibronectin (FN) is an important adhesion molecule in mediating mammalian cell migration.', '1809.06726-4-8-4': 'Cell-secreted FNs assemble into a fibrillar form and bind to collagen, which promotes a mesenchymal cell phenotype[CITATION].', '1809.06726-4-8-5': 'In the mesenchymal mode, cells wider than the matrix pore size extend protrusions at the leading edge to probe the surrounding ECM fibers, form stable adhesions at the poles of the elongated cell.', '1809.06726-4-8-6': 'Further polarization and strengthening of adhesions is accompanied by a rise in actomyosin contractility exerting traction force and proteolytic degradation of ECM fibers at cell-ECM junctions, and rupture the adhesions at the trailing edge of the cell, thus migrating in a ‘‘path-generating’’ manner[CITATION].', '1809.06726-4-9-0': 'Amoeboid migration is characterized by a rounded morphology, formation of bleb-like protrusions, restriction of actomyosin contractility to the cell cortex, and transient, short-lived adhesions with the ECM, and squeezing through the matrix pore when finding a suitable path[CITATION].', '1809.06726-4-9-1': 'The inhibition of proteolysis or integrin-dependent adhesion can be compensated by a weak or non-adhesive amoeboid mode of migration in which the cell adopts a rounded morphology and changes its shape by generating hydrostatic pressure at the cell cortex, thus forcibly extruding processes through available spaces in the porous matrix and eventually deforming the cell body.', '1809.06726-4-9-2': 'In the case of flow, the interstitial-flow-induced amoeboid cell motility was likely caused by the lack of assembled endogenous adhesion molecules such as FN.', '1809.06726-4-9-3': 'More specifically, the flows carried away the cell-secreted adhesion molecules before they were assembled into fibrils and anchored to the collagen fibers[CITATION].', '1809.06726-4-10-0': '# Theory', '1809.06726-4-11-0': 'In the absence of flow, cells exhibit mesenchymal motility phenotype in 3D collagen matrix.', '1809.06726-4-11-1': 'Cells secreted fibronectin molecules into a fibrillar form, and form long-lived adhesions with the collagen fibers, which trigger the downstream signaling that activates actin-network expansion and thus exhibit cell migration[CITATION].', '1809.06726-4-11-2': 'We consider the dynamics of a colony of cells in a dissipative environment where inertial effects are negligible.', '1809.06726-4-11-3': 'Each cell experiences systematic forces arising from mechanical interactions, and a Gaussian random force with a white noise spectrum.', '1809.06726-4-11-4': 'The equation of motion for a single mesenchymal cell [MATH] is [MATH] where [MATH] contains repulsive interactions with range [MATH], adhesion interactions with collagen matrix with range [MATH], and favorable attractive interactions between cells with range [MATH], and with strengths [MATH], [MATH] and [MATH] respectively.', '1809.06726-4-11-5': 'We use Gaussian potentials (see the appendix A1 for details) in order to obtain analytical solutions.', '1809.06726-4-11-6': 'Needless to say that the conclusions would be valid for any short-ranged [MATH].', '1809.06726-4-11-7': 'The highly deformable cells with flexible cell boundaries connected through cell-cell junctions, which can be ruptured and reconnected.', '1809.06726-4-11-8': 'Cell structural rearrangement for changing nearest-neighbor cells has been seen during the migration of cells in movie S1 in [CITATION].', '1809.06726-4-11-9': 'Therefore we model the cell-cell interaction as time-dependent potential.', '1809.06726-4-11-10': 'We assume that the adhesion strength is changing during the topological rearrangement via [MATH][CITATION].', '1809.06726-4-11-11': '[MATH] and [MATH] are initial and final interaction strengths ([MATH] stands for [MATH] and [MATH]) and the time scale for changing the receptor-ligand interaction is given by [MATH].', '1809.06726-4-11-12': 'In addition to short-range interactions, the cells that are well separated, interact with each other via the matrix.', '1809.06726-4-11-13': 'The cells in the collagen matrix can be modeled as force concentration dipoles.', '1809.06726-4-11-14': 'The cells interact via long-ranged elastic interaction potentials ([MATH] with distance [MATH])[CITATION].', '1809.06726-4-11-15': 'The interaction potentials depend on elastic constants, geometry, and cellular orientations.', '1809.06726-4-11-16': 'For simplicity of the calculation, we consider the interaction is isotropic and repulsive.', '1809.06726-4-11-17': 'The potential [MATH], where [MATH] is the magnitude of the force dipole, [MATH], and [MATH], where [MATH] and [MATH] are the Lam[MATH] coefficients of the isotropic elastic medium[CITATION].', '1809.06726-4-11-18': 'The Gaussian white noise, satisfies [MATH].', '1809.06726-4-11-19': 'The mesenchymal cells are subject to a self-generated force of actin-network remodeling.', '1809.06726-4-11-20': 'The cells are, thus in addition subject to a random self-generated force with amplitude [MATH].', '1809.06726-4-11-21': 'The randomness is modeled by an athermal noise [MATH], which is exponentially correlated over a time scale [MATH].', '1809.06726-4-11-22': 'The statistics of the [MATH] is given by [MATH], [MATH] and [MATH].', '1809.06726-4-11-23': 'Where [MATH] is the dimensionless constant.', '1809.06726-4-11-24': 'The athermal noise in general does not obey the FDT.', '1809.06726-4-12-0': 'For the mesenchymal phenotype, the cells form long-lived adhesion with the matrix and exhibit matrix mediated long-range elastic interactions when they are well separated.', '1809.06726-4-12-1': 'On the other hand, the amoeboid cells form short-lived adhesion with the matrix and migrate squeezing through the pore.', '1809.06726-4-12-2': 'Therefore, we assume amoeboid cells do not feel long-range elastic interactions in the long-time limit.', '1809.06726-4-13-0': 'In the absence of flow, cell-secreted FNs assemble into a fibrillar form and bind to collagen, which promotes a mesenchymal cell phenotype.', '1809.06726-4-13-1': 'In the case of flow, the interstitial-flow carried away the cell-secreted adhesion molecules before they were assembled into fibrils and anchored to the collagen fibers and thus amoeboid cell motility was likely caused by the lack of assembled endogenous adhesion molecules such as FN.', '1809.06726-4-13-2': 'Therefore, in the case of flow cells form short-lived adhesion with matrix and form rounded morphology and thus amoeboid motility[CITATION].', '1809.06726-4-13-3': 'The experiment in [CITATION] investigates the role of exogenous FN in cell morphology and motility in the absence/presence of the flows.', '1809.06726-4-13-4': 'In the presence of flow how the time scale of adhesion is reduced by sweeping away the cell-secreted adhesion molecules and therefore the cells exhibit amoeboid motility, can be understood in the following way.', '1809.06726-4-13-5': 'We write the concentration equation for the FN molecules which is advected by the flow, [MATH].', '1809.06726-4-13-6': 'Where [MATH] is the rate of FN molecules assembled into fibrilar form and the fluid velocity follows, [MATH], with Gaussian random noise [MATH].', '1809.06726-4-13-7': 'Assume the strength [MATH] depends on the concentration of FN molecules.', '1809.06726-4-13-8': 'We write [MATH].', '1809.06726-4-13-9': 'From the concentration equation for FN molecules and velocity equation, we obtain the average adhesion strength [MATH].', '1809.06726-4-13-10': 'The average adhesion strength can be written in the form [MATH], where, [MATH] and [MATH].', '1809.06726-4-13-11': 'Now the time scale of adhesion, i.e, [MATH] is reduced by factor of 2.', '1809.06726-4-13-12': 'The cell becomes shortlived and exhibit ameboid motility phenotype.', '1809.06726-4-13-13': 'In the presence of flow we use the time scale of adhesion is [MATH].', '1809.06726-4-13-14': 'The amoeboid cells migrate through squeezing the matrix pore when finding a suitable path.', '1809.06726-4-13-15': 'Therefore we consider the long-range hydrodynamic interactions between amoeboid cells leading to collective behavior of cell motility.', '1809.06726-4-13-16': 'We begin by considering the dynamics of a colony of cells in the presence of flow in a dissipative environment where inertial effects are negligible.', '1809.06726-4-13-17': 'Each cell experiences mechanical forces, such as adhesion, excluded volume interactions due to neighbors, and a random force characterized by Gaussian white noise.', '1809.06726-4-13-18': 'The equation of motion for single cell [MATH] is [CITATION] [EQUATION]', '1809.06726-4-13-19': 'The first term on the r.h.s. of Eq. ([REF]) is the effect of force acting on cell [MATH] creates a hydrodynamic flow-field in the fluid, thereby entraining cell [MATH].', '1809.06726-4-13-20': 'Where [MATH] contains repulsive interactions with range [MATH], adhesion interactions with collagen matrix with range [MATH], and favorable attractive interactions between cells with range [MATH], and with strengths [MATH], [MATH], and [MATH] respectively (see the appendix for details).', '1809.06726-4-14-0': '[MATH] is assumed to be Gaussian random vectors exhibiting hydrodynamic correlations according to fluctuation-dissipation theorem (FDT), [MATH].', '1809.06726-4-14-1': 'The dynamics of the cell is described by [MATH], where [MATH] is the position of the cell.', '1809.06726-4-14-2': 'The flow velocity of the fluid at [MATH] is determined by the solution of the Stokes equation, [MATH], where, [MATH].', '1809.06726-4-14-3': 'We assume that the fluid is incompressible, i.e., [MATH].', '1809.06726-4-14-4': 'From these set of equations, the Stokes equation is readily solved to obtain a fluid velocity at the center of the cell at [MATH] for a point force [MATH] as, u(r)=[MATH], where [MATH] is the mobility matrix with the relative coordinate between cell [MATH] and [MATH], [MATH].', '1809.06726-4-14-5': 'The long-ranged nature of hydrodynamic interaction apparent from [MATH] suggests that a collection of cells that dynamically exert forces on the fluid medium they are immersed in could influence each other very strongly, leading to the possibility of collective behaviors.', '1809.06726-4-14-6': 'The protrusive flowing of the anterior actin network of the cell and squeezing actomyosin contractions of the trailing edge are modeled as the cells are subjected to a self-generated force with amplitude [MATH] during their pathfinding motion in the collagen matrix.', '1809.06726-4-14-7': 'The statistics of the [MATH] is the same as for mesenchymal motility.', '1809.06726-4-15-0': 'We consider the evolution of the density function for a single cell [MATH].', '1809.06726-4-15-1': 'A closed form Langevin equation for the density, [MATH] can be obtained using standard approach [CITATION].', '1809.06726-4-15-2': 'The time evolution of [MATH] is given by, [EQUATION]', '1809.06726-4-15-3': 'Note that the density equation contains the same information as N-body stochastic Langevin equations.', '1809.06726-4-15-4': 'This is an out of equilibrium problem characterized by the absence of fluctuation-dissipation theorem due to long-range hydrodynamic term and self-generated force due to actin remodeling.', '1809.06726-4-15-5': 'Eq. ([REF]) can be studied analytically by treating the non-linear terms using a perturbative approach, based on the stochastic quantization scheme [CITATION].', '1809.06726-4-16-0': '## Stochastic quantization approach', '1809.06726-4-17-0': 'To understand the dynamics of collections of cells, we use the stochastic quantization method developed by Parisi-Wu in another context.', '1809.06726-4-17-1': 'The collective migration of cells described by Eq. ([REF]) is an out of equilibrium problem characterized by the absence of FDT, which relates the correlation and response function in momentum space as [MATH].', '1809.06726-4-17-2': 'The usual analytic route to get the scaling solution of this problem, one can introduce a response field [MATH].', '1809.06726-4-17-3': 'We need to calculate both the response function ([MATH]) and correlation function ([MATH]) because of the absence of a fluctuation-dissipation relation.', '1809.06726-4-17-4': 'The key advantage of the present method is that we do not need to calculate both the correlation and response functions.', '1809.06726-4-17-5': 'The FDT is constructed in fictitious time introduced in the problem.', '1809.06726-4-17-6': 'The FDT relation enables us to obtain the scaling of the correlation function, once the scaling of the response function is known.', '1809.06726-4-17-7': 'By taking the infinite limit in fictitious time, one can obtain the correlation function in real-time.', '1809.06726-4-17-8': 'The scaling solution of the problem can be obtained by power counting analysis instead of doing renormalization group calculation.', '1809.06726-4-18-0': ""We now exploit the Parisi-Wu stochastic quantization scheme [CITATION], and introduce a fictitious time [MATH]', and consider all variables to be functions of [MATH] in addition to k and [MATH]."", '1809.06726-4-18-1': ""A Langevin equation in [MATH]' space is, [EQUATION] with [MATH]."", '1809.06726-4-18-2': 'This ensures that as [MATH], the distribution function will be given by the action [MATH], because in the [MATH]-space a fluctuation dissipation theorem (FDT) is preserved.', '1809.06726-4-18-3': 'The action [MATH] can be obtained by writing down the probability distribution [MATH] corresponding to the noise term [MATH], and the action [MATH] in terms of [MATH] with the help of Eq. ([REF]), where [MATH].', '1809.06726-4-18-4': 'The expression for the [MATH] is in the appendix.', '1809.06726-4-19-0': 'We follow the procedure of obtaining scaling laws of the problem, which has been demonstrated in earlier works [CITATION].', '1809.06726-4-19-1': 'The dynamics of Eq. ([REF]) requires only the calculation of response functions, the correlation functions in this dynamics are related to response function through FDT relation, i.e., in Fourier space, [MATH].', '1809.06726-4-19-2': 'We can obtain the scaling laws in real space and time in a straight forward fashion from the solution in the fictitious time [MATH] space.', '1809.06726-4-20-0': 'We obtain the following self-consistent equation for the self-energy from the calculation of response function using Eq.([REF]): [EQUATION] where, [MATH], [MATH], and [MATH] is the self-energy term, a two-loop contribution from the first order term ( containing two [MATH] fields) in Eq. ([REF]) (first term in Fig. [REF]) will contribute in the scaling laws for the cell in the finite time.', '1809.06726-4-20-1': 'We use Eq.([REF]) for getting the scaling laws of both the amoeboid and mesenchymal cells phenotype.', '1809.06726-4-21-0': '## Mesenchymal motility', '1809.06726-4-22-0': 'The mesenchymal cell phenotype forms long-lived integrin-based adhesions with the collagen matrix and migrate via either remodeling of actin network or degrading the matrix.', '1809.06726-4-22-1': 'The non-linear term i.e. the adhesion interaction plays an important role in the complex dynamics of collective migration of mesenchymal cell phenotype.', '1809.06726-4-22-2': 'In a self consistent mode coupling theory, we now replace [MATH] by [MATH] in the self energy term [MATH] in the first term in Fig. ([REF]), use [MATH] as from Eq. ([REF]), and [MATH], which follows from the FDT.', '1809.06726-4-22-3': 'In the absence of flow, [MATH].', '1809.06726-4-22-4': 'According to scale transformation, we know that [MATH], [MATH], [MATH], [MATH] and the vertex factor [MATH].', '1809.06726-4-22-5': 'The self energy term in Fig. ([REF]) can be written as [MATH].', '1809.06726-4-22-6': 'By carrying out the momentum count of [MATH], and using [MATH], we find that [MATH].', '1809.06726-4-22-7': 'Using Eq. ([REF]) and [MATH], we have [MATH], which leads to [MATH].', '1809.06726-4-22-8': 'MSD exponent [MATH].', '1809.06726-4-22-9': 'In 3D, [MATH], i.e., the mesenchymal cells undergo super-diffusion.', '1809.06726-4-22-10': 'The non-linear term arising from cell-cell adhesion that determines the dynamical shape change during cell motion and self-generated force, produce super-diffusive motion.', '1809.06726-4-22-11': 'The theoretical result is in good agreement with the recent experimental result ([MATH]) using the microfluidic model[CITATION].', '1809.06726-4-22-12': 'If we wait longer than experimental observation time, the cells become apart from each other.', '1809.06726-4-22-13': 'The short-range interactions do not play any role in cell motility.', '1809.06726-4-22-14': 'The long-range elastic interactions dictate the dynamics of the mesenchymal phenotype.', '1809.06726-4-22-15': 'The similar scaling analysis shows the super-diffusive behavior of cells characterized by the single-cell MSD exponent [MATH].', '1809.06726-4-22-16': 'The degree of superdiffusion decreases in the long-time where long-range elastic interactions play a role in cell motility.', '1809.06726-4-23-0': 'Interstitial flow induced amoeboid motility In the presence of interstitial flow, cells exhibit amoeboid motility.', '1809.06726-4-23-1': 'The amoeboid cells form short-lived adhesion with the collagen matrix.', '1809.06726-4-23-2': 'The time scale [MATH] is small compared to the mesenchymal motility phenotype.', '1809.06726-4-23-3': 'In the case of flow, the cells exhibit long-range hydrodynamic interactions that determine the complex spatiotemporal dynamics of amoeboid cell phenotype.', '1809.06726-4-23-4': 'The self-generating force of actin remodeling helps to propel in pathfinding fashion through the collagen matrix.', '1809.06726-4-23-5': 'In a self consistent mode coupling theory, we now replace [MATH] by [MATH] in the self energy term [MATH] in the first term in Fig. ([REF]), use [MATH] as from Eq. ([REF]) and [MATH], which follows from the FDT.', '1809.06726-4-23-6': 'According to scale transformation, we know that [MATH], [MATH], [MATH], [MATH] and the vertex factor [MATH].', '1809.06726-4-23-7': 'The self energy term in Fig. ([REF]) can be written as [MATH].', '1809.06726-4-23-8': 'By carrying out the momentum count of [MATH], and using [MATH], we find that [MATH].', '1809.06726-4-23-9': 'Using Eq. ([REF]) and [MATH], we have [MATH], which leads to [MATH].', '1809.06726-4-23-10': 'MSD exponent [MATH].', '1809.06726-4-23-11': 'In 3D, [MATH], i.e., the amoeboid cells undergo superdiffusion.', '1809.06726-4-23-12': 'The non-linear term due to long-range hydrodynamic interactions among cells besides self-generated force, produce super-diffusive motion for amoeboid cell phenotype.', '1809.06726-4-23-13': 'The decrease of MSD exponent determines the dynamical slowing down of initial collective migration of the mesenchymal cell phenotype.', '1809.06726-4-23-14': ""Therefore, interstitial flows impair the cell's ability to spread by sweeping away the adhesion molecules with the flow and making the cells as amoeboid phenotype with short-lived adhesion with the collagen fiber[CITATION]."", '1809.06726-4-23-15': 'The cells migrate via squeezing through the pore of the collagen fiber when finding a suitable path[CITATION].', '1809.06726-4-23-16': 'The theoretical result is in good agreement with the recent experimental result ([MATH]) using the microfluidic model[CITATION].', '1809.06726-4-23-17': 'If we wait longer than experimental observation time, the cells become apart from each other.', '1809.06726-4-23-18': 'The short-range interactions do not play any role in cell motility.', '1809.06726-4-23-19': 'The cells undergo normal diffusion for the amoeboid motility phenotype characterized by the single-cell MSD exponent [MATH].', '1809.06726-4-24-0': 'We calculate velocity auto-correlation function [MATH], and time-dependent diffusion coefficient, [MATH], where [MATH] is the single cell MSD.', '1809.06726-4-24-1': 'The blue line dictates the mesenchymal motility and red line shows the amoeboid motility.', '1809.06726-4-24-2': 'The slower decay of velocity correlation function shows that the mesenchymal cells are more persistent than amoeboid cells.', '1809.06726-4-24-3': 'Similarly, the time-dependent diffusion coefficient for a mesenchymal cell is higher than the amoeboid cells, i.e, the mesenchymal cell is more diffusive than amoeboid cells.', '1809.06726-4-24-4': 'The theoretical result is in agreement with the experiment[CITATION].', '1809.06726-4-25-0': 'The MSD for a single cell in the experiment in [CITATION] shows the super-diffusion with the exponent [MATH] for both with and without the flow.', '1809.06726-4-25-1': 'Average for MSD calculation has been taken over all cell trajectories (approximate over 60 motile cells).', '1809.06726-4-25-2': 'The movies (S1 and S2) in [CITATION] show that cells are interacting through adhesion interactions that influence each other very strongly leading to the possibility of collective behaviors.', '1809.06726-4-25-3': 'That implies that the super-diffusion arises due to the collective motion of interacting cells for both with and without flow.', '1809.06726-4-25-4': 'Although it is a single-cell measurement, the super-diffusive motion of a cell arises due to the collective migration of interacting cells.', '1809.06726-4-26-0': 'In theory, the flow effect that we model through long-range hydrodynamic interactions among cells.', '1809.06726-4-26-1': 'That induces the super-diffusion for motile amoeboid cells.', '1809.06726-4-26-2': 'On the other hand, the time-dependent short-range interactions with the coupling with self-motility produce super-diffusive motion in mesenchymal cells.', '1809.06726-4-26-3': 'The single-cell MSD shows super-diffusion.', '1809.06726-4-26-4': 'The time-dependent short-range interactions and long-range hydrodynamic interactions between cells influence each other very strongly and produce non-linear terms in the density equations, leading to the possibility of novel collective behaviors.', '1809.06726-4-26-5': 'Therefore the super-diffusion effect is a collective phenomenon.', '1809.06726-4-27-0': '# Conclusion', '1809.06726-4-28-0': 'In the present contribution, using a new theoretical framework, we provide insight into the dynamics of a colony of cancer cells driven by biophysical cues.', '1809.06726-4-28-1': 'The theory reveals that the interstitial flows regulate cancer cell morphology and motility phenotypes, emphasizing the role of fluid flows in regulating cancer cell migration heterogeneity.', '1809.06726-4-28-2': 'The conventional practice in dealing with this out of equilibrium problems is to use a set of fictitious fields called response fields, which provide a field-theoretic prescription for the response function.', '1809.06726-4-28-3': 'In contrast, we propose the introduction of a fictitious time in which an FDT is valid, and thereby only correlation functions need to be calculated.', '1809.06726-4-28-4': 'Our approach greatly simplifies the evaluation of scaling exponents.', '1809.06726-4-28-5': 'We find that the non-linear term in the density evolution equation arising from mechanical interactions along with self-generating force due to actin remodeling determines the scaling behavior for the collective migration of cells.', '1809.06726-4-28-6': 'In the absence of flow, cells exhibit collective migration of mesenchymal motility phenotype induced by time-dependent interaction potentials that determine the structural rearrangement during their path generating migration through the collagen matrix.', '1809.06726-4-28-7': 'In contrast, the cells exhibit the amoeboid motility in the presence of flow and exhibit a dynamical slowing down of directed migration, with a gradually decreasing degree of super-diffusion.', '1809.06726-4-28-8': 'The long-range hydrodynamic interactions among cells in the presence of interstitial flow determine the collective migration of cells through the matrix pore in a pathfinding fashion.', '1809.06726-4-28-9': 'The mesenchymal cells are more persistent and diffusive compared to amoeboid cells.', '1809.06726-4-28-10': 'The theoretical framework introduced here provides evidence of interstitial flow directed collective motion heterogeneity and could explain the invasion of cancer cells under interstitial flow, observed in a recent experiment[CITATION].', '1809.06726-4-28-11': 'The theory introduced here could help us understand how cancer cells spread by invading adjacent tissues involved in metastasis [CITATION].', '1809.06726-4-29-0': '# acknowledgements', '1809.06726-4-30-0': 'I wish to express my gratitude to Prof. D. Thirumalai for supporting me as a postdoc at the department of chemistry at UT, Austin.', '1809.06726-4-30-1': 'I am indebted to Sumit Sinha and Dr. Debayan Chakraborty for valuable comments.', '1809.06726-4-31-0': '# Short-range interaction', '1809.06726-4-32-0': 'To obtain the dynamics of an evolving collection of cells, we use the following simplified form for cell-cell interaction, [EQUATION] where [MATH] and [MATH] are the strengths of excluded volume and attractive interactions, respectively.', '1809.06726-4-33-0': 'In addition, the cell surface-ECM interactions [MATH], determining the configuration-dependent forces experienced by the cells: [MATH].', '1809.06726-4-33-1': 'The potential term [MATH] describes the cell surface-collagen interactions as a function of [MATH], the average distance between cell and collagen.', '1809.06726-4-33-2': 'Cells interact with the collagen through receptor-ligand interactions, described by short range potential.', '1809.06726-4-33-3': 'Mesenchymal cells form long-lived adhesion with collagen.', '1809.06726-4-33-4': 'We assume the adhesion strength is changing during the topological rearrangement via, [MATH].', '1809.06726-4-33-5': '[MATH] and [MATH] are initial and final interaction strengths and the time scale of changing the receptor-ligand interaction is given by [MATH].', '1809.06726-4-33-6': 'Where [MATH] stands for [MATH] and [MATH].', '1809.06726-4-33-7': 'In contrast, amoeboid cells form short lived adhesion with the collagen, i.e., the time scale for the adhesive interaction [MATH], is small compared to mesenchymal cells.', '1809.06726-4-34-0': '# Density equation', '1809.06726-4-35-0': 'To simplify the multiplicative noise term (last term in Eq. (2) in the main text), we assume that the density fluctuates around a constant value.', '1809.06726-4-35-1': 'Hence, we define the density using [MATH], and expand Eq. (1) in the main text in [MATH] up to the lowest order in non-linearity.', '1809.06726-4-35-2': 'In Fourier space, the equation for the density fluctuation becomes, [EQUATION] with [MATH].', '1809.06726-4-35-3': 'Where, [MATH].', '1809.06726-4-35-4': 'The Greens function [MATH] is given by, [EQUATION] where, [MATH], and [MATH] is the self energy term contributed from non-linear adhesion interactions.', '1809.06726-4-36-0': ""# Stochastic quantization technique We now exploit the Parisi-Wu stochastic quantization scheme [CITATION], and introduce a fictitious time [MATH]', and consider all the variables to be functions of [MATH]."", '1809.06726-4-36-1': ""A Langevin equation in [MATH]' space is, [EQUATION] with [MATH]."", '1809.06726-4-36-2': 'Because FDT is valid in the fictitious time it follows that as [MATH], the distribution function will be given by the action [MATH].', '1809.06726-4-36-3': 'The action [MATH] can be obtained by writing down the probability distribution [MATH] corresponding to the noise term [MATH] in Eq. ([REF]), and the action [MATH] in terms of [MATH] using Eq. ([REF]).', '1809.06726-4-36-4': 'The expression for the action [MATH] obtained using Eq. ([REF]) is,', '1809.06726-4-37-0': '[MATH]', '1809.06726-4-38-0': '# Greens function for density equation', '1809.06726-4-39-0': 'The correlation functions, calculated from Eq. ([REF]), lead to the required correlation functions of the original theory, in the [MATH] limit.', '1809.06726-4-39-1': 'In order to obtain the scaling laws, it suffices to work at arbitrary [MATH].', '1809.06726-4-39-2': 'It is obvious from Eq. ([REF]) that in the absence of the non-linear terms, the Greens function [MATH] is given by, [EQUATION] where [MATH] is the frequency corresponding to the fictitious time [MATH].', '1809.06726-4-39-3': ""As is customary, the effect of non-linear terms, can be included perturbatively leading to the Dyson's' equation [EQUATION]"", '1809.06726-4-39-4': 'Here, we are concerned with the behavior of [MATH], which becomes non-linear when expanded to second order.', '1809.06726-4-39-5': 'We note that the contribution comes from two different sources (1) a one-loop contribution from the second order term (containing three [MATH] fields) in Eq. ([REF]) (second term in Fig. [REF]) and (2) a two-loop contribution from the first order term ( containing two [MATH] fields) in Eq. ([REF]) (first term in Fig. [REF]).', '1809.06726-4-39-6': 'The contribution arising coming from the term containing three [MATH] fields, in Eq. ([REF]) can be readily obtained by contracting two of the [MATH] fields.', '1809.06726-4-39-7': 'The second order term coming from the one loop contribution in Eq. ([REF]) does not have any new momentum dependance.', '1809.06726-4-39-8': 'Hence it is the second-order contribution (first term in Fig. ([REF])), coming from the two-loop contribution in Eq. ([REF]), which is significant.', '1809.06726-4-39-9': 'The correlation function is given by the FDT as [MATH].', '1809.06726-4-39-10': 'With these observations, Eq. ([REF]) can be written as, [EQUATION] where [MATH], and [MATH] is defined by [EQUATION] with [MATH].', '1809.06726-4-39-11': 'Expanding [MATH], [MATH] about [MATH] and [MATH], respectively, and noting that the renormalization of [MATH] dominates, we get [EQUATION]', '1809.06726-4-39-12': 'Collagen gels are used extensively for studying cell-matrix mechanical interactions.', '1809.06726-4-39-13': 'The cells are interacting with ECM.', '1809.06726-4-39-14': 'The Reynolds number is small (an estimate: density [MATH], viscosity [MATH], L=[MATH], velocity [MATH], the Reynolds number [MATH]), an overdamped approximation is appropriate implying that the neglect of the inertial term [MATH] is justified.', '1809.06726-4-39-15': 'Since additional adhesive forces are also present, cell movement is further damped.', '1809.06726-4-39-16': 'The friction is high (an estimate:[MATH], radius R=[MATH], then friction [MATH]), compared to water ([MATH], for viscosity [MATH]).', '1809.06726-4-39-17': 'By changing the viscosity we can change the friction and hence the diffusion coefficient ([MATH]).', '1809.06726-4-39-18': 'The cells are self motile.', '1809.06726-4-39-19': 'The estimate for the diffusion due to self motility: [MATH] (where the values of [MATH] and [MATH] are taken from the fig(1F) in [CITATION]) is high compared to brownian diffusion [MATH] in the matrix.', '1809.06726-4-39-20': 'The Brownian diffusion in water is [MATH].', '1809.06726-4-39-21': 'The viscosity of gel is greater than the water.', '1809.06726-4-39-22': 'Therefore the diffusion due to self motility of cell is always greater than the Brownian diffusion.', '1809.06726-4-39-23': 'The scaling is determined by the equation (4).', '1809.06726-4-39-24': 'In the numerator, [MATH] term is dominated by self-motility term [MATH] by following the above argument.', '1809.06726-4-39-25': 'The resulting superdiffusion exponents are reported in the main text.', '1809.06726-4-39-26': 'The non-linear contributions are included in the self-energy term [MATH].', '1809.06726-4-39-27': 'The non-linear interactions determine the dynamical behavior.', '1809.06726-4-39-28': 'The time-dependent adhesion interactions and long-range elastic interactions determine the superdiffusive behavior for the mesenchymal phenotype.', '1809.06726-4-39-29': 'On the other hand, long-range hydrodynamic interactions determine superdiffusive behavior.', '1809.06726-4-39-30': 'We can tune the strength of the interactions that will change the magnitude of the coefficient of the non-linear term.', '1809.06726-4-39-31': 'The scaling of the non-linear term remains unchanged.', '1809.06726-4-39-32': 'Therefore the degree of superdiffusion is unchanged.', '1809.06726-4-40-0': '# Scaling exponents', '1809.06726-4-41-0': 'In a self consistent mode coupling theory, we now replace [MATH] by [MATH] in the self energy term [MATH] in the first term in Fig. ([REF]), use [MATH] as from Eq. ([REF]) and [MATH], which follows from the FDT.', '1809.06726-4-41-1': 'According to scale transformation, we know that [MATH], [MATH], [MATH], [MATH] and the vertex factor [MATH].', '1809.06726-4-41-2': 'The self energy term in Fig. ([REF]) can be written as [MATH].', '1809.06726-4-41-3': 'By carrying out the momentum count of [MATH], and using [MATH], we find that [MATH].', '1809.06726-4-41-4': 'Using Eq. ([REF]) and [MATH], we have [MATH], which leads to [MATH].', '1809.06726-4-41-5': 'Where, [MATH] and use [MATH] is small because in the case of flow, cells are amoeboid phenotype with short lived adhesion with collagen fiber.', '1809.06726-4-41-6': 'MSD exponent [MATH].', '1809.06726-4-41-7': 'In 3D, [MATH], i.e., the amoeboid cells undergo super-diffusion.', '1809.06726-4-42-0': '# The expression for [MATH]', '1809.06726-4-43-0': '[EQUATION] where veterx term, [MATH]'}",
cond-mat-9809115,"{'cond-mat-9809115-1-0-0': 'Using large scale Monte Carlo simulations on a uniformly frustrated 3D-XY model, we report a first order vortex crystal melting transition in clean, isotropic extreme type-II ([MATH]) superconductors.', 'cond-mat-9809115-1-0-1': 'This work clarifies an important issue: the unpinned vortex liquid is always incoherent with no phase coherence in any directions for all anisotropies.', 'cond-mat-9809115-1-0-2': 'Previous claims of a disentangled vortex liquid for isotropic superconductors based on simulations, are due to finite size effects.', 'cond-mat-9809115-1-1-0': '2', 'cond-mat-9809115-1-2-0': '# Introduction', 'cond-mat-9809115-1-3-0': 'Since the discovery of copper oxide based high temperature superconductors (HTSC)[CITATION], which are extreme type-II, there has been great interest in their phenomenological phase diagrams.', 'cond-mat-9809115-1-3-1': ""Abrikosov's mean-field description[CITATION], which is valid for conventional low temperature superconductors, is expected to be modified by the strong thermal fluctuations in HTSC."", 'cond-mat-9809115-1-3-2': 'Extensive research in both theory, numerical simulations, and experiments over the years have resulted in a general consensus on some of the fundamental issues.', 'cond-mat-9809115-1-3-3': 'The current understanding of HTSC in a uniform magnetic field [MATH] is as follows: the low temperature Abrikosov phase melts into a vortex liquid via a first order transition at the temperature [MATH].', 'cond-mat-9809115-1-3-4': 'Above [MATH], the mean-field transition [MATH] is now renormalized by the thermal fluctuations into a large crossover region which marks the onset of strong local diamagnetic fluctuations.', 'cond-mat-9809115-1-3-5': 'The Abrikosov vortex crystal phase is characterized by a transverse triangular crystalline order and a finite longitudinal phase coherence.', 'cond-mat-9809115-1-4-0': 'However, recently there has been some debate about the nature of the vortex liquid which the Abrikosov crystal melts into as temperature increases.', 'cond-mat-9809115-1-4-1': 'Numerous simulations using the 3D XY model[CITATION], London[CITATION] and Lowest Landau Level approximation[CITATION] have indicated that the melted vortex liquid is incoherent, i.e. phase coherence in any direction is zero.', 'cond-mat-9809115-1-4-2': 'The crystalline order and phase coherence are destroyed simultaneously at the melting.', 'cond-mat-9809115-1-4-3': 'This scenario has been supported by experiments on high quality YBCO crystals[CITATION].', 'cond-mat-9809115-1-4-4': 'Other simulations using the 3D-XY model [CITATION] have suggested that the longitudinal phase coherence persists above the melting transition and only vanishes at a higher ""entanglement"" temperature [MATH].', 'cond-mat-9809115-1-4-5': 'In this scenario, the vortex liquid at [MATH] would be the disentangled one with relatively straight vortex lines.', 'cond-mat-9809115-1-4-6': 'For [MATH], it was claimed that larger thermal fluctuations caused the vortex lines to entangled with a concomitant loss of global phase coherence[CITATION].', 'cond-mat-9809115-1-5-0': 'In previous simulations, a disentangled vortex liquid regime was found in simulations on an isotropic system[CITATION].', 'cond-mat-9809115-1-5-1': 'Recently, Ryu and Stroud[CITATION] have proposed that for large [MATH], [MATH] and [MATH] merge into a single transition whereas for small [MATH] they are well separated.', 'cond-mat-9809115-1-5-2': 'The authors found that for an isotropic system, the crossover between the two regimes is at a filling fraction [MATH].', 'cond-mat-9809115-1-5-3': '(The precise definition of [MATH] will be given shortly.)', 'cond-mat-9809115-1-5-4': 'However, these simulations were performed on rather small systems.', 'cond-mat-9809115-1-5-5': 'In the present work, we have performed similar simulations on a much larger system, and found an exclusively incoherent vortex liquid down to [MATH].', 'cond-mat-9809115-1-5-6': 'This implies that the only thermodynamic vortex liquid is an incoherent one, with full translational and rotational symmetries and zero phase coherence in all directions.', 'cond-mat-9809115-1-5-7': ""It is therefore clear that the recently discussed disentangled vortex liquid and Ryu and Stroud's crossover between the disentangled and incoherent vortex liquids are numerical artifacts."", 'cond-mat-9809115-1-6-0': 'The organization of this paper is as follows: we first briefly review the model used in the simulations and then discuss our results and their implications.', 'cond-mat-9809115-1-6-1': 'Finally, we provide a summary and conclusion.', 'cond-mat-9809115-1-7-0': '# Model', 'cond-mat-9809115-1-8-0': 'We use a uniformly frustrated 3D-XY model on a cubic lattice to describe an isotropic, extreme type-II superconductors in a magnetic field.', 'cond-mat-9809115-1-8-1': 'The London model for superconductors can be readily derived from the phenomenological Ginzburg-Landau (GL) model with the approximation that the amplitude of the local complex order parameter [MATH], is fixed.', 'cond-mat-9809115-1-8-2': 'The Hamiltonian [MATH] of the London model consists of degrees of freedom in the phase [MATH] and the gauge vector potential [MATH] associated with the magnetic induction [MATH] inside the system, i.e. [MATH].', 'cond-mat-9809115-1-9-0': 'For an isotropic extreme type-II superconductors, the penetration depth [MATH] is much larger than the coherence length [MATH] such that the GL parameter [MATH].', 'cond-mat-9809115-1-9-1': 'This means that the magnetic fields from the vortex lines strongly overlap with one another giving a spatially smooth [MATH].', 'cond-mat-9809115-1-9-2': '(This condition is ensured in the regime where [MATH].)', 'cond-mat-9809115-1-9-3': 'In other words, the fluctuations of [MATH] on the length scale of [MATH] is negligible compared to the fluctuations of [MATH].', 'cond-mat-9809115-1-9-4': 'Therefore, we can further simplify the problem by dropping the degrees of freedom of [MATH] from the Hamiltonian [MATH], and fix [MATH] equal to the external applied magnetic field.', 'cond-mat-9809115-1-9-5': 'The resulting Hamiltonian [MATH] is the 3D-XY model.', 'cond-mat-9809115-1-9-6': 'This [MATH] (or frozen gauge) approximation has been widely used in phenomenological models for superconductors.', 'cond-mat-9809115-1-10-0': 'In order to perform numerical simulations on the resulting model, we discretize the model on a 3D cubic lattice with grid spacing [MATH].', 'cond-mat-9809115-1-10-1': 'The dimensionless Hamiltonian of this lattice model is given by[CITATION] [EQUATION] where [MATH] is the isotropic coupling energy, [MATH] is the flux quanta and [MATH] is the permeability of a vacuum in SI units.', 'cond-mat-9809115-1-10-2': 'The dimensionless vector [MATH] labels the position of an arbitrary grid point.', 'cond-mat-9809115-1-10-3': 'We defined the gauge-invariant phase difference [MATH] as [EQUATION] where [MATH] is the unit vector along the [MATH]-axis respectively.', 'cond-mat-9809115-1-10-4': 'The convention is that [MATH] ""flows"" from the grid at [MATH] to [MATH].', 'cond-mat-9809115-1-10-5': 'In this model, it is natural to define a dimensionless temperature [MATH].', 'cond-mat-9809115-1-10-6': 'The magnetic induction [MATH] is conveniently represented by the filling fraction [EQUATION]', 'cond-mat-9809115-1-10-7': 'In this paper, we perform a Monte Carlo simulation on an isotropic system with [MATH] number of grid points.', 'cond-mat-9809115-1-10-8': 'The aim in our numerical simulations is to identify possible thermodynamic phases (i.e. vortex crystal or liquid, etc.) and phase transitions (first order or continuous) or crossover effects associated with them.', 'cond-mat-9809115-1-10-9': 'More specifically, we are interested in calculating the internal energy, specific heat, structure factor and helicity moduli.', 'cond-mat-9809115-1-10-10': 'These thermodynamic quantities and their physical significance will be discussed next.', 'cond-mat-9809115-1-11-0': 'A first order transition is indicated by a [MATH]-function anomaly in the specific heat, equivalently a discontinuous jump in the internal energy.', 'cond-mat-9809115-1-11-1': 'On the order hand, a continuous transition has a signature of a jump in the specific heat and a continuous internal energy.', 'cond-mat-9809115-1-11-2': 'The internal energy per site is obtained by averaging [MATH] in Eq. ([MATH]) over the thermal equilibrium states, normalized by the total number of grids, i.e. [MATH].', 'cond-mat-9809115-1-11-3': 'We define a dimensionless specific heat per site [MATH] using the standard fluctuation theorem of a classical system with Gibbs distribution[CITATION]: [EQUATION]', 'cond-mat-9809115-1-11-4': 'A convenient and widely used quantity to probe the global phase coherence of the system is the helicity modulus, which is proportional to the second derivative of the free energy associated with Eq. ([MATH]) with respect to an infinitesimal phase twist[CITATION].', 'cond-mat-9809115-1-11-5': 'On a square lattice, the dimensionless helicity modulus [MATH] can be written as [EQUATION] [MATH] measures the ""stiffness"" of [MATH] with respect to twisting along the [MATH]-direction.', 'cond-mat-9809115-1-11-6': 'If [MATH] then [MATH] is stiff in the [MATH]-direction, or more precisely, there is global phase coherence or superconducting response in that direction.', 'cond-mat-9809115-1-11-7': 'By tracking the temperature at which [MATH] goes to zero, one can determine the superconducting-normal metal transition temperature of the system.', 'cond-mat-9809115-1-11-8': 'In the mixed phase [MATH] and [MATH], where the subscripts [MATH] and [MATH] denote the directions longitudinal and transverse to [MATH], respectively.', 'cond-mat-9809115-1-12-0': 'The structure factor probes the transverse crystalline order of the vortex phase.', 'cond-mat-9809115-1-12-1': 'We adopt a conventional definition [CITATION]: [EQUATION] where [MATH] is a two-dimensional reciprocal vector and [MATH] is the local vorticity measured on the dual square lattice grid [CITATION](which is composed of the centers of every direct unit cell).', 'cond-mat-9809115-1-12-2': 'A crystalline ordered phase is characterized by [MATH] (or Bragg peaks) where [MATH] are the discrete set of reciprocal lattice vectors associated with the crystal structure.', 'cond-mat-9809115-1-12-3': 'On the other hand, [MATH] for a phase with full rotational invariance exhibits ring patterns.', 'cond-mat-9809115-1-13-0': 'At a fixed temperature, the equilibrium configurations are generated by making random changes to [MATH] at each grid subjected to the Metropolis algorithm.', 'cond-mat-9809115-1-13-1': '(This is equivalent to randomly changing all the six [MATH] attached to each grid points.)', 'cond-mat-9809115-1-13-2': 'To ensure conservations of vorticity in each unit cell, [MATH] has to be in the range [MATH].', 'cond-mat-9809115-1-13-3': 'Addition of [MATH] shall be administrated to bring [MATH] back into range at every Monte Carlo step.', 'cond-mat-9809115-1-13-4': 'This procedure introduces into the system vortex-loops, which are the elementary topological excitation of the model.', 'cond-mat-9809115-1-13-5': 'Periodic boundary condition is imposed on [MATH] such that [MATH].', 'cond-mat-9809115-1-13-6': 'In terms of simulations, the thermal average of a quantity [MATH] is evaluated by sampling and averaging [MATH] over many different equilibrium configurations.', 'cond-mat-9809115-1-13-7': 'More detail about the numerical procedure may be found in Refs. [CITATION].', 'cond-mat-9809115-1-14-0': '# Result and Discussion', 'cond-mat-9809115-1-15-0': 'In this section, we discuss the results of our simulations on an isotropic system with size of [MATH] grid points, which is the largest to date on an isolated system.', 'cond-mat-9809115-1-15-1': '(For [MATH], the system has 720 vortices.)', 'cond-mat-9809115-1-15-2': 'The system is subdivided into multiple sections and Monte Carlo procedure is implemented in parallel across the sections using 3D ""black and white"" decomposition[CITATION].', 'cond-mat-9809115-1-15-3': 'The filling fraction considered are [MATH]=20,40 and 60.', 'cond-mat-9809115-1-15-4': 'The system is cooled from high temperatures above the crossover [MATH].', 'cond-mat-9809115-1-15-5': 'For each temperature, a typical run consists of 120000 Monte Carlo sweeps across the whole lattice, 1/3 of that were used for equilibration.', 'cond-mat-9809115-1-15-6': 'Near the transition, up to 600000 sweeps were used.', 'cond-mat-9809115-1-16-0': 'The results for the quantities [MATH], [MATH] and [MATH] in the case of [MATH] is shown in Fig. ([MATH]).', 'cond-mat-9809115-1-16-1': 'Similar qualitative features in this plot are also found in the case of [MATH].', 'cond-mat-9809115-1-16-2': 'First of all, we found that [MATH] exhibits six-fold Bragg peaks at low temperatures (not shown).', 'cond-mat-9809115-1-16-3': 'This indicates that the low temperature phase is a triangular vortex crystal phase.', 'cond-mat-9809115-1-16-4': 'The destruction of this vortex crystal structure is marked by a melting transition at [MATH] where [MATH] drops sharply to zero ([MATH] is the wavevector of one of the six first order Bragg peaks).', 'cond-mat-9809115-1-16-5': 'The sharpness of the drop in [MATH] strongly suggests that the transition is first order.', 'cond-mat-9809115-1-16-6': 'This is confirmed by the appearance of a [MATH]-function like peak in [MATH] at the same temperature [MATH].', 'cond-mat-9809115-1-16-7': 'Coincidentally, [MATH] which is finite at low temperatures, also drops sharply to zero at [MATH].', 'cond-mat-9809115-1-16-8': 'So the system exhibits longitudinal superconductivity below [MATH], but not above.', 'cond-mat-9809115-1-17-0': 'Based on this evidence, one can conclude that a first order melting transition of the Abrikosov vortex crystal exists in an isotropic system.', 'cond-mat-9809115-1-17-1': 'The entire vortex liquid phase is incoherent, i.e. a vortex liquid phase with no longitudinal superconductivity.', 'cond-mat-9809115-1-17-2': 'Note that the same conclusion has been reached in earlier simulations on a moderately anisotropic system[CITATION].', 'cond-mat-9809115-1-17-3': 'There has been earlier report of disentangled vortex liquid, i.e. vortex liquid with longitudinal phases coherence using similar simulation[CITATION].', 'cond-mat-9809115-1-17-4': 'Ryu and Stroud[CITATION] have argued that the disentangled vortex liquid should be most easily observed in isotropic system for [MATH].', 'cond-mat-9809115-1-17-5': 'However, these simulations were on small system.', 'cond-mat-9809115-1-17-6': '(Typically no more than [MATH].)', 'cond-mat-9809115-1-17-7': 'In this work, our results are based on much larger systems ([MATH]).', 'cond-mat-9809115-1-17-8': 'We have observed a slight difference in the temperatures at which [MATH] and [MATH] vanish in simulations on smaller system or lower number of sweeps.', 'cond-mat-9809115-1-17-9': 'Therefore, it may be concluded that the existence of disentangled vortex liquid is a numerical artifact of small system size and/or insufficient simulation time.', 'cond-mat-9809115-1-17-10': 'Our results also do not support the existence of a so called ""supersolid"", which has a no global phase coherence but a substantial degrees of transverse crystalline order[CITATION].', 'cond-mat-9809115-1-17-11': 'This is, however, to be expected in the isotropic case.', 'cond-mat-9809115-1-18-0': 'The wide bump at [MATH] marks the onset of a wide crossover where strong local diamagnetic fluctuations start to form.', 'cond-mat-9809115-1-18-1': 'For the zero magnetic field case, [MATH], the crossover at [MATH] becomes the continuous [MATH]-transition at [MATH] belonging to the 3D-XY universality class[CITATION].', 'cond-mat-9809115-1-18-2': 'It is well understood that the [MATH]-transition is driven by proliferation of thermally excited loops of all sizes.', 'cond-mat-9809115-1-18-3': 'In recent work by some of us, it is now understood that the same mechanism is driving the first order melting transition at low [MATH] and a newly discovered continuous transition involving the breaking of U(1) symmetry at large [MATH][CITATION].', 'cond-mat-9809115-1-19-0': 'Similar results for [MATH] and [MATH] enables us to propose a simple phase diagram for an isotropic extreme type-II superconductor in the absence of disorder (see Fig. ([MATH])).', 'cond-mat-9809115-1-19-1': 'Immediately below and above the [MATH] line are the Abrikosov vortex crystal and the incoherent vortex liquid respectively.', 'cond-mat-9809115-1-19-2': '[MATH] marks the crossover region.', 'cond-mat-9809115-1-19-3': 'In this work, we have not considered the U(1) symmetry-breaking discussed in Ref. [CITATION], where it was proposed that a new transition separating phases with zero and finite line tension within the vortex liquid phase would occur.', 'cond-mat-9809115-1-19-4': 'Preliminary results have shown that the new transition also exists in the isotropic systems[CITATION].', 'cond-mat-9809115-1-20-0': 'We now highlight a minor artifact of this simulation, which exhibits itself as a finite [MATH] at very low temperature.', 'cond-mat-9809115-1-20-1': 'In a pin-free system, one would expect [MATH] to be zero at all temperature.', 'cond-mat-9809115-1-20-2': 'In this case, the numerical lattice, on which simulation is performed, effectively pins the vortex lines from moving in the transverse plane and counteract the Lorentz forces on them in the presence of a transverse applied current.', 'cond-mat-9809115-1-20-3': 'However, the pinning is overcome by thermal fluctuations at higher temperature and the vortex lines are depinned at temperature [MATH].', 'cond-mat-9809115-1-20-4': 'Fortunately, we see that [MATH], which means that near [MATH], the vortex lines, and the melting process, are completely free from the effect of the numerical grids.', 'cond-mat-9809115-1-20-5': 'Therefore, the features of [MATH], [MATH] and [MATH] at [MATH] are genuine thermodynamic effects.', 'cond-mat-9809115-1-21-0': '# Conclusion', 'cond-mat-9809115-1-22-0': 'We have performed simulations of the uniformly frustrated 3D-XY model on a large isotropic system ([MATH] grid points) for a variety of filling fraction [MATH] and [MATH].', 'cond-mat-9809115-1-22-1': 'We found a first order melting transition in this isotropic system in all the three values of [MATH] considered.', 'cond-mat-9809115-1-22-2': 'The longitudinal phase coherence and triangular crystalline order of the Abrikosov vortex crystal are simultaneously destroyed at melting.', 'cond-mat-9809115-1-22-3': 'After melting the incoherent vortex liquid is the only thermodynamic phase.', 'cond-mat-9809115-1-22-4': 'We have demonstrated that previous claims of the existence of disentangled vortex liquid[CITATION] is due to performing simulation on insufficiently large system size and insufficient long simulation time.', 'cond-mat-9809115-1-23-0': 'The 3D-XY model provides a simple and accurate description of both isotropic and anisotropic extreme type-II superconductors in the absence of disorder.', 'cond-mat-9809115-1-23-1': 'The intrinsic phase diagram above the Meissner phase is simple: the Abrikosov vortex phase at low temperature and the incoherent vortex liquid at high temperature.', 'cond-mat-9809115-1-23-2': 'The melting transition between them is first order.', 'cond-mat-9809115-1-23-3': 'There is no intermediate phase-coherent vortex liquid at any value of anisotropy.'}","{'cond-mat-9809115-2-0-0': 'Using large scale Monte Carlo simulations on a uniformly frustrated 3DXY model, we report a first order vortex lattice melting transition in clean, isotropic extreme type-II ([MATH]) superconductors.', 'cond-mat-9809115-2-0-1': 'This work clarifies an important issue: the unpinned vortex liquid is always incoherent with no phase coherence in any directions for all anisotropies.', 'cond-mat-9809115-2-0-2': 'Previous claims of a disentangled vortex liquid for isotropic superconductors based on simulations, are due to finite size effects.', 'cond-mat-9809115-2-0-3': 'We explicitly show that the effective vortex-line tension vanishes precisely at the superconducting phase transition in zero magnetic field.', 'cond-mat-9809115-2-0-4': 'This loss of line tension is accompanied by an abrupt change in the connectivity of the vortex tangle across the superconductor.', 'cond-mat-9809115-2-0-5': 'We also obtain results indicating that the connectivity of the vortex tangle changes in a similar way even in finite magnetic field, and suggest that this could also be associated with a genuine phase-transition.', 'cond-mat-9809115-2-1-0': '2', 'cond-mat-9809115-2-2-0': '2', 'cond-mat-9809115-2-3-0': '# Introduction', 'cond-mat-9809115-2-4-0': 'Since the discovery of copper oxide based high temperature superconductors (HTSC) [CITATION], which are of the extreme type-II variety, there has been great interest in their phenomenological phase diagrams.', 'cond-mat-9809115-2-4-1': ""Abrikosov's mean-field description[CITATION], which is valid for conventional low temperature superconductors, is expected to be modified by the strong thermal fluctuations in HTSC."", 'cond-mat-9809115-2-4-2': 'Extensive research in both theory, numerical simulations, and experiments over the years have resulted in a general consensus on some of the fundamental issues.', 'cond-mat-9809115-2-4-3': 'The current understanding of HTSC in a uniform magnetic field [MATH] is as follows.', 'cond-mat-9809115-2-4-4': 'In the absence of any pinning disorder, the low temperature Abrikosov vortex lattice phase melts into a vortex liquid via a first order transition at the temperature [MATH].', 'cond-mat-9809115-2-4-5': 'The Abrikosov vortex lattice phase is characterized by a transverse triangular crystalline order and a finite longitudinal phase coherence.', 'cond-mat-9809115-2-5-0': 'However, recently there has been some debate about the nature of the vortex liquid which the Abrikosov vortex lattice melts into as temperature increases.', 'cond-mat-9809115-2-5-1': 'Numerous simulations using the 3DXY model [CITATION], London[CITATION] and Lowest Landau Level approximations[CITATION] have indicated that the vortex liquid is incoherent, i.e. the phase coherence or superfluid density in any direction, is zero.', 'cond-mat-9809115-2-5-2': 'The crystalline order and phase coherence are destroyed simultaneously at the melting transition.', 'cond-mat-9809115-2-5-3': 'This scenario has been supported by experiments on high quality YBCO crystals[CITATION].', 'cond-mat-9809115-2-5-4': 'Other simulations using the 3DXY model [CITATION] have suggested that the longitudinal phase coherence persists above the melting transition and only vanishes at a higher ""entanglement"" temperature [MATH].', 'cond-mat-9809115-2-5-5': 'In this scenario, the vortex liquid at [MATH] would be disentangled with relatively straight vortex lines.', 'cond-mat-9809115-2-5-6': 'For [MATH], it was claimed that larger thermal fluctuations causes the vortex lines to be entangled with a concomitant loss of global phase coherence[CITATION].', 'cond-mat-9809115-2-6-0': 'Previously, a vortex liquid with nonzero longitudinal phase-coherence, or superfluid density, was found in simulations on an isotropic system[CITATION].', 'cond-mat-9809115-2-6-1': 'More recently [CITATION], it has been proposed that for large [MATH], [MATH] and [MATH] merge into a single transition whereas for small [MATH] they are well separated.', 'cond-mat-9809115-2-6-2': 'The authors found that for an isotropic system, the crossover between the two regimes is at a filling fraction [MATH], where the precise definition of [MATH] will be given shortly.', 'cond-mat-9809115-2-6-3': 'However, these simulations were performed on rather small systems.', 'cond-mat-9809115-2-6-4': 'In the present work, we have performed similar simulations on a much larger system, and found an exclusively incoherent vortex liquid down to [MATH].', 'cond-mat-9809115-2-6-5': 'This implies that the only thermodynamically stable vortex liquid phase, is one which has zero longitudinal superfluid density, with full translational and rotational symmetries and zero phase coherence in all directions.', 'cond-mat-9809115-2-6-6': 'We are therefore led to the conclusion that the recently discussed [CITATION] crossover between the phase-coherent vortex liquid and the phase-incoherent vortex liquid is a numerical artifact.', 'cond-mat-9809115-2-7-0': 'The organization of this paper is as follows.', 'cond-mat-9809115-2-7-1': 'In Section II we first briefly review the model used in the simulations, and the quantities we calculate.', 'cond-mat-9809115-2-7-2': 'In Section III we discuss our results and their implications.', 'cond-mat-9809115-2-7-3': 'In Section IV we provide a summary and conclusion.', 'cond-mat-9809115-2-8-0': '# Model', 'cond-mat-9809115-2-9-0': 'We use a uniformly frustrated 3DXY model on a cubic lattice to describe an isotropic, extreme type-II superconductors in a magnetic field.', 'cond-mat-9809115-2-9-1': 'The London model for superconductors can be readily derived from the phenomenological Ginzburg-Landau (GL) model with the approximation that the amplitude of the local complex order parameter [MATH], is fixed.', 'cond-mat-9809115-2-9-2': 'The Hamiltonian [MATH] of the London model consists of degrees of freedom in the phase [MATH] and the gauge vector potential [MATH] associated with the magnetic induction [MATH] inside the system, i.e. [MATH].', 'cond-mat-9809115-2-10-0': 'For an isotropic extreme type-II superconductor, the penetration depth [MATH] is much larger than the coherence length [MATH] such that the GL parameter [MATH].', 'cond-mat-9809115-2-10-1': 'This means that the magnetic fields surrounding the vortex lines strongly overlap with one another giving a spatially smooth [MATH].', 'cond-mat-9809115-2-10-2': 'This condition is ensured in the regime where [MATH].', 'cond-mat-9809115-2-10-3': 'In other words, the fluctuations of [MATH] on the length scale of [MATH] are negligible compared to the fluctuations of [MATH].', 'cond-mat-9809115-2-10-4': 'Therefore, we can further simplify the problem by dropping the degrees of freedom of [MATH] from the Hamiltonian [MATH], and fix [MATH] equal to the external applied magnetic field.', 'cond-mat-9809115-2-10-5': 'The resulting Hamiltonian [MATH] is the 3DXY model.', 'cond-mat-9809115-2-10-6': 'This [MATH] (or frozen gauge) approximation has been widely used as a phenomenological model for superconductors.', 'cond-mat-9809115-2-10-7': 'Note that, within this model, the system has no magnetic flux lines, since there can be no tubes of confined magnetic flux when [MATH].', 'cond-mat-9809115-2-10-8': 'The system only exhibits vortex lines.', 'cond-mat-9809115-2-11-0': 'In order to perform numerical simulations on the resulting model, we discretize the model on a 3D cubic lattice with grid spacing [MATH].', 'cond-mat-9809115-2-11-1': 'The dimensionless Hamiltonian of this lattice model is given by [CITATION] [EQUATION] where [MATH] is the isotropic coupling energy, [MATH] is the flux quanta and [MATH] is the permeability of a vacuum in SI units.', 'cond-mat-9809115-2-11-2': 'The dimensionless vector [MATH] labels the position of an arbitrary grid point.', 'cond-mat-9809115-2-11-3': 'We define the gauge-invariant phase difference [MATH] as [EQUATION] where [MATH] is the unit vector along the [MATH]-axis.', 'cond-mat-9809115-2-11-4': 'The convention is that [MATH] ""flows"" from the grid at [MATH] to [MATH].', 'cond-mat-9809115-2-11-5': 'In this model, it is natural to define a dimensionless temperature [MATH].', 'cond-mat-9809115-2-11-6': 'The magnetic induction [MATH] is conveniently represented by the filling fraction [EQUATION]', 'cond-mat-9809115-2-11-7': 'In this paper, we perform a Monte Carlo simulation on an isotropic system with [MATH] number of grid points.', 'cond-mat-9809115-2-11-8': 'The aim of our numerical simulations is to identify possible thermodynamic phases (i.e. vortex lattice or liquid, etc.) and phase transitions (first order or continuous) or crossovers associated with them.', 'cond-mat-9809115-2-11-9': 'Specifically, we are interested in calculating the internal energy, specific heat, structure factor, and helicity moduli.', 'cond-mat-9809115-2-11-10': 'These thermodynamic quantities and their physical significance will be discussed next.', 'cond-mat-9809115-2-11-11': 'In addition, we introduce and define a quantity [MATH] which we denote the vortex-path probability, and discuss some of its physical implications.', 'cond-mat-9809115-2-12-0': 'A first order transition is indicated by a [MATH]-function anomaly in the specific heat, equivalently a discontinuous jump in the internal energy.', 'cond-mat-9809115-2-12-1': 'On the other hand, the hallmark of a continuous transition is a jump in the specific heat, modified by fluctuation contributions to the anomaly, and a continuous internal energy.', 'cond-mat-9809115-2-12-2': 'The internal energy per site is obtained by averaging [MATH] in Eq. ([MATH]) over the thermal equilibrium states, normalized by the total number of grids, i.e. [MATH].', 'cond-mat-9809115-2-12-3': 'We define a dimensionless specific heat per site [MATH] using the standard fluctuation theorem of a classical system with Gibbs distribution[CITATION]: [EQUATION]', 'cond-mat-9809115-2-12-4': 'A convenient and widely used quantity to probe the global phase coherence of the system is the helicity modulus, which is proportional to the second derivative of the free energy associated with Eq. ([MATH]) with respect to an infinitesimal phase twist[CITATION].', 'cond-mat-9809115-2-12-5': 'On a square lattice, the dimensionless helicity modulus [MATH] can be written as [EQUATION] [MATH] measures the ""stiffness"" of [MATH] with respect to twisting along the [MATH]-direction.', 'cond-mat-9809115-2-12-6': 'If [MATH] then [MATH] is stiff in the [MATH]-direction, or more precisely, there is global phase coherence or superconducting response in that direction.', 'cond-mat-9809115-2-12-7': 'By tracking the temperature at which [MATH] goes to zero, one can determine the superconducting-normal metal transition temperature of the system.', 'cond-mat-9809115-2-12-8': 'In the mixed phase [MATH] and [MATH], where the subscripts [MATH] and [MATH] denote the directions longitudinal and transverse to [MATH], respectively.', 'cond-mat-9809115-2-13-0': 'The structure factor probes the transverse crystalline order of the vortex system.', 'cond-mat-9809115-2-13-1': 'We adopt a conventional definition [CITATION]: [EQUATION] where [MATH] is a two-dimensional reciprocal vector and [MATH] is the local vorticity measured on the dual square lattice grid [CITATION], composed of the centers of every direct unit cell.', 'cond-mat-9809115-2-13-2': 'A crystalline ordered phase is characterized by [MATH] (or Bragg peaks) where [MATH] are the discrete set of reciprocal lattice vectors associated with the crystal structure.', 'cond-mat-9809115-2-13-3': 'On the other hand, [MATH] for a phase with full rotational invariance exhibits ring patterns.', 'cond-mat-9809115-2-14-0': 'At a fixed temperature, the equilibrium configurations are generated by making random changes to [MATH] at each grid-point via the the Metropolis algorithm.', 'cond-mat-9809115-2-14-1': 'This is equivalent to randomly changing all the six [MATH] attached to each grid-point.', 'cond-mat-9809115-2-14-2': 'To ensure conservation of vorticity in each unit cell, [MATH] has to be in the range [MATH].', 'cond-mat-9809115-2-14-3': 'Addition of [MATH] shall be administrated to bring [MATH] back into range at every Monte Carlo step.', 'cond-mat-9809115-2-14-4': 'This procedure introduces vortex-loops into the system.', 'cond-mat-9809115-2-14-5': 'Such loops are the elementary topological excitation of the model.', 'cond-mat-9809115-2-14-6': 'Periodic boundary conditions (PBC) are imposed on [MATH] such that [MATH] for [MATH].', 'cond-mat-9809115-2-14-7': 'More details about the Monte Carlo procedure may be found in Refs. [CITATION].', 'cond-mat-9809115-2-15-0': 'We define [MATH] as the probability of finding a directed vortex path threading the entire system transverse to the induction [MATH], without using the PBC along the field direction [CITATION].', 'cond-mat-9809115-2-15-1': 'It is obtained by computing the number [MATH] of times we find at least one such path threading the system in any direction [MATH] in [MATH] different phase-configurations, normalized by [MATH], i.e. [MATH].', 'cond-mat-9809115-2-15-2': 'The fact that [MATH] implies that there is no connected path of vortex segments that threads the entire system in the transverse direction, without using PBC along the field direction several times.', 'cond-mat-9809115-2-15-3': 'Now, let [MATH]) denote the areal density of connected vortex paths threading the system in any direction, including the direction parallel to the induction.', 'cond-mat-9809115-2-15-4': 'It is clear that in the Abrikosov vortex lattice phase [MATH], and [MATH], while [MATH].', 'cond-mat-9809115-2-15-5': 'Thus, [MATH] is a conserved quantity at fixed induction [MATH].', 'cond-mat-9809115-2-15-6': 'On the other hand, [MATH] implies that [MATH], and the total number of vortex paths threading the system in any direction scales with system size, but undergoes thermal fluctuations.', 'cond-mat-9809115-2-15-7': 'Therefore, [MATH] is no longer a conserved quantity.', 'cond-mat-9809115-2-16-0': 'We propose the following scenario to interpret the change in [MATH] and [MATH].', 'cond-mat-9809115-2-16-1': 'Number conservation uniquely identifies a [MATH]-symmetry, and hence the low-temperature phase of the vortex-system (the dual of the phase-representation of the superconductor) exhibits explicit [MATH]-symmetry, since [MATH].', 'cond-mat-9809115-2-16-2': 'At high temperatures [MATH], [MATH] is not conserved, and the [MATH]-symmetry is broken.', 'cond-mat-9809115-2-16-3': 'A [MATH]-symmetric phase cannot be analytically continued to a [MATH]-nonsymmetric one.', 'cond-mat-9809115-2-16-4': 'The change in [MATH] from [MATH] to [MATH] could therefore signal a phase-transition, in this case involving breaking a global [MATH]-symmetry, in finite as well as zero magnetic field.', 'cond-mat-9809115-2-16-5': 'However, to substantiate such a claim, one needs to argue that [MATH] is related to a local order parameter of the system.', 'cond-mat-9809115-2-16-6': 'To this end, we note that it is possible to transcribe the vortex-part of the Ginzburg-Landay theory in the phase-only approximation in such a way that the vortex-part of the theory is specified in terms of a local, complex matter field [MATH], and that the theory then explicitly exhibits a [MATH]-symmetry [CITATION].', 'cond-mat-9809115-2-16-7': 'In the Lattice London Model, corresponding to the Villain-approximation to the Ginzburg-Landau theory, this symmetry is therefore only implicit, or ""hidden"".', 'cond-mat-9809115-2-16-8': 'The probability of finding a connected vortex-path starting at a point [MATH] and ending at point [MATH], [MATH], is given in terms of the two-point correlation function of the matter field [MATH], [MATH] [CITATION].', 'cond-mat-9809115-2-16-9': '[MATH] may be viewed as a special case of [MATH], and in the thermodynamic limit corresponds to [MATH].', 'cond-mat-9809115-2-16-10': 'If [MATH] is non-zero in this limit, this suggests the possibility of having [MATH], and hence a broken [MATH]-symmetry.', 'cond-mat-9809115-2-16-11': 'Although this does not constitute a proof that [MATH] is connected to a local order parameter whose expectation value is associated with a broken [MATH]-symmetry, it seems to be suggestive of such a phase-transition existing even at finite magnetic field.', 'cond-mat-9809115-2-16-12': 'Note that the above local matter field [MATH] appears to be the dual field of a complex order parameter appearing in a somewhat different independent approach to the same problem by Tesanovic [CITATION].', 'cond-mat-9809115-2-17-0': '# Results and Discussion', 'cond-mat-9809115-2-18-0': 'In this section, we discuss the results of our simulations on an isotropic system with size of [MATH] grid points, which is the largest to date on an isotropic system.', 'cond-mat-9809115-2-18-1': 'The system is subdivided into multiple sections, and the Monte Carlo procedure is implemented in parallel across the sections using 3D ""black and white"" decomposition[CITATION].', 'cond-mat-9809115-2-18-2': 'The filling fraction considered are [MATH], and [MATH].', 'cond-mat-9809115-2-18-3': 'The system is cooled from high temperatures.', 'cond-mat-9809115-2-18-4': 'For each temperature, a typical run consists of 120000 Monte Carlo sweeps across the whole lattice, [MATH] of that were used for equilibration.', 'cond-mat-9809115-2-18-5': 'Near phase-transitions, up to 600000 sweeps were used.', 'cond-mat-9809115-2-19-0': 'In Fig. ([MATH]), we present the results for the zero-field case, [MATH], where we show the specific heat [MATH], the superfluid stiffness [MATH], and the vortex-path probability [MATH] as functions of temperature.', 'cond-mat-9809115-2-19-1': 'Note how the specific heat anomaly, the vanishing of the superfluid stiffness and [MATH], all coincide with [MATH].', 'cond-mat-9809115-2-19-2': 'The physical interpretation of [MATH] is that the effective long-wavelength vortex-line tension vanishes [CITATION].', 'cond-mat-9809115-2-19-3': 'This claim is substantiated by calculating the probability distribution [MATH] of vortex loops as a function of perimeter [MATH], at various temperatures.', 'cond-mat-9809115-2-19-4': 'We may fit this distribution to the form [CITATION] [EQUATION] where [MATH] is an effective, temperature dependent, long-wavelength line tension, [MATH], and [MATH].', 'cond-mat-9809115-2-19-5': 'The results are shown in Fig. ([MATH]), demonstrating that at [MATH], a purely algebraic decay is realized, implying [MATH].', 'cond-mat-9809115-2-19-6': 'Below [MATH], the exponential decay is well fitted, showing that [MATH].', 'cond-mat-9809115-2-19-7': 'The inset of Fig.([MATH]) shows [MATH] as a function of temperature, as obtained in our simulations.', 'cond-mat-9809115-2-19-8': 'A similar method of extracting [MATH] for the zero-field case using a similar form for [MATH] (without he power-law prefactor) has previously been used by Li and Teitel, in Ref. Li93.', 'cond-mat-9809115-2-20-0': 'The results for the quantities [MATH], [MATH] and [MATH] for [MATH] are shown in Fig.([MATH]).', 'cond-mat-9809115-2-20-1': 'Similar qualitative features are also found for the cases [MATH].', 'cond-mat-9809115-2-20-2': 'First, [MATH] exhibits six-fold Bragg peaks at low temperatures (not shown).', 'cond-mat-9809115-2-20-3': 'This indicates that the low temperature phase is a triangular vortex lattice phase.', 'cond-mat-9809115-2-20-4': 'The destruction of the vortex lattice structure is marked by a melting transition at [MATH] where [MATH] drops sharply to zero, where [MATH] is the wavevector of one of the six first order Bragg peaks.', 'cond-mat-9809115-2-20-5': 'The sharpness of the drop in [MATH] strongly suggests that the transition is first order.', 'cond-mat-9809115-2-20-6': 'This is confirmed by the appearance of a [MATH]-function like peak in [MATH] at the same temperature [MATH].', 'cond-mat-9809115-2-20-7': 'Coincidentally, [MATH] which is finite at low temperatures, also drops sharply to zero at [MATH].', 'cond-mat-9809115-2-20-8': 'The isotropic system exhibits longitudinal superconductivity below [MATH], but not above.', 'cond-mat-9809115-2-20-9': 'Moreover, based on our above discussion in Section II, we interpret the rise in the quantity [MATH] from [MATH] to [MATH] as signalling that the effective long-wavelength vortex-line tension vanishes.', 'cond-mat-9809115-2-20-10': 'Thus, the vortex liquid phase is divided into two regions in phase-space.', 'cond-mat-9809115-2-20-11': 'In one region, the vortex liquid is phase-incoherent i.e. has no longitudinal superfluid density but has finite vortex-line tension.', 'cond-mat-9809115-2-20-12': 'We propose that this phase exhibits a vortex-associated [MATH]-symmetry.', 'cond-mat-9809115-2-20-13': 'In the other region the vortex liquid is phase-incoherent but has zero vortex-line tension.', 'cond-mat-9809115-2-20-14': 'We propose that this phase exhibits a broken [MATH]-symmetry.', 'cond-mat-9809115-2-21-0': 'Based on this, one can conclude that a first order melting transition of the Abrikosov vortex lattice exists in an isotropic system in the absence of pinning.', 'cond-mat-9809115-2-21-1': 'The entire vortex liquid phase is incoherent, i.e. a vortex liquid phase with no longitudinal superconductivity.', 'cond-mat-9809115-2-21-2': 'Note that the same conclusion has been reached in earlier simulations on anisotropic systems[CITATION].', 'cond-mat-9809115-2-21-3': 'There have been earlier reports of a disentangled vortex liquid, i.e. vortex liquid with non-zero longitudinal phase coherence.', 'cond-mat-9809115-2-21-4': 'These results have been obtained in similar simulation[CITATION].', 'cond-mat-9809115-2-21-5': 'In Ref. [CITATION], it was argued that a phase-coherent vortex liquid should be most easily observed in isotropic systems for [MATH].', 'cond-mat-9809115-2-21-6': 'However, these simulations were performed on comparatively small systems, typically of sizes no larger than [MATH].', 'cond-mat-9809115-2-21-7': 'Our present results are based on much larger systems, [MATH].', 'cond-mat-9809115-2-21-8': 'We have observed a slight difference in the temperatures at which [MATH] and [MATH] vanish in simulations on smaller system or lower number of sweeps.', 'cond-mat-9809115-2-21-9': 'Therefore, it may be concluded that the existence of a vortex liquid with a nonzero longitudinal superfluid density is a numerical artifact of small system sizes and/or insufficient simulation time [CITATION].', 'cond-mat-9809115-2-21-10': 'It is well understood that the [MATH]-transition is driven by proliferation of thermally excited loops of all sizes.', 'cond-mat-9809115-2-21-11': 'In recent work by some of us, it was proposed that the same mechanism is driving the first order melting transition at low [MATH] and a newly discovered continuous transition involving the breaking of U(1) symmetry at large [MATH][CITATION].', 'cond-mat-9809115-2-22-0': 'Similar results for [MATH] and [MATH] enable us to propose a simple phase diagram for an isotropic extreme type-II superconductor in the absence of disorder (see Fig. ([MATH])).', 'cond-mat-9809115-2-22-1': 'Immediately below and above the [MATH] line, the phases of the vortex system are identified as the Abrikosov vortex lattice and the incoherent vortex liquid, respectively.', 'cond-mat-9809115-2-23-0': 'In a pin-free system, one would expect [MATH] to be zero at all temperature.', 'cond-mat-9809115-2-23-1': 'In this case, the numerical lattice, on which simulations are performed, effectively pins the vortex lines from moving in the transverse plane and counteracts the Lorentz forces on them in the presence of a transverse applied current.', 'cond-mat-9809115-2-23-2': 'However, the pinning is overcome by thermal fluctuations at higher temperature and the vortex lines are depinned at temperature [MATH].', 'cond-mat-9809115-2-23-3': 'Fortunately, we see that [MATH], which means that near [MATH], the vortex lines, and the melting process, are completely free from the effects of the numerical grid.', 'cond-mat-9809115-2-23-4': 'Therefore, the features of [MATH], [MATH] and [MATH] at [MATH] are genuine thermodynamic effects.', 'cond-mat-9809115-2-24-0': '# Conclusion', 'cond-mat-9809115-2-25-0': 'We have performed simulations of the uniformly frustrated 3DXY model on a large isotropic system ([MATH] grid points) for a variety of filling fractions [MATH] and [MATH].', 'cond-mat-9809115-2-25-1': 'We found a first order melting transition in this isotropic system for all the three non-zero values of [MATH] considered.', 'cond-mat-9809115-2-25-2': 'The longitudinal phase coherence and triangular crystalline order of the Abrikosov vortex crystal are simultaneously destroyed at the melting transition.', 'cond-mat-9809115-2-25-3': 'Above the melting temperature, the incoherent vortex liquid is the only thermodynamic phase.', 'cond-mat-9809115-2-25-4': 'We have demonstrated that previous claims of the existence of disentangled vortex liquid[CITATION] is due to performing simulations using insufficient system sizes and simulation times.', 'cond-mat-9809115-2-26-0': 'We have shown that the effective vortex-line tension vanishes precisely at the zero-field superconducting transition.', 'cond-mat-9809115-2-26-1': 'The loss of superfluid stiffness, the loss of line tension, and the abrupt change in the connectivity of the vortex tangle, as signalled by the change in the quantitity [MATH] across the system, all coincide in this case.', 'cond-mat-9809115-2-26-2': 'A similar change in connectivity across the vortex system takes place at finite magnetic field.', 'cond-mat-9809115-2-26-3': 'The results of the Ref. [CITATION], the present paper, and in particular those of Ref. [CITATION] strongly indicate that this change in connectivity is sharp in the limit of large systems, thus indicating the loss of number-conservation of connected vortex paths threading the system.', 'cond-mat-9809115-2-26-4': 'Since the finite probability of finding a connected vortex path threading the system in a direction other than the magnetic field may be tied to the finite expectation value of a local complex matter field [CITATION], this lends further support to the argument that the change in [MATH] signals the breaking of a [MATH]-symmetry [CITATION].', 'cond-mat-9809115-2-26-5': 'At the very least this proposition appears to be intriguing enough to warrant further investigation.', 'cond-mat-9809115-2-27-0': 'We finally caution the reader that we so far have not been able to detect any anomaly in specific heat at the suggested new finite-field transition inside the vortex liquid, for the isotropic case.', 'cond-mat-9809115-2-27-1': 'Even the anomaly at the first order melting transition is weak in the isotropic case, and is difficult to bring out in simulations.', 'cond-mat-9809115-2-27-2': 'It may be that considerably larger systems are needed for the isotropic case in order to see signals in the specific heat of the suggested new transition due to the small amount of entropy in the transition.', 'cond-mat-9809115-2-27-3': 'This is the reason why the anomaly at the first order vortex lattice melting transition is difficult to observe in simulations.', 'cond-mat-9809115-2-27-4': 'It is conceivable that increasing the anisotropy of the system should bring out the anomaly clearer, if it exists.', 'cond-mat-9809115-2-27-5': 'This indeed is the case for the anomaly at the vortex lattice melting transition.', 'cond-mat-9809115-2-27-6': 'A weak anomaly associated with the putative [MATH]-transition, may in fact have been observed for the anisotropic case in Ref. [CITATION].'}","[['cond-mat-9809115-1-13-3', 'cond-mat-9809115-2-14-3'], ['cond-mat-9809115-1-11-0', 'cond-mat-9809115-2-12-0'], ['cond-mat-9809115-1-11-2', 'cond-mat-9809115-2-12-2'], ['cond-mat-9809115-1-11-3', 'cond-mat-9809115-2-12-3'], ['cond-mat-9809115-1-11-4', 'cond-mat-9809115-2-12-4'], ['cond-mat-9809115-1-11-5', 'cond-mat-9809115-2-12-5'], ['cond-mat-9809115-1-11-6', 'cond-mat-9809115-2-12-6'], ['cond-mat-9809115-1-11-7', 'cond-mat-9809115-2-12-7'], ['cond-mat-9809115-1-11-8', 'cond-mat-9809115-2-12-8'], ['cond-mat-9809115-1-10-0', 'cond-mat-9809115-2-11-0'], ['cond-mat-9809115-1-10-2', 'cond-mat-9809115-2-11-2'], ['cond-mat-9809115-1-10-4', 'cond-mat-9809115-2-11-4'], ['cond-mat-9809115-1-10-5', 'cond-mat-9809115-2-11-5'], ['cond-mat-9809115-1-10-6', 'cond-mat-9809115-2-11-6'], ['cond-mat-9809115-1-10-7', 'cond-mat-9809115-2-11-7'], ['cond-mat-9809115-1-10-10', 'cond-mat-9809115-2-11-10'], ['cond-mat-9809115-1-3-1', 'cond-mat-9809115-2-4-1'], ['cond-mat-9809115-1-3-2', 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1909.07547,"{'1909.07547-1-0-0': 'Open-domain dialog generation is a challenging problem; maximum likelihood training can lead to repetitive outputs, models have difficulty tracking long-term conversational goals, and training on standard movie or online datasets may lead to the generation of inappropriate, biased, or offensive text.', '1909.07547-1-0-1': 'Reinforcement Learning (RL) is a powerful framework that could potentially address these issues, for example by allowing a dialog model to optimize for reducing toxicity and repetitiveness.', '1909.07547-1-0-2': 'However, previous approaches which apply RL to open-domain dialog generation do so at the word level, making it difficult for the model to learn proper credit assignment for long-term conversational rewards.', '1909.07547-1-0-3': 'In this paper, we propose a novel approach to hierarchical reinforcement learning, VHRL, which uses policy gradients to tune the utterance-level embedding of a variational sequence model.', '1909.07547-1-0-4': 'This hierarchical approach provides greater flexibility for learning long-term, conversational rewards.', '1909.07547-1-0-5': 'We use self-play and RL to optimize for a set of human-centered conversation metrics, and show that our approach provides significant improvements - in terms of both human evaluation and automatic metrics - over state-of-the-art dialog models, including Transformers.', '1909.07547-1-1-0': '# Introduction', '1909.07547-1-2-0': 'Since the inception of the Turing test, generating convincing open-ended (open-domain) dialog has been a fundamental challenge in artificial intelligence (AI).', '1909.07547-1-2-1': 'A successful open-domain dialog system could provide enormous value enabling more natural human-computer interaction through a wide variety of open-ended voice interfaces.', '1909.07547-1-2-2': 'Successful dialog models could also unlock new, beneficial applications of AI, such as companion chatbots for therapy applications.', '1909.07547-1-3-0': 'However, current generative models for dialog suffer from several shortcomings that limit their usefulness in the real world.', '1909.07547-1-3-1': 'Training on existing standard dialog datasets collected online or from movie scripts often leads to models that produce malicious, aggressive, biased, or offensive responses [CITATION].', '1909.07547-1-3-2': 'There are no guarantees on the quality and sensitivity of the generated text, often preventing open-domain dialog systems from being deployed for safety-critical applications such as mental health, or even for customer service.', '1909.07547-1-3-3': 'Further, maximum likelihood estimation (MLE) training of such models often leads to the generation of dull and repetitive text [CITATION].', '1909.07547-1-3-4': 'Finally, models may have difficulty tracking long-term aspects of the conversation, and evidence has shown that they do not adequately condition on the conversation history in generating responses [CITATION].', '1909.07547-1-4-0': 'Reinforcement Learning (RL) is a powerful paradigm that allows dialog models to optimize for non-differentiable metrics of conversation quality, and thereby helps overcome the above problems.', '1909.07547-1-4-1': 'In this paper, we use RL to learn from self-play; the model talks to a fixed copy of itself, and computes reward functions on the generated conversation.', '1909.07547-1-4-2': 'We propose novel, human-centered reward functions, such as minimizing the estimated toxicity of a conversation, in order to limit inappropriate, biased, and offensive responses.', '1909.07547-1-4-3': 'We also design rewards based on the psychology of good conversation (e.g. [CITATION] [CITATION]), and reward recently proposed conversation metrics that are associated with improved human judgments of conversation quality [CITATION].', '1909.07547-1-4-4': 'Table [REF] shows how RL can be used to optimize our proposed metrics, shifting the conversation towards more positive and engaging conversation, with less repetition and toxicity.', '1909.07547-1-5-0': 'Applying RL to open-domain dialog generation is a challenging problem, however.', '1909.07547-1-5-1': 'Most prior approaches (e.g. [CITATION] [CITATION]) learn to model rewards at the word-level, meaning that the reward is applied to affect the probability of generating each word in the response.', '1909.07547-1-5-2': 'Such low-level control makes determining which words lead to higher rewards that are computed based on multiple conversation turns (i.e. word-level credit assignment) excessively challenging.', '1909.07547-1-5-3': ""Further, this approach makes the number of actions available to the RL agent equivalent to the number of words in the dialog model's vocabulary (20,000 in our case)."", '1909.07547-1-5-4': 'Thus there are several orders of magnitude more actions than in typical successful applications of deep RL, such as learning to play an Atari game with around 20 actions (e.g. [CITATION] [CITATION]).', '1909.07547-1-6-0': 'To overcome these challenges, we leverage hierarchical reinforcement learning (HRL) to learn to model rewards at the utterance level, improving the flexibility of dialog models to learn long-term, conversational rewards.', '1909.07547-1-6-1': 'Specifically, we propose a novel approach, Variational Hierarchical Reinforcement Learning (or VHRL), which uses policy gradients to adjust the prior probability distribution of the latent variable learned at the utterance level of a hierarchical variational model.', '1909.07547-1-6-2': 'We show that this approach allows for improved learning of conversational rewards that are not modeled well at the word level.', '1909.07547-1-7-0': 'To evaluate our models, we not only compute automatic metrics, but conduct an interactive human evaluation using the https://neural.chat/ platform [CITATION], in which humans chat live with our bots about anything they choose.', '1909.07547-1-7-1': 'This represents a more realistic test of real-world generalization performance than is typically employed when testing RL models in the same environment in which they were trained.', '1909.07547-1-7-2': 'Our evaluation reveals that VHRL improves human judgments of conversational quality above state-of-the-art dialog architectures, including Transformer-based models.', '1909.07547-1-8-0': 'In summary, the paper makes the following contributions: a) Develops a new technique, VHRL, for hierarchical control of variational dialog models; b) Demonstrates the effectiveness of training open-domain dialog models with VHRL and self-play, showing improvements over state-of-the-art dialog architectures with both human evaluation and automatic analysis; and c) Introduces and compares several reward functions for guiding conversations to be less toxic and repetitive, and more engaging, positive, contingent on user input.', '1909.07547-1-8-1': 'We also release code for our evaluation platform and models at https://github.com/natashamjaques/neural_chat.', '1909.07547-1-9-0': '# Related Work', '1909.07547-1-10-0': 'The difficulty of training open-domain dialog systems is exacerbated by the fact that we lack reliable automatic metrics with which to evaluate dialog quality [CITATION].', '1909.07547-1-10-1': 'Recently, authors have begun to propose new metrics of conversation quality (e.g. [CITATION] [CITATION]), and have even proposed evaluating metrics on conversations generated with self-play [CITATION].', '1909.07547-1-10-2': 'However, these works have not attempted to directly optimize for these non-differentiable metrics with RL.', '1909.07547-1-11-0': 'There has been significant progress in improving dialog generation outside of RL.', '1909.07547-1-11-1': 'The most popular of these approaches is conditional training, where a learned embedding vector is associated with a desired output feature and fed into the decoder to control generation (e.g. [CITATION] [CITATION]).', '1909.07547-1-11-2': 'This approach has multiple limitations.', '1909.07547-1-11-3': 'First, the model can only learn associations present in the training data, and cannot explore to discover improved methods for optimizing the desired features.', '1909.07547-1-11-4': 'Second, the conditional embeddings are learned at training time with teacher forcing and thus suffer from exposure bias [CITATION].', '1909.07547-1-11-5': 'Using RL avoids these limitations as it allows exploring regions of space not present in the training data and directly optimizing for rewards at inference time.', '1909.07547-1-11-6': 'In addition, RL learns the total expected future reward of taking some action now, given how the rest of the conversation will unfold in the future.', '1909.07547-1-11-7': 'This allows RL models to make long-term trade-offs about the benefits of generating words and utterances in the conversation context.', '1909.07547-1-11-8': 'Finally, our approach does not require the addition of any new parameters or complex components.', '1909.07547-1-11-9': 'Instead, it can be use to tune pre-existing models to output better, more appropriate responses.', '1909.07547-1-12-0': '## Reinforcement Learning for Dialog', '1909.07547-1-13-0': 'Improving dialog models with RL is a difficult problem, and past work has largely been restricted to task-oriented dialog systems, which have a limited number of task-specific actions (e.g. [CITATION] [CITATION]).', '1909.07547-1-13-1': 'Attempts to apply RL to open-domain dialog generation are less common.', '1909.07547-1-13-2': ""Even in this setting, authors often use a highly restricted action space; for example, using RL to choose dialog acts for conditional generation [CITATION] or to choose which models to invoke to answer a user's query [CITATION]."", '1909.07547-1-14-0': '[CITATION] [CITATION] applied deep RL to the full vocabulary-sized action space, optimizing for scripted rewards such as ease of answering.', '1909.07547-1-14-1': 'RL has also been used to optimize for rewards based on a GAN-like discriminator trained to distinguish human-generated from model-generated text [CITATION].', '1909.07547-1-14-2': 'Note that it is difficult to apply the traditional GAN approach of backpropagating discriminator gradients directly into the generator, because the sampling procedure used to generate each word in an utterance does not preserve gradients.', '1909.07547-1-14-3': 'Therefore even differentiable models which predict conversation metrics from a sampled utterance cannot be directly applied to optimize a model without RL.', '1909.07547-1-15-0': 'Sentiment has been used as a reward in an RL framework, for example to improve a seq2seq model [CITATION].a', '1909.07547-1-15-1': 'authorjaques2019way jaques2019way optimize for sentiment and several other conversation metrics by learning from a static batch of human-bot conversation data using Batch RL.', '1909.07547-1-15-2': 'We believe we are the first to propose using RL to reduce toxicity in an open-domain dialog setting, in order to ensure the model produces more appropriate and safe conversations.', '1909.07547-1-16-0': 'Hierarchical models have been investigated extensively in the context of MLE language modeling [CITATION].', '1909.07547-1-16-1': 'These architectures take advantage of the natural hierarchical structure of language, decomposing input into utterances at one level, and words at another.', '1909.07547-1-16-2': 'However, attempts to apply hierarchical RL (HRL) to dialog generation have so far been limited to task-oriented dialog systems [CITATION].', '1909.07547-1-16-3': 'To the best of our knowledge, we are the first to apply HRL to open-domain dialog generation.', '1909.07547-1-17-0': '## Hierarchical Reinforcement Learning', '1909.07547-1-18-0': 'Many approaches have been proposed for building hierarchical agents within the context of reinforcement learning for games and robotics [CITATION].', '1909.07547-1-18-1': 'The options framework proposed by [CITATION] [CITATION] is one popular approach for HRL.', '1909.07547-1-18-2': 'At the bottom level of the hierarchy, a set of options (or workers) which are policies over actions interact with the environment until terminated by the agent.', '1909.07547-1-18-3': 'At the top level, a policy over options (or manager) selects options to be executed until termination, at which point another option is picked and the process is repeated.', '1909.07547-1-18-4': 'The different levels of temporal abstraction introduced by this hierarchy allows for better long-term planning relative to traditional, flat reinforcement learning techniques.', '1909.07547-1-19-0': 'A major focus of HRL has been on sub-goal or option discovery for training worker policies.', '1909.07547-1-19-1': 'Bottom-level policies are often learned using handcrafted sub-goals [CITATION], intrinsic rewards [CITATION], or pseudo-rewards [CITATION], while the manager policy is learned using extrinsic rewards from the environment.', '1909.07547-1-19-2': 'Our approach also allows for optimizing different rewards at different levels of the hierarchy, thus creating distinct goals for the worker and the manager.', '1909.07547-1-19-3': 'However, unlike other HRL approaches we expose both the worker and manager policies to extrinsic rewards and add weight hyper-parameters to regulate the effect of the rewards at each level.', '1909.07547-1-19-4': 'This remedies a weakness of pseudo-reward methods where a worker only focuses on achieving its sub-goals while disregarding the effect on the extrinsic environment reward.', '1909.07547-1-20-0': '# Background', '1909.07547-1-21-0': 'A common approach to dialog modeling is to use a hierarchical seq2seq architecture, such as the Variational Hierarchical Recurrent Encoder Decoder (VHRED) [CITATION].', '1909.07547-1-21-1': 'We adopt VHRED here, following previous work which has found it to be the most effective version of several related architectures [CITATION].', '1909.07547-1-22-0': 'As shown in Figure [REF], VHRED uses three recurrent networks to generate the next utterance in a conversation.', '1909.07547-1-22-1': 'The word-level encoder RNN operates on the words (tokens) of the input utterance [MATH], and encodes them into a representation [MATH].', '1909.07547-1-22-2': 'This is fed into a context RNN, which forms the upper level of the hierarchy - it is updated only after each utterance, rather than each token.', '1909.07547-1-22-3': 'Because it updates less frequently, the context RNN is potentially better able to track longer-term aspects of the conversation.', '1909.07547-1-22-4': 'The context RNN outputs [MATH], which is used to produce an utterance embedding [MATH].', '1909.07547-1-22-5': 'This is fed into the word-level decoder RNN [MATH], which produces the output utterance [MATH], one token at a time.', '1909.07547-1-23-0': 'The model is similar to a variational autoencoder; [MATH] is fed into fully connected layers that predict the mean [MATH] and variance [MATH] of a multivariate normal distribution.', '1909.07547-1-23-1': 'Through a KL-divergence constraint and the reparameterization trick, the model learns a probability distribution over the embedding vector [MATH] of each utterance, [MATH].', '1909.07547-1-23-2': 'Formally, the model can be described as follows: [EQUATION]', '1909.07547-1-24-0': '## Reinforcement Learning', '1909.07547-1-25-0': 'We adopt the standard reinforcement learning framework where given the environment state [MATH], an agent takes an action [MATH] according to its policy [MATH], and receives a reward [MATH].', '1909.07547-1-25-1': 'The environment then transitions to the next state according to the transition function [MATH].', '1909.07547-1-25-2': 'The agent seeks to maximize the total expected future reward (long-term return): [EQUATION] given a starting state [MATH] and a discount factor [MATH].', '1909.07547-1-26-0': '## Policy Gradient Methods', '1909.07547-1-27-0': 'Policy gradient methods learn parameterized policies [MATH] for solving RL problems with [MATH] being a learned parameter vector.', '1909.07547-1-27-1': 'The policy gradient theorem [CITATION] derives the gradient of the expected return with respect to the policy parameters.', '1909.07547-1-27-2': 'In this paper, we use REINFORCE [CITATION] which approximates the gradient at each time step [MATH] using [EQUATION] where [MATH] is the observed future reward for an episode that ends at [MATH].', '1909.07547-1-27-3': 'The expected return is maximized with gradient ascent.', '1909.07547-1-27-4': 'This is equivalent to minimizing the loss function [MATH].', '1909.07547-1-28-0': 'In continuous action spaces, actions [MATH] are sampled from a continuous probability distribution, such as a multivariate normal distribution.', '1909.07547-1-29-0': 'In this case, the policy [MATH] can be parameterized as a probability density function over actions, [EQUATION] where the actions are sampled from a multivariate normal [MATH].', '1909.07547-1-29-1': 'Here the mean [MATH] and covariance matrix [MATH] are defined in terms of the current state [MATH] and the policy parameters [MATH].', '1909.07547-1-29-2': 'The density of the probability of actions, rather than the probability, is learned in the continuous case.', '1909.07547-1-29-3': 'We refer the reader to [CITATION] [CITATION] for more details on extending policy gradient methods to the continuous case.', '1909.07547-1-30-0': '# Approach', '1909.07547-1-31-0': 'We pose dialog generation as an RL problem where the state, [MATH], is all the previous dialog turns read by the model up to utterance [MATH], and the rewards are calculated based on the dialog history and generated utterance.', '1909.07547-1-32-0': 'Previous approaches which have applied RL to language generation have done so at the word level, where the policy [MATH] models the distribution over outputting the next word [CITATION].', '1909.07547-1-32-1': 'Instead, we cast our problem in the hierarchical reinforcement learning framework by considering the context RNN as the manager responsible for utterance-level decisions, and the decoder RNN as the worker responsible for word-level decisions.', '1909.07547-1-33-0': 'We leverage the fact that VHRED learns a probability distribution over latent variable [MATH] as a decision making component at the utterance level.', '1909.07547-1-33-1': 'Starting with a pre-trained VHRED model, we apply REINFORCE to tune the variational component, treating [MATH] as a continuous action.', '1909.07547-1-33-2': 'Thus, the manager policy is defined by the distribution of the prior latent variable [MATH], while the worker policy is the distribution of the output words [MATH], which is parameterized by the manager decisions.', '1909.07547-1-34-0': ""More specifically, the probability of a worker action [MATH] is the joint probability of the generated utterance conditioned on the manager's decision [MATH], [EQUATION] while the probability of a manager action is given by the multivariate normal probability density function in Eq. [REF]."", '1909.07547-1-35-0': ""We propose a new approach which allows both the worker and manager to jointly optimize total expected future return by minimizing the following loss: [EQUATION] where [MATH] is the manager's observed future reward and [MATH] is the worker's observed future reward."", '1909.07547-1-35-1': ""This formulation is analogous to REINFORCE as it shifts the model's decisions towards actions associated with positive rewards and discourages actions associated with negative rewards."", '1909.07547-1-35-2': 'The scalars [MATH] are hyperparameters used to regulate the effect of the rewards at each level of the hierarchy.', '1909.07547-1-35-3': 'We call our approach Variational Hierarchical Reinforcement Learning (VHRL).', '1909.07547-1-36-0': 'Unlike recently proposed HRL approaches which train the worker and manager separately as decoupled components [CITATION], we train our entire model jointly, end-to-end.', '1909.07547-1-36-1': 'This implies that the worker (decoder RNN) gradients flow through the manager (context RNN), and both flow through the encoder RNN.', '1909.07547-1-36-2': 'We make this decision for two reasons.', '1909.07547-1-36-3': 'First, [MATH] lives in a continuous high dimensional action space, making it difficult to learn a good policy [MATH] without a signal from the decoder.', '1909.07547-1-36-4': 'Second, this gives the decoder control over the representations learned by the encoder, facilitating optimization.', '1909.07547-1-36-5': 'As an ablation study, we experiment with decoupled decoder and encoder training, and find that the joint approach performs better.', '1909.07547-1-37-0': 'The proposed loss allows for optimizing different rewards at different levels, which can be used to incorporate prior knowledge about the problem.', '1909.07547-1-37-1': 'For example, rewards relevant to the global dialog history could be considered only by the manager through [MATH], rather than the worker.', '1909.07547-1-37-2': 'Conversely, rewards relevant to the word-by-word output could be considered by the worker through [MATH] and not the manager.', '1909.07547-1-37-3': 'For simplicity, we optimize for all rewards at both levels (i.e. [MATH]) and achieve promising results.', '1909.07547-1-38-0': 'Similar to previous work applying RL to dialog [CITATION] we use self-play to simulate the interactive environment in which the agent learns.', '1909.07547-1-38-1': 'We initialize conversations with randomly sampled starting sentences from the training set and let our model interact with a user simulator which is a fixed copy of itself.', '1909.07547-1-38-2': 'We limit each model to 3 turns for a total conversation length of 7 utterances.', '1909.07547-1-38-3': 'Limiting the length of simulated interactions is important since we found that long conversations are more likely to degenerate and go off-topic.', '1909.07547-1-39-0': '# Conversation Metrics', '1909.07547-1-40-0': 'Here we introduce several metrics for improving the quality of a conversation, which can be optimized using RL by treating them as rewards.', '1909.07547-1-40-1': 'Several metrics are inspired by previous work, but we also propose novel metrics such as toxicity.', '1909.07547-1-41-0': 'Sentiment: Emotion is important for creating a sense of understanding in human conversation [CITATION].', '1909.07547-1-41-1': 'Building on previous work which used sentiment as a reward (e.g.authorshin2019happybot,jaques2019way shin2019happybot,jaques2019way, we leverage a state-of-the-art sentiment detector, DeepMoji [CITATION], to reward generated utterances associated with positive sentiment emojis.', '1909.07547-1-41-2': 'The sentiment score is computed using the weights proposed byauthorghandeharioun2019approximating ghandeharioun2019approximating.', '1909.07547-1-42-0': 'Question: Asking questions is an important active listening skill, and can improve the quality of interactions [CITATION].', '1909.07547-1-42-1': 'Thus, we provide a positive reward when both a question word and a question mark are present in a generated response to encourage asking questions.', '1909.07547-1-43-0': 'Repetition: Repetitiveness has been frequently identified as a shortcoming of dialog models trained with MLE [CITATION].', '1909.07547-1-43-1': 'We adopt a measure of repetitiveness recently proposed by [CITATION] [CITATION], which was shown to be highly related to human judgments of conversation quality and engagement.', '1909.07547-1-43-2': 'Unlike previous work, we directly optimize for this metric using RL, rather than relying on conditional generation.', '1909.07547-1-43-3': 'To discourage repetition, our model receives a negative reward for repeating words it has produced in previous turns, excluding stop words and question words.', '1909.07547-1-44-0': 'Semantic Similarity: Paraphrasing and style matching are important in facilitating good conversation [CITATION], however most dialog models are not good at conditioning effectively on the conversation context [CITATION].', '1909.07547-1-44-1': 'Therefore, we reward the cosine similarity between the simulated user and bot utterances in embedding space, as in [CITATION] [CITATION].', '1909.07547-1-44-2': 'However, instead of using word2vec embeddings we make use of the Universal Sentence Encoder [CITATION] as it better correlates with human judgment when evaluating dialog quality [CITATION].', '1909.07547-1-45-0': 'Toxicity: Open-domain dialog systems generate malicious, offensive, and biased language when trained on standard datasets scraped from online forums and movie scripts [CITATION].', '1909.07547-1-45-1': 'We address this issue by penalizing our model for producing toxic responses as determined by a Naive Bayes-Logistic Regression classifier [CITATION] trained on a dataset of 160K comments from the Toxic Comment Classification Challenge.', '1909.07547-1-45-2': 'The comments are labeled for toxicity, identity hate, obscenity, threats, and insults.', '1909.07547-1-45-3': 'We provide the probability of toxicity as a negative reward to penalize our dialog model for producing toxic responses.', '1909.07547-1-46-0': '# Experiments', '1909.07547-1-47-0': 'The goal of our experiments is to evaluate the effectiveness of REINFORCE and VHRL for optimizing utterance level metrics like reducing toxicity and improving interaction quality.', '1909.07547-1-47-1': 'We carry out our analysis relying on both interactive human evaluation and automatic metrics.', '1909.07547-1-48-0': 'All of our models are trained on a corpus of 109K conversations scraped from www.reddit.com/r/CasualConversations, which was shown to result in higher conversation quality than traditional datasets such as Cornell movie dialogs [CITATION].', '1909.07547-1-48-1': 'We create two language model baselines by training on this dataset.', '1909.07547-1-48-2': 'The first is VHRED [CITATION], described in Section [REF].', '1909.07547-1-48-3': 'The second is a Transformer model [CITATION] of comparable capacity for reference.', '1909.07547-1-48-4': 'We base our implementation of the Transformer on ParlAI [CITATION], but incorporate the variable learning rate schedule originally proposed by [CITATION] [CITATION], which is important for state-of-the-art performance.', '1909.07547-1-49-0': 'We test these language models against three RL techniques.', '1909.07547-1-49-1': 'We incorporate transfer learning by initializing all of our RL models with the pre-trained weights of the VHRED model.', '1909.07547-1-49-2': 'Our REINFORCE model applies the REINFORCE algorithm described in Section [REF] at the word-level, to affect the probability of generating each word in the output.', '1909.07547-1-49-3': 'In contrast, VHRL incorporates additional rewards at the utterance-level to improve the continuous utterance embedding [MATH].', '1909.07547-1-49-4': 'We compare these methods with a recently proposed approach for learning offline from a static batch of conversation data, Batch [MATH]-learning [CITATION].', '1909.07547-1-49-5': 'Finally, we include an ablated version of the VHRL model that uses decoupled training; i.e. training alternates between optimizing the different levels of the hierarchy (manager and worker), with the crucial difference that the worker gradients are stopped so they do not update the manager.', '1909.07547-1-49-6': 'This Decoupled VHRL ablation is more typical of standard HRL approaches used in maze and Atari games [CITATION].', '1909.07547-1-49-7': 'All of the code for our models is open-source at https://github.com/natashamjaques/neural_chat, and the hyperparameters for the models are given in the Supplemental Material.', '1909.07547-1-50-0': '## Human Evaluation', '1909.07547-1-51-0': 'In addition to computing automatic measures, we conduct an interactive human evaluation, in which human users are able to chat with each of our bots in an open-ended, free-form setting.', '1909.07547-1-51-1': ""After chatting for at least three turns with each bot, users can end the conversation and rate the overall conversation quality, fluency, the diversity of the responses, and the degree to which the bot's responses were contingent on the user's input."", '1909.07547-1-51-2': 'Because users can choose to chat as long as they like with any particular bot, we also measure chat length as a sign of engagement, following prior work [CITATION].', '1909.07547-1-52-0': 'We conduct two human evaluations by recruiting 100 Mechanical Turk workers to evaluate models on an open-source online platform at https://neural.chat/ [CITATION].', '1909.07547-1-52-1': 'We are releasing all of the models for both studies publicly; rather than cherry picking samples from the models, we encourage the reader to chat with each of the bots.', '1909.07547-1-52-2': 'The first study compares the quality of the proposed reward functions: https://neural.chat/vhrl_rewards/.', '1909.07547-1-52-3': 'The second study assesses the efficacy of the six proposed techniques when optimizing for all of the rewards jointly: https://neural.chat/vhrl_techniques/.', '1909.07547-1-52-4': 'We argue that this form of evaluation represents a more ambitious test of generalization than is typically attempted when deep RL algorithms are tested in the same environment in which they were trained, since human users are free to type any text they choose, about any topic.', '1909.07547-1-53-0': '# Results and Discussion', '1909.07547-1-54-0': 'We first assess whether RL training can allow dialog agents to learn to optimize arbitrary, non-differential metrics of conversation quality.', '1909.07547-1-54-1': 'Table [REF] in Section [REF] shows samples of conversations from VHRL trained with each of the rewards, and Figure [REF] shows the performance of the RL and baseline models on those five metrics.', '1909.07547-1-54-2': 'The RL models are able to dramatically improve generated conversations above the baseline VHRED model with which they are initialized, improving sentiment and semantic similarity, asking more questions, and reducing repetition and toxicity.', '1909.07547-1-54-3': 'The RL models also improve significantly over the performance of the Transformer model on these metrics, with the exception of the semantic similarity metric.', '1909.07547-1-54-4': 'We note that compared to the VHRED architecture, the Transformer has higher similarity but is also much more repetitive.', '1909.07547-1-55-0': 'While both REINFORCE and VHRL are equally able to learn to optimize toxicity and sentiment, VHRL out-performs REINFORCE on repetition and semantic similarity.', '1909.07547-1-55-1': 'We believe this is because sentiment and toxicity are more closely linked to the choice of words used to form a sentence, and thus are able to be learned at the word-level.', '1909.07547-1-55-2': 'In contrast, modeling whether a sentence has occurred earlier in the conversation and is thus being repeated is much harder to learn at a word-level granularity, and can be optimized more easily at the utterance-level using VHRL.', '1909.07547-1-55-3': 'Similarly, making a response more similar to the previous response is also better modeled at the utterance-level.', '1909.07547-1-56-0': 'Note that REINFORCE outperforms VHRL on the question metric.', '1909.07547-1-56-1': 'This is because the model quickly learns to ask a single, repeated question, allowing it to trivially optimize the reward function.', '1909.07547-1-56-2': 'Using reward functions which are too easily exploitable can limit the effectiveness of RL for dialog, a finding also noted in [CITATION] [CITATION].', '1909.07547-1-56-3': 'Here we propose new reward functions, such as toxicity, that are less easy to exploit.', '1909.07547-1-56-4': 'By optimizing a combination of these rewards with sometimes conflicting objectives (as we explore in Section [REF]), we can show that the reward function is difficult to trivially exploit, as suggested by [CITATION] [CITATION].', '1909.07547-1-57-0': 'As an additional post-hoc test of whether reducing our toxicity metric actually reduces undesirable behaviors in dialog models, we count the number of inappropriate and swear words used by each model in response to the 10,000 utterances in the test set.', '1909.07547-1-57-1': 'Figure [REF] shows the results.', '1909.07547-1-57-2': 'The baseline VHRED model uses a swear word in 1.5% of the responses, while using RL to lower the toxicity reduces swearing to less than one third of the original amount.', '1909.07547-1-58-0': 'We conducted an interactive human evaluation, as described in Section [REF], in order to assess how well the proposed reward functions relate to human judgments of conversation quality when optimized with REINFORCE or VHRL; the results are presented in Table [REF].', '1909.07547-1-58-1': 'Each bot was trained with respect to one of the 5 reward functions, and the results are ordered from least to highest scoring rewards in terms of summed human ratings.', '1909.07547-1-58-2': 'As is evident in the table, the VHRL model trained to optimize for asking questions achieved the highest ratings, followed by VHRL minimizing repetition, and VHRL minimizing toxicity.', '1909.07547-1-58-3': 'This provides evidence that our proposed rewards lead to enhanced conversation quality as judged by human users, and that VHRL provides the most effective method for learning them.', '1909.07547-1-59-0': '## Comparison of RL techniques for learning combined reward', '1909.07547-1-60-0': 'As described in the previous section, optimizing for an overly simplistic metric (such as asking questions) can lead algorithms such as REINFORCE to trivially exploit the reward function at the expense of conversation quality.', '1909.07547-1-60-1': 'Here, we train a variety of models to optimize for a combination of all five proposed rewards, making the reward function more complex and less easily exploited.', '1909.07547-1-60-2': 'However, it is clear that the five metrics proposed here do not fully encompass what it means to have a good conversation with a human user when optimized separately.', '1909.07547-1-60-3': 'Previous work found that optimizing for a set of conversation metrics with RL can actually reduce human judgments of conversation quality below the score of the MLE baseline [CITATION].', '1909.07547-1-61-0': 'Therefore, we test whether our RL models can improve on the VHRED baseline and Transformer in terms of human judgment by optimizing for a weighted sum of the proposed rewards.', '1909.07547-1-61-1': 'The results are shown in Table [REF], which is ordered from least to highest summed human ratings.', '1909.07547-1-61-2': 'All models proposed here, including the MLE baselines, outperform prior work by [CITATION] [CITATION].', '1909.07547-1-61-3': 'The ablated version of our approach, Decoupled VHRL, performs poorly, suggesting our proposed joint training scheme for VHRL is an important component of the algorithm.', '1909.07547-1-62-0': 'VHRED, REINFORCE, and VHRL all exceed the performance of Transformer in terms of human judgments of conversation quality.', '1909.07547-1-62-1': 'While recent advances in language modeling with Transformer architectures (e.g. [CITATION] [CITATION]) are quite promising, translating these successes into conversational AI is still an ongoing area of research.', '1909.07547-1-62-2': 'Models such as VHRED have an architecture designed for the dyadic nature of the conversation (with an explicit update based on each conversation turn); in contrast, the Transformer has no special architectural features to denote conversation turns beyond the [MATH]end of utterance[MATH] token.', '1909.07547-1-62-3': 'Recurrent models have been shown to be more adept at capturing and exploiting hierarchical information in language [CITATION].', '1909.07547-1-62-4': 'Further, Transformers have been shown to be less sensitive to the conversation history relative to recurrent models [CITATION].', '1909.07547-1-62-5': ""We have observed that the Transformer is highly repetitive, and vulnerable to monologuing rather than generating answers contingent on the user's input."", '1909.07547-1-62-6': 'Although it has a high score on the semantic similarity metric in Figure [REF], Table [REF] demonstrates that this does not translate into improved contingency ratings.', '1909.07547-1-63-0': 'Finally, in comparing the RL techniques to the VHRED baseline on which they were based, we see that a naive application of the REINFORCE algorithm does not lead to overall improvements in human ratings of conversation quality.', '1909.07547-1-63-1': 'While the language generated by the REINFORCE model is less toxic and more positive, this comes at the cost of a slight reduction in overall conversation quality.', '1909.07547-1-63-2': 'In contrast, VHRL is the only technique that allows the model to optimize for reducing toxicity, improving sentiment, etc., while increasing the overall human judgments of the quality of the conversation.', '1909.07547-1-63-3': 'Note that the chat length is higher with VHRL, suggesting users are more interested and engaged when chatting with VHRL versus the other models.', '1909.07547-1-63-4': 'Thus, VHRL can be used to optimize for metrics that make the dialog model more safe and appropriate for a particular application domain, while maintaining the ability to have an enjoyable and engaging conversation with human users.', '1909.07547-1-64-0': '# Conclusion', '1909.07547-1-65-0': 'We have demonstrated that RL can be used to improve the outputs of an open-domain dialog model with respect to human-centered metrics of conversation quality.', '1909.07547-1-65-1': 'For example, RL can reduce the toxicity of the generated language, a problem that has previously hindered deployment of these systems to the real world.', '1909.07547-1-65-2': 'By developing metrics tailored to a particular application domain (for example, increasing politeness for a technical-support system), these techniques could be used to help open-domain dialog models integrate with real-world products.', '1909.07547-1-65-3': 'We have shown that our proposed VHRL technique is most effective for optimizing long-term conversation rewards, and for improving conversation quality while improving metrics like toxicity.', '1909.07547-1-66-0': '9.0pt'}","{'1909.07547-2-0-0': 'Open-domain dialog generation is a challenging problem; maximum likelihood training can lead to repetitive outputs, models have difficulty tracking long-term conversational goals, and training on standard movie or online datasets may lead to the generation of inappropriate, biased, or offensive text.', '1909.07547-2-0-1': 'Reinforcement Learning (RL) is a powerful framework that could potentially address these issues, for example by allowing a dialog model to optimize for reducing toxicity and repetitiveness.', '1909.07547-2-0-2': 'However, previous approaches which apply RL to open-domain dialog generation do so at the word level, making it difficult for the model to learn proper credit assignment for long-term conversational rewards.', '1909.07547-2-0-3': 'In this paper, we propose a novel approach to hierarchical reinforcement learning, VHRL, which uses policy gradients to tune the utterance-level embedding of a variational sequence model.', '1909.07547-2-0-4': 'This hierarchical approach provides greater flexibility for learning long-term, conversational rewards.', '1909.07547-2-0-5': 'We use self-play and RL to optimize for a set of human-centered conversation metrics, and show that our approach provides significant improvements - in terms of both human evaluation and automatic metrics - over state-of-the-art dialog models, including Transformers.', '1909.07547-2-1-0': '# Introduction', '1909.07547-2-2-0': 'Since the inception of the Turing test, generating convincing open-ended (open-domain) dialog has been a fundamental challenge in artificial intelligence (AI).', '1909.07547-2-2-1': 'A successful open-domain dialog system could provide enormous value enabling more natural human-computer interaction through a wide variety of open-ended voice interfaces.', '1909.07547-2-2-2': 'Successful dialog models could also unlock new, beneficial applications of AI, such as companion chatbots for therapy applications.', '1909.07547-2-3-0': 'However, current generative models for dialog suffer from several shortcomings that limit their usefulness in the real world.', '1909.07547-2-3-1': 'Training on existing standard dialog datasets collected online or from movie scripts often leads to models that produce malicious, aggressive, biased, or offensive responses [CITATION].', '1909.07547-2-3-2': 'There are no guarantees on the quality and sensitivity of the generated text, often preventing open-domain dialog systems from being deployed for safety-critical applications such as mental health, or even for customer service.', '1909.07547-2-3-3': 'Further, maximum likelihood estimation (MLE) training of such models often leads to the generation of dull and repetitive text [CITATION].', '1909.07547-2-3-4': 'Finally, models may have difficulty tracking long-term aspects of the conversation, and evidence has shown that they do not adequately condition on the conversation history in generating responses [CITATION].', '1909.07547-2-4-0': 'Reinforcement Learning (RL) is a powerful paradigm that allows dialog models to optimize for non-differentiable metrics of conversation quality, and thereby helps overcome the above problems.', '1909.07547-2-4-1': 'In this paper, we use RL to learn from self-play; the model talks to a fixed copy of itself, and computes reward functions on the generated conversation.', '1909.07547-2-4-2': 'We propose novel, human-centered reward functions, such as minimizing the estimated toxicity of a conversation, in order to limit inappropriate, biased, and offensive responses.', '1909.07547-2-4-3': 'We also design rewards based on the psychology of good conversation (e.g. [CITATION] [CITATION]), and reward recently proposed conversation metrics that are associated with improved human judgments of conversation quality [CITATION].', '1909.07547-2-4-4': 'Table [REF] shows how RL can be used to optimize our proposed metrics, shifting the conversation towards more positive and engaging conversation, with less repetition and toxicity.', '1909.07547-2-5-0': 'Applying RL to open-domain dialog generation is a challenging problem, however.', '1909.07547-2-5-1': 'Most prior approaches (e.g. [CITATION] [CITATION]) learn to model rewards at the word-level, meaning that the reward is applied to affect the probability of generating each word in the response.', '1909.07547-2-5-2': 'Such low-level control makes determining which words lead to higher rewards that are computed based on multiple conversation turns (i.e. word-level credit assignment) excessively challenging.', '1909.07547-2-5-3': ""Further, this approach makes the number of actions available to the RL agent equivalent to the number of words in the dialog model's vocabulary (20,000 in our case)."", '1909.07547-2-5-4': 'Thus there are several orders of magnitude more actions than in typical successful applications of deep RL, such as learning to play an Atari game with around 20 actions (e.g. [CITATION] [CITATION]).', '1909.07547-2-6-0': 'To overcome these challenges, we leverage hierarchical reinforcement learning (HRL) to learn to model rewards at the utterance level, improving the flexibility of dialog models to learn long-term, conversational rewards.', '1909.07547-2-6-1': 'Specifically, we propose a novel approach, Variational Hierarchical Reinforcement Learning (or VHRL), which uses policy gradients to adjust the prior probability distribution of the latent variable learned at the utterance level of a hierarchical variational model.', '1909.07547-2-6-2': 'We show that this approach allows for improved learning of conversational rewards that are not modeled well at the word level.', '1909.07547-2-7-0': 'To evaluate our models, we not only compute automatic metrics, but conduct an interactive human evaluation using the https://neural.chat/ platform [CITATION], in which humans chat live with our bots about anything they choose.', '1909.07547-2-7-1': 'This represents a more realistic test of real-world generalization performance than is typically employed when testing RL models in the same environment in which they were trained.', '1909.07547-2-7-2': 'Our evaluation reveals that VHRL improves human judgments of conversational quality above state-of-the-art dialog architectures, including Transformer-based models.', '1909.07547-2-8-0': 'In summary, the paper makes the following contributions: a) Develops a new technique, VHRL, for hierarchical control of variational dialog models; b) Demonstrates the effectiveness of training open-domain dialog models with VHRL and self-play, showing improvements over state-of-the-art dialog architectures with both human evaluation and automatic analysis; and c) Introduces and compares several reward functions for guiding conversations to be less toxic and repetitive, and more engaging, positive, contingent on user input.', '1909.07547-2-8-1': 'We also release code for our evaluation platform and models at https://github.com/natashamjaques/neural_chat.', '1909.07547-2-9-0': '# Related Work', '1909.07547-2-10-0': 'The difficulty of training open-domain dialog systems is exacerbated by the fact that we lack reliable automatic metrics with which to evaluate dialog quality [CITATION].', '1909.07547-2-10-1': 'Recently, authors have begun to propose new metrics of conversation quality (e.g. [CITATION] [CITATION]), and have even proposed evaluating metrics on conversations generated with self-play [CITATION].', '1909.07547-2-10-2': 'However, these works have not attempted to directly optimize for these non-differentiable metrics with RL.', '1909.07547-2-11-0': 'There has been significant progress in improving dialog generation outside of RL.', '1909.07547-2-11-1': 'The most popular of these approaches is conditional training, where a learned embedding vector is associated with a desired output feature and fed into the decoder to control generation (e.g. [CITATION] [CITATION]).', '1909.07547-2-11-2': 'This approach has multiple limitations.', '1909.07547-2-11-3': 'First, the model can only learn associations present in the training data, and cannot explore to discover improved methods for optimizing the desired features.', '1909.07547-2-11-4': 'Second, the conditional embeddings are learned at training time with teacher forcing and thus suffer from exposure bias [CITATION].', '1909.07547-2-11-5': 'Using RL avoids these limitations as it allows exploring regions of space not present in the training data and directly optimizing for rewards at inference time.', '1909.07547-2-11-6': 'In addition, RL learns the total expected future reward of taking some action now, given how the rest of the conversation will unfold in the future.', '1909.07547-2-11-7': 'This allows RL models to make long-term trade-offs about the benefits of generating words and utterances in the conversation context.', '1909.07547-2-11-8': 'Finally, our approach does not require the addition of any new parameters or complex components.', '1909.07547-2-11-9': 'Instead, it can be use to tune pre-existing models to output better, more appropriate responses.', '1909.07547-2-12-0': '## Reinforcement Learning for Dialog', '1909.07547-2-13-0': 'Improving dialog models with RL is a difficult problem, and past work has largely been restricted to task-oriented dialog systems, which have a limited number of task-specific actions (e.g. [CITATION] [CITATION]).', '1909.07547-2-13-1': 'Attempts to apply RL to open-domain dialog generation are less common.', '1909.07547-2-13-2': ""Even in this setting, authors often use a highly restricted action space; for example, using RL to choose dialog acts for conditional generation [CITATION] or to choose which models to invoke to answer a user's query [CITATION]."", '1909.07547-2-14-0': '[CITATION] [CITATION] applied deep RL to the full vocabulary-sized action space, optimizing for scripted rewards such as ease of answering.', '1909.07547-2-14-1': 'RL has also been used to optimize for rewards based on a GAN-like discriminator trained to distinguish human-generated from model-generated text [CITATION].', '1909.07547-2-14-2': 'Note that it is difficult to apply the traditional GAN approach of backpropagating discriminator gradients directly into the generator, because the sampling procedure used to generate each word in an utterance does not preserve gradients.', '1909.07547-2-14-3': 'Therefore even differentiable models which predict conversation metrics from a sampled utterance cannot be directly applied to optimize a model without RL.', '1909.07547-2-15-0': 'Sentiment has been used as a reward in an RL framework, for example to improve a seq2seq model [CITATION].a', '1909.07547-2-15-1': 'authorjaques2019way jaques2019way optimize for sentiment and several other conversation metrics by learning from a static batch of human-bot conversation data using Batch RL.', '1909.07547-2-15-2': 'We believe we are the first to propose using RL to reduce toxicity in an open-domain dialog setting, in order to ensure the model produces more appropriate and safe conversations.', '1909.07547-2-16-0': 'Hierarchical models have been investigated extensively in the context of MLE language modeling [CITATION].', '1909.07547-2-16-1': 'These architectures take advantage of the natural hierarchical structure of language, decomposing input into utterances at one level, and words at another.', '1909.07547-2-16-2': 'However, attempts to apply hierarchical RL (HRL) to dialog generation have so far been limited to task-oriented dialog systems [CITATION].', '1909.07547-2-16-3': 'To the best of our knowledge, we are the first to apply HRL to open-domain dialog generation.', '1909.07547-2-17-0': '## Hierarchical Reinforcement Learning', '1909.07547-2-18-0': 'Many approaches have been proposed for building hierarchical agents within the context of reinforcement learning for games and robotics [CITATION].', '1909.07547-2-18-1': 'The options framework proposed by [CITATION] [CITATION] is one popular approach for HRL.', '1909.07547-2-18-2': 'At the bottom level of the hierarchy, a set of options (or workers) which are policies over actions interact with the environment until terminated by the agent.', '1909.07547-2-18-3': 'At the top level, a policy over options (or manager) selects options to be executed until termination, at which point another option is picked and the process is repeated.', '1909.07547-2-18-4': 'The different levels of temporal abstraction introduced by this hierarchy allows for better long-term planning relative to traditional, flat reinforcement learning techniques.', '1909.07547-2-19-0': 'A major focus of HRL has been on sub-goal or option discovery for training worker policies.', '1909.07547-2-19-1': 'Bottom-level policies are often learned using handcrafted sub-goals [CITATION], intrinsic rewards [CITATION], or pseudo-rewards [CITATION], while the manager policy is learned using extrinsic rewards from the environment.', '1909.07547-2-19-2': 'Our approach also allows for optimizing different rewards at different levels of the hierarchy, thus creating distinct goals for the worker and the manager.', '1909.07547-2-19-3': 'However, unlike other HRL approaches we expose both the worker and manager policies to extrinsic rewards and add weight hyper-parameters to regulate the effect of the rewards at each level.', '1909.07547-2-19-4': 'This remedies a weakness of pseudo-reward methods where a worker only focuses on achieving its sub-goals while disregarding the effect on the extrinsic environment reward.', '1909.07547-2-20-0': '# Background', '1909.07547-2-21-0': 'A common approach to dialog modeling is to use a hierarchical seq2seq architecture, such as the Variational Hierarchical Recurrent Encoder Decoder (VHRED) [CITATION].', '1909.07547-2-21-1': 'We adopt VHRED here, following previous work which has found it to be the most effective version of several related architectures [CITATION].', '1909.07547-2-22-0': 'As shown in Figure [REF], VHRED uses three recurrent networks to generate the next utterance in a conversation.', '1909.07547-2-22-1': 'The word-level encoder RNN operates on the words (tokens) of the input utterance [MATH], and encodes them into a representation [MATH].', '1909.07547-2-22-2': 'This is fed into a context RNN, which forms the upper level of the hierarchy - it is updated only after each utterance, rather than each token.', '1909.07547-2-22-3': 'Because it updates less frequently, the context RNN is potentially better able to track longer-term aspects of the conversation.', '1909.07547-2-22-4': 'The context RNN outputs [MATH], which is used to produce an utterance embedding [MATH].', '1909.07547-2-22-5': 'This is fed into the word-level decoder RNN [MATH], which produces the output utterance [MATH], one token at a time.', '1909.07547-2-23-0': 'The model is similar to a variational autoencoder; [MATH] is fed into fully connected layers that predict the mean [MATH] and variance [MATH] of a multivariate normal distribution.', '1909.07547-2-23-1': 'Through a KL-divergence constraint and the reparameterization trick, the model learns a probability distribution over the embedding vector [MATH] of each utterance, [MATH].', '1909.07547-2-23-2': 'Formally, the model can be described as follows: [EQUATION]', '1909.07547-2-24-0': '## Reinforcement Learning', '1909.07547-2-25-0': 'We adopt the standard reinforcement learning framework where given the environment state [MATH], an agent takes an action [MATH] according to its policy [MATH], and receives a reward [MATH].', '1909.07547-2-25-1': 'The environment then transitions to the next state according to the transition function [MATH].', '1909.07547-2-25-2': 'The agent seeks to maximize the total expected future reward (long-term return): [EQUATION] given a starting state [MATH] and a discount factor [MATH].', '1909.07547-2-26-0': '## Policy Gradient Methods', '1909.07547-2-27-0': 'Policy gradient methods learn parameterized policies [MATH] for solving RL problems with [MATH] being a learned parameter vector.', '1909.07547-2-27-1': 'The policy gradient theorem [CITATION] derives the gradient of the expected return with respect to the policy parameters.', '1909.07547-2-27-2': 'In this paper, we use REINFORCE [CITATION] which approximates the gradient at each time step [MATH] using [EQUATION] where [MATH] is the observed future reward for an episode that ends at [MATH].', '1909.07547-2-27-3': 'The expected return is maximized with gradient ascent.', '1909.07547-2-27-4': 'This is equivalent to minimizing the loss function [MATH].', '1909.07547-2-28-0': 'In continuous action spaces, actions [MATH] are sampled from a continuous probability distribution, such as a multivariate normal distribution.', '1909.07547-2-29-0': 'In this case, the policy [MATH] can be parameterized as a probability density function over actions, [EQUATION] where the actions are sampled from a multivariate normal [MATH].', '1909.07547-2-29-1': 'Here the mean [MATH] and covariance matrix [MATH] are defined in terms of the current state [MATH] and the policy parameters [MATH].', '1909.07547-2-29-2': 'The density of the probability of actions, rather than the probability, is learned in the continuous case.', '1909.07547-2-29-3': 'We refer the reader to [CITATION] [CITATION] for more details on extending policy gradient methods to the continuous case.', '1909.07547-2-30-0': '# Approach', '1909.07547-2-31-0': 'We pose dialog generation as an RL problem where the state, [MATH], is all the previous dialog turns read by the model up to utterance [MATH], and the rewards are calculated based on the dialog history and generated utterance.', '1909.07547-2-32-0': 'Previous approaches which have applied RL to language generation have done so at the word level, where the policy [MATH] models the distribution over outputting the next word [CITATION].', '1909.07547-2-32-1': 'Instead, we cast our problem in the hierarchical reinforcement learning framework by considering the context RNN as the manager responsible for utterance-level decisions, and the decoder RNN as the worker responsible for word-level decisions.', '1909.07547-2-33-0': 'We leverage the fact that VHRED learns a probability distribution over latent variable [MATH] as a decision making component at the utterance level.', '1909.07547-2-33-1': 'Starting with a pre-trained VHRED model, we apply REINFORCE to tune the variational component, treating [MATH] as a continuous action.', '1909.07547-2-33-2': 'Thus, the manager policy is defined by the distribution of the prior latent variable [MATH], while the worker policy is the distribution of the output words [MATH], which is parameterized by the manager decisions.', '1909.07547-2-34-0': ""More specifically, the probability of a worker action [MATH] is the joint probability of the generated utterance conditioned on the manager's decision [MATH], [EQUATION] while the probability of a manager action is given by the multivariate normal probability density function in Eq. [REF]."", '1909.07547-2-35-0': ""We propose a new approach which allows both the worker and manager to jointly optimize total expected future return by minimizing the following loss: [EQUATION] where [MATH] is the manager's observed future reward and [MATH] is the worker's observed future reward."", '1909.07547-2-35-1': ""This formulation is analogous to REINFORCE as it shifts the model's decisions towards actions associated with positive rewards and discourages actions associated with negative rewards."", '1909.07547-2-35-2': 'The scalars [MATH] are hyperparameters used to regulate the effect of the rewards at each level of the hierarchy.', '1909.07547-2-35-3': 'We call our approach Variational Hierarchical Reinforcement Learning (VHRL).', '1909.07547-2-36-0': 'Unlike recently proposed HRL approaches which train the worker and manager separately as decoupled components [CITATION], we train our entire model jointly, end-to-end.', '1909.07547-2-36-1': 'This implies that the worker (decoder RNN) gradients flow through the manager (context RNN), and both flow through the encoder RNN.', '1909.07547-2-36-2': 'We make this decision for two reasons.', '1909.07547-2-36-3': 'First, [MATH] lives in a continuous high dimensional action space, making it difficult to learn a good policy [MATH] without a signal from the decoder.', '1909.07547-2-36-4': 'Second, this gives the decoder control over the representations learned by the encoder, facilitating optimization.', '1909.07547-2-36-5': 'As an ablation study, we experiment with decoupled decoder and encoder training, and find that the joint approach performs better.', '1909.07547-2-37-0': 'The proposed loss allows for optimizing different rewards at different levels, which can be used to incorporate prior knowledge about the problem.', '1909.07547-2-37-1': 'For example, rewards relevant to the global dialog history could be considered only by the manager through [MATH], rather than the worker.', '1909.07547-2-37-2': 'Conversely, rewards relevant to the word-by-word output could be considered by the worker through [MATH] and not the manager.', '1909.07547-2-37-3': 'For simplicity, we optimize for all rewards at both levels (i.e. [MATH]) and achieve promising results.', '1909.07547-2-38-0': 'Similar to previous work applying RL to dialog [CITATION] we use self-play to simulate the interactive environment in which the agent learns.', '1909.07547-2-38-1': 'We initialize conversations with randomly sampled starting sentences from the training set and let our model interact with a user simulator which is a fixed copy of itself.', '1909.07547-2-38-2': 'We limit each model to 3 turns for a total conversation length of 7 utterances.', '1909.07547-2-38-3': 'Limiting the length of simulated interactions is important since we found that long conversations are more likely to degenerate and go off-topic.', '1909.07547-2-39-0': '# Conversation Metrics', '1909.07547-2-40-0': 'Here we introduce several metrics for improving the quality of a conversation, which can be optimized using RL by treating them as rewards.', '1909.07547-2-40-1': 'Several metrics are inspired by previous work, but we also propose novel metrics such as toxicity.', '1909.07547-2-41-0': 'Sentiment: Emotion is important for creating a sense of understanding in human conversation [CITATION].', '1909.07547-2-41-1': 'Building on previous work which used sentiment as a reward (e.g.authorshin2019happybot,jaques2019way shin2019happybot,jaques2019way, we leverage a state-of-the-art sentiment detector, DeepMoji [CITATION], to reward generated utterances associated with positive sentiment emojis.', '1909.07547-2-41-2': 'The sentiment score is computed using the weights proposed byauthorghandeharioun2019approximating ghandeharioun2019approximating.', '1909.07547-2-42-0': 'Question: Asking questions is an important active listening skill, and can improve the quality of interactions [CITATION].', '1909.07547-2-42-1': 'Thus, we provide a positive reward when both a question word and a question mark are present in a generated response to encourage asking questions.', '1909.07547-2-43-0': 'Repetition: Repetitiveness has been frequently identified as a shortcoming of dialog models trained with MLE [CITATION].', '1909.07547-2-43-1': 'We adopt a measure of repetitiveness recently proposed by [CITATION] [CITATION], which was shown to be highly related to human judgments of conversation quality and engagement.', '1909.07547-2-43-2': 'Unlike previous work, we directly optimize for this metric using RL, rather than relying on conditional generation.', '1909.07547-2-43-3': 'To discourage repetition, our model receives a negative reward for repeating words it has produced in previous turns, excluding stop words and question words.', '1909.07547-2-44-0': 'Semantic Similarity: Paraphrasing and style matching are important in facilitating good conversation [CITATION], however most dialog models are not good at conditioning effectively on the conversation context [CITATION].', '1909.07547-2-44-1': 'Therefore, we reward the cosine similarity between the simulated user and bot utterances in embedding space, as in [CITATION] [CITATION].', '1909.07547-2-44-2': 'However, instead of using word2vec embeddings we make use of the Universal Sentence Encoder [CITATION] as it better correlates with human judgment when evaluating dialog quality [CITATION].', '1909.07547-2-45-0': 'Toxicity: Open-domain dialog systems generate malicious, offensive, and biased language when trained on standard datasets scraped from online forums and movie scripts [CITATION].', '1909.07547-2-45-1': 'We address this issue by penalizing our model for producing toxic responses as determined by a Naive Bayes-Logistic Regression classifier [CITATION] trained on a dataset of 160K comments from the Toxic Comment Classification Challenge.', '1909.07547-2-45-2': 'The comments are labeled for toxicity, identity hate, obscenity, threats, and insults.', '1909.07547-2-45-3': 'We provide the probability of toxicity as a negative reward to penalize our dialog model for producing toxic responses.', '1909.07547-2-46-0': '# Experiments', '1909.07547-2-47-0': 'The goal of our experiments is to evaluate the effectiveness of REINFORCE and VHRL for optimizing utterance level metrics like reducing toxicity and improving interaction quality.', '1909.07547-2-47-1': 'We carry out our analysis relying on both interactive human evaluation and automatic metrics.', '1909.07547-2-48-0': 'All of our models are trained on a corpus of 109K conversations scraped from www.reddit.com/r/CasualConversations, which was shown to result in higher conversation quality than traditional datasets such as Cornell movie dialogs [CITATION].', '1909.07547-2-48-1': 'We create two language model baselines by training on this dataset.', '1909.07547-2-48-2': 'The first is VHRED [CITATION], described in Section [REF].', '1909.07547-2-48-3': 'The second is a Transformer model [CITATION] of comparable capacity for reference.', '1909.07547-2-48-4': 'We base our implementation of the Transformer on ParlAI [CITATION], but incorporate the variable learning rate schedule originally proposed by [CITATION] [CITATION], which is important for state-of-the-art performance.', '1909.07547-2-49-0': 'We test these language models against three RL techniques.', '1909.07547-2-49-1': 'We incorporate transfer learning by initializing all of our RL models with the pre-trained weights of the VHRED model.', '1909.07547-2-49-2': 'Our REINFORCE model applies the REINFORCE algorithm described in Section [REF] at the word-level, to affect the probability of generating each word in the output.', '1909.07547-2-49-3': 'In contrast, VHRL incorporates additional rewards at the utterance-level to improve the continuous utterance embedding [MATH].', '1909.07547-2-49-4': 'We compare these methods with a recently proposed approach for learning offline from a static batch of conversation data, Batch [MATH]-learning [CITATION].', '1909.07547-2-49-5': 'Finally, we include an ablated version of the VHRL model that uses decoupled training; i.e. training alternates between optimizing the different levels of the hierarchy (manager and worker), with the crucial difference that the worker gradients are stopped so they do not update the manager.', '1909.07547-2-49-6': 'This Decoupled VHRL ablation is more typical of standard HRL approaches used in maze and Atari games [CITATION].', '1909.07547-2-49-7': 'All of the code for our models is open-source at https://github.com/natashamjaques/neural_chat, and the hyperparameters for the models are given in the Supplemental Material.', '1909.07547-2-50-0': '## Human Evaluation', '1909.07547-2-51-0': 'In addition to computing automatic measures, we conduct an interactive human evaluation, in which human users are able to chat with each of our bots in an open-ended, free-form setting.', '1909.07547-2-51-1': ""After chatting for at least three turns with each bot, users can end the conversation and rate the overall conversation quality, fluency, the diversity of the responses, and the degree to which the bot's responses were contingent on the user's input."", '1909.07547-2-51-2': 'Because users can choose to chat as long as they like with any particular bot, we also measure chat length as a sign of engagement, following prior work [CITATION].', '1909.07547-2-52-0': 'We conduct two human evaluations by recruiting 100 Mechanical Turk workers to evaluate models on an open-source online platform at https://neural.chat/ [CITATION].', '1909.07547-2-52-1': 'We are releasing all of the models for both studies publicly; rather than cherry picking samples from the models, we encourage the reader to chat with each of the bots.', '1909.07547-2-52-2': 'The first study compares the quality of the proposed reward functions: https://neural.chat/vhrl_rewards/.', '1909.07547-2-52-3': 'The second study assesses the efficacy of the six proposed techniques when optimizing for all of the rewards jointly: https://neural.chat/vhrl_techniques/.', '1909.07547-2-52-4': 'We argue that this form of evaluation represents a more ambitious test of generalization than is typically attempted when deep RL algorithms are tested in the same environment in which they were trained, since human users are free to type any text they choose, about any topic.', '1909.07547-2-53-0': '# Results and Discussion', '1909.07547-2-54-0': 'We first assess whether RL training can allow dialog agents to learn to optimize arbitrary, non-differential metrics of conversation quality.', '1909.07547-2-54-1': 'Table [REF] in Section [REF] shows samples of conversations from VHRL trained with each of the rewards, and Figure [REF] shows the performance of the RL and baseline models on those five metrics.', '1909.07547-2-54-2': 'The RL models are able to dramatically improve generated conversations above the baseline VHRED model with which they are initialized, improving sentiment and semantic similarity, asking more questions, and reducing repetition and toxicity.', '1909.07547-2-54-3': 'The RL models also improve significantly over the performance of the Transformer model on these metrics, with the exception of the semantic similarity metric.', '1909.07547-2-54-4': 'We note that compared to the VHRED architecture, the Transformer has higher similarity but is also much more repetitive.', '1909.07547-2-55-0': 'While both REINFORCE and VHRL are equally able to learn to optimize toxicity and sentiment, VHRL out-performs REINFORCE on repetition and semantic similarity.', '1909.07547-2-55-1': 'We believe this is because sentiment and toxicity are more closely linked to the choice of words used to form a sentence, and thus are able to be learned at the word-level.', '1909.07547-2-55-2': 'In contrast, modeling whether a sentence has occurred earlier in the conversation and is thus being repeated is much harder to learn at a word-level granularity, and can be optimized more easily at the utterance-level using VHRL.', '1909.07547-2-55-3': 'Similarly, making a response more similar to the previous response is also better modeled at the utterance-level.', '1909.07547-2-56-0': 'Note that REINFORCE outperforms VHRL on the question metric.', '1909.07547-2-56-1': 'This is because the model quickly learns to ask a single, repeated question, allowing it to trivially optimize the reward function.', '1909.07547-2-56-2': 'Using reward functions which are too easily exploitable can limit the effectiveness of RL for dialog, a finding also noted in [CITATION] [CITATION].', '1909.07547-2-56-3': 'Here we propose new reward functions, such as toxicity, that are less easy to exploit.', '1909.07547-2-56-4': 'By optimizing a combination of these rewards with sometimes conflicting objectives (as we explore in Section [REF]), we can show that the reward function is difficult to trivially exploit, as suggested by [CITATION] [CITATION].', '1909.07547-2-57-0': 'As an additional post-hoc test of whether reducing our toxicity metric actually reduces undesirable behaviors in dialog models, we count the number of inappropriate and swear words used by each model in response to the 10,000 utterances in the test set.', '1909.07547-2-57-1': 'Figure [REF] shows the results.', '1909.07547-2-57-2': 'The baseline VHRED model uses a swear word in 1.5% of the responses, while using RL to lower the toxicity reduces swearing to less than one third of the original amount.', '1909.07547-2-58-0': 'We conducted an interactive human evaluation, as described in Section [REF], in order to assess how well the proposed reward functions relate to human judgments of conversation quality when optimized with REINFORCE or VHRL; the results are presented in Table [REF].', '1909.07547-2-58-1': 'Each bot was trained with respect to one of the 5 reward functions, and the results are ordered from least to highest scoring rewards in terms of summed human ratings.', '1909.07547-2-58-2': 'As is evident in the table, the VHRL model trained to optimize for asking questions achieved the highest ratings, followed by VHRL minimizing repetition, and VHRL minimizing toxicity.', '1909.07547-2-58-3': 'This provides evidence that our proposed rewards lead to enhanced conversation quality as judged by human users, and that VHRL provides the most effective method for learning them.', '1909.07547-2-59-0': '## Comparison of RL techniques for learning combined reward', '1909.07547-2-60-0': 'As described in the previous section, optimizing for an overly simplistic metric (such as asking questions) can lead algorithms such as REINFORCE to trivially exploit the reward function at the expense of conversation quality.', '1909.07547-2-60-1': 'Here, we train a variety of models to optimize for a combination of all five proposed rewards, making the reward function more complex and less easily exploited.', '1909.07547-2-60-2': 'However, it is clear that the five metrics proposed here do not fully encompass what it means to have a good conversation with a human user when optimized separately.', '1909.07547-2-60-3': 'Previous work found that optimizing for a set of conversation metrics with RL can actually reduce human judgments of conversation quality below the score of the MLE baseline [CITATION].', '1909.07547-2-61-0': 'Therefore, we test whether our RL models can improve on the VHRED baseline and Transformer in terms of human judgment by optimizing for a weighted sum of the proposed rewards.', '1909.07547-2-61-1': 'The results are shown in Table [REF], which is ordered from least to highest summed human ratings.', '1909.07547-2-61-2': 'All models proposed here, including the MLE baselines, outperform prior work by [CITATION] [CITATION].', '1909.07547-2-61-3': 'The ablated version of our approach, Decoupled VHRL, performs poorly, suggesting our proposed joint training scheme for VHRL is an important component of the algorithm.', '1909.07547-2-62-0': 'VHRED, REINFORCE, and VHRL all exceed the performance of Transformer in terms of human judgments of conversation quality.', '1909.07547-2-62-1': 'While recent advances in language modeling with Transformer architectures (e.g. [CITATION] [CITATION]) are quite promising, translating these successes into conversational AI is still an ongoing area of research.', '1909.07547-2-62-2': 'Models such as VHRED have an architecture designed for the dyadic nature of the conversation (with an explicit update based on each conversation turn); in contrast, the Transformer has no special architectural features to denote conversation turns beyond the [MATH]end of utterance[MATH] token.', '1909.07547-2-62-3': 'Recurrent models have been shown to be more adept at capturing and exploiting hierarchical information in language [CITATION].', '1909.07547-2-62-4': 'Further, Transformers have been shown to be less sensitive to the conversation history relative to recurrent models [CITATION].', '1909.07547-2-62-5': ""We have observed that the Transformer is highly repetitive, and vulnerable to monologuing rather than generating answers contingent on the user's input."", '1909.07547-2-62-6': 'Although it has a high score on the semantic similarity metric in Figure [REF], Table [REF] demonstrates that this does not translate into improved contingency ratings.', '1909.07547-2-63-0': 'Finally, in comparing the RL techniques to the VHRED baseline on which they were based, we see that a naive application of the REINFORCE algorithm does not lead to overall improvements in human ratings of conversation quality.', '1909.07547-2-63-1': 'While the language generated by the REINFORCE model is less toxic and more positive, this comes at the cost of a slight reduction in overall conversation quality.', '1909.07547-2-63-2': 'In contrast, VHRL is the only technique that allows the model to optimize for reducing toxicity, improving sentiment, etc., while increasing the overall human judgments of the quality of the conversation.', '1909.07547-2-63-3': 'Note that the chat length is higher with VHRL, suggesting users are more interested and engaged when chatting with VHRL versus the other models.', '1909.07547-2-63-4': 'Thus, VHRL can be used to optimize for metrics that make the dialog model more safe and appropriate for a particular application domain, while maintaining the ability to have an enjoyable and engaging conversation with human users.', '1909.07547-2-64-0': '# Conclusion', '1909.07547-2-65-0': 'We have demonstrated that RL can be used to improve the outputs of an open-domain dialog model with respect to human-centered metrics of conversation quality.', '1909.07547-2-65-1': 'For example, RL can reduce the toxicity of the generated language, a problem that has previously hindered deployment of these systems to the real world.', '1909.07547-2-65-2': 'By developing metrics tailored to a particular application domain (for example, increasing politeness for a technical-support system), these techniques could be used to help open-domain dialog models integrate with real-world products.', '1909.07547-2-65-3': 'We have shown that our proposed VHRL technique is most effective for optimizing long-term conversation rewards, and for improving conversation quality while improving metrics like toxicity.', '1909.07547-2-66-0': '9.0pt'}","[['1909.07547-1-27-0', '1909.07547-2-27-0'], ['1909.07547-1-27-1', '1909.07547-2-27-1'], ['1909.07547-1-27-2', '1909.07547-2-27-2'], ['1909.07547-1-27-3', '1909.07547-2-27-3'], ['1909.07547-1-27-4', '1909.07547-2-27-4'], ['1909.07547-1-58-0', '1909.07547-2-58-0'], ['1909.07547-1-58-1', '1909.07547-2-58-1'], ['1909.07547-1-58-2', '1909.07547-2-58-2'], ['1909.07547-1-58-3', '1909.07547-2-58-3'], ['1909.07547-1-7-0', '1909.07547-2-7-0'], ['1909.07547-1-7-1', '1909.07547-2-7-1'], ['1909.07547-1-7-2', '1909.07547-2-7-2'], ['1909.07547-1-21-0', '1909.07547-2-21-0'], ['1909.07547-1-21-1', '1909.07547-2-21-1'], 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1507.08797,"{'1507.08797-1-0-0': '# Introduction', '1507.08797-1-1-0': 'Precision measurements at the Large Hadron Collider provide detailed information about the nature of the strong interaction and its role within the Standard Model.', '1507.08797-1-1-1': 'With new data already arriving from Run II, there is a growing need for higher-precision theoretical predictions for a variety of different observables.', '1507.08797-1-1-2': 'Recent years have seen considerable progress in this respect.', '1507.08797-1-1-3': 'Our ability to make predictions in perturbative QCD now covers most of the relevant [MATH] scattering processes at next-to-next-to-leading order (NNLO) precision [CITATION], as well as the example of inclusive Higgs production at N[MATH]LO [CITATION].', '1507.08797-1-1-4': 'Despite this, processes with more than two particles in the final state remain beyond the reach of current NNLO methods.', '1507.08797-1-2-0': 'Next-to-leading order (NLO) corrections to high multiplicity final states are by now commonplace in phenomenological studies.', '1507.08797-1-2-1': 'Such computations are possible thanks to automated techniques, which make use of integrand reduction [CITATION], recursive techniques [CITATION], (generalized) unitarity cuts [CITATION] and the known basis of scalar integrals.', '1507.08797-1-2-2': 'Processes with up to five coloured partons [CITATION] in the final state are feasible using on-shell methods that, by only working with the physical degrees of freedom, are efficient at controlling the complexity.', '1507.08797-1-3-0': 'Multi-loop calculations of [MATH] scattering processes [CITATION] have been quite successful using the more traditional approaches of Feynman diagrams and integration-by-parts identities (IBPs) [CITATION], though there are notable exceptions using unitarity cutting techniques [CITATION].', '1507.08797-1-3-1': 'The main focus for these processes has been in the evaluation of the resulting master integrals.', '1507.08797-1-3-2': 'At higher multiplicity, rapid growth in the complexity of the Feynman diagram representation motivates an alternative approach that makes use of the lessons learned during the automation of one-loop computations.', '1507.08797-1-4-0': 'Two methods have already been explored in this direction.', '1507.08797-1-4-1': 'The first of these is maximal unitarity [CITATION], which generalises the cutting techniques of Britto, Cachazo and Feng [CITATION] and Forde [CITATION] to compute the rational coefficients of the master integrals, incorporating information from IBPs.', '1507.08797-1-4-2': 'Maximal unitarity has been used to look at maximal cuts for a variety of high-multiplicity examples in four dimensions [CITATION].', '1507.08797-1-4-3': 'The second approach extends the integrand reduction program of Ossola, Papadopoulos and Pittau [CITATION].', '1507.08797-1-4-4': 'The initial steps in this direction [CITATION] have now developed into a deeper understanding using the language of computational algebraic geometry [CITATION].', '1507.08797-1-4-5': 'The [MATH]-dimensional extension of this method has also been understood and applied in the context of the planar two-loop, five-gluon amplitude in QCD with all helicities positive (all plus) [CITATION].', '1507.08797-1-5-0': 'In the context of supersymmetric theories, computational methods based on an analysis of unitarity cuts have enabled a large number of high-loop computations.', '1507.08797-1-5-1': 'Other methods have also been developed, mainly in the context of these simplified theories.', '1507.08797-1-5-2': 'For example, the colour-kinematics duality of Bern, Carrasco and Johansson (BCJ) [CITATION] has been successfully exploited to find the complete colour-dressed four-loop, four-gluon amplitude in [MATH] supersymmetric-Yang-Mills (sYM) [CITATION].', '1507.08797-1-5-3': 'The two-loop, five-gluon amplitude, computed in [MATH] sYM in ref. [CITATION], has since been extended to the non-planar sector and cast into a complete set of numerators satisfying colour-kinematics duality [CITATION].', '1507.08797-1-5-4': 'The integrands in planar [MATH] sYM are known to all loop orders [CITATION], and recent studies indicate that this simplicity may extend to the non-planar sector [CITATION].', '1507.08797-1-6-0': 'The observation that [MATH] sYM theory and the all-plus sector of QCD are related by a dimension-shifting relation [CITATION] suggests that the all-plus amplitude at two loops could be a useful testing ground for new techniques.', '1507.08797-1-6-1': 'Indeed, at two loops the planar sector of the all-plus amplitude was observed to be related to the [MATH] amplitude at the integrand level in a similar pattern to the one-loop story, although additional corrections to the [MATH] sector appeared in the form of one-loop squared (or butterfly) topologies [CITATION].', '1507.08797-1-6-2': 'This fact prompts the question as to how much the techniques applied in supersymmetric cases may help to simplify QCD applications.', '1507.08797-1-7-0': 'In this article, we complete the computation of the two-loop, five-gluon, all-plus helicity amplitude including the non-planar sector.', '1507.08797-1-7-1': 'In order to deal with the increase in complexity of the full colour amplitude, we introduce a method to find compact colour decompositions that make full use of the underlying Kleiss-Kuijf (KK) relations [CITATION] in a similar way to the previous treatment at tree level and one loop by Del Duca, Dixon and Maltoni (DDM) [CITATION].', '1507.08797-1-7-2': 'We then further exploit the on-shell construction of the irreducible numerators to show that all of the non-planar cuts can be obtained from the planar cuts.', '1507.08797-1-7-3': 'This is reminiscent of the colour-kinematics duality, and indeed we employ the BCJ relations [CITATION] at tree level to relate planar and non-planar cuts.', '1507.08797-1-8-0': 'Our paper is organised as follows.', '1507.08797-1-8-1': 'We first present the colour decomposition of the all-plus, two-loop amplitude, exploiting the multi-peripheral decomposition of the underlying tree-level amplitudes.', '1507.08797-1-8-2': 'In the next section we describe how the complete kinematic structure can be constructed using knowledge from the planar sector and tree-level identities.', '1507.08797-1-8-3': 'After describing a worked example, we present compact results for the full integrand.', '1507.08797-1-8-4': 'We perform checks of the universal soft behaviour of the amplitude by evaluating the leading [MATH] poles of the integrals in the dimensional regularisation parameter [MATH].', '1507.08797-1-8-5': 'Finally, we draw some conclusions and discuss some future directions.', '1507.08797-1-9-0': '# Review of irreducible numerators', '1507.08797-1-10-0': 'We follow a multi-loop integrand reduction algorithm [CITATION] which uses multivariate polynomial division to find an integrand representation of the two-loop amplitude.', '1507.08797-1-10-1': 'This section is intended as a brief overview of the approach; we encourage the reader to refer to the literature for more detailed information.', '1507.08797-1-11-0': 'An integrand-reduced two-loop amplitude has the form [EQUATION] where the sum over runs over graphs [MATH], which are defined by a specific set of denominators [MATH] (the set [MATH] labels the propagators in the graph [MATH].)', '1507.08797-1-11-1': 'Associated with each graph in the sum is a colour-dressed irreducible numerator [MATH]; these are functions of the external momenta [MATH], the loop momenta [MATH] and [MATH], and also of the external colour indices [MATH].', '1507.08797-1-11-2': 'Each of these numerators has a colour decomposition [EQUATION] where we must explicitly determine the permutation sum [MATH] and the associated colour factors [MATH].', '1507.08797-1-11-3': 'We will present an algorithm to find a simple colour decomposition for our Yang-Mills amplitudes in the next section.', '1507.08797-1-12-0': 'An irreducible numerator [MATH] can be written in terms of monomials of irreducible scalar products (ISPs).', '1507.08797-1-12-1': 'To determine a set of ISPs, we first choose a spanning set of momenta to expand the scalar products along the lines of the van Neerven-Vermaseren basis [CITATION].', '1507.08797-1-12-2': 'By re-expressing the propagators in terms of this spanning set of scalar products, we can see that many can be written as linear combinations of propagators and can therefore be removed and pushed down into simpler topologies.', '1507.08797-1-12-3': 'The remaining scalar products are the ISPs; in general the propagators will be quadratic functions of them.', '1507.08797-1-12-4': 'To find a basis set of ISP monomials, these additional quadratic relations are removed using polynomial division with respect to a Grobner basis of the relations.', '1507.08797-1-12-5': 'This technique has been developed in the public code BasisDet [CITATION].', '1507.08797-1-12-6': 'When working in [MATH] dimensions we also include three extra-dimensional ISPs [EQUATION]', '1507.08797-1-12-7': 'Once a basis set of ISP monomials is identified, their rational coefficients are computed from the generalized unitarity cuts of the amplitude.', '1507.08797-1-12-8': 'As we take all the propagators contained in a particular graph on shell, the cut amplitude factorises into a product of tree-level amplitudes summed over internal helicity states.', '1507.08797-1-12-9': 'Following our schematic notation we can write this as [EQUATION] where the trees [MATH] are those associated with each vertex in the graph [MATH].', '1507.08797-1-12-10': 'Making the distinction between the cut associated with a graph [MATH] and the irreducible numerator associated with the same graph is crucial for understanding this construction.', '1507.08797-1-12-11': 'The irreducible numerator contains only that information which is required on the cut associated with [MATH], and which is not captured by irreducible numerators of graphs [MATH] that are ""larger"" than [MATH], in the sense that the propagators contained in [MATH] are a proper superset of the propagators contained in [MATH].', '1507.08797-1-12-12': 'In other words, by applying the cuts in a top-down approach we can isolate each topology systematically subtracting the higher-point singularities.', '1507.08797-1-13-0': 'We will frequently specify the irreducible numerator associated with a graph [MATH] as [MATH] for clarity; one should remember that the function [MATH] depends on loop and external momenta.', '1507.08797-1-13-1': 'Furthermore, throughout this paper we will adopt an index notation for the graph labels which lists the number of propagators in each of the three two-loop branches [MATH], [MATH] and [MATH].', '1507.08797-1-13-2': 'In addition, we add extra labels to distinguish between topologies of this type.', '1507.08797-1-13-3': 'We follow the convention that the right branch is first index, the left branch the second and finally the central branch in the last entry.', '1507.08797-1-13-4': 'For example, the planar pentagon-box reads while double box with five legs in a non-planar ordering is written A complete dictionary between this nomenclature and the graphs relevant for our two-loop, five-point calculation is given in Table [REF].', '1507.08797-1-14-0': 'As shown in ref. [CITATION], there is only ever a single branch to the set of solutions to the on-shell equation in [MATH] dimensions, which simplifies the inversion of the system in eq. [REF] to find the coefficients of the ISP monomials in [MATH], though at the cost of an increased number of monomials with respect to the four-dimensional case.', '1507.08797-1-15-0': 'There are two important remarks about this construction.', '1507.08797-1-15-1': 'The integrand representation of eq. [REF] is not unique, and there are different choices for both the set of ISPs and the set of monomials.', '1507.08797-1-15-2': 'Different sets of spanning vectors will result in different ISPs and the polynomial division requires a choice of monomial ordering.', '1507.08797-1-16-0': 'In the following sections we will exploit two important consequences of this approach.', '1507.08797-1-16-1': 'Firstly, we will restrict the form of our irreducible numerators to ensure that the choice of ISPs and monomials satisfy the basic symmetries required by our colour decompositions.', '1507.08797-1-16-2': 'Secondly, we will make use of the factorisation of irreducible numerators into ordered tree-level amplitudes.', '1507.08797-1-16-3': 'These tree amplitudes satisfy a number of relationships among different orderings.', '1507.08797-1-16-4': 'This fact will allow us to determine all non-planar cuts of the two-loop all-plus amplitude from the planar irreducible numerators computed in ref. [CITATION].', '1507.08797-1-17-0': '# Colour decomposition', '1507.08797-1-18-0': 'The main result of this work is the construction of the complete five-point, two-loop, all-plus amplitude in Yang-Mills theory.', '1507.08797-1-18-1': 'As we mentioned in Section [REF], it is necessary to choose a particular colour decomposition.', '1507.08797-1-18-2': 'This decomposition picks a set of colour tensors describing the colour structure of the amplitude.', '1507.08797-1-18-3': 'At the same time, it specifies an associated set of cut diagrams which must be computed.', '1507.08797-1-18-4': 'Each of these cut diagrams is, in turn, associated with a unique irreducible numerator.', '1507.08797-1-18-5': 'Thus the colour decomposition that we pick is of central importance, because it determines the set of irreducible numerators that we need to calculate.', '1507.08797-1-19-0': 'In this section, we describe the general algorithm that we used for constructing an appropriate colour decomposition of the amplitude, before applying this algorithm to the specific case of the two-loop five-point amplitude.', '1507.08797-1-20-0': '## Multi-peripheral colour decomposition', '1507.08797-1-21-0': 'Our algorithm is applicable to the general case of an [MATH]-loop Yang-Mills amplitude.', '1507.08797-1-21-1': 'Following the generalized unitarity principle, we begin by writing the amplitude as a sum over all colour-dressed cuts.', '1507.08797-1-21-2': 'Diagrammatically, these cuts consist of vertices formed from colour-dressed tree amplitudes which are joined by on-shell propagators.', '1507.08797-1-21-3': 'At two loops the set of colour-dressed cuts can be classified by two basic topologies: the genuine two-loop topologies are shown in figure [REF], and the one-loop squared (or butterfly) topologies in figure [REF].', '1507.08797-1-22-0': 'The central idea is to build the loop-level colour decomposition using knowledge of the underlying tree-level amplitudes.', '1507.08797-1-22-1': 'There are a variety of well-known presentations of these trees.', '1507.08797-1-22-2': 'We find it convenient to use the DDM form [CITATION] for the tree amplitudes: [EQUATION] where [MATH] are proportional to the standard structure constants in [MATH], the gauge group of our Yang-Mills theory.', '1507.08797-1-22-3': 'The main advantage of this form of the amplitude is that it contains [MATH] colour structures, as compared to [MATH] in the standard trace-based decomposition, for example.', '1507.08797-1-22-4': 'This fact helps to reduce the number of generated diagrams; in particular, an algorithm based directly on the trace decomposition of tree amplitudes generates a larger set of diagrams, some of which are rather obscure.', '1507.08797-1-23-0': 'Each of the colour structures in the tree decomposition is a string of group theoretic structure constants [MATH].', '1507.08797-1-23-1': 'For an [MATH]-gluon amplitude the decomposition is constructed by fixing the position of any two gluons at either end of this string.', '1507.08797-1-23-2': 'The [MATH] permutations of the remaining gluons between the ends of the string form the set of colour factors each of which is associated with a colour-ordered tree of the same ordering.', '1507.08797-1-23-3': 'Pictorially, the colour structures look like combs, as shown in figure [REF].', '1507.08797-1-24-0': 'It is straightforward to build the loop colour structure from these DDM tree colour structures.', '1507.08797-1-24-1': 'The loop structure follows directly from the cut diagram: one simply inserts the DDM trees at the vertices; propagator lines connecting trees indicate that the ends of the DDM combs at either end of the propagator have the same colour index to be summed over.', '1507.08797-1-24-2': 'Notice, however, that we must pick two special lines in the DDM form of the tree amplitudes (corresponding to lines [MATH] and [MATH] in figure [REF].)', '1507.08797-1-24-3': 'These lines are on opposite ends of the DDM colour strings, so one can informally think of this choice as picking two lines and ""stretching"" the colour ordered tree between these two ends.', '1507.08797-1-24-4': 'We make canonical choices of which legs to pick as special, depending on the number of propagators that connect to the three-point amplitude.', '1507.08797-1-24-5': 'These choices are:', '1507.08797-1-25-0': 'In this way, we build a set of colour structures.', '1507.08797-1-25-1': 'The kinematic structure associated with each colour structure is easily understood.', '1507.08797-1-25-2': 'Each time we insert a particular DDM colour trace, we also pick up a factor of the associated colour-ordered tree amplitude.', '1507.08797-1-25-3': 'Thus, the orientation of the legs in the kinematic diagram, associated to an irreducible numerator, is the same as in the colour structure; of course, the ""stretching"" procedure does not produce new propagators in the irreducible numerator.', '1507.08797-1-26-0': 'One advantage of using the DDM presentation of the amplitude at tree level is that the KK relations are automatically satisfied.', '1507.08797-1-26-1': 'Our procedure recycles this property to loop level: we automatically generate a set of colour diagrams that is KK-independent.', '1507.08797-1-26-2': 'Along the way, we generate ordered diagrams for the kinematics.', '1507.08797-1-26-3': 'The same procedure works at [MATH] loops; the amplitude is expressed as [EQUATION] where [MATH] are the symmetry factors of the graphs, [MATH] denote the products of the (inverse scalar) propagators, and the [MATH] are irreducible numerators for appropriate colour factors [MATH] generated through our algorithm.', '1507.08797-1-26-4': 'Now let us see this procedure at work in the context of the five-point, two-loop amplitude, which is our main focus.', '1507.08797-1-27-0': '## Five-point, two-loop amplitude', '1507.08797-1-28-0': 'Now we describe the colour structure of the five-point, two-loop amplitude.', '1507.08797-1-28-1': 'We concentrate on the diagrams that do not vanish in the all-plus case according to ref. [CITATION] and our calculations in section [REF].', '1507.08797-1-28-2': 'A generic five-point two-loop amplitude can be constructed by a straightforward extension of the present discussion.', '1507.08797-1-29-0': 'Let us first write the amplitude and then explain its content in more detail.', '1507.08797-1-29-1': 'We label [MATH] and their colour factors by their diagrams directly in the formula: [EQUATION] where the integration operator [MATH] acts on every [MATH] as [EQUATION]', '1507.08797-1-29-2': 'The explicit symmetry factors compensate for the over-counts introduced by the overall sum over permutations of external legs.', '1507.08797-1-29-3': 'For convenience, we recapitulate all these [MATH] in table [REF], where for each irreducible numerator [MATH] we also show its diagram, that of its colour factor, as well as the set of its non-equivalent permutations.', '1507.08797-1-30-0': 'The first three graphs in table [REF], [MATH], [MATH] and [MATH], are the master diagrams corresponding to the maximal cuts.', '1507.08797-1-30-1': 'They are purely trivalent, thus their colour factors are unambiguously defined by their proper graphs.', '1507.08797-1-31-0': 'The next three graphs, [MATH], [MATH] and [MATH], have a four-point vertex with two external and two internal edges.', '1507.08797-1-31-1': 'The two external legs automatically enter in the permutation sum with two possible orderings, hence multi-peripheral subgraphs are naturally obtained by fixing the internal lines, as in the insert [MATH] of figure [REF].', '1507.08797-1-31-2': '""Stretching"" the four-point vertex by these lines gives a master graph for each colour factor.', '1507.08797-1-32-0': 'The following two diagrams, [MATH] and [MATH], share the same graph structure, up to the ordering of the four-point vertex.', '1507.08797-1-32-1': 'The apparent asymmetry introduced by our selecting these two diagrams, and omitting the graph with the external leg in the right loop, is an artefact of our colour decomposition.', '1507.08797-1-32-2': 'One could make other choices; the KK relations satisfied by the trees and their symmetries ensure that any choice is valid.', '1507.08797-1-33-0': 'To expand the four-point vertex in the ninth graph, [MATH], we fixed the internal lines of the ""diamond"" subdiagram, hence its colour factor is [MATH], but with a minus sign due to one flipped vertex.', '1507.08797-1-33-1': 'The other permutation of the four-point vertex corresponds to the same topology and is present in the overall permutation sum with the right permuted colour diagram.', '1507.08797-1-34-0': 'The colour factor of the planar graph [MATH] follows in a straightforward manner from our algorithm (see figure [REF]), yielding [MATH] as its colour factor.', '1507.08797-1-34-1': 'Its descendant [MATH] is more interesting, since it is the only graph in the all-plus case with two four-point vertices.', '1507.08797-1-34-2': 'They can be treated independently by linearity of colour decomposition.', '1507.08797-1-34-3': 'The external one is thus expanded in the same way as in [MATH], producing [MATH] as an intermediate step.', '1507.08797-1-34-4': 'Expanding the internal four-point vertex gives [MATH] again.', '1507.08797-1-35-0': 'The last two graphs, [MATH] and [MATH], share a five-point vertex.', '1507.08797-1-35-1': 'To explain their colour factors, let us consider the corresponding colour-dressed cut: [EQUATION]', '1507.08797-1-35-2': 'We obtain the multi-peripheral decomposition of the five-point vertex by fixing the two right-hand loop edges and permuting the other three edges.', '1507.08797-1-35-3': 'The graphs in the second line can be vertically flipped to put leg [MATH] downstairs to match the presentation in table [REF].', '1507.08797-1-35-4': 'Obviously, an equivalent decomposition could be achieved by fixing the loop edges on the left, which would change the orientation of leg [MATH] in the superficially non-planar graphs [MATH].', '1507.08797-1-35-5': 'The [MATH]-summation in eq. [REF] effectively symmetrises the colour structure over the two choices of multi-peripheral decompositions.', '1507.08797-1-36-0': 'In the present paper we can avoid lower levels of the graph hierarchy thanks to the simplicity of the fully symmetric helicity configuration, but it already incorporates all the key elements of the general colour structure.', '1507.08797-1-37-0': '# Kinematic structure', '1507.08797-1-38-0': 'With our colour decomposition at hand, we turn our attention to the kinematic structure of the amplitude.', '1507.08797-1-38-1': 'We need to compute an irreducible numerator associated to each diagram in eq. [REF]; as Frellesvig, Zhang and one of the current authors have already computed all the planar irreducible numerators [CITATION], our task is to determine the remaining non-planar numerators.', '1507.08797-1-38-2': 'Of course, these numerators can be computed directly from their cuts.', '1507.08797-1-38-3': 'However, as we will see, it is easy to determine the complete set of non-planar irreducible numerators for this amplitude from its planar numerators and the knowledge of tree-level amplitude relations.', '1507.08797-1-39-0': '## Non-planar from planar', '1507.08797-1-40-0': 'However, we can avoid calculating this non-planar cut explicitly by relating it to a planar cut.', '1507.08797-1-40-1': 'We do so in two steps: first, we coalesce two (ordered) three-point trees into a limit of an ordered four-point tree; then we use the BCJ relations [CITATION] satisfied by the ordered four-point tree to reorder the legs until the complete diagram becomes planar.', '1507.08797-1-41-0': 'In more detail, we use the following well-known relation, which is satisfied by on-shell amplitudes in the cuts: [EQUATION] where [MATH] are the standard Mandelstam invariants.', '1507.08797-1-41-1': 'Since this identity is of central importance for us, we present a short proof.', '1507.08797-1-41-2': 'A four-point tree amplitude can be constructed from two three-point amplitudes using the BCFW recursion relation [CITATION]: [EQUATION] where hat signs on the right-hand side indicate that the amplitudes are evaluated on complex kinematics for some BCFW shift of external legs.', '1507.08797-1-41-3': 'The exact complex value of the shifted internal momentum [MATH] is defined by the on-shell condition [EQUATION] where the precise expression for [MATH] depends on the particular BCFW shift.', '1507.08797-1-41-4': 'The key point is that [MATH] is a linear function of [MATH], with the property that in the limit [MATH], [MATH].', '1507.08797-1-41-5': 'In this limit eq. [REF] becomes [EQUATION]', '1507.08797-1-41-6': 'Notice that the left-hand side contains a nonzero contribution due to the pole in [MATH].', '1507.08797-1-41-7': 'Now we can remove the factor of [MATH] on the left-hand side of eq. [REF] by using the BCJ amplitude relation [CITATION], [EQUATION]', '1507.08797-1-41-8': 'This proves the identity [REF].', '1507.08797-1-42-0': 'We proceed by applying our identity [REF] to tree amplitudes inside the non-planar cut, rearranging the diagram until it becomes planar.', '1507.08797-1-42-1': 'It is simplest to begin with maximal diagrams, and then to continue to topologies with fewer propagators.', '1507.08797-1-43-0': 'We will work through the calculation of [MATH], displayed in Figure [REF], as an example; we computed all non-planar irreducible numerators using the same technique.', '1507.08797-1-44-0': 'The calculation starts at the level of the cuts: [EQUATION] where [MATH], and we understand that all internal helicities are implicitly summed over while all exposed propagators are cut.', '1507.08797-1-44-1': 'These cuts are decomposed into irreducible numerators as [EQUATION]', '1507.08797-1-44-2': 'Using the fact that [MATH] on this cut, we see that [EQUATION]', '1507.08797-1-44-3': 'A similar calculation for [MATH] leads to [EQUATION]', '1507.08797-1-44-4': 'So far the obtained non-planar numerators are valid only on their cuts, but they can be extended off shell.', '1507.08797-1-44-5': 'We may simply express the numerators in terms of a given set of ISPs and then define off-shell numerators unambiguously through these ISPs.', '1507.08797-1-44-6': 'The value of a given numerator depends on the choice of ISP basis off shell (in contrast to the situation on shell, of course).', '1507.08797-1-44-7': 'In this way, we determine a valid set of non-planar irreducible numerators.', '1507.08797-1-44-8': 'Notice that the ISP monomial choices made in the planar sector, such as the higher powers of [MATH] preferred over high powers of [MATH], will then be easily translated to the non-planar numerators.', '1507.08797-1-45-0': 'It is very useful to maintain the symmetries of the underlying graphs in this off-shell continuation.', '1507.08797-1-45-1': 'We achieve this by choosing an appropriate basis of ISPs on a graph-by-graph basis.', '1507.08797-1-45-2': 'One engineers the ISP basis such that the loop momentum-dependence of each irreducible numerator is captured by a set of ISPs, which map into one another under the graph symmetries without using any cut conditions.', '1507.08797-1-45-3': 'The symmetries of the maximal non-planar graphs, for example, are [EQUATION]', '1507.08797-1-45-4': 'These symmetries motivate our choices of ISPs, given in Table [REF].', '1507.08797-1-45-5': 'For instance, the second symmetry of [MATH] in [REF] leads to a map of ISPs [EQUATION]', '1507.08797-1-45-6': 'After we express loop-momentum dependence in eqs. [REF] and [REF] in terms of the ISPs of Table [REF], using the fact that the RSPs (cut propagators) are zero on shell, we are left with appropriate off-shell irreducible numerators.', '1507.08797-1-45-7': 'These are listed in Section [REF].', '1507.08797-1-45-8': 'Note that the function [MATH] is not the same as the function [MATH] despite the on-shell equation ([REF]): different ISPs are chosen to make different off-shell symmetries manifest.', '1507.08797-1-46-0': 'We obtained irreducible numerators for lower-level non-planar diagrams in the same way, using the BCJ relations on cuts and extending the results off shell.', '1507.08797-1-46-1': 'For the all-plus amplitude at hand we find that many lower-level irreducible numerators vanish.', '1507.08797-1-46-2': 'In other words, the higher-level numerators capture the unitarity cut structure of the full amplitude, which is given below.', '1507.08797-1-47-0': '# The full-colour five-gluon all-plus amplitude', '1507.08797-1-48-0': 'In this section we present a complete summary of all kinematic numerators contributing to the colour decomposition in eq. [REF].', '1507.08797-1-48-1': 'We include both planar [CITATION] and non-planar irreducible numerators computed using the technique described in Section [REF].', '1507.08797-1-48-2': 'The result is presented unrenormalized including the dependence on the spin dimension [MATH] of the gluon, which is equal to [MATH] in the FDH scheme and [MATH] in CDR [CITATION].', '1507.08797-1-48-3': 'The dependence on the extra dimensional ISPs [MATH] can be collected into three general functions, [EQUATION]', '1507.08797-1-48-4': 'The remaining coefficients are expressed using the standard spinor-helicity formalism.', '1507.08797-1-48-5': 'In particular, we denote [EQUATION] since [MATH].', '1507.08797-1-48-6': 'We also make use of ""spurious"" directions in order to find compact representations of the integrands, [EQUATION]', '1507.08797-1-48-7': 'The full amplitude reads [EQUATION]', '1507.08797-1-48-8': 'The three maximal graphs are [EQUATION]', '1507.08797-1-48-9': 'Meanwhile, the graphs at level 1 are [EQUATION]', '1507.08797-1-48-10': 'Finally, the graphs at level 2 are [EQUATION]', '1507.08797-1-48-11': 'We have found a representation of the full amplitude with no topologies with fewer than six propagators.', '1507.08797-1-48-12': 'We note that there are nonzero cuts at the integrand level, but the resulting integrals are scaleless and hence zero in dimensional regularisation.', '1507.08797-1-48-13': 'We have checked additional cuts at levels 2 and 3 to ensure that no nonzero topologies remain.', '1507.08797-1-48-14': 'To find an integrand with this property, the ISPs [MATH] and [MATH] in the numerators [MATH] and [MATH] were upgraded to include terms proportional to [MATH] and [MATH].', '1507.08797-1-49-0': '# Checking the soft divergences', '1507.08797-1-50-0': 'Since the all-plus helicity configuration is zero at tree level, the universal infrared (IR) structure is the same as that of an ordinary one-loop amplitude.', '1507.08797-1-50-1': 'The poles of our two-loop amplitude should therefore be equivalent to those of the finite one-loop amplitude multiplied by the infrared pole operator including the sum over colour correlations [CITATION] [EQUATION] where we have used the [MATH] symbol to indicate that the colour matrices (in this purely gluonic case they will all be structure constants) should be inserted into the colour factors of the one-loop amplitude.', '1507.08797-1-50-2': 'The standard loop prefactor is given by [EQUATION]', '1507.08797-1-50-3': 'The one-loop amplitude to all orders in [MATH] can be found in ref. [CITATION].', '1507.08797-1-51-0': 'There are a number of difficulties in checking eq. [REF] in full since the five-point planar and non-planar integrals required are still unknown at this time.', '1507.08797-1-51-1': 'Resorting to numerical evaluation, as has been done in the planar case [CITATION], is computationally prohibitive for two reasons.', '1507.08797-1-51-2': 'Firstly, the full colour expansion contains a large number of dimension-shifted integrals ([MATH]) - an order of magnitude more than the leading colour terms.', '1507.08797-1-51-3': 'Secondly, there is no Euclidean region for the complete amplitude, and so contour deformation must be performed for many of these integrals, making them more complicated than the planar cases.', '1507.08797-1-51-4': 'This task is probably achievable with public tools like FIESTA [CITATION] and SecDec [CITATION].', '1507.08797-1-52-0': 'There is, however, a much simpler method to check the leading soft singularities up to [MATH], which can be done analytically.', '1507.08797-1-52-1': 'In the leading soft limit, the colour correlations drop out of eq. [REF]: [EQUATION]', '1507.08797-1-52-2': 'Clearly this is a weaker check than the full IR poles, but it does require non-trivial properties of the non-planar sector.', '1507.08797-1-52-3': 'The butterfly (one-loop squared) topologies, are all finite and therefore not relevant for the IR properties.', '1507.08797-1-52-4': 'Scattering amplitudes in the soft or eikonal limit have many remarkable structures and universal properties.', '1507.08797-1-52-5': 'The interested reader may like to turn to ref. [CITATION] for a recent introduction to the subject.', '1507.08797-1-53-0': '## Evaluating the massless double box in the soft limit', '1507.08797-1-54-0': 'The extra simplicity in our all-plus loop amplitude that sets it aside from most two-loop amplitudes is that the integrals contain at most a single soft divergence, rather than the maximum double soft divergence.', '1507.08797-1-54-1': 'We can therefore break our loop amplitudes up into sums of regions with soft singularities and evaluate the amplitude in the limit.', '1507.08797-1-54-2': 'In this limit the integral factorises into a product of two one-loop integrals and can be evaluated to extract the leading [MATH] divergence.', '1507.08797-1-55-0': 'All of the poles of our amplitude are contained in the topologies proportional to the same dimension shifting numerator: [EQUATION] which also has a simple behaviour, [EQUATION]', '1507.08797-1-55-1': 'Taking the example of the two-loop double box, we find two soft regions by taking the limit of either loop.', '1507.08797-1-55-2': 'We find a soft singularity whenever we have two adjacent massless legs in one of the loop integrations.', '1507.08797-1-55-3': 'In each case, we factorise into an IR divergent triangle and a dimension-shifted box: [EQUATION]', '1507.08797-1-55-4': 'Recalling the one-loop integrals, [EQUATION] and summing the two regions, we quickly arrive at the result: [EQUATION]', '1507.08797-1-56-0': '## Soft divergences of the five-point integrals', '1507.08797-1-57-0': 'By following the method described in the previous section, we have derived the complete set of integrals required for the [MATH] part of the amplitude.', '1507.08797-1-57-1': 'All of the integrals have been checked numerically using the sector decomposition methods implemented in FIESTA [CITATION] and SecDec [CITATION].', '1507.08797-1-57-2': 'Some of these integrals have been computed long ago in [MATH] dimensions and can be used to write the full integrals including finite terms via the dimensional reduction identities implemented in LiteRed [CITATION] and IBP relations from FIRE5 [CITATION].', '1507.08797-1-57-3': 'We have performed this task for the planar double box with an off-shell leg.', '1507.08797-1-57-4': 'Thus we arrived at the following soft limits for the integrals: [EQUATION]', '1507.08797-1-58-0': '[EQUATION]', '1507.08797-1-59-0': '[EQUATION]', '1507.08797-1-59-1': 'Using these results, we have checked that eq. [REF] does hold as expected for our amplitude [REF].', '1507.08797-1-60-0': '# Conclusions', '1507.08797-1-61-0': 'In this paper we have explored the impact of tree-level amplitude relations in multi-loop integrand computations.', '1507.08797-1-61-1': 'There were two major aspects to our work.', '1507.08797-1-61-2': 'Firstly, we exploited the Kleiss-Kuijf relations to find a compact colour decomposition for the two-loop amplitude in terms of multi-peripheral colour factors in an analogous way to the tree-level and one-loop decompositions of Del Duca, Dixon and Maltoni [CITATION].', '1507.08797-1-62-0': 'Secondly, we applied the BCJ amplitude relations [CITATION] to relate all non-planar generalized unitarity cuts to the previously computed planar ones.', '1507.08797-1-62-1': 'This allowed us to easily generate a compact representation the full colour two-loop, five-gluon, all-plus integrand building on previous planar work [CITATION].', '1507.08797-1-62-2': 'The soft infrared poles of the full amplitude were checked against the well-known universal pole structure.', '1507.08797-1-63-0': 'We hope that the computational methods developed here will be of good use in the necessary extension to more general helicity configurations and other [MATH] scattering processes at two loops.', '1507.08797-1-63-1': 'They highlight some advantages of relating two-loop integrands to tree-level amplitudes via generalized unitarity cuts.', '1507.08797-1-63-2': 'As well as avoiding the large intermediate steps that make Feynman diagram computations at this loop order and multiplicity extremely computationally intensive, we are able to build known on-shell symmetries and relations into the amplitude by construction.', '1507.08797-1-64-0': 'Another interesting aspect of the all-plus amplitude is the continuing connection to the previously known amplitudes in [MATH] sYM.', '1507.08797-1-64-1': 'Though the dimension shifting relation observed at one loop no longer holds, the integrands of our full all-plus amplitude and the expressions of Carrasco and Johansson [CITATION] are related by the same dimension shifting operator seen in the planar case.', '1507.08797-1-64-2': 'For example, we find that [EQUATION]', '1507.08797-1-64-3': 'The one-loop squared topologies have the form [EQUATION] where [MATH] and [MATH] are some functions of the external kinematics and loop momenta.', '1507.08797-1-64-4': 'The second term is proportional to [MATH] and is a genuine contribution in QCD not related to [MATH].', '1507.08797-1-64-5': 'This additional numerator structure is enough to make the off-shell BCJ symmetries non-trivial to satisfy, even though the [MATH] integrand has been cast in such a form.', '1507.08797-1-64-6': 'It is an interesting question as to whether this would be possible for the amplitudes presented here and one that we intend to explore in the future.', '1507.08797-1-65-0': 'We would like to thank JJ Carrasco, Claude Duhr, Einan Gardi, and Henrik Johansson for useful discussions.', '1507.08797-1-65-1': 'Particular thanks go to Chris White for enlightening remarks on the soft singularities of loop integrals.', '1507.08797-1-65-2': 'SB is supported by an STFC Rutherford Fellowship ST/L004925/1.', '1507.08797-1-65-3': 'GM is supported by an STFC Studentship ST/K501980/1.', '1507.08797-1-65-4': 'AO is supported by the EU via a Marie Curie Actions grant FP7-PEOPLE-2013-CIG 631370.', '1507.08797-1-65-5': 'DOC is supported in part by the STFC consolidated grant ""Particle Physics at the Higgs Centre"", by the National Science Foundation under grant NSF PHY11-25915, and by the Marie Curie FP7 grant 631370.'}","{'1507.08797-2-0-0': '# Introduction', '1507.08797-2-1-0': 'Precision measurements at the Large Hadron Collider provide detailed information about the nature of the strong interaction and its role within the Standard Model.', '1507.08797-2-1-1': 'With new data already arriving from Run II, there is a growing need for higher-precision theoretical predictions for a variety of different observables.', '1507.08797-2-1-2': 'Recent years have seen considerable progress in this respect.', '1507.08797-2-1-3': 'Our ability to make predictions in perturbative QCD now covers most of the relevant [MATH] scattering processes at next-to-next-to-leading order (NNLO) precision [CITATION], as well as the example of inclusive Higgs production at N[MATH]LO [CITATION].', '1507.08797-2-1-4': 'Despite this, processes with more than two particles in the final state remain beyond the reach of current NNLO methods.', '1507.08797-2-2-0': 'Next-to-leading order (NLO) corrections to high multiplicity final states are by now commonplace in phenomenological studies.', '1507.08797-2-2-1': 'Such computations are possible thanks to automated techniques, which make use of integrand reduction [CITATION], recursive techniques [CITATION], (generalized) unitarity cuts [CITATION] and the known basis of scalar integrals.', '1507.08797-2-2-2': 'Processes with up to five coloured partons [CITATION] in the final state are feasible using on-shell methods that, by only working with the physical degrees of freedom, are efficient at controlling the complexity.', '1507.08797-2-3-0': 'Multi-loop calculations of [MATH] scattering processes [CITATION] have been quite successful using the more traditional approaches of Feynman diagrams and integration-by-parts identities (IBPs) [CITATION], though there are notable exceptions using unitarity cutting techniques [CITATION].', '1507.08797-2-3-1': 'The main focus for these processes has been in the evaluation of the resulting master integrals.', '1507.08797-2-3-2': 'At higher multiplicity, rapid growth in the complexity of the Feynman diagram representation motivates an alternative approach that makes use of the lessons learned during the automation of one-loop computations.', '1507.08797-2-4-0': 'Two methods have already been explored in this direction.', '1507.08797-2-4-1': 'The first of these is maximal unitarity [CITATION], which generalises the cutting techniques of Britto, Cachazo and Feng [CITATION] and Forde [CITATION] to compute the rational coefficients of the master integrals, incorporating information from IBPs.', '1507.08797-2-4-2': 'Maximal unitarity has been used to look at maximal cuts for a variety of high-multiplicity examples in four dimensions [CITATION].', '1507.08797-2-4-3': 'The second approach extends the integrand reduction program of Ossola, Papadopoulos and Pittau [CITATION].', '1507.08797-2-4-4': 'The initial steps in this direction [CITATION] have now developed into a deeper understanding using the language of computational algebraic geometry [CITATION].', '1507.08797-2-4-5': 'The [MATH]-dimensional extension of this method has also been understood and applied in the context of the planar two-loop, five-gluon amplitude in QCD with all helicities positive (all plus) [CITATION].', '1507.08797-2-5-0': 'In the context of supersymmetric theories, computational methods based on an analysis of unitarity cuts have enabled a large number of high-loop computations.', '1507.08797-2-5-1': 'Other methods have also been developed, mainly in the context of these simplified theories.', '1507.08797-2-5-2': 'For example, the colour-kinematics duality of Bern, Carrasco and Johansson (BCJ) [CITATION] has been successfully exploited to find the complete colour-dressed four-loop, four-gluon amplitude in [MATH] supersymmetric-Yang-Mills (sYM) [CITATION].', '1507.08797-2-5-3': 'The two-loop, five-gluon amplitude, computed in [MATH] sYM in ref. [CITATION], has since been extended to the non-planar sector and cast into a complete set of numerators satisfying colour-kinematics duality [CITATION].', '1507.08797-2-5-4': 'The integrands in planar [MATH] sYM are known to all loop orders [CITATION], and recent studies indicate that this simplicity may extend to the non-planar sector [CITATION].', '1507.08797-2-6-0': 'The observation that [MATH] sYM theory and the all-plus sector of QCD are related by a dimension-shifting relation [CITATION] suggests that the all-plus amplitude at two loops could be a useful testing ground for new techniques.', '1507.08797-2-6-1': 'Indeed, at two loops the planar sector of the all-plus amplitude was observed to be related to the [MATH] amplitude at the integrand level in a similar pattern to the one-loop story, although additional corrections to the [MATH] sector appeared in the form of one-loop squared (or butterfly) topologies [CITATION].', '1507.08797-2-6-2': 'This fact prompts the question as to how much the techniques applied in supersymmetric cases may help to simplify QCD applications.', '1507.08797-2-7-0': 'In this article, we complete the computation of the two-loop, five-gluon, all-plus helicity amplitude including the non-planar sector.', '1507.08797-2-7-1': 'In order to deal with the increase in complexity of the full colour amplitude, we introduce a method to find compact colour decompositions that make full use of the underlying Kleiss-Kuijf (KK) relations [CITATION] in a similar way to the previous treatment at tree level and one loop by Del Duca, Dixon and Maltoni (DDM) [CITATION].', '1507.08797-2-7-2': 'We then further exploit the on-shell construction of the irreducible numerators to show that all of the non-planar cuts can be obtained from the planar cuts.', '1507.08797-2-7-3': 'This is reminiscent of the colour-kinematics duality, and indeed we employ the BCJ relations [CITATION] at tree level to relate planar and non-planar cuts.', '1507.08797-2-8-0': 'Our paper is organised as follows.', '1507.08797-2-8-1': 'We first present the colour decomposition of the all-plus, two-loop amplitude, exploiting the multi-peripheral decomposition of the underlying tree-level amplitudes.', '1507.08797-2-8-2': 'In the next section we describe how the complete kinematic structure can be constructed using knowledge from the planar sector and tree-level identities.', '1507.08797-2-8-3': 'After describing a worked example, we present compact results for the full integrand.', '1507.08797-2-8-4': 'We perform checks of the universal soft behaviour of the amplitude by evaluating the leading [MATH] poles of the integrals in the dimensional regularisation parameter [MATH].', '1507.08797-2-8-5': 'Finally, we draw some conclusions and discuss some future directions.', '1507.08797-2-9-0': '# Review of irreducible numerators', '1507.08797-2-10-0': 'We follow a multi-loop integrand reduction algorithm [CITATION] which uses multivariate polynomial division to find an integrand representation of the two-loop amplitude.', '1507.08797-2-10-1': 'This section is intended as a brief overview of the approach; we encourage the reader to refer to the literature for more detailed information.', '1507.08797-2-11-0': 'An integrand-reduced two-loop amplitude has the form [EQUATION] where the sum over runs over graphs [MATH], which are defined by a specific set of denominators [MATH] (the set [MATH] labels the propagators in the graph [MATH].)', '1507.08797-2-11-1': 'Associated with each graph in the sum is a colour-dressed irreducible numerator [MATH]; these are functions of the external momenta [MATH], the loop momenta [MATH] and [MATH], and also of the external colour indices [MATH].', '1507.08797-2-11-2': 'Each of these numerators has a colour decomposition [EQUATION] where we must explicitly determine the permutation sum [MATH] and the associated colour factors [MATH].', '1507.08797-2-11-3': 'We will present an algorithm to find a simple colour decomposition for our Yang-Mills amplitudes in the next section.', '1507.08797-2-12-0': 'An irreducible numerator [MATH] can be written in terms of monomials of irreducible scalar products (ISPs).', '1507.08797-2-12-1': 'To determine a set of ISPs, we first choose a spanning set of momenta to expand the scalar products along the lines of the van Neerven-Vermaseren basis [CITATION].', '1507.08797-2-12-2': 'By re-expressing the propagators in terms of this spanning set of scalar products, we can see that many can be written as linear combinations of propagators and can therefore be removed and pushed down into simpler topologies.', '1507.08797-2-12-3': 'These scalar products are known as reducible scalar products (RSPs.)', '1507.08797-2-12-4': 'The remaining scalar products are the ISPs; in general the propagators will be quadratic functions of them.', '1507.08797-2-12-5': 'To find a basis set of ISP monomials, these additional quadratic relations are removed using polynomial division with respect to a Grobner basis of the relations.', '1507.08797-2-12-6': 'This technique has been developed in the public code BasisDet [CITATION].', '1507.08797-2-12-7': 'When working in [MATH] dimensions we also include three extra-dimensional ISPs [EQUATION]', '1507.08797-2-12-8': 'Once a basis set of ISP monomials is identified, their rational coefficients are computed from the generalized unitarity cuts of the amplitude.', '1507.08797-2-12-9': 'As we take all the propagators contained in a particular graph on shell, the cut amplitude factorises into a product of tree-level amplitudes summed over internal helicity states.', '1507.08797-2-12-10': 'Following our schematic notation we can write this as [EQUATION] where the trees [MATH] are those associated with each vertex in the graph [MATH].', '1507.08797-2-12-11': 'Making the distinction between the cut associated with a graph [MATH] and the irreducible numerator associated with the same graph is crucial for understanding this construction.', '1507.08797-2-12-12': 'The irreducible numerator contains only that information which is required on the cut associated with [MATH], and which is not captured by irreducible numerators of graphs [MATH] that are ""larger"" than [MATH], in the sense that the propagators contained in [MATH] are a proper superset of the propagators contained in [MATH].', '1507.08797-2-12-13': 'In other words, by applying the cuts in a top-down approach we can isolate each topology systematically subtracting the higher-point singularities.', '1507.08797-2-13-0': 'We will frequently specify the irreducible numerator associated with a graph [MATH] as [MATH] for clarity; one should remember that the function [MATH] depends on loop and external momenta.', '1507.08797-2-13-1': 'Furthermore, throughout this paper we will adopt an index notation for the graph labels which lists the number of propagators in each of the three two-loop branches [MATH], [MATH] and [MATH].', '1507.08797-2-13-2': 'In addition, we add extra labels to distinguish between topologies of this type.', '1507.08797-2-13-3': 'We follow the convention that the right branch is first index, the left branch the second and finally the central branch in the last entry.', '1507.08797-2-13-4': 'For example, the planar pentagon-box reads while double box with five legs in a non-planar ordering is written A complete dictionary between this nomenclature and the graphs relevant for our two-loop, five-point calculation is given in Table [REF].', '1507.08797-2-14-0': 'As shown in ref. [CITATION], there is only ever a single branch to the set of solutions to the on-shell equation in [MATH] dimensions, which simplifies the inversion of the system in eq. [REF] to find the coefficients of the ISP monomials in [MATH], though at the cost of an increased number of monomials with respect to the four-dimensional case.', '1507.08797-2-15-0': 'There are two important remarks about this construction.', '1507.08797-2-15-1': 'The integrand representation of eq. [REF] is not unique, and there are different choices for both the set of ISPs and the set of monomials.', '1507.08797-2-15-2': 'Different sets of spanning vectors will result in different ISPs and the polynomial division requires a choice of monomial ordering.', '1507.08797-2-16-0': 'In the following sections we will exploit two important consequences of this approach.', '1507.08797-2-16-1': 'Firstly, we will restrict the form of our irreducible numerators to ensure that the choice of ISPs and monomials satisfy the basic symmetries required by our colour decompositions.', '1507.08797-2-16-2': 'Secondly, we will make use of the factorisation of irreducible numerators into ordered tree-level amplitudes.', '1507.08797-2-16-3': 'These tree amplitudes satisfy a number of relationships among different orderings.', '1507.08797-2-16-4': 'This fact will allow us to determine all non-planar cuts of the two-loop all-plus amplitude from the planar irreducible numerators computed in ref. [CITATION].', '1507.08797-2-17-0': '# Colour decomposition', '1507.08797-2-18-0': 'The main result of this work is the construction of the complete five-point, two-loop, all-plus amplitude in Yang-Mills theory.', '1507.08797-2-18-1': 'As we mentioned in Section [REF], it is necessary to choose a particular colour decomposition.', '1507.08797-2-18-2': 'This decomposition picks a set of colour tensors describing the colour structure of the amplitude.', '1507.08797-2-18-3': 'At the same time, it specifies an associated set of cut diagrams which must be computed.', '1507.08797-2-18-4': 'Each of these cut diagrams is, in turn, associated with a unique irreducible numerator.', '1507.08797-2-18-5': 'Thus the colour decomposition that we pick is of central importance, because it determines the set of irreducible numerators that we need to calculate.', '1507.08797-2-19-0': 'In this section, we describe the general algorithm that we used for constructing an appropriate colour decomposition of the amplitude, before applying this algorithm to the specific case of the two-loop five-point amplitude.', '1507.08797-2-20-0': '## Multi-peripheral colour decomposition', '1507.08797-2-21-0': 'Our algorithm is applicable to the general case of an [MATH]-loop Yang-Mills amplitude.', '1507.08797-2-21-1': 'Following the generalized unitarity principle, we begin by writing the amplitude as a sum over all colour-dressed cuts.', '1507.08797-2-21-2': 'Diagrammatically, these cuts consist of vertices formed from colour-dressed tree amplitudes which are joined by on-shell propagators.', '1507.08797-2-21-3': 'At two loops the set of colour-dressed cuts can be classified by two basic topologies: the genuine two-loop topologies are shown in figure [REF], and the one-loop squared (or butterfly) topologies in figure [REF].', '1507.08797-2-22-0': 'The central idea is to build the loop-level colour decomposition using knowledge of the underlying tree-level amplitudes.', '1507.08797-2-22-1': 'There are a variety of well-known presentations of these trees.', '1507.08797-2-22-2': 'We find it convenient to use the DDM form [CITATION] for the tree amplitudes: [EQUATION] where [MATH] are proportional to the standard structure constants in [MATH], the gauge group of our Yang-Mills theory.', '1507.08797-2-22-3': 'The main advantage of this form of the amplitude is that it contains [MATH] colour structures, as compared to [MATH] in the standard trace-based decomposition, for example.', '1507.08797-2-22-4': 'This fact helps to reduce the number of generated diagrams; in particular, an algorithm based directly on the trace decomposition of tree amplitudes generates a larger set of diagrams, some of which are rather obscure.', '1507.08797-2-23-0': 'Each of the colour structures in the tree decomposition is a string of group theoretic structure constants [MATH].', '1507.08797-2-23-1': 'For an [MATH]-gluon amplitude the decomposition is constructed by fixing the position of any two gluons at either end of this string.', '1507.08797-2-23-2': 'The [MATH] permutations of the remaining gluons between the ends of the string form the set of colour factors each of which is associated with a colour-ordered tree of the same ordering.', '1507.08797-2-23-3': 'Pictorially, the colour structures look like combs, as shown in figure [REF].', '1507.08797-2-24-0': 'It is straightforward to build the loop colour structure from these DDM tree colour structures.', '1507.08797-2-24-1': 'The loop structure follows directly from the cut diagram: one simply inserts the DDM trees at the vertices; propagator lines connecting trees indicate that the ends of the DDM combs at either end of the propagator have the same colour index to be summed over.', '1507.08797-2-24-2': 'Notice, however, that we must pick two special lines in the DDM form of the tree amplitudes (corresponding to lines [MATH] and [MATH] in figure [REF].)', '1507.08797-2-24-3': 'These lines are on opposite ends of the DDM colour strings, so one can informally think of this choice as picking two lines and ""stretching"" the colour ordered tree between these two ends.', '1507.08797-2-24-4': 'We make canonical choices of which legs to pick as special, depending on the number of propagators that connect to the three-point amplitude.', '1507.08797-2-24-5': 'These choices are:', '1507.08797-2-25-0': 'In this way, we build a set of colour structures.', '1507.08797-2-25-1': 'The kinematic structure associated with each colour structure is easily understood.', '1507.08797-2-25-2': 'Each time we insert a particular DDM colour trace, we also pick up a factor of the associated colour-ordered tree amplitude.', '1507.08797-2-25-3': 'Thus, the orientation of the legs in the kinematic diagram, associated to an irreducible numerator, is the same as in the colour structure; of course, the ""stretching"" procedure does not produce new propagators in the irreducible numerator.', '1507.08797-2-26-0': 'One advantage of using the DDM presentation of the amplitude at tree level is that the KK relations are automatically satisfied.', '1507.08797-2-26-1': 'Our procedure recycles this property to loop level: we automatically generate a set of colour diagrams that is KK-independent.', '1507.08797-2-26-2': 'Along the way, we generate ordered diagrams for the kinematics.', '1507.08797-2-26-3': 'The same procedure works at [MATH] loops; the amplitude is expressed as [EQUATION] where [MATH] are the symmetry factors of the graphs, [MATH] denote the products of the (inverse scalar) propagators, and the [MATH] are irreducible numerators for appropriate colour factors [MATH] generated through our algorithm.', '1507.08797-2-26-4': 'Now let us see this procedure at work in the context of the five-point, two-loop amplitude, which is our main focus.', '1507.08797-2-27-0': '## Five-point, two-loop amplitude', '1507.08797-2-28-0': 'Now we describe the colour structure of the five-point, two-loop amplitude.', '1507.08797-2-28-1': 'We concentrate on the diagrams that do not vanish in the all-plus case according to ref. [CITATION] and our calculations in section [REF].', '1507.08797-2-28-2': 'A generic five-point two-loop amplitude can be constructed by a straightforward extension of the present discussion.', '1507.08797-2-29-0': 'Let us first write the amplitude and then explain its content in more detail.', '1507.08797-2-29-1': 'We label [MATH] and their colour factors by their diagrams directly in the formula: [EQUATION] where the integration operator [MATH] acts on every [MATH] as [EQUATION]', '1507.08797-2-29-2': 'The explicit symmetry factors compensate for the over-counts introduced by the overall sum over permutations of external legs.', '1507.08797-2-29-3': 'For convenience, we recapitulate all these [MATH] in table [REF], where for each irreducible numerator [MATH] we also show its diagram, that of its colour factor, as well as the set of its non-equivalent permutations.', '1507.08797-2-30-0': 'The first three graphs in table [REF], [MATH], [MATH] and [MATH], are the master diagrams corresponding to the maximal cuts.', '1507.08797-2-30-1': 'They are purely trivalent, thus their colour factors are unambiguously defined by their proper graphs.', '1507.08797-2-31-0': 'The next three graphs, [MATH], [MATH] and [MATH], have a four-point vertex with two external and two internal edges.', '1507.08797-2-31-1': 'The two external legs automatically enter in the permutation sum with two possible orderings, hence multi-peripheral subgraphs are naturally obtained by fixing the internal lines, as in the insert [MATH] of figure [REF].', '1507.08797-2-31-2': '""Stretching"" the four-point vertex by these lines gives a master graph for each colour factor.', '1507.08797-2-32-0': 'The following two diagrams, [MATH] and [MATH], share the same graph structure, up to the ordering of the four-point vertex.', '1507.08797-2-32-1': 'The apparent asymmetry introduced by our selecting these two diagrams, and omitting the graph with the external leg in the right loop, is an artefact of our colour decomposition.', '1507.08797-2-32-2': 'One could make other choices; the KK relations satisfied by the trees and their symmetries ensure that any choice is valid.', '1507.08797-2-33-0': 'To expand the four-point vertex in the ninth graph, [MATH], we fixed the internal lines of the ""diamond"" subdiagram, hence its colour factor is [MATH], but with a minus sign due to one flipped vertex.', '1507.08797-2-33-1': 'The other permutation of the four-point vertex corresponds to the same topology and is present in the overall permutation sum with the right permuted colour diagram.', '1507.08797-2-34-0': 'The colour factor of the planar graph [MATH] follows in a straightforward manner from our algorithm (see figure [REF]), yielding [MATH] as its colour factor.', '1507.08797-2-34-1': 'Its descendant [MATH] is more interesting, since it is the only graph in the all-plus case with two four-point vertices.', '1507.08797-2-34-2': 'They can be treated independently by linearity of colour decomposition.', '1507.08797-2-34-3': 'The external one is thus expanded in the same way as in [MATH], producing [MATH] as an intermediate step.', '1507.08797-2-34-4': 'Expanding the internal four-point vertex gives [MATH] again.', '1507.08797-2-35-0': 'The last two graphs, [MATH] and [MATH], share a five-point vertex.', '1507.08797-2-35-1': 'To explain their colour factors, let us consider the corresponding colour-dressed cut: [EQUATION]', '1507.08797-2-35-2': 'We obtain the multi-peripheral decomposition of the five-point vertex by fixing the two right-hand loop edges and permuting the other three edges.', '1507.08797-2-35-3': 'The graphs in the second line can be vertically flipped to put leg [MATH] downstairs to match the presentation in table [REF].', '1507.08797-2-35-4': 'Obviously, an equivalent decomposition could be achieved by fixing the loop edges on the left, which would change the orientation of leg [MATH] in the superficially non-planar graphs [MATH].', '1507.08797-2-35-5': 'The [MATH]-summation in eq. [REF] effectively symmetrises the colour structure over the two choices of multi-peripheral decompositions.', '1507.08797-2-36-0': 'In the present paper we can avoid lower levels of the graph hierarchy thanks to the simplicity of the fully symmetric helicity configuration, but it already incorporates all the key elements of the general colour structure.', '1507.08797-2-37-0': '# Kinematic structure', '1507.08797-2-38-0': 'With our colour decomposition at hand, we turn our attention to the kinematic structure of the amplitude.', '1507.08797-2-38-1': 'We need to compute an irreducible numerator associated to each diagram in eq. [REF]; as Frellesvig, Zhang and one of the current authors have already computed all the planar irreducible numerators [CITATION], our task is to determine the remaining non-planar numerators.', '1507.08797-2-38-2': 'Of course, these numerators can be computed directly from their cuts.', '1507.08797-2-38-3': 'However, as we will see, it is easy to determine the complete set of non-planar irreducible numerators for this amplitude from its planar numerators and the knowledge of tree-level amplitude relations.', '1507.08797-2-39-0': '## Non-planar from planar', '1507.08797-2-40-0': 'However, we can avoid calculating this non-planar cut explicitly by relating it to a planar cut.', '1507.08797-2-40-1': 'We do so in two steps: first, we coalesce two (ordered) three-point trees into a limit of an ordered four-point tree; then we use the BCJ relations [CITATION] satisfied by the ordered four-point tree to reorder the legs until the complete diagram becomes planar.', '1507.08797-2-41-0': 'In more detail, we use the following well-known relation, which is satisfied by on-shell amplitudes in the cuts: [EQUATION] where [MATH] are the standard Mandelstam invariants.', '1507.08797-2-41-1': 'Since this identity is of central importance for us, we present a short proof.', '1507.08797-2-41-2': 'A four-point tree amplitude can be constructed from two three-point amplitudes using the BCFW recursion relation [CITATION]: [EQUATION] where hat signs on the right-hand side indicate that the amplitudes are evaluated on complex kinematics for some BCFW shift of external legs.', '1507.08797-2-41-3': 'The exact complex value of the shifted internal momentum [MATH] is defined by the on-shell condition [EQUATION] where the precise expression for [MATH] depends on the particular BCFW shift.', '1507.08797-2-41-4': 'The key point is that [MATH] is a linear function of [MATH], with the property that in the limit [MATH], [MATH].', '1507.08797-2-41-5': 'In this limit eq. [REF] becomes [EQUATION]', '1507.08797-2-41-6': 'Notice that the left-hand side contains a nonzero contribution due to the pole in [MATH].', '1507.08797-2-41-7': 'Now we can remove the factor of [MATH] on the left-hand side of eq. [REF] by using the BCJ amplitude relation [CITATION], [EQUATION]', '1507.08797-2-41-8': 'This proves the identity [REF].', '1507.08797-2-42-0': 'We proceed by applying our identity [REF] to tree amplitudes inside the non-planar cut, rearranging the diagram until it becomes planar.', '1507.08797-2-42-1': 'It is simplest to begin with maximal diagrams, and then to continue to topologies with fewer propagators.', '1507.08797-2-43-0': 'We will work through the calculation of [MATH], displayed in Figure [REF], as an example; we computed all non-planar irreducible numerators using the same technique.', '1507.08797-2-44-0': 'The calculation starts at the level of the cuts: [EQUATION] where [MATH], and we understand that all internal helicities are implicitly summed over while all exposed propagators are cut.', '1507.08797-2-44-1': 'These cuts are decomposed into irreducible numerators as [EQUATION]', '1507.08797-2-44-2': 'Using the fact that [MATH] on this cut, we see that [EQUATION]', '1507.08797-2-44-3': 'A similar calculation for [MATH] leads to [EQUATION]', '1507.08797-2-44-4': 'So far the obtained non-planar numerators are valid only on their cuts, but they can be extended off shell.', '1507.08797-2-44-5': 'We may simply express the numerators in terms of a given set of ISPs and then define off-shell numerators unambiguously through these ISPs.', '1507.08797-2-44-6': 'The value of a given numerator depends on the choice of ISP basis off shell (in contrast to the situation on shell, of course).', '1507.08797-2-44-7': 'In this way, we determine a valid set of non-planar irreducible numerators.', '1507.08797-2-44-8': 'Notice that the ISP monomial choices made in the planar sector, such as the higher powers of [MATH] preferred over high powers of [MATH], will then be easily translated to the non-planar numerators.', '1507.08797-2-45-0': 'It is very useful to maintain the symmetries of the underlying graphs in this off-shell continuation.', '1507.08797-2-45-1': 'We achieve this by choosing an appropriate basis of ISPs on a graph-by-graph basis.', '1507.08797-2-45-2': 'One engineers the ISP basis such that the loop momentum-dependence of each irreducible numerator is captured by a set of ISPs, which map into one another under the graph symmetries without using any cut conditions.', '1507.08797-2-45-3': 'The symmetries of the maximal non-planar graphs, for example, are [EQUATION]', '1507.08797-2-45-4': 'These symmetries motivate our choices of ISPs, given in Table [REF].', '1507.08797-2-45-5': 'For instance, the second symmetry of [MATH] in [REF] leads to a map of ISPs [EQUATION]', '1507.08797-2-45-6': 'After we express loop-momentum dependence in eqs. [REF] and [REF] in terms of the ISPs of Table [REF], using the fact that the RSPs (cut propagators) are zero on shell, we are left with appropriate off-shell irreducible numerators.', '1507.08797-2-45-7': 'These are listed in Section [REF].', '1507.08797-2-45-8': 'Note that the function [MATH] is not the same as the function [MATH] despite the on-shell equation ([REF]): different ISPs are chosen to make different off-shell symmetries manifest.', '1507.08797-2-46-0': 'We obtained irreducible numerators for lower-level non-planar diagrams in the same way, using the BCJ relations on cuts and extending the results off shell.', '1507.08797-2-46-1': 'For the all-plus amplitude at hand we find that many lower-level irreducible numerators vanish.', '1507.08797-2-46-2': 'In other words, the higher-level numerators capture the unitarity cut structure of the full amplitude, which is given below.', '1507.08797-2-47-0': '# The full-colour five-gluon all-plus amplitude', '1507.08797-2-48-0': 'In this section we present a complete summary of all kinematic numerators contributing to the colour decomposition in eq. [REF].', '1507.08797-2-48-1': 'We include both planar [CITATION] and non-planar irreducible numerators computed using the technique described in Section [REF].', '1507.08797-2-48-2': 'The result is presented unrenormalized including the dependence on the spin dimension [MATH] of the gluon, which is equal to [MATH] in the FDH scheme and [MATH] in CDR [CITATION].', '1507.08797-2-48-3': 'The dependence on the extra dimensional ISPs [MATH] can be collected into three general functions, [EQUATION]', '1507.08797-2-48-4': 'The remaining coefficients are expressed using the standard spinor-helicity formalism.', '1507.08797-2-48-5': 'In particular, we denote [EQUATION] since [MATH].', '1507.08797-2-48-6': 'We also make use of ""spurious"" directions in order to find compact representations of the integrands, [EQUATION]', '1507.08797-2-48-7': 'The full amplitude reads [EQUATION]', '1507.08797-2-48-8': 'The three maximal graphs are [EQUATION]', '1507.08797-2-48-9': 'Meanwhile, the graphs at level 1 are [EQUATION]', '1507.08797-2-48-10': 'Finally, the graphs at level 2 are [EQUATION]', '1507.08797-2-48-11': 'We have found a representation of the full amplitude with no topologies with fewer than six propagators.', '1507.08797-2-48-12': 'We note that there are nonzero cuts at the integrand level, but the resulting integrals are scaleless and hence zero in dimensional regularisation.', '1507.08797-2-48-13': 'We have checked additional cuts at levels 2 and 3 to ensure that no nonzero topologies remain.', '1507.08797-2-48-14': 'To find an integrand with this property, the ISPs [MATH] and [MATH] in the numerators [MATH] and [MATH] were upgraded to include terms proportional to [MATH] and [MATH].', '1507.08797-2-49-0': '# Checking the soft divergences', '1507.08797-2-50-0': 'Since the all-plus helicity configuration is zero at tree level, the universal infrared (IR) structure is the same as that of an ordinary one-loop amplitude.', '1507.08797-2-50-1': 'The poles of our two-loop amplitude should therefore be equivalent to those of the finite one-loop amplitude multiplied by the infrared pole operator including the sum over colour correlations [CITATION] [EQUATION] where we have used the [MATH] symbol to indicate that the colour matrices (in this purely gluonic case they will all be structure constants) should be inserted into the colour factors of the one-loop amplitude.', '1507.08797-2-50-2': 'The standard loop prefactor is given by [EQUATION]', '1507.08797-2-50-3': 'The one-loop amplitude to all orders in [MATH] can be found in ref. [CITATION].', '1507.08797-2-51-0': 'There are a number of difficulties in checking eq. [REF] in full since the five-point planar and non-planar integrals required are still unknown at this time.', '1507.08797-2-51-1': 'Resorting to numerical evaluation, as has been done in the planar case [CITATION], is computationally prohibitive for two reasons.', '1507.08797-2-51-2': 'Firstly, the full colour expansion contains a large number of dimension-shifted integrals ([MATH]) - an order of magnitude more than the leading colour terms.', '1507.08797-2-51-3': 'Secondly, there is no Euclidean region for the complete amplitude, and so contour deformation must be performed for many of these integrals, making them more complicated than the planar cases.', '1507.08797-2-51-4': 'This task is probably achievable with public tools like FIESTA [CITATION] and SecDec [CITATION].', '1507.08797-2-52-0': 'There is, however, a much simpler method to check the leading soft singularities up to [MATH], which can be done analytically.', '1507.08797-2-52-1': 'In the leading soft limit, the colour correlations drop out of eq. [REF]: [EQUATION]', '1507.08797-2-52-2': 'Clearly this is a weaker check than the full IR poles, but it does require non-trivial properties of the non-planar sector.', '1507.08797-2-52-3': 'The butterfly (one-loop squared) topologies, are all finite and therefore not relevant for the IR properties.', '1507.08797-2-52-4': 'Scattering amplitudes in the soft or eikonal limit have many remarkable structures and universal properties.', '1507.08797-2-52-5': 'The interested reader may like to turn to ref. [CITATION] for a recent introduction to the subject.', '1507.08797-2-53-0': '## Evaluating the massless double box in the soft limit', '1507.08797-2-54-0': 'The extra simplicity in our all-plus loop amplitude that sets it aside from most two-loop amplitudes is that the integrals contain at most a single soft divergence, rather than the maximum double soft divergence.', '1507.08797-2-54-1': 'We can therefore break our loop amplitudes up into sums of regions with soft singularities and evaluate the amplitude in the limit.', '1507.08797-2-54-2': 'In this limit the integral factorises into a product of two one-loop integrals and can be evaluated to extract the leading [MATH] divergence.', '1507.08797-2-55-0': 'All of the poles of our amplitude are contained in the topologies proportional to the same dimension shifting numerator: [EQUATION] which also has a simple behaviour, [EQUATION]', '1507.08797-2-55-1': 'Taking the example of the two-loop double box, we find two soft regions by taking the limit of either loop.', '1507.08797-2-55-2': 'We find a soft singularity whenever we have two adjacent massless legs in one of the loop integrations.', '1507.08797-2-55-3': 'In each case, we factorise into an IR divergent triangle and a dimension-shifted box: [EQUATION]', '1507.08797-2-55-4': 'Recalling the one-loop integrals, [EQUATION] and summing the two regions, we quickly arrive at the result: [EQUATION]', '1507.08797-2-56-0': '## Soft divergences of the five-point integrals', '1507.08797-2-57-0': 'By following the method described in the previous section, we have derived the complete set of integrals required for the [MATH] part of the amplitude.', '1507.08797-2-57-1': 'All of the integrals have been checked numerically using the sector decomposition methods implemented in FIESTA [CITATION] and SecDec [CITATION].', '1507.08797-2-57-2': 'Some of these integrals have been computed long ago in [MATH] dimensions and can be used to write the full integrals including finite terms via the dimensional reduction identities implemented in LiteRed [CITATION] and IBP relations from FIRE5 [CITATION].', '1507.08797-2-57-3': 'We have performed this task for the planar double box with an off-shell leg.', '1507.08797-2-57-4': 'Thus we arrived at the following soft limits for the integrals: [EQUATION]', '1507.08797-2-58-0': '[EQUATION]', '1507.08797-2-59-0': '[EQUATION]', '1507.08797-2-59-1': 'Using these results, we have checked that eq. [REF] does hold as expected for our amplitude [REF].', '1507.08797-2-60-0': '# Conclusions', '1507.08797-2-61-0': 'In this paper we have explored the impact of tree-level amplitude relations in multi-loop integrand computations.', '1507.08797-2-61-1': 'There were two major aspects to our work.', '1507.08797-2-61-2': 'Firstly, we exploited the Kleiss-Kuijf relations to find a compact colour decomposition for the two-loop amplitude in terms of multi-peripheral colour factors in an analogous way to the tree-level and one-loop decompositions of Del Duca, Dixon and Maltoni [CITATION].', '1507.08797-2-62-0': 'Secondly, we applied the BCJ amplitude relations [CITATION] to relate all non-planar generalized unitarity cuts to the previously computed planar ones.', '1507.08797-2-62-1': 'This allowed us to easily generate a compact representation the full colour two-loop, five-gluon, all-plus integrand building on previous planar work [CITATION].', '1507.08797-2-62-2': 'The soft infrared poles of the full amplitude were checked against the well-known universal pole structure.', '1507.08797-2-63-0': 'We hope that the computational methods developed here will be of good use in the necessary extension to more general helicity configurations and other [MATH] scattering processes at two loops.', '1507.08797-2-63-1': 'They highlight some advantages of relating two-loop integrands to tree-level amplitudes via generalized unitarity cuts.', '1507.08797-2-63-2': 'As well as avoiding the large intermediate steps that make Feynman diagram computations at this loop order and multiplicity extremely computationally intensive, we are able to build known on-shell symmetries and relations into the amplitude by construction.', '1507.08797-2-64-0': 'Another interesting aspect of the all-plus amplitude is the continuing connection to the previously known amplitudes in [MATH] sYM.', '1507.08797-2-64-1': 'Though the dimension shifting relation observed at one loop no longer holds, the integrands of our full all-plus amplitude and the expressions of Carrasco and Johansson [CITATION] are related by the same dimension shifting operator seen in the planar case.', '1507.08797-2-64-2': 'For example, we find that [EQUATION]', '1507.08797-2-64-3': 'The one-loop squared topologies have the form [EQUATION] where [MATH] and [MATH] are some functions of the external kinematics and loop momenta.', '1507.08797-2-64-4': 'The second term is proportional to [MATH] and is a genuine contribution in QCD not related to [MATH].', '1507.08797-2-64-5': 'This additional numerator structure is enough to make the off-shell BCJ symmetries non-trivial to satisfy, even though the [MATH] integrand has been cast in such a form.', '1507.08797-2-64-6': 'It is an interesting question as to whether this would be possible for the amplitudes presented here and one that we intend to explore in the future.', '1507.08797-2-65-0': 'We would like to thank JJ Carrasco, Claude Duhr, Einan Gardi, and Henrik Johansson for useful discussions.', '1507.08797-2-65-1': 'Particular thanks go to Chris White for enlightening remarks on the soft singularities of loop integrals.', '1507.08797-2-65-2': 'SB is supported by an STFC Rutherford Fellowship ST/L004925/1.', '1507.08797-2-65-3': 'GM is supported by an STFC Studentship ST/K501980/1.', '1507.08797-2-65-4': 'AO is supported by the EU via a Marie Curie Actions grant FP7-PEOPLE-2013-CIG 631370.', '1507.08797-2-65-5': 'DOC is supported in part by the STFC consolidated grant ""Particle Physics at the Higgs Centre"", by the National Science Foundation under grant NSF PHY11-25915, and by the Marie Curie FP7 grant 631370.'}","[['1507.08797-1-5-0', '1507.08797-2-5-0'], ['1507.08797-1-5-1', '1507.08797-2-5-1'], ['1507.08797-1-5-2', '1507.08797-2-5-2'], ['1507.08797-1-5-3', '1507.08797-2-5-3'], ['1507.08797-1-5-4', '1507.08797-2-5-4'], ['1507.08797-1-21-0', '1507.08797-2-21-0'], ['1507.08797-1-21-1', '1507.08797-2-21-1'], ['1507.08797-1-21-2', '1507.08797-2-21-2'], 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